[{"data":1,"prerenderedAt":88410},["ShallowReactive",2],{"blog-post-expert-notes\u002Fwigner-functions-single-qubit-gates":3,"sibling-dives-expert-notes\u002Fwigner-functions-single-qubit-gates":20123,"learn-track-expert-notes\u002Fwigner-functions-single-qubit-gates":88409},{"id":4,"title":5,"authors":6,"body":8,"breadcrumb":20087,"builders":20092,"byline":20103,"challenge":20105,"courseAuthor":20105,"courseLead":20105,"dek":20105,"description":20106,"draft":20107,"extension":20108,"eyebrow":20109,"featured":20107,"finish":20105,"fork":20105,"hero":20105,"heroAlt":20105,"heroCta":20105,"heroImage":20105,"homepageFeatured":20107,"kind":20110,"lessonCount":20105,"meta":20111,"navigation":10300,"newsItems":20105,"next":20105,"ogImage":20112,"order":20105,"outcomes":20105,"path":20113,"publishDate":20114,"readingTime":20115,"related":20116,"relatedProjects":20105,"seo":20117,"stem":20118,"tags":20119,"track":20105,"trackName":20105,"__hash__":20122},"blog\u002Fblog\u002Fexpert-notes\u002Fwigner-functions-single-qubit-gates.md","Wigner Functions for Single-Qubit Quantum Gates",[7],"onri-jay-benally",{"type":9,"value":10,"toc":20072},"minimark",[11,24,27,32,133,911,1015,1483,1629,1779,1990,1992,1996,1999,2339,2408,2946,3011,3651,3657,4030,4110,4112,4147,4179,4833,5122,5124,5128,5233,5557,5691,5931,5933,5968,6215,6964,6967,7474,7579,8074,8076,8134,8290,8705,8712,8985,8988,8990,8994,9057,9733,9736,9738,9742,10072,10074,10078,10081,10091,10093,10097,10267,10514,12793,13978,13984,13990,13996,14001,14005,14009,15001,15007,15011,15014,17691,17697,17703,17709,17713,17717,17721,20058,20064,20068],[12,13,14],"p",{},[15,16,17,18,23],"em",{},"Republished with the author's permission from ",[19,20,22],"a",{"href":21},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FWigner_Functions_for_Single_Qubit_Quantum_Gates_.ipynb","the original notebook"," and shared under CC-BY-4.0. The text, code, and figures below are Onri's, as published.",[25,26],"hr",{},[28,29,31],"h2",{"id":30},"_1-core-objects-and-symbols","1. Core Objects and Symbols",[12,33,34,38,39,132],{},[35,36,37],"strong",{},"Pauli matrices"," (in the computational basis (",[40,41,44,86],"span",{"className":42},[43],"katex",[40,45,48],{"className":46},[47],"katex-mathml",[49,50,52],"math",{"xmlns":51},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[53,54,55,81],"semantics",{},[56,57,58,63,67,70,74,76,79],"mrow",{},[59,60,62],"mo",{"stretchy":61},"false","∣",[64,65,66],"mn",{},"0",[59,68,69],{"stretchy":61},"⟩",[59,71,73],{"separator":72},"true",",",[59,75,62],{"stretchy":61},[64,77,78],{},"1",[59,80,69],{"stretchy":61},[82,83,85],"annotation",{"encoding":84},"application\u002Fx-tex","{\\lvert 0\\rangle,\\lvert 1\\rangle}",[40,87,90],{"className":88,"ariaHidden":72},[89],"katex-html",[40,91,94,99],{"className":92},[93],"base",[40,95],{"className":96,"style":98},[97],"strut","height:1em;vertical-align:-0.25em;",[40,100,103,107,110,114,118,123,126,129],{"className":101},[102],"mord",[40,104,62],{"className":105},[106],"mopen",[40,108,66],{"className":109},[102],[40,111,69],{"className":112},[113],"mclose",[40,115,73],{"className":116},[117],"mpunct",[40,119],{"className":120,"style":122},[121],"mspace","margin-right:0.1667em;",[40,124,62],{"className":125},[106],[40,127,78],{"className":128},[102],[40,130,69],{"className":131},[113],")):",[40,134,137],{"className":135},[136],"katex-display",[40,138,140,319],{"className":139},[43],[40,141,143],{"className":142},[47],[49,144,146],{"xmlns":51,"display":145},"block",[53,147,148,316],{},[56,149,150,160,163,208,210,212,219,221,263,265,267,274,276],{},[151,152,153,157],"msub",{},[154,155,156],"mi",{},"σ",[154,158,159],{},"x",[59,161,162],{},"=",[56,164,165,168,205],{},[59,166,167],{"fence":72},"(",[169,170,174,191],"mtable",{"rowspacing":171,"columnalign":172,"columnspacing":173},"0.16em","center center","1em",[175,176,177,185],"mtr",{},[178,179,180],"mtd",{},[181,182,183],"mstyle",{"scriptlevel":66,"displaystyle":61},[64,184,66],{},[178,186,187],{},[181,188,189],{"scriptlevel":66,"displaystyle":61},[64,190,78],{},[175,192,193,199],{},[178,194,195],{},[181,196,197],{"scriptlevel":66,"displaystyle":61},[64,198,78],{},[178,200,201],{},[181,202,203],{"scriptlevel":66,"displaystyle":61},[64,204,66],{},[59,206,207],{"fence":72},")",[59,209,73],{"separator":72},[121,211],{"width":173},[151,213,214,216],{},[154,215,156],{},[154,217,218],{},"y",[59,220,162],{},[56,222,223,225,261],{},[59,224,167],{"fence":72},[169,226,227,247],{"rowspacing":171,"columnalign":172,"columnspacing":173},[175,228,229,235],{},[178,230,231],{},[181,232,233],{"scriptlevel":66,"displaystyle":61},[64,234,66],{},[178,236,237],{},[181,238,239],{"scriptlevel":66,"displaystyle":61},[56,240,241,244],{},[59,242,243],{},"−",[154,245,246],{},"i",[175,248,249,255],{},[178,250,251],{},[181,252,253],{"scriptlevel":66,"displaystyle":61},[154,254,246],{},[178,256,257],{},[181,258,259],{"scriptlevel":66,"displaystyle":61},[64,260,66],{},[59,262,207],{"fence":72},[59,264,73],{"separator":72},[121,266],{"width":173},[151,268,269,271],{},[154,270,156],{},[154,272,273],{},"z",[59,275,162],{},[56,277,278,280,314],{},[59,279,167],{"fence":72},[169,281,282,296],{"rowspacing":171,"columnalign":172,"columnspacing":173},[175,283,284,290],{},[178,285,286],{},[181,287,288],{"scriptlevel":66,"displaystyle":61},[64,289,78],{},[178,291,292],{},[181,293,294],{"scriptlevel":66,"displaystyle":61},[64,295,66],{},[175,297,298,304],{},[178,299,300],{},[181,301,302],{"scriptlevel":66,"displaystyle":61},[64,303,66],{},[178,305,306],{},[181,307,308],{"scriptlevel":66,"displaystyle":61},[56,309,310,312],{},[59,311,243],{},[64,313,78],{},[59,315,207],{"fence":72},[82,317,318],{"encoding":84},"\\sigma_x = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}, \\quad\n\\sigma_y = \\begin{pmatrix} 0 & -i \\\\ i & 0 \\end{pmatrix}, \\quad\n\\sigma_z = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}",[40,320,322,398,597,785],{"className":321,"ariaHidden":72},[89],[40,323,325,329,387,391,395],{"className":324},[93],[40,326],{"className":327,"style":328},[97],"height:0.5806em;vertical-align:-0.15em;",[40,330,332,337],{"className":331},[102],[40,333,156],{"className":334,"style":336},[102,335],"mathnormal","margin-right:0.0359em;",[40,338,341],{"className":339},[340],"msupsub",[40,342,346,378],{"className":343},[344,345],"vlist-t","vlist-t2",[40,347,350,373],{"className":348},[349],"vlist-r",[40,351,355],{"className":352,"style":354},[353],"vlist","height:0.1514em;",[40,356,358,363],{"style":357},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[40,359],{"className":360,"style":362},[361],"pstrut","height:2.7em;",[40,364,370],{"className":365},[366,367,368,369],"sizing","reset-size6","size3","mtight",[40,371,159],{"className":372},[102,335,369],[40,374,377],{"className":375},[376],"vlist-s","​",[40,379,381],{"className":380},[349],[40,382,385],{"className":383,"style":384},[353],"height:0.15em;",[40,386],{},[40,388],{"className":389,"style":390},[121],"margin-right:0.2778em;",[40,392,162],{"className":393},[394],"mrel",[40,396],{"className":397,"style":390},[121],[40,399,401,405,534,537,540,544,547,588,591,594],{"className":400},[93],[40,402],{"className":403,"style":404},[97],"height:2.4em;vertical-align:-0.95em;",[40,406,409,418,528],{"className":407},[408],"minner",[40,410,414],{"className":411,"style":413},[106,412],"delimcenter","top:0em;",[40,415,167],{"className":416},[417,368],"delimsizing",[40,419,421],{"className":420},[102],[40,422,424,475,480,483],{"className":423},[169],[40,425,428],{"className":426},[427],"col-align-c",[40,429,431,466],{"className":430},[344,345],[40,432,434,463],{"className":433},[349],[40,435,438,451],{"className":436,"style":437},[353],"height:1.45em;",[40,439,441,445],{"style":440},"top:-3.61em;",[40,442],{"className":443,"style":444},[361],"height:3em;",[40,446,448],{"className":447},[102],[40,449,66],{"className":450},[102],[40,452,454,457],{"style":453},"top:-2.41em;",[40,455],{"className":456,"style":444},[361],[40,458,460],{"className":459},[102],[40,461,78],{"className":462},[102],[40,464,377],{"className":465},[376],[40,467,469],{"className":468},[349],[40,470,473],{"className":471,"style":472},[353],"height:0.95em;",[40,474],{},[40,476],{"className":477,"style":479},[478],"arraycolsep","width:0.5em;",[40,481],{"className":482,"style":479},[478],[40,484,486],{"className":485},[427],[40,487,489,520],{"className":488},[344,345],[40,490,492,517],{"className":491},[349],[40,493,495,506],{"className":494,"style":437},[353],[40,496,497,500],{"style":440},[40,498],{"className":499,"style":444},[361],[40,501,503],{"className":502},[102],[40,504,78],{"className":505},[102],[40,507,508,511],{"style":453},[40,509],{"className":510,"style":444},[361],[40,512,514],{"className":513},[102],[40,515,66],{"className":516},[102],[40,518,377],{"className":519},[376],[40,521,523],{"className":522},[349],[40,524,526],{"className":525,"style":472},[353],[40,527],{},[40,529,531],{"className":530,"style":413},[113,412],[40,532,207],{"className":533},[417,368],[40,535],{"className":536,"style":122},[121],[40,538,73],{"className":539},[117],[40,541],{"className":542,"style":543},[121],"margin-right:1em;",[40,545],{"className":546,"style":122},[121],[40,548,550,553],{"className":549},[102],[40,551,156],{"className":552,"style":336},[102,335],[40,554,556],{"className":555},[340],[40,557,559,579],{"className":558},[344,345],[40,560,562,576],{"className":561},[349],[40,563,565],{"className":564,"style":354},[353],[40,566,567,570],{"style":357},[40,568],{"className":569,"style":362},[361],[40,571,573],{"className":572},[366,367,368,369],[40,574,218],{"className":575,"style":336},[102,335,369],[40,577,377],{"className":578},[376],[40,580,582],{"className":581},[349],[40,583,586],{"className":584,"style":585},[353],"height:0.2861em;",[40,587],{},[40,589],{"className":590,"style":390},[121],[40,592,162],{"className":593},[394],[40,595],{"className":596,"style":390},[121],[40,598,600,603,723,726,729,732,735,776,779,782],{"className":599},[93],[40,601],{"className":602,"style":404},[97],[40,604,606,612,717],{"className":605},[408],[40,607,609],{"className":608,"style":413},[106,412],[40,610,167],{"className":611},[417,368],[40,613,615],{"className":614},[102],[40,616,618,663,666,669],{"className":617},[169],[40,619,621],{"className":620},[427],[40,622,624,655],{"className":623},[344,345],[40,625,627,652],{"className":626},[349],[40,628,630,641],{"className":629,"style":437},[353],[40,631,632,635],{"style":440},[40,633],{"className":634,"style":444},[361],[40,636,638],{"className":637},[102],[40,639,66],{"className":640},[102],[40,642,643,646],{"style":453},[40,644],{"className":645,"style":444},[361],[40,647,649],{"className":648},[102],[40,650,246],{"className":651},[102,335],[40,653,377],{"className":654},[376],[40,656,658],{"className":657},[349],[40,659,661],{"className":660,"style":472},[353],[40,662],{},[40,664],{"className":665,"style":479},[478],[40,667],{"className":668,"style":479},[478],[40,670,672],{"className":671},[427],[40,673,675,709],{"className":674},[344,345],[40,676,678,706],{"className":677},[349],[40,679,681,695],{"className":680,"style":437},[353],[40,682,683,686],{"style":440},[40,684],{"className":685,"style":444},[361],[40,687,689,692],{"className":688},[102],[40,690,243],{"className":691},[102],[40,693,246],{"className":694},[102,335],[40,696,697,700],{"style":453},[40,698],{"className":699,"style":444},[361],[40,701,703],{"className":702},[102],[40,704,66],{"className":705},[102],[40,707,377],{"className":708},[376],[40,710,712],{"className":711},[349],[40,713,715],{"className":714,"style":472},[353],[40,716],{},[40,718,720],{"className":719,"style":413},[113,412],[40,721,207],{"className":722},[417,368],[40,724],{"className":725,"style":122},[121],[40,727,73],{"className":728},[117],[40,730],{"className":731,"style":543},[121],[40,733],{"className":734,"style":122},[121],[40,736,738,741],{"className":737},[102],[40,739,156],{"className":740,"style":336},[102,335],[40,742,744],{"className":743},[340],[40,745,747,768],{"className":746},[344,345],[40,748,750,765],{"className":749},[349],[40,751,753],{"className":752,"style":354},[353],[40,754,755,758],{"style":357},[40,756],{"className":757,"style":362},[361],[40,759,761],{"className":760},[366,367,368,369],[40,762,273],{"className":763,"style":764},[102,335,369],"margin-right:0.044em;",[40,766,377],{"className":767},[376],[40,769,771],{"className":770},[349],[40,772,774],{"className":773,"style":384},[353],[40,775],{},[40,777],{"className":778,"style":390},[121],[40,780,162],{"className":781},[394],[40,783],{"className":784,"style":390},[121],[40,786,788,791],{"className":787},[93],[40,789],{"className":790,"style":404},[97],[40,792,794,800,905],{"className":793},[408],[40,795,797],{"className":796,"style":413},[106,412],[40,798,167],{"className":799},[417,368],[40,801,803],{"className":802},[102],[40,804,806,851,854,857],{"className":805},[169],[40,807,809],{"className":808},[427],[40,810,812,843],{"className":811},[344,345],[40,813,815,840],{"className":814},[349],[40,816,818,829],{"className":817,"style":437},[353],[40,819,820,823],{"style":440},[40,821],{"className":822,"style":444},[361],[40,824,826],{"className":825},[102],[40,827,78],{"className":828},[102],[40,830,831,834],{"style":453},[40,832],{"className":833,"style":444},[361],[40,835,837],{"className":836},[102],[40,838,66],{"className":839},[102],[40,841,377],{"className":842},[376],[40,844,846],{"className":845},[349],[40,847,849],{"className":848,"style":472},[353],[40,850],{},[40,852],{"className":853,"style":479},[478],[40,855],{"className":856,"style":479},[478],[40,858,860],{"className":859},[427],[40,861,863,897],{"className":862},[344,345],[40,864,866,894],{"className":865},[349],[40,867,869,880],{"className":868,"style":437},[353],[40,870,871,874],{"style":440},[40,872],{"className":873,"style":444},[361],[40,875,877],{"className":876},[102],[40,878,66],{"className":879},[102],[40,881,882,885],{"style":453},[40,883],{"className":884,"style":444},[361],[40,886,888,891],{"className":887},[102],[40,889,243],{"className":890},[102],[40,892,78],{"className":893},[102],[40,895,377],{"className":896},[376],[40,898,900],{"className":899},[349],[40,901,903],{"className":902,"style":472},[353],[40,904],{},[40,906,908],{"className":907,"style":413},[113,412],[40,909,207],{"className":910},[417,368],[12,912,913,916,917,920,921,1014],{},[35,914,915],{},"Density operator"," and ",[35,918,919],{},"Bloch vector"," (",[40,922,924,953],{"className":923},[43],[40,925,927],{"className":926},[47],[49,928,929],{"xmlns":51},[53,930,931,950],{},[56,932,933,937,940],{},[154,934,936],{"mathvariant":935},"bold","r",[59,938,939],{},"∈",[941,942,943,947],"msup",{},[154,944,946],{"mathvariant":945},"double-struck","R",[64,948,949],{},"3",[82,951,952],{"encoding":84},"\\mathbf r\\in\\mathbb R^3",[40,954,956,976],{"className":955,"ariaHidden":72},[89],[40,957,959,963,967,970,973],{"className":958},[93],[40,960],{"className":961,"style":962},[97],"height:0.5782em;vertical-align:-0.0391em;",[40,964,936],{"className":965},[102,966],"mathbf",[40,968],{"className":969,"style":390},[121],[40,971,939],{"className":972},[394],[40,974],{"className":975,"style":390},[121],[40,977,979,983],{"className":978},[93],[40,980],{"className":981,"style":982},[97],"height:0.8141em;",[40,984,986,990],{"className":985},[102],[40,987,946],{"className":988},[102,989],"mathbb",[40,991,993],{"className":992},[340],[40,994,996],{"className":995},[344],[40,997,999],{"className":998},[349],[40,1000,1002],{"className":1001,"style":982},[353],[40,1003,1005,1008],{"style":1004},"top:-3.063em;margin-right:0.05em;",[40,1006],{"className":1007,"style":362},[361],[40,1009,1011],{"className":1010},[366,367,368,369],[40,1012,949],{"className":1013},[102,369],"):",[40,1016,1018],{"className":1017},[136],[40,1019,1021,1127],{"className":1020},[43],[40,1022,1024],{"className":1023},[47],[49,1025,1026],{"xmlns":51,"display":145},[53,1027,1028,1124],{},[56,1029,1030,1033,1035,1045,1048,1051,1054,1056,1059,1062,1064,1066,1069,1076,1078,1082,1085,1087,1089,1091,1097,1099,1101,1105,1107,1109,1121],{},[154,1031,1032],{},"ρ",[59,1034,162],{},[181,1036,1037],{"scriptlevel":66,"displaystyle":61},[1038,1039,1040,1042],"mfrac",{},[64,1041,78],{},[64,1043,1044],{},"2",[59,1046,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},"1.2em",[154,1049,1050],{},"I",[59,1052,1053],{},"+",[154,1055,936],{"mathvariant":935},[59,1057,1058],{},"⋅",[154,1060,156],{"mathvariant":1061},"bold-italic",[59,1063,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,1065,73],{"separator":72},[121,1067],{"width":1068},"2em",[151,1070,1071,1073],{},[154,1072,936],{},[154,1074,1075],{},"j",[59,1077,162],{},[154,1079,1081],{"mathvariant":1080},"normal","Tr",[59,1083,1084],{},"⁡",[59,1086,167],{"stretchy":61},[154,1088,1032],{},[59,1090,73],{"separator":72},[151,1092,1093,1095],{},[154,1094,156],{},[154,1096,1075],{},[59,1098,207],{"stretchy":61},[59,1100,73],{"separator":72},[1102,1103,1104],"mtext",{}," ",[154,1106,1075],{},[59,1108,939],{},[56,1110,1111,1113,1115,1117,1119],{},[154,1112,159],{},[59,1114,73],{"separator":72},[154,1116,218],{},[59,1118,73],{"separator":72},[154,1120,273],{},[154,1122,1123],{"mathvariant":1080},".",[82,1125,1126],{"encoding":84},"\\rho = \\tfrac{1}{2}\\big(I + \\mathbf r\\cdot\\boldsymbol{\\sigma}\\big),\\qquad\nr_j = \\operatorname{Tr}(\\rho,\\sigma_j),\\ j\\in{x,y,z}.",[40,1128,1130,1149,1254,1273,1359,1450],{"className":1129,"ariaHidden":72},[89],[40,1131,1133,1137,1140,1143,1146],{"className":1132},[93],[40,1134],{"className":1135,"style":1136},[97],"height:0.625em;vertical-align:-0.1944em;",[40,1138,1032],{"className":1139},[102,335],[40,1141],{"className":1142,"style":390},[121],[40,1144,162],{"className":1145},[394],[40,1147],{"className":1148,"style":390},[121],[40,1150,1152,1156,1232,1239,1243,1247,1251],{"className":1151},[93],[40,1153],{"className":1154,"style":1155},[97],"height:1.2em;vertical-align:-0.35em;",[40,1157,1159,1163,1229],{"className":1158},[102],[40,1160],{"className":1161},[106,1162],"nulldelimiter",[40,1164,1166],{"className":1165},[1038],[40,1167,1169,1220],{"className":1168},[344,345],[40,1170,1172,1217],{"className":1171},[349],[40,1173,1176,1191,1202],{"className":1174,"style":1175},[353],"height:0.8451em;",[40,1177,1179,1182],{"style":1178},"top:-2.655em;",[40,1180],{"className":1181,"style":444},[361],[40,1183,1185],{"className":1184},[366,367,368,369],[40,1186,1188],{"className":1187},[102,369],[40,1189,1044],{"className":1190},[102,369],[40,1192,1194,1197],{"style":1193},"top:-3.23em;",[40,1195],{"className":1196,"style":444},[361],[40,1198],{"className":1199,"style":1201},[1200],"frac-line","border-bottom-width:0.04em;",[40,1203,1205,1208],{"style":1204},"top:-3.394em;",[40,1206],{"className":1207,"style":444},[361],[40,1209,1211],{"className":1210},[366,367,368,369],[40,1212,1214],{"className":1213},[102,369],[40,1215,78],{"className":1216},[102,369],[40,1218,377],{"className":1219},[376],[40,1221,1223],{"className":1222},[349],[40,1224,1227],{"className":1225,"style":1226},[353],"height:0.345em;",[40,1228],{},[40,1230],{"className":1231},[113,1162],[40,1233,1235],{"className":1234},[102],[40,1236,167],{"className":1237},[417,1238],"size1",[40,1240,1050],{"className":1241,"style":1242},[102,335],"margin-right:0.0785em;",[40,1244],{"className":1245,"style":1246},[121],"margin-right:0.2222em;",[40,1248,1053],{"className":1249},[1250],"mbin",[40,1252],{"className":1253,"style":1246},[121],[40,1255,1257,1261,1264,1267,1270],{"className":1256},[93],[40,1258],{"className":1259,"style":1260},[97],"height:0.4445em;",[40,1262,936],{"className":1263},[102,966],[40,1265],{"className":1266,"style":1246},[121],[40,1268,1058],{"className":1269},[1250],[40,1271],{"className":1272,"style":1246},[121],[40,1274,1276,1279,1290,1296,1299,1303,1306,1350,1353,1356],{"className":1275},[93],[40,1277],{"className":1278,"style":1155},[97],[40,1280,1282],{"className":1281},[102],[40,1283,1285],{"className":1284},[102],[40,1286,156],{"className":1287,"style":1289},[102,1288],"boldsymbol","margin-right:0.037em;",[40,1291,1293],{"className":1292},[102],[40,1294,207],{"className":1295},[417,1238],[40,1297,73],{"className":1298},[117],[40,1300],{"className":1301,"style":1302},[121],"margin-right:2em;",[40,1304],{"className":1305,"style":122},[121],[40,1307,1309,1313],{"className":1308},[102],[40,1310,936],{"className":1311,"style":1312},[102,335],"margin-right:0.0278em;",[40,1314,1316],{"className":1315},[340],[40,1317,1319,1342],{"className":1318},[344,345],[40,1320,1322,1339],{"className":1321},[349],[40,1323,1326],{"className":1324,"style":1325},[353],"height:0.3117em;",[40,1327,1329,1332],{"style":1328},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[40,1330],{"className":1331,"style":362},[361],[40,1333,1335],{"className":1334},[366,367,368,369],[40,1336,1075],{"className":1337,"style":1338},[102,335,369],"margin-right:0.0572em;",[40,1340,377],{"className":1341},[376],[40,1343,1345],{"className":1344},[349],[40,1346,1348],{"className":1347,"style":585},[353],[40,1349],{},[40,1351],{"className":1352,"style":390},[121],[40,1354,162],{"className":1355},[394],[40,1357],{"className":1358,"style":390},[121],[40,1360,1362,1366,1374,1377,1380,1383,1386,1426,1429,1432,1435,1438,1441,1444,1447],{"className":1361},[93],[40,1363],{"className":1364,"style":1365},[97],"height:1.0361em;vertical-align:-0.2861em;",[40,1367,1370],{"className":1368},[1369],"mop",[40,1371,1081],{"className":1372},[102,1373],"mathrm",[40,1375,167],{"className":1376},[106],[40,1378,1032],{"className":1379},[102,335],[40,1381,73],{"className":1382},[117],[40,1384],{"className":1385,"style":122},[121],[40,1387,1389,1392],{"className":1388},[102],[40,1390,156],{"className":1391,"style":336},[102,335],[40,1393,1395],{"className":1394},[340],[40,1396,1398,1418],{"className":1397},[344,345],[40,1399,1401,1415],{"className":1400},[349],[40,1402,1404],{"className":1403,"style":1325},[353],[40,1405,1406,1409],{"style":357},[40,1407],{"className":1408,"style":362},[361],[40,1410,1412],{"className":1411},[366,367,368,369],[40,1413,1075],{"className":1414,"style":1338},[102,335,369],[40,1416,377],{"className":1417},[376],[40,1419,1421],{"className":1420},[349],[40,1422,1424],{"className":1423,"style":585},[353],[40,1425],{},[40,1427,207],{"className":1428},[113],[40,1430,73],{"className":1431},[117],[40,1433,1104],{"className":1434},[121],[40,1436],{"className":1437,"style":122},[121],[40,1439,1075],{"className":1440,"style":1338},[102,335],[40,1442],{"className":1443,"style":390},[121],[40,1445,939],{"className":1446},[394],[40,1448],{"className":1449,"style":390},[121],[40,1451,1453,1456,1480],{"className":1452},[93],[40,1454],{"className":1455,"style":1136},[97],[40,1457,1459,1462,1465,1468,1471,1474,1477],{"className":1458},[102],[40,1460,159],{"className":1461},[102,335],[40,1463,73],{"className":1464},[117],[40,1466],{"className":1467,"style":122},[121],[40,1469,218],{"className":1470,"style":336},[102,335],[40,1472,73],{"className":1473},[117],[40,1475],{"className":1476,"style":122},[121],[40,1478,273],{"className":1479,"style":764},[102,335],[40,1481,1123],{"className":1482},[102],[12,1484,1485,1486,1564,1565,1628],{},"For a pure state (",[40,1487,1489,1519],{"className":1488},[43],[40,1490,1492],{"className":1491},[47],[49,1493,1494],{"xmlns":51},[53,1495,1496,1516],{},[56,1497,1498,1500,1502,1504,1507,1509,1512,1514],{},[154,1499,1032],{},[59,1501,162],{},[59,1503,62],{"stretchy":61},[154,1505,1506],{},"ψ",[59,1508,69],{"stretchy":61},[59,1510,1511],{"stretchy":61},"⟨",[154,1513,1506],{},[59,1515,62],{"stretchy":61},[82,1517,1518],{"encoding":84},"\\rho = \\lvert\\psi\\rangle\\langle\\psi\\rvert",[40,1520,1522,1540],{"className":1521,"ariaHidden":72},[89],[40,1523,1525,1528,1531,1534,1537],{"className":1524},[93],[40,1526],{"className":1527,"style":1136},[97],[40,1529,1032],{"className":1530},[102,335],[40,1532],{"className":1533,"style":390},[121],[40,1535,162],{"className":1536},[394],[40,1538],{"className":1539,"style":390},[121],[40,1541,1543,1546,1549,1552,1555,1558,1561],{"className":1542},[93],[40,1544],{"className":1545,"style":98},[97],[40,1547,62],{"className":1548},[106],[40,1550,1506],{"className":1551,"style":336},[102,335],[40,1553,69],{"className":1554},[113],[40,1556,1511],{"className":1557},[106],[40,1559,1506],{"className":1560,"style":336},[102,335],[40,1562,62],{"className":1563},[113],"), one has (",[40,1566,1568,1591],{"className":1567},[43],[40,1569,1571],{"className":1570},[47],[49,1572,1573],{"xmlns":51},[53,1574,1575,1588],{},[56,1576,1577,1580,1582,1584,1586],{},[59,1578,1579],{"stretchy":61},"∥",[154,1581,936],{"mathvariant":935},[59,1583,1579],{"stretchy":61},[59,1585,162],{},[64,1587,78],{},[82,1589,1590],{"encoding":84},"\\lVert\\mathbf r\\rVert = 1",[40,1592,1594,1618],{"className":1593,"ariaHidden":72},[89],[40,1595,1597,1600,1603,1606,1609,1612,1615],{"className":1596},[93],[40,1598],{"className":1599,"style":98},[97],[40,1601,1579],{"className":1602},[106],[40,1604,936],{"className":1605},[102,966],[40,1607,1579],{"className":1608},[113],[40,1610],{"className":1611,"style":390},[121],[40,1613,162],{"className":1614},[394],[40,1616],{"className":1617,"style":390},[121],[40,1619,1621,1625],{"className":1620},[93],[40,1622],{"className":1623,"style":1624},[97],"height:0.6444em;",[40,1626,78],{"className":1627},[102],").",[12,1630,1631,1634,1635,1778],{},[35,1632,1633],{},"Unit vector on the sphere"," (spherical angles (",[40,1636,1638,1687],{"className":1637},[43],[40,1639,1641],{"className":1640},[47],[49,1642,1643],{"xmlns":51},[53,1644,1645,1684],{},[56,1646,1647,1650,1652,1655,1657,1659,1662,1665,1667,1669,1672,1674,1676,1678,1680,1682],{},[154,1648,1649],{},"θ",[59,1651,939],{},[59,1653,1654],{"stretchy":61},"[",[64,1656,66],{},[59,1658,73],{"separator":72},[154,1660,1661],{},"π",[59,1663,1664],{"stretchy":61},"]",[59,1666,73],{"separator":72},[1102,1668,1104],{},[154,1670,1671],{},"ϕ",[59,1673,939],{},[59,1675,1654],{"stretchy":61},[64,1677,66],{},[59,1679,73],{"separator":72},[64,1681,1044],{},[154,1683,1661],{},[82,1685,1686],{"encoding":84},"\\theta\\in[0,\\pi],\\ \\phi\\in[0,2\\pi",[40,1688,1690,1709,1754],{"className":1689,"ariaHidden":72},[89],[40,1691,1693,1697,1700,1703,1706],{"className":1692},[93],[40,1694],{"className":1695,"style":1696},[97],"height:0.7335em;vertical-align:-0.0391em;",[40,1698,1649],{"className":1699,"style":1312},[102,335],[40,1701],{"className":1702,"style":390},[121],[40,1704,939],{"className":1705},[394],[40,1707],{"className":1708,"style":390},[121],[40,1710,1712,1715,1718,1721,1724,1727,1730,1733,1736,1739,1742,1745,1748,1751],{"className":1711},[93],[40,1713],{"className":1714,"style":98},[97],[40,1716,1654],{"className":1717},[106],[40,1719,66],{"className":1720},[102],[40,1722,73],{"className":1723},[117],[40,1725],{"className":1726,"style":122},[121],[40,1728,1661],{"className":1729,"style":336},[102,335],[40,1731,1664],{"className":1732},[113],[40,1734,73],{"className":1735},[117],[40,1737,1104],{"className":1738},[121],[40,1740],{"className":1741,"style":122},[121],[40,1743,1671],{"className":1744},[102,335],[40,1746],{"className":1747,"style":390},[121],[40,1749,939],{"className":1750},[394],[40,1752],{"className":1753,"style":390},[121],[40,1755,1757,1760,1763,1766,1769,1772,1775],{"className":1756},[93],[40,1758],{"className":1759,"style":98},[97],[40,1761,1654],{"className":1762},[106],[40,1764,66],{"className":1765},[102],[40,1767,73],{"className":1768},[117],[40,1770],{"className":1771,"style":122},[121],[40,1773,1044],{"className":1774},[102],[40,1776,1661],{"className":1777,"style":336},[102,335],"))):",[40,1780,1782],{"className":1781},[136],[40,1783,1785,1858],{"className":1784},[43],[40,1786,1788],{"className":1787},[47],[49,1789,1790],{"xmlns":51,"display":145},[53,1791,1792,1855],{},[56,1793,1794,1797,1799,1801,1803,1805,1807,1809,1811,1814,1816,1818,1821,1823,1825,1827,1829,1831,1833,1835,1837,1839,1841,1843,1845,1847,1849,1851,1853],{},[154,1795,1796],{"mathvariant":935},"n",[59,1798,167],{"stretchy":61},[154,1800,1649],{},[59,1802,73],{"separator":72},[154,1804,1671],{},[59,1806,207],{"stretchy":61},[59,1808,162],{},[59,1810,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,1812,1813],{},"sin",[59,1815,1084],{},[154,1817,1649],{},[154,1819,1820],{},"cos",[59,1822,1084],{},[154,1824,1671],{},[59,1826,73],{"separator":72},[1102,1828,1104],{},[154,1830,1813],{},[59,1832,1084],{},[154,1834,1649],{},[154,1836,1813],{},[59,1838,1084],{},[154,1840,1671],{},[59,1842,73],{"separator":72},[1102,1844,1104],{},[154,1846,1820],{},[59,1848,1084],{},[154,1850,1649],{},[59,1852,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,1854,1123],{"mathvariant":1080},[82,1856,1857],{"encoding":84},"\\mathbf n(\\theta,\\phi) = \\big(\\sin\\theta\\cos\\phi,\\ \\sin\\theta\\sin\\phi,\\ \\cos\\theta\\big).",[40,1859,1861,1897],{"className":1860,"ariaHidden":72},[89],[40,1862,1864,1867,1870,1873,1876,1879,1882,1885,1888,1891,1894],{"className":1863},[93],[40,1865],{"className":1866,"style":98},[97],[40,1868,1796],{"className":1869},[102,966],[40,1871,167],{"className":1872},[106],[40,1874,1649],{"className":1875,"style":1312},[102,335],[40,1877,73],{"className":1878},[117],[40,1880],{"className":1881,"style":122},[121],[40,1883,1671],{"className":1884},[102,335],[40,1886,207],{"className":1887},[113],[40,1889],{"className":1890,"style":390},[121],[40,1892,162],{"className":1893},[394],[40,1895],{"className":1896,"style":390},[121],[40,1898,1900,1903,1909,1912,1915,1918,1921,1924,1927,1930,1933,1936,1939,1942,1945,1948,1951,1954,1957,1960,1963,1966,1969,1972,1975,1978,1981,1987],{"className":1899},[93],[40,1901],{"className":1902,"style":1155},[97],[40,1904,1906],{"className":1905},[102],[40,1907,167],{"className":1908},[417,1238],[40,1910],{"className":1911,"style":122},[121],[40,1913,1813],{"className":1914},[1369],[40,1916],{"className":1917,"style":122},[121],[40,1919,1649],{"className":1920,"style":1312},[102,335],[40,1922],{"className":1923,"style":122},[121],[40,1925,1820],{"className":1926},[1369],[40,1928],{"className":1929,"style":122},[121],[40,1931,1671],{"className":1932},[102,335],[40,1934,73],{"className":1935},[117],[40,1937,1104],{"className":1938},[121],[40,1940],{"className":1941,"style":122},[121],[40,1943,1813],{"className":1944},[1369],[40,1946],{"className":1947,"style":122},[121],[40,1949,1649],{"className":1950,"style":1312},[102,335],[40,1952],{"className":1953,"style":122},[121],[40,1955,1813],{"className":1956},[1369],[40,1958],{"className":1959,"style":122},[121],[40,1961,1671],{"className":1962},[102,335],[40,1964,73],{"className":1965},[117],[40,1967,1104],{"className":1968},[121],[40,1970],{"className":1971,"style":122},[121],[40,1973,1820],{"className":1974},[1369],[40,1976],{"className":1977,"style":122},[121],[40,1979,1649],{"className":1980,"style":1312},[102,335],[40,1982,1984],{"className":1983},[102],[40,1985,207],{"className":1986},[417,1238],[40,1988,1123],{"className":1989},[102],[25,1991],{},[28,1993,1995],{"id":1994},"_2-stratonovichweyl-sw-kernel-and-qubit-wigner-function","2. Stratonovich–Weyl (SW) Kernel and Qubit Wigner Function",[12,1997,1998],{},"A convenient SW kernel for a qubit is",[40,2000,2002],{"className":2001},[136],[40,2003,2005,2075],{"className":2004},[43],[40,2006,2008],{"className":2007},[47],[49,2009,2010],{"xmlns":51,"display":145},[53,2011,2012,2072],{},[56,2013,2014,2017,2019,2021,2023,2025,2027,2029,2037,2040,2042,2044,2049,2052,2054,2056,2058,2060,2062,2064,2066,2068,2070],{},[154,2015,2016],{"mathvariant":1080},"Δ",[59,2018,167],{"stretchy":61},[154,2020,1649],{},[59,2022,73],{"separator":72},[154,2024,1671],{},[59,2026,207],{"stretchy":61},[59,2028,162],{},[181,2030,2031],{"scriptlevel":66,"displaystyle":61},[1038,2032,2033,2035],{},[64,2034,78],{},[64,2036,1044],{},[59,2038,167],{"fence":61,"stretchy":72,"minsize":2039,"maxsize":2039},"1.8em",[154,2041,1050],{},[59,2043,1053],{},[2045,2046,2047],"msqrt",{},[64,2048,949],{},[59,2050,2051],{"separator":72},";",[154,2053,1796],{"mathvariant":935},[59,2055,167],{"stretchy":61},[154,2057,1649],{},[59,2059,73],{"separator":72},[154,2061,1671],{},[59,2063,207],{"stretchy":61},[59,2065,1058],{},[154,2067,156],{"mathvariant":1061},[59,2069,207],{"fence":61,"stretchy":72,"minsize":2039,"maxsize":2039},[59,2071,73],{"separator":72},[82,2073,2074],{"encoding":84},"\\Delta(\\theta,\\phi) = \\tfrac{1}{2}\\Big(I + \\sqrt{3};\\mathbf n(\\theta,\\phi)\\cdot\\boldsymbol{\\sigma}\\Big),",[40,2076,2078,2114,2208,2315],{"className":2077,"ariaHidden":72},[89],[40,2079,2081,2084,2087,2090,2093,2096,2099,2102,2105,2108,2111],{"className":2080},[93],[40,2082],{"className":2083,"style":98},[97],[40,2085,2016],{"className":2086},[102],[40,2088,167],{"className":2089},[106],[40,2091,1649],{"className":2092,"style":1312},[102,335],[40,2094,73],{"className":2095},[117],[40,2097],{"className":2098,"style":122},[121],[40,2100,1671],{"className":2101},[102,335],[40,2103,207],{"className":2104},[113],[40,2106],{"className":2107,"style":390},[121],[40,2109,162],{"className":2110},[394],[40,2112],{"className":2113,"style":390},[121],[40,2115,2117,2121,2189,2196,2199,2202,2205],{"className":2116},[93],[40,2118],{"className":2119,"style":2120},[97],"height:1.8em;vertical-align:-0.65em;",[40,2122,2124,2127,2186],{"className":2123},[102],[40,2125],{"className":2126},[106,1162],[40,2128,2130],{"className":2129},[1038],[40,2131,2133,2178],{"className":2132},[344,345],[40,2134,2136,2175],{"className":2135},[349],[40,2137,2139,2153,2161],{"className":2138,"style":1175},[353],[40,2140,2141,2144],{"style":1178},[40,2142],{"className":2143,"style":444},[361],[40,2145,2147],{"className":2146},[366,367,368,369],[40,2148,2150],{"className":2149},[102,369],[40,2151,1044],{"className":2152},[102,369],[40,2154,2155,2158],{"style":1193},[40,2156],{"className":2157,"style":444},[361],[40,2159],{"className":2160,"style":1201},[1200],[40,2162,2163,2166],{"style":1204},[40,2164],{"className":2165,"style":444},[361],[40,2167,2169],{"className":2168},[366,367,368,369],[40,2170,2172],{"className":2171},[102,369],[40,2173,78],{"className":2174},[102,369],[40,2176,377],{"className":2177},[376],[40,2179,2181],{"className":2180},[349],[40,2182,2184],{"className":2183,"style":1226},[353],[40,2185],{},[40,2187],{"className":2188},[113,1162],[40,2190,2192],{"className":2191},[102],[40,2193,167],{"className":2194},[417,2195],"size2",[40,2197,1050],{"className":2198,"style":1242},[102,335],[40,2200],{"className":2201,"style":1246},[121],[40,2203,1053],{"className":2204},[1250],[40,2206],{"className":2207,"style":1246},[121],[40,2209,2211,2215,2279,2282,2285,2288,2291,2294,2297,2300,2303,2306,2309,2312],{"className":2210},[93],[40,2212],{"className":2213,"style":2214},[97],"height:1.2061em;vertical-align:-0.25em;",[40,2216,2219],{"className":2217},[102,2218],"sqrt",[40,2220,2222,2270],{"className":2221},[344,345],[40,2223,2225,2267],{"className":2224},[349],[40,2226,2229,2244],{"className":2227,"style":2228},[353],"height:0.9561em;",[40,2230,2234,2237],{"className":2231,"style":2233},[2232],"svg-align","top:-3em;",[40,2235],{"className":2236,"style":444},[361],[40,2238,2241],{"className":2239,"style":2240},[102],"padding-left:0.833em;",[40,2242,949],{"className":2243},[102],[40,2245,2247,2250],{"style":2246},"top:-2.9161em;",[40,2248],{"className":2249,"style":444},[361],[40,2251,2255],{"className":2252,"style":2254},[2253],"hide-tail","min-width:0.853em;height:1.08em;",[2256,2257,2263],"svg",{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[2264,2265],"path",{"d":2266},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[40,2268,377],{"className":2269},[376],[40,2271,2273],{"className":2272},[349],[40,2274,2277],{"className":2275,"style":2276},[353],"height:0.0839em;",[40,2278],{},[40,2280,2051],{"className":2281},[117],[40,2283],{"className":2284,"style":122},[121],[40,2286,1796],{"className":2287},[102,966],[40,2289,167],{"className":2290},[106],[40,2292,1649],{"className":2293,"style":1312},[102,335],[40,2295,73],{"className":2296},[117],[40,2298],{"className":2299,"style":122},[121],[40,2301,1671],{"className":2302},[102,335],[40,2304,207],{"className":2305},[113],[40,2307],{"className":2308,"style":1246},[121],[40,2310,1058],{"className":2311},[1250],[40,2313],{"className":2314,"style":1246},[121],[40,2316,2318,2321,2330,2336],{"className":2317},[93],[40,2319],{"className":2320,"style":2120},[97],[40,2322,2324],{"className":2323},[102],[40,2325,2327],{"className":2326},[102],[40,2328,156],{"className":2329,"style":1289},[102,1288],[40,2331,2333],{"className":2332},[102],[40,2334,207],{"className":2335},[417,2195],[40,2337,73],{"className":2338},[117],[12,2340,2341,2342,2345,2346,2407],{},"which yields the ",[35,2343,2344],{},"spin Wigner function"," (on (",[40,2347,2349,2368],{"className":2348},[43],[40,2350,2352],{"className":2351},[47],[49,2353,2354],{"xmlns":51},[53,2355,2356,2365],{},[56,2357,2358],{},[941,2359,2360,2363],{},[154,2361,2362],{},"S",[64,2364,1044],{},[82,2366,2367],{"encoding":84},"S^2",[40,2369,2371],{"className":2370,"ariaHidden":72},[89],[40,2372,2374,2377],{"className":2373},[93],[40,2375],{"className":2376,"style":982},[97],[40,2378,2380,2384],{"className":2379},[102],[40,2381,2362],{"className":2382,"style":2383},[102,335],"margin-right:0.0576em;",[40,2385,2387],{"className":2386},[340],[40,2388,2390],{"className":2389},[344],[40,2391,2393],{"className":2392},[349],[40,2394,2396],{"className":2395,"style":982},[353],[40,2397,2398,2401],{"style":1004},[40,2399],{"className":2400,"style":362},[361],[40,2402,2404],{"className":2403},[366,367,368,369],[40,2405,1044],{"className":2406},[102,369],"))",[40,2409,2411],{"className":2410},[136],[40,2412,2414,2526],{"className":2413},[43],[40,2415,2417],{"className":2416},[47],[49,2418,2419],{"xmlns":51,"display":145},[53,2420,2421,2523],{},[169,2422,2424],{"width":2423},"100%",[175,2425,2426,2429,2516,2518],{},[178,2427],{"width":2428},"50%",[178,2430,2431],{},[56,2432,2433,2440,2442,2444,2446,2448,2450,2452,2454,2456,2458,2460,2462,2464,2466,2468,2470,2472,2474,2476,2478,2486,2488,2496,2498,2500,2502,2504,2506,2508,2510,2512,2514],{},[151,2434,2435,2438],{},[154,2436,2437],{},"W",[154,2439,1032],{},[59,2441,167],{"stretchy":61},[154,2443,1649],{},[59,2445,73],{"separator":72},[154,2447,1671],{},[59,2449,207],{"stretchy":61},[59,2451,162],{},[154,2453,1081],{"mathvariant":1080},[59,2455,1084],{},[59,2457,1654],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,2459,1032],{},[59,2461,73],{"separator":72},[154,2463,2016],{"mathvariant":1080},[59,2465,167],{"stretchy":61},[154,2467,1649],{},[59,2469,73],{"separator":72},[154,2471,1671],{},[59,2473,207],{"stretchy":61},[59,2475,1664],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,2477,162],{},[181,2479,2480],{"scriptlevel":66,"displaystyle":61},[1038,2481,2482,2484],{},[64,2483,78],{},[64,2485,1044],{},[59,2487,1053],{},[1038,2489,2490,2494],{},[2045,2491,2492],{},[64,2493,949],{},[64,2495,1044],{},[59,2497,2051],{"separator":72},[154,2499,936],{"mathvariant":935},[59,2501,1058],{},[154,2503,1796],{"mathvariant":935},[59,2505,167],{"stretchy":61},[154,2507,1649],{},[59,2509,73],{"separator":72},[154,2511,1671],{},[59,2513,207],{"stretchy":61},[154,2515,1123],{"mathvariant":1080},[178,2517],{"width":2428},[178,2519,2520],{},[1102,2521,2522],{},"(SW–Qubit)",[82,2524,2525],{"encoding":84},"W_\\rho(\\theta,\\phi) = \\operatorname{Tr}\\big[\\rho,\\Delta(\\theta,\\phi)\\big]\n= \\tfrac{1}{2} + \\frac{\\sqrt{3}}{2}; \\mathbf r\\cdot\\mathbf n(\\theta,\\phi).\n\\tag{SW–Qubit}",[40,2527,2529,2604,2670,2754,2892,2922],{"className":2528,"ariaHidden":72},[89],[40,2530,2532,2535,2577,2580,2583,2586,2589,2592,2595,2598,2601],{"className":2531},[93],[40,2533],{"className":2534,"style":1365},[97],[40,2536,2538,2542],{"className":2537},[102],[40,2539,2437],{"className":2540,"style":2541},[102,335],"margin-right:0.1389em;",[40,2543,2545],{"className":2544},[340],[40,2546,2548,2569],{"className":2547},[344,345],[40,2549,2551,2566],{"className":2550},[349],[40,2552,2554],{"className":2553,"style":354},[353],[40,2555,2557,2560],{"style":2556},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[40,2558],{"className":2559,"style":362},[361],[40,2561,2563],{"className":2562},[366,367,368,369],[40,2564,1032],{"className":2565},[102,335,369],[40,2567,377],{"className":2568},[376],[40,2570,2572],{"className":2571},[349],[40,2573,2575],{"className":2574,"style":585},[353],[40,2576],{},[40,2578,167],{"className":2579},[106],[40,2581,1649],{"className":2582,"style":1312},[102,335],[40,2584,73],{"className":2585},[117],[40,2587],{"className":2588,"style":122},[121],[40,2590,1671],{"className":2591},[102,335],[40,2593,207],{"className":2594},[113],[40,2596],{"className":2597,"style":390},[121],[40,2599,162],{"className":2600},[394],[40,2602],{"className":2603,"style":390},[121],[40,2605,2607,2610,2616,2619,2625,2628,2631,2634,2637,2640,2643,2646,2649,2652,2655,2661,2664,2667],{"className":2606},[93],[40,2608],{"className":2609,"style":1155},[97],[40,2611,2613],{"className":2612},[1369],[40,2614,1081],{"className":2615},[102,1373],[40,2617],{"className":2618,"style":122},[121],[40,2620,2622],{"className":2621},[102],[40,2623,1654],{"className":2624},[417,1238],[40,2626,1032],{"className":2627},[102,335],[40,2629,73],{"className":2630},[117],[40,2632],{"className":2633,"style":122},[121],[40,2635,2016],{"className":2636},[102],[40,2638,167],{"className":2639},[106],[40,2641,1649],{"className":2642,"style":1312},[102,335],[40,2644,73],{"className":2645},[117],[40,2647],{"className":2648,"style":122},[121],[40,2650,1671],{"className":2651},[102,335],[40,2653,207],{"className":2654},[113],[40,2656,2658],{"className":2657},[102],[40,2659,1664],{"className":2660},[417,1238],[40,2662],{"className":2663,"style":390},[121],[40,2665,162],{"className":2666},[394],[40,2668],{"className":2669,"style":390},[121],[40,2671,2673,2677,2745,2748,2751],{"className":2672},[93],[40,2674],{"className":2675,"style":2676},[97],"height:1.1901em;vertical-align:-0.345em;",[40,2678,2680,2683,2742],{"className":2679},[102],[40,2681],{"className":2682},[106,1162],[40,2684,2686],{"className":2685},[1038],[40,2687,2689,2734],{"className":2688},[344,345],[40,2690,2692,2731],{"className":2691},[349],[40,2693,2695,2709,2717],{"className":2694,"style":1175},[353],[40,2696,2697,2700],{"style":1178},[40,2698],{"className":2699,"style":444},[361],[40,2701,2703],{"className":2702},[366,367,368,369],[40,2704,2706],{"className":2705},[102,369],[40,2707,1044],{"className":2708},[102,369],[40,2710,2711,2714],{"style":1193},[40,2712],{"className":2713,"style":444},[361],[40,2715],{"className":2716,"style":1201},[1200],[40,2718,2719,2722],{"style":1204},[40,2720],{"className":2721,"style":444},[361],[40,2723,2725],{"className":2724},[366,367,368,369],[40,2726,2728],{"className":2727},[102,369],[40,2729,78],{"className":2730},[102,369],[40,2732,377],{"className":2733},[376],[40,2735,2737],{"className":2736},[349],[40,2738,2740],{"className":2739,"style":1226},[353],[40,2741],{},[40,2743],{"className":2744},[113,1162],[40,2746],{"className":2747,"style":1246},[121],[40,2749,1053],{"className":2750},[1250],[40,2752],{"className":2753,"style":1246},[121],[40,2755,2757,2761,2874,2877,2880,2883,2886,2889],{"className":2756},[93],[40,2758],{"className":2759,"style":2760},[97],"height:2.2702em;vertical-align:-0.686em;",[40,2762,2764,2767,2871],{"className":2763},[102],[40,2765],{"className":2766},[106,1162],[40,2768,2770],{"className":2769},[1038],[40,2771,2773,2862],{"className":2772},[344,345],[40,2774,2776,2859],{"className":2775},[349],[40,2777,2780,2792,2800],{"className":2778,"style":2779},[353],"height:1.5842em;",[40,2781,2783,2786],{"style":2782},"top:-2.314em;",[40,2784],{"className":2785,"style":444},[361],[40,2787,2789],{"className":2788},[102],[40,2790,1044],{"className":2791},[102],[40,2793,2794,2797],{"style":1193},[40,2795],{"className":2796,"style":444},[361],[40,2798],{"className":2799,"style":1201},[1200],[40,2801,2803,2806],{"style":2802},"top:-3.677em;",[40,2804],{"className":2805,"style":444},[361],[40,2807,2809],{"className":2808},[102],[40,2810,2812],{"className":2811},[102,2218],[40,2813,2815,2850],{"className":2814},[344,345],[40,2816,2818,2847],{"className":2817},[349],[40,2819,2822,2834],{"className":2820,"style":2821},[353],"height:0.9072em;",[40,2823,2825,2828],{"className":2824,"style":2233},[2232],[40,2826],{"className":2827,"style":444},[361],[40,2829,2831],{"className":2830,"style":2240},[102],[40,2832,949],{"className":2833},[102],[40,2835,2837,2840],{"style":2836},"top:-2.8672em;",[40,2838],{"className":2839,"style":444},[361],[40,2841,2843],{"className":2842,"style":2254},[2253],[2256,2844,2845],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,2846],{"d":2266},[40,2848,377],{"className":2849},[376],[40,2851,2853],{"className":2852},[349],[40,2854,2857],{"className":2855,"style":2856},[353],"height:0.1328em;",[40,2858],{},[40,2860,377],{"className":2861},[376],[40,2863,2865],{"className":2864},[349],[40,2866,2869],{"className":2867,"style":2868},[353],"height:0.686em;",[40,2870],{},[40,2872],{"className":2873},[113,1162],[40,2875,2051],{"className":2876},[117],[40,2878],{"className":2879,"style":122},[121],[40,2881,936],{"className":2882},[102,966],[40,2884],{"className":2885,"style":1246},[121],[40,2887,1058],{"className":2888},[1250],[40,2890],{"className":2891,"style":1246},[121],[40,2893,2895,2898,2901,2904,2907,2910,2913,2916,2919],{"className":2894},[93],[40,2896],{"className":2897,"style":98},[97],[40,2899,1796],{"className":2900},[102,966],[40,2902,167],{"className":2903},[106],[40,2905,1649],{"className":2906,"style":1312},[102,335],[40,2908,73],{"className":2909},[117],[40,2911],{"className":2912,"style":122},[121],[40,2914,1671],{"className":2915},[102,335],[40,2917,207],{"className":2918},[113],[40,2920,1123],{"className":2921},[102],[40,2923,2926,2929],{"className":2924},[2925],"tag",[40,2927],{"className":2928,"style":2760},[97],[40,2930,2933,2936,2943],{"className":2931},[102,2932],"text",[40,2934,167],{"className":2935},[102],[40,2937,2939],{"className":2938},[102],[40,2940,2942],{"className":2941},[102],"SW–Qubit",[40,2944,207],{"className":2945},[102],[12,2947,2948,2949,3010],{},"For a pure state ((",[40,2950,2952,2974],{"className":2951},[43],[40,2953,2955],{"className":2954},[47],[49,2956,2957],{"xmlns":51},[53,2958,2959,2971],{},[56,2960,2961,2963,2965,2967,2969],{},[59,2962,1579],{"stretchy":61},[154,2964,936],{"mathvariant":935},[59,2966,1579],{"stretchy":61},[59,2968,162],{},[64,2970,78],{},[82,2972,2973],{"encoding":84},"\\lVert\\mathbf r\\rVert=1",[40,2975,2977,3001],{"className":2976,"ariaHidden":72},[89],[40,2978,2980,2983,2986,2989,2992,2995,2998],{"className":2979},[93],[40,2981],{"className":2982,"style":98},[97],[40,2984,1579],{"className":2985},[106],[40,2987,936],{"className":2988},[102,966],[40,2990,1579],{"className":2991},[113],[40,2993],{"className":2994,"style":390},[121],[40,2996,162],{"className":2997},[394],[40,2999],{"className":3000,"style":390},[121],[40,3002,3004,3007],{"className":3003},[93],[40,3005],{"className":3006,"style":1624},[97],[40,3008,78],{"className":3009},[102],")), the extrema are",[40,3012,3014],{"className":3013},[136],[40,3015,3017,3097],{"className":3016},[43],[40,3018,3020],{"className":3019},[47],[49,3021,3022],{"xmlns":51,"display":145},[53,3023,3024,3094],{},[56,3025,3026,3035,3037,3045,3047,3057,3059,3061,3070,3072,3080,3082,3092],{},[151,3027,3028,3030,3033],{},[154,3029,2437],{},[154,3031,3032],{},"max",[59,3034,1084],{},[59,3036,162],{},[181,3038,3039],{"scriptlevel":66,"displaystyle":61},[1038,3040,3041,3043],{},[64,3042,78],{},[64,3044,1044],{},[59,3046,1053],{},[181,3048,3049],{"scriptlevel":66,"displaystyle":61},[1038,3050,3051,3055],{},[2045,3052,3053],{},[64,3054,949],{},[64,3056,1044],{},[59,3058,73],{"separator":72},[121,3060],{"width":1068},[151,3062,3063,3065,3068],{},[154,3064,2437],{},[154,3066,3067],{},"min",[59,3069,1084],{},[59,3071,162],{},[181,3073,3074],{"scriptlevel":66,"displaystyle":61},[1038,3075,3076,3078],{},[64,3077,78],{},[64,3079,1044],{},[59,3081,243],{},[181,3083,3084],{"scriptlevel":66,"displaystyle":61},[1038,3085,3086,3090],{},[2045,3087,3088],{},[64,3089,949],{},[64,3091,1044],{},[154,3093,1123],{"mathvariant":1080},[82,3095,3096],{"encoding":84},"W_{\\max}=\\tfrac{1}{2}+\\tfrac{\\sqrt{3}}{2},\\qquad W_{\\min}=\\tfrac{1}{2}-\\tfrac{\\sqrt{3}}{2}.",[40,3098,3100,3169,3252,3447,3530],{"className":3099,"ariaHidden":72},[89],[40,3101,3103,3107,3160,3163,3166],{"className":3102},[93],[40,3104],{"className":3105,"style":3106},[97],"height:0.8333em;vertical-align:-0.15em;",[40,3108,3110,3113],{"className":3109},[102],[40,3111,2437],{"className":3112,"style":2541},[102,335],[40,3114,3116],{"className":3115},[340],[40,3117,3119,3152],{"className":3118},[344,345],[40,3120,3122,3149],{"className":3121},[349],[40,3123,3125],{"className":3124,"style":354},[353],[40,3126,3127,3130],{"style":2556},[40,3128],{"className":3129,"style":362},[361],[40,3131,3133],{"className":3132},[366,367,368,369],[40,3134,3136],{"className":3135},[102,369],[40,3137,3139,3143,3146],{"className":3138},[1369,369],[40,3140,3142],{"className":3141},[369],"m",[40,3144,19],{"className":3145},[369],[40,3147,159],{"className":3148},[369],[40,3150,377],{"className":3151},[376],[40,3153,3155],{"className":3154},[349],[40,3156,3158],{"className":3157,"style":384},[353],[40,3159],{},[40,3161],{"className":3162,"style":390},[121],[40,3164,162],{"className":3165},[394],[40,3167],{"className":3168,"style":390},[121],[40,3170,3172,3175,3243,3246,3249],{"className":3171},[93],[40,3173],{"className":3174,"style":2676},[97],[40,3176,3178,3181,3240],{"className":3177},[102],[40,3179],{"className":3180},[106,1162],[40,3182,3184],{"className":3183},[1038],[40,3185,3187,3232],{"className":3186},[344,345],[40,3188,3190,3229],{"className":3189},[349],[40,3191,3193,3207,3215],{"className":3192,"style":1175},[353],[40,3194,3195,3198],{"style":1178},[40,3196],{"className":3197,"style":444},[361],[40,3199,3201],{"className":3200},[366,367,368,369],[40,3202,3204],{"className":3203},[102,369],[40,3205,1044],{"className":3206},[102,369],[40,3208,3209,3212],{"style":1193},[40,3210],{"className":3211,"style":444},[361],[40,3213],{"className":3214,"style":1201},[1200],[40,3216,3217,3220],{"style":1204},[40,3218],{"className":3219,"style":444},[361],[40,3221,3223],{"className":3222},[366,367,368,369],[40,3224,3226],{"className":3225},[102,369],[40,3227,78],{"className":3228},[102,369],[40,3230,377],{"className":3231},[376],[40,3233,3235],{"className":3234},[349],[40,3236,3238],{"className":3237,"style":1226},[353],[40,3239],{},[40,3241],{"className":3242},[113,1162],[40,3244],{"className":3245,"style":1246},[121],[40,3247,1053],{"className":3248},[1250],[40,3250],{"className":3251,"style":1246},[121],[40,3253,3255,3259,3376,3379,3382,3385,3438,3441,3444],{"className":3254},[93],[40,3256],{"className":3257,"style":3258},[97],"height:1.383em;vertical-align:-0.345em;",[40,3260,3262,3265,3373],{"className":3261},[102],[40,3263],{"className":3264},[106,1162],[40,3266,3268],{"className":3267},[1038],[40,3269,3271,3365],{"className":3270},[344,345],[40,3272,3274,3362],{"className":3273},[349],[40,3275,3278,3292,3300],{"className":3276,"style":3277},[353],"height:1.038em;",[40,3279,3280,3283],{"style":1178},[40,3281],{"className":3282,"style":444},[361],[40,3284,3286],{"className":3285},[366,367,368,369],[40,3287,3289],{"className":3288},[102,369],[40,3290,1044],{"className":3291},[102,369],[40,3293,3294,3297],{"style":1193},[40,3295],{"className":3296,"style":444},[361],[40,3298],{"className":3299,"style":1201},[1200],[40,3301,3303,3306],{"style":3302},"top:-3.399em;",[40,3304],{"className":3305,"style":444},[361],[40,3307,3309],{"className":3308},[366,367,368,369],[40,3310,3312],{"className":3311},[102,369],[40,3313,3315],{"className":3314},[102,2218,369],[40,3316,3318,3353],{"className":3317},[344,345],[40,3319,3321,3350],{"className":3320},[349],[40,3322,3325,3337],{"className":3323,"style":3324},[353],"height:0.9128em;",[40,3326,3328,3331],{"className":3327,"style":2233},[2232],[40,3329],{"className":3330,"style":444},[361],[40,3332,3334],{"className":3333,"style":2240},[102,369],[40,3335,949],{"className":3336},[102,369],[40,3338,3340,3343],{"style":3339},"top:-2.8728em;",[40,3341],{"className":3342,"style":444},[361],[40,3344,3346],{"className":3345,"style":2254},[2253,369],[2256,3347,3348],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,3349],{"d":2266},[40,3351,377],{"className":3352},[376],[40,3354,3356],{"className":3355},[349],[40,3357,3360],{"className":3358,"style":3359},[353],"height:0.1272em;",[40,3361],{},[40,3363,377],{"className":3364},[376],[40,3366,3368],{"className":3367},[349],[40,3369,3371],{"className":3370,"style":1226},[353],[40,3372],{},[40,3374],{"className":3375},[113,1162],[40,3377,73],{"className":3378},[117],[40,3380],{"className":3381,"style":1302},[121],[40,3383],{"className":3384,"style":122},[121],[40,3386,3388,3391],{"className":3387},[102],[40,3389,2437],{"className":3390,"style":2541},[102,335],[40,3392,3394],{"className":3393},[340],[40,3395,3397,3430],{"className":3396},[344,345],[40,3398,3400,3427],{"className":3399},[349],[40,3401,3404],{"className":3402,"style":3403},[353],"height:0.3175em;",[40,3405,3406,3409],{"style":2556},[40,3407],{"className":3408,"style":362},[361],[40,3410,3412],{"className":3411},[366,367,368,369],[40,3413,3415],{"className":3414},[102,369],[40,3416,3418,3421,3424],{"className":3417},[1369,369],[40,3419,3142],{"className":3420},[369],[40,3422,246],{"className":3423},[369],[40,3425,1796],{"className":3426},[369],[40,3428,377],{"className":3429},[376],[40,3431,3433],{"className":3432},[349],[40,3434,3436],{"className":3435,"style":384},[353],[40,3437],{},[40,3439],{"className":3440,"style":390},[121],[40,3442,162],{"className":3443},[394],[40,3445],{"className":3446,"style":390},[121],[40,3448,3450,3453,3521,3524,3527],{"className":3449},[93],[40,3451],{"className":3452,"style":2676},[97],[40,3454,3456,3459,3518],{"className":3455},[102],[40,3457],{"className":3458},[106,1162],[40,3460,3462],{"className":3461},[1038],[40,3463,3465,3510],{"className":3464},[344,345],[40,3466,3468,3507],{"className":3467},[349],[40,3469,3471,3485,3493],{"className":3470,"style":1175},[353],[40,3472,3473,3476],{"style":1178},[40,3474],{"className":3475,"style":444},[361],[40,3477,3479],{"className":3478},[366,367,368,369],[40,3480,3482],{"className":3481},[102,369],[40,3483,1044],{"className":3484},[102,369],[40,3486,3487,3490],{"style":1193},[40,3488],{"className":3489,"style":444},[361],[40,3491],{"className":3492,"style":1201},[1200],[40,3494,3495,3498],{"style":1204},[40,3496],{"className":3497,"style":444},[361],[40,3499,3501],{"className":3500},[366,367,368,369],[40,3502,3504],{"className":3503},[102,369],[40,3505,78],{"className":3506},[102,369],[40,3508,377],{"className":3509},[376],[40,3511,3513],{"className":3512},[349],[40,3514,3516],{"className":3515,"style":1226},[353],[40,3517],{},[40,3519],{"className":3520},[113,1162],[40,3522],{"className":3523,"style":1246},[121],[40,3525,243],{"className":3526},[1250],[40,3528],{"className":3529,"style":1246},[121],[40,3531,3533,3536,3648],{"className":3532},[93],[40,3534],{"className":3535,"style":3258},[97],[40,3537,3539,3542,3645],{"className":3538},[102],[40,3540],{"className":3541},[106,1162],[40,3543,3545],{"className":3544},[1038],[40,3546,3548,3637],{"className":3547},[344,345],[40,3549,3551,3634],{"className":3550},[349],[40,3552,3554,3568,3576],{"className":3553,"style":3277},[353],[40,3555,3556,3559],{"style":1178},[40,3557],{"className":3558,"style":444},[361],[40,3560,3562],{"className":3561},[366,367,368,369],[40,3563,3565],{"className":3564},[102,369],[40,3566,1044],{"className":3567},[102,369],[40,3569,3570,3573],{"style":1193},[40,3571],{"className":3572,"style":444},[361],[40,3574],{"className":3575,"style":1201},[1200],[40,3577,3578,3581],{"style":3302},[40,3579],{"className":3580,"style":444},[361],[40,3582,3584],{"className":3583},[366,367,368,369],[40,3585,3587],{"className":3586},[102,369],[40,3588,3590],{"className":3589},[102,2218,369],[40,3591,3593,3626],{"className":3592},[344,345],[40,3594,3596,3623],{"className":3595},[349],[40,3597,3599,3611],{"className":3598,"style":3324},[353],[40,3600,3602,3605],{"className":3601,"style":2233},[2232],[40,3603],{"className":3604,"style":444},[361],[40,3606,3608],{"className":3607,"style":2240},[102,369],[40,3609,949],{"className":3610},[102,369],[40,3612,3613,3616],{"style":3339},[40,3614],{"className":3615,"style":444},[361],[40,3617,3619],{"className":3618,"style":2254},[2253,369],[2256,3620,3621],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,3622],{"d":2266},[40,3624,377],{"className":3625},[376],[40,3627,3629],{"className":3628},[349],[40,3630,3632],{"className":3631,"style":3359},[353],[40,3633],{},[40,3635,377],{"className":3636},[376],[40,3638,3640],{"className":3639},[349],[40,3641,3643],{"className":3642,"style":1226},[353],[40,3644],{},[40,3646],{"className":3647},[113,1162],[40,3649,1123],{"className":3650},[102],[12,3652,3653,3656],{},[35,3654,3655],{},"Covariance (rigid rotation):"," for any unitary (U),",[40,3658,3660],{"className":3659},[136],[40,3661,3663,3758],{"className":3662},[43],[40,3664,3666],{"className":3665},[47],[49,3667,3668],{"xmlns":51,"display":145},[53,3669,3670,3755],{},[169,3671,3672],{"width":2423},[175,3673,3674,3676,3748,3750],{},[178,3675],{"width":2428},[178,3677,3678],{},[56,3679,3680,3699,3701,3703,3705,3707,3709,3711,3717,3720,3722,3732,3734,3736,3738,3740,3742,3744,3746],{},[151,3681,3682,3684],{},[154,3683,2437],{},[56,3685,3686,3689,3691],{},[154,3687,3688],{},"U",[154,3690,1032],{},[941,3692,3693,3695],{},[154,3694,3688],{},[59,3696,3698],{"lspace":3697,"rspace":3697},"0em","†",[59,3700,167],{"stretchy":61},[154,3702,1649],{},[59,3704,73],{"separator":72},[154,3706,1671],{},[59,3708,207],{"stretchy":61},[59,3710,162],{},[151,3712,3713,3715],{},[154,3714,2437],{},[154,3716,1032],{},[59,3718,3719],{"stretchy":61},"!",[59,3721,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[941,3723,3724,3726],{},[154,3725,946],{},[56,3727,3728,3730],{},[59,3729,243],{},[64,3731,78],{},[154,3733,1796],{"mathvariant":935},[59,3735,167],{"stretchy":61},[154,3737,1649],{},[59,3739,73],{"separator":72},[154,3741,1671],{},[59,3743,207],{"stretchy":61},[59,3745,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,3747,73],{"separator":72},[178,3749],{"width":2428},[178,3751,3752],{},[1102,3753,3754],{},"(Covariance)",[82,3756,3757],{"encoding":84},"W_{U\\rho U^{\\dagger}}(\\theta,\\phi) = W_{\\rho}!\\big(R^{-1}\\mathbf n(\\theta,\\phi)\\big),\n\\tag{Covariance}",[40,3759,3761,3881,4008],{"className":3760,"ariaHidden":72},[89],[40,3762,3764,3768,3854,3857,3860,3863,3866,3869,3872,3875,3878],{"className":3763},[93],[40,3765],{"className":3766,"style":3767},[97],"height:1.0887em;vertical-align:-0.3387em;",[40,3769,3771,3774],{"className":3770},[102],[40,3772,2437],{"className":3773,"style":2541},[102,335],[40,3775,3777],{"className":3776},[340],[40,3778,3780,3845],{"className":3779},[344,345],[40,3781,3783,3842],{"className":3782},[349],[40,3784,3787],{"className":3785,"style":3786},[353],"height:0.3448em;",[40,3788,3790,3793],{"style":3789},"top:-2.4974em;margin-left:-0.1389em;margin-right:0.05em;",[40,3791],{"className":3792,"style":362},[361],[40,3794,3796],{"className":3795},[366,367,368,369],[40,3797,3799,3803,3806],{"className":3798},[102,369],[40,3800,3688],{"className":3801,"style":3802},[102,335,369],"margin-right:0.109em;",[40,3804,1032],{"className":3805},[102,335,369],[40,3807,3809,3812],{"className":3808},[102,369],[40,3810,3688],{"className":3811,"style":3802},[102,335,369],[40,3813,3815],{"className":3814},[340],[40,3816,3818],{"className":3817},[344],[40,3819,3821],{"className":3820},[349],[40,3822,3825],{"className":3823,"style":3824},[353],"height:0.782em;",[40,3826,3828,3832],{"style":3827},"top:-2.786em;margin-right:0.0714em;",[40,3829],{"className":3830,"style":3831},[361],"height:2.5em;",[40,3833,3836],{"className":3834},[366,3835,1238,369],"reset-size3",[40,3837,3839],{"className":3838},[102,369],[40,3840,3698],{"className":3841},[102,369],[40,3843,377],{"className":3844},[376],[40,3846,3848],{"className":3847},[349],[40,3849,3852],{"className":3850,"style":3851},[353],"height:0.3387em;",[40,3853],{},[40,3855,167],{"className":3856},[106],[40,3858,1649],{"className":3859,"style":1312},[102,335],[40,3861,73],{"className":3862},[117],[40,3864],{"className":3865,"style":122},[121],[40,3867,1671],{"className":3868},[102,335],[40,3870,207],{"className":3871},[113],[40,3873],{"className":3874,"style":390},[121],[40,3876,162],{"className":3877},[394],[40,3879],{"className":3880,"style":390},[121],[40,3882,3884,3888,3931,3934,3940,3978,3981,3984,3987,3990,3993,3996,3999,4005],{"className":3883},[93],[40,3885],{"className":3886,"style":3887},[97],"height:1.2141em;vertical-align:-0.35em;",[40,3889,3891,3894],{"className":3890},[102],[40,3892,2437],{"className":3893,"style":2541},[102,335],[40,3895,3897],{"className":3896},[340],[40,3898,3900,3923],{"className":3899},[344,345],[40,3901,3903,3920],{"className":3902},[349],[40,3904,3906],{"className":3905,"style":354},[353],[40,3907,3908,3911],{"style":2556},[40,3909],{"className":3910,"style":362},[361],[40,3912,3914],{"className":3913},[366,367,368,369],[40,3915,3917],{"className":3916},[102,369],[40,3918,1032],{"className":3919},[102,335,369],[40,3921,377],{"className":3922},[376],[40,3924,3926],{"className":3925},[349],[40,3927,3929],{"className":3928,"style":585},[353],[40,3930],{},[40,3932,3719],{"className":3933},[113],[40,3935,3937],{"className":3936},[102],[40,3938,167],{"className":3939},[417,1238],[40,3941,3943,3947],{"className":3942},[102],[40,3944,946],{"className":3945,"style":3946},[102,335],"margin-right:0.0077em;",[40,3948,3950],{"className":3949},[340],[40,3951,3953],{"className":3952},[344],[40,3954,3956],{"className":3955},[349],[40,3957,3960],{"className":3958,"style":3959},[353],"height:0.8641em;",[40,3961,3963,3966],{"style":3962},"top:-3.113em;margin-right:0.05em;",[40,3964],{"className":3965,"style":362},[361],[40,3967,3969],{"className":3968},[366,367,368,369],[40,3970,3972,3975],{"className":3971},[102,369],[40,3973,243],{"className":3974},[102,369],[40,3976,78],{"className":3977},[102,369],[40,3979,1796],{"className":3980},[102,966],[40,3982,167],{"className":3983},[106],[40,3985,1649],{"className":3986,"style":1312},[102,335],[40,3988,73],{"className":3989},[117],[40,3991],{"className":3992,"style":122},[121],[40,3994,1671],{"className":3995},[102,335],[40,3997,207],{"className":3998},[113],[40,4000,4002],{"className":4001},[102],[40,4003,207],{"className":4004},[417,1238],[40,4006,73],{"className":4007},[117],[40,4009,4011,4014],{"className":4010},[2925],[40,4012],{"className":4013,"style":3887},[97],[40,4015,4017,4020,4027],{"className":4016},[102,2932],[40,4018,167],{"className":4019},[102],[40,4021,4023],{"className":4022},[102],[40,4024,4026],{"className":4025},[102],"Covariance",[40,4028,207],{"className":4029},[102],[12,4031,4032,4033,4109],{},"where (",[40,4034,4036,4065],{"className":4035},[43],[40,4037,4039],{"className":4038},[47],[49,4040,4041],{"xmlns":51},[53,4042,4043,4062],{},[56,4044,4045,4047,4049,4056,4058,4060],{},[154,4046,946],{},[59,4048,939],{},[56,4050,4051,4053],{},[154,4052,2362],{"mathvariant":1080},[154,4054,4055],{"mathvariant":1080},"O",[59,4057,167],{"stretchy":61},[64,4059,949],{},[59,4061,207],{"stretchy":61},[82,4063,4064],{"encoding":84},"R\\in\\mathrm{SO}(3)",[40,4066,4068,4087],{"className":4067,"ariaHidden":72},[89],[40,4069,4071,4075,4078,4081,4084],{"className":4070},[93],[40,4072],{"className":4073,"style":4074},[97],"height:0.7224em;vertical-align:-0.0391em;",[40,4076,946],{"className":4077,"style":3946},[102,335],[40,4079],{"className":4080,"style":390},[121],[40,4082,939],{"className":4083},[394],[40,4085],{"className":4086,"style":390},[121],[40,4088,4090,4093,4100,4103,4106],{"className":4089},[93],[40,4091],{"className":4092,"style":98},[97],[40,4094,4096],{"className":4095},[102],[40,4097,4099],{"className":4098},[102,1373],"SO",[40,4101,167],{"className":4102},[106],[40,4104,949],{"className":4105},[102],[40,4107,207],{"className":4108},[113],") is the 3D rotation associated with (U) (see §3–§4).",[25,4111],{},[28,4113,4115,4116,4146],{"id":4114},"_3-pauli-gates-as-πpiπ-rotations","3. Pauli Gates as (",[40,4117,4119,4133],{"className":4118},[43],[40,4120,4122],{"className":4121},[47],[49,4123,4124],{"xmlns":51},[53,4125,4126,4130],{},[56,4127,4128],{},[154,4129,1661],{},[82,4131,4132],{"encoding":84},"\\pi",[40,4134,4136],{"className":4135,"ariaHidden":72},[89],[40,4137,4139,4143],{"className":4138},[93],[40,4140],{"className":4141,"style":4142},[97],"height:0.4306em;",[40,4144,1661],{"className":4145,"style":336},[102,335],")-Rotations",[12,4148,4149,4150,4178],{},"The single‑qubit Pauli gates are equivalent (up to a global phase) to (",[40,4151,4153,4166],{"className":4152},[43],[40,4154,4156],{"className":4155},[47],[49,4157,4158],{"xmlns":51},[53,4159,4160,4164],{},[56,4161,4162],{},[154,4163,1661],{},[82,4165,4132],{"encoding":84},[40,4167,4169],{"className":4168,"ariaHidden":72},[89],[40,4170,4172,4175],{"className":4171},[93],[40,4173],{"className":4174,"style":4142},[97],[40,4176,1661],{"className":4177,"style":336},[102,335],")-rotations about Cartesian axes:",[40,4180,4182],{"className":4181},[136],[40,4183,4185,4295],{"className":4184},[43],[40,4186,4188],{"className":4187},[47],[49,4189,4190],{"xmlns":51,"display":145},[53,4191,4192,4292],{},[56,4193,4194,4197,4199,4224,4226,4228,4231,4233,4257,4259,4261,4264,4266,4290],{},[154,4195,4196],{},"X",[59,4198,162],{},[941,4200,4201,4204],{},[154,4202,4203],{},"e",[56,4205,4206,4208,4210,4216,4218],{},[59,4207,243],{},[154,4209,246],{},[1038,4211,4212,4214],{},[154,4213,1661],{},[64,4215,1044],{},[59,4217,73],{"separator":72},[151,4219,4220,4222],{},[154,4221,156],{},[154,4223,159],{},[59,4225,73],{"separator":72},[121,4227],{"width":1068},[154,4229,4230],{},"Y",[59,4232,162],{},[941,4234,4235,4237],{},[154,4236,4203],{},[56,4238,4239,4241,4243,4249,4251],{},[59,4240,243],{},[154,4242,246],{},[1038,4244,4245,4247],{},[154,4246,1661],{},[64,4248,1044],{},[59,4250,73],{"separator":72},[151,4252,4253,4255],{},[154,4254,156],{},[154,4256,218],{},[59,4258,73],{"separator":72},[121,4260],{"width":1068},[154,4262,4263],{},"Z",[59,4265,162],{},[941,4267,4268,4270],{},[154,4269,4203],{},[56,4271,4272,4274,4276,4282,4284],{},[59,4273,243],{},[154,4275,246],{},[1038,4277,4278,4280],{},[154,4279,1661],{},[64,4281,1044],{},[59,4283,73],{"separator":72},[151,4285,4286,4288],{},[154,4287,156],{},[154,4289,273],{},[154,4291,1123],{"mathvariant":1080},[82,4293,4294],{"encoding":84},"X = e^{-i\\frac{\\pi}{2},\\sigma_x},\\qquad\nY = e^{-i\\frac{\\pi}{2},\\sigma_y},\\qquad\nZ = e^{-i\\frac{\\pi}{2},\\sigma_z}.",[40,4296,4298,4317,4503,4678],{"className":4297,"ariaHidden":72},[89],[40,4299,4301,4305,4308,4311,4314],{"className":4300},[93],[40,4302],{"className":4303,"style":4304},[97],"height:0.6833em;",[40,4306,4196],{"className":4307,"style":1242},[102,335],[40,4309],{"className":4310,"style":390},[121],[40,4312,162],{"className":4313},[394],[40,4315],{"className":4316,"style":390},[121],[40,4318,4320,4324,4482,4485,4488,4491,4494,4497,4500],{"className":4319},[93],[40,4321],{"className":4322,"style":4323},[97],"height:1.0915em;vertical-align:-0.1944em;",[40,4325,4327,4330],{"className":4326},[102],[40,4328,4203],{"className":4329},[102,335],[40,4331,4333],{"className":4332},[340],[40,4334,4336],{"className":4335},[344],[40,4337,4339],{"className":4338},[349],[40,4340,4343],{"className":4341,"style":4342},[353],"height:0.8971em;",[40,4344,4346,4349],{"style":4345},"top:-3.413em;margin-right:0.05em;",[40,4347],{"className":4348,"style":444},[361],[40,4350,4352],{"className":4351},[366,367,368,369],[40,4353,4355,4358,4361,4436,4439],{"className":4354},[102,369],[40,4356,243],{"className":4357},[102,369],[40,4359,246],{"className":4360},[102,335,369],[40,4362,4364,4368,4433],{"className":4363},[102,369],[40,4365],{"className":4366},[106,1162,366,3835,4367],"size6",[40,4369,4371],{"className":4370},[1038],[40,4372,4374,4424],{"className":4373},[344,345],[40,4375,4377,4421],{"className":4376},[349],[40,4378,4381,4396,4406],{"className":4379,"style":4380},[353],"height:0.6915em;",[40,4382,4384,4387],{"style":4383},"top:-2.656em;",[40,4385],{"className":4386,"style":444},[361],[40,4388,4390],{"className":4389},[366,3835,1238,369],[40,4391,4393],{"className":4392},[102,369],[40,4394,1044],{"className":4395},[102,369],[40,4397,4399,4402],{"style":4398},"top:-3.2255em;",[40,4400],{"className":4401,"style":444},[361],[40,4403],{"className":4404,"style":4405},[1200,369],"border-bottom-width:0.049em;",[40,4407,4409,4412],{"style":4408},"top:-3.384em;",[40,4410],{"className":4411,"style":444},[361],[40,4413,4415],{"className":4414},[366,3835,1238,369],[40,4416,4418],{"className":4417},[102,369],[40,4419,1661],{"className":4420,"style":336},[102,335,369],[40,4422,377],{"className":4423},[376],[40,4425,4427],{"className":4426},[349],[40,4428,4431],{"className":4429,"style":4430},[353],"height:0.344em;",[40,4432],{},[40,4434],{"className":4435},[113,1162,366,3835,4367],[40,4437,73],{"className":4438},[117,369],[40,4440,4442,4445],{"className":4441},[102,369],[40,4443,156],{"className":4444,"style":336},[102,335,369],[40,4446,4448],{"className":4447},[340],[40,4449,4451,4473],{"className":4450},[344,345],[40,4452,4454,4470],{"className":4453},[349],[40,4455,4458],{"className":4456,"style":4457},[353],"height:0.1645em;",[40,4459,4461,4464],{"style":4460},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[40,4462],{"className":4463,"style":3831},[361],[40,4465,4467],{"className":4466},[366,3835,1238,369],[40,4468,159],{"className":4469},[102,335,369],[40,4471,377],{"className":4472},[376],[40,4474,4476],{"className":4475},[349],[40,4477,4480],{"className":4478,"style":4479},[353],"height:0.143em;",[40,4481],{},[40,4483,73],{"className":4484},[117],[40,4486],{"className":4487,"style":1302},[121],[40,4489],{"className":4490,"style":122},[121],[40,4492,4230],{"className":4493,"style":1246},[102,335],[40,4495],{"className":4496,"style":390},[121],[40,4498,162],{"className":4499},[394],[40,4501],{"className":4502,"style":390},[121],[40,4504,4506,4509,4656,4659,4662,4665,4669,4672,4675],{"className":4505},[93],[40,4507],{"className":4508,"style":4323},[97],[40,4510,4512,4515],{"className":4511},[102],[40,4513,4203],{"className":4514},[102,335],[40,4516,4518],{"className":4517},[340],[40,4519,4521],{"className":4520},[344],[40,4522,4524],{"className":4523},[349],[40,4525,4527],{"className":4526,"style":4342},[353],[40,4528,4529,4532],{"style":4345},[40,4530],{"className":4531,"style":444},[361],[40,4533,4535],{"className":4534},[366,367,368,369],[40,4536,4538,4541,4544,4612,4615],{"className":4537},[102,369],[40,4539,243],{"className":4540},[102,369],[40,4542,246],{"className":4543},[102,335,369],[40,4545,4547,4550,4609],{"className":4546},[102,369],[40,4548],{"className":4549},[106,1162,366,3835,4367],[40,4551,4553],{"className":4552},[1038],[40,4554,4556,4601],{"className":4555},[344,345],[40,4557,4559,4598],{"className":4558},[349],[40,4560,4562,4576,4584],{"className":4561,"style":4380},[353],[40,4563,4564,4567],{"style":4383},[40,4565],{"className":4566,"style":444},[361],[40,4568,4570],{"className":4569},[366,3835,1238,369],[40,4571,4573],{"className":4572},[102,369],[40,4574,1044],{"className":4575},[102,369],[40,4577,4578,4581],{"style":4398},[40,4579],{"className":4580,"style":444},[361],[40,4582],{"className":4583,"style":4405},[1200,369],[40,4585,4586,4589],{"style":4408},[40,4587],{"className":4588,"style":444},[361],[40,4590,4592],{"className":4591},[366,3835,1238,369],[40,4593,4595],{"className":4594},[102,369],[40,4596,1661],{"className":4597,"style":336},[102,335,369],[40,4599,377],{"className":4600},[376],[40,4602,4604],{"className":4603},[349],[40,4605,4607],{"className":4606,"style":4430},[353],[40,4608],{},[40,4610],{"className":4611},[113,1162,366,3835,4367],[40,4613,73],{"className":4614},[117,369],[40,4616,4618,4621],{"className":4617},[102,369],[40,4619,156],{"className":4620,"style":336},[102,335,369],[40,4622,4624],{"className":4623},[340],[40,4625,4627,4647],{"className":4626},[344,345],[40,4628,4630,4644],{"className":4629},[349],[40,4631,4633],{"className":4632,"style":4457},[353],[40,4634,4635,4638],{"style":4460},[40,4636],{"className":4637,"style":3831},[361],[40,4639,4641],{"className":4640},[366,3835,1238,369],[40,4642,218],{"className":4643,"style":336},[102,335,369],[40,4645,377],{"className":4646},[376],[40,4648,4650],{"className":4649},[349],[40,4651,4654],{"className":4652,"style":4653},[353],"height:0.2819em;",[40,4655],{},[40,4657,73],{"className":4658},[117],[40,4660],{"className":4661,"style":1302},[121],[40,4663],{"className":4664,"style":122},[121],[40,4666,4263],{"className":4667,"style":4668},[102,335],"margin-right:0.0715em;",[40,4670],{"className":4671,"style":390},[121],[40,4673,162],{"className":4674},[394],[40,4676],{"className":4677,"style":390},[121],[40,4679,4681,4684,4830],{"className":4680},[93],[40,4682],{"className":4683,"style":4342},[97],[40,4685,4687,4690],{"className":4686},[102],[40,4688,4203],{"className":4689},[102,335],[40,4691,4693],{"className":4692},[340],[40,4694,4696],{"className":4695},[344],[40,4697,4699],{"className":4698},[349],[40,4700,4702],{"className":4701,"style":4342},[353],[40,4703,4704,4707],{"style":4345},[40,4705],{"className":4706,"style":444},[361],[40,4708,4710],{"className":4709},[366,367,368,369],[40,4711,4713,4716,4719,4787,4790],{"className":4712},[102,369],[40,4714,243],{"className":4715},[102,369],[40,4717,246],{"className":4718},[102,335,369],[40,4720,4722,4725,4784],{"className":4721},[102,369],[40,4723],{"className":4724},[106,1162,366,3835,4367],[40,4726,4728],{"className":4727},[1038],[40,4729,4731,4776],{"className":4730},[344,345],[40,4732,4734,4773],{"className":4733},[349],[40,4735,4737,4751,4759],{"className":4736,"style":4380},[353],[40,4738,4739,4742],{"style":4383},[40,4740],{"className":4741,"style":444},[361],[40,4743,4745],{"className":4744},[366,3835,1238,369],[40,4746,4748],{"className":4747},[102,369],[40,4749,1044],{"className":4750},[102,369],[40,4752,4753,4756],{"style":4398},[40,4754],{"className":4755,"style":444},[361],[40,4757],{"className":4758,"style":4405},[1200,369],[40,4760,4761,4764],{"style":4408},[40,4762],{"className":4763,"style":444},[361],[40,4765,4767],{"className":4766},[366,3835,1238,369],[40,4768,4770],{"className":4769},[102,369],[40,4771,1661],{"className":4772,"style":336},[102,335,369],[40,4774,377],{"className":4775},[376],[40,4777,4779],{"className":4778},[349],[40,4780,4782],{"className":4781,"style":4430},[353],[40,4783],{},[40,4785],{"className":4786},[113,1162,366,3835,4367],[40,4788,73],{"className":4789},[117,369],[40,4791,4793,4796],{"className":4792},[102,369],[40,4794,156],{"className":4795,"style":336},[102,335,369],[40,4797,4799],{"className":4798},[340],[40,4800,4802,4822],{"className":4801},[344,345],[40,4803,4805,4819],{"className":4804},[349],[40,4806,4808],{"className":4807,"style":4457},[353],[40,4809,4810,4813],{"style":4460},[40,4811],{"className":4812,"style":3831},[361],[40,4814,4816],{"className":4815},[366,3835,1238,369],[40,4817,273],{"className":4818,"style":764},[102,335,369],[40,4820,377],{"className":4821},[376],[40,4823,4825],{"className":4824},[349],[40,4826,4828],{"className":4827,"style":4479},[353],[40,4829],{},[40,4831,1123],{"className":4832},[102],[12,4834,4835,4836,5060,5061,5063,5064,1628],{},"They act on the Bloch vector by the corresponding rotation matrices (",[40,4837,4839,4891],{"className":4838},[43],[40,4840,4842],{"className":4841},[47],[49,4843,4844],{"xmlns":51},[53,4845,4846,4888],{},[56,4847,4848,4854,4856,4858,4860,4862,4868,4870,4872,4874,4876,4882,4884,4886],{},[151,4849,4850,4852],{},[154,4851,946],{},[154,4853,159],{},[59,4855,167],{"stretchy":61},[154,4857,1661],{},[59,4859,207],{"stretchy":61},[59,4861,73],{"separator":72},[151,4863,4864,4866],{},[154,4865,946],{},[154,4867,218],{},[59,4869,167],{"stretchy":61},[154,4871,1661],{},[59,4873,207],{"stretchy":61},[59,4875,73],{"separator":72},[151,4877,4878,4880],{},[154,4879,946],{},[154,4881,273],{},[59,4883,167],{"stretchy":61},[154,4885,1661],{},[59,4887,207],{"stretchy":61},[82,4889,4890],{"encoding":84},"R_x(\\pi),R_y(\\pi),R_z(\\pi)",[40,4892,4894],{"className":4893,"ariaHidden":72},[89],[40,4895,4897,4900,4941,4944,4947,4950,4953,4956,4996,4999,5002,5005,5008,5011,5051,5054,5057],{"className":4896},[93],[40,4898],{"className":4899,"style":1365},[97],[40,4901,4903,4906],{"className":4902},[102],[40,4904,946],{"className":4905,"style":3946},[102,335],[40,4907,4909],{"className":4908},[340],[40,4910,4912,4933],{"className":4911},[344,345],[40,4913,4915,4930],{"className":4914},[349],[40,4916,4918],{"className":4917,"style":354},[353],[40,4919,4921,4924],{"style":4920},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[40,4922],{"className":4923,"style":362},[361],[40,4925,4927],{"className":4926},[366,367,368,369],[40,4928,159],{"className":4929},[102,335,369],[40,4931,377],{"className":4932},[376],[40,4934,4936],{"className":4935},[349],[40,4937,4939],{"className":4938,"style":384},[353],[40,4940],{},[40,4942,167],{"className":4943},[106],[40,4945,1661],{"className":4946,"style":336},[102,335],[40,4948,207],{"className":4949},[113],[40,4951,73],{"className":4952},[117],[40,4954],{"className":4955,"style":122},[121],[40,4957,4959,4962],{"className":4958},[102],[40,4960,946],{"className":4961,"style":3946},[102,335],[40,4963,4965],{"className":4964},[340],[40,4966,4968,4988],{"className":4967},[344,345],[40,4969,4971,4985],{"className":4970},[349],[40,4972,4974],{"className":4973,"style":354},[353],[40,4975,4976,4979],{"style":4920},[40,4977],{"className":4978,"style":362},[361],[40,4980,4982],{"className":4981},[366,367,368,369],[40,4983,218],{"className":4984,"style":336},[102,335,369],[40,4986,377],{"className":4987},[376],[40,4989,4991],{"className":4990},[349],[40,4992,4994],{"className":4993,"style":585},[353],[40,4995],{},[40,4997,167],{"className":4998},[106],[40,5000,1661],{"className":5001,"style":336},[102,335],[40,5003,207],{"className":5004},[113],[40,5006,73],{"className":5007},[117],[40,5009],{"className":5010,"style":122},[121],[40,5012,5014,5017],{"className":5013},[102],[40,5015,946],{"className":5016,"style":3946},[102,335],[40,5018,5020],{"className":5019},[340],[40,5021,5023,5043],{"className":5022},[344,345],[40,5024,5026,5040],{"className":5025},[349],[40,5027,5029],{"className":5028,"style":354},[353],[40,5030,5031,5034],{"style":4920},[40,5032],{"className":5033,"style":362},[361],[40,5035,5037],{"className":5036},[366,367,368,369],[40,5038,273],{"className":5039,"style":764},[102,335,369],[40,5041,377],{"className":5042},[376],[40,5044,5046],{"className":5045},[349],[40,5047,5049],{"className":5048,"style":384},[353],[40,5050],{},[40,5052,167],{"className":5053},[106],[40,5055,1661],{"className":5056,"style":336},[102,335],[40,5058,207],{"className":5059},[113],"), so ",[35,5062,3754],{}," implies the Wigner pattern simply rotates rigidly on (",[40,5065,5067,5084],{"className":5066},[43],[40,5068,5070],{"className":5069},[47],[49,5071,5072],{"xmlns":51},[53,5073,5074,5082],{},[56,5075,5076],{},[941,5077,5078,5080],{},[154,5079,2362],{},[64,5081,1044],{},[82,5083,2367],{"encoding":84},[40,5085,5087],{"className":5086,"ariaHidden":72},[89],[40,5088,5090,5093],{"className":5089},[93],[40,5091],{"className":5092,"style":982},[97],[40,5094,5096,5099],{"className":5095},[102],[40,5097,2362],{"className":5098,"style":2383},[102,335],[40,5100,5102],{"className":5101},[340],[40,5103,5105],{"className":5104},[344],[40,5106,5108],{"className":5107},[349],[40,5109,5111],{"className":5110,"style":982},[353],[40,5112,5113,5116],{"style":1004},[40,5114],{"className":5115,"style":362},[361],[40,5117,5119],{"className":5118},[366,367,368,369],[40,5120,1044],{"className":5121},[102,369],[25,5123],{},[28,5125,5127],{"id":5126},"_4-general-su2-rotation-and-the-so3-map","4. General SU(2) Rotation and the SO(3) Map",[12,5129,5130,5131,5200,5201,5232],{},"A general single‑qubit rotation about a unit axis (",[40,5132,5134,5155],{"className":5133},[43],[40,5135,5137],{"className":5136},[47],[49,5138,5139],{"xmlns":51},[53,5140,5141,5152],{},[56,5142,5143],{},[5144,5145,5146,5149],"mover",{"accent":72},[154,5147,5148],{"mathvariant":935},"u",[59,5150,5151],{},"^",[82,5153,5154],{"encoding":84},"\\hat{\\mathbf u}",[40,5156,5158],{"className":5157,"ariaHidden":72},[89],[40,5159,5161,5165],{"className":5160},[93],[40,5162],{"className":5163,"style":5164},[97],"height:0.7079em;",[40,5166,5169],{"className":5167},[102,5168],"accent",[40,5170,5172],{"className":5171},[344],[40,5173,5175],{"className":5174},[349],[40,5176,5178,5186],{"className":5177,"style":5164},[353],[40,5179,5180,5183],{"style":2233},[40,5181],{"className":5182,"style":444},[361],[40,5184,5148],{"className":5185},[102,966],[40,5187,5189,5192],{"style":5188},"top:-3.0134em;",[40,5190],{"className":5191,"style":444},[361],[40,5193,5197],{"className":5194,"style":5196},[5195],"accent-body","left:-0.25em;",[40,5198,5151],{"className":5199},[102],") by angle (",[40,5202,5204,5219],{"className":5203},[43],[40,5205,5207],{"className":5206},[47],[49,5208,5209],{"xmlns":51},[53,5210,5211,5216],{},[56,5212,5213],{},[154,5214,5215],{},"α",[82,5217,5218],{"encoding":84},"\\alpha",[40,5220,5222],{"className":5221,"ariaHidden":72},[89],[40,5223,5225,5228],{"className":5224},[93],[40,5226],{"className":5227,"style":4142},[97],[40,5229,5215],{"className":5230,"style":5231},[102,335],"margin-right:0.0037em;",") is",[40,5234,5236],{"className":5235},[136],[40,5237,5239,5308],{"className":5238},[43],[40,5240,5242],{"className":5241},[47],[49,5243,5244],{"xmlns":51,"display":145},[53,5245,5246,5305],{},[56,5247,5248,5250,5252,5254,5256,5262,5264,5266,5269,5271,5273,5275,5277,5279,5281,5289,5291,5297,5299,5301,5303],{},[154,5249,3688],{},[59,5251,167],{"stretchy":61},[154,5253,5215],{},[59,5255,73],{"separator":72},[5144,5257,5258,5260],{"accent":72},[154,5259,5148],{"mathvariant":935},[59,5261,5151],{},[59,5263,207],{"stretchy":61},[59,5265,162],{},[154,5267,5268],{},"exp",[59,5270,1084],{},[59,5272,3719],{"stretchy":61},[59,5274,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,5276,243],{},[154,5278,246],{},[59,5280,73],{"separator":72},[181,5282,5283],{"scriptlevel":66,"displaystyle":61},[1038,5284,5285,5287],{},[154,5286,5215],{},[64,5288,1044],{},[59,5290,73],{"separator":72},[5144,5292,5293,5295],{"accent":72},[154,5294,5148],{"mathvariant":935},[59,5296,5151],{},[59,5298,1058],{},[154,5300,156],{"mathvariant":1061},[59,5302,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,5304,1123],{"mathvariant":1080},[82,5306,5307],{"encoding":84},"U(\\alpha,\\hat{\\mathbf u}) = \\exp!\\big(-i,\\tfrac{\\alpha}{2},\\hat{\\mathbf u}\\cdot\\boldsymbol{\\sigma}\\big).",[40,5309,5311,5375,5402,5533],{"className":5310,"ariaHidden":72},[89],[40,5312,5314,5317,5320,5323,5326,5329,5332,5363,5366,5369,5372],{"className":5313},[93],[40,5315],{"className":5316,"style":98},[97],[40,5318,3688],{"className":5319,"style":3802},[102,335],[40,5321,167],{"className":5322},[106],[40,5324,5215],{"className":5325,"style":5231},[102,335],[40,5327,73],{"className":5328},[117],[40,5330],{"className":5331,"style":122},[121],[40,5333,5335],{"className":5334},[102,5168],[40,5336,5338],{"className":5337},[344],[40,5339,5341],{"className":5340},[349],[40,5342,5344,5352],{"className":5343,"style":5164},[353],[40,5345,5346,5349],{"style":2233},[40,5347],{"className":5348,"style":444},[361],[40,5350,5148],{"className":5351},[102,966],[40,5353,5354,5357],{"style":5188},[40,5355],{"className":5356,"style":444},[361],[40,5358,5360],{"className":5359,"style":5196},[5195],[40,5361,5151],{"className":5362},[102],[40,5364,207],{"className":5365},[113],[40,5367],{"className":5368,"style":390},[121],[40,5370,162],{"className":5371},[394],[40,5373],{"className":5374,"style":390},[121],[40,5376,5378,5381,5384,5387,5393,5396,5399],{"className":5377},[93],[40,5379],{"className":5380,"style":1155},[97],[40,5382,5268],{"className":5383},[1369],[40,5385,3719],{"className":5386},[113],[40,5388,5390],{"className":5389},[102],[40,5391,167],{"className":5392},[417,1238],[40,5394],{"className":5395,"style":1246},[121],[40,5397,243],{"className":5398},[1250],[40,5400],{"className":5401,"style":1246},[121],[40,5403,5405,5409,5412,5415,5418,5487,5490,5493,5524,5527,5530],{"className":5404},[93],[40,5406],{"className":5407,"style":5408},[97],"height:1.0529em;vertical-align:-0.345em;",[40,5410,246],{"className":5411},[102,335],[40,5413,73],{"className":5414},[117],[40,5416],{"className":5417,"style":122},[121],[40,5419,5421,5424,5484],{"className":5420},[102],[40,5422],{"className":5423},[106,1162],[40,5425,5427],{"className":5426},[1038],[40,5428,5430,5476],{"className":5429},[344,345],[40,5431,5433,5473],{"className":5432},[349],[40,5434,5437,5451,5459],{"className":5435,"style":5436},[353],"height:0.6954em;",[40,5438,5439,5442],{"style":1178},[40,5440],{"className":5441,"style":444},[361],[40,5443,5445],{"className":5444},[366,367,368,369],[40,5446,5448],{"className":5447},[102,369],[40,5449,1044],{"className":5450},[102,369],[40,5452,5453,5456],{"style":1193},[40,5454],{"className":5455,"style":444},[361],[40,5457],{"className":5458,"style":1201},[1200],[40,5460,5461,5464],{"style":1204},[40,5462],{"className":5463,"style":444},[361],[40,5465,5467],{"className":5466},[366,367,368,369],[40,5468,5470],{"className":5469},[102,369],[40,5471,5215],{"className":5472,"style":5231},[102,335,369],[40,5474,377],{"className":5475},[376],[40,5477,5479],{"className":5478},[349],[40,5480,5482],{"className":5481,"style":1226},[353],[40,5483],{},[40,5485],{"className":5486},[113,1162],[40,5488,73],{"className":5489},[117],[40,5491],{"className":5492,"style":122},[121],[40,5494,5496],{"className":5495},[102,5168],[40,5497,5499],{"className":5498},[344],[40,5500,5502],{"className":5501},[349],[40,5503,5505,5513],{"className":5504,"style":5164},[353],[40,5506,5507,5510],{"style":2233},[40,5508],{"className":5509,"style":444},[361],[40,5511,5148],{"className":5512},[102,966],[40,5514,5515,5518],{"style":5188},[40,5516],{"className":5517,"style":444},[361],[40,5519,5521],{"className":5520,"style":5196},[5195],[40,5522,5151],{"className":5523},[102],[40,5525],{"className":5526,"style":1246},[121],[40,5528,1058],{"className":5529},[1250],[40,5531],{"className":5532,"style":1246},[121],[40,5534,5536,5539,5548,5554],{"className":5535},[93],[40,5537],{"className":5538,"style":1155},[97],[40,5540,5542],{"className":5541},[102],[40,5543,5545],{"className":5544},[102],[40,5546,156],{"className":5547,"style":1289},[102,1288],[40,5549,5551],{"className":5550},[102],[40,5552,207],{"className":5553},[417,1238],[40,5555,1123],{"className":5556},[102],[12,5558,5559,5560,5590,5591,920,5594,1014],{},"Its action on (",[40,5561,5563,5577],{"className":5562},[43],[40,5564,5566],{"className":5565},[47],[49,5567,5568],{"xmlns":51},[53,5569,5570,5574],{},[56,5571,5572],{},[154,5573,936],{"mathvariant":935},[82,5575,5576],{"encoding":84},"\\mathbf r",[40,5578,5580],{"className":5579,"ariaHidden":72},[89],[40,5581,5583,5587],{"className":5582},[93],[40,5584],{"className":5585,"style":5586},[97],"height:0.4444em;",[40,5588,936],{"className":5589},[102,966],") is the ",[35,5592,5593],{},"adjoint map",[40,5595,5597,5636],{"className":5596},[43],[40,5598,5600],{"className":5599},[47],[49,5601,5602],{"xmlns":51},[53,5603,5604,5633],{},[56,5605,5606,5612,5614,5616,5618,5621,5627,5629,5631],{},[56,5607,5608,5610],{},[154,5609,2362],{"mathvariant":1080},[154,5611,3688],{"mathvariant":1080},[59,5613,167],{"stretchy":61},[64,5615,1044],{},[59,5617,207],{"stretchy":61},[59,5619,5620],{},"→",[56,5622,5623,5625],{},[154,5624,2362],{"mathvariant":1080},[154,5626,4055],{"mathvariant":1080},[59,5628,167],{"stretchy":61},[64,5630,949],{},[59,5632,207],{"stretchy":61},[82,5634,5635],{"encoding":84},"\\mathrm{SU}(2)\\to\\mathrm{SO}(3)",[40,5637,5639,5670],{"className":5638,"ariaHidden":72},[89],[40,5640,5642,5645,5652,5655,5658,5661,5664,5667],{"className":5641},[93],[40,5643],{"className":5644,"style":98},[97],[40,5646,5648],{"className":5647},[102],[40,5649,5651],{"className":5650},[102,1373],"SU",[40,5653,167],{"className":5654},[106],[40,5656,1044],{"className":5657},[102],[40,5659,207],{"className":5660},[113],[40,5662],{"className":5663,"style":390},[121],[40,5665,5620],{"className":5666},[394],[40,5668],{"className":5669,"style":390},[121],[40,5671,5673,5676,5682,5685,5688],{"className":5672},[93],[40,5674],{"className":5675,"style":98},[97],[40,5677,5679],{"className":5678},[102],[40,5680,4099],{"className":5681},[102,1373],[40,5683,167],{"className":5684},[106],[40,5686,949],{"className":5687},[102],[40,5689,207],{"className":5690},[113],[40,5692,5694],{"className":5693},[136],[40,5695,5697,5764],{"className":5696},[43],[40,5698,5700],{"className":5699},[47],[49,5701,5702],{"xmlns":51,"display":145},[53,5703,5704,5761],{},[169,5705,5706],{"width":2423},[175,5707,5708,5710,5754,5756],{},[178,5709],{"width":2428},[178,5711,5712],{},[56,5713,5714,5716,5718,5721,5723,5730,5732,5734,5736,5738,5740,5746,5748,5750,5752],{},[154,5715,936],{"mathvariant":935},[59,5717,2051],{"separator":72},[59,5719,5720],{},"⟼",[59,5722,2051],{"separator":72},[941,5724,5725,5727],{},[154,5726,936],{"mathvariant":935},[59,5728,5729],{"mathvariant":1080,"lspace":3697,"rspace":3697},"′",[59,5731,162],{},[154,5733,946],{},[59,5735,167],{"stretchy":61},[154,5737,5215],{},[59,5739,73],{"separator":72},[5144,5741,5742,5744],{"accent":72},[154,5743,5148],{"mathvariant":935},[59,5745,5151],{},[59,5747,207],{"stretchy":61},[59,5749,73],{"separator":72},[154,5751,936],{"mathvariant":935},[154,5753,1123],{"mathvariant":1080},[178,5755],{"width":2428},[178,5757,5758],{},[1102,5759,5760],{},"(Adjoint)",[82,5762,5763],{"encoding":84},"\\mathbf r;\\longmapsto;\\mathbf r' = R(\\alpha,\\hat{\\mathbf u}),\\mathbf r.\n\\tag{Adjoint}",[40,5765,5767,5786,5841,5908],{"className":5766,"ariaHidden":72},[89],[40,5768,5770,5774,5777,5780,5783],{"className":5769},[93],[40,5771],{"className":5772,"style":5773},[97],"height:0.7054em;vertical-align:-0.1944em;",[40,5775,936],{"className":5776},[102,966],[40,5778,2051],{"className":5779},[117],[40,5781],{"className":5782,"style":390},[121],[40,5784,5720],{"className":5785},[394],[40,5787,5789,5793,5796,5799,5832,5835,5838],{"className":5788},[93],[40,5790],{"className":5791,"style":5792},[97],"height:0.9963em;vertical-align:-0.1944em;",[40,5794,2051],{"className":5795},[117],[40,5797],{"className":5798,"style":122},[121],[40,5800,5802,5805],{"className":5801},[102],[40,5803,936],{"className":5804},[102,966],[40,5806,5808],{"className":5807},[340],[40,5809,5811],{"className":5810},[344],[40,5812,5814],{"className":5813},[349],[40,5815,5818],{"className":5816,"style":5817},[353],"height:0.8019em;",[40,5819,5820,5823],{"style":3962},[40,5821],{"className":5822,"style":362},[361],[40,5824,5826],{"className":5825},[366,367,368,369],[40,5827,5829],{"className":5828},[102,369],[40,5830,5729],{"className":5831},[102,369],[40,5833],{"className":5834,"style":390},[121],[40,5836,162],{"className":5837},[394],[40,5839],{"className":5840,"style":390},[121],[40,5842,5844,5847,5850,5853,5856,5859,5862,5893,5896,5899,5902,5905],{"className":5843},[93],[40,5845],{"className":5846,"style":98},[97],[40,5848,946],{"className":5849,"style":3946},[102,335],[40,5851,167],{"className":5852},[106],[40,5854,5215],{"className":5855,"style":5231},[102,335],[40,5857,73],{"className":5858},[117],[40,5860],{"className":5861,"style":122},[121],[40,5863,5865],{"className":5864},[102,5168],[40,5866,5868],{"className":5867},[344],[40,5869,5871],{"className":5870},[349],[40,5872,5874,5882],{"className":5873,"style":5164},[353],[40,5875,5876,5879],{"style":2233},[40,5877],{"className":5878,"style":444},[361],[40,5880,5148],{"className":5881},[102,966],[40,5883,5884,5887],{"style":5188},[40,5885],{"className":5886,"style":444},[361],[40,5888,5890],{"className":5889,"style":5196},[5195],[40,5891,5151],{"className":5892},[102],[40,5894,207],{"className":5895},[113],[40,5897,73],{"className":5898},[117],[40,5900],{"className":5901,"style":122},[121],[40,5903,936],{"className":5904},[102,966],[40,5906,1123],{"className":5907},[102],[40,5909,5911,5915],{"className":5910},[2925],[40,5912],{"className":5913,"style":5914},[97],"height:1.0519em;vertical-align:-0.25em;",[40,5916,5918,5921,5928],{"className":5917},[102,2932],[40,5919,167],{"className":5920},[102],[40,5922,5924],{"className":5923},[102],[40,5925,5927],{"className":5926},[102],"Adjoint",[40,5929,207],{"className":5930},[102],[25,5932],{},[28,5934,5936,5937,5967],{"id":5935},"_5-rodrigues-formula-axisangle-to-rotation-matrix","5. Rodrigues’ Formula (Axis–Angle (",[40,5938,5940,5954],{"className":5939},[43],[40,5941,5943],{"className":5942},[47],[49,5944,5945],{"xmlns":51},[53,5946,5947,5951],{},[56,5948,5949],{},[59,5950,5620],{},[82,5952,5953],{"encoding":84},"\\to",[40,5955,5957],{"className":5956,"ariaHidden":72},[89],[40,5958,5960,5964],{"className":5959},[93],[40,5961],{"className":5962,"style":5963},[97],"height:0.3669em;",[40,5965,5620],{"className":5966},[394],") Rotation Matrix)",[12,5969,5970,5971,6214],{},"Let (",[40,5972,5974,6020],{"className":5973},[43],[40,5975,5977],{"className":5976},[47],[49,5978,5979],{"xmlns":51},[53,5980,5981,6017],{},[56,5982,5983,5989,5991,5993,5999,6001,6007,6009,6015],{},[5144,5984,5985,5987],{"accent":72},[154,5986,5148],{"mathvariant":935},[59,5988,5151],{},[59,5990,162],{},[59,5992,167],{"stretchy":61},[151,5994,5995,5997],{},[154,5996,5148],{},[154,5998,159],{},[59,6000,73],{"separator":72},[151,6002,6003,6005],{},[154,6004,5148],{},[154,6006,218],{},[59,6008,73],{"separator":72},[151,6010,6011,6013],{},[154,6012,5148],{},[154,6014,273],{},[59,6016,207],{"stretchy":61},[82,6018,6019],{"encoding":84},"\\hat{\\mathbf u}=(u_x,u_y,u_z)",[40,6021,6023,6069],{"className":6022,"ariaHidden":72},[89],[40,6024,6026,6029,6060,6063,6066],{"className":6025},[93],[40,6027],{"className":6028,"style":5164},[97],[40,6030,6032],{"className":6031},[102,5168],[40,6033,6035],{"className":6034},[344],[40,6036,6038],{"className":6037},[349],[40,6039,6041,6049],{"className":6040,"style":5164},[353],[40,6042,6043,6046],{"style":2233},[40,6044],{"className":6045,"style":444},[361],[40,6047,5148],{"className":6048},[102,966],[40,6050,6051,6054],{"style":5188},[40,6052],{"className":6053,"style":444},[361],[40,6055,6057],{"className":6056,"style":5196},[5195],[40,6058,5151],{"className":6059},[102],[40,6061],{"className":6062,"style":390},[121],[40,6064,162],{"className":6065},[394],[40,6067],{"className":6068,"style":390},[121],[40,6070,6072,6075,6078,6119,6122,6125,6165,6168,6171,6211],{"className":6071},[93],[40,6073],{"className":6074,"style":1365},[97],[40,6076,167],{"className":6077},[106],[40,6079,6081,6084],{"className":6080},[102],[40,6082,5148],{"className":6083},[102,335],[40,6085,6087],{"className":6086},[340],[40,6088,6090,6111],{"className":6089},[344,345],[40,6091,6093,6108],{"className":6092},[349],[40,6094,6096],{"className":6095,"style":354},[353],[40,6097,6099,6102],{"style":6098},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[40,6100],{"className":6101,"style":362},[361],[40,6103,6105],{"className":6104},[366,367,368,369],[40,6106,159],{"className":6107},[102,335,369],[40,6109,377],{"className":6110},[376],[40,6112,6114],{"className":6113},[349],[40,6115,6117],{"className":6116,"style":384},[353],[40,6118],{},[40,6120,73],{"className":6121},[117],[40,6123],{"className":6124,"style":122},[121],[40,6126,6128,6131],{"className":6127},[102],[40,6129,5148],{"className":6130},[102,335],[40,6132,6134],{"className":6133},[340],[40,6135,6137,6157],{"className":6136},[344,345],[40,6138,6140,6154],{"className":6139},[349],[40,6141,6143],{"className":6142,"style":354},[353],[40,6144,6145,6148],{"style":6098},[40,6146],{"className":6147,"style":362},[361],[40,6149,6151],{"className":6150},[366,367,368,369],[40,6152,218],{"className":6153,"style":336},[102,335,369],[40,6155,377],{"className":6156},[376],[40,6158,6160],{"className":6159},[349],[40,6161,6163],{"className":6162,"style":585},[353],[40,6164],{},[40,6166,73],{"className":6167},[117],[40,6169],{"className":6170,"style":122},[121],[40,6172,6174,6177],{"className":6173},[102],[40,6175,5148],{"className":6176},[102,335],[40,6178,6180],{"className":6179},[340],[40,6181,6183,6203],{"className":6182},[344,345],[40,6184,6186,6200],{"className":6185},[349],[40,6187,6189],{"className":6188,"style":354},[353],[40,6190,6191,6194],{"style":6098},[40,6192],{"className":6193,"style":362},[361],[40,6195,6197],{"className":6196},[366,367,368,369],[40,6198,273],{"className":6199,"style":764},[102,335,369],[40,6201,377],{"className":6202},[376],[40,6204,6206],{"className":6205},[349],[40,6207,6209],{"className":6208,"style":384},[353],[40,6210],{},[40,6212,207],{"className":6213},[113],") be a unit axis and define the cross‑product matrix",[40,6216,6218],{"className":6217},[136],[40,6219,6221,6347],{"className":6220},[43],[40,6222,6224],{"className":6223},[47],[49,6225,6226],{"xmlns":51,"display":145},[53,6227,6228,6344],{},[56,6229,6230,6233,6235,6241,6243,6245,6342],{},[154,6231,6232],{},"K",[59,6234,167],{"stretchy":61},[5144,6236,6237,6239],{"accent":72},[154,6238,5148],{"mathvariant":935},[59,6240,5151],{},[59,6242,207],{"stretchy":61},[59,6244,162],{},[56,6246,6247,6249,6340],{},[59,6248,167],{"fence":72},[169,6250,6252],{"rowspacing":171,"columnalign":6251,"columnspacing":173},"center center center center center center center",[175,6253,6254,6260,6274,6294,6300,6324,6334],{},[178,6255,6256],{},[181,6257,6258],{"scriptlevel":66,"displaystyle":61},[64,6259,66],{},[178,6261,6262],{},[181,6263,6264],{"scriptlevel":66,"displaystyle":61},[56,6265,6266,6268],{},[59,6267,243],{},[151,6269,6270,6272],{},[154,6271,5148],{},[154,6273,273],{},[178,6275,6276],{},[181,6277,6278],{"scriptlevel":66,"displaystyle":61},[56,6279,6280,6286,6288],{},[151,6281,6282,6284],{},[154,6283,5148],{},[154,6285,218],{},[1102,6287,1104],{},[151,6289,6290,6292],{},[154,6291,5148],{},[154,6293,273],{},[178,6295,6296],{},[181,6297,6298],{"scriptlevel":66,"displaystyle":61},[64,6299,66],{},[178,6301,6302],{},[181,6303,6304],{"scriptlevel":66,"displaystyle":61},[56,6305,6306,6308,6314,6316,6318],{},[59,6307,243],{},[151,6309,6310,6312],{},[154,6311,5148],{},[154,6313,159],{},[1102,6315,1104],{},[59,6317,243],{},[151,6319,6320,6322],{},[154,6321,5148],{},[154,6323,218],{},[178,6325,6326],{},[181,6327,6328],{"scriptlevel":66,"displaystyle":61},[151,6329,6330,6332],{},[154,6331,5148],{},[154,6333,159],{},[178,6335,6336],{},[181,6337,6338],{"scriptlevel":66,"displaystyle":61},[64,6339,66],{},[59,6341,207],{"fence":72},[154,6343,1123],{"mathvariant":1080},[82,6345,6346],{"encoding":84},"K(\\hat{\\mathbf u}) = \\begin{pmatrix}\n0 & -u_z & u_y\\\nu_z & 0 & -u_x\\\n-u_y & u_x & 0\n\\end{pmatrix}.",[40,6348,6350,6405],{"className":6349,"ariaHidden":72},[89],[40,6351,6353,6356,6359,6362,6393,6396,6399,6402],{"className":6352},[93],[40,6354],{"className":6355,"style":98},[97],[40,6357,6232],{"className":6358,"style":4668},[102,335],[40,6360,167],{"className":6361},[106],[40,6363,6365],{"className":6364},[102,5168],[40,6366,6368],{"className":6367},[344],[40,6369,6371],{"className":6370},[349],[40,6372,6374,6382],{"className":6373,"style":5164},[353],[40,6375,6376,6379],{"style":2233},[40,6377],{"className":6378,"style":444},[361],[40,6380,5148],{"className":6381},[102,966],[40,6383,6384,6387],{"style":5188},[40,6385],{"className":6386,"style":444},[361],[40,6388,6390],{"className":6389,"style":5196},[5195],[40,6391,5151],{"className":6392},[102],[40,6394,207],{"className":6395},[113],[40,6397],{"className":6398,"style":390},[121],[40,6400,162],{"className":6401},[394],[40,6403],{"className":6404,"style":390},[121],[40,6406,6408,6411,6958,6961],{"className":6407},[93],[40,6409],{"className":6410,"style":1155},[97],[40,6412,6414,6420,6952],{"className":6413},[408],[40,6415,6417],{"className":6416,"style":413},[106,412],[40,6418,167],{"className":6419},[417,1238],[40,6421,6423],{"className":6422},[102],[40,6424,6426,6463,6466,6469,6543,6546,6549,6663,6666,6669,6703,6706,6709,6835,6838,6841,6912,6915,6918],{"className":6425},[169],[40,6427,6429],{"className":6428},[427],[40,6430,6432,6454],{"className":6431},[344,345],[40,6433,6435,6451],{"className":6434},[349],[40,6436,6439],{"className":6437,"style":6438},[353],"height:0.85em;",[40,6440,6442,6445],{"style":6441},"top:-3.01em;",[40,6443],{"className":6444,"style":444},[361],[40,6446,6448],{"className":6447},[102],[40,6449,66],{"className":6450},[102],[40,6452,377],{"className":6453},[376],[40,6455,6457],{"className":6456},[349],[40,6458,6461],{"className":6459,"style":6460},[353],"height:0.35em;",[40,6462],{},[40,6464],{"className":6465,"style":479},[478],[40,6467],{"className":6468,"style":479},[478],[40,6470,6472],{"className":6471},[427],[40,6473,6475,6535],{"className":6474},[344,345],[40,6476,6478,6532],{"className":6477},[349],[40,6479,6481],{"className":6480,"style":6438},[353],[40,6482,6483,6486],{"style":6441},[40,6484],{"className":6485,"style":444},[361],[40,6487,6489,6492],{"className":6488},[102],[40,6490,243],{"className":6491},[102],[40,6493,6495,6498],{"className":6494},[102],[40,6496,5148],{"className":6497},[102,335],[40,6499,6501],{"className":6500},[340],[40,6502,6504,6524],{"className":6503},[344,345],[40,6505,6507,6521],{"className":6506},[349],[40,6508,6510],{"className":6509,"style":354},[353],[40,6511,6512,6515],{"style":6098},[40,6513],{"className":6514,"style":362},[361],[40,6516,6518],{"className":6517},[366,367,368,369],[40,6519,273],{"className":6520,"style":764},[102,335,369],[40,6522,377],{"className":6523},[376],[40,6525,6527],{"className":6526},[349],[40,6528,6530],{"className":6529,"style":384},[353],[40,6531],{},[40,6533,377],{"className":6534},[376],[40,6536,6538],{"className":6537},[349],[40,6539,6541],{"className":6540,"style":6460},[353],[40,6542],{},[40,6544],{"className":6545,"style":479},[478],[40,6547],{"className":6548,"style":479},[478],[40,6550,6552],{"className":6551},[427],[40,6553,6555,6655],{"className":6554},[344,345],[40,6556,6558,6652],{"className":6557},[349],[40,6559,6561],{"className":6560,"style":6438},[353],[40,6562,6563,6566],{"style":6441},[40,6564],{"className":6565,"style":444},[361],[40,6567,6569,6609,6612],{"className":6568},[102],[40,6570,6572,6575],{"className":6571},[102],[40,6573,5148],{"className":6574},[102,335],[40,6576,6578],{"className":6577},[340],[40,6579,6581,6601],{"className":6580},[344,345],[40,6582,6584,6598],{"className":6583},[349],[40,6585,6587],{"className":6586,"style":354},[353],[40,6588,6589,6592],{"style":6098},[40,6590],{"className":6591,"style":362},[361],[40,6593,6595],{"className":6594},[366,367,368,369],[40,6596,218],{"className":6597,"style":336},[102,335,369],[40,6599,377],{"className":6600},[376],[40,6602,6604],{"className":6603},[349],[40,6605,6607],{"className":6606,"style":585},[353],[40,6608],{},[40,6610,1104],{"className":6611},[121],[40,6613,6615,6618],{"className":6614},[102],[40,6616,5148],{"className":6617},[102,335],[40,6619,6621],{"className":6620},[340],[40,6622,6624,6644],{"className":6623},[344,345],[40,6625,6627,6641],{"className":6626},[349],[40,6628,6630],{"className":6629,"style":354},[353],[40,6631,6632,6635],{"style":6098},[40,6633],{"className":6634,"style":362},[361],[40,6636,6638],{"className":6637},[366,367,368,369],[40,6639,273],{"className":6640,"style":764},[102,335,369],[40,6642,377],{"className":6643},[376],[40,6645,6647],{"className":6646},[349],[40,6648,6650],{"className":6649,"style":384},[353],[40,6651],{},[40,6653,377],{"className":6654},[376],[40,6656,6658],{"className":6657},[349],[40,6659,6661],{"className":6660,"style":6460},[353],[40,6662],{},[40,6664],{"className":6665,"style":479},[478],[40,6667],{"className":6668,"style":479},[478],[40,6670,6672],{"className":6671},[427],[40,6673,6675,6695],{"className":6674},[344,345],[40,6676,6678,6692],{"className":6677},[349],[40,6679,6681],{"className":6680,"style":6438},[353],[40,6682,6683,6686],{"style":6441},[40,6684],{"className":6685,"style":444},[361],[40,6687,6689],{"className":6688},[102],[40,6690,66],{"className":6691},[102],[40,6693,377],{"className":6694},[376],[40,6696,6698],{"className":6697},[349],[40,6699,6701],{"className":6700,"style":6460},[353],[40,6702],{},[40,6704],{"className":6705,"style":479},[478],[40,6707],{"className":6708,"style":479},[478],[40,6710,6712],{"className":6711},[427],[40,6713,6715,6827],{"className":6714},[344,345],[40,6716,6718,6824],{"className":6717},[349],[40,6719,6721],{"className":6720,"style":6438},[353],[40,6722,6723,6726],{"style":6441},[40,6724],{"className":6725,"style":444},[361],[40,6727,6729,6732,6772,6775,6778,6781,6784],{"className":6728},[102],[40,6730,243],{"className":6731},[102],[40,6733,6735,6738],{"className":6734},[102],[40,6736,5148],{"className":6737},[102,335],[40,6739,6741],{"className":6740},[340],[40,6742,6744,6764],{"className":6743},[344,345],[40,6745,6747,6761],{"className":6746},[349],[40,6748,6750],{"className":6749,"style":354},[353],[40,6751,6752,6755],{"style":6098},[40,6753],{"className":6754,"style":362},[361],[40,6756,6758],{"className":6757},[366,367,368,369],[40,6759,159],{"className":6760},[102,335,369],[40,6762,377],{"className":6763},[376],[40,6765,6767],{"className":6766},[349],[40,6768,6770],{"className":6769,"style":384},[353],[40,6771],{},[40,6773,1104],{"className":6774},[121],[40,6776],{"className":6777,"style":1246},[121],[40,6779,243],{"className":6780},[1250],[40,6782],{"className":6783,"style":1246},[121],[40,6785,6787,6790],{"className":6786},[102],[40,6788,5148],{"className":6789},[102,335],[40,6791,6793],{"className":6792},[340],[40,6794,6796,6816],{"className":6795},[344,345],[40,6797,6799,6813],{"className":6798},[349],[40,6800,6802],{"className":6801,"style":354},[353],[40,6803,6804,6807],{"style":6098},[40,6805],{"className":6806,"style":362},[361],[40,6808,6810],{"className":6809},[366,367,368,369],[40,6811,218],{"className":6812,"style":336},[102,335,369],[40,6814,377],{"className":6815},[376],[40,6817,6819],{"className":6818},[349],[40,6820,6822],{"className":6821,"style":585},[353],[40,6823],{},[40,6825,377],{"className":6826},[376],[40,6828,6830],{"className":6829},[349],[40,6831,6833],{"className":6832,"style":6460},[353],[40,6834],{},[40,6836],{"className":6837,"style":479},[478],[40,6839],{"className":6840,"style":479},[478],[40,6842,6844],{"className":6843},[427],[40,6845,6847,6904],{"className":6846},[344,345],[40,6848,6850,6901],{"className":6849},[349],[40,6851,6853],{"className":6852,"style":6438},[353],[40,6854,6855,6858],{"style":6441},[40,6856],{"className":6857,"style":444},[361],[40,6859,6861],{"className":6860},[102],[40,6862,6864,6867],{"className":6863},[102],[40,6865,5148],{"className":6866},[102,335],[40,6868,6870],{"className":6869},[340],[40,6871,6873,6893],{"className":6872},[344,345],[40,6874,6876,6890],{"className":6875},[349],[40,6877,6879],{"className":6878,"style":354},[353],[40,6880,6881,6884],{"style":6098},[40,6882],{"className":6883,"style":362},[361],[40,6885,6887],{"className":6886},[366,367,368,369],[40,6888,159],{"className":6889},[102,335,369],[40,6891,377],{"className":6892},[376],[40,6894,6896],{"className":6895},[349],[40,6897,6899],{"className":6898,"style":384},[353],[40,6900],{},[40,6902,377],{"className":6903},[376],[40,6905,6907],{"className":6906},[349],[40,6908,6910],{"className":6909,"style":6460},[353],[40,6911],{},[40,6913],{"className":6914,"style":479},[478],[40,6916],{"className":6917,"style":479},[478],[40,6919,6921],{"className":6920},[427],[40,6922,6924,6944],{"className":6923},[344,345],[40,6925,6927,6941],{"className":6926},[349],[40,6928,6930],{"className":6929,"style":6438},[353],[40,6931,6932,6935],{"style":6441},[40,6933],{"className":6934,"style":444},[361],[40,6936,6938],{"className":6937},[102],[40,6939,66],{"className":6940},[102],[40,6942,377],{"className":6943},[376],[40,6945,6947],{"className":6946},[349],[40,6948,6950],{"className":6949,"style":6460},[353],[40,6951],{},[40,6953,6955],{"className":6954,"style":413},[113,412],[40,6956,207],{"className":6957},[417,1238],[40,6959],{"className":6960,"style":122},[121],[40,6962,1123],{"className":6963},[102],[12,6965,6966],{},"Then the corresponding rotation matrix is",[40,6968,6970],{"className":6969},[136],[40,6971,6973,7094],{"className":6972},[43],[40,6974,6976],{"className":6975},[47],[49,6977,6978],{"xmlns":51,"display":145},[53,6979,6980,7091],{},[169,6981,6982],{"width":2423},[175,6983,6984,6986,7084,7086],{},[178,6985],{"width":2428},[178,6987,6988],{},[56,6989,6990,6992,6994,6996,6998,7004,7006,7008,7010,7012,7014,7016,7022,7024,7026,7028,7030,7032,7034,7036,7038,7040,7046,7048,7060,7062,7064,7066,7068,7070,7072,7074,7080,7082],{},[154,6991,946],{},[59,6993,167],{"stretchy":61},[154,6995,5215],{},[59,6997,73],{"separator":72},[5144,6999,7000,7002],{"accent":72},[154,7001,5148],{"mathvariant":935},[59,7003,5151],{},[59,7005,207],{"stretchy":61},[59,7007,162],{},[154,7009,1820],{},[59,7011,1084],{},[154,7013,5215],{},[59,7015,73],{"separator":72},[151,7017,7018,7020],{},[154,7019,1050],{},[64,7021,949],{},[59,7023,1053],{},[59,7025,167],{"stretchy":61},[64,7027,78],{},[59,7029,243],{},[154,7031,1820],{},[59,7033,1084],{},[154,7035,5215],{},[59,7037,207],{"stretchy":61},[59,7039,73],{"separator":72},[5144,7041,7042,7044],{"accent":72},[154,7043,5148],{"mathvariant":935},[59,7045,5151],{},[59,7047,73],{"separator":72},[941,7049,7050,7056],{},[5144,7051,7052,7054],{"accent":72},[154,7053,5148],{"mathvariant":935},[59,7055,5151],{},[154,7057,7059],{"mathvariant":7058},"sans-serif","T",[59,7061,1053],{},[154,7063,1813],{},[59,7065,1084],{},[154,7067,5215],{},[59,7069,73],{"separator":72},[154,7071,6232],{},[59,7073,167],{"stretchy":61},[5144,7075,7076,7078],{"accent":72},[154,7077,5148],{"mathvariant":935},[59,7079,5151],{},[59,7081,207],{"stretchy":61},[154,7083,1123],{"mathvariant":1080},[178,7085],{"width":2428},[178,7087,7088],{},[1102,7089,7090],{},"(Rodrigues)",[82,7092,7093],{"encoding":84},"R(\\alpha,\\hat{\\mathbf u}) = \\cos\\alpha,I_3 + (1-\\cos\\alpha),\\hat{\\mathbf u},\\hat{\\mathbf u}^{\\mathsf T} + \\sin\\alpha,K(\\hat{\\mathbf u}).\n\\tag{Rodrigues}",[40,7095,7097,7161,7234,7255,7388,7452],{"className":7096,"ariaHidden":72},[89],[40,7098,7100,7103,7106,7109,7112,7115,7118,7149,7152,7155,7158],{"className":7099},[93],[40,7101],{"className":7102,"style":98},[97],[40,7104,946],{"className":7105,"style":3946},[102,335],[40,7107,167],{"className":7108},[106],[40,7110,5215],{"className":7111,"style":5231},[102,335],[40,7113,73],{"className":7114},[117],[40,7116],{"className":7117,"style":122},[121],[40,7119,7121],{"className":7120},[102,5168],[40,7122,7124],{"className":7123},[344],[40,7125,7127],{"className":7126},[349],[40,7128,7130,7138],{"className":7129,"style":5164},[353],[40,7131,7132,7135],{"style":2233},[40,7133],{"className":7134,"style":444},[361],[40,7136,5148],{"className":7137},[102,966],[40,7139,7140,7143],{"style":5188},[40,7141],{"className":7142,"style":444},[361],[40,7144,7146],{"className":7145,"style":5196},[5195],[40,7147,5151],{"className":7148},[102],[40,7150,207],{"className":7151},[113],[40,7153],{"className":7154,"style":390},[121],[40,7156,162],{"className":7157},[394],[40,7159],{"className":7160,"style":390},[121],[40,7162,7164,7168,7171,7174,7177,7180,7183,7225,7228,7231],{"className":7163},[93],[40,7165],{"className":7166,"style":7167},[97],"height:0.8778em;vertical-align:-0.1944em;",[40,7169,1820],{"className":7170},[1369],[40,7172],{"className":7173,"style":122},[121],[40,7175,5215],{"className":7176,"style":5231},[102,335],[40,7178,73],{"className":7179},[117],[40,7181],{"className":7182,"style":122},[121],[40,7184,7186,7189],{"className":7185},[102],[40,7187,1050],{"className":7188,"style":1242},[102,335],[40,7190,7192],{"className":7191},[340],[40,7193,7195,7217],{"className":7194},[344,345],[40,7196,7198,7214],{"className":7197},[349],[40,7199,7202],{"className":7200,"style":7201},[353],"height:0.3011em;",[40,7203,7205,7208],{"style":7204},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[40,7206],{"className":7207,"style":362},[361],[40,7209,7211],{"className":7210},[366,367,368,369],[40,7212,949],{"className":7213},[102,369],[40,7215,377],{"className":7216},[376],[40,7218,7220],{"className":7219},[349],[40,7221,7223],{"className":7222,"style":384},[353],[40,7224],{},[40,7226],{"className":7227,"style":1246},[121],[40,7229,1053],{"className":7230},[1250],[40,7232],{"className":7233,"style":1246},[121],[40,7235,7237,7240,7243,7246,7249,7252],{"className":7236},[93],[40,7238],{"className":7239,"style":98},[97],[40,7241,167],{"className":7242},[106],[40,7244,78],{"className":7245},[102],[40,7247],{"className":7248,"style":1246},[121],[40,7250,243],{"className":7251},[1250],[40,7253],{"className":7254,"style":1246},[121],[40,7256,7258,7262,7265,7268,7271,7274,7277,7280,7311,7314,7317,7379,7382,7385],{"className":7257},[93],[40,7259],{"className":7260,"style":7261},[97],"height:1.1491em;vertical-align:-0.25em;",[40,7263,1820],{"className":7264},[1369],[40,7266],{"className":7267,"style":122},[121],[40,7269,5215],{"className":7270,"style":5231},[102,335],[40,7272,207],{"className":7273},[113],[40,7275,73],{"className":7276},[117],[40,7278],{"className":7279,"style":122},[121],[40,7281,7283],{"className":7282},[102,5168],[40,7284,7286],{"className":7285},[344],[40,7287,7289],{"className":7288},[349],[40,7290,7292,7300],{"className":7291,"style":5164},[353],[40,7293,7294,7297],{"style":2233},[40,7295],{"className":7296,"style":444},[361],[40,7298,5148],{"className":7299},[102,966],[40,7301,7302,7305],{"style":5188},[40,7303],{"className":7304,"style":444},[361],[40,7306,7308],{"className":7307,"style":5196},[5195],[40,7309,5151],{"className":7310},[102],[40,7312,73],{"className":7313},[117],[40,7315],{"className":7316,"style":122},[121],[40,7318,7320,7351],{"className":7319},[102],[40,7321,7323],{"className":7322},[102,5168],[40,7324,7326],{"className":7325},[344],[40,7327,7329],{"className":7328},[349],[40,7330,7332,7340],{"className":7331,"style":5164},[353],[40,7333,7334,7337],{"style":2233},[40,7335],{"className":7336,"style":444},[361],[40,7338,5148],{"className":7339},[102,966],[40,7341,7342,7345],{"style":5188},[40,7343],{"className":7344,"style":444},[361],[40,7346,7348],{"className":7347,"style":5196},[5195],[40,7349,5151],{"className":7350},[102],[40,7352,7354],{"className":7353},[340],[40,7355,7357],{"className":7356},[344],[40,7358,7360],{"className":7359},[349],[40,7361,7364],{"className":7362,"style":7363},[353],"height:0.8991em;",[40,7365,7366,7369],{"style":3962},[40,7367],{"className":7368,"style":362},[361],[40,7370,7372],{"className":7371},[366,367,368,369],[40,7373,7375],{"className":7374},[102,369],[40,7376,7059],{"className":7377},[102,7378,369],"mathsf",[40,7380],{"className":7381,"style":1246},[121],[40,7383,1053],{"className":7384},[1250],[40,7386],{"className":7387,"style":1246},[121],[40,7389,7391,7394,7397,7400,7403,7406,7409,7412,7415,7446,7449],{"className":7390},[93],[40,7392],{"className":7393,"style":98},[97],[40,7395,1813],{"className":7396},[1369],[40,7398],{"className":7399,"style":122},[121],[40,7401,5215],{"className":7402,"style":5231},[102,335],[40,7404,73],{"className":7405},[117],[40,7407],{"className":7408,"style":122},[121],[40,7410,6232],{"className":7411,"style":4668},[102,335],[40,7413,167],{"className":7414},[106],[40,7416,7418],{"className":7417},[102,5168],[40,7419,7421],{"className":7420},[344],[40,7422,7424],{"className":7423},[349],[40,7425,7427,7435],{"className":7426,"style":5164},[353],[40,7428,7429,7432],{"style":2233},[40,7430],{"className":7431,"style":444},[361],[40,7433,5148],{"className":7434},[102,966],[40,7436,7437,7440],{"style":5188},[40,7438],{"className":7439,"style":444},[361],[40,7441,7443],{"className":7442,"style":5196},[5195],[40,7444,5151],{"className":7445},[102],[40,7447,207],{"className":7448},[113],[40,7450,1123],{"className":7451},[102],[40,7453,7455,7458],{"className":7454},[2925],[40,7456],{"className":7457,"style":7261},[97],[40,7459,7461,7464,7471],{"className":7460},[102,2932],[40,7462,167],{"className":7463},[102],[40,7465,7467],{"className":7466},[102],[40,7468,7470],{"className":7469},[102],"Rodrigues",[40,7472,207],{"className":7473},[102],[12,7475,7476,7477,7575,7576,7578],{},"This matrix implements the evolution (",[40,7478,7480,7506],{"className":7479},[43],[40,7481,7483],{"className":7482},[47],[49,7484,7485],{"xmlns":51},[53,7486,7487,7503],{},[56,7488,7489,7495,7497,7499,7501],{},[941,7490,7491,7493],{},[154,7492,936],{"mathvariant":935},[59,7494,5729],{"mathvariant":1080,"lspace":3697,"rspace":3697},[59,7496,162],{},[154,7498,946],{},[59,7500,73],{"separator":72},[154,7502,936],{"mathvariant":935},[82,7504,7505],{"encoding":84},"\\mathbf r' = R,\\mathbf r",[40,7507,7509,7557],{"className":7508,"ariaHidden":72},[89],[40,7510,7512,7516,7548,7551,7554],{"className":7511},[93],[40,7513],{"className":7514,"style":7515},[97],"height:0.7519em;",[40,7517,7519,7522],{"className":7518},[102],[40,7520,936],{"className":7521},[102,966],[40,7523,7525],{"className":7524},[340],[40,7526,7528],{"className":7527},[344],[40,7529,7531],{"className":7530},[349],[40,7532,7534],{"className":7533,"style":7515},[353],[40,7535,7536,7539],{"style":1004},[40,7537],{"className":7538,"style":362},[361],[40,7540,7542],{"className":7541},[366,367,368,369],[40,7543,7545],{"className":7544},[102,369],[40,7546,5729],{"className":7547},[102,369],[40,7549],{"className":7550,"style":390},[121],[40,7552,162],{"className":7553},[394],[40,7555],{"className":7556,"style":390},[121],[40,7558,7560,7563,7566,7569,7572],{"className":7559},[93],[40,7561],{"className":7562,"style":7167},[97],[40,7564,946],{"className":7565,"style":3946},[102,335],[40,7567,73],{"className":7568},[117],[40,7570],{"className":7571,"style":122},[121],[40,7573,936],{"className":7574},[102,966],"), and by ",[35,7577,3754],{}," the Wigner function obeys",[40,7580,7582],{"className":7581},[136],[40,7583,7585,7679],{"className":7584},[43],[40,7586,7588],{"className":7587},[47],[49,7589,7590],{"xmlns":51,"display":145},[53,7591,7592,7676],{},[56,7593,7594,7600,7602,7604,7606,7608,7610,7612,7614,7616,7624,7626,7634,7636,7638,7640,7642,7644,7646,7652,7654,7656,7658,7660,7662,7664,7666,7668,7670,7672,7674],{},[151,7595,7596,7598],{},[154,7597,2437],{},[154,7599,1032],{},[59,7601,167],{"stretchy":61},[154,7603,1649],{},[59,7605,73],{"separator":72},[154,7607,1671],{},[59,7609,2051],{"separator":72},[154,7611,5215],{},[59,7613,207],{"stretchy":61},[59,7615,162],{},[181,7617,7618],{"scriptlevel":66,"displaystyle":61},[1038,7619,7620,7622],{},[64,7621,78],{},[64,7623,1044],{},[59,7625,1053],{},[1038,7627,7628,7632],{},[2045,7629,7630],{},[64,7631,949],{},[64,7633,1044],{},[59,7635,2051],{"separator":72},[59,7637,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,7639,946],{},[59,7641,167],{"stretchy":61},[154,7643,5215],{},[59,7645,73],{"separator":72},[5144,7647,7648,7650],{"accent":72},[154,7649,5148],{"mathvariant":935},[59,7651,5151],{},[59,7653,207],{"stretchy":61},[59,7655,73],{"separator":72},[154,7657,936],{"mathvariant":935},[59,7659,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,7661,1058],{},[154,7663,1796],{"mathvariant":935},[59,7665,167],{"stretchy":61},[154,7667,1649],{},[59,7669,73],{"separator":72},[154,7671,1671],{},[59,7673,207],{"stretchy":61},[154,7675,1123],{"mathvariant":1080},[82,7677,7678],{"encoding":84},"W_\\rho(\\theta,\\phi;\\alpha) = \\tfrac{1}{2} + \\frac{\\sqrt{3}}{2}; \\big(R(\\alpha,\\hat{\\mathbf u}),\\mathbf r\\big)\\cdot\\mathbf n(\\theta,\\phi).",[40,7680,7682,7764,7847,8044],{"className":7681,"ariaHidden":72},[89],[40,7683,7685,7688,7728,7731,7734,7737,7740,7743,7746,7749,7752,7755,7758,7761],{"className":7684},[93],[40,7686],{"className":7687,"style":1365},[97],[40,7689,7691,7694],{"className":7690},[102],[40,7692,2437],{"className":7693,"style":2541},[102,335],[40,7695,7697],{"className":7696},[340],[40,7698,7700,7720],{"className":7699},[344,345],[40,7701,7703,7717],{"className":7702},[349],[40,7704,7706],{"className":7705,"style":354},[353],[40,7707,7708,7711],{"style":2556},[40,7709],{"className":7710,"style":362},[361],[40,7712,7714],{"className":7713},[366,367,368,369],[40,7715,1032],{"className":7716},[102,335,369],[40,7718,377],{"className":7719},[376],[40,7721,7723],{"className":7722},[349],[40,7724,7726],{"className":7725,"style":585},[353],[40,7727],{},[40,7729,167],{"className":7730},[106],[40,7732,1649],{"className":7733,"style":1312},[102,335],[40,7735,73],{"className":7736},[117],[40,7738],{"className":7739,"style":122},[121],[40,7741,1671],{"className":7742},[102,335],[40,7744,2051],{"className":7745},[117],[40,7747],{"className":7748,"style":122},[121],[40,7750,5215],{"className":7751,"style":5231},[102,335],[40,7753,207],{"className":7754},[113],[40,7756],{"className":7757,"style":390},[121],[40,7759,162],{"className":7760},[394],[40,7762],{"className":7763,"style":390},[121],[40,7765,7767,7770,7838,7841,7844],{"className":7766},[93],[40,7768],{"className":7769,"style":2676},[97],[40,7771,7773,7776,7835],{"className":7772},[102],[40,7774],{"className":7775},[106,1162],[40,7777,7779],{"className":7778},[1038],[40,7780,7782,7827],{"className":7781},[344,345],[40,7783,7785,7824],{"className":7784},[349],[40,7786,7788,7802,7810],{"className":7787,"style":1175},[353],[40,7789,7790,7793],{"style":1178},[40,7791],{"className":7792,"style":444},[361],[40,7794,7796],{"className":7795},[366,367,368,369],[40,7797,7799],{"className":7798},[102,369],[40,7800,1044],{"className":7801},[102,369],[40,7803,7804,7807],{"style":1193},[40,7805],{"className":7806,"style":444},[361],[40,7808],{"className":7809,"style":1201},[1200],[40,7811,7812,7815],{"style":1204},[40,7813],{"className":7814,"style":444},[361],[40,7816,7818],{"className":7817},[366,367,368,369],[40,7819,7821],{"className":7820},[102,369],[40,7822,78],{"className":7823},[102,369],[40,7825,377],{"className":7826},[376],[40,7828,7830],{"className":7829},[349],[40,7831,7833],{"className":7832,"style":1226},[353],[40,7834],{},[40,7836],{"className":7837},[113,1162],[40,7839],{"className":7840,"style":1246},[121],[40,7842,1053],{"className":7843},[1250],[40,7845],{"className":7846,"style":1246},[121],[40,7848,7850,7853,7959,7962,7965,7971,7974,7977,7980,7983,7986,8017,8020,8023,8026,8029,8035,8038,8041],{"className":7849},[93],[40,7851],{"className":7852,"style":2760},[97],[40,7854,7856,7859,7956],{"className":7855},[102],[40,7857],{"className":7858},[106,1162],[40,7860,7862],{"className":7861},[1038],[40,7863,7865,7948],{"className":7864},[344,345],[40,7866,7868,7945],{"className":7867},[349],[40,7869,7871,7882,7890],{"className":7870,"style":2779},[353],[40,7872,7873,7876],{"style":2782},[40,7874],{"className":7875,"style":444},[361],[40,7877,7879],{"className":7878},[102],[40,7880,1044],{"className":7881},[102],[40,7883,7884,7887],{"style":1193},[40,7885],{"className":7886,"style":444},[361],[40,7888],{"className":7889,"style":1201},[1200],[40,7891,7892,7895],{"style":2802},[40,7893],{"className":7894,"style":444},[361],[40,7896,7898],{"className":7897},[102],[40,7899,7901],{"className":7900},[102,2218],[40,7902,7904,7937],{"className":7903},[344,345],[40,7905,7907,7934],{"className":7906},[349],[40,7908,7910,7922],{"className":7909,"style":2821},[353],[40,7911,7913,7916],{"className":7912,"style":2233},[2232],[40,7914],{"className":7915,"style":444},[361],[40,7917,7919],{"className":7918,"style":2240},[102],[40,7920,949],{"className":7921},[102],[40,7923,7924,7927],{"style":2836},[40,7925],{"className":7926,"style":444},[361],[40,7928,7930],{"className":7929,"style":2254},[2253],[2256,7931,7932],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,7933],{"d":2266},[40,7935,377],{"className":7936},[376],[40,7938,7940],{"className":7939},[349],[40,7941,7943],{"className":7942,"style":2856},[353],[40,7944],{},[40,7946,377],{"className":7947},[376],[40,7949,7951],{"className":7950},[349],[40,7952,7954],{"className":7953,"style":2868},[353],[40,7955],{},[40,7957],{"className":7958},[113,1162],[40,7960,2051],{"className":7961},[117],[40,7963],{"className":7964,"style":122},[121],[40,7966,7968],{"className":7967},[102],[40,7969,167],{"className":7970},[417,1238],[40,7972,946],{"className":7973,"style":3946},[102,335],[40,7975,167],{"className":7976},[106],[40,7978,5215],{"className":7979,"style":5231},[102,335],[40,7981,73],{"className":7982},[117],[40,7984],{"className":7985,"style":122},[121],[40,7987,7989],{"className":7988},[102,5168],[40,7990,7992],{"className":7991},[344],[40,7993,7995],{"className":7994},[349],[40,7996,7998,8006],{"className":7997,"style":5164},[353],[40,7999,8000,8003],{"style":2233},[40,8001],{"className":8002,"style":444},[361],[40,8004,5148],{"className":8005},[102,966],[40,8007,8008,8011],{"style":5188},[40,8009],{"className":8010,"style":444},[361],[40,8012,8014],{"className":8013,"style":5196},[5195],[40,8015,5151],{"className":8016},[102],[40,8018,207],{"className":8019},[113],[40,8021,73],{"className":8022},[117],[40,8024],{"className":8025,"style":122},[121],[40,8027,936],{"className":8028},[102,966],[40,8030,8032],{"className":8031},[102],[40,8033,207],{"className":8034},[417,1238],[40,8036],{"className":8037,"style":1246},[121],[40,8039,1058],{"className":8040},[1250],[40,8042],{"className":8043,"style":1246},[121],[40,8045,8047,8050,8053,8056,8059,8062,8065,8068,8071],{"className":8046},[93],[40,8048],{"className":8049,"style":98},[97],[40,8051,1796],{"className":8052},[102,966],[40,8054,167],{"className":8055},[106],[40,8057,1649],{"className":8058,"style":1312},[102,335],[40,8060,73],{"className":8061},[117],[40,8063],{"className":8064,"style":122},[121],[40,8066,1671],{"className":8067},[102,335],[40,8069,207],{"className":8070},[113],[40,8072,1123],{"className":8073},[102],[25,8075],{},[28,8077,8079,8080,8133],{"id":8078},"_6-optional-discrete-222times222-finitestate-wigner","6. Optional: Discrete (",[40,8081,8083,8102],{"className":8082},[43],[40,8084,8086],{"className":8085},[47],[49,8087,8088],{"xmlns":51},[53,8089,8090,8099],{},[56,8091,8092,8094,8097],{},[64,8093,1044],{},[59,8095,8096],{},"×",[64,8098,1044],{},[82,8100,8101],{"encoding":84},"2\\times2",[40,8103,8105,8124],{"className":8104,"ariaHidden":72},[89],[40,8106,8108,8112,8115,8118,8121],{"className":8107},[93],[40,8109],{"className":8110,"style":8111},[97],"height:0.7278em;vertical-align:-0.0833em;",[40,8113,1044],{"className":8114},[102],[40,8116],{"className":8117,"style":1246},[121],[40,8119,8096],{"className":8120},[1250],[40,8122],{"className":8123,"style":1246},[121],[40,8125,8127,8130],{"className":8126},[93],[40,8128],{"className":8129,"style":1624},[97],[40,8131,1044],{"className":8132},[102],") (Finite‑State) Wigner",[12,8135,8136,8137,8217,8218,8289],{},"On a 4‑point discrete phase space (",[40,8138,8140,8169],{"className":8139},[43],[40,8141,8143],{"className":8142},[47],[49,8144,8145],{"xmlns":51},[53,8146,8147,8166],{},[56,8148,8149,8151,8153],{},[154,8150,5215],{},[59,8152,939],{},[56,8154,8155,8157,8160,8162,8164],{},[59,8156,167],{"stretchy":61},[154,8158,8159],{},"q",[59,8161,73],{"separator":72},[154,8163,12],{},[59,8165,207],{"stretchy":61},[82,8167,8168],{"encoding":84},"\\alpha\\in{(q,p)}",[40,8170,8172,8190],{"className":8171,"ariaHidden":72},[89],[40,8173,8175,8178,8181,8184,8187],{"className":8174},[93],[40,8176],{"className":8177,"style":962},[97],[40,8179,5215],{"className":8180,"style":5231},[102,335],[40,8182],{"className":8183,"style":390},[121],[40,8185,939],{"className":8186},[394],[40,8188],{"className":8189,"style":390},[121],[40,8191,8193,8196],{"className":8192},[93],[40,8194],{"className":8195,"style":98},[97],[40,8197,8199,8202,8205,8208,8211,8214],{"className":8198},[102],[40,8200,167],{"className":8201},[106],[40,8203,8159],{"className":8204,"style":336},[102,335],[40,8206,73],{"className":8207},[117],[40,8209],{"className":8210,"style":122},[121],[40,8212,12],{"className":8213},[102,335],[40,8215,207],{"className":8216},[113],"), define phase‑point operators (",[40,8219,8221,8240],{"className":8220},[43],[40,8222,8224],{"className":8223},[47],[49,8225,8226],{"xmlns":51},[53,8227,8228,8237],{},[56,8229,8230],{},[151,8231,8232,8235],{},[154,8233,8234],{},"A",[154,8236,5215],{},[82,8238,8239],{"encoding":84},"A_\\alpha",[40,8241,8243],{"className":8242,"ariaHidden":72},[89],[40,8244,8246,8249],{"className":8245},[93],[40,8247],{"className":8248,"style":3106},[97],[40,8250,8252,8255],{"className":8251},[102],[40,8253,8234],{"className":8254},[102,335],[40,8256,8258],{"className":8257},[340],[40,8259,8261,8281],{"className":8260},[344,345],[40,8262,8264,8278],{"className":8263},[349],[40,8265,8267],{"className":8266,"style":354},[353],[40,8268,8269,8272],{"style":6098},[40,8270],{"className":8271,"style":362},[361],[40,8273,8275],{"className":8274},[366,367,368,369],[40,8276,5215],{"className":8277,"style":5231},[102,335,369],[40,8279,377],{"className":8280},[376],[40,8282,8284],{"className":8283},[349],[40,8285,8287],{"className":8286,"style":384},[353],[40,8288],{},") with",[40,8291,8293],{"className":8292},[136],[40,8294,8296,8373],{"className":8295},[43],[40,8297,8299],{"className":8298},[47],[49,8300,8301],{"xmlns":51,"display":145},[53,8302,8303,8370],{},[56,8304,8305,8313,8319,8321,8323,8325,8327,8329,8331,8333,8335,8341,8343,8345,8347,8349,8351,8357,8359,8368],{},[8306,8307,8308,8311],"munder",{},[59,8309,8310],{},"∑",[154,8312,5215],{},[151,8314,8315,8317],{},[154,8316,8234],{},[154,8318,5215],{},[59,8320,162],{},[64,8322,1044],{},[154,8324,1050],{},[59,8326,73],{"separator":72},[121,8328],{"width":173},[154,8330,1081],{"mathvariant":1080},[59,8332,1084],{},[59,8334,167],{"stretchy":61},[151,8336,8337,8339],{},[154,8338,8234],{},[154,8340,5215],{},[59,8342,207],{"stretchy":61},[59,8344,162],{},[64,8346,78],{},[59,8348,73],{"separator":72},[121,8350],{"width":173},[151,8352,8353,8355],{},[154,8354,8234],{},[154,8356,5215],{},[59,8358,162],{},[8360,8361,8362,8364,8366],"msubsup",{},[154,8363,8234],{},[154,8365,5215],{},[59,8367,3698],{"lspace":3697,"rspace":3697},[154,8369,1123],{"mathvariant":1080},[82,8371,8372],{"encoding":84},"\\sum_{\\alpha} A_\\alpha = 2I,\\quad \\operatorname{Tr}(A_\\alpha)=1,\\quad A_\\alpha = A_\\alpha^{\\dagger}.",[40,8374,8376,8490,8572,8639],{"className":8375,"ariaHidden":72},[89],[40,8377,8379,8383,8438,8441,8481,8484,8487],{"className":8378},[93],[40,8380],{"className":8381,"style":8382},[97],"height:2.3em;vertical-align:-1.25em;",[40,8384,8387],{"className":8385},[1369,8386],"op-limits",[40,8388,8390,8429],{"className":8389},[344,345],[40,8391,8393,8426],{"className":8392},[349],[40,8394,8397,8413],{"className":8395,"style":8396},[353],"height:1.05em;",[40,8398,8400,8404],{"style":8399},"top:-1.9em;margin-left:0em;",[40,8401],{"className":8402,"style":8403},[361],"height:3.05em;",[40,8405,8407],{"className":8406},[366,367,368,369],[40,8408,8410],{"className":8409},[102,369],[40,8411,5215],{"className":8412,"style":5231},[102,335,369],[40,8414,8416,8419],{"style":8415},"top:-3.05em;",[40,8417],{"className":8418,"style":8403},[361],[40,8420,8421],{},[40,8422,8310],{"className":8423},[1369,8424,8425],"op-symbol","large-op",[40,8427,377],{"className":8428},[376],[40,8430,8432],{"className":8431},[349],[40,8433,8436],{"className":8434,"style":8435},[353],"height:1.25em;",[40,8437],{},[40,8439],{"className":8440,"style":122},[121],[40,8442,8444,8447],{"className":8443},[102],[40,8445,8234],{"className":8446},[102,335],[40,8448,8450],{"className":8449},[340],[40,8451,8453,8473],{"className":8452},[344,345],[40,8454,8456,8470],{"className":8455},[349],[40,8457,8459],{"className":8458,"style":354},[353],[40,8460,8461,8464],{"style":6098},[40,8462],{"className":8463,"style":362},[361],[40,8465,8467],{"className":8466},[366,367,368,369],[40,8468,5215],{"className":8469,"style":5231},[102,335,369],[40,8471,377],{"className":8472},[376],[40,8474,8476],{"className":8475},[349],[40,8477,8479],{"className":8478,"style":384},[353],[40,8480],{},[40,8482],{"className":8483,"style":390},[121],[40,8485,162],{"className":8486},[394],[40,8488],{"className":8489,"style":390},[121],[40,8491,8493,8496,8499,8502,8505,8508,8511,8517,8520,8560,8563,8566,8569],{"className":8492},[93],[40,8494],{"className":8495,"style":98},[97],[40,8497,1044],{"className":8498},[102],[40,8500,1050],{"className":8501,"style":1242},[102,335],[40,8503,73],{"className":8504},[117],[40,8506],{"className":8507,"style":543},[121],[40,8509],{"className":8510,"style":122},[121],[40,8512,8514],{"className":8513},[1369],[40,8515,1081],{"className":8516},[102,1373],[40,8518,167],{"className":8519},[106],[40,8521,8523,8526],{"className":8522},[102],[40,8524,8234],{"className":8525},[102,335],[40,8527,8529],{"className":8528},[340],[40,8530,8532,8552],{"className":8531},[344,345],[40,8533,8535,8549],{"className":8534},[349],[40,8536,8538],{"className":8537,"style":354},[353],[40,8539,8540,8543],{"style":6098},[40,8541],{"className":8542,"style":362},[361],[40,8544,8546],{"className":8545},[366,367,368,369],[40,8547,5215],{"className":8548,"style":5231},[102,335,369],[40,8550,377],{"className":8551},[376],[40,8553,8555],{"className":8554},[349],[40,8556,8558],{"className":8557,"style":384},[353],[40,8559],{},[40,8561,207],{"className":8562},[113],[40,8564],{"className":8565,"style":390},[121],[40,8567,162],{"className":8568},[394],[40,8570],{"className":8571,"style":390},[121],[40,8573,8575,8578,8581,8584,8587,8590,8630,8633,8636],{"className":8574},[93],[40,8576],{"className":8577,"style":7167},[97],[40,8579,78],{"className":8580},[102],[40,8582,73],{"className":8583},[117],[40,8585],{"className":8586,"style":543},[121],[40,8588],{"className":8589,"style":122},[121],[40,8591,8593,8596],{"className":8592},[102],[40,8594,8234],{"className":8595},[102,335],[40,8597,8599],{"className":8598},[340],[40,8600,8602,8622],{"className":8601},[344,345],[40,8603,8605,8619],{"className":8604},[349],[40,8606,8608],{"className":8607,"style":354},[353],[40,8609,8610,8613],{"style":6098},[40,8611],{"className":8612,"style":362},[361],[40,8614,8616],{"className":8615},[366,367,368,369],[40,8617,5215],{"className":8618,"style":5231},[102,335,369],[40,8620,377],{"className":8621},[376],[40,8623,8625],{"className":8624},[349],[40,8626,8628],{"className":8627,"style":384},[353],[40,8629],{},[40,8631],{"className":8632,"style":390},[121],[40,8634,162],{"className":8635},[394],[40,8637],{"className":8638,"style":390},[121],[40,8640,8642,8646,8702],{"className":8641},[93],[40,8643],{"className":8644,"style":8645},[97],"height:1.1461em;vertical-align:-0.247em;",[40,8647,8649,8652],{"className":8648},[102],[40,8650,8234],{"className":8651},[102,335],[40,8653,8655],{"className":8654},[340],[40,8656,8658,8693],{"className":8657},[344,345],[40,8659,8661,8690],{"className":8660},[349],[40,8662,8664,8676],{"className":8663,"style":7363},[353],[40,8665,8667,8670],{"style":8666},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[40,8668],{"className":8669,"style":362},[361],[40,8671,8673],{"className":8672},[366,367,368,369],[40,8674,5215],{"className":8675,"style":5231},[102,335,369],[40,8677,8678,8681],{"style":3962},[40,8679],{"className":8680,"style":362},[361],[40,8682,8684],{"className":8683},[366,367,368,369],[40,8685,8687],{"className":8686},[102,369],[40,8688,3698],{"className":8689},[102,369],[40,8691,377],{"className":8692},[376],[40,8694,8696],{"className":8695},[349],[40,8697,8700],{"className":8698,"style":8699},[353],"height:0.247em;",[40,8701],{},[40,8703,1123],{"className":8704},[102],[12,8706,8707,8708,8711],{},"The ",[35,8709,8710],{},"discrete Wigner function"," is",[40,8713,8715],{"className":8714},[136],[40,8716,8718,8774],{"className":8717},[43],[40,8719,8721],{"className":8720},[47],[49,8722,8723],{"xmlns":51,"display":145},[53,8724,8725,8771],{},[56,8726,8727,8733,8735,8737,8739,8741,8749,8751,8753,8755,8757,8759,8761,8767,8769],{},[151,8728,8729,8731],{},[154,8730,2437],{},[154,8732,1032],{},[59,8734,167],{"stretchy":61},[154,8736,5215],{},[59,8738,207],{"stretchy":61},[59,8740,162],{},[181,8742,8743],{"scriptlevel":66,"displaystyle":61},[1038,8744,8745,8747],{},[64,8746,78],{},[64,8748,1044],{},[59,8750,73],{"separator":72},[154,8752,1081],{"mathvariant":1080},[59,8754,1084],{},[59,8756,167],{"stretchy":61},[154,8758,1032],{},[59,8760,73],{"separator":72},[151,8762,8763,8765],{},[154,8764,8234],{},[154,8766,5215],{},[59,8768,207],{"stretchy":61},[59,8770,73],{"separator":72},[82,8772,8773],{"encoding":84},"W_\\rho(\\alpha) = \\tfrac{1}{2},\\operatorname{Tr}(\\rho,A_\\alpha),",[40,8775,8777,8841],{"className":8776,"ariaHidden":72},[89],[40,8778,8780,8783,8823,8826,8829,8832,8835,8838],{"className":8779},[93],[40,8781],{"className":8782,"style":1365},[97],[40,8784,8786,8789],{"className":8785},[102],[40,8787,2437],{"className":8788,"style":2541},[102,335],[40,8790,8792],{"className":8791},[340],[40,8793,8795,8815],{"className":8794},[344,345],[40,8796,8798,8812],{"className":8797},[349],[40,8799,8801],{"className":8800,"style":354},[353],[40,8802,8803,8806],{"style":2556},[40,8804],{"className":8805,"style":362},[361],[40,8807,8809],{"className":8808},[366,367,368,369],[40,8810,1032],{"className":8811},[102,335,369],[40,8813,377],{"className":8814},[376],[40,8816,8818],{"className":8817},[349],[40,8819,8821],{"className":8820,"style":585},[353],[40,8822],{},[40,8824,167],{"className":8825},[106],[40,8827,5215],{"className":8828,"style":5231},[102,335],[40,8830,207],{"className":8831},[113],[40,8833],{"className":8834,"style":390},[121],[40,8836,162],{"className":8837},[394],[40,8839],{"className":8840,"style":390},[121],[40,8842,8844,8847,8915,8918,8921,8927,8930,8933,8936,8939,8979,8982],{"className":8843},[93],[40,8845],{"className":8846,"style":2676},[97],[40,8848,8850,8853,8912],{"className":8849},[102],[40,8851],{"className":8852},[106,1162],[40,8854,8856],{"className":8855},[1038],[40,8857,8859,8904],{"className":8858},[344,345],[40,8860,8862,8901],{"className":8861},[349],[40,8863,8865,8879,8887],{"className":8864,"style":1175},[353],[40,8866,8867,8870],{"style":1178},[40,8868],{"className":8869,"style":444},[361],[40,8871,8873],{"className":8872},[366,367,368,369],[40,8874,8876],{"className":8875},[102,369],[40,8877,1044],{"className":8878},[102,369],[40,8880,8881,8884],{"style":1193},[40,8882],{"className":8883,"style":444},[361],[40,8885],{"className":8886,"style":1201},[1200],[40,8888,8889,8892],{"style":1204},[40,8890],{"className":8891,"style":444},[361],[40,8893,8895],{"className":8894},[366,367,368,369],[40,8896,8898],{"className":8897},[102,369],[40,8899,78],{"className":8900},[102,369],[40,8902,377],{"className":8903},[376],[40,8905,8907],{"className":8906},[349],[40,8908,8910],{"className":8909,"style":1226},[353],[40,8911],{},[40,8913],{"className":8914},[113,1162],[40,8916,73],{"className":8917},[117],[40,8919],{"className":8920,"style":122},[121],[40,8922,8924],{"className":8923},[1369],[40,8925,1081],{"className":8926},[102,1373],[40,8928,167],{"className":8929},[106],[40,8931,1032],{"className":8932},[102,335],[40,8934,73],{"className":8935},[117],[40,8937],{"className":8938,"style":122},[121],[40,8940,8942,8945],{"className":8941},[102],[40,8943,8234],{"className":8944},[102,335],[40,8946,8948],{"className":8947},[340],[40,8949,8951,8971],{"className":8950},[344,345],[40,8952,8954,8968],{"className":8953},[349],[40,8955,8957],{"className":8956,"style":354},[353],[40,8958,8959,8962],{"style":6098},[40,8960],{"className":8961,"style":362},[361],[40,8963,8965],{"className":8964},[366,367,368,369],[40,8966,5215],{"className":8967,"style":5231},[102,335,369],[40,8969,377],{"className":8970},[376],[40,8972,8974],{"className":8973},[349],[40,8975,8977],{"className":8976,"style":384},[353],[40,8978],{},[40,8980,207],{"className":8981},[113],[40,8983,73],{"className":8984},[117],[12,8986,8987],{},"with line sums reproducing measurement probabilities in mutually unbiased bases.",[25,8989],{},[28,8991,8993],{"id":8992},"_7-color-scale-purestate-bounds","7. Color Scale (Pure‑State Bounds)",[12,8995,8996,8997,1014],{},"For (",[40,8998,9000,9021],{"className":8999},[43],[40,9001,9003],{"className":9002},[47],[49,9004,9005],{"xmlns":51},[53,9006,9007,9019],{},[56,9008,9009,9011,9013,9015,9017],{},[59,9010,1579],{"stretchy":61},[154,9012,936],{"mathvariant":935},[59,9014,1579],{"stretchy":61},[59,9016,162],{},[64,9018,78],{},[82,9020,2973],{"encoding":84},[40,9022,9024,9048],{"className":9023,"ariaHidden":72},[89],[40,9025,9027,9030,9033,9036,9039,9042,9045],{"className":9026},[93],[40,9028],{"className":9029,"style":98},[97],[40,9031,1579],{"className":9032},[106],[40,9034,936],{"className":9035},[102,966],[40,9037,1579],{"className":9038},[113],[40,9040],{"className":9041,"style":390},[121],[40,9043,162],{"className":9044},[394],[40,9046],{"className":9047,"style":390},[121],[40,9049,9051,9054],{"className":9050},[93],[40,9052],{"className":9053,"style":1624},[97],[40,9055,78],{"className":9056},[102],[40,9058,9060],{"className":9059},[136],[40,9061,9063,9152],{"className":9062},[43],[40,9064,9066],{"className":9065},[47],[49,9067,9068],{"xmlns":51,"display":145},[53,9069,9070,9149],{},[56,9071,9072,9080,9082,9090,9092,9102,9105,9107,9110,9112,9114,9122,9124,9132,9134,9144,9146],{},[151,9073,9074,9076,9078],{},[154,9075,2437],{},[154,9077,3067],{},[59,9079,1084],{},[59,9081,162],{},[181,9083,9084],{"scriptlevel":66,"displaystyle":61},[1038,9085,9086,9088],{},[64,9087,78],{},[64,9089,1044],{},[59,9091,243],{},[181,9093,9094],{"scriptlevel":66,"displaystyle":61},[1038,9095,9096,9100],{},[2045,9097,9098],{},[64,9099,949],{},[64,9101,1044],{},[59,9103,9104],{},"≈",[59,9106,243],{},[64,9108,9109],{},"0.3660254",[59,9111,73],{"separator":72},[121,9113],{"width":1068},[151,9115,9116,9118,9120],{},[154,9117,2437],{},[154,9119,3032],{},[59,9121,1084],{},[59,9123,162],{},[181,9125,9126],{"scriptlevel":66,"displaystyle":61},[1038,9127,9128,9130],{},[64,9129,78],{},[64,9131,1044],{},[59,9133,1053],{},[181,9135,9136],{"scriptlevel":66,"displaystyle":61},[1038,9137,9138,9142],{},[2045,9139,9140],{},[64,9141,949],{},[64,9143,1044],{},[59,9145,9104],{},[64,9147,9148],{},"1.3660254.",[82,9150,9151],{"encoding":84},"W_{\\min}=\\tfrac{1}{2}-\\tfrac{\\sqrt{3}}{2}\\approx -0.3660254,\\qquad\nW_{\\max}=\\tfrac{1}{2}+\\tfrac{\\sqrt{3}}{2}\\approx 1.3660254.",[40,9153,9155,9222,9305,9432,9514,9597,9724],{"className":9154,"ariaHidden":72},[89],[40,9156,9158,9161,9213,9216,9219],{"className":9157},[93],[40,9159],{"className":9160,"style":3106},[97],[40,9162,9164,9167],{"className":9163},[102],[40,9165,2437],{"className":9166,"style":2541},[102,335],[40,9168,9170],{"className":9169},[340],[40,9171,9173,9205],{"className":9172},[344,345],[40,9174,9176,9202],{"className":9175},[349],[40,9177,9179],{"className":9178,"style":3403},[353],[40,9180,9181,9184],{"style":2556},[40,9182],{"className":9183,"style":362},[361],[40,9185,9187],{"className":9186},[366,367,368,369],[40,9188,9190],{"className":9189},[102,369],[40,9191,9193,9196,9199],{"className":9192},[1369,369],[40,9194,3142],{"className":9195},[369],[40,9197,246],{"className":9198},[369],[40,9200,1796],{"className":9201},[369],[40,9203,377],{"className":9204},[376],[40,9206,9208],{"className":9207},[349],[40,9209,9211],{"className":9210,"style":384},[353],[40,9212],{},[40,9214],{"className":9215,"style":390},[121],[40,9217,162],{"className":9218},[394],[40,9220],{"className":9221,"style":390},[121],[40,9223,9225,9228,9296,9299,9302],{"className":9224},[93],[40,9226],{"className":9227,"style":2676},[97],[40,9229,9231,9234,9293],{"className":9230},[102],[40,9232],{"className":9233},[106,1162],[40,9235,9237],{"className":9236},[1038],[40,9238,9240,9285],{"className":9239},[344,345],[40,9241,9243,9282],{"className":9242},[349],[40,9244,9246,9260,9268],{"className":9245,"style":1175},[353],[40,9247,9248,9251],{"style":1178},[40,9249],{"className":9250,"style":444},[361],[40,9252,9254],{"className":9253},[366,367,368,369],[40,9255,9257],{"className":9256},[102,369],[40,9258,1044],{"className":9259},[102,369],[40,9261,9262,9265],{"style":1193},[40,9263],{"className":9264,"style":444},[361],[40,9266],{"className":9267,"style":1201},[1200],[40,9269,9270,9273],{"style":1204},[40,9271],{"className":9272,"style":444},[361],[40,9274,9276],{"className":9275},[366,367,368,369],[40,9277,9279],{"className":9278},[102,369],[40,9280,78],{"className":9281},[102,369],[40,9283,377],{"className":9284},[376],[40,9286,9288],{"className":9287},[349],[40,9289,9291],{"className":9290,"style":1226},[353],[40,9292],{},[40,9294],{"className":9295},[113,1162],[40,9297],{"className":9298,"style":1246},[121],[40,9300,243],{"className":9301},[1250],[40,9303],{"className":9304,"style":1246},[121],[40,9306,9308,9311,9423,9426,9429],{"className":9307},[93],[40,9309],{"className":9310,"style":3258},[97],[40,9312,9314,9317,9420],{"className":9313},[102],[40,9315],{"className":9316},[106,1162],[40,9318,9320],{"className":9319},[1038],[40,9321,9323,9412],{"className":9322},[344,345],[40,9324,9326,9409],{"className":9325},[349],[40,9327,9329,9343,9351],{"className":9328,"style":3277},[353],[40,9330,9331,9334],{"style":1178},[40,9332],{"className":9333,"style":444},[361],[40,9335,9337],{"className":9336},[366,367,368,369],[40,9338,9340],{"className":9339},[102,369],[40,9341,1044],{"className":9342},[102,369],[40,9344,9345,9348],{"style":1193},[40,9346],{"className":9347,"style":444},[361],[40,9349],{"className":9350,"style":1201},[1200],[40,9352,9353,9356],{"style":3302},[40,9354],{"className":9355,"style":444},[361],[40,9357,9359],{"className":9358},[366,367,368,369],[40,9360,9362],{"className":9361},[102,369],[40,9363,9365],{"className":9364},[102,2218,369],[40,9366,9368,9401],{"className":9367},[344,345],[40,9369,9371,9398],{"className":9370},[349],[40,9372,9374,9386],{"className":9373,"style":3324},[353],[40,9375,9377,9380],{"className":9376,"style":2233},[2232],[40,9378],{"className":9379,"style":444},[361],[40,9381,9383],{"className":9382,"style":2240},[102,369],[40,9384,949],{"className":9385},[102,369],[40,9387,9388,9391],{"style":3339},[40,9389],{"className":9390,"style":444},[361],[40,9392,9394],{"className":9393,"style":2254},[2253,369],[2256,9395,9396],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,9397],{"d":2266},[40,9399,377],{"className":9400},[376],[40,9402,9404],{"className":9403},[349],[40,9405,9407],{"className":9406,"style":3359},[353],[40,9408],{},[40,9410,377],{"className":9411},[376],[40,9413,9415],{"className":9414},[349],[40,9416,9418],{"className":9417,"style":1226},[353],[40,9419],{},[40,9421],{"className":9422},[113,1162],[40,9424],{"className":9425,"style":390},[121],[40,9427,9104],{"className":9428},[394],[40,9430],{"className":9431,"style":390},[121],[40,9433,9435,9438,9441,9444,9447,9450,9453,9505,9508,9511],{"className":9434},[93],[40,9436],{"className":9437,"style":7167},[97],[40,9439,243],{"className":9440},[102],[40,9442,9109],{"className":9443},[102],[40,9445,73],{"className":9446},[117],[40,9448],{"className":9449,"style":1302},[121],[40,9451],{"className":9452,"style":122},[121],[40,9454,9456,9459],{"className":9455},[102],[40,9457,2437],{"className":9458,"style":2541},[102,335],[40,9460,9462],{"className":9461},[340],[40,9463,9465,9497],{"className":9464},[344,345],[40,9466,9468,9494],{"className":9467},[349],[40,9469,9471],{"className":9470,"style":354},[353],[40,9472,9473,9476],{"style":2556},[40,9474],{"className":9475,"style":362},[361],[40,9477,9479],{"className":9478},[366,367,368,369],[40,9480,9482],{"className":9481},[102,369],[40,9483,9485,9488,9491],{"className":9484},[1369,369],[40,9486,3142],{"className":9487},[369],[40,9489,19],{"className":9490},[369],[40,9492,159],{"className":9493},[369],[40,9495,377],{"className":9496},[376],[40,9498,9500],{"className":9499},[349],[40,9501,9503],{"className":9502,"style":384},[353],[40,9504],{},[40,9506],{"className":9507,"style":390},[121],[40,9509,162],{"className":9510},[394],[40,9512],{"className":9513,"style":390},[121],[40,9515,9517,9520,9588,9591,9594],{"className":9516},[93],[40,9518],{"className":9519,"style":2676},[97],[40,9521,9523,9526,9585],{"className":9522},[102],[40,9524],{"className":9525},[106,1162],[40,9527,9529],{"className":9528},[1038],[40,9530,9532,9577],{"className":9531},[344,345],[40,9533,9535,9574],{"className":9534},[349],[40,9536,9538,9552,9560],{"className":9537,"style":1175},[353],[40,9539,9540,9543],{"style":1178},[40,9541],{"className":9542,"style":444},[361],[40,9544,9546],{"className":9545},[366,367,368,369],[40,9547,9549],{"className":9548},[102,369],[40,9550,1044],{"className":9551},[102,369],[40,9553,9554,9557],{"style":1193},[40,9555],{"className":9556,"style":444},[361],[40,9558],{"className":9559,"style":1201},[1200],[40,9561,9562,9565],{"style":1204},[40,9563],{"className":9564,"style":444},[361],[40,9566,9568],{"className":9567},[366,367,368,369],[40,9569,9571],{"className":9570},[102,369],[40,9572,78],{"className":9573},[102,369],[40,9575,377],{"className":9576},[376],[40,9578,9580],{"className":9579},[349],[40,9581,9583],{"className":9582,"style":1226},[353],[40,9584],{},[40,9586],{"className":9587},[113,1162],[40,9589],{"className":9590,"style":1246},[121],[40,9592,1053],{"className":9593},[1250],[40,9595],{"className":9596,"style":1246},[121],[40,9598,9600,9603,9715,9718,9721],{"className":9599},[93],[40,9601],{"className":9602,"style":3258},[97],[40,9604,9606,9609,9712],{"className":9605},[102],[40,9607],{"className":9608},[106,1162],[40,9610,9612],{"className":9611},[1038],[40,9613,9615,9704],{"className":9614},[344,345],[40,9616,9618,9701],{"className":9617},[349],[40,9619,9621,9635,9643],{"className":9620,"style":3277},[353],[40,9622,9623,9626],{"style":1178},[40,9624],{"className":9625,"style":444},[361],[40,9627,9629],{"className":9628},[366,367,368,369],[40,9630,9632],{"className":9631},[102,369],[40,9633,1044],{"className":9634},[102,369],[40,9636,9637,9640],{"style":1193},[40,9638],{"className":9639,"style":444},[361],[40,9641],{"className":9642,"style":1201},[1200],[40,9644,9645,9648],{"style":3302},[40,9646],{"className":9647,"style":444},[361],[40,9649,9651],{"className":9650},[366,367,368,369],[40,9652,9654],{"className":9653},[102,369],[40,9655,9657],{"className":9656},[102,2218,369],[40,9658,9660,9693],{"className":9659},[344,345],[40,9661,9663,9690],{"className":9662},[349],[40,9664,9666,9678],{"className":9665,"style":3324},[353],[40,9667,9669,9672],{"className":9668,"style":2233},[2232],[40,9670],{"className":9671,"style":444},[361],[40,9673,9675],{"className":9674,"style":2240},[102,369],[40,9676,949],{"className":9677},[102,369],[40,9679,9680,9683],{"style":3339},[40,9681],{"className":9682,"style":444},[361],[40,9684,9686],{"className":9685,"style":2254},[2253,369],[2256,9687,9688],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,9689],{"d":2266},[40,9691,377],{"className":9692},[376],[40,9694,9696],{"className":9695},[349],[40,9697,9699],{"className":9698,"style":3359},[353],[40,9700],{},[40,9702,377],{"className":9703},[376],[40,9705,9707],{"className":9706},[349],[40,9708,9710],{"className":9709,"style":1226},[353],[40,9711],{},[40,9713],{"className":9714},[113,1162],[40,9716],{"className":9717,"style":390},[121],[40,9719,9104],{"className":9720},[394],[40,9722],{"className":9723,"style":390},[121],[40,9725,9727,9730],{"className":9726},[93],[40,9728],{"className":9729,"style":1624},[97],[40,9731,9148],{"className":9732},[102],[12,9734,9735],{},"These constants are useful for fixing a consistent color scale across frames.",[25,9737],{},[28,9739,9741],{"id":9740},"_8-acronymnotation-glossary","8. Acronym\u002FNotation Glossary",[9743,9744,9745,9752,9758,9764,9770,9888],"ul",{},[9746,9747,9748,9751],"li",{},[35,9749,9750],{},"SW"," — Stratonovich–Weyl (kernel\u002Fcorrespondence).",[9746,9753,9754,9757],{},[35,9755,9756],{},"SU(2)"," — Special Unitary group of degree two (spin‑(1\u002F2) rotations).",[9746,9759,9760,9763],{},[35,9761,9762],{},"SO(3)"," — Special Orthogonal group in 3D (real rotation matrices).",[9746,9765,9766,9769],{},[35,9767,9768],{},"WF"," — Wigner function.",[9746,9771,9772,9813,9814,9855,9856,9887],{},[35,9773,167,9774,207],{},[40,9775,9777,9795],{"className":9776},[43],[40,9778,9780],{"className":9779},[47],[49,9781,9782],{"xmlns":51},[53,9783,9784,9792],{},[56,9785,9786,9788,9790],{},[59,9787,62],{"stretchy":61},[154,9789,1506],{},[59,9791,69],{"stretchy":61},[82,9793,9794],{"encoding":84},"\\lvert\\psi\\rangle",[40,9796,9798],{"className":9797,"ariaHidden":72},[89],[40,9799,9801,9804,9807,9810],{"className":9800},[93],[40,9802],{"className":9803,"style":98},[97],[40,9805,62],{"className":9806},[106],[40,9808,1506],{"className":9809,"style":336},[102,335],[40,9811,69],{"className":9812},[113]," — State vector (ket); ",[35,9815,167,9816,207],{},[40,9817,9819,9837],{"className":9818},[43],[40,9820,9822],{"className":9821},[47],[49,9823,9824],{"xmlns":51},[53,9825,9826,9834],{},[56,9827,9828,9830,9832],{},[59,9829,1511],{"stretchy":61},[154,9831,1506],{},[59,9833,62],{"stretchy":61},[82,9835,9836],{"encoding":84},"\\langle\\psi\\rvert",[40,9838,9840],{"className":9839,"ariaHidden":72},[89],[40,9841,9843,9846,9849,9852],{"className":9842},[93],[40,9844],{"className":9845,"style":98},[97],[40,9847,1511],{"className":9848},[106],[40,9850,1506],{"className":9851,"style":336},[102,335],[40,9853,62],{"className":9854},[113]," — bra; ",[35,9857,167,9858,207],{},[40,9859,9861,9875],{"className":9860},[43],[40,9862,9864],{"className":9863},[47],[49,9865,9866],{"xmlns":51},[53,9867,9868,9872],{},[56,9869,9870],{},[154,9871,1032],{},[82,9873,9874],{"encoding":84},"\\rho",[40,9876,9878],{"className":9877,"ariaHidden":72},[89],[40,9879,9881,9884],{"className":9880},[93],[40,9882],{"className":9883,"style":1136},[97],[40,9885,1032],{"className":9886},[102,335]," — density operator.",[9746,9889,9890,9893,9894,10014,10015,1628],{},[35,9891,9892],{},"Adjoint map"," — Conjugation action (",[40,9895,9897,9929],{"className":9896},[43],[40,9898,9900],{"className":9899},[47],[49,9901,9902],{"xmlns":51},[53,9903,9904,9926],{},[56,9905,9906,9908,9911,9913,9916,9918,9920],{},[154,9907,3688],{},[59,9909,9910],{},":",[154,9912,1032],{},[59,9914,9915],{},"↦",[154,9917,3688],{},[154,9919,1032],{},[941,9921,9922,9924],{},[154,9923,3688],{},[59,9925,3698],{"lspace":3697,"rspace":3697},[82,9927,9928],{"encoding":84},"U: \\rho\\mapsto U\\rho U^{\\dagger}",[40,9930,9932,9950,9968],{"className":9931,"ariaHidden":72},[89],[40,9933,9935,9938,9941,9944,9947],{"className":9934},[93],[40,9936],{"className":9937,"style":4304},[97],[40,9939,3688],{"className":9940,"style":3802},[102,335],[40,9942],{"className":9943,"style":390},[121],[40,9945,9910],{"className":9946},[394],[40,9948],{"className":9949,"style":390},[121],[40,9951,9953,9956,9959,9962,9965],{"className":9952},[93],[40,9954],{"className":9955,"style":5773},[97],[40,9957,1032],{"className":9958},[102,335],[40,9960],{"className":9961,"style":390},[121],[40,9963,9915],{"className":9964},[394],[40,9966],{"className":9967,"style":390},[121],[40,9969,9971,9975,9978,9981],{"className":9970},[93],[40,9972],{"className":9973,"style":9974},[97],"height:1.0435em;vertical-align:-0.1944em;",[40,9976,3688],{"className":9977,"style":3802},[102,335],[40,9979,1032],{"className":9980},[102,335],[40,9982,9984,9987],{"className":9983},[102],[40,9985,3688],{"className":9986,"style":3802},[102,335],[40,9988,9990],{"className":9989},[340],[40,9991,9993],{"className":9992},[344],[40,9994,9996],{"className":9995},[349],[40,9997,10000],{"className":9998,"style":9999},[353],"height:0.8491em;",[40,10001,10002,10005],{"style":1004},[40,10003],{"className":10004,"style":362},[361],[40,10006,10008],{"className":10007},[366,367,368,369],[40,10009,10011],{"className":10010},[102,369],[40,10012,3698],{"className":10013},[102,369],") inducing (",[40,10016,10018,10038],{"className":10017},[43],[40,10019,10021],{"className":10020},[47],[49,10022,10023],{"xmlns":51},[53,10024,10025,10035],{},[56,10026,10027,10029,10031,10033],{},[154,10028,936],{"mathvariant":935},[59,10030,9915],{},[154,10032,946],{},[154,10034,936],{"mathvariant":935},[82,10036,10037],{"encoding":84},"\\mathbf r\\mapsto R\\mathbf r",[40,10039,10041,10060],{"className":10040,"ariaHidden":72},[89],[40,10042,10044,10048,10051,10054,10057],{"className":10043},[93],[40,10045],{"className":10046,"style":10047},[97],"height:0.522em;vertical-align:-0.011em;",[40,10049,936],{"className":10050},[102,966],[40,10052],{"className":10053,"style":390},[121],[40,10055,9915],{"className":10056},[394],[40,10058],{"className":10059,"style":390},[121],[40,10061,10063,10066,10069],{"className":10062},[93],[40,10064],{"className":10065,"style":4304},[97],[40,10067,946],{"className":10068,"style":3946},[102,335],[40,10070,936],{"className":10071},[102,966],[25,10073],{},[28,10075,10077],{"id":10076},"_9-mindmap","9. Mind‑Map",[12,10079,10080],{},"Wigner for Qubits",[10082,10083,10087],"pre",{"className":10084,"code":10086,"language":2932},[10085],"language-text","Wigner (spin, S^2)\n├─ SW kernel Δ(θ,φ) → W(θ,φ) = Tr[ρΔ]\n│  └─ Explicit: ½ + (√3\u002F2) r·n(θ,φ)\n├─ State ρ ↔ Bloch vector r\n├─ Gates U = exp[-i(α\u002F2) u·σ]\n│  ├─ Pauli X\u002FY\u002FZ: α = π, axes x\u002Fy\u002Fz\n│  └─ Custom axis–angle (Rodrigues)\n└─ Covariance: W → W∘R^{-1}\n",[10088,10089,10086],"code",{"__ignoreMap":10090},"",[25,10092],{},[28,10094,10096],{"id":10095},"_10-minimal-substitutions-if-variations-are-needed","10. Minimal Substitutions (if variations are needed)",[9743,10098,10099,10161],{},[9746,10100,10101,10102,10131,10132,10160],{},"If a different SW kernel normalization is chosen, the prefactors in (",[40,10103,10105,10119],{"className":10104},[43],[40,10106,10108],{"className":10107},[47],[49,10109,10110],{"xmlns":51},[53,10111,10112,10116],{},[56,10113,10114],{},[154,10115,2016],{"mathvariant":1080},[82,10117,10118],{"encoding":84},"\\Delta",[40,10120,10122],{"className":10121,"ariaHidden":72},[89],[40,10123,10125,10128],{"className":10124},[93],[40,10126],{"className":10127,"style":4304},[97],[40,10129,2016],{"className":10130},[102],") and hence (",[40,10133,10135,10148],{"className":10134},[43],[40,10136,10138],{"className":10137},[47],[49,10139,10140],{"xmlns":51},[53,10141,10142,10146],{},[56,10143,10144],{},[154,10145,2437],{},[82,10147,2437],{"encoding":84},[40,10149,10151],{"className":10150,"ariaHidden":72},[89],[40,10152,10154,10157],{"className":10153},[93],[40,10155],{"className":10156,"style":4304},[97],[40,10158,2437],{"className":10159,"style":2541},[102,335],") adjust accordingly, while covariance and linearity remain.",[9746,10162,10163,10164,10226,10227,10266],{},"Mixed states use the same formulas with (",[40,10165,10167,10190],{"className":10166},[43],[40,10168,10170],{"className":10169},[47],[49,10171,10172],{"xmlns":51},[53,10173,10174,10187],{},[56,10175,10176,10178,10180,10182,10185],{},[59,10177,1579],{"stretchy":61},[154,10179,936],{"mathvariant":935},[59,10181,1579],{"stretchy":61},[59,10183,10184],{},"\u003C",[64,10186,78],{},[82,10188,10189],{"encoding":84},"\\lVert\\mathbf r\\rVert\u003C1",[40,10191,10193,10217],{"className":10192,"ariaHidden":72},[89],[40,10194,10196,10199,10202,10205,10208,10211,10214],{"className":10195},[93],[40,10197],{"className":10198,"style":98},[97],[40,10200,1579],{"className":10201},[106],[40,10203,936],{"className":10204},[102,966],[40,10206,1579],{"className":10207},[113],[40,10209],{"className":10210,"style":390},[121],[40,10212,10184],{"className":10213},[394],[40,10215],{"className":10216,"style":390},[121],[40,10218,10220,10223],{"className":10219},[93],[40,10221],{"className":10222,"style":1624},[97],[40,10224,78],{"className":10225},[102],") (the extrema shrink linearly with (",[40,10228,10230,10248],{"className":10229},[43],[40,10231,10233],{"className":10232},[47],[49,10234,10235],{"xmlns":51},[53,10236,10237,10245],{},[56,10238,10239,10241,10243],{},[59,10240,1579],{"stretchy":61},[154,10242,936],{"mathvariant":935},[59,10244,1579],{"stretchy":61},[82,10246,10247],{"encoding":84},"\\lVert\\mathbf r\\rVert",[40,10249,10251],{"className":10250,"ariaHidden":72},[89],[40,10252,10254,10257,10260,10263],{"className":10253},[93],[40,10255],{"className":10256,"style":98},[97],[40,10258,1579],{"className":10259},[106],[40,10261,936],{"className":10262},[102,966],[40,10264,1579],{"className":10265},[113],")).",[10082,10268,10272],{"className":10269,"code":10270,"language":10271,"meta":10090,"style":10090},"language-python shiki shiki-themes one-dark-pro","# @title Cell 1 — Imports & global plotting config (DPI=200)\n\"\"\"\nSpin (SU(2)) Wigner-function animations for single-qubit gates.\n\nThis notebook renders the qubit Wigner function\n    W(θ, φ) = 1\u002F2 + (√3\u002F2) * r · n(θ, φ)\nand animates its rigid rotation under Pauli and custom axis–angle gates.\n\nAll titles and dynamic text are drawn *inside* the plotted region so they\nnever clip in the saved GIFs. Use Cells 3 and 4 to generate animations.\n\"\"\"\n\nfrom __future__ import annotations\n\nimport numpy as np\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n\n# Global Matplotlib configuration: DPI = 200 everywhere\nmpl.rcParams.update({\n    \"figure.dpi\": 200,       # on-screen DPI\n    \"savefig.dpi\": 200,      # saved file DPI\n    \"figure.figsize\": (6.4, 3.6),  # 1280×720 at DPI=200\n    \"axes.grid\": False,\n})\n","python",[10088,10273,10274,10282,10289,10295,10302,10308,10314,10320,10325,10331,10337,10342,10347,10365,10370,10385,10398,10411,10416,10422,10435,10454,10470,10494,10508],{"__ignoreMap":10090},[40,10275,10278],{"class":10276,"line":10277},"line",1,[40,10279,10281],{"class":10280},"sV9Aq","# @title Cell 1 — Imports & global plotting config (DPI=200)\n",[40,10283,10285],{"class":10276,"line":10284},2,[40,10286,10288],{"class":10287},"subq3","\"\"\"\n",[40,10290,10292],{"class":10276,"line":10291},3,[40,10293,10294],{"class":10287},"Spin (SU(2)) Wigner-function animations for single-qubit gates.\n",[40,10296,10298],{"class":10276,"line":10297},4,[40,10299,10301],{"emptyLinePlaceholder":10300},true,"\n",[40,10303,10305],{"class":10276,"line":10304},5,[40,10306,10307],{"class":10287},"This notebook renders the qubit Wigner function\n",[40,10309,10311],{"class":10276,"line":10310},6,[40,10312,10313],{"class":10287},"    W(θ, φ) = 1\u002F2 + (√3\u002F2) * r · n(θ, φ)\n",[40,10315,10317],{"class":10276,"line":10316},7,[40,10318,10319],{"class":10287},"and animates its rigid rotation under Pauli and custom axis–angle gates.\n",[40,10321,10323],{"class":10276,"line":10322},8,[40,10324,10301],{"emptyLinePlaceholder":10300},[40,10326,10328],{"class":10276,"line":10327},9,[40,10329,10330],{"class":10287},"All titles and dynamic text are drawn *inside* the plotted region so they\n",[40,10332,10334],{"class":10276,"line":10333},10,[40,10335,10336],{"class":10287},"never clip in the saved GIFs. Use Cells 3 and 4 to generate animations.\n",[40,10338,10340],{"class":10276,"line":10339},11,[40,10341,10288],{"class":10287},[40,10343,10345],{"class":10276,"line":10344},12,[40,10346,10301],{"emptyLinePlaceholder":10300},[40,10348,10350,10354,10358,10361],{"class":10276,"line":10349},13,[40,10351,10353],{"class":10352},"seHd6","from",[40,10355,10357],{"class":10356},"sVyAn"," __future__",[40,10359,10360],{"class":10352}," import",[40,10362,10364],{"class":10363},"sn6KH"," annotations\n",[40,10366,10368],{"class":10276,"line":10367},14,[40,10369,10301],{"emptyLinePlaceholder":10300},[40,10371,10373,10376,10379,10382],{"class":10276,"line":10372},15,[40,10374,10375],{"class":10352},"import",[40,10377,10378],{"class":10363}," numpy ",[40,10380,10381],{"class":10352},"as",[40,10383,10384],{"class":10363}," np\n",[40,10386,10388,10390,10393,10395],{"class":10276,"line":10387},16,[40,10389,10375],{"class":10352},[40,10391,10392],{"class":10363}," matplotlib ",[40,10394,10381],{"class":10352},[40,10396,10397],{"class":10363}," mpl\n",[40,10399,10401,10403,10406,10408],{"class":10276,"line":10400},17,[40,10402,10375],{"class":10352},[40,10404,10405],{"class":10363}," matplotlib.pyplot ",[40,10407,10381],{"class":10352},[40,10409,10410],{"class":10363}," plt\n",[40,10412,10414],{"class":10276,"line":10413},18,[40,10415,10301],{"emptyLinePlaceholder":10300},[40,10417,10419],{"class":10276,"line":10418},19,[40,10420,10421],{"class":10280},"# Global Matplotlib configuration: DPI = 200 everywhere\n",[40,10423,10425,10428,10432],{"class":10276,"line":10424},20,[40,10426,10427],{"class":10363},"mpl.rcParams.",[40,10429,10431],{"class":10430},"sVbv2","update",[40,10433,10434],{"class":10363},"({\n",[40,10436,10438,10441,10444,10448,10451],{"class":10276,"line":10437},21,[40,10439,10440],{"class":10287},"    \"figure.dpi\"",[40,10442,10443],{"class":10363},": ",[40,10445,10447],{"class":10446},"sVC51","200",[40,10449,10450],{"class":10363},",       ",[40,10452,10453],{"class":10280},"# on-screen DPI\n",[40,10455,10457,10460,10462,10464,10467],{"class":10276,"line":10456},22,[40,10458,10459],{"class":10287},"    \"savefig.dpi\"",[40,10461,10443],{"class":10363},[40,10463,10447],{"class":10446},[40,10465,10466],{"class":10363},",      ",[40,10468,10469],{"class":10280},"# saved file DPI\n",[40,10471,10473,10476,10479,10482,10485,10488,10491],{"class":10276,"line":10472},23,[40,10474,10475],{"class":10287},"    \"figure.figsize\"",[40,10477,10478],{"class":10363},": (",[40,10480,10481],{"class":10446},"6.4",[40,10483,10484],{"class":10363},", ",[40,10486,10487],{"class":10446},"3.6",[40,10489,10490],{"class":10363},"),  ",[40,10492,10493],{"class":10280},"# 1280×720 at DPI=200\n",[40,10495,10497,10500,10502,10505],{"class":10276,"line":10496},24,[40,10498,10499],{"class":10287},"    \"axes.grid\"",[40,10501,10443],{"class":10363},[40,10503,10504],{"class":10446},"False",[40,10506,10507],{"class":10363},",\n",[40,10509,10511],{"class":10276,"line":10510},25,[40,10512,10513],{"class":10363},"})\n",[10082,10515,10517],{"className":10269,"code":10516,"language":10271,"meta":10090,"style":10090},"# @title Cell 2 — Math & utilities (PEP 8 \u002F PEP 257)\nfrom dataclasses import dataclass\nfrom pathlib import Path\nfrom typing import Tuple, Iterable\nimport textwrap\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib.animation import FuncAnimation, PillowWriter\nfrom IPython.display import Image, display\n\n\n@dataclass\nclass Knobs:\n    \"\"\"User controls for Wigner animations.\n\n    Attributes:\n        gate_axes: Pauli axes to animate, each in {\"x\", \"y\", \"z\"}.\n        initial_state: Starting Bloch eigenstate, or \"custom\".\n        custom_r: Custom Bloch vector if initial_state == \"custom\".\n        frames: Number of frames (≥ 2).\n        n_theta: Polar samples (θ ∈ [0, π]).\n        n_phi: Azimuth samples (φ ∈ [0, 2π]).\n        fps: GIF framerate.\n        out_dir: Output directory (e.g., \"\u002Fcontent\" in Colab).\n        figsize: Figure size in inches (DPI set globally).\n        info_loc: In-axes anchor (\"ul\", \"ur\", \"ll\", \"lr\").\n        info_fontsize: Font size for in-axes info text.\n        wrap_chars: Manual wrap width for info text.\n    \"\"\"\n    gate_axes: Tuple[str, ...] = (\"x\", \"y\", \"z\")\n    initial_state: str = \"+z\"\n    custom_r: Tuple[float, float, float] = (0.0, 0.0, 1.0)\n    frames: int = 24\n    n_theta: int = 80\n    n_phi: int = 160\n    fps: int = 20\n    out_dir: str = \"\u002Fcontent\"\n    figsize: Tuple[float, float] = (6.4, 3.6)\n    info_loc: str = \"ul\"\n    info_fontsize: int = 9\n    wrap_chars: int = 42\n\n\n# ------------------------------ Bloch helpers ---------------------------------\ndef bloch_vector(initial_state: str,\n                 custom_r: Tuple[float, float, float]) -> np.ndarray:\n    \"\"\"Return a unit Bloch vector for a named eigenstate or custom input.\"\"\"\n    mapping = {\n        \"+z\": (0.0, 0.0, 1.0), \"-z\": (0.0, 0.0, -1.0),\n        \"+x\": (1.0, 0.0, 0.0), \"-x\": (-1.0, 0.0, 0.0),\n        \"+y\": (0.0, 1.0, 0.0), \"-y\": (0.0, -1.0, 0.0),\n    }\n    r = np.asarray(mapping.get(initial_state.lower(), custom_r), float)\n    nrm = np.linalg.norm(r)\n    return np.array([0.0, 0.0, 1.0]) if nrm == 0 else r \u002F nrm\n\n\n# ---------------------------- Axis–angle rotations ----------------------------\ndef rotation_matrix(axis: str, angle: float) -> np.ndarray:\n    \"\"\"Rodrigues rotation matrix for axis in {'x','y','z'} and angle (radians).\"\"\"\n    c, s = np.cos(angle), np.sin(angle)\n    if axis == \"x\":\n        return np.array([[1, 0, 0], [0, c, -s], [0, s, c]], float)\n    if axis == \"y\":\n        return np.array([[c, 0, s], [0, 1, 0], [-s, 0, c]], float)\n    if axis == \"z\":\n        return np.array([[c, -s, 0], [s, c, 0], [0, 0, 1]], float)\n    raise ValueError(\"axis must be one of {'x','y','z'}\")\n\n\ndef rotation_matrix_axis(axis_vec: Tuple[float, float, float],\n                         angle: float) -> np.ndarray:\n    \"\"\"Rodrigues matrix for arbitrary axis vector and angle (radians).\"\"\"\n    v = np.asarray(axis_vec, float)\n    nrm = np.linalg.norm(v)\n    if nrm == 0:\n        raise ValueError(\"axis_vec must be nonzero\")\n    k = v \u002F nrm\n    K = np.array([[0.0, -k[2], k[1]], [k[2], 0.0, -k[0]], [-k[1], k[0], 0.0]], float)\n    I = np.eye(3)\n    c, s = np.cos(angle), np.sin(angle)\n    return c * I + (1 - c) * np.outer(k, k) + s * K\n\n\n# ------------------------------ Wigner on S^2 ---------------------------------\ndef n_grid(n_theta: int, n_phi: int) -> np.ndarray:\n    \"\"\"Return unit-vector grid n(θ,φ) stacked as (3, n_theta, n_phi).\"\"\"\n    theta = np.linspace(0.0, np.pi, n_theta)\n    phi = np.linspace(0.0, 2.0 * np.pi, n_phi)\n    th, ph = np.meshgrid(theta, phi, indexing=\"ij\")\n    nx = np.sin(th) * np.cos(ph)\n    ny = np.sin(th) * np.sin(ph)\n    nz = np.cos(th)\n    return np.stack((nx, ny, nz), axis=0)\n\n\ndef spin_wigner_qubit(r_vec: np.ndarray, ngrid: np.ndarray) -> np.ndarray:\n    \"\"\"Spin SW Wigner for a qubit: W = 1\u002F2 + (√3\u002F2) * r·n on the (θ,φ) grid.\"\"\"\n    return 0.5 + (np.sqrt(3.0) \u002F 2.0) * (r_vec.reshape(3, 1, 1) * ngrid).sum(axis=0)\n\n\n# ----------------------- In-axes title & info placement -----------------------\ndef _info_xy(loc: str) -> Tuple[float, float, dict]:\n    \"\"\"Map location code to (x, y, kwargs) in axes-fraction coordinates.\"\"\"\n    loc = loc.lower()\n    if loc == \"ul\":\n        return 0.015, 0.985, dict(ha=\"left\", va=\"top\")\n    if loc == \"ur\":\n        return 0.985, 0.985, dict(ha=\"right\", va=\"top\")\n    if loc == \"ll\":\n        return 0.015, 0.015, dict(ha=\"left\", va=\"bottom\")\n    if loc == \"lr\":\n        return 0.985, 0.015, dict(ha=\"right\", va=\"bottom\")\n    return 0.015, 0.985, dict(ha=\"left\", va=\"top\")\n\n\ndef add_info_box(ax: plt.Axes,\n                 text: str,\n                 loc: str = \"ul\",\n                 fontsize: int = 9,\n                 wrap: int = 42) -> plt.Text:\n    \"\"\"Create a wrapped, padded text box inside the axes (never clips).\"\"\"\n    x, y, kw = _info_xy(loc)\n    return ax.text(\n        x, y, textwrap.fill(text, wrap),\n        transform=ax.transAxes, fontsize=fontsize, zorder=5,\n        bbox=dict(boxstyle=\"round,pad=0.3\", fc=\"white\", alpha=0.75, ec=\"none\"),\n        **kw,\n    )\n\n\ndef add_title_inside(ax: plt.Axes, text: str) -> plt.Text:\n    \"\"\"Draw a title inside the Axes at the very top center (never clips).\"\"\"\n    return ax.text(\n        0.5, 0.985, text,\n        transform=ax.transAxes, ha=\"center\", va=\"top\",\n        fontsize=11,\n        bbox=dict(boxstyle=\"round,pad=0.3\", fc=\"white\", alpha=0.75, ec=\"none\"),\n        zorder=6,\n    )\n",[10088,10518,10519,10524,10536,10548,10560,10567,10571,10581,10591,10603,10615,10619,10623,10628,10640,10645,10649,10654,10659,10664,10669,10674,10679,10684,10689,10694,10700,10706,10712,10718,10724,10762,10776,10813,10827,10840,10853,10866,10879,10905,10918,10931,10944,10949,10954,10960,10981,11003,11009,11020,11062,11101,11140,11146,11179,11196,11247,11252,11257,11263,11292,11298,11318,11334,11379,11393,11436,11450,11491,11505,11510,11515,11543,11555,11561,11580,11594,11607,11620,11635,11706,11725,11742,11788,11793,11798,11804,11832,11838,11858,11885,11912,11936,11958,11973,11994,11999,12004,12026,12032,12103,12108,12113,12119,12153,12159,12175,12190,12229,12243,12278,12292,12327,12341,12374,12407,12412,12417,12433,12445,12461,12478,12496,12502,12515,12528,12540,12569,12620,12626,12632,12637,12642,12665,12671,12682,12695,12721,12734,12775,12788],{"__ignoreMap":10090},[40,10520,10521],{"class":10276,"line":10277},[40,10522,10523],{"class":10280},"# @title Cell 2 — Math & utilities (PEP 8 \u002F PEP 257)\n",[40,10525,10526,10528,10531,10533],{"class":10276,"line":10284},[40,10527,10353],{"class":10352},[40,10529,10530],{"class":10363}," dataclasses ",[40,10532,10375],{"class":10352},[40,10534,10535],{"class":10363}," dataclass\n",[40,10537,10538,10540,10543,10545],{"class":10276,"line":10291},[40,10539,10353],{"class":10352},[40,10541,10542],{"class":10363}," pathlib ",[40,10544,10375],{"class":10352},[40,10546,10547],{"class":10363}," Path\n",[40,10549,10550,10552,10555,10557],{"class":10276,"line":10297},[40,10551,10353],{"class":10352},[40,10553,10554],{"class":10363}," typing ",[40,10556,10375],{"class":10352},[40,10558,10559],{"class":10363}," Tuple, Iterable\n",[40,10561,10562,10564],{"class":10276,"line":10304},[40,10563,10375],{"class":10352},[40,10565,10566],{"class":10363}," textwrap\n",[40,10568,10569],{"class":10276,"line":10310},[40,10570,10301],{"emptyLinePlaceholder":10300},[40,10572,10573,10575,10577,10579],{"class":10276,"line":10316},[40,10574,10375],{"class":10352},[40,10576,10378],{"class":10363},[40,10578,10381],{"class":10352},[40,10580,10384],{"class":10363},[40,10582,10583,10585,10587,10589],{"class":10276,"line":10322},[40,10584,10375],{"class":10352},[40,10586,10405],{"class":10363},[40,10588,10381],{"class":10352},[40,10590,10410],{"class":10363},[40,10592,10593,10595,10598,10600],{"class":10276,"line":10327},[40,10594,10353],{"class":10352},[40,10596,10597],{"class":10363}," matplotlib.animation ",[40,10599,10375],{"class":10352},[40,10601,10602],{"class":10363}," FuncAnimation, PillowWriter\n",[40,10604,10605,10607,10610,10612],{"class":10276,"line":10333},[40,10606,10353],{"class":10352},[40,10608,10609],{"class":10363}," IPython.display ",[40,10611,10375],{"class":10352},[40,10613,10614],{"class":10363}," Image, display\n",[40,10616,10617],{"class":10276,"line":10339},[40,10618,10301],{"emptyLinePlaceholder":10300},[40,10620,10621],{"class":10276,"line":10344},[40,10622,10301],{"emptyLinePlaceholder":10300},[40,10624,10625],{"class":10276,"line":10349},[40,10626,10627],{"class":10430},"@dataclass\n",[40,10629,10630,10633,10637],{"class":10276,"line":10367},[40,10631,10632],{"class":10352},"class",[40,10634,10636],{"class":10635},"sU0A5"," Knobs",[40,10638,10639],{"class":10363},":\n",[40,10641,10642],{"class":10276,"line":10372},[40,10643,10644],{"class":10287},"    \"\"\"User controls for Wigner animations.\n",[40,10646,10647],{"class":10276,"line":10387},[40,10648,10301],{"emptyLinePlaceholder":10300},[40,10650,10651],{"class":10276,"line":10400},[40,10652,10653],{"class":10287},"    Attributes:\n",[40,10655,10656],{"class":10276,"line":10413},[40,10657,10658],{"class":10287},"        gate_axes: Pauli axes to animate, each in {\"x\", \"y\", \"z\"}.\n",[40,10660,10661],{"class":10276,"line":10418},[40,10662,10663],{"class":10287},"        initial_state: Starting Bloch eigenstate, or \"custom\".\n",[40,10665,10666],{"class":10276,"line":10424},[40,10667,10668],{"class":10287},"        custom_r: Custom Bloch vector if initial_state == \"custom\".\n",[40,10670,10671],{"class":10276,"line":10437},[40,10672,10673],{"class":10287},"        frames: Number of frames (≥ 2).\n",[40,10675,10676],{"class":10276,"line":10456},[40,10677,10678],{"class":10287},"        n_theta: Polar samples (θ ∈ [0, π]).\n",[40,10680,10681],{"class":10276,"line":10472},[40,10682,10683],{"class":10287},"        n_phi: Azimuth samples (φ ∈ [0, 2π]).\n",[40,10685,10686],{"class":10276,"line":10496},[40,10687,10688],{"class":10287},"        fps: GIF framerate.\n",[40,10690,10691],{"class":10276,"line":10510},[40,10692,10693],{"class":10287},"        out_dir: Output directory (e.g., \"\u002Fcontent\" in Colab).\n",[40,10695,10697],{"class":10276,"line":10696},26,[40,10698,10699],{"class":10287},"        figsize: Figure size in inches (DPI set globally).\n",[40,10701,10703],{"class":10276,"line":10702},27,[40,10704,10705],{"class":10287},"        info_loc: In-axes anchor (\"ul\", \"ur\", \"ll\", \"lr\").\n",[40,10707,10709],{"class":10276,"line":10708},28,[40,10710,10711],{"class":10287},"        info_fontsize: Font size for in-axes info text.\n",[40,10713,10715],{"class":10276,"line":10714},29,[40,10716,10717],{"class":10287},"        wrap_chars: Manual wrap width for info text.\n",[40,10719,10721],{"class":10276,"line":10720},30,[40,10722,10723],{"class":10287},"    \"\"\"\n",[40,10725,10727,10730,10734,10736,10739,10742,10744,10746,10749,10751,10754,10756,10759],{"class":10276,"line":10726},31,[40,10728,10729],{"class":10363},"    gate_axes: Tuple[",[40,10731,10733],{"class":10732},"sjrmR","str",[40,10735,10484],{"class":10363},[40,10737,10738],{"class":10446},"...",[40,10740,10741],{"class":10363},"] ",[40,10743,162],{"class":10732},[40,10745,920],{"class":10363},[40,10747,10748],{"class":10287},"\"x\"",[40,10750,10484],{"class":10363},[40,10752,10753],{"class":10287},"\"y\"",[40,10755,10484],{"class":10363},[40,10757,10758],{"class":10287},"\"z\"",[40,10760,10761],{"class":10363},")\n",[40,10763,10765,10768,10770,10773],{"class":10276,"line":10764},32,[40,10766,10767],{"class":10363},"    initial_state: ",[40,10769,10733],{"class":10732},[40,10771,10772],{"class":10732}," =",[40,10774,10775],{"class":10287}," \"+z\"\n",[40,10777,10779,10782,10785,10787,10789,10791,10793,10795,10797,10799,10802,10804,10806,10808,10811],{"class":10276,"line":10778},33,[40,10780,10781],{"class":10363},"    custom_r: Tuple[",[40,10783,10784],{"class":10732},"float",[40,10786,10484],{"class":10363},[40,10788,10784],{"class":10732},[40,10790,10484],{"class":10363},[40,10792,10784],{"class":10732},[40,10794,10741],{"class":10363},[40,10796,162],{"class":10732},[40,10798,920],{"class":10363},[40,10800,10801],{"class":10446},"0.0",[40,10803,10484],{"class":10363},[40,10805,10801],{"class":10446},[40,10807,10484],{"class":10363},[40,10809,10810],{"class":10446},"1.0",[40,10812,10761],{"class":10363},[40,10814,10816,10819,10822,10824],{"class":10276,"line":10815},34,[40,10817,10818],{"class":10363},"    frames: ",[40,10820,10821],{"class":10732},"int",[40,10823,10772],{"class":10732},[40,10825,10826],{"class":10446}," 24\n",[40,10828,10830,10833,10835,10837],{"class":10276,"line":10829},35,[40,10831,10832],{"class":10363},"    n_theta: ",[40,10834,10821],{"class":10732},[40,10836,10772],{"class":10732},[40,10838,10839],{"class":10446}," 80\n",[40,10841,10843,10846,10848,10850],{"class":10276,"line":10842},36,[40,10844,10845],{"class":10363},"    n_phi: ",[40,10847,10821],{"class":10732},[40,10849,10772],{"class":10732},[40,10851,10852],{"class":10446}," 160\n",[40,10854,10856,10859,10861,10863],{"class":10276,"line":10855},37,[40,10857,10858],{"class":10363},"    fps: ",[40,10860,10821],{"class":10732},[40,10862,10772],{"class":10732},[40,10864,10865],{"class":10446}," 20\n",[40,10867,10869,10872,10874,10876],{"class":10276,"line":10868},38,[40,10870,10871],{"class":10363},"    out_dir: ",[40,10873,10733],{"class":10732},[40,10875,10772],{"class":10732},[40,10877,10878],{"class":10287}," \"\u002Fcontent\"\n",[40,10880,10882,10885,10887,10889,10891,10893,10895,10897,10899,10901,10903],{"class":10276,"line":10881},39,[40,10883,10884],{"class":10363},"    figsize: Tuple[",[40,10886,10784],{"class":10732},[40,10888,10484],{"class":10363},[40,10890,10784],{"class":10732},[40,10892,10741],{"class":10363},[40,10894,162],{"class":10732},[40,10896,920],{"class":10363},[40,10898,10481],{"class":10446},[40,10900,10484],{"class":10363},[40,10902,10487],{"class":10446},[40,10904,10761],{"class":10363},[40,10906,10908,10911,10913,10915],{"class":10276,"line":10907},40,[40,10909,10910],{"class":10363},"    info_loc: ",[40,10912,10733],{"class":10732},[40,10914,10772],{"class":10732},[40,10916,10917],{"class":10287}," \"ul\"\n",[40,10919,10921,10924,10926,10928],{"class":10276,"line":10920},41,[40,10922,10923],{"class":10363},"    info_fontsize: ",[40,10925,10821],{"class":10732},[40,10927,10772],{"class":10732},[40,10929,10930],{"class":10446}," 9\n",[40,10932,10934,10937,10939,10941],{"class":10276,"line":10933},42,[40,10935,10936],{"class":10363},"    wrap_chars: ",[40,10938,10821],{"class":10732},[40,10940,10772],{"class":10732},[40,10942,10943],{"class":10446}," 42\n",[40,10945,10947],{"class":10276,"line":10946},43,[40,10948,10301],{"emptyLinePlaceholder":10300},[40,10950,10952],{"class":10276,"line":10951},44,[40,10953,10301],{"emptyLinePlaceholder":10300},[40,10955,10957],{"class":10276,"line":10956},45,[40,10958,10959],{"class":10280},"# ------------------------------ Bloch helpers ---------------------------------\n",[40,10961,10963,10966,10969,10971,10975,10977,10979],{"class":10276,"line":10962},46,[40,10964,10965],{"class":10352},"def",[40,10967,10968],{"class":10430}," bloch_vector",[40,10970,167],{"class":10363},[40,10972,10974],{"class":10973},"sb9H8","initial_state",[40,10976,10443],{"class":10363},[40,10978,10733],{"class":10732},[40,10980,10507],{"class":10363},[40,10982,10984,10987,10990,10992,10994,10996,10998,11000],{"class":10276,"line":10983},47,[40,10985,10986],{"class":10973},"                 custom_r",[40,10988,10989],{"class":10363},": Tuple[",[40,10991,10784],{"class":10732},[40,10993,10484],{"class":10363},[40,10995,10784],{"class":10732},[40,10997,10484],{"class":10363},[40,10999,10784],{"class":10732},[40,11001,11002],{"class":10363},"]) -> np.ndarray:\n",[40,11004,11006],{"class":10276,"line":11005},48,[40,11007,11008],{"class":10287},"    \"\"\"Return a unit Bloch vector for a named eigenstate or custom input.\"\"\"\n",[40,11010,11012,11015,11017],{"class":10276,"line":11011},49,[40,11013,11014],{"class":10363},"    mapping ",[40,11016,162],{"class":10732},[40,11018,11019],{"class":10363}," {\n",[40,11021,11023,11026,11028,11030,11032,11034,11036,11038,11041,11044,11046,11048,11050,11052,11054,11057,11059],{"class":10276,"line":11022},50,[40,11024,11025],{"class":10287},"        \"+z\"",[40,11027,10478],{"class":10363},[40,11029,10801],{"class":10446},[40,11031,10484],{"class":10363},[40,11033,10801],{"class":10446},[40,11035,10484],{"class":10363},[40,11037,10810],{"class":10446},[40,11039,11040],{"class":10363},"), ",[40,11042,11043],{"class":10287},"\"-z\"",[40,11045,10478],{"class":10363},[40,11047,10801],{"class":10446},[40,11049,10484],{"class":10363},[40,11051,10801],{"class":10446},[40,11053,10484],{"class":10363},[40,11055,11056],{"class":10732},"-",[40,11058,10810],{"class":10446},[40,11060,11061],{"class":10363},"),\n",[40,11063,11065,11068,11070,11072,11074,11076,11078,11080,11082,11085,11087,11089,11091,11093,11095,11097,11099],{"class":10276,"line":11064},51,[40,11066,11067],{"class":10287},"        \"+x\"",[40,11069,10478],{"class":10363},[40,11071,10810],{"class":10446},[40,11073,10484],{"class":10363},[40,11075,10801],{"class":10446},[40,11077,10484],{"class":10363},[40,11079,10801],{"class":10446},[40,11081,11040],{"class":10363},[40,11083,11084],{"class":10287},"\"-x\"",[40,11086,10478],{"class":10363},[40,11088,11056],{"class":10732},[40,11090,10810],{"class":10446},[40,11092,10484],{"class":10363},[40,11094,10801],{"class":10446},[40,11096,10484],{"class":10363},[40,11098,10801],{"class":10446},[40,11100,11061],{"class":10363},[40,11102,11104,11107,11109,11111,11113,11115,11117,11119,11121,11124,11126,11128,11130,11132,11134,11136,11138],{"class":10276,"line":11103},52,[40,11105,11106],{"class":10287},"        \"+y\"",[40,11108,10478],{"class":10363},[40,11110,10801],{"class":10446},[40,11112,10484],{"class":10363},[40,11114,10810],{"class":10446},[40,11116,10484],{"class":10363},[40,11118,10801],{"class":10446},[40,11120,11040],{"class":10363},[40,11122,11123],{"class":10287},"\"-y\"",[40,11125,10478],{"class":10363},[40,11127,10801],{"class":10446},[40,11129,10484],{"class":10363},[40,11131,11056],{"class":10732},[40,11133,10810],{"class":10446},[40,11135,10484],{"class":10363},[40,11137,10801],{"class":10446},[40,11139,11061],{"class":10363},[40,11141,11143],{"class":10276,"line":11142},53,[40,11144,11145],{"class":10363},"    }\n",[40,11147,11149,11152,11154,11157,11160,11163,11166,11169,11172,11175,11177],{"class":10276,"line":11148},54,[40,11150,11151],{"class":10363},"    r ",[40,11153,162],{"class":10732},[40,11155,11156],{"class":10363}," np.",[40,11158,11159],{"class":10430},"asarray",[40,11161,11162],{"class":10363},"(mapping.",[40,11164,11165],{"class":10430},"get",[40,11167,11168],{"class":10363},"(initial_state.",[40,11170,11171],{"class":10430},"lower",[40,11173,11174],{"class":10363},"(), custom_r), ",[40,11176,10784],{"class":10732},[40,11178,10761],{"class":10363},[40,11180,11182,11185,11187,11190,11193],{"class":10276,"line":11181},55,[40,11183,11184],{"class":10363},"    nrm ",[40,11186,162],{"class":10732},[40,11188,11189],{"class":10363}," np.linalg.",[40,11191,11192],{"class":10430},"norm",[40,11194,11195],{"class":10363},"(r)\n",[40,11197,11199,11202,11204,11207,11210,11212,11214,11216,11218,11220,11223,11226,11229,11232,11235,11238,11241,11244],{"class":10276,"line":11198},56,[40,11200,11201],{"class":10352},"    return",[40,11203,11156],{"class":10363},[40,11205,11206],{"class":10430},"array",[40,11208,11209],{"class":10363},"([",[40,11211,10801],{"class":10446},[40,11213,10484],{"class":10363},[40,11215,10801],{"class":10446},[40,11217,10484],{"class":10363},[40,11219,10810],{"class":10446},[40,11221,11222],{"class":10363},"]) ",[40,11224,11225],{"class":10352},"if",[40,11227,11228],{"class":10363}," nrm ",[40,11230,11231],{"class":10732},"==",[40,11233,11234],{"class":10446}," 0",[40,11236,11237],{"class":10352}," else",[40,11239,11240],{"class":10363}," r ",[40,11242,11243],{"class":10732},"\u002F",[40,11245,11246],{"class":10363}," nrm\n",[40,11248,11250],{"class":10276,"line":11249},57,[40,11251,10301],{"emptyLinePlaceholder":10300},[40,11253,11255],{"class":10276,"line":11254},58,[40,11256,10301],{"emptyLinePlaceholder":10300},[40,11258,11260],{"class":10276,"line":11259},59,[40,11261,11262],{"class":10280},"# ---------------------------- Axis–angle rotations ----------------------------\n",[40,11264,11266,11268,11271,11273,11276,11278,11280,11282,11285,11287,11289],{"class":10276,"line":11265},60,[40,11267,10965],{"class":10352},[40,11269,11270],{"class":10430}," rotation_matrix",[40,11272,167],{"class":10363},[40,11274,11275],{"class":10973},"axis",[40,11277,10443],{"class":10363},[40,11279,10733],{"class":10732},[40,11281,10484],{"class":10363},[40,11283,11284],{"class":10973},"angle",[40,11286,10443],{"class":10363},[40,11288,10784],{"class":10732},[40,11290,11291],{"class":10363},") -> np.ndarray:\n",[40,11293,11295],{"class":10276,"line":11294},61,[40,11296,11297],{"class":10287},"    \"\"\"Rodrigues rotation matrix for axis in {'x','y','z'} and angle (radians).\"\"\"\n",[40,11299,11301,11304,11306,11308,11310,11313,11315],{"class":10276,"line":11300},62,[40,11302,11303],{"class":10363},"    c, s ",[40,11305,162],{"class":10732},[40,11307,11156],{"class":10363},[40,11309,1820],{"class":10430},[40,11311,11312],{"class":10363},"(angle), np.",[40,11314,1813],{"class":10430},[40,11316,11317],{"class":10363},"(angle)\n",[40,11319,11321,11324,11327,11329,11332],{"class":10276,"line":11320},63,[40,11322,11323],{"class":10352},"    if",[40,11325,11326],{"class":10363}," axis ",[40,11328,11231],{"class":10732},[40,11330,11331],{"class":10287}," \"x\"",[40,11333,10639],{"class":10363},[40,11335,11337,11340,11342,11344,11347,11349,11351,11353,11355,11357,11360,11362,11365,11367,11370,11372,11375,11377],{"class":10276,"line":11336},64,[40,11338,11339],{"class":10352},"        return",[40,11341,11156],{"class":10363},[40,11343,11206],{"class":10430},[40,11345,11346],{"class":10363},"([[",[40,11348,78],{"class":10446},[40,11350,10484],{"class":10363},[40,11352,66],{"class":10446},[40,11354,10484],{"class":10363},[40,11356,66],{"class":10446},[40,11358,11359],{"class":10363},"], [",[40,11361,66],{"class":10446},[40,11363,11364],{"class":10363},", c, ",[40,11366,11056],{"class":10732},[40,11368,11369],{"class":10363},"s], [",[40,11371,66],{"class":10446},[40,11373,11374],{"class":10363},", s, c]], ",[40,11376,10784],{"class":10732},[40,11378,10761],{"class":10363},[40,11380,11382,11384,11386,11388,11391],{"class":10276,"line":11381},65,[40,11383,11323],{"class":10352},[40,11385,11326],{"class":10363},[40,11387,11231],{"class":10732},[40,11389,11390],{"class":10287}," \"y\"",[40,11392,10639],{"class":10363},[40,11394,11396,11398,11400,11402,11405,11407,11410,11412,11414,11416,11418,11420,11422,11424,11427,11429,11432,11434],{"class":10276,"line":11395},66,[40,11397,11339],{"class":10352},[40,11399,11156],{"class":10363},[40,11401,11206],{"class":10430},[40,11403,11404],{"class":10363},"([[c, ",[40,11406,66],{"class":10446},[40,11408,11409],{"class":10363},", s], [",[40,11411,66],{"class":10446},[40,11413,10484],{"class":10363},[40,11415,78],{"class":10446},[40,11417,10484],{"class":10363},[40,11419,66],{"class":10446},[40,11421,11359],{"class":10363},[40,11423,11056],{"class":10732},[40,11425,11426],{"class":10363},"s, ",[40,11428,66],{"class":10446},[40,11430,11431],{"class":10363},", c]], ",[40,11433,10784],{"class":10732},[40,11435,10761],{"class":10363},[40,11437,11439,11441,11443,11445,11448],{"class":10276,"line":11438},67,[40,11440,11323],{"class":10352},[40,11442,11326],{"class":10363},[40,11444,11231],{"class":10732},[40,11446,11447],{"class":10287}," \"z\"",[40,11449,10639],{"class":10363},[40,11451,11453,11455,11457,11459,11461,11463,11465,11467,11470,11472,11474,11476,11478,11480,11482,11484,11487,11489],{"class":10276,"line":11452},68,[40,11454,11339],{"class":10352},[40,11456,11156],{"class":10363},[40,11458,11206],{"class":10430},[40,11460,11404],{"class":10363},[40,11462,11056],{"class":10732},[40,11464,11426],{"class":10363},[40,11466,66],{"class":10446},[40,11468,11469],{"class":10363},"], [s, c, ",[40,11471,66],{"class":10446},[40,11473,11359],{"class":10363},[40,11475,66],{"class":10446},[40,11477,10484],{"class":10363},[40,11479,66],{"class":10446},[40,11481,10484],{"class":10363},[40,11483,78],{"class":10446},[40,11485,11486],{"class":10363},"]], ",[40,11488,10784],{"class":10732},[40,11490,10761],{"class":10363},[40,11492,11494,11497,11500,11503],{"class":10276,"line":11493},69,[40,11495,11496],{"class":10352},"    raise",[40,11498,11499],{"class":10363}," ValueError(",[40,11501,11502],{"class":10287},"\"axis must be one of {'x','y','z'}\"",[40,11504,10761],{"class":10363},[40,11506,11508],{"class":10276,"line":11507},70,[40,11509,10301],{"emptyLinePlaceholder":10300},[40,11511,11513],{"class":10276,"line":11512},71,[40,11514,10301],{"emptyLinePlaceholder":10300},[40,11516,11518,11520,11523,11525,11528,11530,11532,11534,11536,11538,11540],{"class":10276,"line":11517},72,[40,11519,10965],{"class":10352},[40,11521,11522],{"class":10430}," rotation_matrix_axis",[40,11524,167],{"class":10363},[40,11526,11527],{"class":10973},"axis_vec",[40,11529,10989],{"class":10363},[40,11531,10784],{"class":10732},[40,11533,10484],{"class":10363},[40,11535,10784],{"class":10732},[40,11537,10484],{"class":10363},[40,11539,10784],{"class":10732},[40,11541,11542],{"class":10363},"],\n",[40,11544,11546,11549,11551,11553],{"class":10276,"line":11545},73,[40,11547,11548],{"class":10973},"                         angle",[40,11550,10443],{"class":10363},[40,11552,10784],{"class":10732},[40,11554,11291],{"class":10363},[40,11556,11558],{"class":10276,"line":11557},74,[40,11559,11560],{"class":10287},"    \"\"\"Rodrigues matrix for arbitrary axis vector and angle (radians).\"\"\"\n",[40,11562,11564,11567,11569,11571,11573,11576,11578],{"class":10276,"line":11563},75,[40,11565,11566],{"class":10363},"    v ",[40,11568,162],{"class":10732},[40,11570,11156],{"class":10363},[40,11572,11159],{"class":10430},[40,11574,11575],{"class":10363},"(axis_vec, ",[40,11577,10784],{"class":10732},[40,11579,10761],{"class":10363},[40,11581,11583,11585,11587,11589,11591],{"class":10276,"line":11582},76,[40,11584,11184],{"class":10363},[40,11586,162],{"class":10732},[40,11588,11189],{"class":10363},[40,11590,11192],{"class":10430},[40,11592,11593],{"class":10363},"(v)\n",[40,11595,11597,11599,11601,11603,11605],{"class":10276,"line":11596},77,[40,11598,11323],{"class":10352},[40,11600,11228],{"class":10363},[40,11602,11231],{"class":10732},[40,11604,11234],{"class":10446},[40,11606,10639],{"class":10363},[40,11608,11610,11613,11615,11618],{"class":10276,"line":11609},78,[40,11611,11612],{"class":10352},"        raise",[40,11614,11499],{"class":10363},[40,11616,11617],{"class":10287},"\"axis_vec must be nonzero\"",[40,11619,10761],{"class":10363},[40,11621,11623,11626,11628,11631,11633],{"class":10276,"line":11622},79,[40,11624,11625],{"class":10363},"    k ",[40,11627,162],{"class":10732},[40,11629,11630],{"class":10363}," v ",[40,11632,11243],{"class":10732},[40,11634,11246],{"class":10363},[40,11636,11638,11641,11643,11645,11647,11649,11651,11653,11655,11658,11660,11663,11665,11668,11670,11673,11675,11677,11679,11681,11683,11686,11688,11690,11692,11694,11696,11698,11700,11702,11704],{"class":10276,"line":11637},80,[40,11639,11640],{"class":10363},"    K ",[40,11642,162],{"class":10732},[40,11644,11156],{"class":10363},[40,11646,11206],{"class":10430},[40,11648,11346],{"class":10363},[40,11650,10801],{"class":10446},[40,11652,10484],{"class":10363},[40,11654,11056],{"class":10732},[40,11656,11657],{"class":10363},"k[",[40,11659,1044],{"class":10446},[40,11661,11662],{"class":10363},"], k[",[40,11664,78],{"class":10446},[40,11666,11667],{"class":10363},"]], [k[",[40,11669,1044],{"class":10446},[40,11671,11672],{"class":10363},"], ",[40,11674,10801],{"class":10446},[40,11676,10484],{"class":10363},[40,11678,11056],{"class":10732},[40,11680,11657],{"class":10363},[40,11682,66],{"class":10446},[40,11684,11685],{"class":10363},"]], [",[40,11687,11056],{"class":10732},[40,11689,11657],{"class":10363},[40,11691,78],{"class":10446},[40,11693,11662],{"class":10363},[40,11695,66],{"class":10446},[40,11697,11672],{"class":10363},[40,11699,10801],{"class":10446},[40,11701,11486],{"class":10363},[40,11703,10784],{"class":10732},[40,11705,10761],{"class":10363},[40,11707,11709,11712,11714,11716,11719,11721,11723],{"class":10276,"line":11708},81,[40,11710,11711],{"class":10363},"    I ",[40,11713,162],{"class":10732},[40,11715,11156],{"class":10363},[40,11717,11718],{"class":10430},"eye",[40,11720,167],{"class":10363},[40,11722,949],{"class":10446},[40,11724,10761],{"class":10363},[40,11726,11728,11730,11732,11734,11736,11738,11740],{"class":10276,"line":11727},82,[40,11729,11303],{"class":10363},[40,11731,162],{"class":10732},[40,11733,11156],{"class":10363},[40,11735,1820],{"class":10430},[40,11737,11312],{"class":10363},[40,11739,1813],{"class":10430},[40,11741,11317],{"class":10363},[40,11743,11745,11747,11750,11753,11756,11758,11760,11762,11765,11768,11770,11772,11775,11778,11780,11783,11785],{"class":10276,"line":11744},83,[40,11746,11201],{"class":10352},[40,11748,11749],{"class":10363}," c ",[40,11751,11752],{"class":10732},"*",[40,11754,11755],{"class":10363}," I ",[40,11757,1053],{"class":10732},[40,11759,920],{"class":10363},[40,11761,78],{"class":10446},[40,11763,11764],{"class":10732}," -",[40,11766,11767],{"class":10363}," c) ",[40,11769,11752],{"class":10732},[40,11771,11156],{"class":10363},[40,11773,11774],{"class":10430},"outer",[40,11776,11777],{"class":10363},"(k, k) ",[40,11779,1053],{"class":10732},[40,11781,11782],{"class":10363}," s ",[40,11784,11752],{"class":10732},[40,11786,11787],{"class":10363}," K\n",[40,11789,11791],{"class":10276,"line":11790},84,[40,11792,10301],{"emptyLinePlaceholder":10300},[40,11794,11796],{"class":10276,"line":11795},85,[40,11797,10301],{"emptyLinePlaceholder":10300},[40,11799,11801],{"class":10276,"line":11800},86,[40,11802,11803],{"class":10280},"# ------------------------------ Wigner on S^2 ---------------------------------\n",[40,11805,11807,11809,11812,11814,11817,11819,11821,11823,11826,11828,11830],{"class":10276,"line":11806},87,[40,11808,10965],{"class":10352},[40,11810,11811],{"class":10430}," n_grid",[40,11813,167],{"class":10363},[40,11815,11816],{"class":10973},"n_theta",[40,11818,10443],{"class":10363},[40,11820,10821],{"class":10732},[40,11822,10484],{"class":10363},[40,11824,11825],{"class":10973},"n_phi",[40,11827,10443],{"class":10363},[40,11829,10821],{"class":10732},[40,11831,11291],{"class":10363},[40,11833,11835],{"class":10276,"line":11834},88,[40,11836,11837],{"class":10287},"    \"\"\"Return unit-vector grid n(θ,φ) stacked as (3, n_theta, n_phi).\"\"\"\n",[40,11839,11841,11844,11846,11848,11851,11853,11855],{"class":10276,"line":11840},89,[40,11842,11843],{"class":10363},"    theta ",[40,11845,162],{"class":10732},[40,11847,11156],{"class":10363},[40,11849,11850],{"class":10430},"linspace",[40,11852,167],{"class":10363},[40,11854,10801],{"class":10446},[40,11856,11857],{"class":10363},", np.pi, n_theta)\n",[40,11859,11861,11864,11866,11868,11870,11872,11874,11876,11879,11882],{"class":10276,"line":11860},90,[40,11862,11863],{"class":10363},"    phi ",[40,11865,162],{"class":10732},[40,11867,11156],{"class":10363},[40,11869,11850],{"class":10430},[40,11871,167],{"class":10363},[40,11873,10801],{"class":10446},[40,11875,10484],{"class":10363},[40,11877,11878],{"class":10446},"2.0",[40,11880,11881],{"class":10732}," *",[40,11883,11884],{"class":10363}," np.pi, n_phi)\n",[40,11886,11888,11891,11893,11895,11898,11901,11905,11907,11910],{"class":10276,"line":11887},91,[40,11889,11890],{"class":10363},"    th, ph ",[40,11892,162],{"class":10732},[40,11894,11156],{"class":10363},[40,11896,11897],{"class":10430},"meshgrid",[40,11899,11900],{"class":10363},"(theta, phi, ",[40,11902,11904],{"class":11903},"s_ZVi","indexing",[40,11906,162],{"class":10732},[40,11908,11909],{"class":10287},"\"ij\"",[40,11911,10761],{"class":10363},[40,11913,11915,11918,11920,11922,11924,11927,11929,11931,11933],{"class":10276,"line":11914},92,[40,11916,11917],{"class":10363},"    nx ",[40,11919,162],{"class":10732},[40,11921,11156],{"class":10363},[40,11923,1813],{"class":10430},[40,11925,11926],{"class":10363},"(th) ",[40,11928,11752],{"class":10732},[40,11930,11156],{"class":10363},[40,11932,1820],{"class":10430},[40,11934,11935],{"class":10363},"(ph)\n",[40,11937,11939,11942,11944,11946,11948,11950,11952,11954,11956],{"class":10276,"line":11938},93,[40,11940,11941],{"class":10363},"    ny ",[40,11943,162],{"class":10732},[40,11945,11156],{"class":10363},[40,11947,1813],{"class":10430},[40,11949,11926],{"class":10363},[40,11951,11752],{"class":10732},[40,11953,11156],{"class":10363},[40,11955,1813],{"class":10430},[40,11957,11935],{"class":10363},[40,11959,11961,11964,11966,11968,11970],{"class":10276,"line":11960},94,[40,11962,11963],{"class":10363},"    nz ",[40,11965,162],{"class":10732},[40,11967,11156],{"class":10363},[40,11969,1820],{"class":10430},[40,11971,11972],{"class":10363},"(th)\n",[40,11974,11976,11978,11980,11983,11986,11988,11990,11992],{"class":10276,"line":11975},95,[40,11977,11201],{"class":10352},[40,11979,11156],{"class":10363},[40,11981,11982],{"class":10430},"stack",[40,11984,11985],{"class":10363},"((nx, ny, nz), ",[40,11987,11275],{"class":11903},[40,11989,162],{"class":10732},[40,11991,66],{"class":10446},[40,11993,10761],{"class":10363},[40,11995,11997],{"class":10276,"line":11996},96,[40,11998,10301],{"emptyLinePlaceholder":10300},[40,12000,12002],{"class":10276,"line":12001},97,[40,12003,10301],{"emptyLinePlaceholder":10300},[40,12005,12007,12009,12012,12014,12017,12020,12023],{"class":10276,"line":12006},98,[40,12008,10965],{"class":10352},[40,12010,12011],{"class":10430}," spin_wigner_qubit",[40,12013,167],{"class":10363},[40,12015,12016],{"class":10973},"r_vec",[40,12018,12019],{"class":10363},": np.ndarray, ",[40,12021,12022],{"class":10973},"ngrid",[40,12024,12025],{"class":10363},": np.ndarray) -> np.ndarray:\n",[40,12027,12029],{"class":10276,"line":12028},99,[40,12030,12031],{"class":10287},"    \"\"\"Spin SW Wigner for a qubit: W = 1\u002F2 + (√3\u002F2) * r·n on the (θ,φ) grid.\"\"\"\n",[40,12033,12035,12037,12040,12043,12046,12048,12050,12053,12056,12058,12061,12063,12065,12068,12071,12073,12075,12077,12079,12081,12083,12085,12087,12090,12093,12095,12097,12099,12101],{"class":10276,"line":12034},100,[40,12036,11201],{"class":10352},[40,12038,12039],{"class":10446}," 0.5",[40,12041,12042],{"class":10732}," +",[40,12044,12045],{"class":10363}," (np.",[40,12047,2218],{"class":10430},[40,12049,167],{"class":10363},[40,12051,12052],{"class":10446},"3.0",[40,12054,12055],{"class":10363},") ",[40,12057,11243],{"class":10732},[40,12059,12060],{"class":10446}," 2.0",[40,12062,12055],{"class":10363},[40,12064,11752],{"class":10732},[40,12066,12067],{"class":10363}," (r_vec.",[40,12069,12070],{"class":10430},"reshape",[40,12072,167],{"class":10363},[40,12074,949],{"class":10446},[40,12076,10484],{"class":10363},[40,12078,78],{"class":10446},[40,12080,10484],{"class":10363},[40,12082,78],{"class":10446},[40,12084,12055],{"class":10363},[40,12086,11752],{"class":10732},[40,12088,12089],{"class":10363}," ngrid).",[40,12091,12092],{"class":10430},"sum",[40,12094,167],{"class":10363},[40,12096,11275],{"class":11903},[40,12098,162],{"class":10732},[40,12100,66],{"class":10446},[40,12102,10761],{"class":10363},[40,12104,12106],{"class":10276,"line":12105},101,[40,12107,10301],{"emptyLinePlaceholder":10300},[40,12109,12111],{"class":10276,"line":12110},102,[40,12112,10301],{"emptyLinePlaceholder":10300},[40,12114,12116],{"class":10276,"line":12115},103,[40,12117,12118],{"class":10280},"# ----------------------- In-axes title & info placement -----------------------\n",[40,12120,12122,12124,12127,12129,12132,12134,12136,12139,12141,12143,12145,12147,12150],{"class":10276,"line":12121},104,[40,12123,10965],{"class":10352},[40,12125,12126],{"class":10430}," _info_xy",[40,12128,167],{"class":10363},[40,12130,12131],{"class":10973},"loc",[40,12133,10443],{"class":10363},[40,12135,10733],{"class":10732},[40,12137,12138],{"class":10363},") -> Tuple[",[40,12140,10784],{"class":10732},[40,12142,10484],{"class":10363},[40,12144,10784],{"class":10732},[40,12146,10484],{"class":10363},[40,12148,12149],{"class":10732},"dict",[40,12151,12152],{"class":10363},"]:\n",[40,12154,12156],{"class":10276,"line":12155},105,[40,12157,12158],{"class":10287},"    \"\"\"Map location code to (x, y, kwargs) in axes-fraction coordinates.\"\"\"\n",[40,12160,12162,12165,12167,12170,12172],{"class":10276,"line":12161},106,[40,12163,12164],{"class":10363},"    loc ",[40,12166,162],{"class":10732},[40,12168,12169],{"class":10363}," loc.",[40,12171,11171],{"class":10430},[40,12173,12174],{"class":10363},"()\n",[40,12176,12178,12180,12183,12185,12188],{"class":10276,"line":12177},107,[40,12179,11323],{"class":10352},[40,12181,12182],{"class":10363}," loc ",[40,12184,11231],{"class":10732},[40,12186,12187],{"class":10287}," \"ul\"",[40,12189,10639],{"class":10363},[40,12191,12193,12195,12198,12200,12203,12205,12207,12209,12212,12214,12217,12219,12222,12224,12227],{"class":10276,"line":12192},108,[40,12194,11339],{"class":10352},[40,12196,12197],{"class":10446}," 0.015",[40,12199,10484],{"class":10363},[40,12201,12202],{"class":10446},"0.985",[40,12204,10484],{"class":10363},[40,12206,12149],{"class":10732},[40,12208,167],{"class":10363},[40,12210,12211],{"class":11903},"ha",[40,12213,162],{"class":10732},[40,12215,12216],{"class":10287},"\"left\"",[40,12218,10484],{"class":10363},[40,12220,12221],{"class":11903},"va",[40,12223,162],{"class":10732},[40,12225,12226],{"class":10287},"\"top\"",[40,12228,10761],{"class":10363},[40,12230,12232,12234,12236,12238,12241],{"class":10276,"line":12231},109,[40,12233,11323],{"class":10352},[40,12235,12182],{"class":10363},[40,12237,11231],{"class":10732},[40,12239,12240],{"class":10287}," \"ur\"",[40,12242,10639],{"class":10363},[40,12244,12246,12248,12251,12253,12255,12257,12259,12261,12263,12265,12268,12270,12272,12274,12276],{"class":10276,"line":12245},110,[40,12247,11339],{"class":10352},[40,12249,12250],{"class":10446}," 0.985",[40,12252,10484],{"class":10363},[40,12254,12202],{"class":10446},[40,12256,10484],{"class":10363},[40,12258,12149],{"class":10732},[40,12260,167],{"class":10363},[40,12262,12211],{"class":11903},[40,12264,162],{"class":10732},[40,12266,12267],{"class":10287},"\"right\"",[40,12269,10484],{"class":10363},[40,12271,12221],{"class":11903},[40,12273,162],{"class":10732},[40,12275,12226],{"class":10287},[40,12277,10761],{"class":10363},[40,12279,12281,12283,12285,12287,12290],{"class":10276,"line":12280},111,[40,12282,11323],{"class":10352},[40,12284,12182],{"class":10363},[40,12286,11231],{"class":10732},[40,12288,12289],{"class":10287}," \"ll\"",[40,12291,10639],{"class":10363},[40,12293,12295,12297,12299,12301,12304,12306,12308,12310,12312,12314,12316,12318,12320,12322,12325],{"class":10276,"line":12294},112,[40,12296,11339],{"class":10352},[40,12298,12197],{"class":10446},[40,12300,10484],{"class":10363},[40,12302,12303],{"class":10446},"0.015",[40,12305,10484],{"class":10363},[40,12307,12149],{"class":10732},[40,12309,167],{"class":10363},[40,12311,12211],{"class":11903},[40,12313,162],{"class":10732},[40,12315,12216],{"class":10287},[40,12317,10484],{"class":10363},[40,12319,12221],{"class":11903},[40,12321,162],{"class":10732},[40,12323,12324],{"class":10287},"\"bottom\"",[40,12326,10761],{"class":10363},[40,12328,12330,12332,12334,12336,12339],{"class":10276,"line":12329},113,[40,12331,11323],{"class":10352},[40,12333,12182],{"class":10363},[40,12335,11231],{"class":10732},[40,12337,12338],{"class":10287}," \"lr\"",[40,12340,10639],{"class":10363},[40,12342,12344,12346,12348,12350,12352,12354,12356,12358,12360,12362,12364,12366,12368,12370,12372],{"class":10276,"line":12343},114,[40,12345,11339],{"class":10352},[40,12347,12250],{"class":10446},[40,12349,10484],{"class":10363},[40,12351,12303],{"class":10446},[40,12353,10484],{"class":10363},[40,12355,12149],{"class":10732},[40,12357,167],{"class":10363},[40,12359,12211],{"class":11903},[40,12361,162],{"class":10732},[40,12363,12267],{"class":10287},[40,12365,10484],{"class":10363},[40,12367,12221],{"class":11903},[40,12369,162],{"class":10732},[40,12371,12324],{"class":10287},[40,12373,10761],{"class":10363},[40,12375,12377,12379,12381,12383,12385,12387,12389,12391,12393,12395,12397,12399,12401,12403,12405],{"class":10276,"line":12376},115,[40,12378,11201],{"class":10352},[40,12380,12197],{"class":10446},[40,12382,10484],{"class":10363},[40,12384,12202],{"class":10446},[40,12386,10484],{"class":10363},[40,12388,12149],{"class":10732},[40,12390,167],{"class":10363},[40,12392,12211],{"class":11903},[40,12394,162],{"class":10732},[40,12396,12216],{"class":10287},[40,12398,10484],{"class":10363},[40,12400,12221],{"class":11903},[40,12402,162],{"class":10732},[40,12404,12226],{"class":10287},[40,12406,10761],{"class":10363},[40,12408,12410],{"class":10276,"line":12409},116,[40,12411,10301],{"emptyLinePlaceholder":10300},[40,12413,12415],{"class":10276,"line":12414},117,[40,12416,10301],{"emptyLinePlaceholder":10300},[40,12418,12420,12422,12425,12427,12430],{"class":10276,"line":12419},118,[40,12421,10965],{"class":10352},[40,12423,12424],{"class":10430}," add_info_box",[40,12426,167],{"class":10363},[40,12428,12429],{"class":10973},"ax",[40,12431,12432],{"class":10363},": plt.Axes,\n",[40,12434,12436,12439,12441,12443],{"class":10276,"line":12435},119,[40,12437,12438],{"class":10973},"                 text",[40,12440,10443],{"class":10363},[40,12442,10733],{"class":10732},[40,12444,10507],{"class":10363},[40,12446,12448,12451,12453,12455,12457,12459],{"class":10276,"line":12447},120,[40,12449,12450],{"class":10973},"                 loc",[40,12452,10443],{"class":10363},[40,12454,10733],{"class":10732},[40,12456,10772],{"class":10732},[40,12458,12187],{"class":10287},[40,12460,10507],{"class":10363},[40,12462,12464,12467,12469,12471,12473,12476],{"class":10276,"line":12463},121,[40,12465,12466],{"class":10973},"                 fontsize",[40,12468,10443],{"class":10363},[40,12470,10821],{"class":10732},[40,12472,10772],{"class":10732},[40,12474,12475],{"class":10446}," 9",[40,12477,10507],{"class":10363},[40,12479,12481,12484,12486,12488,12490,12493],{"class":10276,"line":12480},122,[40,12482,12483],{"class":10973},"                 wrap",[40,12485,10443],{"class":10363},[40,12487,10821],{"class":10732},[40,12489,10772],{"class":10732},[40,12491,12492],{"class":10446}," 42",[40,12494,12495],{"class":10363},") -> plt.Text:\n",[40,12497,12499],{"class":10276,"line":12498},123,[40,12500,12501],{"class":10287},"    \"\"\"Create a wrapped, padded text box inside the axes (never clips).\"\"\"\n",[40,12503,12505,12508,12510,12512],{"class":10276,"line":12504},124,[40,12506,12507],{"class":10363},"    x, y, kw ",[40,12509,162],{"class":10732},[40,12511,12126],{"class":10430},[40,12513,12514],{"class":10363},"(loc)\n",[40,12516,12518,12520,12523,12525],{"class":10276,"line":12517},125,[40,12519,11201],{"class":10352},[40,12521,12522],{"class":10363}," ax.",[40,12524,2932],{"class":10430},[40,12526,12527],{"class":10363},"(\n",[40,12529,12531,12534,12537],{"class":10276,"line":12530},126,[40,12532,12533],{"class":10363},"        x, y, textwrap.",[40,12535,12536],{"class":10430},"fill",[40,12538,12539],{"class":10363},"(text, wrap),\n",[40,12541,12543,12546,12548,12551,12554,12556,12559,12562,12564,12567],{"class":10276,"line":12542},127,[40,12544,12545],{"class":11903},"        transform",[40,12547,162],{"class":10732},[40,12549,12550],{"class":10363},"ax.transAxes, ",[40,12552,12553],{"class":11903},"fontsize",[40,12555,162],{"class":10732},[40,12557,12558],{"class":10363},"fontsize, ",[40,12560,12561],{"class":11903},"zorder",[40,12563,162],{"class":10732},[40,12565,12566],{"class":10446},"5",[40,12568,10507],{"class":10363},[40,12570,12572,12575,12578,12580,12583,12585,12588,12590,12593,12595,12598,12600,12603,12605,12608,12610,12613,12615,12618],{"class":10276,"line":12571},128,[40,12573,12574],{"class":11903},"        bbox",[40,12576,12577],{"class":10732},"=dict",[40,12579,167],{"class":10363},[40,12581,12582],{"class":11903},"boxstyle",[40,12584,162],{"class":10732},[40,12586,12587],{"class":10287},"\"round,pad=0.3\"",[40,12589,10484],{"class":10363},[40,12591,12592],{"class":11903},"fc",[40,12594,162],{"class":10732},[40,12596,12597],{"class":10287},"\"white\"",[40,12599,10484],{"class":10363},[40,12601,12602],{"class":11903},"alpha",[40,12604,162],{"class":10732},[40,12606,12607],{"class":10446},"0.75",[40,12609,10484],{"class":10363},[40,12611,12612],{"class":11903},"ec",[40,12614,162],{"class":10732},[40,12616,12617],{"class":10287},"\"none\"",[40,12619,11061],{"class":10363},[40,12621,12623],{"class":10276,"line":12622},129,[40,12624,12625],{"class":10363},"        **kw,\n",[40,12627,12629],{"class":10276,"line":12628},130,[40,12630,12631],{"class":10363},"    )\n",[40,12633,12635],{"class":10276,"line":12634},131,[40,12636,10301],{"emptyLinePlaceholder":10300},[40,12638,12640],{"class":10276,"line":12639},132,[40,12641,10301],{"emptyLinePlaceholder":10300},[40,12643,12645,12647,12650,12652,12654,12657,12659,12661,12663],{"class":10276,"line":12644},133,[40,12646,10965],{"class":10352},[40,12648,12649],{"class":10430}," add_title_inside",[40,12651,167],{"class":10363},[40,12653,12429],{"class":10973},[40,12655,12656],{"class":10363},": plt.Axes, ",[40,12658,2932],{"class":10973},[40,12660,10443],{"class":10363},[40,12662,10733],{"class":10732},[40,12664,12495],{"class":10363},[40,12666,12668],{"class":10276,"line":12667},134,[40,12669,12670],{"class":10287},"    \"\"\"Draw a title inside the Axes at the very top center (never clips).\"\"\"\n",[40,12672,12674,12676,12678,12680],{"class":10276,"line":12673},135,[40,12675,11201],{"class":10352},[40,12677,12522],{"class":10363},[40,12679,2932],{"class":10430},[40,12681,12527],{"class":10363},[40,12683,12685,12688,12690,12692],{"class":10276,"line":12684},136,[40,12686,12687],{"class":10446},"        0.5",[40,12689,10484],{"class":10363},[40,12691,12202],{"class":10446},[40,12693,12694],{"class":10363},", text,\n",[40,12696,12698,12700,12702,12704,12706,12708,12711,12713,12715,12717,12719],{"class":10276,"line":12697},137,[40,12699,12545],{"class":11903},[40,12701,162],{"class":10732},[40,12703,12550],{"class":10363},[40,12705,12211],{"class":11903},[40,12707,162],{"class":10732},[40,12709,12710],{"class":10287},"\"center\"",[40,12712,10484],{"class":10363},[40,12714,12221],{"class":11903},[40,12716,162],{"class":10732},[40,12718,12226],{"class":10287},[40,12720,10507],{"class":10363},[40,12722,12724,12727,12729,12732],{"class":10276,"line":12723},138,[40,12725,12726],{"class":11903},"        fontsize",[40,12728,162],{"class":10732},[40,12730,12731],{"class":10446},"11",[40,12733,10507],{"class":10363},[40,12735,12737,12739,12741,12743,12745,12747,12749,12751,12753,12755,12757,12759,12761,12763,12765,12767,12769,12771,12773],{"class":10276,"line":12736},139,[40,12738,12574],{"class":11903},[40,12740,12577],{"class":10732},[40,12742,167],{"class":10363},[40,12744,12582],{"class":11903},[40,12746,162],{"class":10732},[40,12748,12587],{"class":10287},[40,12750,10484],{"class":10363},[40,12752,12592],{"class":11903},[40,12754,162],{"class":10732},[40,12756,12597],{"class":10287},[40,12758,10484],{"class":10363},[40,12760,12602],{"class":11903},[40,12762,162],{"class":10732},[40,12764,12607],{"class":10446},[40,12766,10484],{"class":10363},[40,12768,12612],{"class":11903},[40,12770,162],{"class":10732},[40,12772,12617],{"class":10287},[40,12774,11061],{"class":10363},[40,12776,12778,12781,12783,12786],{"class":10276,"line":12777},140,[40,12779,12780],{"class":11903},"        zorder",[40,12782,162],{"class":10732},[40,12784,12785],{"class":10446},"6",[40,12787,10507],{"class":10363},[40,12789,12791],{"class":10276,"line":12790},141,[40,12792,12631],{"class":10363},[10082,12794,12796],{"className":10269,"code":12795,"language":10271,"meta":10090,"style":10090},"# @title Cell 3 — Pauli X\u002FY\u002FZ Wigner animations (titles *inside* the plot)\nfrom pathlib import Path\n\ndef animate_wigner_pauli(axis: str,\n                         r0: np.ndarray,\n                         frames: int,\n                         n_theta: int,\n                         n_phi: int,\n                         fps: int,\n                         figsize: Tuple[float, float],\n                         info_loc: str,\n                         info_fontsize: int,\n                         wrap_chars: int,\n                         out_path: Path) -> Path:\n    \"\"\"Animate Wigner under a Pauli rotation U = exp(-i σ_axis * π * t \u002F 2).\"\"\"\n    assert frames >= 2, \"frames must be ≥ 2\"\n    ngrid = n_grid(n_theta, n_phi)\n    w_min = 0.5 - (np.sqrt(3.0) \u002F 2.0)\n    w_max = 0.5 + (np.sqrt(3.0) \u002F 2.0)\n    extent = (0.0, 2.0 * np.pi, 0.0, np.pi)\n\n    fig, ax = plt.subplots(figsize=figsize, constrained_layout=True)\n    im = ax.imshow(np.zeros((n_theta, n_phi)), origin=\"lower\", extent=extent,\n                   vmin=w_min, vmax=w_max, aspect=\"auto\")\n    cbar = fig.colorbar(im, ax=ax, fraction=0.046, pad=0.04)\n    cbar.set_label(\"Wigner value\")\n\n    ax.set_xlabel(\"φ (radians)\")\n    ax.set_ylabel(\"θ (radians)\")\n    add_title_inside(ax, f\"Wigner — {axis.upper()} rotation\")\n\n    info = add_info_box(ax, \"\", loc=info_loc, fontsize=info_fontsize,\n                        wrap=wrap_chars)\n\n    def update(k: int):\n        ang = (k \u002F (frames - 1)) * np.pi  # 0 → π\n        r = rotation_matrix(axis.lower(), ang) @ r0\n        im.set_data(spin_wigner_qubit(r, ngrid))\n        info.set_text(f\"θ = {ang:.3f} rad\\nframe {k+1}\u002F{frames}\")\n        return (im, info)\n\n    anim = FuncAnimation(fig, update, frames=frames, interval=60, blit=False)\n    out_path.parent.mkdir(parents=True, exist_ok=True)\n    anim.save(out_path.as_posix(), writer=PillowWriter(fps=fps),\n              savefig_kwargs=dict(facecolor=\"white\"))\n    plt.close(fig)\n    return out_path\n\n\n# ---------------------------- Run X, Y, Z animations --------------------------\nknobs = Knobs(  # tweak if desired\n    gate_axes=(\"x\", \"y\", \"z\"),\n    initial_state=\"+z\",\n    frames=24,                # raise to 48 for smoother motion\n    n_theta=80, n_phi=160,    # raise to 120×240 for more detail\n    fps=20,\n    out_dir=\"\u002Fcontent\",\n    figsize=(6.4, 3.6),\n    info_loc=\"ul\", info_fontsize=9, wrap_chars=42,\n)\n\nr0 = bloch_vector(knobs.initial_state, knobs.custom_r)\nouts = []\nfor axname in knobs.gate_axes:\n    path = Path(knobs.out_dir) \u002F f\"wigner_pauli_{axname}.gif\"\n    outs.append(\n        animate_wigner_pauli(axname, r0, knobs.frames,\n                             knobs.n_theta, knobs.n_phi, knobs.fps,\n                             knobs.figsize, knobs.info_loc,\n                             knobs.info_fontsize, knobs.wrap_chars,\n                             path)\n    )\n\n# Display inline\nfor p in outs:\n    display(Image(filename=str(p)))\n    print(\"Saved:\", p)\n",[10088,12797,12798,12803,12813,12817,12834,12842,12853,12864,12875,12886,12901,12912,12923,12934,12942,12947,12966,12978,13005,13032,13057,13061,13094,13133,13161,13204,13219,13223,13238,13252,13286,13290,13320,13330,13334,13354,13385,13408,13424,13481,13488,13492,13531,13559,13594,13613,13624,13631,13635,13639,13644,13659,13680,13692,13708,13733,13745,13757,13774,13806,13810,13814,13826,13836,13850,13881,13891,13899,13904,13909,13914,13919,13923,13927,13932,13944,13965],{"__ignoreMap":10090},[40,12799,12800],{"class":10276,"line":10277},[40,12801,12802],{"class":10280},"# @title Cell 3 — Pauli X\u002FY\u002FZ Wigner animations (titles *inside* the plot)\n",[40,12804,12805,12807,12809,12811],{"class":10276,"line":10284},[40,12806,10353],{"class":10352},[40,12808,10542],{"class":10363},[40,12810,10375],{"class":10352},[40,12812,10547],{"class":10363},[40,12814,12815],{"class":10276,"line":10291},[40,12816,10301],{"emptyLinePlaceholder":10300},[40,12818,12819,12821,12824,12826,12828,12830,12832],{"class":10276,"line":10297},[40,12820,10965],{"class":10352},[40,12822,12823],{"class":10430}," animate_wigner_pauli",[40,12825,167],{"class":10363},[40,12827,11275],{"class":10973},[40,12829,10443],{"class":10363},[40,12831,10733],{"class":10732},[40,12833,10507],{"class":10363},[40,12835,12836,12839],{"class":10276,"line":10304},[40,12837,12838],{"class":10973},"                         r0",[40,12840,12841],{"class":10363},": np.ndarray,\n",[40,12843,12844,12847,12849,12851],{"class":10276,"line":10310},[40,12845,12846],{"class":10973},"                         frames",[40,12848,10443],{"class":10363},[40,12850,10821],{"class":10732},[40,12852,10507],{"class":10363},[40,12854,12855,12858,12860,12862],{"class":10276,"line":10316},[40,12856,12857],{"class":10973},"                         n_theta",[40,12859,10443],{"class":10363},[40,12861,10821],{"class":10732},[40,12863,10507],{"class":10363},[40,12865,12866,12869,12871,12873],{"class":10276,"line":10322},[40,12867,12868],{"class":10973},"                         n_phi",[40,12870,10443],{"class":10363},[40,12872,10821],{"class":10732},[40,12874,10507],{"class":10363},[40,12876,12877,12880,12882,12884],{"class":10276,"line":10327},[40,12878,12879],{"class":10973},"                         fps",[40,12881,10443],{"class":10363},[40,12883,10821],{"class":10732},[40,12885,10507],{"class":10363},[40,12887,12888,12891,12893,12895,12897,12899],{"class":10276,"line":10333},[40,12889,12890],{"class":10973},"                         figsize",[40,12892,10989],{"class":10363},[40,12894,10784],{"class":10732},[40,12896,10484],{"class":10363},[40,12898,10784],{"class":10732},[40,12900,11542],{"class":10363},[40,12902,12903,12906,12908,12910],{"class":10276,"line":10339},[40,12904,12905],{"class":10973},"                         info_loc",[40,12907,10443],{"class":10363},[40,12909,10733],{"class":10732},[40,12911,10507],{"class":10363},[40,12913,12914,12917,12919,12921],{"class":10276,"line":10344},[40,12915,12916],{"class":10973},"                         info_fontsize",[40,12918,10443],{"class":10363},[40,12920,10821],{"class":10732},[40,12922,10507],{"class":10363},[40,12924,12925,12928,12930,12932],{"class":10276,"line":10349},[40,12926,12927],{"class":10973},"                         wrap_chars",[40,12929,10443],{"class":10363},[40,12931,10821],{"class":10732},[40,12933,10507],{"class":10363},[40,12935,12936,12939],{"class":10276,"line":10367},[40,12937,12938],{"class":10973},"                         out_path",[40,12940,12941],{"class":10363},": Path) -> Path:\n",[40,12943,12944],{"class":10276,"line":10372},[40,12945,12946],{"class":10287},"    \"\"\"Animate Wigner under a Pauli rotation U = exp(-i σ_axis * π * t \u002F 2).\"\"\"\n",[40,12948,12949,12952,12955,12958,12961,12963],{"class":10276,"line":10387},[40,12950,12951],{"class":10352},"    assert",[40,12953,12954],{"class":10363}," frames ",[40,12956,12957],{"class":10732},">=",[40,12959,12960],{"class":10446}," 2",[40,12962,10484],{"class":10363},[40,12964,12965],{"class":10287},"\"frames must be ≥ 2\"\n",[40,12967,12968,12971,12973,12975],{"class":10276,"line":10400},[40,12969,12970],{"class":10363},"    ngrid ",[40,12972,162],{"class":10732},[40,12974,11811],{"class":10430},[40,12976,12977],{"class":10363},"(n_theta, n_phi)\n",[40,12979,12980,12983,12985,12987,12989,12991,12993,12995,12997,12999,13001,13003],{"class":10276,"line":10413},[40,12981,12982],{"class":10363},"    w_min ",[40,12984,162],{"class":10732},[40,12986,12039],{"class":10446},[40,12988,11764],{"class":10732},[40,12990,12045],{"class":10363},[40,12992,2218],{"class":10430},[40,12994,167],{"class":10363},[40,12996,12052],{"class":10446},[40,12998,12055],{"class":10363},[40,13000,11243],{"class":10732},[40,13002,12060],{"class":10446},[40,13004,10761],{"class":10363},[40,13006,13007,13010,13012,13014,13016,13018,13020,13022,13024,13026,13028,13030],{"class":10276,"line":10418},[40,13008,13009],{"class":10363},"    w_max ",[40,13011,162],{"class":10732},[40,13013,12039],{"class":10446},[40,13015,12042],{"class":10732},[40,13017,12045],{"class":10363},[40,13019,2218],{"class":10430},[40,13021,167],{"class":10363},[40,13023,12052],{"class":10446},[40,13025,12055],{"class":10363},[40,13027,11243],{"class":10732},[40,13029,12060],{"class":10446},[40,13031,10761],{"class":10363},[40,13033,13034,13037,13039,13041,13043,13045,13047,13049,13052,13054],{"class":10276,"line":10424},[40,13035,13036],{"class":10363},"    extent ",[40,13038,162],{"class":10732},[40,13040,920],{"class":10363},[40,13042,10801],{"class":10446},[40,13044,10484],{"class":10363},[40,13046,11878],{"class":10446},[40,13048,11881],{"class":10732},[40,13050,13051],{"class":10363}," np.pi, ",[40,13053,10801],{"class":10446},[40,13055,13056],{"class":10363},", np.pi)\n",[40,13058,13059],{"class":10276,"line":10437},[40,13060,10301],{"emptyLinePlaceholder":10300},[40,13062,13063,13066,13068,13071,13074,13076,13079,13081,13084,13087,13089,13092],{"class":10276,"line":10456},[40,13064,13065],{"class":10363},"    fig, ax ",[40,13067,162],{"class":10732},[40,13069,13070],{"class":10363}," plt.",[40,13072,13073],{"class":10430},"subplots",[40,13075,167],{"class":10363},[40,13077,13078],{"class":11903},"figsize",[40,13080,162],{"class":10732},[40,13082,13083],{"class":10363},"figsize, ",[40,13085,13086],{"class":11903},"constrained_layout",[40,13088,162],{"class":10732},[40,13090,13091],{"class":10446},"True",[40,13093,10761],{"class":10363},[40,13095,13096,13099,13101,13103,13106,13109,13112,13115,13118,13120,13123,13125,13128,13130],{"class":10276,"line":10472},[40,13097,13098],{"class":10363},"    im ",[40,13100,162],{"class":10732},[40,13102,12522],{"class":10363},[40,13104,13105],{"class":10430},"imshow",[40,13107,13108],{"class":10363},"(np.",[40,13110,13111],{"class":10430},"zeros",[40,13113,13114],{"class":10363},"((n_theta, n_phi)), ",[40,13116,13117],{"class":11903},"origin",[40,13119,162],{"class":10732},[40,13121,13122],{"class":10287},"\"lower\"",[40,13124,10484],{"class":10363},[40,13126,13127],{"class":11903},"extent",[40,13129,162],{"class":10732},[40,13131,13132],{"class":10363},"extent,\n",[40,13134,13135,13138,13140,13143,13146,13148,13151,13154,13156,13159],{"class":10276,"line":10496},[40,13136,13137],{"class":11903},"                   vmin",[40,13139,162],{"class":10732},[40,13141,13142],{"class":10363},"w_min, ",[40,13144,13145],{"class":11903},"vmax",[40,13147,162],{"class":10732},[40,13149,13150],{"class":10363},"w_max, ",[40,13152,13153],{"class":11903},"aspect",[40,13155,162],{"class":10732},[40,13157,13158],{"class":10287},"\"auto\"",[40,13160,10761],{"class":10363},[40,13162,13163,13166,13168,13171,13174,13177,13179,13181,13184,13187,13189,13192,13194,13197,13199,13202],{"class":10276,"line":10510},[40,13164,13165],{"class":10363},"    cbar ",[40,13167,162],{"class":10732},[40,13169,13170],{"class":10363}," fig.",[40,13172,13173],{"class":10430},"colorbar",[40,13175,13176],{"class":10363},"(im, ",[40,13178,12429],{"class":11903},[40,13180,162],{"class":10732},[40,13182,13183],{"class":10363},"ax, ",[40,13185,13186],{"class":11903},"fraction",[40,13188,162],{"class":10732},[40,13190,13191],{"class":10446},"0.046",[40,13193,10484],{"class":10363},[40,13195,13196],{"class":11903},"pad",[40,13198,162],{"class":10732},[40,13200,13201],{"class":10446},"0.04",[40,13203,10761],{"class":10363},[40,13205,13206,13209,13212,13214,13217],{"class":10276,"line":10696},[40,13207,13208],{"class":10363},"    cbar.",[40,13210,13211],{"class":10430},"set_label",[40,13213,167],{"class":10363},[40,13215,13216],{"class":10287},"\"Wigner value\"",[40,13218,10761],{"class":10363},[40,13220,13221],{"class":10276,"line":10702},[40,13222,10301],{"emptyLinePlaceholder":10300},[40,13224,13225,13228,13231,13233,13236],{"class":10276,"line":10708},[40,13226,13227],{"class":10363},"    ax.",[40,13229,13230],{"class":10430},"set_xlabel",[40,13232,167],{"class":10363},[40,13234,13235],{"class":10287},"\"φ (radians)\"",[40,13237,10761],{"class":10363},[40,13239,13240,13242,13245,13247,13250],{"class":10276,"line":10714},[40,13241,13227],{"class":10363},[40,13243,13244],{"class":10430},"set_ylabel",[40,13246,167],{"class":10363},[40,13248,13249],{"class":10287},"\"θ (radians)\"",[40,13251,10761],{"class":10363},[40,13253,13254,13257,13260,13263,13266,13269,13272,13275,13278,13281,13284],{"class":10276,"line":10720},[40,13255,13256],{"class":10430},"    add_title_inside",[40,13258,13259],{"class":10363},"(ax, ",[40,13261,13262],{"class":10352},"f",[40,13264,13265],{"class":10287},"\"Wigner — ",[40,13267,13268],{"class":10446},"{",[40,13270,13271],{"class":10363},"axis.",[40,13273,13274],{"class":10430},"upper",[40,13276,13277],{"class":10363},"()",[40,13279,13280],{"class":10446},"}",[40,13282,13283],{"class":10287}," rotation\"",[40,13285,10761],{"class":10363},[40,13287,13288],{"class":10276,"line":10726},[40,13289,10301],{"emptyLinePlaceholder":10300},[40,13291,13292,13295,13297,13299,13301,13304,13306,13308,13310,13313,13315,13317],{"class":10276,"line":10764},[40,13293,13294],{"class":10363},"    info ",[40,13296,162],{"class":10732},[40,13298,12424],{"class":10430},[40,13300,13259],{"class":10363},[40,13302,13303],{"class":10287},"\"\"",[40,13305,10484],{"class":10363},[40,13307,12131],{"class":11903},[40,13309,162],{"class":10732},[40,13311,13312],{"class":10363},"info_loc, ",[40,13314,12553],{"class":11903},[40,13316,162],{"class":10732},[40,13318,13319],{"class":10363},"info_fontsize,\n",[40,13321,13322,13325,13327],{"class":10276,"line":10778},[40,13323,13324],{"class":11903},"                        wrap",[40,13326,162],{"class":10732},[40,13328,13329],{"class":10363},"wrap_chars)\n",[40,13331,13332],{"class":10276,"line":10815},[40,13333,10301],{"emptyLinePlaceholder":10300},[40,13335,13336,13339,13342,13344,13347,13349,13351],{"class":10276,"line":10829},[40,13337,13338],{"class":10352},"    def",[40,13340,13341],{"class":10430}," update",[40,13343,167],{"class":10363},[40,13345,13346],{"class":10973},"k",[40,13348,10443],{"class":10363},[40,13350,10821],{"class":10732},[40,13352,13353],{"class":10363},"):\n",[40,13355,13356,13359,13361,13364,13366,13369,13371,13374,13377,13379,13382],{"class":10276,"line":10842},[40,13357,13358],{"class":10363},"        ang ",[40,13360,162],{"class":10732},[40,13362,13363],{"class":10363}," (k ",[40,13365,11243],{"class":10732},[40,13367,13368],{"class":10363}," (frames ",[40,13370,11056],{"class":10732},[40,13372,13373],{"class":10446}," 1",[40,13375,13376],{"class":10363},")) ",[40,13378,11752],{"class":10732},[40,13380,13381],{"class":10363}," np.pi  ",[40,13383,13384],{"class":10280},"# 0 → π\n",[40,13386,13387,13390,13392,13394,13397,13399,13402,13405],{"class":10276,"line":10855},[40,13388,13389],{"class":10363},"        r ",[40,13391,162],{"class":10732},[40,13393,11270],{"class":10430},[40,13395,13396],{"class":10363},"(axis.",[40,13398,11171],{"class":10430},[40,13400,13401],{"class":10363},"(), ang) ",[40,13403,13404],{"class":10732},"@",[40,13406,13407],{"class":10363}," r0\n",[40,13409,13410,13413,13416,13418,13421],{"class":10276,"line":10868},[40,13411,13412],{"class":10363},"        im.",[40,13414,13415],{"class":10430},"set_data",[40,13417,167],{"class":10363},[40,13419,13420],{"class":10430},"spin_wigner_qubit",[40,13422,13423],{"class":10363},"(r, ngrid))\n",[40,13425,13426,13429,13432,13434,13436,13439,13441,13444,13447,13449,13452,13455,13458,13460,13462,13464,13467,13469,13471,13474,13476,13479],{"class":10276,"line":10881},[40,13427,13428],{"class":10363},"        info.",[40,13430,13431],{"class":10430},"set_text",[40,13433,167],{"class":10363},[40,13435,13262],{"class":10352},[40,13437,13438],{"class":10287},"\"θ = ",[40,13440,13268],{"class":10446},[40,13442,13443],{"class":10363},"ang",[40,13445,13446],{"class":10352},":.3f",[40,13448,13280],{"class":10446},[40,13450,13451],{"class":10287}," rad",[40,13453,13454],{"class":10732},"\\n",[40,13456,13457],{"class":10287},"frame ",[40,13459,13268],{"class":10446},[40,13461,13346],{"class":10363},[40,13463,1053],{"class":10732},[40,13465,13466],{"class":10446},"1}",[40,13468,11243],{"class":10287},[40,13470,13268],{"class":10446},[40,13472,13473],{"class":10363},"frames",[40,13475,13280],{"class":10446},[40,13477,13478],{"class":10287},"\"",[40,13480,10761],{"class":10363},[40,13482,13483,13485],{"class":10276,"line":10907},[40,13484,11339],{"class":10352},[40,13486,13487],{"class":10363}," (im, info)\n",[40,13489,13490],{"class":10276,"line":10920},[40,13491,10301],{"emptyLinePlaceholder":10300},[40,13493,13494,13497,13499,13502,13505,13507,13509,13512,13515,13517,13520,13522,13525,13527,13529],{"class":10276,"line":10933},[40,13495,13496],{"class":10363},"    anim ",[40,13498,162],{"class":10732},[40,13500,13501],{"class":10430}," FuncAnimation",[40,13503,13504],{"class":10363},"(fig, update, ",[40,13506,13473],{"class":11903},[40,13508,162],{"class":10732},[40,13510,13511],{"class":10363},"frames, ",[40,13513,13514],{"class":11903},"interval",[40,13516,162],{"class":10732},[40,13518,13519],{"class":10446},"60",[40,13521,10484],{"class":10363},[40,13523,13524],{"class":11903},"blit",[40,13526,162],{"class":10732},[40,13528,10504],{"class":10446},[40,13530,10761],{"class":10363},[40,13532,13533,13536,13539,13541,13544,13546,13548,13550,13553,13555,13557],{"class":10276,"line":10946},[40,13534,13535],{"class":10363},"    out_path.parent.",[40,13537,13538],{"class":10430},"mkdir",[40,13540,167],{"class":10363},[40,13542,13543],{"class":11903},"parents",[40,13545,162],{"class":10732},[40,13547,13091],{"class":10446},[40,13549,10484],{"class":10363},[40,13551,13552],{"class":11903},"exist_ok",[40,13554,162],{"class":10732},[40,13556,13091],{"class":10446},[40,13558,10761],{"class":10363},[40,13560,13561,13564,13567,13570,13573,13576,13579,13581,13584,13586,13589,13591],{"class":10276,"line":10951},[40,13562,13563],{"class":10363},"    anim.",[40,13565,13566],{"class":10430},"save",[40,13568,13569],{"class":10363},"(out_path.",[40,13571,13572],{"class":10430},"as_posix",[40,13574,13575],{"class":10363},"(), ",[40,13577,13578],{"class":11903},"writer",[40,13580,162],{"class":10732},[40,13582,13583],{"class":10430},"PillowWriter",[40,13585,167],{"class":10363},[40,13587,13588],{"class":11903},"fps",[40,13590,162],{"class":10732},[40,13592,13593],{"class":10363},"fps),\n",[40,13595,13596,13599,13601,13603,13606,13608,13610],{"class":10276,"line":10956},[40,13597,13598],{"class":11903},"              savefig_kwargs",[40,13600,12577],{"class":10732},[40,13602,167],{"class":10363},[40,13604,13605],{"class":11903},"facecolor",[40,13607,162],{"class":10732},[40,13609,12597],{"class":10287},[40,13611,13612],{"class":10363},"))\n",[40,13614,13615,13618,13621],{"class":10276,"line":10962},[40,13616,13617],{"class":10363},"    plt.",[40,13619,13620],{"class":10430},"close",[40,13622,13623],{"class":10363},"(fig)\n",[40,13625,13626,13628],{"class":10276,"line":10983},[40,13627,11201],{"class":10352},[40,13629,13630],{"class":10363}," out_path\n",[40,13632,13633],{"class":10276,"line":11005},[40,13634,10301],{"emptyLinePlaceholder":10300},[40,13636,13637],{"class":10276,"line":11011},[40,13638,10301],{"emptyLinePlaceholder":10300},[40,13640,13641],{"class":10276,"line":11022},[40,13642,13643],{"class":10280},"# ---------------------------- Run X, Y, Z animations --------------------------\n",[40,13645,13646,13649,13651,13653,13656],{"class":10276,"line":11064},[40,13647,13648],{"class":10363},"knobs ",[40,13650,162],{"class":10732},[40,13652,10636],{"class":10430},[40,13654,13655],{"class":10363},"(  ",[40,13657,13658],{"class":10280},"# tweak if desired\n",[40,13660,13661,13664,13666,13668,13670,13672,13674,13676,13678],{"class":10276,"line":11103},[40,13662,13663],{"class":11903},"    gate_axes",[40,13665,162],{"class":10732},[40,13667,167],{"class":10363},[40,13669,10748],{"class":10287},[40,13671,10484],{"class":10363},[40,13673,10753],{"class":10287},[40,13675,10484],{"class":10363},[40,13677,10758],{"class":10287},[40,13679,11061],{"class":10363},[40,13681,13682,13685,13687,13690],{"class":10276,"line":11142},[40,13683,13684],{"class":11903},"    initial_state",[40,13686,162],{"class":10732},[40,13688,13689],{"class":10287},"\"+z\"",[40,13691,10507],{"class":10363},[40,13693,13694,13697,13699,13702,13705],{"class":10276,"line":11148},[40,13695,13696],{"class":11903},"    frames",[40,13698,162],{"class":10732},[40,13700,13701],{"class":10446},"24",[40,13703,13704],{"class":10363},",                ",[40,13706,13707],{"class":10280},"# raise to 48 for smoother motion\n",[40,13709,13710,13713,13715,13718,13720,13722,13724,13727,13730],{"class":10276,"line":11181},[40,13711,13712],{"class":11903},"    n_theta",[40,13714,162],{"class":10732},[40,13716,13717],{"class":10446},"80",[40,13719,10484],{"class":10363},[40,13721,11825],{"class":11903},[40,13723,162],{"class":10732},[40,13725,13726],{"class":10446},"160",[40,13728,13729],{"class":10363},",    ",[40,13731,13732],{"class":10280},"# raise to 120×240 for more detail\n",[40,13734,13735,13738,13740,13743],{"class":10276,"line":11198},[40,13736,13737],{"class":11903},"    fps",[40,13739,162],{"class":10732},[40,13741,13742],{"class":10446},"20",[40,13744,10507],{"class":10363},[40,13746,13747,13750,13752,13755],{"class":10276,"line":11249},[40,13748,13749],{"class":11903},"    out_dir",[40,13751,162],{"class":10732},[40,13753,13754],{"class":10287},"\"\u002Fcontent\"",[40,13756,10507],{"class":10363},[40,13758,13759,13762,13764,13766,13768,13770,13772],{"class":10276,"line":11254},[40,13760,13761],{"class":11903},"    figsize",[40,13763,162],{"class":10732},[40,13765,167],{"class":10363},[40,13767,10481],{"class":10446},[40,13769,10484],{"class":10363},[40,13771,10487],{"class":10446},[40,13773,11061],{"class":10363},[40,13775,13776,13779,13781,13784,13786,13789,13791,13794,13796,13799,13801,13804],{"class":10276,"line":11259},[40,13777,13778],{"class":11903},"    info_loc",[40,13780,162],{"class":10732},[40,13782,13783],{"class":10287},"\"ul\"",[40,13785,10484],{"class":10363},[40,13787,13788],{"class":11903},"info_fontsize",[40,13790,162],{"class":10732},[40,13792,13793],{"class":10446},"9",[40,13795,10484],{"class":10363},[40,13797,13798],{"class":11903},"wrap_chars",[40,13800,162],{"class":10732},[40,13802,13803],{"class":10446},"42",[40,13805,10507],{"class":10363},[40,13807,13808],{"class":10276,"line":11265},[40,13809,10761],{"class":10363},[40,13811,13812],{"class":10276,"line":11294},[40,13813,10301],{"emptyLinePlaceholder":10300},[40,13815,13816,13819,13821,13823],{"class":10276,"line":11300},[40,13817,13818],{"class":10363},"r0 ",[40,13820,162],{"class":10732},[40,13822,10968],{"class":10430},[40,13824,13825],{"class":10363},"(knobs.initial_state, knobs.custom_r)\n",[40,13827,13828,13831,13833],{"class":10276,"line":11320},[40,13829,13830],{"class":10363},"outs ",[40,13832,162],{"class":10732},[40,13834,13835],{"class":10363}," []\n",[40,13837,13838,13841,13844,13847],{"class":10276,"line":11336},[40,13839,13840],{"class":10352},"for",[40,13842,13843],{"class":10363}," axname ",[40,13845,13846],{"class":10352},"in",[40,13848,13849],{"class":10363}," knobs.gate_axes:\n",[40,13851,13852,13855,13857,13860,13863,13865,13868,13871,13873,13876,13878],{"class":10276,"line":11381},[40,13853,13854],{"class":10363},"    path ",[40,13856,162],{"class":10732},[40,13858,13859],{"class":10430}," Path",[40,13861,13862],{"class":10363},"(knobs.out_dir) ",[40,13864,11243],{"class":10732},[40,13866,13867],{"class":10352}," f",[40,13869,13870],{"class":10287},"\"wigner_pauli_",[40,13872,13268],{"class":10446},[40,13874,13875],{"class":10363},"axname",[40,13877,13280],{"class":10446},[40,13879,13880],{"class":10287},".gif\"\n",[40,13882,13883,13886,13889],{"class":10276,"line":11395},[40,13884,13885],{"class":10363},"    outs.",[40,13887,13888],{"class":10430},"append",[40,13890,12527],{"class":10363},[40,13892,13893,13896],{"class":10276,"line":11438},[40,13894,13895],{"class":10430},"        animate_wigner_pauli",[40,13897,13898],{"class":10363},"(axname, r0, knobs.frames,\n",[40,13900,13901],{"class":10276,"line":11452},[40,13902,13903],{"class":10363},"                             knobs.n_theta, knobs.n_phi, knobs.fps,\n",[40,13905,13906],{"class":10276,"line":11493},[40,13907,13908],{"class":10363},"                             knobs.figsize, knobs.info_loc,\n",[40,13910,13911],{"class":10276,"line":11507},[40,13912,13913],{"class":10363},"                             knobs.info_fontsize, knobs.wrap_chars,\n",[40,13915,13916],{"class":10276,"line":11512},[40,13917,13918],{"class":10363},"                             path)\n",[40,13920,13921],{"class":10276,"line":11517},[40,13922,12631],{"class":10363},[40,13924,13925],{"class":10276,"line":11545},[40,13926,10301],{"emptyLinePlaceholder":10300},[40,13928,13929],{"class":10276,"line":11557},[40,13930,13931],{"class":10280},"# Display inline\n",[40,13933,13934,13936,13939,13941],{"class":10276,"line":11563},[40,13935,13840],{"class":10352},[40,13937,13938],{"class":10363}," p ",[40,13940,13846],{"class":10352},[40,13942,13943],{"class":10363}," outs:\n",[40,13945,13946,13949,13951,13954,13956,13959,13962],{"class":10276,"line":11582},[40,13947,13948],{"class":10430},"    display",[40,13950,167],{"class":10363},[40,13952,13953],{"class":10430},"Image",[40,13955,167],{"class":10363},[40,13957,13958],{"class":11903},"filename",[40,13960,13961],{"class":10732},"=str",[40,13963,13964],{"class":10363},"(p)))\n",[40,13966,13967,13970,13972,13975],{"class":10276,"line":11596},[40,13968,13969],{"class":10732},"    print",[40,13971,167],{"class":10363},[40,13973,13974],{"class":10287},"\"Saved:\"",[40,13976,13977],{"class":10363},", p)\n",[10082,13979,13982],{"className":13980,"code":13981,"language":2932,"meta":10090},[10085],"Saved: \u002Fcontent\u002Fwigner_pauli_x.gif\n",[10088,13983,13981],{"__ignoreMap":10090},[10082,13985,13988],{"className":13986,"code":13987,"language":2932,"meta":10090},[10085],"Saved: \u002Fcontent\u002Fwigner_pauli_y.gif\n",[10088,13989,13987],{"__ignoreMap":10090},[10082,13991,13994],{"className":13992,"code":13993,"language":2932,"meta":10090},[10085],"Saved: \u002Fcontent\u002Fwigner_pauli_z.gif\n",[10088,13995,13993],{"__ignoreMap":10090},[13997,13998],"blog-figure",{"alt":13999,"src":14000},"Output 1 of the notebook","\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-01.webp",[13997,14002],{"alt":14003,"src":14004},"Output 2 of the notebook","\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-02.webp",[13997,14006],{"alt":14007,"src":14008},"Output 3 of the notebook","\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-03.webp",[10082,14010,14012],{"className":10269,"code":14011,"language":10271,"meta":10090,"style":10090},"# @title Cell 4 — Custom axis–angle gate (example: 45° about y)\nfrom pathlib import Path\n\ndef animate_wigner_custom(axis_vec: Tuple[float, float, float],\n                          angle_deg: float,\n                          r0: np.ndarray,\n                          frames: int,\n                          n_theta: int,\n                          n_phi: int,\n                          fps: int,\n                          figsize: Tuple[float, float],\n                          info_loc: str,\n                          info_fontsize: int,\n                          wrap_chars: int,\n                          out_path: Path) -> Path:\n    \"\"\"Animate Wigner under an arbitrary axis–angle rotation 0 → angle_deg.\"\"\"\n    assert frames >= 2, \"frames must be ≥ 2\"\n    ngrid = n_grid(n_theta, n_phi)\n    w_min = 0.5 - (np.sqrt(3.0) \u002F 2.0)\n    w_max = 0.5 + (np.sqrt(3.0) \u002F 2.0)\n    extent = (0.0, 2.0 * np.pi, 0.0, np.pi)\n\n    fig, ax = plt.subplots(figsize=figsize, constrained_layout=True)\n    im = ax.imshow(np.zeros((n_theta, n_phi)), origin=\"lower\", extent=extent,\n                   vmin=w_min, vmax=w_max, aspect=\"auto\")\n    cbar = fig.colorbar(im, ax=ax, fraction=0.046, pad=0.04)\n    cbar.set_label(\"Wigner value\")\n\n    ax.set_xlabel(\"φ (radians)\")\n    ax.set_ylabel(\"θ (radians)\")\n    add_title_inside(ax, \"Custom rotation\")\n\n    info = add_info_box(ax, \"\", loc=info_loc, fontsize=info_fontsize,\n                        wrap=wrap_chars)\n\n    def update(k: int):\n        ang = (k \u002F (frames - 1)) * np.deg2rad(angle_deg)\n        R = rotation_matrix_axis(axis_vec, ang)\n        r = R @ r0\n        im.set_data(spin_wigner_qubit(r, ngrid))\n        info.set_text(\n            f\"axis = ({axis_vec[0]:.2f}, {axis_vec[1]:.2f}, {axis_vec[2]:.2f})\\n\"\n            f\"θ = {np.rad2deg(ang):.1f}° \u002F {angle_deg:.1f}°\"\n        )\n        return (im, info)\n\n    anim = FuncAnimation(fig, update, frames=frames, interval=60, blit=False)\n    out_path.parent.mkdir(parents=True, exist_ok=True)\n    anim.save(out_path.as_posix(), writer=PillowWriter(fps=fps),\n              savefig_kwargs=dict(facecolor=\"white\"))\n    plt.close(fig)\n    return out_path\n\n\n# Example: 45° about the y-axis, starting from |+z⟩\ncustom_out = Path(knobs.out_dir) \u002F \"wigner_custom_y45.gif\"\nanimate_wigner_custom(\n    axis_vec=(0.0, 1.0, 0.0),\n    angle_deg=45.0,\n    r0=r0,\n    frames=24,                 # raise to 48 if you prefer\n    n_theta=80, n_phi=160,\n    fps=20,\n    figsize=knobs.figsize,\n    info_loc=knobs.info_loc, info_fontsize=knobs.info_fontsize,\n    wrap_chars=knobs.wrap_chars,\n    out_path=custom_out,\n)\n\ndisplay(Image(filename=str(custom_out)))\nprint(\"Saved:\", custom_out)\n",[10088,14013,14014,14019,14029,14033,14058,14069,14076,14087,14098,14109,14120,14135,14146,14157,14168,14175,14180,14194,14204,14230,14256,14278,14282,14308,14338,14360,14394,14406,14410,14422,14434,14445,14449,14475,14483,14487,14503,14531,14543,14556,14568,14576,14633,14670,14675,14681,14685,14717,14741,14767,14783,14791,14797,14801,14805,14810,14826,14833,14854,14866,14876,14890,14908,14918,14927,14943,14953,14963,14967,14971,14989],{"__ignoreMap":10090},[40,14015,14016],{"class":10276,"line":10277},[40,14017,14018],{"class":10280},"# @title Cell 4 — Custom axis–angle gate (example: 45° about y)\n",[40,14020,14021,14023,14025,14027],{"class":10276,"line":10284},[40,14022,10353],{"class":10352},[40,14024,10542],{"class":10363},[40,14026,10375],{"class":10352},[40,14028,10547],{"class":10363},[40,14030,14031],{"class":10276,"line":10291},[40,14032,10301],{"emptyLinePlaceholder":10300},[40,14034,14035,14037,14040,14042,14044,14046,14048,14050,14052,14054,14056],{"class":10276,"line":10297},[40,14036,10965],{"class":10352},[40,14038,14039],{"class":10430}," animate_wigner_custom",[40,14041,167],{"class":10363},[40,14043,11527],{"class":10973},[40,14045,10989],{"class":10363},[40,14047,10784],{"class":10732},[40,14049,10484],{"class":10363},[40,14051,10784],{"class":10732},[40,14053,10484],{"class":10363},[40,14055,10784],{"class":10732},[40,14057,11542],{"class":10363},[40,14059,14060,14063,14065,14067],{"class":10276,"line":10304},[40,14061,14062],{"class":10973},"                          angle_deg",[40,14064,10443],{"class":10363},[40,14066,10784],{"class":10732},[40,14068,10507],{"class":10363},[40,14070,14071,14074],{"class":10276,"line":10310},[40,14072,14073],{"class":10973},"                          r0",[40,14075,12841],{"class":10363},[40,14077,14078,14081,14083,14085],{"class":10276,"line":10316},[40,14079,14080],{"class":10973},"                          frames",[40,14082,10443],{"class":10363},[40,14084,10821],{"class":10732},[40,14086,10507],{"class":10363},[40,14088,14089,14092,14094,14096],{"class":10276,"line":10322},[40,14090,14091],{"class":10973},"                          n_theta",[40,14093,10443],{"class":10363},[40,14095,10821],{"class":10732},[40,14097,10507],{"class":10363},[40,14099,14100,14103,14105,14107],{"class":10276,"line":10327},[40,14101,14102],{"class":10973},"                          n_phi",[40,14104,10443],{"class":10363},[40,14106,10821],{"class":10732},[40,14108,10507],{"class":10363},[40,14110,14111,14114,14116,14118],{"class":10276,"line":10333},[40,14112,14113],{"class":10973},"                          fps",[40,14115,10443],{"class":10363},[40,14117,10821],{"class":10732},[40,14119,10507],{"class":10363},[40,14121,14122,14125,14127,14129,14131,14133],{"class":10276,"line":10339},[40,14123,14124],{"class":10973},"                          figsize",[40,14126,10989],{"class":10363},[40,14128,10784],{"class":10732},[40,14130,10484],{"class":10363},[40,14132,10784],{"class":10732},[40,14134,11542],{"class":10363},[40,14136,14137,14140,14142,14144],{"class":10276,"line":10344},[40,14138,14139],{"class":10973},"                          info_loc",[40,14141,10443],{"class":10363},[40,14143,10733],{"class":10732},[40,14145,10507],{"class":10363},[40,14147,14148,14151,14153,14155],{"class":10276,"line":10349},[40,14149,14150],{"class":10973},"                          info_fontsize",[40,14152,10443],{"class":10363},[40,14154,10821],{"class":10732},[40,14156,10507],{"class":10363},[40,14158,14159,14162,14164,14166],{"class":10276,"line":10367},[40,14160,14161],{"class":10973},"                          wrap_chars",[40,14163,10443],{"class":10363},[40,14165,10821],{"class":10732},[40,14167,10507],{"class":10363},[40,14169,14170,14173],{"class":10276,"line":10372},[40,14171,14172],{"class":10973},"                          out_path",[40,14174,12941],{"class":10363},[40,14176,14177],{"class":10276,"line":10387},[40,14178,14179],{"class":10287},"    \"\"\"Animate Wigner under an arbitrary axis–angle rotation 0 → angle_deg.\"\"\"\n",[40,14181,14182,14184,14186,14188,14190,14192],{"class":10276,"line":10400},[40,14183,12951],{"class":10352},[40,14185,12954],{"class":10363},[40,14187,12957],{"class":10732},[40,14189,12960],{"class":10446},[40,14191,10484],{"class":10363},[40,14193,12965],{"class":10287},[40,14195,14196,14198,14200,14202],{"class":10276,"line":10413},[40,14197,12970],{"class":10363},[40,14199,162],{"class":10732},[40,14201,11811],{"class":10430},[40,14203,12977],{"class":10363},[40,14205,14206,14208,14210,14212,14214,14216,14218,14220,14222,14224,14226,14228],{"class":10276,"line":10418},[40,14207,12982],{"class":10363},[40,14209,162],{"class":10732},[40,14211,12039],{"class":10446},[40,14213,11764],{"class":10732},[40,14215,12045],{"class":10363},[40,14217,2218],{"class":10430},[40,14219,167],{"class":10363},[40,14221,12052],{"class":10446},[40,14223,12055],{"class":10363},[40,14225,11243],{"class":10732},[40,14227,12060],{"class":10446},[40,14229,10761],{"class":10363},[40,14231,14232,14234,14236,14238,14240,14242,14244,14246,14248,14250,14252,14254],{"class":10276,"line":10424},[40,14233,13009],{"class":10363},[40,14235,162],{"class":10732},[40,14237,12039],{"class":10446},[40,14239,12042],{"class":10732},[40,14241,12045],{"class":10363},[40,14243,2218],{"class":10430},[40,14245,167],{"class":10363},[40,14247,12052],{"class":10446},[40,14249,12055],{"class":10363},[40,14251,11243],{"class":10732},[40,14253,12060],{"class":10446},[40,14255,10761],{"class":10363},[40,14257,14258,14260,14262,14264,14266,14268,14270,14272,14274,14276],{"class":10276,"line":10437},[40,14259,13036],{"class":10363},[40,14261,162],{"class":10732},[40,14263,920],{"class":10363},[40,14265,10801],{"class":10446},[40,14267,10484],{"class":10363},[40,14269,11878],{"class":10446},[40,14271,11881],{"class":10732},[40,14273,13051],{"class":10363},[40,14275,10801],{"class":10446},[40,14277,13056],{"class":10363},[40,14279,14280],{"class":10276,"line":10456},[40,14281,10301],{"emptyLinePlaceholder":10300},[40,14283,14284,14286,14288,14290,14292,14294,14296,14298,14300,14302,14304,14306],{"class":10276,"line":10472},[40,14285,13065],{"class":10363},[40,14287,162],{"class":10732},[40,14289,13070],{"class":10363},[40,14291,13073],{"class":10430},[40,14293,167],{"class":10363},[40,14295,13078],{"class":11903},[40,14297,162],{"class":10732},[40,14299,13083],{"class":10363},[40,14301,13086],{"class":11903},[40,14303,162],{"class":10732},[40,14305,13091],{"class":10446},[40,14307,10761],{"class":10363},[40,14309,14310,14312,14314,14316,14318,14320,14322,14324,14326,14328,14330,14332,14334,14336],{"class":10276,"line":10496},[40,14311,13098],{"class":10363},[40,14313,162],{"class":10732},[40,14315,12522],{"class":10363},[40,14317,13105],{"class":10430},[40,14319,13108],{"class":10363},[40,14321,13111],{"class":10430},[40,14323,13114],{"class":10363},[40,14325,13117],{"class":11903},[40,14327,162],{"class":10732},[40,14329,13122],{"class":10287},[40,14331,10484],{"class":10363},[40,14333,13127],{"class":11903},[40,14335,162],{"class":10732},[40,14337,13132],{"class":10363},[40,14339,14340,14342,14344,14346,14348,14350,14352,14354,14356,14358],{"class":10276,"line":10510},[40,14341,13137],{"class":11903},[40,14343,162],{"class":10732},[40,14345,13142],{"class":10363},[40,14347,13145],{"class":11903},[40,14349,162],{"class":10732},[40,14351,13150],{"class":10363},[40,14353,13153],{"class":11903},[40,14355,162],{"class":10732},[40,14357,13158],{"class":10287},[40,14359,10761],{"class":10363},[40,14361,14362,14364,14366,14368,14370,14372,14374,14376,14378,14380,14382,14384,14386,14388,14390,14392],{"class":10276,"line":10696},[40,14363,13165],{"class":10363},[40,14365,162],{"class":10732},[40,14367,13170],{"class":10363},[40,14369,13173],{"class":10430},[40,14371,13176],{"class":10363},[40,14373,12429],{"class":11903},[40,14375,162],{"class":10732},[40,14377,13183],{"class":10363},[40,14379,13186],{"class":11903},[40,14381,162],{"class":10732},[40,14383,13191],{"class":10446},[40,14385,10484],{"class":10363},[40,14387,13196],{"class":11903},[40,14389,162],{"class":10732},[40,14391,13201],{"class":10446},[40,14393,10761],{"class":10363},[40,14395,14396,14398,14400,14402,14404],{"class":10276,"line":10702},[40,14397,13208],{"class":10363},[40,14399,13211],{"class":10430},[40,14401,167],{"class":10363},[40,14403,13216],{"class":10287},[40,14405,10761],{"class":10363},[40,14407,14408],{"class":10276,"line":10708},[40,14409,10301],{"emptyLinePlaceholder":10300},[40,14411,14412,14414,14416,14418,14420],{"class":10276,"line":10714},[40,14413,13227],{"class":10363},[40,14415,13230],{"class":10430},[40,14417,167],{"class":10363},[40,14419,13235],{"class":10287},[40,14421,10761],{"class":10363},[40,14423,14424,14426,14428,14430,14432],{"class":10276,"line":10720},[40,14425,13227],{"class":10363},[40,14427,13244],{"class":10430},[40,14429,167],{"class":10363},[40,14431,13249],{"class":10287},[40,14433,10761],{"class":10363},[40,14435,14436,14438,14440,14443],{"class":10276,"line":10726},[40,14437,13256],{"class":10430},[40,14439,13259],{"class":10363},[40,14441,14442],{"class":10287},"\"Custom rotation\"",[40,14444,10761],{"class":10363},[40,14446,14447],{"class":10276,"line":10764},[40,14448,10301],{"emptyLinePlaceholder":10300},[40,14450,14451,14453,14455,14457,14459,14461,14463,14465,14467,14469,14471,14473],{"class":10276,"line":10778},[40,14452,13294],{"class":10363},[40,14454,162],{"class":10732},[40,14456,12424],{"class":10430},[40,14458,13259],{"class":10363},[40,14460,13303],{"class":10287},[40,14462,10484],{"class":10363},[40,14464,12131],{"class":11903},[40,14466,162],{"class":10732},[40,14468,13312],{"class":10363},[40,14470,12553],{"class":11903},[40,14472,162],{"class":10732},[40,14474,13319],{"class":10363},[40,14476,14477,14479,14481],{"class":10276,"line":10815},[40,14478,13324],{"class":11903},[40,14480,162],{"class":10732},[40,14482,13329],{"class":10363},[40,14484,14485],{"class":10276,"line":10829},[40,14486,10301],{"emptyLinePlaceholder":10300},[40,14488,14489,14491,14493,14495,14497,14499,14501],{"class":10276,"line":10842},[40,14490,13338],{"class":10352},[40,14492,13341],{"class":10430},[40,14494,167],{"class":10363},[40,14496,13346],{"class":10973},[40,14498,10443],{"class":10363},[40,14500,10821],{"class":10732},[40,14502,13353],{"class":10363},[40,14504,14505,14507,14509,14511,14513,14515,14517,14519,14521,14523,14525,14528],{"class":10276,"line":10855},[40,14506,13358],{"class":10363},[40,14508,162],{"class":10732},[40,14510,13363],{"class":10363},[40,14512,11243],{"class":10732},[40,14514,13368],{"class":10363},[40,14516,11056],{"class":10732},[40,14518,13373],{"class":10446},[40,14520,13376],{"class":10363},[40,14522,11752],{"class":10732},[40,14524,11156],{"class":10363},[40,14526,14527],{"class":10430},"deg2rad",[40,14529,14530],{"class":10363},"(angle_deg)\n",[40,14532,14533,14536,14538,14540],{"class":10276,"line":10868},[40,14534,14535],{"class":10363},"        R ",[40,14537,162],{"class":10732},[40,14539,11522],{"class":10430},[40,14541,14542],{"class":10363},"(axis_vec, ang)\n",[40,14544,14545,14547,14549,14552,14554],{"class":10276,"line":10881},[40,14546,13389],{"class":10363},[40,14548,162],{"class":10732},[40,14550,14551],{"class":10363}," R ",[40,14553,13404],{"class":10732},[40,14555,13407],{"class":10363},[40,14557,14558,14560,14562,14564,14566],{"class":10276,"line":10907},[40,14559,13412],{"class":10363},[40,14561,13415],{"class":10430},[40,14563,167],{"class":10363},[40,14565,13420],{"class":10430},[40,14567,13423],{"class":10363},[40,14569,14570,14572,14574],{"class":10276,"line":10920},[40,14571,13428],{"class":10363},[40,14573,13431],{"class":10430},[40,14575,12527],{"class":10363},[40,14577,14578,14581,14584,14586,14589,14591,14593,14596,14598,14600,14602,14604,14606,14608,14610,14612,14614,14616,14618,14620,14622,14624,14626,14628,14630],{"class":10276,"line":10933},[40,14579,14580],{"class":10352},"            f",[40,14582,14583],{"class":10287},"\"axis = (",[40,14585,13268],{"class":10446},[40,14587,14588],{"class":10363},"axis_vec[",[40,14590,66],{"class":10446},[40,14592,1664],{"class":10363},[40,14594,14595],{"class":10352},":.2f",[40,14597,13280],{"class":10446},[40,14599,10484],{"class":10287},[40,14601,13268],{"class":10446},[40,14603,14588],{"class":10363},[40,14605,78],{"class":10446},[40,14607,1664],{"class":10363},[40,14609,14595],{"class":10352},[40,14611,13280],{"class":10446},[40,14613,10484],{"class":10287},[40,14615,13268],{"class":10446},[40,14617,14588],{"class":10363},[40,14619,1044],{"class":10446},[40,14621,1664],{"class":10363},[40,14623,14595],{"class":10352},[40,14625,13280],{"class":10446},[40,14627,207],{"class":10287},[40,14629,13454],{"class":10732},[40,14631,14632],{"class":10287},"\"\n",[40,14634,14635,14637,14639,14641,14644,14647,14650,14653,14655,14658,14660,14663,14665,14667],{"class":10276,"line":10946},[40,14636,14580],{"class":10352},[40,14638,13438],{"class":10287},[40,14640,13268],{"class":10446},[40,14642,14643],{"class":10363},"np.",[40,14645,14646],{"class":10430},"rad2deg",[40,14648,14649],{"class":10363},"(ang)",[40,14651,14652],{"class":10352},":.1f",[40,14654,13280],{"class":10446},[40,14656,14657],{"class":10287},"° \u002F ",[40,14659,13268],{"class":10446},[40,14661,14662],{"class":10363},"angle_deg",[40,14664,14652],{"class":10352},[40,14666,13280],{"class":10446},[40,14668,14669],{"class":10287},"°\"\n",[40,14671,14672],{"class":10276,"line":10951},[40,14673,14674],{"class":10363},"        )\n",[40,14676,14677,14679],{"class":10276,"line":10956},[40,14678,11339],{"class":10352},[40,14680,13487],{"class":10363},[40,14682,14683],{"class":10276,"line":10962},[40,14684,10301],{"emptyLinePlaceholder":10300},[40,14686,14687,14689,14691,14693,14695,14697,14699,14701,14703,14705,14707,14709,14711,14713,14715],{"class":10276,"line":10983},[40,14688,13496],{"class":10363},[40,14690,162],{"class":10732},[40,14692,13501],{"class":10430},[40,14694,13504],{"class":10363},[40,14696,13473],{"class":11903},[40,14698,162],{"class":10732},[40,14700,13511],{"class":10363},[40,14702,13514],{"class":11903},[40,14704,162],{"class":10732},[40,14706,13519],{"class":10446},[40,14708,10484],{"class":10363},[40,14710,13524],{"class":11903},[40,14712,162],{"class":10732},[40,14714,10504],{"class":10446},[40,14716,10761],{"class":10363},[40,14718,14719,14721,14723,14725,14727,14729,14731,14733,14735,14737,14739],{"class":10276,"line":11005},[40,14720,13535],{"class":10363},[40,14722,13538],{"class":10430},[40,14724,167],{"class":10363},[40,14726,13543],{"class":11903},[40,14728,162],{"class":10732},[40,14730,13091],{"class":10446},[40,14732,10484],{"class":10363},[40,14734,13552],{"class":11903},[40,14736,162],{"class":10732},[40,14738,13091],{"class":10446},[40,14740,10761],{"class":10363},[40,14742,14743,14745,14747,14749,14751,14753,14755,14757,14759,14761,14763,14765],{"class":10276,"line":11011},[40,14744,13563],{"class":10363},[40,14746,13566],{"class":10430},[40,14748,13569],{"class":10363},[40,14750,13572],{"class":10430},[40,14752,13575],{"class":10363},[40,14754,13578],{"class":11903},[40,14756,162],{"class":10732},[40,14758,13583],{"class":10430},[40,14760,167],{"class":10363},[40,14762,13588],{"class":11903},[40,14764,162],{"class":10732},[40,14766,13593],{"class":10363},[40,14768,14769,14771,14773,14775,14777,14779,14781],{"class":10276,"line":11022},[40,14770,13598],{"class":11903},[40,14772,12577],{"class":10732},[40,14774,167],{"class":10363},[40,14776,13605],{"class":11903},[40,14778,162],{"class":10732},[40,14780,12597],{"class":10287},[40,14782,13612],{"class":10363},[40,14784,14785,14787,14789],{"class":10276,"line":11064},[40,14786,13617],{"class":10363},[40,14788,13620],{"class":10430},[40,14790,13623],{"class":10363},[40,14792,14793,14795],{"class":10276,"line":11103},[40,14794,11201],{"class":10352},[40,14796,13630],{"class":10363},[40,14798,14799],{"class":10276,"line":11142},[40,14800,10301],{"emptyLinePlaceholder":10300},[40,14802,14803],{"class":10276,"line":11148},[40,14804,10301],{"emptyLinePlaceholder":10300},[40,14806,14807],{"class":10276,"line":11181},[40,14808,14809],{"class":10280},"# Example: 45° about the y-axis, starting from |+z⟩\n",[40,14811,14812,14815,14817,14819,14821,14823],{"class":10276,"line":11198},[40,14813,14814],{"class":10363},"custom_out ",[40,14816,162],{"class":10732},[40,14818,13859],{"class":10430},[40,14820,13862],{"class":10363},[40,14822,11243],{"class":10732},[40,14824,14825],{"class":10287}," \"wigner_custom_y45.gif\"\n",[40,14827,14828,14831],{"class":10276,"line":11249},[40,14829,14830],{"class":10430},"animate_wigner_custom",[40,14832,12527],{"class":10363},[40,14834,14835,14838,14840,14842,14844,14846,14848,14850,14852],{"class":10276,"line":11254},[40,14836,14837],{"class":11903},"    axis_vec",[40,14839,162],{"class":10732},[40,14841,167],{"class":10363},[40,14843,10801],{"class":10446},[40,14845,10484],{"class":10363},[40,14847,10810],{"class":10446},[40,14849,10484],{"class":10363},[40,14851,10801],{"class":10446},[40,14853,11061],{"class":10363},[40,14855,14856,14859,14861,14864],{"class":10276,"line":11259},[40,14857,14858],{"class":11903},"    angle_deg",[40,14860,162],{"class":10732},[40,14862,14863],{"class":10446},"45.0",[40,14865,10507],{"class":10363},[40,14867,14868,14871,14873],{"class":10276,"line":11265},[40,14869,14870],{"class":11903},"    r0",[40,14872,162],{"class":10732},[40,14874,14875],{"class":10363},"r0,\n",[40,14877,14878,14880,14882,14884,14887],{"class":10276,"line":11294},[40,14879,13696],{"class":11903},[40,14881,162],{"class":10732},[40,14883,13701],{"class":10446},[40,14885,14886],{"class":10363},",                 ",[40,14888,14889],{"class":10280},"# raise to 48 if you prefer\n",[40,14891,14892,14894,14896,14898,14900,14902,14904,14906],{"class":10276,"line":11300},[40,14893,13712],{"class":11903},[40,14895,162],{"class":10732},[40,14897,13717],{"class":10446},[40,14899,10484],{"class":10363},[40,14901,11825],{"class":11903},[40,14903,162],{"class":10732},[40,14905,13726],{"class":10446},[40,14907,10507],{"class":10363},[40,14909,14910,14912,14914,14916],{"class":10276,"line":11320},[40,14911,13737],{"class":11903},[40,14913,162],{"class":10732},[40,14915,13742],{"class":10446},[40,14917,10507],{"class":10363},[40,14919,14920,14922,14924],{"class":10276,"line":11336},[40,14921,13761],{"class":11903},[40,14923,162],{"class":10732},[40,14925,14926],{"class":10363},"knobs.figsize,\n",[40,14928,14929,14931,14933,14936,14938,14940],{"class":10276,"line":11381},[40,14930,13778],{"class":11903},[40,14932,162],{"class":10732},[40,14934,14935],{"class":10363},"knobs.info_loc, ",[40,14937,13788],{"class":11903},[40,14939,162],{"class":10732},[40,14941,14942],{"class":10363},"knobs.info_fontsize,\n",[40,14944,14945,14948,14950],{"class":10276,"line":11395},[40,14946,14947],{"class":11903},"    wrap_chars",[40,14949,162],{"class":10732},[40,14951,14952],{"class":10363},"knobs.wrap_chars,\n",[40,14954,14955,14958,14960],{"class":10276,"line":11438},[40,14956,14957],{"class":11903},"    out_path",[40,14959,162],{"class":10732},[40,14961,14962],{"class":10363},"custom_out,\n",[40,14964,14965],{"class":10276,"line":11452},[40,14966,10761],{"class":10363},[40,14968,14969],{"class":10276,"line":11493},[40,14970,10301],{"emptyLinePlaceholder":10300},[40,14972,14973,14976,14978,14980,14982,14984,14986],{"class":10276,"line":11507},[40,14974,14975],{"class":10430},"display",[40,14977,167],{"class":10363},[40,14979,13953],{"class":10430},[40,14981,167],{"class":10363},[40,14983,13958],{"class":11903},[40,14985,13961],{"class":10732},[40,14987,14988],{"class":10363},"(custom_out)))\n",[40,14990,14991,14994,14996,14998],{"class":10276,"line":11512},[40,14992,14993],{"class":10732},"print",[40,14995,167],{"class":10363},[40,14997,13974],{"class":10287},[40,14999,15000],{"class":10363},", custom_out)\n",[10082,15002,15005],{"className":15003,"code":15004,"language":2932,"meta":10090},[10085],"Saved: \u002Fcontent\u002Fwigner_custom_y45.gif\n",[10088,15006,15004],{"__ignoreMap":10090},[13997,15008],{"alt":15009,"src":15010},"Output 4 of the notebook","\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-04.webp",[12,15012,15013],{},"The next following scripts are for the 3D Wigner plots.",[10082,15015,15017],{"className":10269,"code":15016,"language":10271,"meta":10090,"style":10090},"# @title Cell 5 — 3D Wigner animations for ALL Pauli gates (X, Y, Z)\n# PEP 8\u002F257 compliant. Stable: updates facecolors in place; no deleting artists.\n\nfrom __future__ import annotations\n\nfrom dataclasses import dataclass\nfrom pathlib import Path\nfrom typing import Tuple, Iterable\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib.animation import FuncAnimation, PillowWriter\nfrom matplotlib.colors import Normalize\nfrom matplotlib.cm import ScalarMappable\nfrom IPython.display import Image, display\n\n\n@dataclass\nclass Knobs3D:\n    \"\"\"User controls for 3D Pauli-gate Wigner animations on a colored sphere.\n\n    Attributes:\n        initial_state: Starting Bloch eigenstate label or \"custom\".\n        custom_r: Custom Bloch vector if initial_state == \"custom\".\n        frames: Number of frames (≥ 2) for 0 → π rotation.\n        n_theta: Polar grid size for the sphere.\n        n_phi: Azimuth grid size for the sphere.\n        fps: GIF framerate.\n        elev: Camera elevation in degrees.\n        azim: Camera azimuth in degrees.\n        out_dir: Output directory (e.g., \"\u002Fcontent\" on Colab).\n        figsize: Figure size in inches (DPI set globally by rcParams).\n        info_fontsize: In-axes info text font size.\n    \"\"\"\n    initial_state: str = \"+z\"\n    custom_r: Tuple[float, float, float] = (0.0, 0.0, 1.0)\n    frames: int = 24\n    n_theta: int = 64\n    n_phi: int = 128\n    fps: int = 20\n    elev: float = 25.0\n    azim: float = -60.0\n    out_dir: str = \"\u002Fcontent\"\n    figsize: Tuple[float, float] = (6.4, 4.8)\n    info_fontsize: int = 9\n\n\n# ----------------------------- Helpers (self-contained) -----------------------------\ndef _bloch_vector(label: str, custom=(0.0, 0.0, 1.0)) -> np.ndarray:\n    mapping = {\n        \"+z\": (0.0, 0.0, 1.0), \"-z\": (0.0, 0.0, -1.0),\n        \"+x\": (1.0, 0.0, 0.0), \"-x\": (-1.0, 0.0, 0.0),\n        \"+y\": (0.0, 1.0, 0.0), \"-y\": (0.0, -1.0, 0.0),\n    }\n    r = np.asarray(mapping.get(label.lower(), custom), float)\n    nrm = np.linalg.norm(r) or 1.0\n    return r \u002F nrm\n\n\ndef _rotation_matrix(axis: str, angle: float) -> np.ndarray:\n    c, s = np.cos(angle), np.sin(angle)\n    if axis == \"x\":\n        return np.array([[1, 0, 0], [0, c, -s], [0, s, c]], float)\n    if axis == \"y\":\n        return np.array([[c, 0, s], [0, 1, 0], [-s, 0, c]], float)\n    if axis == \"z\":\n        return np.array([[c, -s, 0], [s, c, 0], [0, 0, 1]], float)\n    raise ValueError(\"axis must be 'x', 'y', or 'z'\")\n\n\ndef _n_grid(n_theta: int, n_phi: int):\n    \"\"\"Return (θ, φ, n) with n stacked as (3, n_theta, n_phi).\"\"\"\n    theta = np.linspace(0.0, np.pi, n_theta)\n    phi = np.linspace(0.0, 2.0 * np.pi, n_phi)\n    th, ph = np.meshgrid(theta, phi, indexing=\"ij\")\n    nx = np.sin(th) * np.cos(ph)\n    ny = np.sin(th) * np.sin(ph)\n    nz = np.cos(th)\n    n = np.stack((nx, ny, nz), axis=0)\n    return th, ph, n\n\n\ndef _wigner_qubit(r_vec: np.ndarray, n: np.ndarray) -> np.ndarray:\n    \"\"\"W(θ, φ) = 1\u002F2 + (√3\u002F2) * r·n on the (θ, φ) grid.\"\"\"\n    return 0.5 + (np.sqrt(3.0) \u002F 2.0) * (r_vec.reshape(3, 1, 1) * n).sum(axis=0)\n\n\ndef _animate_one_pauli(axis: str, k: Knobs3D) -> Path:\n    \"\"\"Render one 3D Wigner animation for a chosen Pauli axis; return output path.\"\"\"\n    r0 = _bloch_vector(k.initial_state, k.custom_r)\n    th, ph, n = _n_grid(k.n_theta, k.n_phi)\n\n    # Unit sphere geometry\n    X = np.sin(th) * np.cos(ph)\n    Y = np.sin(th) * np.sin(ph)\n    Z = np.cos(th)\n\n    # Fixed normalization (pure-state bounds) for constant color scale\n    w_min = 0.5 - (np.sqrt(3.0) \u002F 2.0)\n    w_max = 0.5 + (np.sqrt(3.0) \u002F 2.0)\n    norm = Normalize(vmin=w_min, vmax=w_max)\n    cmap = plt.get_cmap()\n    sm = ScalarMappable(norm=norm, cmap=cmap)\n    sm.set_array([])  # required for colorbar from a ScalarMappable\n\n    fig = plt.figure(figsize=k.figsize, constrained_layout=True)\n    ax = fig.add_subplot(111, projection=\"3d\")\n    ax.view_init(elev=k.elev, azim=k.azim)\n    if hasattr(ax, \"set_box_aspect\"):\n        ax.set_box_aspect((1, 1, 1))\n\n    # Initial facecolors (use cell-centered (M-1, N-1) colors for quads)\n    W0 = _wigner_qubit(_rotation_matrix(axis, 0.0) @ r0, n)\n    FC0 = cmap(norm(W0[:-1, :-1]))  # shape: (n_theta-1, n_phi-1, 4)\n\n    surf = ax.plot_surface(\n        X, Y, Z,\n        facecolors=FC0,\n        rstride=1, cstride=1,\n        antialiased=False, linewidth=0,\n        shade=False,  # use given facecolors directly\n    )\n\n    # Colorbar (bound to ScalarMappable using same norm+cmap)\n    cbar = fig.colorbar(sm, ax=ax, fraction=0.046, pad=0.04)\n    cbar.set_label(\"Wigner value\")\n\n    # Labels and in-axes title and info\n    ax.set_xlabel(\"x\")\n    ax.set_ylabel(\"y\")\n    ax.set_zlabel(\"z\")\n    title_txt = ax.text2D(\n        0.5, 0.98, f\"Wigner — {axis.upper()} rotation (3D sphere)\",\n        transform=ax.transAxes, ha=\"center\", va=\"top\", fontsize=11,\n        bbox=dict(boxstyle=\"round,pad=0.3\", fc=\"white\", alpha=0.75, ec=\"none\"),\n        zorder=6,\n    )\n    info_txt = ax.text2D(\n        0.02, 0.96, \"\", transform=ax.transAxes, ha=\"left\", va=\"top\",\n        fontsize=k.info_fontsize,\n        bbox=dict(boxstyle=\"round,pad=0.3\", fc=\"white\", alpha=0.75, ec=\"none\"),\n        zorder=6,\n    )\n\n    def update(frame: int):\n        angle = (frame \u002F (k.frames - 1)) * np.pi  # 0 → π\n        r_now = _rotation_matrix(axis, angle) @ r0\n        W_now = _wigner_qubit(r_now, n)\n        FC_now = cmap(norm(W_now[:-1, :-1]))        # (M-1, N-1, 4)\n        surf.set_facecolors(FC_now.reshape(-1, 4))  # update in place\n        info_txt.set_text(f\"θ = {angle:.3f} rad\\nframe {frame+1}\u002F{k.frames}\")\n        return (surf, info_txt, title_txt)\n\n    anim = FuncAnimation(fig, update, frames=k.frames, interval=60, blit=False)\n    out = Path(k.out_dir) \u002F f\"wigner3d_pauli_{axis}.gif\"\n    out.parent.mkdir(parents=True, exist_ok=True)\n    anim.save(out.as_posix(), writer=PillowWriter(fps=k.fps),\n              savefig_kwargs=dict(facecolor=\"white\"))\n    plt.close(fig)\n    return out\n\n\ndef animate_all_paulis_3d(k: Knobs3D,\n                          axes: Tuple[str, ...] = (\"x\", \"y\", \"z\")) -> Iterable[Path]:\n    \"\"\"Generate and display GIFs for all requested Pauli axes; yield file paths.\"\"\"\n    for axname in axes:\n        path = _animate_one_pauli(axname.lower(), k)\n        display(Image(filename=str(path)))\n        print(f\"Saved: {path}\")\n        yield path\n\n\n# ------------------------------ Run all three gates ------------------------------\nK3D = Knobs3D(  # adjust as desired\n    initial_state=\"+z\",\n    frames=24, n_theta=64, n_phi=128, fps=20,\n    elev=25.0, azim=-60.0,\n    out_dir=\"\u002Fcontent\",\n    figsize=(6.4, 4.8),\n    info_fontsize=9,\n)\n\nlist(animate_all_paulis_3d(K3D, axes=(\"x\", \"y\", \"z\")))\n",[10088,15018,15019,15024,15029,15033,15043,15047,15057,15067,15077,15081,15091,15101,15111,15123,15135,15145,15149,15153,15157,15166,15171,15175,15179,15184,15188,15193,15198,15203,15207,15212,15217,15222,15227,15232,15236,15246,15278,15288,15299,15310,15320,15332,15346,15356,15381,15391,15395,15399,15404,15441,15449,15485,15521,15557,15561,15587,15606,15616,15620,15624,15649,15665,15677,15715,15727,15765,15777,15815,15826,15830,15834,15859,15864,15880,15902,15922,15942,15962,15974,15995,16002,16006,16010,16027,16032,16093,16097,16101,16123,16128,16140,16152,16156,16161,16182,16203,16216,16220,16225,16251,16277,16303,16317,16344,16358,16362,16391,16420,16445,16459,16482,16486,16491,16517,16551,16555,16569,16574,16586,16606,16626,16641,16645,16649,16654,16689,16701,16705,16710,16722,16734,16747,16761,16791,16823,16863,16873,16877,16890,16929,16938,16978,16989,16994,16999,17017,17045,17062,17075,17108,17143,17192,17200,17205,17239,17267,17293,17322,17339,17348,17356,17361,17366,17381,17414,17420,17433,17451,17470,17493,17502,17507,17512,17518,17533,17544,17581,17604,17615,17632,17644,17649,17654],{"__ignoreMap":10090},[40,15020,15021],{"class":10276,"line":10277},[40,15022,15023],{"class":10280},"# @title Cell 5 — 3D Wigner animations for ALL Pauli gates (X, Y, Z)\n",[40,15025,15026],{"class":10276,"line":10284},[40,15027,15028],{"class":10280},"# PEP 8\u002F257 compliant. Stable: updates facecolors in place; no deleting artists.\n",[40,15030,15031],{"class":10276,"line":10291},[40,15032,10301],{"emptyLinePlaceholder":10300},[40,15034,15035,15037,15039,15041],{"class":10276,"line":10297},[40,15036,10353],{"class":10352},[40,15038,10357],{"class":10356},[40,15040,10360],{"class":10352},[40,15042,10364],{"class":10363},[40,15044,15045],{"class":10276,"line":10304},[40,15046,10301],{"emptyLinePlaceholder":10300},[40,15048,15049,15051,15053,15055],{"class":10276,"line":10310},[40,15050,10353],{"class":10352},[40,15052,10530],{"class":10363},[40,15054,10375],{"class":10352},[40,15056,10535],{"class":10363},[40,15058,15059,15061,15063,15065],{"class":10276,"line":10316},[40,15060,10353],{"class":10352},[40,15062,10542],{"class":10363},[40,15064,10375],{"class":10352},[40,15066,10547],{"class":10363},[40,15068,15069,15071,15073,15075],{"class":10276,"line":10322},[40,15070,10353],{"class":10352},[40,15072,10554],{"class":10363},[40,15074,10375],{"class":10352},[40,15076,10559],{"class":10363},[40,15078,15079],{"class":10276,"line":10327},[40,15080,10301],{"emptyLinePlaceholder":10300},[40,15082,15083,15085,15087,15089],{"class":10276,"line":10333},[40,15084,10375],{"class":10352},[40,15086,10378],{"class":10363},[40,15088,10381],{"class":10352},[40,15090,10384],{"class":10363},[40,15092,15093,15095,15097,15099],{"class":10276,"line":10339},[40,15094,10375],{"class":10352},[40,15096,10405],{"class":10363},[40,15098,10381],{"class":10352},[40,15100,10410],{"class":10363},[40,15102,15103,15105,15107,15109],{"class":10276,"line":10344},[40,15104,10353],{"class":10352},[40,15106,10597],{"class":10363},[40,15108,10375],{"class":10352},[40,15110,10602],{"class":10363},[40,15112,15113,15115,15118,15120],{"class":10276,"line":10349},[40,15114,10353],{"class":10352},[40,15116,15117],{"class":10363}," matplotlib.colors ",[40,15119,10375],{"class":10352},[40,15121,15122],{"class":10363}," Normalize\n",[40,15124,15125,15127,15130,15132],{"class":10276,"line":10367},[40,15126,10353],{"class":10352},[40,15128,15129],{"class":10363}," matplotlib.cm ",[40,15131,10375],{"class":10352},[40,15133,15134],{"class":10363}," ScalarMappable\n",[40,15136,15137,15139,15141,15143],{"class":10276,"line":10372},[40,15138,10353],{"class":10352},[40,15140,10609],{"class":10363},[40,15142,10375],{"class":10352},[40,15144,10614],{"class":10363},[40,15146,15147],{"class":10276,"line":10387},[40,15148,10301],{"emptyLinePlaceholder":10300},[40,15150,15151],{"class":10276,"line":10400},[40,15152,10301],{"emptyLinePlaceholder":10300},[40,15154,15155],{"class":10276,"line":10413},[40,15156,10627],{"class":10430},[40,15158,15159,15161,15164],{"class":10276,"line":10418},[40,15160,10632],{"class":10352},[40,15162,15163],{"class":10635}," Knobs3D",[40,15165,10639],{"class":10363},[40,15167,15168],{"class":10276,"line":10424},[40,15169,15170],{"class":10287},"    \"\"\"User controls for 3D Pauli-gate Wigner animations on a colored sphere.\n",[40,15172,15173],{"class":10276,"line":10437},[40,15174,10301],{"emptyLinePlaceholder":10300},[40,15176,15177],{"class":10276,"line":10456},[40,15178,10653],{"class":10287},[40,15180,15181],{"class":10276,"line":10472},[40,15182,15183],{"class":10287},"        initial_state: Starting Bloch eigenstate label or \"custom\".\n",[40,15185,15186],{"class":10276,"line":10496},[40,15187,10668],{"class":10287},[40,15189,15190],{"class":10276,"line":10510},[40,15191,15192],{"class":10287},"        frames: Number of frames (≥ 2) for 0 → π rotation.\n",[40,15194,15195],{"class":10276,"line":10696},[40,15196,15197],{"class":10287},"        n_theta: Polar grid size for the sphere.\n",[40,15199,15200],{"class":10276,"line":10702},[40,15201,15202],{"class":10287},"        n_phi: Azimuth grid size for the sphere.\n",[40,15204,15205],{"class":10276,"line":10708},[40,15206,10688],{"class":10287},[40,15208,15209],{"class":10276,"line":10714},[40,15210,15211],{"class":10287},"        elev: Camera elevation in degrees.\n",[40,15213,15214],{"class":10276,"line":10720},[40,15215,15216],{"class":10287},"        azim: Camera azimuth in degrees.\n",[40,15218,15219],{"class":10276,"line":10726},[40,15220,15221],{"class":10287},"        out_dir: Output directory (e.g., \"\u002Fcontent\" on Colab).\n",[40,15223,15224],{"class":10276,"line":10764},[40,15225,15226],{"class":10287},"        figsize: Figure size in inches (DPI set globally by rcParams).\n",[40,15228,15229],{"class":10276,"line":10778},[40,15230,15231],{"class":10287},"        info_fontsize: In-axes info text font size.\n",[40,15233,15234],{"class":10276,"line":10815},[40,15235,10723],{"class":10287},[40,15237,15238,15240,15242,15244],{"class":10276,"line":10829},[40,15239,10767],{"class":10363},[40,15241,10733],{"class":10732},[40,15243,10772],{"class":10732},[40,15245,10775],{"class":10287},[40,15247,15248,15250,15252,15254,15256,15258,15260,15262,15264,15266,15268,15270,15272,15274,15276],{"class":10276,"line":10842},[40,15249,10781],{"class":10363},[40,15251,10784],{"class":10732},[40,15253,10484],{"class":10363},[40,15255,10784],{"class":10732},[40,15257,10484],{"class":10363},[40,15259,10784],{"class":10732},[40,15261,10741],{"class":10363},[40,15263,162],{"class":10732},[40,15265,920],{"class":10363},[40,15267,10801],{"class":10446},[40,15269,10484],{"class":10363},[40,15271,10801],{"class":10446},[40,15273,10484],{"class":10363},[40,15275,10810],{"class":10446},[40,15277,10761],{"class":10363},[40,15279,15280,15282,15284,15286],{"class":10276,"line":10855},[40,15281,10818],{"class":10363},[40,15283,10821],{"class":10732},[40,15285,10772],{"class":10732},[40,15287,10826],{"class":10446},[40,15289,15290,15292,15294,15296],{"class":10276,"line":10868},[40,15291,10832],{"class":10363},[40,15293,10821],{"class":10732},[40,15295,10772],{"class":10732},[40,15297,15298],{"class":10446}," 64\n",[40,15300,15301,15303,15305,15307],{"class":10276,"line":10881},[40,15302,10845],{"class":10363},[40,15304,10821],{"class":10732},[40,15306,10772],{"class":10732},[40,15308,15309],{"class":10446}," 128\n",[40,15311,15312,15314,15316,15318],{"class":10276,"line":10907},[40,15313,10858],{"class":10363},[40,15315,10821],{"class":10732},[40,15317,10772],{"class":10732},[40,15319,10865],{"class":10446},[40,15321,15322,15325,15327,15329],{"class":10276,"line":10920},[40,15323,15324],{"class":10363},"    elev: ",[40,15326,10784],{"class":10732},[40,15328,10772],{"class":10732},[40,15330,15331],{"class":10446}," 25.0\n",[40,15333,15334,15337,15339,15341,15343],{"class":10276,"line":10933},[40,15335,15336],{"class":10363},"    azim: ",[40,15338,10784],{"class":10732},[40,15340,10772],{"class":10732},[40,15342,11764],{"class":10732},[40,15344,15345],{"class":10446},"60.0\n",[40,15347,15348,15350,15352,15354],{"class":10276,"line":10946},[40,15349,10871],{"class":10363},[40,15351,10733],{"class":10732},[40,15353,10772],{"class":10732},[40,15355,10878],{"class":10287},[40,15357,15358,15360,15362,15364,15366,15368,15370,15372,15374,15376,15379],{"class":10276,"line":10951},[40,15359,10884],{"class":10363},[40,15361,10784],{"class":10732},[40,15363,10484],{"class":10363},[40,15365,10784],{"class":10732},[40,15367,10741],{"class":10363},[40,15369,162],{"class":10732},[40,15371,920],{"class":10363},[40,15373,10481],{"class":10446},[40,15375,10484],{"class":10363},[40,15377,15378],{"class":10446},"4.8",[40,15380,10761],{"class":10363},[40,15382,15383,15385,15387,15389],{"class":10276,"line":10956},[40,15384,10923],{"class":10363},[40,15386,10821],{"class":10732},[40,15388,10772],{"class":10732},[40,15390,10930],{"class":10446},[40,15392,15393],{"class":10276,"line":10962},[40,15394,10301],{"emptyLinePlaceholder":10300},[40,15396,15397],{"class":10276,"line":10983},[40,15398,10301],{"emptyLinePlaceholder":10300},[40,15400,15401],{"class":10276,"line":11005},[40,15402,15403],{"class":10280},"# ----------------------------- Helpers (self-contained) -----------------------------\n",[40,15405,15406,15408,15411,15413,15416,15418,15420,15422,15425,15428,15430,15432,15434,15436,15438],{"class":10276,"line":11011},[40,15407,10965],{"class":10352},[40,15409,15410],{"class":10430}," _bloch_vector",[40,15412,167],{"class":10363},[40,15414,15415],{"class":10973},"label",[40,15417,10443],{"class":10363},[40,15419,10733],{"class":10732},[40,15421,10484],{"class":10363},[40,15423,15424],{"class":10973},"custom",[40,15426,15427],{"class":10363},"=(",[40,15429,10801],{"class":10446},[40,15431,10484],{"class":10363},[40,15433,10801],{"class":10446},[40,15435,10484],{"class":10363},[40,15437,10810],{"class":10446},[40,15439,15440],{"class":10363},")) -> np.ndarray:\n",[40,15442,15443,15445,15447],{"class":10276,"line":11022},[40,15444,11014],{"class":10363},[40,15446,162],{"class":10732},[40,15448,11019],{"class":10363},[40,15450,15451,15453,15455,15457,15459,15461,15463,15465,15467,15469,15471,15473,15475,15477,15479,15481,15483],{"class":10276,"line":11064},[40,15452,11025],{"class":10287},[40,15454,10478],{"class":10363},[40,15456,10801],{"class":10446},[40,15458,10484],{"class":10363},[40,15460,10801],{"class":10446},[40,15462,10484],{"class":10363},[40,15464,10810],{"class":10446},[40,15466,11040],{"class":10363},[40,15468,11043],{"class":10287},[40,15470,10478],{"class":10363},[40,15472,10801],{"class":10446},[40,15474,10484],{"class":10363},[40,15476,10801],{"class":10446},[40,15478,10484],{"class":10363},[40,15480,11056],{"class":10732},[40,15482,10810],{"class":10446},[40,15484,11061],{"class":10363},[40,15486,15487,15489,15491,15493,15495,15497,15499,15501,15503,15505,15507,15509,15511,15513,15515,15517,15519],{"class":10276,"line":11103},[40,15488,11067],{"class":10287},[40,15490,10478],{"class":10363},[40,15492,10810],{"class":10446},[40,15494,10484],{"class":10363},[40,15496,10801],{"class":10446},[40,15498,10484],{"class":10363},[40,15500,10801],{"class":10446},[40,15502,11040],{"class":10363},[40,15504,11084],{"class":10287},[40,15506,10478],{"class":10363},[40,15508,11056],{"class":10732},[40,15510,10810],{"class":10446},[40,15512,10484],{"class":10363},[40,15514,10801],{"class":10446},[40,15516,10484],{"class":10363},[40,15518,10801],{"class":10446},[40,15520,11061],{"class":10363},[40,15522,15523,15525,15527,15529,15531,15533,15535,15537,15539,15541,15543,15545,15547,15549,15551,15553,15555],{"class":10276,"line":11142},[40,15524,11106],{"class":10287},[40,15526,10478],{"class":10363},[40,15528,10801],{"class":10446},[40,15530,10484],{"class":10363},[40,15532,10810],{"class":10446},[40,15534,10484],{"class":10363},[40,15536,10801],{"class":10446},[40,15538,11040],{"class":10363},[40,15540,11123],{"class":10287},[40,15542,10478],{"class":10363},[40,15544,10801],{"class":10446},[40,15546,10484],{"class":10363},[40,15548,11056],{"class":10732},[40,15550,10810],{"class":10446},[40,15552,10484],{"class":10363},[40,15554,10801],{"class":10446},[40,15556,11061],{"class":10363},[40,15558,15559],{"class":10276,"line":11148},[40,15560,11145],{"class":10363},[40,15562,15563,15565,15567,15569,15571,15573,15575,15578,15580,15583,15585],{"class":10276,"line":11181},[40,15564,11151],{"class":10363},[40,15566,162],{"class":10732},[40,15568,11156],{"class":10363},[40,15570,11159],{"class":10430},[40,15572,11162],{"class":10363},[40,15574,11165],{"class":10430},[40,15576,15577],{"class":10363},"(label.",[40,15579,11171],{"class":10430},[40,15581,15582],{"class":10363},"(), custom), ",[40,15584,10784],{"class":10732},[40,15586,10761],{"class":10363},[40,15588,15589,15591,15593,15595,15597,15600,15603],{"class":10276,"line":11198},[40,15590,11184],{"class":10363},[40,15592,162],{"class":10732},[40,15594,11189],{"class":10363},[40,15596,11192],{"class":10430},[40,15598,15599],{"class":10363},"(r) ",[40,15601,15602],{"class":10352},"or",[40,15604,15605],{"class":10446}," 1.0\n",[40,15607,15608,15610,15612,15614],{"class":10276,"line":11249},[40,15609,11201],{"class":10352},[40,15611,11240],{"class":10363},[40,15613,11243],{"class":10732},[40,15615,11246],{"class":10363},[40,15617,15618],{"class":10276,"line":11254},[40,15619,10301],{"emptyLinePlaceholder":10300},[40,15621,15622],{"class":10276,"line":11259},[40,15623,10301],{"emptyLinePlaceholder":10300},[40,15625,15626,15628,15631,15633,15635,15637,15639,15641,15643,15645,15647],{"class":10276,"line":11265},[40,15627,10965],{"class":10352},[40,15629,15630],{"class":10430}," _rotation_matrix",[40,15632,167],{"class":10363},[40,15634,11275],{"class":10973},[40,15636,10443],{"class":10363},[40,15638,10733],{"class":10732},[40,15640,10484],{"class":10363},[40,15642,11284],{"class":10973},[40,15644,10443],{"class":10363},[40,15646,10784],{"class":10732},[40,15648,11291],{"class":10363},[40,15650,15651,15653,15655,15657,15659,15661,15663],{"class":10276,"line":11294},[40,15652,11303],{"class":10363},[40,15654,162],{"class":10732},[40,15656,11156],{"class":10363},[40,15658,1820],{"class":10430},[40,15660,11312],{"class":10363},[40,15662,1813],{"class":10430},[40,15664,11317],{"class":10363},[40,15666,15667,15669,15671,15673,15675],{"class":10276,"line":11300},[40,15668,11323],{"class":10352},[40,15670,11326],{"class":10363},[40,15672,11231],{"class":10732},[40,15674,11331],{"class":10287},[40,15676,10639],{"class":10363},[40,15678,15679,15681,15683,15685,15687,15689,15691,15693,15695,15697,15699,15701,15703,15705,15707,15709,15711,15713],{"class":10276,"line":11320},[40,15680,11339],{"class":10352},[40,15682,11156],{"class":10363},[40,15684,11206],{"class":10430},[40,15686,11346],{"class":10363},[40,15688,78],{"class":10446},[40,15690,10484],{"class":10363},[40,15692,66],{"class":10446},[40,15694,10484],{"class":10363},[40,15696,66],{"class":10446},[40,15698,11359],{"class":10363},[40,15700,66],{"class":10446},[40,15702,11364],{"class":10363},[40,15704,11056],{"class":10732},[40,15706,11369],{"class":10363},[40,15708,66],{"class":10446},[40,15710,11374],{"class":10363},[40,15712,10784],{"class":10732},[40,15714,10761],{"class":10363},[40,15716,15717,15719,15721,15723,15725],{"class":10276,"line":11336},[40,15718,11323],{"class":10352},[40,15720,11326],{"class":10363},[40,15722,11231],{"class":10732},[40,15724,11390],{"class":10287},[40,15726,10639],{"class":10363},[40,15728,15729,15731,15733,15735,15737,15739,15741,15743,15745,15747,15749,15751,15753,15755,15757,15759,15761,15763],{"class":10276,"line":11381},[40,15730,11339],{"class":10352},[40,15732,11156],{"class":10363},[40,15734,11206],{"class":10430},[40,15736,11404],{"class":10363},[40,15738,66],{"class":10446},[40,15740,11409],{"class":10363},[40,15742,66],{"class":10446},[40,15744,10484],{"class":10363},[40,15746,78],{"class":10446},[40,15748,10484],{"class":10363},[40,15750,66],{"class":10446},[40,15752,11359],{"class":10363},[40,15754,11056],{"class":10732},[40,15756,11426],{"class":10363},[40,15758,66],{"class":10446},[40,15760,11431],{"class":10363},[40,15762,10784],{"class":10732},[40,15764,10761],{"class":10363},[40,15766,15767,15769,15771,15773,15775],{"class":10276,"line":11395},[40,15768,11323],{"class":10352},[40,15770,11326],{"class":10363},[40,15772,11231],{"class":10732},[40,15774,11447],{"class":10287},[40,15776,10639],{"class":10363},[40,15778,15779,15781,15783,15785,15787,15789,15791,15793,15795,15797,15799,15801,15803,15805,15807,15809,15811,15813],{"class":10276,"line":11438},[40,15780,11339],{"class":10352},[40,15782,11156],{"class":10363},[40,15784,11206],{"class":10430},[40,15786,11404],{"class":10363},[40,15788,11056],{"class":10732},[40,15790,11426],{"class":10363},[40,15792,66],{"class":10446},[40,15794,11469],{"class":10363},[40,15796,66],{"class":10446},[40,15798,11359],{"class":10363},[40,15800,66],{"class":10446},[40,15802,10484],{"class":10363},[40,15804,66],{"class":10446},[40,15806,10484],{"class":10363},[40,15808,78],{"class":10446},[40,15810,11486],{"class":10363},[40,15812,10784],{"class":10732},[40,15814,10761],{"class":10363},[40,15816,15817,15819,15821,15824],{"class":10276,"line":11452},[40,15818,11496],{"class":10352},[40,15820,11499],{"class":10363},[40,15822,15823],{"class":10287},"\"axis must be 'x', 'y', or 'z'\"",[40,15825,10761],{"class":10363},[40,15827,15828],{"class":10276,"line":11493},[40,15829,10301],{"emptyLinePlaceholder":10300},[40,15831,15832],{"class":10276,"line":11507},[40,15833,10301],{"emptyLinePlaceholder":10300},[40,15835,15836,15838,15841,15843,15845,15847,15849,15851,15853,15855,15857],{"class":10276,"line":11512},[40,15837,10965],{"class":10352},[40,15839,15840],{"class":10430}," _n_grid",[40,15842,167],{"class":10363},[40,15844,11816],{"class":10973},[40,15846,10443],{"class":10363},[40,15848,10821],{"class":10732},[40,15850,10484],{"class":10363},[40,15852,11825],{"class":10973},[40,15854,10443],{"class":10363},[40,15856,10821],{"class":10732},[40,15858,13353],{"class":10363},[40,15860,15861],{"class":10276,"line":11517},[40,15862,15863],{"class":10287},"    \"\"\"Return (θ, φ, n) with n stacked as (3, n_theta, n_phi).\"\"\"\n",[40,15865,15866,15868,15870,15872,15874,15876,15878],{"class":10276,"line":11545},[40,15867,11843],{"class":10363},[40,15869,162],{"class":10732},[40,15871,11156],{"class":10363},[40,15873,11850],{"class":10430},[40,15875,167],{"class":10363},[40,15877,10801],{"class":10446},[40,15879,11857],{"class":10363},[40,15881,15882,15884,15886,15888,15890,15892,15894,15896,15898,15900],{"class":10276,"line":11557},[40,15883,11863],{"class":10363},[40,15885,162],{"class":10732},[40,15887,11156],{"class":10363},[40,15889,11850],{"class":10430},[40,15891,167],{"class":10363},[40,15893,10801],{"class":10446},[40,15895,10484],{"class":10363},[40,15897,11878],{"class":10446},[40,15899,11881],{"class":10732},[40,15901,11884],{"class":10363},[40,15903,15904,15906,15908,15910,15912,15914,15916,15918,15920],{"class":10276,"line":11563},[40,15905,11890],{"class":10363},[40,15907,162],{"class":10732},[40,15909,11156],{"class":10363},[40,15911,11897],{"class":10430},[40,15913,11900],{"class":10363},[40,15915,11904],{"class":11903},[40,15917,162],{"class":10732},[40,15919,11909],{"class":10287},[40,15921,10761],{"class":10363},[40,15923,15924,15926,15928,15930,15932,15934,15936,15938,15940],{"class":10276,"line":11582},[40,15925,11917],{"class":10363},[40,15927,162],{"class":10732},[40,15929,11156],{"class":10363},[40,15931,1813],{"class":10430},[40,15933,11926],{"class":10363},[40,15935,11752],{"class":10732},[40,15937,11156],{"class":10363},[40,15939,1820],{"class":10430},[40,15941,11935],{"class":10363},[40,15943,15944,15946,15948,15950,15952,15954,15956,15958,15960],{"class":10276,"line":11596},[40,15945,11941],{"class":10363},[40,15947,162],{"class":10732},[40,15949,11156],{"class":10363},[40,15951,1813],{"class":10430},[40,15953,11926],{"class":10363},[40,15955,11752],{"class":10732},[40,15957,11156],{"class":10363},[40,15959,1813],{"class":10430},[40,15961,11935],{"class":10363},[40,15963,15964,15966,15968,15970,15972],{"class":10276,"line":11609},[40,15965,11963],{"class":10363},[40,15967,162],{"class":10732},[40,15969,11156],{"class":10363},[40,15971,1820],{"class":10430},[40,15973,11972],{"class":10363},[40,15975,15976,15979,15981,15983,15985,15987,15989,15991,15993],{"class":10276,"line":11622},[40,15977,15978],{"class":10363},"    n ",[40,15980,162],{"class":10732},[40,15982,11156],{"class":10363},[40,15984,11982],{"class":10430},[40,15986,11985],{"class":10363},[40,15988,11275],{"class":11903},[40,15990,162],{"class":10732},[40,15992,66],{"class":10446},[40,15994,10761],{"class":10363},[40,15996,15997,15999],{"class":10276,"line":11637},[40,15998,11201],{"class":10352},[40,16000,16001],{"class":10363}," th, ph, n\n",[40,16003,16004],{"class":10276,"line":11708},[40,16005,10301],{"emptyLinePlaceholder":10300},[40,16007,16008],{"class":10276,"line":11727},[40,16009,10301],{"emptyLinePlaceholder":10300},[40,16011,16012,16014,16017,16019,16021,16023,16025],{"class":10276,"line":11744},[40,16013,10965],{"class":10352},[40,16015,16016],{"class":10430}," _wigner_qubit",[40,16018,167],{"class":10363},[40,16020,12016],{"class":10973},[40,16022,12019],{"class":10363},[40,16024,1796],{"class":10973},[40,16026,12025],{"class":10363},[40,16028,16029],{"class":10276,"line":11790},[40,16030,16031],{"class":10287},"    \"\"\"W(θ, φ) = 1\u002F2 + (√3\u002F2) * r·n on the (θ, φ) grid.\"\"\"\n",[40,16033,16034,16036,16038,16040,16042,16044,16046,16048,16050,16052,16054,16056,16058,16060,16062,16064,16066,16068,16070,16072,16074,16076,16078,16081,16083,16085,16087,16089,16091],{"class":10276,"line":11795},[40,16035,11201],{"class":10352},[40,16037,12039],{"class":10446},[40,16039,12042],{"class":10732},[40,16041,12045],{"class":10363},[40,16043,2218],{"class":10430},[40,16045,167],{"class":10363},[40,16047,12052],{"class":10446},[40,16049,12055],{"class":10363},[40,16051,11243],{"class":10732},[40,16053,12060],{"class":10446},[40,16055,12055],{"class":10363},[40,16057,11752],{"class":10732},[40,16059,12067],{"class":10363},[40,16061,12070],{"class":10430},[40,16063,167],{"class":10363},[40,16065,949],{"class":10446},[40,16067,10484],{"class":10363},[40,16069,78],{"class":10446},[40,16071,10484],{"class":10363},[40,16073,78],{"class":10446},[40,16075,12055],{"class":10363},[40,16077,11752],{"class":10732},[40,16079,16080],{"class":10363}," n).",[40,16082,12092],{"class":10430},[40,16084,167],{"class":10363},[40,16086,11275],{"class":11903},[40,16088,162],{"class":10732},[40,16090,66],{"class":10446},[40,16092,10761],{"class":10363},[40,16094,16095],{"class":10276,"line":11800},[40,16096,10301],{"emptyLinePlaceholder":10300},[40,16098,16099],{"class":10276,"line":11806},[40,16100,10301],{"emptyLinePlaceholder":10300},[40,16102,16103,16105,16108,16110,16112,16114,16116,16118,16120],{"class":10276,"line":11834},[40,16104,10965],{"class":10352},[40,16106,16107],{"class":10430}," _animate_one_pauli",[40,16109,167],{"class":10363},[40,16111,11275],{"class":10973},[40,16113,10443],{"class":10363},[40,16115,10733],{"class":10732},[40,16117,10484],{"class":10363},[40,16119,13346],{"class":10973},[40,16121,16122],{"class":10363},": Knobs3D) -> Path:\n",[40,16124,16125],{"class":10276,"line":11840},[40,16126,16127],{"class":10287},"    \"\"\"Render one 3D Wigner animation for a chosen Pauli axis; return output path.\"\"\"\n",[40,16129,16130,16133,16135,16137],{"class":10276,"line":11860},[40,16131,16132],{"class":10363},"    r0 ",[40,16134,162],{"class":10732},[40,16136,15410],{"class":10430},[40,16138,16139],{"class":10363},"(k.initial_state, k.custom_r)\n",[40,16141,16142,16145,16147,16149],{"class":10276,"line":11887},[40,16143,16144],{"class":10363},"    th, ph, n ",[40,16146,162],{"class":10732},[40,16148,15840],{"class":10430},[40,16150,16151],{"class":10363},"(k.n_theta, k.n_phi)\n",[40,16153,16154],{"class":10276,"line":11914},[40,16155,10301],{"emptyLinePlaceholder":10300},[40,16157,16158],{"class":10276,"line":11938},[40,16159,16160],{"class":10280},"    # Unit sphere geometry\n",[40,16162,16163,16166,16168,16170,16172,16174,16176,16178,16180],{"class":10276,"line":11960},[40,16164,16165],{"class":10363},"    X ",[40,16167,162],{"class":10732},[40,16169,11156],{"class":10363},[40,16171,1813],{"class":10430},[40,16173,11926],{"class":10363},[40,16175,11752],{"class":10732},[40,16177,11156],{"class":10363},[40,16179,1820],{"class":10430},[40,16181,11935],{"class":10363},[40,16183,16184,16187,16189,16191,16193,16195,16197,16199,16201],{"class":10276,"line":11975},[40,16185,16186],{"class":10363},"    Y ",[40,16188,162],{"class":10732},[40,16190,11156],{"class":10363},[40,16192,1813],{"class":10430},[40,16194,11926],{"class":10363},[40,16196,11752],{"class":10732},[40,16198,11156],{"class":10363},[40,16200,1813],{"class":10430},[40,16202,11935],{"class":10363},[40,16204,16205,16208,16210,16212,16214],{"class":10276,"line":11996},[40,16206,16207],{"class":10363},"    Z ",[40,16209,162],{"class":10732},[40,16211,11156],{"class":10363},[40,16213,1820],{"class":10430},[40,16215,11972],{"class":10363},[40,16217,16218],{"class":10276,"line":12001},[40,16219,10301],{"emptyLinePlaceholder":10300},[40,16221,16222],{"class":10276,"line":12006},[40,16223,16224],{"class":10280},"    # Fixed normalization (pure-state bounds) for constant color scale\n",[40,16226,16227,16229,16231,16233,16235,16237,16239,16241,16243,16245,16247,16249],{"class":10276,"line":12028},[40,16228,12982],{"class":10363},[40,16230,162],{"class":10732},[40,16232,12039],{"class":10446},[40,16234,11764],{"class":10732},[40,16236,12045],{"class":10363},[40,16238,2218],{"class":10430},[40,16240,167],{"class":10363},[40,16242,12052],{"class":10446},[40,16244,12055],{"class":10363},[40,16246,11243],{"class":10732},[40,16248,12060],{"class":10446},[40,16250,10761],{"class":10363},[40,16252,16253,16255,16257,16259,16261,16263,16265,16267,16269,16271,16273,16275],{"class":10276,"line":12034},[40,16254,13009],{"class":10363},[40,16256,162],{"class":10732},[40,16258,12039],{"class":10446},[40,16260,12042],{"class":10732},[40,16262,12045],{"class":10363},[40,16264,2218],{"class":10430},[40,16266,167],{"class":10363},[40,16268,12052],{"class":10446},[40,16270,12055],{"class":10363},[40,16272,11243],{"class":10732},[40,16274,12060],{"class":10446},[40,16276,10761],{"class":10363},[40,16278,16279,16282,16284,16287,16289,16292,16294,16296,16298,16300],{"class":10276,"line":12105},[40,16280,16281],{"class":10363},"    norm ",[40,16283,162],{"class":10732},[40,16285,16286],{"class":10430}," Normalize",[40,16288,167],{"class":10363},[40,16290,16291],{"class":11903},"vmin",[40,16293,162],{"class":10732},[40,16295,13142],{"class":10363},[40,16297,13145],{"class":11903},[40,16299,162],{"class":10732},[40,16301,16302],{"class":10363},"w_max)\n",[40,16304,16305,16308,16310,16312,16315],{"class":10276,"line":12110},[40,16306,16307],{"class":10363},"    cmap ",[40,16309,162],{"class":10732},[40,16311,13070],{"class":10363},[40,16313,16314],{"class":10430},"get_cmap",[40,16316,12174],{"class":10363},[40,16318,16319,16322,16324,16327,16329,16331,16333,16336,16339,16341],{"class":10276,"line":12115},[40,16320,16321],{"class":10363},"    sm ",[40,16323,162],{"class":10732},[40,16325,16326],{"class":10430}," ScalarMappable",[40,16328,167],{"class":10363},[40,16330,11192],{"class":11903},[40,16332,162],{"class":10732},[40,16334,16335],{"class":10363},"norm, ",[40,16337,16338],{"class":11903},"cmap",[40,16340,162],{"class":10732},[40,16342,16343],{"class":10363},"cmap)\n",[40,16345,16346,16349,16352,16355],{"class":10276,"line":12121},[40,16347,16348],{"class":10363},"    sm.",[40,16350,16351],{"class":10430},"set_array",[40,16353,16354],{"class":10363},"([])  ",[40,16356,16357],{"class":10280},"# required for colorbar from a ScalarMappable\n",[40,16359,16360],{"class":10276,"line":12155},[40,16361,10301],{"emptyLinePlaceholder":10300},[40,16363,16364,16367,16369,16371,16374,16376,16378,16380,16383,16385,16387,16389],{"class":10276,"line":12161},[40,16365,16366],{"class":10363},"    fig ",[40,16368,162],{"class":10732},[40,16370,13070],{"class":10363},[40,16372,16373],{"class":10430},"figure",[40,16375,167],{"class":10363},[40,16377,13078],{"class":11903},[40,16379,162],{"class":10732},[40,16381,16382],{"class":10363},"k.figsize, ",[40,16384,13086],{"class":11903},[40,16386,162],{"class":10732},[40,16388,13091],{"class":10446},[40,16390,10761],{"class":10363},[40,16392,16393,16396,16398,16400,16403,16405,16408,16410,16413,16415,16418],{"class":10276,"line":12177},[40,16394,16395],{"class":10363},"    ax ",[40,16397,162],{"class":10732},[40,16399,13170],{"class":10363},[40,16401,16402],{"class":10430},"add_subplot",[40,16404,167],{"class":10363},[40,16406,16407],{"class":10446},"111",[40,16409,10484],{"class":10363},[40,16411,16412],{"class":11903},"projection",[40,16414,162],{"class":10732},[40,16416,16417],{"class":10287},"\"3d\"",[40,16419,10761],{"class":10363},[40,16421,16422,16424,16427,16429,16432,16434,16437,16440,16442],{"class":10276,"line":12192},[40,16423,13227],{"class":10363},[40,16425,16426],{"class":10430},"view_init",[40,16428,167],{"class":10363},[40,16430,16431],{"class":11903},"elev",[40,16433,162],{"class":10732},[40,16435,16436],{"class":10363},"k.elev, ",[40,16438,16439],{"class":11903},"azim",[40,16441,162],{"class":10732},[40,16443,16444],{"class":10363},"k.azim)\n",[40,16446,16447,16449,16452,16454,16457],{"class":10276,"line":12231},[40,16448,11323],{"class":10352},[40,16450,16451],{"class":10732}," hasattr",[40,16453,13259],{"class":10363},[40,16455,16456],{"class":10287},"\"set_box_aspect\"",[40,16458,13353],{"class":10363},[40,16460,16461,16464,16467,16470,16472,16474,16476,16478,16480],{"class":10276,"line":12245},[40,16462,16463],{"class":10363},"        ax.",[40,16465,16466],{"class":10430},"set_box_aspect",[40,16468,16469],{"class":10363},"((",[40,16471,78],{"class":10446},[40,16473,10484],{"class":10363},[40,16475,78],{"class":10446},[40,16477,10484],{"class":10363},[40,16479,78],{"class":10446},[40,16481,13612],{"class":10363},[40,16483,16484],{"class":10276,"line":12280},[40,16485,10301],{"emptyLinePlaceholder":10300},[40,16487,16488],{"class":10276,"line":12294},[40,16489,16490],{"class":10280},"    # Initial facecolors (use cell-centered (M-1, N-1) colors for quads)\n",[40,16492,16493,16496,16498,16500,16502,16505,16508,16510,16512,16514],{"class":10276,"line":12329},[40,16494,16495],{"class":10363},"    W0 ",[40,16497,162],{"class":10732},[40,16499,16016],{"class":10430},[40,16501,167],{"class":10363},[40,16503,16504],{"class":10430},"_rotation_matrix",[40,16506,16507],{"class":10363},"(axis, ",[40,16509,10801],{"class":10446},[40,16511,12055],{"class":10363},[40,16513,13404],{"class":10732},[40,16515,16516],{"class":10363}," r0, n)\n",[40,16518,16519,16522,16524,16527,16529,16531,16534,16536,16538,16541,16543,16545,16548],{"class":10276,"line":12343},[40,16520,16521],{"class":10446},"    FC0",[40,16523,10772],{"class":10732},[40,16525,16526],{"class":10430}," cmap",[40,16528,167],{"class":10363},[40,16530,11192],{"class":10430},[40,16532,16533],{"class":10363},"(W0[:",[40,16535,11056],{"class":10732},[40,16537,78],{"class":10446},[40,16539,16540],{"class":10363},", :",[40,16542,11056],{"class":10732},[40,16544,78],{"class":10446},[40,16546,16547],{"class":10363},"]))  ",[40,16549,16550],{"class":10280},"# shape: (n_theta-1, n_phi-1, 4)\n",[40,16552,16553],{"class":10276,"line":12376},[40,16554,10301],{"emptyLinePlaceholder":10300},[40,16556,16557,16560,16562,16564,16567],{"class":10276,"line":12409},[40,16558,16559],{"class":10363},"    surf ",[40,16561,162],{"class":10732},[40,16563,12522],{"class":10363},[40,16565,16566],{"class":10430},"plot_surface",[40,16568,12527],{"class":10363},[40,16570,16571],{"class":10276,"line":12414},[40,16572,16573],{"class":10363},"        X, Y, Z,\n",[40,16575,16576,16579,16581,16584],{"class":10276,"line":12419},[40,16577,16578],{"class":11903},"        facecolors",[40,16580,162],{"class":10732},[40,16582,16583],{"class":10446},"FC0",[40,16585,10507],{"class":10363},[40,16587,16588,16591,16593,16595,16597,16600,16602,16604],{"class":10276,"line":12435},[40,16589,16590],{"class":11903},"        rstride",[40,16592,162],{"class":10732},[40,16594,78],{"class":10446},[40,16596,10484],{"class":10363},[40,16598,16599],{"class":11903},"cstride",[40,16601,162],{"class":10732},[40,16603,78],{"class":10446},[40,16605,10507],{"class":10363},[40,16607,16608,16611,16613,16615,16617,16620,16622,16624],{"class":10276,"line":12447},[40,16609,16610],{"class":11903},"        antialiased",[40,16612,162],{"class":10732},[40,16614,10504],{"class":10446},[40,16616,10484],{"class":10363},[40,16618,16619],{"class":11903},"linewidth",[40,16621,162],{"class":10732},[40,16623,66],{"class":10446},[40,16625,10507],{"class":10363},[40,16627,16628,16631,16633,16635,16638],{"class":10276,"line":12463},[40,16629,16630],{"class":11903},"        shade",[40,16632,162],{"class":10732},[40,16634,10504],{"class":10446},[40,16636,16637],{"class":10363},",  ",[40,16639,16640],{"class":10280},"# use given facecolors directly\n",[40,16642,16643],{"class":10276,"line":12480},[40,16644,12631],{"class":10363},[40,16646,16647],{"class":10276,"line":12498},[40,16648,10301],{"emptyLinePlaceholder":10300},[40,16650,16651],{"class":10276,"line":12504},[40,16652,16653],{"class":10280},"    # Colorbar (bound to ScalarMappable using same norm+cmap)\n",[40,16655,16656,16658,16660,16662,16664,16667,16669,16671,16673,16675,16677,16679,16681,16683,16685,16687],{"class":10276,"line":12517},[40,16657,13165],{"class":10363},[40,16659,162],{"class":10732},[40,16661,13170],{"class":10363},[40,16663,13173],{"class":10430},[40,16665,16666],{"class":10363},"(sm, ",[40,16668,12429],{"class":11903},[40,16670,162],{"class":10732},[40,16672,13183],{"class":10363},[40,16674,13186],{"class":11903},[40,16676,162],{"class":10732},[40,16678,13191],{"class":10446},[40,16680,10484],{"class":10363},[40,16682,13196],{"class":11903},[40,16684,162],{"class":10732},[40,16686,13201],{"class":10446},[40,16688,10761],{"class":10363},[40,16690,16691,16693,16695,16697,16699],{"class":10276,"line":12530},[40,16692,13208],{"class":10363},[40,16694,13211],{"class":10430},[40,16696,167],{"class":10363},[40,16698,13216],{"class":10287},[40,16700,10761],{"class":10363},[40,16702,16703],{"class":10276,"line":12542},[40,16704,10301],{"emptyLinePlaceholder":10300},[40,16706,16707],{"class":10276,"line":12571},[40,16708,16709],{"class":10280},"    # Labels and in-axes title and info\n",[40,16711,16712,16714,16716,16718,16720],{"class":10276,"line":12622},[40,16713,13227],{"class":10363},[40,16715,13230],{"class":10430},[40,16717,167],{"class":10363},[40,16719,10748],{"class":10287},[40,16721,10761],{"class":10363},[40,16723,16724,16726,16728,16730,16732],{"class":10276,"line":12628},[40,16725,13227],{"class":10363},[40,16727,13244],{"class":10430},[40,16729,167],{"class":10363},[40,16731,10753],{"class":10287},[40,16733,10761],{"class":10363},[40,16735,16736,16738,16741,16743,16745],{"class":10276,"line":12634},[40,16737,13227],{"class":10363},[40,16739,16740],{"class":10430},"set_zlabel",[40,16742,167],{"class":10363},[40,16744,10758],{"class":10287},[40,16746,10761],{"class":10363},[40,16748,16749,16752,16754,16756,16759],{"class":10276,"line":12639},[40,16750,16751],{"class":10363},"    title_txt ",[40,16753,162],{"class":10732},[40,16755,12522],{"class":10363},[40,16757,16758],{"class":10430},"text2D",[40,16760,12527],{"class":10363},[40,16762,16763,16765,16767,16770,16772,16774,16776,16778,16780,16782,16784,16786,16789],{"class":10276,"line":12644},[40,16764,12687],{"class":10446},[40,16766,10484],{"class":10363},[40,16768,16769],{"class":10446},"0.98",[40,16771,10484],{"class":10363},[40,16773,13262],{"class":10352},[40,16775,13265],{"class":10287},[40,16777,13268],{"class":10446},[40,16779,13271],{"class":10363},[40,16781,13274],{"class":10430},[40,16783,13277],{"class":10363},[40,16785,13280],{"class":10446},[40,16787,16788],{"class":10287}," rotation (3D sphere)\"",[40,16790,10507],{"class":10363},[40,16792,16793,16795,16797,16799,16801,16803,16805,16807,16809,16811,16813,16815,16817,16819,16821],{"class":10276,"line":12667},[40,16794,12545],{"class":11903},[40,16796,162],{"class":10732},[40,16798,12550],{"class":10363},[40,16800,12211],{"class":11903},[40,16802,162],{"class":10732},[40,16804,12710],{"class":10287},[40,16806,10484],{"class":10363},[40,16808,12221],{"class":11903},[40,16810,162],{"class":10732},[40,16812,12226],{"class":10287},[40,16814,10484],{"class":10363},[40,16816,12553],{"class":11903},[40,16818,162],{"class":10732},[40,16820,12731],{"class":10446},[40,16822,10507],{"class":10363},[40,16824,16825,16827,16829,16831,16833,16835,16837,16839,16841,16843,16845,16847,16849,16851,16853,16855,16857,16859,16861],{"class":10276,"line":12673},[40,16826,12574],{"class":11903},[40,16828,12577],{"class":10732},[40,16830,167],{"class":10363},[40,16832,12582],{"class":11903},[40,16834,162],{"class":10732},[40,16836,12587],{"class":10287},[40,16838,10484],{"class":10363},[40,16840,12592],{"class":11903},[40,16842,162],{"class":10732},[40,16844,12597],{"class":10287},[40,16846,10484],{"class":10363},[40,16848,12602],{"class":11903},[40,16850,162],{"class":10732},[40,16852,12607],{"class":10446},[40,16854,10484],{"class":10363},[40,16856,12612],{"class":11903},[40,16858,162],{"class":10732},[40,16860,12617],{"class":10287},[40,16862,11061],{"class":10363},[40,16864,16865,16867,16869,16871],{"class":10276,"line":12684},[40,16866,12780],{"class":11903},[40,16868,162],{"class":10732},[40,16870,12785],{"class":10446},[40,16872,10507],{"class":10363},[40,16874,16875],{"class":10276,"line":12697},[40,16876,12631],{"class":10363},[40,16878,16879,16882,16884,16886,16888],{"class":10276,"line":12723},[40,16880,16881],{"class":10363},"    info_txt ",[40,16883,162],{"class":10732},[40,16885,12522],{"class":10363},[40,16887,16758],{"class":10430},[40,16889,12527],{"class":10363},[40,16891,16892,16895,16897,16900,16902,16904,16906,16909,16911,16913,16915,16917,16919,16921,16923,16925,16927],{"class":10276,"line":12736},[40,16893,16894],{"class":10446},"        0.02",[40,16896,10484],{"class":10363},[40,16898,16899],{"class":10446},"0.96",[40,16901,10484],{"class":10363},[40,16903,13303],{"class":10287},[40,16905,10484],{"class":10363},[40,16907,16908],{"class":11903},"transform",[40,16910,162],{"class":10732},[40,16912,12550],{"class":10363},[40,16914,12211],{"class":11903},[40,16916,162],{"class":10732},[40,16918,12216],{"class":10287},[40,16920,10484],{"class":10363},[40,16922,12221],{"class":11903},[40,16924,162],{"class":10732},[40,16926,12226],{"class":10287},[40,16928,10507],{"class":10363},[40,16930,16931,16933,16935],{"class":10276,"line":12777},[40,16932,12726],{"class":11903},[40,16934,162],{"class":10732},[40,16936,16937],{"class":10363},"k.info_fontsize,\n",[40,16939,16940,16942,16944,16946,16948,16950,16952,16954,16956,16958,16960,16962,16964,16966,16968,16970,16972,16974,16976],{"class":10276,"line":12790},[40,16941,12574],{"class":11903},[40,16943,12577],{"class":10732},[40,16945,167],{"class":10363},[40,16947,12582],{"class":11903},[40,16949,162],{"class":10732},[40,16951,12587],{"class":10287},[40,16953,10484],{"class":10363},[40,16955,12592],{"class":11903},[40,16957,162],{"class":10732},[40,16959,12597],{"class":10287},[40,16961,10484],{"class":10363},[40,16963,12602],{"class":11903},[40,16965,162],{"class":10732},[40,16967,12607],{"class":10446},[40,16969,10484],{"class":10363},[40,16971,12612],{"class":11903},[40,16973,162],{"class":10732},[40,16975,12617],{"class":10287},[40,16977,11061],{"class":10363},[40,16979,16981,16983,16985,16987],{"class":10276,"line":16980},142,[40,16982,12780],{"class":11903},[40,16984,162],{"class":10732},[40,16986,12785],{"class":10446},[40,16988,10507],{"class":10363},[40,16990,16992],{"class":10276,"line":16991},143,[40,16993,12631],{"class":10363},[40,16995,16997],{"class":10276,"line":16996},144,[40,16998,10301],{"emptyLinePlaceholder":10300},[40,17000,17002,17004,17006,17008,17011,17013,17015],{"class":10276,"line":17001},145,[40,17003,13338],{"class":10352},[40,17005,13341],{"class":10430},[40,17007,167],{"class":10363},[40,17009,17010],{"class":10973},"frame",[40,17012,10443],{"class":10363},[40,17014,10821],{"class":10732},[40,17016,13353],{"class":10363},[40,17018,17020,17023,17025,17028,17030,17033,17035,17037,17039,17041,17043],{"class":10276,"line":17019},146,[40,17021,17022],{"class":10363},"        angle ",[40,17024,162],{"class":10732},[40,17026,17027],{"class":10363}," (frame ",[40,17029,11243],{"class":10732},[40,17031,17032],{"class":10363}," (k.frames ",[40,17034,11056],{"class":10732},[40,17036,13373],{"class":10446},[40,17038,13376],{"class":10363},[40,17040,11752],{"class":10732},[40,17042,13381],{"class":10363},[40,17044,13384],{"class":10280},[40,17046,17048,17051,17053,17055,17058,17060],{"class":10276,"line":17047},147,[40,17049,17050],{"class":10363},"        r_now ",[40,17052,162],{"class":10732},[40,17054,15630],{"class":10430},[40,17056,17057],{"class":10363},"(axis, angle) ",[40,17059,13404],{"class":10732},[40,17061,13407],{"class":10363},[40,17063,17065,17068,17070,17072],{"class":10276,"line":17064},148,[40,17066,17067],{"class":10363},"        W_now ",[40,17069,162],{"class":10732},[40,17071,16016],{"class":10430},[40,17073,17074],{"class":10363},"(r_now, n)\n",[40,17076,17078,17081,17083,17085,17087,17089,17092,17094,17096,17098,17100,17102,17105],{"class":10276,"line":17077},149,[40,17079,17080],{"class":10446},"        FC_now",[40,17082,10772],{"class":10732},[40,17084,16526],{"class":10430},[40,17086,167],{"class":10363},[40,17088,11192],{"class":10430},[40,17090,17091],{"class":10363},"(W_now[:",[40,17093,11056],{"class":10732},[40,17095,78],{"class":10446},[40,17097,16540],{"class":10363},[40,17099,11056],{"class":10732},[40,17101,78],{"class":10446},[40,17103,17104],{"class":10363},"]))        ",[40,17106,17107],{"class":10280},"# (M-1, N-1, 4)\n",[40,17109,17111,17114,17117,17119,17122,17124,17126,17128,17130,17132,17134,17137,17140],{"class":10276,"line":17110},150,[40,17112,17113],{"class":10363},"        surf.",[40,17115,17116],{"class":10430},"set_facecolors",[40,17118,167],{"class":10363},[40,17120,17121],{"class":10446},"FC_now",[40,17123,1123],{"class":10363},[40,17125,12070],{"class":10430},[40,17127,167],{"class":10363},[40,17129,11056],{"class":10732},[40,17131,78],{"class":10446},[40,17133,10484],{"class":10363},[40,17135,17136],{"class":10446},"4",[40,17138,17139],{"class":10363},"))  ",[40,17141,17142],{"class":10280},"# update in place\n",[40,17144,17146,17149,17151,17153,17155,17157,17159,17161,17163,17165,17167,17169,17171,17173,17175,17177,17179,17181,17183,17186,17188,17190],{"class":10276,"line":17145},151,[40,17147,17148],{"class":10363},"        info_txt.",[40,17150,13431],{"class":10430},[40,17152,167],{"class":10363},[40,17154,13262],{"class":10352},[40,17156,13438],{"class":10287},[40,17158,13268],{"class":10446},[40,17160,11284],{"class":10363},[40,17162,13446],{"class":10352},[40,17164,13280],{"class":10446},[40,17166,13451],{"class":10287},[40,17168,13454],{"class":10732},[40,17170,13457],{"class":10287},[40,17172,13268],{"class":10446},[40,17174,17010],{"class":10363},[40,17176,1053],{"class":10732},[40,17178,13466],{"class":10446},[40,17180,11243],{"class":10287},[40,17182,13268],{"class":10446},[40,17184,17185],{"class":10363},"k.frames",[40,17187,13280],{"class":10446},[40,17189,13478],{"class":10287},[40,17191,10761],{"class":10363},[40,17193,17195,17197],{"class":10276,"line":17194},152,[40,17196,11339],{"class":10352},[40,17198,17199],{"class":10363}," (surf, info_txt, title_txt)\n",[40,17201,17203],{"class":10276,"line":17202},153,[40,17204,10301],{"emptyLinePlaceholder":10300},[40,17206,17208,17210,17212,17214,17216,17218,17220,17223,17225,17227,17229,17231,17233,17235,17237],{"class":10276,"line":17207},154,[40,17209,13496],{"class":10363},[40,17211,162],{"class":10732},[40,17213,13501],{"class":10430},[40,17215,13504],{"class":10363},[40,17217,13473],{"class":11903},[40,17219,162],{"class":10732},[40,17221,17222],{"class":10363},"k.frames, ",[40,17224,13514],{"class":11903},[40,17226,162],{"class":10732},[40,17228,13519],{"class":10446},[40,17230,10484],{"class":10363},[40,17232,13524],{"class":11903},[40,17234,162],{"class":10732},[40,17236,10504],{"class":10446},[40,17238,10761],{"class":10363},[40,17240,17242,17245,17247,17249,17252,17254,17256,17259,17261,17263,17265],{"class":10276,"line":17241},155,[40,17243,17244],{"class":10363},"    out ",[40,17246,162],{"class":10732},[40,17248,13859],{"class":10430},[40,17250,17251],{"class":10363},"(k.out_dir) ",[40,17253,11243],{"class":10732},[40,17255,13867],{"class":10352},[40,17257,17258],{"class":10287},"\"wigner3d_pauli_",[40,17260,13268],{"class":10446},[40,17262,11275],{"class":10363},[40,17264,13280],{"class":10446},[40,17266,13880],{"class":10287},[40,17268,17270,17273,17275,17277,17279,17281,17283,17285,17287,17289,17291],{"class":10276,"line":17269},156,[40,17271,17272],{"class":10363},"    out.parent.",[40,17274,13538],{"class":10430},[40,17276,167],{"class":10363},[40,17278,13543],{"class":11903},[40,17280,162],{"class":10732},[40,17282,13091],{"class":10446},[40,17284,10484],{"class":10363},[40,17286,13552],{"class":11903},[40,17288,162],{"class":10732},[40,17290,13091],{"class":10446},[40,17292,10761],{"class":10363},[40,17294,17296,17298,17300,17303,17305,17307,17309,17311,17313,17315,17317,17319],{"class":10276,"line":17295},157,[40,17297,13563],{"class":10363},[40,17299,13566],{"class":10430},[40,17301,17302],{"class":10363},"(out.",[40,17304,13572],{"class":10430},[40,17306,13575],{"class":10363},[40,17308,13578],{"class":11903},[40,17310,162],{"class":10732},[40,17312,13583],{"class":10430},[40,17314,167],{"class":10363},[40,17316,13588],{"class":11903},[40,17318,162],{"class":10732},[40,17320,17321],{"class":10363},"k.fps),\n",[40,17323,17325,17327,17329,17331,17333,17335,17337],{"class":10276,"line":17324},158,[40,17326,13598],{"class":11903},[40,17328,12577],{"class":10732},[40,17330,167],{"class":10363},[40,17332,13605],{"class":11903},[40,17334,162],{"class":10732},[40,17336,12597],{"class":10287},[40,17338,13612],{"class":10363},[40,17340,17342,17344,17346],{"class":10276,"line":17341},159,[40,17343,13617],{"class":10363},[40,17345,13620],{"class":10430},[40,17347,13623],{"class":10363},[40,17349,17351,17353],{"class":10276,"line":17350},160,[40,17352,11201],{"class":10352},[40,17354,17355],{"class":10363}," out\n",[40,17357,17359],{"class":10276,"line":17358},161,[40,17360,10301],{"emptyLinePlaceholder":10300},[40,17362,17364],{"class":10276,"line":17363},162,[40,17365,10301],{"emptyLinePlaceholder":10300},[40,17367,17369,17371,17374,17376,17378],{"class":10276,"line":17368},163,[40,17370,10965],{"class":10352},[40,17372,17373],{"class":10430}," animate_all_paulis_3d",[40,17375,167],{"class":10363},[40,17377,13346],{"class":10973},[40,17379,17380],{"class":10363},": Knobs3D,\n",[40,17382,17384,17387,17389,17391,17393,17395,17397,17399,17401,17403,17405,17407,17409,17411],{"class":10276,"line":17383},164,[40,17385,17386],{"class":10973},"                          axes",[40,17388,10989],{"class":10363},[40,17390,10733],{"class":10732},[40,17392,10484],{"class":10363},[40,17394,10738],{"class":10446},[40,17396,10741],{"class":10363},[40,17398,162],{"class":10732},[40,17400,920],{"class":10363},[40,17402,10748],{"class":10287},[40,17404,10484],{"class":10363},[40,17406,10753],{"class":10287},[40,17408,10484],{"class":10363},[40,17410,10758],{"class":10287},[40,17412,17413],{"class":10363},")) -> Iterable[Path]:\n",[40,17415,17417],{"class":10276,"line":17416},165,[40,17418,17419],{"class":10287},"    \"\"\"Generate and display GIFs for all requested Pauli axes; yield file paths.\"\"\"\n",[40,17421,17423,17426,17428,17430],{"class":10276,"line":17422},166,[40,17424,17425],{"class":10352},"    for",[40,17427,13843],{"class":10363},[40,17429,13846],{"class":10352},[40,17431,17432],{"class":10363}," axes:\n",[40,17434,17436,17439,17441,17443,17446,17448],{"class":10276,"line":17435},167,[40,17437,17438],{"class":10363},"        path ",[40,17440,162],{"class":10732},[40,17442,16107],{"class":10430},[40,17444,17445],{"class":10363},"(axname.",[40,17447,11171],{"class":10430},[40,17449,17450],{"class":10363},"(), k)\n",[40,17452,17454,17457,17459,17461,17463,17465,17467],{"class":10276,"line":17453},168,[40,17455,17456],{"class":10430},"        display",[40,17458,167],{"class":10363},[40,17460,13953],{"class":10430},[40,17462,167],{"class":10363},[40,17464,13958],{"class":11903},[40,17466,13961],{"class":10732},[40,17468,17469],{"class":10363},"(path)))\n",[40,17471,17473,17476,17478,17480,17483,17485,17487,17489,17491],{"class":10276,"line":17472},169,[40,17474,17475],{"class":10732},"        print",[40,17477,167],{"class":10363},[40,17479,13262],{"class":10352},[40,17481,17482],{"class":10287},"\"Saved: ",[40,17484,13268],{"class":10446},[40,17486,2264],{"class":10363},[40,17488,13280],{"class":10446},[40,17490,13478],{"class":10287},[40,17492,10761],{"class":10363},[40,17494,17496,17499],{"class":10276,"line":17495},170,[40,17497,17498],{"class":10352},"        yield",[40,17500,17501],{"class":10363}," path\n",[40,17503,17505],{"class":10276,"line":17504},171,[40,17506,10301],{"emptyLinePlaceholder":10300},[40,17508,17510],{"class":10276,"line":17509},172,[40,17511,10301],{"emptyLinePlaceholder":10300},[40,17513,17515],{"class":10276,"line":17514},173,[40,17516,17517],{"class":10280},"# ------------------------------ Run all three gates ------------------------------\n",[40,17519,17521,17524,17526,17528,17530],{"class":10276,"line":17520},174,[40,17522,17523],{"class":10446},"K3D",[40,17525,10772],{"class":10732},[40,17527,15163],{"class":10430},[40,17529,13655],{"class":10363},[40,17531,17532],{"class":10280},"# adjust as desired\n",[40,17534,17536,17538,17540,17542],{"class":10276,"line":17535},175,[40,17537,13684],{"class":11903},[40,17539,162],{"class":10732},[40,17541,13689],{"class":10287},[40,17543,10507],{"class":10363},[40,17545,17547,17549,17551,17553,17555,17557,17559,17562,17564,17566,17568,17571,17573,17575,17577,17579],{"class":10276,"line":17546},176,[40,17548,13696],{"class":11903},[40,17550,162],{"class":10732},[40,17552,13701],{"class":10446},[40,17554,10484],{"class":10363},[40,17556,11816],{"class":11903},[40,17558,162],{"class":10732},[40,17560,17561],{"class":10446},"64",[40,17563,10484],{"class":10363},[40,17565,11825],{"class":11903},[40,17567,162],{"class":10732},[40,17569,17570],{"class":10446},"128",[40,17572,10484],{"class":10363},[40,17574,13588],{"class":11903},[40,17576,162],{"class":10732},[40,17578,13742],{"class":10446},[40,17580,10507],{"class":10363},[40,17582,17584,17587,17589,17592,17594,17596,17599,17602],{"class":10276,"line":17583},177,[40,17585,17586],{"class":11903},"    elev",[40,17588,162],{"class":10732},[40,17590,17591],{"class":10446},"25.0",[40,17593,10484],{"class":10363},[40,17595,16439],{"class":11903},[40,17597,17598],{"class":10732},"=-",[40,17600,17601],{"class":10446},"60.0",[40,17603,10507],{"class":10363},[40,17605,17607,17609,17611,17613],{"class":10276,"line":17606},178,[40,17608,13749],{"class":11903},[40,17610,162],{"class":10732},[40,17612,13754],{"class":10287},[40,17614,10507],{"class":10363},[40,17616,17618,17620,17622,17624,17626,17628,17630],{"class":10276,"line":17617},179,[40,17619,13761],{"class":11903},[40,17621,162],{"class":10732},[40,17623,167],{"class":10363},[40,17625,10481],{"class":10446},[40,17627,10484],{"class":10363},[40,17629,15378],{"class":10446},[40,17631,11061],{"class":10363},[40,17633,17635,17638,17640,17642],{"class":10276,"line":17634},180,[40,17636,17637],{"class":11903},"    info_fontsize",[40,17639,162],{"class":10732},[40,17641,13793],{"class":10446},[40,17643,10507],{"class":10363},[40,17645,17647],{"class":10276,"line":17646},181,[40,17648,10761],{"class":10363},[40,17650,17652],{"class":10276,"line":17651},182,[40,17653,10301],{"emptyLinePlaceholder":10300},[40,17655,17657,17660,17662,17665,17667,17669,17671,17674,17676,17678,17680,17682,17684,17686,17688],{"class":10276,"line":17656},183,[40,17658,17659],{"class":10732},"list",[40,17661,167],{"class":10363},[40,17663,17664],{"class":10430},"animate_all_paulis_3d",[40,17666,167],{"class":10363},[40,17668,17523],{"class":10446},[40,17670,10484],{"class":10363},[40,17672,17673],{"class":11903},"axes",[40,17675,162],{"class":10732},[40,17677,167],{"class":10363},[40,17679,10748],{"class":10287},[40,17681,10484],{"class":10363},[40,17683,10753],{"class":10287},[40,17685,10484],{"class":10363},[40,17687,10758],{"class":10287},[40,17689,17690],{"class":10363},")))\n",[10082,17692,17695],{"className":17693,"code":17694,"language":2932,"meta":10090},[10085],"Saved: \u002Fcontent\u002Fwigner3d_pauli_x.gif\n",[10088,17696,17694],{"__ignoreMap":10090},[10082,17698,17701],{"className":17699,"code":17700,"language":2932,"meta":10090},[10085],"Saved: \u002Fcontent\u002Fwigner3d_pauli_y.gif\n",[10088,17702,17700],{"__ignoreMap":10090},[10082,17704,17707],{"className":17705,"code":17706,"language":2932,"meta":10090},[10085],"Saved: \u002Fcontent\u002Fwigner3d_pauli_z.gif\n",[10088,17708,17706],{"__ignoreMap":10090},[13997,17710],{"alt":17711,"src":17712},"Output 5 of the notebook","\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-05.webp",[13997,17714],{"alt":17715,"src":17716},"Output 6 of the notebook","\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-06.webp",[13997,17718],{"alt":17719,"src":17720},"Output 7 of the notebook","\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-07.webp",[10082,17722,17724],{"className":10269,"code":17723,"language":10271,"meta":10090,"style":10090},"# @title Cell 6 — 3D Wigner animation (custom axis–angle; colors-only, no geometry edits)\n# PEP 8\u002F257 compliant. Rotates Bloch vector about an arbitrary axis by a target angle.\n\nfrom __future__ import annotations\n\nfrom dataclasses import dataclass\nfrom pathlib import Path\nfrom typing import Tuple\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib.animation import FuncAnimation, PillowWriter\nfrom matplotlib.colors import Normalize\nfrom matplotlib.cm import ScalarMappable\nfrom IPython.display import Image, display\n\n\n@dataclass\nclass Knobs3DCustom:\n    \"\"\"User controls for 3D custom axis–angle Wigner animation.\"\"\"\n    axis_vec: Tuple[float, float, float] = (0.0, 1.0, 0.0)  # default: y-axis\n    angle_deg: float = 45.0\n    initial_state: str = \"+z\"\n    custom_r: Tuple[float, float, float] = (0.0, 0.0, 1.0)\n    frames: int = 24\n    n_theta: int = 64\n    n_phi: int = 128\n    fps: int = 20\n    elev: float = 25.0\n    azim: float = -60.0\n    out_path: str = \"\u002Fcontent\u002Fwigner3d_custom_y45.gif\"\n    figsize: Tuple[float, float] = (6.4, 4.8)\n    info_fontsize: int = 9\n\n\n# ------------- helpers (self-contained; safe for this cell alone) -------------\ndef _bloch_vector(label: str, custom=(0.0, 0.0, 1.0)) -> np.ndarray:\n    mapping = {\n        \"+z\": (0.0, 0.0, 1.0), \"-z\": (0.0, 0.0, -1.0),\n        \"+x\": (1.0, 0.0, 0.0), \"-x\": (-1.0, 0.0, 0.0),\n        \"+y\": (0.0, 1.0, 0.0), \"-y\": (0.0, -1.0, 0.0),\n    }\n    r = np.asarray(mapping.get(label.lower(), custom), float)\n    nrm = np.linalg.norm(r) or 1.0\n    return r \u002F nrm\n\n\ndef _rotation_matrix_axis(axis_vec: Tuple[float, float, float], angle: float) -> np.ndarray:\n    v = np.asarray(axis_vec, float)\n    nrm = np.linalg.norm(v)\n    if nrm == 0:\n        raise ValueError(\"axis_vec must be nonzero\")\n    k = v \u002F nrm\n    K = np.array([[0.0, -k[2], k[1]], [k[2], 0.0, -k[0]], [-k[1], k[0], 0.0]], float)\n    I = np.eye(3)\n    c, s = np.cos(angle), np.sin(angle)\n    return c * I + (1 - c) * np.outer(k, k) + s * K\n\n\ndef _n_grid(n_theta: int, n_phi: int):\n    theta = np.linspace(0.0, np.pi, n_theta)\n    phi = np.linspace(0.0, 2.0 * np.pi, n_phi)\n    th, ph = np.meshgrid(theta, phi, indexing=\"ij\")\n    nx = np.sin(th) * np.cos(ph)\n    ny = np.sin(th) * np.sin(ph)\n    nz = np.cos(th)\n    n = np.stack((nx, ny, nz), axis=0)\n    return th, ph, n\n\n\ndef _wigner_qubit(r_vec: np.ndarray, n: np.ndarray) -> np.ndarray:\n    return 0.5 + (np.sqrt(3.0) \u002F 2.0) * (r_vec.reshape(3, 1, 1) * n).sum(axis=0)\n\n\ndef animate_wigner_custom_3d(k: Knobs3DCustom) -> Path:\n    \"\"\"Animate a 3D sphere with colors driven by Wigner under axis–angle rotation.\"\"\"\n    r0 = _bloch_vector(k.initial_state, k.custom_r)\n    th, ph, n = _n_grid(k.n_theta, k.n_phi)\n\n    X = np.sin(th) * np.cos(ph)\n    Y = np.sin(th) * np.sin(ph)\n    Z = np.cos(th)\n\n    w_min = 0.5 - (np.sqrt(3.0) \u002F 2.0)\n    w_max = 0.5 + (np.sqrt(3.0) \u002F 2.0)\n    norm = Normalize(vmin=w_min, vmax=w_max)\n    cmap = plt.get_cmap()\n    sm = ScalarMappable(norm=norm, cmap=cmap)\n    sm.set_array([])\n\n    fig = plt.figure(figsize=k.figsize, constrained_layout=True)\n    ax = fig.add_subplot(111, projection=\"3d\")\n    ax.view_init(elev=k.elev, azim=k.azim)\n    if hasattr(ax, \"set_box_aspect\"):\n        ax.set_box_aspect((1, 1, 1))\n\n    # Initial facecolors\n    angle0 = 0.0\n    r_init = _rotation_matrix_axis(k.axis_vec, angle0) @ r0\n    W0 = _wigner_qubit(r_init, n)\n    FC0 = cmap(norm(W0[:-1, :-1]))  # (M-1,N-1,4)\n\n    surf = ax.plot_surface(\n        X, Y, Z,\n        facecolors=FC0,\n        rstride=1, cstride=1,\n        antialiased=False, linewidth=0,\n        shade=False,\n    )\n\n    cbar = fig.colorbar(sm, ax=ax, fraction=0.046, pad=0.04)\n    cbar.set_label(\"Wigner value\")\n\n    ax.set_xlabel(\"x\")\n    ax.set_ylabel(\"y\")\n    ax.set_zlabel(\"z\")\n    title_txt = ax.text2D(\n        0.5, 0.98, \"Wigner - custom angle\",\n        transform=ax.transAxes, ha=\"center\", va=\"top\", fontsize=11,\n        bbox=dict(boxstyle=\"round,pad=0.3\", fc=\"white\", alpha=0.75, ec=\"none\"),\n        zorder=6,\n    )\n    info_txt = ax.text2D(\n        0.02, 0.96, \"\", transform=ax.transAxes, ha=\"left\", va=\"top\",\n        fontsize=k.info_fontsize,\n        bbox=dict(boxstyle=\"round,pad=0.3\", fc=\"white\", alpha=0.75, ec=\"none\"),\n        zorder=6,\n    )\n\n    def update(frame: int):\n        ang = (frame \u002F (k.frames - 1)) * np.deg2rad(k.angle_deg)\n        r_now = _rotation_matrix_axis(k.axis_vec, ang) @ r0\n        W_now = _wigner_qubit(r_now, n)\n        FC_now = cmap(norm(W_now[:-1, :-1]))          # (M-1,N-1,4)\n        surf.set_facecolors(FC_now.reshape(-1, 4))    # in-place update\n        info_txt.set_text(\n            f\"axis = ({k.axis_vec[0]:.2f}, {k.axis_vec[1]:.2f}, {k.axis_vec[2]:.2f})\\n\"\n            f\"θ = {np.rad2deg(ang):.1f}° \u002F {k.angle_deg:.1f}°\"\n        )\n        return (surf, info_txt, title_txt)\n\n    anim = FuncAnimation(fig, update, frames=k.frames, interval=60, blit=False)\n    out = Path(k.out_path)\n    out.parent.mkdir(parents=True, exist_ok=True)\n    anim.save(out.as_posix(), writer=PillowWriter(fps=k.fps),\n              savefig_kwargs=dict(facecolor=\"white\"))\n    plt.close(fig)\n    return out\n\n\n# ----------------------------- Example run ------------------------------------\nK3C = Knobs3DCustom(\n    axis_vec=(0.0, 1.0, 0.0), angle_deg=45.0,\n    initial_state=\"+z\", out_path=\"\u002Fcontent\u002Fwigner3d_custom_y45.gif\",\n)\nsaved_custom = animate_wigner_custom_3d(K3C)\ndisplay(Image(filename=saved_custom.as_posix()))\nprint(\"Saved:\", saved_custom)\n",[10088,17725,17726,17731,17736,17740,17750,17754,17764,17774,17785,17789,17799,17809,17819,17829,17839,17849,17853,17857,17861,17870,17875,17912,17924,17934,17966,17976,17986,17996,18006,18016,18028,18040,18064,18074,18078,18082,18087,18119,18127,18163,18199,18235,18239,18263,18279,18289,18293,18297,18330,18346,18358,18370,18380,18392,18456,18472,18488,18524,18528,18532,18556,18572,18594,18614,18634,18654,18666,18686,18692,18696,18700,18716,18776,18780,18784,18798,18803,18813,18823,18827,18847,18867,18879,18883,18909,18935,18957,18969,18991,19000,19004,19030,19054,19074,19086,19106,19110,19115,19125,19141,19152,19181,19185,19197,19201,19211,19229,19247,19257,19261,19265,19299,19311,19315,19327,19339,19351,19363,19378,19410,19450,19460,19464,19476,19512,19520,19560,19570,19574,19578,19594,19621,19636,19646,19675,19705,19713,19766,19797,19801,19807,19811,19843,19854,19878,19904,19920,19928,19934,19938,19942,19947,19958,19986,20006,20010,20025,20047],{"__ignoreMap":10090},[40,17727,17728],{"class":10276,"line":10277},[40,17729,17730],{"class":10280},"# @title Cell 6 — 3D Wigner animation (custom axis–angle; colors-only, no geometry edits)\n",[40,17732,17733],{"class":10276,"line":10284},[40,17734,17735],{"class":10280},"# PEP 8\u002F257 compliant. Rotates Bloch vector about an arbitrary axis by a target angle.\n",[40,17737,17738],{"class":10276,"line":10291},[40,17739,10301],{"emptyLinePlaceholder":10300},[40,17741,17742,17744,17746,17748],{"class":10276,"line":10297},[40,17743,10353],{"class":10352},[40,17745,10357],{"class":10356},[40,17747,10360],{"class":10352},[40,17749,10364],{"class":10363},[40,17751,17752],{"class":10276,"line":10304},[40,17753,10301],{"emptyLinePlaceholder":10300},[40,17755,17756,17758,17760,17762],{"class":10276,"line":10310},[40,17757,10353],{"class":10352},[40,17759,10530],{"class":10363},[40,17761,10375],{"class":10352},[40,17763,10535],{"class":10363},[40,17765,17766,17768,17770,17772],{"class":10276,"line":10316},[40,17767,10353],{"class":10352},[40,17769,10542],{"class":10363},[40,17771,10375],{"class":10352},[40,17773,10547],{"class":10363},[40,17775,17776,17778,17780,17782],{"class":10276,"line":10322},[40,17777,10353],{"class":10352},[40,17779,10554],{"class":10363},[40,17781,10375],{"class":10352},[40,17783,17784],{"class":10363}," Tuple\n",[40,17786,17787],{"class":10276,"line":10327},[40,17788,10301],{"emptyLinePlaceholder":10300},[40,17790,17791,17793,17795,17797],{"class":10276,"line":10333},[40,17792,10375],{"class":10352},[40,17794,10378],{"class":10363},[40,17796,10381],{"class":10352},[40,17798,10384],{"class":10363},[40,17800,17801,17803,17805,17807],{"class":10276,"line":10339},[40,17802,10375],{"class":10352},[40,17804,10405],{"class":10363},[40,17806,10381],{"class":10352},[40,17808,10410],{"class":10363},[40,17810,17811,17813,17815,17817],{"class":10276,"line":10344},[40,17812,10353],{"class":10352},[40,17814,10597],{"class":10363},[40,17816,10375],{"class":10352},[40,17818,10602],{"class":10363},[40,17820,17821,17823,17825,17827],{"class":10276,"line":10349},[40,17822,10353],{"class":10352},[40,17824,15117],{"class":10363},[40,17826,10375],{"class":10352},[40,17828,15122],{"class":10363},[40,17830,17831,17833,17835,17837],{"class":10276,"line":10367},[40,17832,10353],{"class":10352},[40,17834,15129],{"class":10363},[40,17836,10375],{"class":10352},[40,17838,15134],{"class":10363},[40,17840,17841,17843,17845,17847],{"class":10276,"line":10372},[40,17842,10353],{"class":10352},[40,17844,10609],{"class":10363},[40,17846,10375],{"class":10352},[40,17848,10614],{"class":10363},[40,17850,17851],{"class":10276,"line":10387},[40,17852,10301],{"emptyLinePlaceholder":10300},[40,17854,17855],{"class":10276,"line":10400},[40,17856,10301],{"emptyLinePlaceholder":10300},[40,17858,17859],{"class":10276,"line":10413},[40,17860,10627],{"class":10430},[40,17862,17863,17865,17868],{"class":10276,"line":10418},[40,17864,10632],{"class":10352},[40,17866,17867],{"class":10635}," Knobs3DCustom",[40,17869,10639],{"class":10363},[40,17871,17872],{"class":10276,"line":10424},[40,17873,17874],{"class":10287},"    \"\"\"User controls for 3D custom axis–angle Wigner animation.\"\"\"\n",[40,17876,17877,17880,17882,17884,17886,17888,17890,17892,17894,17896,17898,17900,17902,17904,17906,17909],{"class":10276,"line":10437},[40,17878,17879],{"class":10363},"    axis_vec: Tuple[",[40,17881,10784],{"class":10732},[40,17883,10484],{"class":10363},[40,17885,10784],{"class":10732},[40,17887,10484],{"class":10363},[40,17889,10784],{"class":10732},[40,17891,10741],{"class":10363},[40,17893,162],{"class":10732},[40,17895,920],{"class":10363},[40,17897,10801],{"class":10446},[40,17899,10484],{"class":10363},[40,17901,10810],{"class":10446},[40,17903,10484],{"class":10363},[40,17905,10801],{"class":10446},[40,17907,17908],{"class":10363},")  ",[40,17910,17911],{"class":10280},"# default: y-axis\n",[40,17913,17914,17917,17919,17921],{"class":10276,"line":10456},[40,17915,17916],{"class":10363},"    angle_deg: ",[40,17918,10784],{"class":10732},[40,17920,10772],{"class":10732},[40,17922,17923],{"class":10446}," 45.0\n",[40,17925,17926,17928,17930,17932],{"class":10276,"line":10472},[40,17927,10767],{"class":10363},[40,17929,10733],{"class":10732},[40,17931,10772],{"class":10732},[40,17933,10775],{"class":10287},[40,17935,17936,17938,17940,17942,17944,17946,17948,17950,17952,17954,17956,17958,17960,17962,17964],{"class":10276,"line":10496},[40,17937,10781],{"class":10363},[40,17939,10784],{"class":10732},[40,17941,10484],{"class":10363},[40,17943,10784],{"class":10732},[40,17945,10484],{"class":10363},[40,17947,10784],{"class":10732},[40,17949,10741],{"class":10363},[40,17951,162],{"class":10732},[40,17953,920],{"class":10363},[40,17955,10801],{"class":10446},[40,17957,10484],{"class":10363},[40,17959,10801],{"class":10446},[40,17961,10484],{"class":10363},[40,17963,10810],{"class":10446},[40,17965,10761],{"class":10363},[40,17967,17968,17970,17972,17974],{"class":10276,"line":10510},[40,17969,10818],{"class":10363},[40,17971,10821],{"class":10732},[40,17973,10772],{"class":10732},[40,17975,10826],{"class":10446},[40,17977,17978,17980,17982,17984],{"class":10276,"line":10696},[40,17979,10832],{"class":10363},[40,17981,10821],{"class":10732},[40,17983,10772],{"class":10732},[40,17985,15298],{"class":10446},[40,17987,17988,17990,17992,17994],{"class":10276,"line":10702},[40,17989,10845],{"class":10363},[40,17991,10821],{"class":10732},[40,17993,10772],{"class":10732},[40,17995,15309],{"class":10446},[40,17997,17998,18000,18002,18004],{"class":10276,"line":10708},[40,17999,10858],{"class":10363},[40,18001,10821],{"class":10732},[40,18003,10772],{"class":10732},[40,18005,10865],{"class":10446},[40,18007,18008,18010,18012,18014],{"class":10276,"line":10714},[40,18009,15324],{"class":10363},[40,18011,10784],{"class":10732},[40,18013,10772],{"class":10732},[40,18015,15331],{"class":10446},[40,18017,18018,18020,18022,18024,18026],{"class":10276,"line":10720},[40,18019,15336],{"class":10363},[40,18021,10784],{"class":10732},[40,18023,10772],{"class":10732},[40,18025,11764],{"class":10732},[40,18027,15345],{"class":10446},[40,18029,18030,18033,18035,18037],{"class":10276,"line":10726},[40,18031,18032],{"class":10363},"    out_path: ",[40,18034,10733],{"class":10732},[40,18036,10772],{"class":10732},[40,18038,18039],{"class":10287}," \"\u002Fcontent\u002Fwigner3d_custom_y45.gif\"\n",[40,18041,18042,18044,18046,18048,18050,18052,18054,18056,18058,18060,18062],{"class":10276,"line":10764},[40,18043,10884],{"class":10363},[40,18045,10784],{"class":10732},[40,18047,10484],{"class":10363},[40,18049,10784],{"class":10732},[40,18051,10741],{"class":10363},[40,18053,162],{"class":10732},[40,18055,920],{"class":10363},[40,18057,10481],{"class":10446},[40,18059,10484],{"class":10363},[40,18061,15378],{"class":10446},[40,18063,10761],{"class":10363},[40,18065,18066,18068,18070,18072],{"class":10276,"line":10778},[40,18067,10923],{"class":10363},[40,18069,10821],{"class":10732},[40,18071,10772],{"class":10732},[40,18073,10930],{"class":10446},[40,18075,18076],{"class":10276,"line":10815},[40,18077,10301],{"emptyLinePlaceholder":10300},[40,18079,18080],{"class":10276,"line":10829},[40,18081,10301],{"emptyLinePlaceholder":10300},[40,18083,18084],{"class":10276,"line":10842},[40,18085,18086],{"class":10280},"# ------------- helpers (self-contained; safe for this cell alone) -------------\n",[40,18088,18089,18091,18093,18095,18097,18099,18101,18103,18105,18107,18109,18111,18113,18115,18117],{"class":10276,"line":10855},[40,18090,10965],{"class":10352},[40,18092,15410],{"class":10430},[40,18094,167],{"class":10363},[40,18096,15415],{"class":10973},[40,18098,10443],{"class":10363},[40,18100,10733],{"class":10732},[40,18102,10484],{"class":10363},[40,18104,15424],{"class":10973},[40,18106,15427],{"class":10363},[40,18108,10801],{"class":10446},[40,18110,10484],{"class":10363},[40,18112,10801],{"class":10446},[40,18114,10484],{"class":10363},[40,18116,10810],{"class":10446},[40,18118,15440],{"class":10363},[40,18120,18121,18123,18125],{"class":10276,"line":10868},[40,18122,11014],{"class":10363},[40,18124,162],{"class":10732},[40,18126,11019],{"class":10363},[40,18128,18129,18131,18133,18135,18137,18139,18141,18143,18145,18147,18149,18151,18153,18155,18157,18159,18161],{"class":10276,"line":10881},[40,18130,11025],{"class":10287},[40,18132,10478],{"class":10363},[40,18134,10801],{"class":10446},[40,18136,10484],{"class":10363},[40,18138,10801],{"class":10446},[40,18140,10484],{"class":10363},[40,18142,10810],{"class":10446},[40,18144,11040],{"class":10363},[40,18146,11043],{"class":10287},[40,18148,10478],{"class":10363},[40,18150,10801],{"class":10446},[40,18152,10484],{"class":10363},[40,18154,10801],{"class":10446},[40,18156,10484],{"class":10363},[40,18158,11056],{"class":10732},[40,18160,10810],{"class":10446},[40,18162,11061],{"class":10363},[40,18164,18165,18167,18169,18171,18173,18175,18177,18179,18181,18183,18185,18187,18189,18191,18193,18195,18197],{"class":10276,"line":10907},[40,18166,11067],{"class":10287},[40,18168,10478],{"class":10363},[40,18170,10810],{"class":10446},[40,18172,10484],{"class":10363},[40,18174,10801],{"class":10446},[40,18176,10484],{"class":10363},[40,18178,10801],{"class":10446},[40,18180,11040],{"class":10363},[40,18182,11084],{"class":10287},[40,18184,10478],{"class":10363},[40,18186,11056],{"class":10732},[40,18188,10810],{"class":10446},[40,18190,10484],{"class":10363},[40,18192,10801],{"class":10446},[40,18194,10484],{"class":10363},[40,18196,10801],{"class":10446},[40,18198,11061],{"class":10363},[40,18200,18201,18203,18205,18207,18209,18211,18213,18215,18217,18219,18221,18223,18225,18227,18229,18231,18233],{"class":10276,"line":10920},[40,18202,11106],{"class":10287},[40,18204,10478],{"class":10363},[40,18206,10801],{"class":10446},[40,18208,10484],{"class":10363},[40,18210,10810],{"class":10446},[40,18212,10484],{"class":10363},[40,18214,10801],{"class":10446},[40,18216,11040],{"class":10363},[40,18218,11123],{"class":10287},[40,18220,10478],{"class":10363},[40,18222,10801],{"class":10446},[40,18224,10484],{"class":10363},[40,18226,11056],{"class":10732},[40,18228,10810],{"class":10446},[40,18230,10484],{"class":10363},[40,18232,10801],{"class":10446},[40,18234,11061],{"class":10363},[40,18236,18237],{"class":10276,"line":10933},[40,18238,11145],{"class":10363},[40,18240,18241,18243,18245,18247,18249,18251,18253,18255,18257,18259,18261],{"class":10276,"line":10946},[40,18242,11151],{"class":10363},[40,18244,162],{"class":10732},[40,18246,11156],{"class":10363},[40,18248,11159],{"class":10430},[40,18250,11162],{"class":10363},[40,18252,11165],{"class":10430},[40,18254,15577],{"class":10363},[40,18256,11171],{"class":10430},[40,18258,15582],{"class":10363},[40,18260,10784],{"class":10732},[40,18262,10761],{"class":10363},[40,18264,18265,18267,18269,18271,18273,18275,18277],{"class":10276,"line":10951},[40,18266,11184],{"class":10363},[40,18268,162],{"class":10732},[40,18270,11189],{"class":10363},[40,18272,11192],{"class":10430},[40,18274,15599],{"class":10363},[40,18276,15602],{"class":10352},[40,18278,15605],{"class":10446},[40,18280,18281,18283,18285,18287],{"class":10276,"line":10956},[40,18282,11201],{"class":10352},[40,18284,11240],{"class":10363},[40,18286,11243],{"class":10732},[40,18288,11246],{"class":10363},[40,18290,18291],{"class":10276,"line":10962},[40,18292,10301],{"emptyLinePlaceholder":10300},[40,18294,18295],{"class":10276,"line":10983},[40,18296,10301],{"emptyLinePlaceholder":10300},[40,18298,18299,18301,18304,18306,18308,18310,18312,18314,18316,18318,18320,18322,18324,18326,18328],{"class":10276,"line":11005},[40,18300,10965],{"class":10352},[40,18302,18303],{"class":10430}," _rotation_matrix_axis",[40,18305,167],{"class":10363},[40,18307,11527],{"class":10973},[40,18309,10989],{"class":10363},[40,18311,10784],{"class":10732},[40,18313,10484],{"class":10363},[40,18315,10784],{"class":10732},[40,18317,10484],{"class":10363},[40,18319,10784],{"class":10732},[40,18321,11672],{"class":10363},[40,18323,11284],{"class":10973},[40,18325,10443],{"class":10363},[40,18327,10784],{"class":10732},[40,18329,11291],{"class":10363},[40,18331,18332,18334,18336,18338,18340,18342,18344],{"class":10276,"line":11011},[40,18333,11566],{"class":10363},[40,18335,162],{"class":10732},[40,18337,11156],{"class":10363},[40,18339,11159],{"class":10430},[40,18341,11575],{"class":10363},[40,18343,10784],{"class":10732},[40,18345,10761],{"class":10363},[40,18347,18348,18350,18352,18354,18356],{"class":10276,"line":11022},[40,18349,11184],{"class":10363},[40,18351,162],{"class":10732},[40,18353,11189],{"class":10363},[40,18355,11192],{"class":10430},[40,18357,11593],{"class":10363},[40,18359,18360,18362,18364,18366,18368],{"class":10276,"line":11064},[40,18361,11323],{"class":10352},[40,18363,11228],{"class":10363},[40,18365,11231],{"class":10732},[40,18367,11234],{"class":10446},[40,18369,10639],{"class":10363},[40,18371,18372,18374,18376,18378],{"class":10276,"line":11103},[40,18373,11612],{"class":10352},[40,18375,11499],{"class":10363},[40,18377,11617],{"class":10287},[40,18379,10761],{"class":10363},[40,18381,18382,18384,18386,18388,18390],{"class":10276,"line":11142},[40,18383,11625],{"class":10363},[40,18385,162],{"class":10732},[40,18387,11630],{"class":10363},[40,18389,11243],{"class":10732},[40,18391,11246],{"class":10363},[40,18393,18394,18396,18398,18400,18402,18404,18406,18408,18410,18412,18414,18416,18418,18420,18422,18424,18426,18428,18430,18432,18434,18436,18438,18440,18442,18444,18446,18448,18450,18452,18454],{"class":10276,"line":11148},[40,18395,11640],{"class":10363},[40,18397,162],{"class":10732},[40,18399,11156],{"class":10363},[40,18401,11206],{"class":10430},[40,18403,11346],{"class":10363},[40,18405,10801],{"class":10446},[40,18407,10484],{"class":10363},[40,18409,11056],{"class":10732},[40,18411,11657],{"class":10363},[40,18413,1044],{"class":10446},[40,18415,11662],{"class":10363},[40,18417,78],{"class":10446},[40,18419,11667],{"class":10363},[40,18421,1044],{"class":10446},[40,18423,11672],{"class":10363},[40,18425,10801],{"class":10446},[40,18427,10484],{"class":10363},[40,18429,11056],{"class":10732},[40,18431,11657],{"class":10363},[40,18433,66],{"class":10446},[40,18435,11685],{"class":10363},[40,18437,11056],{"class":10732},[40,18439,11657],{"class":10363},[40,18441,78],{"class":10446},[40,18443,11662],{"class":10363},[40,18445,66],{"class":10446},[40,18447,11672],{"class":10363},[40,18449,10801],{"class":10446},[40,18451,11486],{"class":10363},[40,18453,10784],{"class":10732},[40,18455,10761],{"class":10363},[40,18457,18458,18460,18462,18464,18466,18468,18470],{"class":10276,"line":11181},[40,18459,11711],{"class":10363},[40,18461,162],{"class":10732},[40,18463,11156],{"class":10363},[40,18465,11718],{"class":10430},[40,18467,167],{"class":10363},[40,18469,949],{"class":10446},[40,18471,10761],{"class":10363},[40,18473,18474,18476,18478,18480,18482,18484,18486],{"class":10276,"line":11198},[40,18475,11303],{"class":10363},[40,18477,162],{"class":10732},[40,18479,11156],{"class":10363},[40,18481,1820],{"class":10430},[40,18483,11312],{"class":10363},[40,18485,1813],{"class":10430},[40,18487,11317],{"class":10363},[40,18489,18490,18492,18494,18496,18498,18500,18502,18504,18506,18508,18510,18512,18514,18516,18518,18520,18522],{"class":10276,"line":11249},[40,18491,11201],{"class":10352},[40,18493,11749],{"class":10363},[40,18495,11752],{"class":10732},[40,18497,11755],{"class":10363},[40,18499,1053],{"class":10732},[40,18501,920],{"class":10363},[40,18503,78],{"class":10446},[40,18505,11764],{"class":10732},[40,18507,11767],{"class":10363},[40,18509,11752],{"class":10732},[40,18511,11156],{"class":10363},[40,18513,11774],{"class":10430},[40,18515,11777],{"class":10363},[40,18517,1053],{"class":10732},[40,18519,11782],{"class":10363},[40,18521,11752],{"class":10732},[40,18523,11787],{"class":10363},[40,18525,18526],{"class":10276,"line":11254},[40,18527,10301],{"emptyLinePlaceholder":10300},[40,18529,18530],{"class":10276,"line":11259},[40,18531,10301],{"emptyLinePlaceholder":10300},[40,18533,18534,18536,18538,18540,18542,18544,18546,18548,18550,18552,18554],{"class":10276,"line":11265},[40,18535,10965],{"class":10352},[40,18537,15840],{"class":10430},[40,18539,167],{"class":10363},[40,18541,11816],{"class":10973},[40,18543,10443],{"class":10363},[40,18545,10821],{"class":10732},[40,18547,10484],{"class":10363},[40,18549,11825],{"class":10973},[40,18551,10443],{"class":10363},[40,18553,10821],{"class":10732},[40,18555,13353],{"class":10363},[40,18557,18558,18560,18562,18564,18566,18568,18570],{"class":10276,"line":11294},[40,18559,11843],{"class":10363},[40,18561,162],{"class":10732},[40,18563,11156],{"class":10363},[40,18565,11850],{"class":10430},[40,18567,167],{"class":10363},[40,18569,10801],{"class":10446},[40,18571,11857],{"class":10363},[40,18573,18574,18576,18578,18580,18582,18584,18586,18588,18590,18592],{"class":10276,"line":11300},[40,18575,11863],{"class":10363},[40,18577,162],{"class":10732},[40,18579,11156],{"class":10363},[40,18581,11850],{"class":10430},[40,18583,167],{"class":10363},[40,18585,10801],{"class":10446},[40,18587,10484],{"class":10363},[40,18589,11878],{"class":10446},[40,18591,11881],{"class":10732},[40,18593,11884],{"class":10363},[40,18595,18596,18598,18600,18602,18604,18606,18608,18610,18612],{"class":10276,"line":11320},[40,18597,11890],{"class":10363},[40,18599,162],{"class":10732},[40,18601,11156],{"class":10363},[40,18603,11897],{"class":10430},[40,18605,11900],{"class":10363},[40,18607,11904],{"class":11903},[40,18609,162],{"class":10732},[40,18611,11909],{"class":10287},[40,18613,10761],{"class":10363},[40,18615,18616,18618,18620,18622,18624,18626,18628,18630,18632],{"class":10276,"line":11336},[40,18617,11917],{"class":10363},[40,18619,162],{"class":10732},[40,18621,11156],{"class":10363},[40,18623,1813],{"class":10430},[40,18625,11926],{"class":10363},[40,18627,11752],{"class":10732},[40,18629,11156],{"class":10363},[40,18631,1820],{"class":10430},[40,18633,11935],{"class":10363},[40,18635,18636,18638,18640,18642,18644,18646,18648,18650,18652],{"class":10276,"line":11381},[40,18637,11941],{"class":10363},[40,18639,162],{"class":10732},[40,18641,11156],{"class":10363},[40,18643,1813],{"class":10430},[40,18645,11926],{"class":10363},[40,18647,11752],{"class":10732},[40,18649,11156],{"class":10363},[40,18651,1813],{"class":10430},[40,18653,11935],{"class":10363},[40,18655,18656,18658,18660,18662,18664],{"class":10276,"line":11395},[40,18657,11963],{"class":10363},[40,18659,162],{"class":10732},[40,18661,11156],{"class":10363},[40,18663,1820],{"class":10430},[40,18665,11972],{"class":10363},[40,18667,18668,18670,18672,18674,18676,18678,18680,18682,18684],{"class":10276,"line":11438},[40,18669,15978],{"class":10363},[40,18671,162],{"class":10732},[40,18673,11156],{"class":10363},[40,18675,11982],{"class":10430},[40,18677,11985],{"class":10363},[40,18679,11275],{"class":11903},[40,18681,162],{"class":10732},[40,18683,66],{"class":10446},[40,18685,10761],{"class":10363},[40,18687,18688,18690],{"class":10276,"line":11452},[40,18689,11201],{"class":10352},[40,18691,16001],{"class":10363},[40,18693,18694],{"class":10276,"line":11493},[40,18695,10301],{"emptyLinePlaceholder":10300},[40,18697,18698],{"class":10276,"line":11507},[40,18699,10301],{"emptyLinePlaceholder":10300},[40,18701,18702,18704,18706,18708,18710,18712,18714],{"class":10276,"line":11512},[40,18703,10965],{"class":10352},[40,18705,16016],{"class":10430},[40,18707,167],{"class":10363},[40,18709,12016],{"class":10973},[40,18711,12019],{"class":10363},[40,18713,1796],{"class":10973},[40,18715,12025],{"class":10363},[40,18717,18718,18720,18722,18724,18726,18728,18730,18732,18734,18736,18738,18740,18742,18744,18746,18748,18750,18752,18754,18756,18758,18760,18762,18764,18766,18768,18770,18772,18774],{"class":10276,"line":11517},[40,18719,11201],{"class":10352},[40,18721,12039],{"class":10446},[40,18723,12042],{"class":10732},[40,18725,12045],{"class":10363},[40,18727,2218],{"class":10430},[40,18729,167],{"class":10363},[40,18731,12052],{"class":10446},[40,18733,12055],{"class":10363},[40,18735,11243],{"class":10732},[40,18737,12060],{"class":10446},[40,18739,12055],{"class":10363},[40,18741,11752],{"class":10732},[40,18743,12067],{"class":10363},[40,18745,12070],{"class":10430},[40,18747,167],{"class":10363},[40,18749,949],{"class":10446},[40,18751,10484],{"class":10363},[40,18753,78],{"class":10446},[40,18755,10484],{"class":10363},[40,18757,78],{"class":10446},[40,18759,12055],{"class":10363},[40,18761,11752],{"class":10732},[40,18763,16080],{"class":10363},[40,18765,12092],{"class":10430},[40,18767,167],{"class":10363},[40,18769,11275],{"class":11903},[40,18771,162],{"class":10732},[40,18773,66],{"class":10446},[40,18775,10761],{"class":10363},[40,18777,18778],{"class":10276,"line":11545},[40,18779,10301],{"emptyLinePlaceholder":10300},[40,18781,18782],{"class":10276,"line":11557},[40,18783,10301],{"emptyLinePlaceholder":10300},[40,18785,18786,18788,18791,18793,18795],{"class":10276,"line":11563},[40,18787,10965],{"class":10352},[40,18789,18790],{"class":10430}," animate_wigner_custom_3d",[40,18792,167],{"class":10363},[40,18794,13346],{"class":10973},[40,18796,18797],{"class":10363},": Knobs3DCustom) -> Path:\n",[40,18799,18800],{"class":10276,"line":11582},[40,18801,18802],{"class":10287},"    \"\"\"Animate a 3D sphere with colors driven by Wigner under axis–angle rotation.\"\"\"\n",[40,18804,18805,18807,18809,18811],{"class":10276,"line":11596},[40,18806,16132],{"class":10363},[40,18808,162],{"class":10732},[40,18810,15410],{"class":10430},[40,18812,16139],{"class":10363},[40,18814,18815,18817,18819,18821],{"class":10276,"line":11609},[40,18816,16144],{"class":10363},[40,18818,162],{"class":10732},[40,18820,15840],{"class":10430},[40,18822,16151],{"class":10363},[40,18824,18825],{"class":10276,"line":11622},[40,18826,10301],{"emptyLinePlaceholder":10300},[40,18828,18829,18831,18833,18835,18837,18839,18841,18843,18845],{"class":10276,"line":11637},[40,18830,16165],{"class":10363},[40,18832,162],{"class":10732},[40,18834,11156],{"class":10363},[40,18836,1813],{"class":10430},[40,18838,11926],{"class":10363},[40,18840,11752],{"class":10732},[40,18842,11156],{"class":10363},[40,18844,1820],{"class":10430},[40,18846,11935],{"class":10363},[40,18848,18849,18851,18853,18855,18857,18859,18861,18863,18865],{"class":10276,"line":11708},[40,18850,16186],{"class":10363},[40,18852,162],{"class":10732},[40,18854,11156],{"class":10363},[40,18856,1813],{"class":10430},[40,18858,11926],{"class":10363},[40,18860,11752],{"class":10732},[40,18862,11156],{"class":10363},[40,18864,1813],{"class":10430},[40,18866,11935],{"class":10363},[40,18868,18869,18871,18873,18875,18877],{"class":10276,"line":11727},[40,18870,16207],{"class":10363},[40,18872,162],{"class":10732},[40,18874,11156],{"class":10363},[40,18876,1820],{"class":10430},[40,18878,11972],{"class":10363},[40,18880,18881],{"class":10276,"line":11744},[40,18882,10301],{"emptyLinePlaceholder":10300},[40,18884,18885,18887,18889,18891,18893,18895,18897,18899,18901,18903,18905,18907],{"class":10276,"line":11790},[40,18886,12982],{"class":10363},[40,18888,162],{"class":10732},[40,18890,12039],{"class":10446},[40,18892,11764],{"class":10732},[40,18894,12045],{"class":10363},[40,18896,2218],{"class":10430},[40,18898,167],{"class":10363},[40,18900,12052],{"class":10446},[40,18902,12055],{"class":10363},[40,18904,11243],{"class":10732},[40,18906,12060],{"class":10446},[40,18908,10761],{"class":10363},[40,18910,18911,18913,18915,18917,18919,18921,18923,18925,18927,18929,18931,18933],{"class":10276,"line":11795},[40,18912,13009],{"class":10363},[40,18914,162],{"class":10732},[40,18916,12039],{"class":10446},[40,18918,12042],{"class":10732},[40,18920,12045],{"class":10363},[40,18922,2218],{"class":10430},[40,18924,167],{"class":10363},[40,18926,12052],{"class":10446},[40,18928,12055],{"class":10363},[40,18930,11243],{"class":10732},[40,18932,12060],{"class":10446},[40,18934,10761],{"class":10363},[40,18936,18937,18939,18941,18943,18945,18947,18949,18951,18953,18955],{"class":10276,"line":11800},[40,18938,16281],{"class":10363},[40,18940,162],{"class":10732},[40,18942,16286],{"class":10430},[40,18944,167],{"class":10363},[40,18946,16291],{"class":11903},[40,18948,162],{"class":10732},[40,18950,13142],{"class":10363},[40,18952,13145],{"class":11903},[40,18954,162],{"class":10732},[40,18956,16302],{"class":10363},[40,18958,18959,18961,18963,18965,18967],{"class":10276,"line":11806},[40,18960,16307],{"class":10363},[40,18962,162],{"class":10732},[40,18964,13070],{"class":10363},[40,18966,16314],{"class":10430},[40,18968,12174],{"class":10363},[40,18970,18971,18973,18975,18977,18979,18981,18983,18985,18987,18989],{"class":10276,"line":11834},[40,18972,16321],{"class":10363},[40,18974,162],{"class":10732},[40,18976,16326],{"class":10430},[40,18978,167],{"class":10363},[40,18980,11192],{"class":11903},[40,18982,162],{"class":10732},[40,18984,16335],{"class":10363},[40,18986,16338],{"class":11903},[40,18988,162],{"class":10732},[40,18990,16343],{"class":10363},[40,18992,18993,18995,18997],{"class":10276,"line":11840},[40,18994,16348],{"class":10363},[40,18996,16351],{"class":10430},[40,18998,18999],{"class":10363},"([])\n",[40,19001,19002],{"class":10276,"line":11860},[40,19003,10301],{"emptyLinePlaceholder":10300},[40,19005,19006,19008,19010,19012,19014,19016,19018,19020,19022,19024,19026,19028],{"class":10276,"line":11887},[40,19007,16366],{"class":10363},[40,19009,162],{"class":10732},[40,19011,13070],{"class":10363},[40,19013,16373],{"class":10430},[40,19015,167],{"class":10363},[40,19017,13078],{"class":11903},[40,19019,162],{"class":10732},[40,19021,16382],{"class":10363},[40,19023,13086],{"class":11903},[40,19025,162],{"class":10732},[40,19027,13091],{"class":10446},[40,19029,10761],{"class":10363},[40,19031,19032,19034,19036,19038,19040,19042,19044,19046,19048,19050,19052],{"class":10276,"line":11914},[40,19033,16395],{"class":10363},[40,19035,162],{"class":10732},[40,19037,13170],{"class":10363},[40,19039,16402],{"class":10430},[40,19041,167],{"class":10363},[40,19043,16407],{"class":10446},[40,19045,10484],{"class":10363},[40,19047,16412],{"class":11903},[40,19049,162],{"class":10732},[40,19051,16417],{"class":10287},[40,19053,10761],{"class":10363},[40,19055,19056,19058,19060,19062,19064,19066,19068,19070,19072],{"class":10276,"line":11938},[40,19057,13227],{"class":10363},[40,19059,16426],{"class":10430},[40,19061,167],{"class":10363},[40,19063,16431],{"class":11903},[40,19065,162],{"class":10732},[40,19067,16436],{"class":10363},[40,19069,16439],{"class":11903},[40,19071,162],{"class":10732},[40,19073,16444],{"class":10363},[40,19075,19076,19078,19080,19082,19084],{"class":10276,"line":11960},[40,19077,11323],{"class":10352},[40,19079,16451],{"class":10732},[40,19081,13259],{"class":10363},[40,19083,16456],{"class":10287},[40,19085,13353],{"class":10363},[40,19087,19088,19090,19092,19094,19096,19098,19100,19102,19104],{"class":10276,"line":11975},[40,19089,16463],{"class":10363},[40,19091,16466],{"class":10430},[40,19093,16469],{"class":10363},[40,19095,78],{"class":10446},[40,19097,10484],{"class":10363},[40,19099,78],{"class":10446},[40,19101,10484],{"class":10363},[40,19103,78],{"class":10446},[40,19105,13612],{"class":10363},[40,19107,19108],{"class":10276,"line":11996},[40,19109,10301],{"emptyLinePlaceholder":10300},[40,19111,19112],{"class":10276,"line":12001},[40,19113,19114],{"class":10280},"    # Initial facecolors\n",[40,19116,19117,19120,19122],{"class":10276,"line":12006},[40,19118,19119],{"class":10363},"    angle0 ",[40,19121,162],{"class":10732},[40,19123,19124],{"class":10446}," 0.0\n",[40,19126,19127,19130,19132,19134,19137,19139],{"class":10276,"line":12028},[40,19128,19129],{"class":10363},"    r_init ",[40,19131,162],{"class":10732},[40,19133,18303],{"class":10430},[40,19135,19136],{"class":10363},"(k.axis_vec, angle0) ",[40,19138,13404],{"class":10732},[40,19140,13407],{"class":10363},[40,19142,19143,19145,19147,19149],{"class":10276,"line":12034},[40,19144,16495],{"class":10363},[40,19146,162],{"class":10732},[40,19148,16016],{"class":10430},[40,19150,19151],{"class":10363},"(r_init, n)\n",[40,19153,19154,19156,19158,19160,19162,19164,19166,19168,19170,19172,19174,19176,19178],{"class":10276,"line":12105},[40,19155,16521],{"class":10446},[40,19157,10772],{"class":10732},[40,19159,16526],{"class":10430},[40,19161,167],{"class":10363},[40,19163,11192],{"class":10430},[40,19165,16533],{"class":10363},[40,19167,11056],{"class":10732},[40,19169,78],{"class":10446},[40,19171,16540],{"class":10363},[40,19173,11056],{"class":10732},[40,19175,78],{"class":10446},[40,19177,16547],{"class":10363},[40,19179,19180],{"class":10280},"# (M-1,N-1,4)\n",[40,19182,19183],{"class":10276,"line":12110},[40,19184,10301],{"emptyLinePlaceholder":10300},[40,19186,19187,19189,19191,19193,19195],{"class":10276,"line":12115},[40,19188,16559],{"class":10363},[40,19190,162],{"class":10732},[40,19192,12522],{"class":10363},[40,19194,16566],{"class":10430},[40,19196,12527],{"class":10363},[40,19198,19199],{"class":10276,"line":12121},[40,19200,16573],{"class":10363},[40,19202,19203,19205,19207,19209],{"class":10276,"line":12155},[40,19204,16578],{"class":11903},[40,19206,162],{"class":10732},[40,19208,16583],{"class":10446},[40,19210,10507],{"class":10363},[40,19212,19213,19215,19217,19219,19221,19223,19225,19227],{"class":10276,"line":12161},[40,19214,16590],{"class":11903},[40,19216,162],{"class":10732},[40,19218,78],{"class":10446},[40,19220,10484],{"class":10363},[40,19222,16599],{"class":11903},[40,19224,162],{"class":10732},[40,19226,78],{"class":10446},[40,19228,10507],{"class":10363},[40,19230,19231,19233,19235,19237,19239,19241,19243,19245],{"class":10276,"line":12177},[40,19232,16610],{"class":11903},[40,19234,162],{"class":10732},[40,19236,10504],{"class":10446},[40,19238,10484],{"class":10363},[40,19240,16619],{"class":11903},[40,19242,162],{"class":10732},[40,19244,66],{"class":10446},[40,19246,10507],{"class":10363},[40,19248,19249,19251,19253,19255],{"class":10276,"line":12192},[40,19250,16630],{"class":11903},[40,19252,162],{"class":10732},[40,19254,10504],{"class":10446},[40,19256,10507],{"class":10363},[40,19258,19259],{"class":10276,"line":12231},[40,19260,12631],{"class":10363},[40,19262,19263],{"class":10276,"line":12245},[40,19264,10301],{"emptyLinePlaceholder":10300},[40,19266,19267,19269,19271,19273,19275,19277,19279,19281,19283,19285,19287,19289,19291,19293,19295,19297],{"class":10276,"line":12280},[40,19268,13165],{"class":10363},[40,19270,162],{"class":10732},[40,19272,13170],{"class":10363},[40,19274,13173],{"class":10430},[40,19276,16666],{"class":10363},[40,19278,12429],{"class":11903},[40,19280,162],{"class":10732},[40,19282,13183],{"class":10363},[40,19284,13186],{"class":11903},[40,19286,162],{"class":10732},[40,19288,13191],{"class":10446},[40,19290,10484],{"class":10363},[40,19292,13196],{"class":11903},[40,19294,162],{"class":10732},[40,19296,13201],{"class":10446},[40,19298,10761],{"class":10363},[40,19300,19301,19303,19305,19307,19309],{"class":10276,"line":12294},[40,19302,13208],{"class":10363},[40,19304,13211],{"class":10430},[40,19306,167],{"class":10363},[40,19308,13216],{"class":10287},[40,19310,10761],{"class":10363},[40,19312,19313],{"class":10276,"line":12329},[40,19314,10301],{"emptyLinePlaceholder":10300},[40,19316,19317,19319,19321,19323,19325],{"class":10276,"line":12343},[40,19318,13227],{"class":10363},[40,19320,13230],{"class":10430},[40,19322,167],{"class":10363},[40,19324,10748],{"class":10287},[40,19326,10761],{"class":10363},[40,19328,19329,19331,19333,19335,19337],{"class":10276,"line":12376},[40,19330,13227],{"class":10363},[40,19332,13244],{"class":10430},[40,19334,167],{"class":10363},[40,19336,10753],{"class":10287},[40,19338,10761],{"class":10363},[40,19340,19341,19343,19345,19347,19349],{"class":10276,"line":12409},[40,19342,13227],{"class":10363},[40,19344,16740],{"class":10430},[40,19346,167],{"class":10363},[40,19348,10758],{"class":10287},[40,19350,10761],{"class":10363},[40,19352,19353,19355,19357,19359,19361],{"class":10276,"line":12414},[40,19354,16751],{"class":10363},[40,19356,162],{"class":10732},[40,19358,12522],{"class":10363},[40,19360,16758],{"class":10430},[40,19362,12527],{"class":10363},[40,19364,19365,19367,19369,19371,19373,19376],{"class":10276,"line":12419},[40,19366,12687],{"class":10446},[40,19368,10484],{"class":10363},[40,19370,16769],{"class":10446},[40,19372,10484],{"class":10363},[40,19374,19375],{"class":10287},"\"Wigner - custom angle\"",[40,19377,10507],{"class":10363},[40,19379,19380,19382,19384,19386,19388,19390,19392,19394,19396,19398,19400,19402,19404,19406,19408],{"class":10276,"line":12435},[40,19381,12545],{"class":11903},[40,19383,162],{"class":10732},[40,19385,12550],{"class":10363},[40,19387,12211],{"class":11903},[40,19389,162],{"class":10732},[40,19391,12710],{"class":10287},[40,19393,10484],{"class":10363},[40,19395,12221],{"class":11903},[40,19397,162],{"class":10732},[40,19399,12226],{"class":10287},[40,19401,10484],{"class":10363},[40,19403,12553],{"class":11903},[40,19405,162],{"class":10732},[40,19407,12731],{"class":10446},[40,19409,10507],{"class":10363},[40,19411,19412,19414,19416,19418,19420,19422,19424,19426,19428,19430,19432,19434,19436,19438,19440,19442,19444,19446,19448],{"class":10276,"line":12447},[40,19413,12574],{"class":11903},[40,19415,12577],{"class":10732},[40,19417,167],{"class":10363},[40,19419,12582],{"class":11903},[40,19421,162],{"class":10732},[40,19423,12587],{"class":10287},[40,19425,10484],{"class":10363},[40,19427,12592],{"class":11903},[40,19429,162],{"class":10732},[40,19431,12597],{"class":10287},[40,19433,10484],{"class":10363},[40,19435,12602],{"class":11903},[40,19437,162],{"class":10732},[40,19439,12607],{"class":10446},[40,19441,10484],{"class":10363},[40,19443,12612],{"class":11903},[40,19445,162],{"class":10732},[40,19447,12617],{"class":10287},[40,19449,11061],{"class":10363},[40,19451,19452,19454,19456,19458],{"class":10276,"line":12463},[40,19453,12780],{"class":11903},[40,19455,162],{"class":10732},[40,19457,12785],{"class":10446},[40,19459,10507],{"class":10363},[40,19461,19462],{"class":10276,"line":12480},[40,19463,12631],{"class":10363},[40,19465,19466,19468,19470,19472,19474],{"class":10276,"line":12498},[40,19467,16881],{"class":10363},[40,19469,162],{"class":10732},[40,19471,12522],{"class":10363},[40,19473,16758],{"class":10430},[40,19475,12527],{"class":10363},[40,19477,19478,19480,19482,19484,19486,19488,19490,19492,19494,19496,19498,19500,19502,19504,19506,19508,19510],{"class":10276,"line":12504},[40,19479,16894],{"class":10446},[40,19481,10484],{"class":10363},[40,19483,16899],{"class":10446},[40,19485,10484],{"class":10363},[40,19487,13303],{"class":10287},[40,19489,10484],{"class":10363},[40,19491,16908],{"class":11903},[40,19493,162],{"class":10732},[40,19495,12550],{"class":10363},[40,19497,12211],{"class":11903},[40,19499,162],{"class":10732},[40,19501,12216],{"class":10287},[40,19503,10484],{"class":10363},[40,19505,12221],{"class":11903},[40,19507,162],{"class":10732},[40,19509,12226],{"class":10287},[40,19511,10507],{"class":10363},[40,19513,19514,19516,19518],{"class":10276,"line":12517},[40,19515,12726],{"class":11903},[40,19517,162],{"class":10732},[40,19519,16937],{"class":10363},[40,19521,19522,19524,19526,19528,19530,19532,19534,19536,19538,19540,19542,19544,19546,19548,19550,19552,19554,19556,19558],{"class":10276,"line":12530},[40,19523,12574],{"class":11903},[40,19525,12577],{"class":10732},[40,19527,167],{"class":10363},[40,19529,12582],{"class":11903},[40,19531,162],{"class":10732},[40,19533,12587],{"class":10287},[40,19535,10484],{"class":10363},[40,19537,12592],{"class":11903},[40,19539,162],{"class":10732},[40,19541,12597],{"class":10287},[40,19543,10484],{"class":10363},[40,19545,12602],{"class":11903},[40,19547,162],{"class":10732},[40,19549,12607],{"class":10446},[40,19551,10484],{"class":10363},[40,19553,12612],{"class":11903},[40,19555,162],{"class":10732},[40,19557,12617],{"class":10287},[40,19559,11061],{"class":10363},[40,19561,19562,19564,19566,19568],{"class":10276,"line":12542},[40,19563,12780],{"class":11903},[40,19565,162],{"class":10732},[40,19567,12785],{"class":10446},[40,19569,10507],{"class":10363},[40,19571,19572],{"class":10276,"line":12571},[40,19573,12631],{"class":10363},[40,19575,19576],{"class":10276,"line":12622},[40,19577,10301],{"emptyLinePlaceholder":10300},[40,19579,19580,19582,19584,19586,19588,19590,19592],{"class":10276,"line":12628},[40,19581,13338],{"class":10352},[40,19583,13341],{"class":10430},[40,19585,167],{"class":10363},[40,19587,17010],{"class":10973},[40,19589,10443],{"class":10363},[40,19591,10821],{"class":10732},[40,19593,13353],{"class":10363},[40,19595,19596,19598,19600,19602,19604,19606,19608,19610,19612,19614,19616,19618],{"class":10276,"line":12634},[40,19597,13358],{"class":10363},[40,19599,162],{"class":10732},[40,19601,17027],{"class":10363},[40,19603,11243],{"class":10732},[40,19605,17032],{"class":10363},[40,19607,11056],{"class":10732},[40,19609,13373],{"class":10446},[40,19611,13376],{"class":10363},[40,19613,11752],{"class":10732},[40,19615,11156],{"class":10363},[40,19617,14527],{"class":10430},[40,19619,19620],{"class":10363},"(k.angle_deg)\n",[40,19622,19623,19625,19627,19629,19632,19634],{"class":10276,"line":12639},[40,19624,17050],{"class":10363},[40,19626,162],{"class":10732},[40,19628,18303],{"class":10430},[40,19630,19631],{"class":10363},"(k.axis_vec, ang) ",[40,19633,13404],{"class":10732},[40,19635,13407],{"class":10363},[40,19637,19638,19640,19642,19644],{"class":10276,"line":12644},[40,19639,17067],{"class":10363},[40,19641,162],{"class":10732},[40,19643,16016],{"class":10430},[40,19645,17074],{"class":10363},[40,19647,19648,19650,19652,19654,19656,19658,19660,19662,19664,19666,19668,19670,19673],{"class":10276,"line":12667},[40,19649,17080],{"class":10446},[40,19651,10772],{"class":10732},[40,19653,16526],{"class":10430},[40,19655,167],{"class":10363},[40,19657,11192],{"class":10430},[40,19659,17091],{"class":10363},[40,19661,11056],{"class":10732},[40,19663,78],{"class":10446},[40,19665,16540],{"class":10363},[40,19667,11056],{"class":10732},[40,19669,78],{"class":10446},[40,19671,19672],{"class":10363},"]))          ",[40,19674,19180],{"class":10280},[40,19676,19677,19679,19681,19683,19685,19687,19689,19691,19693,19695,19697,19699,19702],{"class":10276,"line":12673},[40,19678,17113],{"class":10363},[40,19680,17116],{"class":10430},[40,19682,167],{"class":10363},[40,19684,17121],{"class":10446},[40,19686,1123],{"class":10363},[40,19688,12070],{"class":10430},[40,19690,167],{"class":10363},[40,19692,11056],{"class":10732},[40,19694,78],{"class":10446},[40,19696,10484],{"class":10363},[40,19698,17136],{"class":10446},[40,19700,19701],{"class":10363},"))    ",[40,19703,19704],{"class":10280},"# in-place update\n",[40,19706,19707,19709,19711],{"class":10276,"line":12684},[40,19708,17148],{"class":10363},[40,19710,13431],{"class":10430},[40,19712,12527],{"class":10363},[40,19714,19715,19717,19719,19721,19724,19726,19728,19730,19732,19734,19736,19738,19740,19742,19744,19746,19748,19750,19752,19754,19756,19758,19760,19762,19764],{"class":10276,"line":12697},[40,19716,14580],{"class":10352},[40,19718,14583],{"class":10287},[40,19720,13268],{"class":10446},[40,19722,19723],{"class":10363},"k.axis_vec[",[40,19725,66],{"class":10446},[40,19727,1664],{"class":10363},[40,19729,14595],{"class":10352},[40,19731,13280],{"class":10446},[40,19733,10484],{"class":10287},[40,19735,13268],{"class":10446},[40,19737,19723],{"class":10363},[40,19739,78],{"class":10446},[40,19741,1664],{"class":10363},[40,19743,14595],{"class":10352},[40,19745,13280],{"class":10446},[40,19747,10484],{"class":10287},[40,19749,13268],{"class":10446},[40,19751,19723],{"class":10363},[40,19753,1044],{"class":10446},[40,19755,1664],{"class":10363},[40,19757,14595],{"class":10352},[40,19759,13280],{"class":10446},[40,19761,207],{"class":10287},[40,19763,13454],{"class":10732},[40,19765,14632],{"class":10287},[40,19767,19768,19770,19772,19774,19776,19778,19780,19782,19784,19786,19788,19791,19793,19795],{"class":10276,"line":12723},[40,19769,14580],{"class":10352},[40,19771,13438],{"class":10287},[40,19773,13268],{"class":10446},[40,19775,14643],{"class":10363},[40,19777,14646],{"class":10430},[40,19779,14649],{"class":10363},[40,19781,14652],{"class":10352},[40,19783,13280],{"class":10446},[40,19785,14657],{"class":10287},[40,19787,13268],{"class":10446},[40,19789,19790],{"class":10363},"k.angle_deg",[40,19792,14652],{"class":10352},[40,19794,13280],{"class":10446},[40,19796,14669],{"class":10287},[40,19798,19799],{"class":10276,"line":12736},[40,19800,14674],{"class":10363},[40,19802,19803,19805],{"class":10276,"line":12777},[40,19804,11339],{"class":10352},[40,19806,17199],{"class":10363},[40,19808,19809],{"class":10276,"line":12790},[40,19810,10301],{"emptyLinePlaceholder":10300},[40,19812,19813,19815,19817,19819,19821,19823,19825,19827,19829,19831,19833,19835,19837,19839,19841],{"class":10276,"line":16980},[40,19814,13496],{"class":10363},[40,19816,162],{"class":10732},[40,19818,13501],{"class":10430},[40,19820,13504],{"class":10363},[40,19822,13473],{"class":11903},[40,19824,162],{"class":10732},[40,19826,17222],{"class":10363},[40,19828,13514],{"class":11903},[40,19830,162],{"class":10732},[40,19832,13519],{"class":10446},[40,19834,10484],{"class":10363},[40,19836,13524],{"class":11903},[40,19838,162],{"class":10732},[40,19840,10504],{"class":10446},[40,19842,10761],{"class":10363},[40,19844,19845,19847,19849,19851],{"class":10276,"line":16991},[40,19846,17244],{"class":10363},[40,19848,162],{"class":10732},[40,19850,13859],{"class":10430},[40,19852,19853],{"class":10363},"(k.out_path)\n",[40,19855,19856,19858,19860,19862,19864,19866,19868,19870,19872,19874,19876],{"class":10276,"line":16996},[40,19857,17272],{"class":10363},[40,19859,13538],{"class":10430},[40,19861,167],{"class":10363},[40,19863,13543],{"class":11903},[40,19865,162],{"class":10732},[40,19867,13091],{"class":10446},[40,19869,10484],{"class":10363},[40,19871,13552],{"class":11903},[40,19873,162],{"class":10732},[40,19875,13091],{"class":10446},[40,19877,10761],{"class":10363},[40,19879,19880,19882,19884,19886,19888,19890,19892,19894,19896,19898,19900,19902],{"class":10276,"line":17001},[40,19881,13563],{"class":10363},[40,19883,13566],{"class":10430},[40,19885,17302],{"class":10363},[40,19887,13572],{"class":10430},[40,19889,13575],{"class":10363},[40,19891,13578],{"class":11903},[40,19893,162],{"class":10732},[40,19895,13583],{"class":10430},[40,19897,167],{"class":10363},[40,19899,13588],{"class":11903},[40,19901,162],{"class":10732},[40,19903,17321],{"class":10363},[40,19905,19906,19908,19910,19912,19914,19916,19918],{"class":10276,"line":17019},[40,19907,13598],{"class":11903},[40,19909,12577],{"class":10732},[40,19911,167],{"class":10363},[40,19913,13605],{"class":11903},[40,19915,162],{"class":10732},[40,19917,12597],{"class":10287},[40,19919,13612],{"class":10363},[40,19921,19922,19924,19926],{"class":10276,"line":17047},[40,19923,13617],{"class":10363},[40,19925,13620],{"class":10430},[40,19927,13623],{"class":10363},[40,19929,19930,19932],{"class":10276,"line":17064},[40,19931,11201],{"class":10352},[40,19933,17355],{"class":10363},[40,19935,19936],{"class":10276,"line":17077},[40,19937,10301],{"emptyLinePlaceholder":10300},[40,19939,19940],{"class":10276,"line":17110},[40,19941,10301],{"emptyLinePlaceholder":10300},[40,19943,19944],{"class":10276,"line":17145},[40,19945,19946],{"class":10280},"# ----------------------------- Example run ------------------------------------\n",[40,19948,19949,19952,19954,19956],{"class":10276,"line":17194},[40,19950,19951],{"class":10446},"K3C",[40,19953,10772],{"class":10732},[40,19955,17867],{"class":10430},[40,19957,12527],{"class":10363},[40,19959,19960,19962,19964,19966,19968,19970,19972,19974,19976,19978,19980,19982,19984],{"class":10276,"line":17202},[40,19961,14837],{"class":11903},[40,19963,162],{"class":10732},[40,19965,167],{"class":10363},[40,19967,10801],{"class":10446},[40,19969,10484],{"class":10363},[40,19971,10810],{"class":10446},[40,19973,10484],{"class":10363},[40,19975,10801],{"class":10446},[40,19977,11040],{"class":10363},[40,19979,14662],{"class":11903},[40,19981,162],{"class":10732},[40,19983,14863],{"class":10446},[40,19985,10507],{"class":10363},[40,19987,19988,19990,19992,19994,19996,19999,20001,20004],{"class":10276,"line":17207},[40,19989,13684],{"class":11903},[40,19991,162],{"class":10732},[40,19993,13689],{"class":10287},[40,19995,10484],{"class":10363},[40,19997,19998],{"class":11903},"out_path",[40,20000,162],{"class":10732},[40,20002,20003],{"class":10287},"\"\u002Fcontent\u002Fwigner3d_custom_y45.gif\"",[40,20005,10507],{"class":10363},[40,20007,20008],{"class":10276,"line":17241},[40,20009,10761],{"class":10363},[40,20011,20012,20015,20017,20019,20021,20023],{"class":10276,"line":17269},[40,20013,20014],{"class":10363},"saved_custom ",[40,20016,162],{"class":10732},[40,20018,18790],{"class":10430},[40,20020,167],{"class":10363},[40,20022,19951],{"class":10446},[40,20024,10761],{"class":10363},[40,20026,20027,20029,20031,20033,20035,20037,20039,20042,20044],{"class":10276,"line":17295},[40,20028,14975],{"class":10430},[40,20030,167],{"class":10363},[40,20032,13953],{"class":10430},[40,20034,167],{"class":10363},[40,20036,13958],{"class":11903},[40,20038,162],{"class":10732},[40,20040,20041],{"class":10363},"saved_custom.",[40,20043,13572],{"class":10430},[40,20045,20046],{"class":10363},"()))\n",[40,20048,20049,20051,20053,20055],{"class":10276,"line":17324},[40,20050,14993],{"class":10732},[40,20052,167],{"class":10363},[40,20054,13974],{"class":10287},[40,20056,20057],{"class":10363},", saved_custom)\n",[10082,20059,20062],{"className":20060,"code":20061,"language":2932,"meta":10090},[10085],"Saved: \u002Fcontent\u002Fwigner3d_custom_y45.gif\n",[10088,20063,20061],{"__ignoreMap":10090},[13997,20065],{"alt":20066,"src":20067},"Output 8 of the notebook","\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-08.webp",[20069,20070,20071],"style",{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .sU0A5, html code.shiki .sU0A5{--shiki-default:#E5C07B}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}",{"title":10090,"searchDepth":10284,"depth":10284,"links":20073},[20074,20075,20076,20078,20079,20081,20083,20084,20085,20086],{"id":30,"depth":10284,"text":31},{"id":1994,"depth":10284,"text":1995},{"id":4114,"depth":10284,"text":20077},"3. Pauli Gates as (π\\piπ)-Rotations",{"id":5126,"depth":10284,"text":5127},{"id":5935,"depth":10284,"text":20080},"5. Rodrigues’ Formula (Axis–Angle (→\\to→) Rotation Matrix)",{"id":8078,"depth":10284,"text":20082},"6. Optional: Discrete (2×22\\times22×2) (Finite‑State) Wigner",{"id":8992,"depth":10284,"text":8993},{"id":9740,"depth":10284,"text":9741},{"id":10076,"depth":10284,"text":10077},{"id":10095,"depth":10284,"text":10096},[20088,20089,20090,20091],"Qollab","Explore","Expert notes","Wigner functions",[20093],{"username":7,"name":20094,"role":20095,"bio":20096,"links":20097},"Onri Jay Benally","Quantum Hardware Engineer · Doctoral Researcher, University of Minnesota","Onri fabricates the hardware he writes about. Eight years in the J.P. Wang group at Minnesota have gone into nanoelectronics fabrication and materials integration: cryogenic MRAM, spin-orbit torque devices, spintronic quantum processor chips. He has trained twenty engineers and physicists on high-yield fabrication in an ISO 4 cleanroom, and has worked as a quantum hardware engineer at IBM's T.J. Watson lab. By his own account what he does best is turn technical jargon into practical layman's terms, which is what these notebooks are for.",[20098,20101],{"label":20099,"href":20100},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":20102,"href":21},"Original notebook ↗",{"username":7,"name":20094,"role":20104},"Quantum Hardware Engineer, University of Minnesota",null,"Pauli rotations rendered as Wigner functions, animated in two and three dimensions, with the density-operator and Bloch-vector groundwork.",false,"md","Deep dive · Visualization","article",{},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fwigner-functions-single-qubit-gates","2026-08-06","17 min read",[],{"title":5,"description":20106},"blog\u002Fexpert-notes\u002Fwigner-functions-single-qubit-gates",[20120,20121],"visualization","quantum","F21gsgnqd_6pE29RCUocKz6jo6vSs6rJlQM17vqwdBU",[20124,27290,39386,57961,83819],{"id":20125,"title":20126,"authors":20127,"body":20128,"breadcrumb":27270,"builders":27272,"byline":27277,"challenge":20105,"courseAuthor":20105,"courseLead":20105,"dek":20105,"description":27278,"draft":20107,"extension":20108,"eyebrow":27279,"featured":20107,"finish":20105,"fork":20105,"hero":20105,"heroAlt":20105,"heroCta":20105,"heroImage":20105,"homepageFeatured":20107,"kind":20110,"lessonCount":20105,"meta":27280,"navigation":10300,"newsItems":20105,"next":20105,"ogImage":27281,"order":20105,"outcomes":20105,"path":27282,"publishDate":20114,"readingTime":27283,"related":27284,"relatedProjects":20105,"seo":27285,"stem":27286,"tags":27287,"track":20105,"trackName":20105,"__hash__":27289},"blog\u002Fblog\u002Fexpert-notes\u002Fcross-resonance-gate-visualization.md","Cross-Resonance Gate: Slices and Heatmaps",[7],{"type":9,"value":20129,"toc":27265},[20130,20137,20141,21673,21675,21680,21758,24574,24577,24580,24583,24586,25565,25568,27259,27262],[12,20131,20132],{},[15,20133,17,20134,23],{},[19,20135,22],{"href":20136},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FCross_Resonance_Gate_Visualization.ipynb",[28,20138,20140],{"id":20139},"cross-resonance-cr-gate","Cross-Resonance (CR) Gate",[20142,20143,20144,20161],"table",{},[20145,20146,20147],"thead",{},[20148,20149,20150,20155,20158],"tr",{},[20151,20152,20154],"th",{"align":20153},"right","#",[20151,20156,20157],{},"Figure",[20151,20159,20160],{},"What to notice",[20162,20163,20164,20399,20631,20874,21072,21345,21516],"tbody",{},[20148,20165,20166,20169,20368],{},[20167,20168,78],"td",{"align":20153},[20167,20170,20171,20253,20254,20337,20338,207],{},[40,20172,20174,20197],{"className":20173},[43],[40,20175,20177],{"className":20176},[47],[49,20178,20179],{"xmlns":51},[53,20180,20181,20194],{},[56,20182,20183],{},[151,20184,20185,20188],{},[154,20186,20187],{},"ω",[56,20189,20190,20192],{},[154,20191,4263],{},[154,20193,4196],{},[82,20195,20196],{"encoding":84},"\\omega_{ZX}",[40,20198,20200],{"className":20199,"ariaHidden":72},[89],[40,20201,20203,20206],{"className":20202},[93],[40,20204],{"className":20205,"style":328},[97],[40,20207,20209,20212],{"className":20208},[102],[40,20210,20187],{"className":20211,"style":336},[102,335],[40,20213,20215],{"className":20214},[340],[40,20216,20218,20245],{"className":20217},[344,345],[40,20219,20221,20242],{"className":20220},[349],[40,20222,20225],{"className":20223,"style":20224},[353],"height:0.3283em;",[40,20226,20227,20230],{"style":357},[40,20228],{"className":20229,"style":362},[361],[40,20231,20233],{"className":20232},[366,367,368,369],[40,20234,20236,20239],{"className":20235},[102,369],[40,20237,4263],{"className":20238,"style":4668},[102,335,369],[40,20240,4196],{"className":20241,"style":1242},[102,335,369],[40,20243,377],{"className":20244},[376],[40,20246,20248],{"className":20247},[349],[40,20249,20251],{"className":20250,"style":384},[353],[40,20252],{}," vs ",[40,20255,20257,20281],{"className":20256},[43],[40,20258,20260],{"className":20259},[47],[49,20261,20262],{"xmlns":51},[53,20263,20264,20278],{},[56,20265,20266],{},[151,20267,20268,20270],{},[154,20269,2016],{"mathvariant":1080},[56,20271,20272,20275],{},[154,20273,20274],{},"c",[154,20276,20277],{},"t",[82,20279,20280],{"encoding":84},"\\Delta_{ct}",[40,20282,20284],{"className":20283,"ariaHidden":72},[89],[40,20285,20287,20290],{"className":20286},[93],[40,20288],{"className":20289,"style":3106},[97],[40,20291,20293,20296],{"className":20292},[102],[40,20294,2016],{"className":20295},[102],[40,20297,20299],{"className":20298},[340],[40,20300,20302,20329],{"className":20301},[344,345],[40,20303,20305,20326],{"className":20304},[349],[40,20306,20309],{"className":20307,"style":20308},[353],"height:0.2806em;",[40,20310,20311,20314],{"style":6098},[40,20312],{"className":20313,"style":362},[361],[40,20315,20317],{"className":20316},[366,367,368,369],[40,20318,20320,20323],{"className":20319},[102,369],[40,20321,20274],{"className":20322},[102,335,369],[40,20324,20277],{"className":20325},[102,335,369],[40,20327,377],{"className":20328},[376],[40,20330,20332],{"className":20331},[349],[40,20333,20335],{"className":20334,"style":384},[353],[40,20336],{}," (slice at fixed ",[40,20339,20341,20356],{"className":20340},[43],[40,20342,20344],{"className":20343},[47],[49,20345,20346],{"xmlns":51},[53,20347,20348,20353],{},[56,20349,20350],{},[154,20351,20352],{"mathvariant":1080},"Ω",[82,20354,20355],{"encoding":84},"\\Omega",[40,20357,20359],{"className":20358,"ariaHidden":72},[89],[40,20360,20362,20365],{"className":20361},[93],[40,20363],{"className":20364,"style":4304},[97],[40,20366,20352],{"className":20367},[102],[20167,20369,20370,20371,1123],{},"Entangling-rate sign flips at poles; magnitude peaks between poles; roughly linear in drive amplitude ",[40,20372,20374,20387],{"className":20373},[43],[40,20375,20377],{"className":20376},[47],[49,20378,20379],{"xmlns":51},[53,20380,20381,20385],{},[56,20382,20383],{},[154,20384,20352],{"mathvariant":1080},[82,20386,20355],{"encoding":84},[40,20388,20390],{"className":20389,"ariaHidden":72},[89],[40,20391,20393,20396],{"className":20392},[93],[40,20394],{"className":20395,"style":4304},[97],[40,20397,20352],{"className":20398},[102],[20148,20400,20401,20403,20564],{},[20167,20402,1044],{"align":20153},[20167,20404,20405,20253,20485],{},[40,20406,20408,20430],{"className":20407},[43],[40,20409,20411],{"className":20410},[47],[49,20412,20413],{"xmlns":51},[53,20414,20415,20427],{},[56,20416,20417],{},[151,20418,20419,20421],{},[154,20420,20187],{},[56,20422,20423,20425],{},[154,20424,4263],{},[154,20426,4263],{},[82,20428,20429],{"encoding":84},"\\omega_{ZZ}",[40,20431,20433],{"className":20432,"ariaHidden":72},[89],[40,20434,20436,20439],{"className":20435},[93],[40,20437],{"className":20438,"style":328},[97],[40,20440,20442,20445],{"className":20441},[102],[40,20443,20187],{"className":20444,"style":336},[102,335],[40,20446,20448],{"className":20447},[340],[40,20449,20451,20477],{"className":20450},[344,345],[40,20452,20454,20474],{"className":20453},[349],[40,20455,20457],{"className":20456,"style":20224},[353],[40,20458,20459,20462],{"style":357},[40,20460],{"className":20461,"style":362},[361],[40,20463,20465],{"className":20464},[366,367,368,369],[40,20466,20468,20471],{"className":20467},[102,369],[40,20469,4263],{"className":20470,"style":4668},[102,335,369],[40,20472,4263],{"className":20473,"style":4668},[102,335,369],[40,20475,377],{"className":20476},[376],[40,20478,20480],{"className":20479},[349],[40,20481,20483],{"className":20482,"style":384},[353],[40,20484],{},[40,20486,20488,20509],{"className":20487},[43],[40,20489,20491],{"className":20490},[47],[49,20492,20493],{"xmlns":51},[53,20494,20495,20507],{},[56,20496,20497],{},[151,20498,20499,20501],{},[154,20500,2016],{"mathvariant":1080},[56,20502,20503,20505],{},[154,20504,20274],{},[154,20506,20277],{},[82,20508,20280],{"encoding":84},[40,20510,20512],{"className":20511,"ariaHidden":72},[89],[40,20513,20515,20518],{"className":20514},[93],[40,20516],{"className":20517,"style":3106},[97],[40,20519,20521,20524],{"className":20520},[102],[40,20522,2016],{"className":20523},[102],[40,20525,20527],{"className":20526},[340],[40,20528,20530,20556],{"className":20529},[344,345],[40,20531,20533,20553],{"className":20532},[349],[40,20534,20536],{"className":20535,"style":20308},[353],[40,20537,20538,20541],{"style":6098},[40,20539],{"className":20540,"style":362},[361],[40,20542,20544],{"className":20543},[366,367,368,369],[40,20545,20547,20550],{"className":20546},[102,369],[40,20548,20274],{"className":20549},[102,335,369],[40,20551,20277],{"className":20552},[102,335,369],[40,20554,377],{"className":20555},[376],[40,20557,20559],{"className":20558},[349],[40,20560,20562],{"className":20561,"style":384},[353],[40,20563],{},[20167,20565,20566,20567,20601,20602,20630],{},"Static ",[40,20568,20570,20586],{"className":20569},[43],[40,20571,20573],{"className":20572},[47],[49,20574,20575],{"xmlns":51},[53,20576,20577,20583],{},[56,20578,20579,20581],{},[154,20580,4263],{},[154,20582,4263],{},[82,20584,20585],{"encoding":84},"ZZ",[40,20587,20589],{"className":20588,"ariaHidden":72},[89],[40,20590,20592,20595,20598],{"className":20591},[93],[40,20593],{"className":20594,"style":4304},[97],[40,20596,4263],{"className":20597,"style":4668},[102,335],[40,20599,4263],{"className":20600,"style":4668},[102,335]," varies strongly near poles; lowest between poles; independent of ",[40,20603,20605,20618],{"className":20604},[43],[40,20606,20608],{"className":20607},[47],[49,20609,20610],{"xmlns":51},[53,20611,20612,20616],{},[56,20613,20614],{},[154,20615,20352],{"mathvariant":1080},[82,20617,20355],{"encoding":84},[40,20619,20621],{"className":20620,"ariaHidden":72},[89],[40,20622,20624,20627],{"className":20623},[93],[40,20625],{"className":20626,"style":4304},[97],[40,20628,20352],{"className":20629},[102]," at leading order.",[20148,20632,20633,20635,20867],{},[20167,20634,949],{"align":20153},[20167,20636,20637,20253,20788],{},[40,20638,20640,20678],{"className":20639},[43],[40,20641,20643],{"className":20642},[47],[49,20644,20645],{"xmlns":51},[53,20646,20647,20675],{},[56,20648,20649,20651,20661,20663,20673],{},[59,20650,62],{"stretchy":61},[151,20652,20653,20655],{},[154,20654,20187],{},[56,20656,20657,20659],{},[154,20658,4263],{},[154,20660,4196],{},[154,20662,11243],{"mathvariant":1080},[151,20664,20665,20667],{},[154,20666,20187],{},[56,20668,20669,20671],{},[154,20670,4263],{},[154,20672,4263],{},[59,20674,62],{"stretchy":61},[82,20676,20677],{"encoding":84},"\\lvert \\omega_{ZX}\u002F\\omega_{ZZ}\\rvert",[40,20679,20681],{"className":20680,"ariaHidden":72},[89],[40,20682,20684,20687,20690,20736,20739,20785],{"className":20683},[93],[40,20685],{"className":20686,"style":98},[97],[40,20688,62],{"className":20689},[106],[40,20691,20693,20696],{"className":20692},[102],[40,20694,20187],{"className":20695,"style":336},[102,335],[40,20697,20699],{"className":20698},[340],[40,20700,20702,20728],{"className":20701},[344,345],[40,20703,20705,20725],{"className":20704},[349],[40,20706,20708],{"className":20707,"style":20224},[353],[40,20709,20710,20713],{"style":357},[40,20711],{"className":20712,"style":362},[361],[40,20714,20716],{"className":20715},[366,367,368,369],[40,20717,20719,20722],{"className":20718},[102,369],[40,20720,4263],{"className":20721,"style":4668},[102,335,369],[40,20723,4196],{"className":20724,"style":1242},[102,335,369],[40,20726,377],{"className":20727},[376],[40,20729,20731],{"className":20730},[349],[40,20732,20734],{"className":20733,"style":384},[353],[40,20735],{},[40,20737,11243],{"className":20738},[102],[40,20740,20742,20745],{"className":20741},[102],[40,20743,20187],{"className":20744,"style":336},[102,335],[40,20746,20748],{"className":20747},[340],[40,20749,20751,20777],{"className":20750},[344,345],[40,20752,20754,20774],{"className":20753},[349],[40,20755,20757],{"className":20756,"style":20224},[353],[40,20758,20759,20762],{"style":357},[40,20760],{"className":20761,"style":362},[361],[40,20763,20765],{"className":20764},[366,367,368,369],[40,20766,20768,20771],{"className":20767},[102,369],[40,20769,4263],{"className":20770,"style":4668},[102,335,369],[40,20772,4263],{"className":20773,"style":4668},[102,335,369],[40,20775,377],{"className":20776},[376],[40,20778,20780],{"className":20779},[349],[40,20781,20783],{"className":20782,"style":384},[353],[40,20784],{},[40,20786,62],{"className":20787},[113],[40,20789,20791,20812],{"className":20790},[43],[40,20792,20794],{"className":20793},[47],[49,20795,20796],{"xmlns":51},[53,20797,20798,20810],{},[56,20799,20800],{},[151,20801,20802,20804],{},[154,20803,2016],{"mathvariant":1080},[56,20805,20806,20808],{},[154,20807,20274],{},[154,20809,20277],{},[82,20811,20280],{"encoding":84},[40,20813,20815],{"className":20814,"ariaHidden":72},[89],[40,20816,20818,20821],{"className":20817},[93],[40,20819],{"className":20820,"style":3106},[97],[40,20822,20824,20827],{"className":20823},[102],[40,20825,2016],{"className":20826},[102],[40,20828,20830],{"className":20829},[340],[40,20831,20833,20859],{"className":20832},[344,345],[40,20834,20836,20856],{"className":20835},[349],[40,20837,20839],{"className":20838,"style":20308},[353],[40,20840,20841,20844],{"style":6098},[40,20842],{"className":20843,"style":362},[361],[40,20845,20847],{"className":20846},[366,367,368,369],[40,20848,20850,20853],{"className":20849},[102,369],[40,20851,20274],{"className":20852},[102,335,369],[40,20854,20277],{"className":20855},[102,335,369],[40,20857,377],{"className":20858},[376],[40,20860,20862],{"className":20861},[349],[40,20863,20865],{"className":20864,"style":384},[353],[40,20866],{},[20167,20868,20869,20870,20873],{},"Heuristic ",[35,20871,20872],{},"speed-vs-static"," figure; tall regions mark friendlier operating points (away from poles).",[20148,20875,20876,20878,21040],{},[20167,20877,17136],{"align":20153},[20167,20879,20880,21039],{},[40,20881,20883,20923],{"className":20882},[43],[40,20884,20886],{"className":20885},[47],[49,20887,20888],{"xmlns":51},[53,20889,20890,20920],{},[56,20891,20892,20902,20904,20914,20916,20918],{},[151,20893,20894,20896],{},[154,20895,20187],{},[56,20897,20898,20900],{},[154,20899,4263],{},[154,20901,4196],{},[59,20903,167],{"stretchy":61},[151,20905,20906,20908],{},[154,20907,2016],{"mathvariant":1080},[56,20909,20910,20912],{},[154,20911,20274],{},[154,20913,20277],{},[59,20915,73],{"separator":72},[154,20917,20352],{"mathvariant":1080},[59,20919,207],{"stretchy":61},[82,20921,20922],{"encoding":84},"\\omega_{ZX}(\\Delta_{ct}, \\Omega)",[40,20924,20926],{"className":20925,"ariaHidden":72},[89],[40,20927,20929,20932,20978,20981,21027,21030,21033,21036],{"className":20928},[93],[40,20930],{"className":20931,"style":98},[97],[40,20933,20935,20938],{"className":20934},[102],[40,20936,20187],{"className":20937,"style":336},[102,335],[40,20939,20941],{"className":20940},[340],[40,20942,20944,20970],{"className":20943},[344,345],[40,20945,20947,20967],{"className":20946},[349],[40,20948,20950],{"className":20949,"style":20224},[353],[40,20951,20952,20955],{"style":357},[40,20953],{"className":20954,"style":362},[361],[40,20956,20958],{"className":20957},[366,367,368,369],[40,20959,20961,20964],{"className":20960},[102,369],[40,20962,4263],{"className":20963,"style":4668},[102,335,369],[40,20965,4196],{"className":20966,"style":1242},[102,335,369],[40,20968,377],{"className":20969},[376],[40,20971,20973],{"className":20972},[349],[40,20974,20976],{"className":20975,"style":384},[353],[40,20977],{},[40,20979,167],{"className":20980},[106],[40,20982,20984,20987],{"className":20983},[102],[40,20985,2016],{"className":20986},[102],[40,20988,20990],{"className":20989},[340],[40,20991,20993,21019],{"className":20992},[344,345],[40,20994,20996,21016],{"className":20995},[349],[40,20997,20999],{"className":20998,"style":20308},[353],[40,21000,21001,21004],{"style":6098},[40,21002],{"className":21003,"style":362},[361],[40,21005,21007],{"className":21006},[366,367,368,369],[40,21008,21010,21013],{"className":21009},[102,369],[40,21011,20274],{"className":21012},[102,335,369],[40,21014,20277],{"className":21015},[102,335,369],[40,21017,377],{"className":21018},[376],[40,21020,21022],{"className":21021},[349],[40,21023,21025],{"className":21024,"style":384},[353],[40,21026],{},[40,21028,73],{"className":21029},[117],[40,21031],{"className":21032,"style":122},[121],[40,21034,20352],{"className":21035},[102],[40,21037,207],{"className":21038},[113]," heatmap",[20167,21041,21042,21043,21071],{},"Linear growth with ",[40,21044,21046,21059],{"className":21045},[43],[40,21047,21049],{"className":21048},[47],[49,21050,21051],{"xmlns":51},[53,21052,21053,21057],{},[56,21054,21055],{},[154,21056,20352],{"mathvariant":1080},[82,21058,20355],{"encoding":84},[40,21060,21062],{"className":21061,"ariaHidden":72},[89],[40,21063,21065,21068],{"className":21064},[93],[40,21066],{"className":21067,"style":4304},[97],[40,21069,20352],{"className":21070},[102],"; detuning structure set by anharmonicities; region boundaries visible as sharp changes.",[20148,21073,21074,21076,21308],{},[20167,21075,12566],{"align":20153},[20167,21077,21078,21039],{},[40,21079,21081,21137],{"className":21080},[43],[40,21082,21084],{"className":21083},[47],[49,21085,21086],{"xmlns":51},[53,21087,21088,21134],{},[56,21089,21090,21092,21102,21104,21114,21116,21118,21128,21130,21132],{},[59,21091,62],{"stretchy":61},[151,21093,21094,21096],{},[154,21095,20187],{},[56,21097,21098,21100],{},[154,21099,4263],{},[154,21101,4196],{},[154,21103,11243],{"mathvariant":1080},[151,21105,21106,21108],{},[154,21107,20187],{},[56,21109,21110,21112],{},[154,21111,4263],{},[154,21113,4263],{},[59,21115,62],{"stretchy":61},[59,21117,167],{"stretchy":61},[151,21119,21120,21122],{},[154,21121,2016],{"mathvariant":1080},[56,21123,21124,21126],{},[154,21125,20274],{},[154,21127,20277],{},[59,21129,73],{"separator":72},[154,21131,20352],{"mathvariant":1080},[59,21133,207],{"stretchy":61},[82,21135,21136],{"encoding":84},"\\lvert \\omega_{ZX}\u002F\\omega_{ZZ}\\rvert(\\Delta_{ct}, \\Omega)",[40,21138,21140],{"className":21139,"ariaHidden":72},[89],[40,21141,21143,21146,21149,21195,21198,21244,21247,21250,21296,21299,21302,21305],{"className":21142},[93],[40,21144],{"className":21145,"style":98},[97],[40,21147,62],{"className":21148},[106],[40,21150,21152,21155],{"className":21151},[102],[40,21153,20187],{"className":21154,"style":336},[102,335],[40,21156,21158],{"className":21157},[340],[40,21159,21161,21187],{"className":21160},[344,345],[40,21162,21164,21184],{"className":21163},[349],[40,21165,21167],{"className":21166,"style":20224},[353],[40,21168,21169,21172],{"style":357},[40,21170],{"className":21171,"style":362},[361],[40,21173,21175],{"className":21174},[366,367,368,369],[40,21176,21178,21181],{"className":21177},[102,369],[40,21179,4263],{"className":21180,"style":4668},[102,335,369],[40,21182,4196],{"className":21183,"style":1242},[102,335,369],[40,21185,377],{"className":21186},[376],[40,21188,21190],{"className":21189},[349],[40,21191,21193],{"className":21192,"style":384},[353],[40,21194],{},[40,21196,11243],{"className":21197},[102],[40,21199,21201,21204],{"className":21200},[102],[40,21202,20187],{"className":21203,"style":336},[102,335],[40,21205,21207],{"className":21206},[340],[40,21208,21210,21236],{"className":21209},[344,345],[40,21211,21213,21233],{"className":21212},[349],[40,21214,21216],{"className":21215,"style":20224},[353],[40,21217,21218,21221],{"style":357},[40,21219],{"className":21220,"style":362},[361],[40,21222,21224],{"className":21223},[366,367,368,369],[40,21225,21227,21230],{"className":21226},[102,369],[40,21228,4263],{"className":21229,"style":4668},[102,335,369],[40,21231,4263],{"className":21232,"style":4668},[102,335,369],[40,21234,377],{"className":21235},[376],[40,21237,21239],{"className":21238},[349],[40,21240,21242],{"className":21241,"style":384},[353],[40,21243],{},[40,21245,62],{"className":21246},[113],[40,21248,167],{"className":21249},[106],[40,21251,21253,21256],{"className":21252},[102],[40,21254,2016],{"className":21255},[102],[40,21257,21259],{"className":21258},[340],[40,21260,21262,21288],{"className":21261},[344,345],[40,21263,21265,21285],{"className":21264},[349],[40,21266,21268],{"className":21267,"style":20308},[353],[40,21269,21270,21273],{"style":6098},[40,21271],{"className":21272,"style":362},[361],[40,21274,21276],{"className":21275},[366,367,368,369],[40,21277,21279,21282],{"className":21278},[102,369],[40,21280,20274],{"className":21281},[102,335,369],[40,21283,20277],{"className":21284},[102,335,369],[40,21286,377],{"className":21287},[376],[40,21289,21291],{"className":21290},[349],[40,21292,21294],{"className":21293,"style":384},[353],[40,21295],{},[40,21297,73],{"className":21298},[117],[40,21300],{"className":21301,"style":122},[121],[40,21303,20352],{"className":21304},[102],[40,21306,207],{"className":21307},[113],[20167,21309,21310,21311,21344],{},"Bright bands away from poles suggest high entangling-rate with low static ",[40,21312,21314,21329],{"className":21313},[43],[40,21315,21317],{"className":21316},[47],[49,21318,21319],{"xmlns":51},[53,21320,21321,21327],{},[56,21322,21323,21325],{},[154,21324,4263],{},[154,21326,4263],{},[82,21328,20585],{"encoding":84},[40,21330,21332],{"className":21331,"ariaHidden":72},[89],[40,21333,21335,21338,21341],{"className":21334},[93],[40,21336],{"className":21337,"style":4304},[97],[40,21339,4263],{"className":21340,"style":4668},[102,335],[40,21342,4263],{"className":21343,"style":4668},[102,335]," (good for echoed-CR).",[20148,21346,21347,21349,21352],{},[20167,21348,12785],{"align":20153},[20167,21350,21351],{},"Echoed-CR coefficients (two-segment echo)",[20167,21353,21354,21355,21383,21384,21418,21419,21447,21448,10484,21482,21515],{},"Echo preserves odd-in-",[40,21356,21358,21371],{"className":21357},[43],[40,21359,21361],{"className":21360},[47],[49,21362,21363],{"xmlns":51},[53,21364,21365,21369],{},[56,21366,21367],{},[154,21368,20352],{"mathvariant":1080},[82,21370,20355],{"encoding":84},[40,21372,21374],{"className":21373,"ariaHidden":72},[89],[40,21375,21377,21380],{"className":21376},[93],[40,21378],{"className":21379,"style":4304},[97],[40,21381,20352],{"className":21382},[102]," ",[40,21385,21387,21403],{"className":21386},[43],[40,21388,21390],{"className":21389},[47],[49,21391,21392],{"xmlns":51},[53,21393,21394,21400],{},[56,21395,21396,21398],{},[154,21397,4263],{},[154,21399,4196],{},[82,21401,21402],{"encoding":84},"ZX",[40,21404,21406],{"className":21405,"ariaHidden":72},[89],[40,21407,21409,21412,21415],{"className":21408},[93],[40,21410],{"className":21411,"style":4304},[97],[40,21413,4263],{"className":21414,"style":4668},[102,335],[40,21416,4196],{"className":21417,"style":1242},[102,335]," and suppresses even-in-",[40,21420,21422,21435],{"className":21421},[43],[40,21423,21425],{"className":21424},[47],[49,21426,21427],{"xmlns":51},[53,21428,21429,21433],{},[56,21430,21431],{},[154,21432,20352],{"mathvariant":1080},[82,21434,20355],{"encoding":84},[40,21436,21438],{"className":21437,"ariaHidden":72},[89],[40,21439,21441,21444],{"className":21440},[93],[40,21442],{"className":21443,"style":4304},[97],[40,21445,20352],{"className":21446},[102]," terms (",[40,21449,21451,21467],{"className":21450},[43],[40,21452,21454],{"className":21453},[47],[49,21455,21456],{"xmlns":51},[53,21457,21458,21464],{},[56,21459,21460,21462],{},[154,21461,1050],{},[154,21463,4263],{},[82,21465,21466],{"encoding":84},"IZ",[40,21468,21470],{"className":21469,"ariaHidden":72},[89],[40,21471,21473,21476,21479],{"className":21472},[93],[40,21474],{"className":21475,"style":4304},[97],[40,21477,1050],{"className":21478,"style":1242},[102,335],[40,21480,4263],{"className":21481,"style":4668},[102,335],[40,21483,21485,21500],{"className":21484},[43],[40,21486,21488],{"className":21487},[47],[49,21489,21490],{"xmlns":51},[53,21491,21492,21498],{},[56,21493,21494,21496],{},[154,21495,4263],{},[154,21497,4263],{},[82,21499,20585],{"encoding":84},[40,21501,21503],{"className":21502,"ariaHidden":72},[89],[40,21504,21506,21509,21512],{"className":21505},[93],[40,21507],{"className":21508,"style":4304},[97],[40,21510,4263],{"className":21511,"style":4668},[102,335],[40,21513,4263],{"className":21514,"style":4668},[102,335],") to first order.",[20148,21517,21518,21521,21602],{},[20167,21519,21520],{"align":20153},"7",[20167,21522,21523,21524,20253,21563,207],{},"Conditional target Rabi (control ",[40,21525,21527,21545],{"className":21526},[43],[40,21528,21530],{"className":21529},[47],[49,21531,21532],{"xmlns":51},[53,21533,21534,21542],{},[56,21535,21536,21538,21540],{},[59,21537,62],{"stretchy":61},[64,21539,66],{},[59,21541,69],{"stretchy":61},[82,21543,21544],{"encoding":84},"\\lvert0\\rangle",[40,21546,21548],{"className":21547,"ariaHidden":72},[89],[40,21549,21551,21554,21557,21560],{"className":21550},[93],[40,21552],{"className":21553,"style":98},[97],[40,21555,62],{"className":21556},[106],[40,21558,66],{"className":21559},[102],[40,21561,69],{"className":21562},[113],[40,21564,21566,21584],{"className":21565},[43],[40,21567,21569],{"className":21568},[47],[49,21570,21571],{"xmlns":51},[53,21572,21573,21581],{},[56,21574,21575,21577,21579],{},[59,21576,62],{"stretchy":61},[64,21578,78],{},[59,21580,69],{"stretchy":61},[82,21582,21583],{"encoding":84},"\\lvert1\\rangle",[40,21585,21587],{"className":21586,"ariaHidden":72},[89],[40,21588,21590,21593,21596,21599],{"className":21589},[93],[40,21591],{"className":21592,"style":98},[97],[40,21594,62],{"className":21595},[106],[40,21597,78],{"className":21598},[102],[40,21600,69],{"className":21601},[113],[20167,21603,21604,21605,21638,21639,1628],{},"Target oscillation frequency\u002Fphase depends on control state due to the ",[40,21606,21608,21623],{"className":21607},[43],[40,21609,21611],{"className":21610},[47],[49,21612,21613],{"xmlns":51},[53,21614,21615,21621],{},[56,21616,21617,21619],{},[154,21618,4263],{},[154,21620,4196],{},[82,21622,21402],{"encoding":84},[40,21624,21626],{"className":21625,"ariaHidden":72},[89],[40,21627,21629,21632,21635],{"className":21628},[93],[40,21630],{"className":21631,"style":4304},[97],[40,21633,4263],{"className":21634,"style":4668},[102,335],[40,21636,4196],{"className":21637,"style":1242},[102,335]," term (and any residual ",[40,21640,21642,21658],{"className":21641},[43],[40,21643,21645],{"className":21644},[47],[49,21646,21647],{"xmlns":51},[53,21648,21649,21655],{},[56,21650,21651,21653],{},[154,21652,1050],{},[154,21654,4196],{},[82,21656,21657],{"encoding":84},"IX",[40,21659,21661],{"className":21660,"ariaHidden":72},[89],[40,21662,21664,21667,21670],{"className":21663},[93],[40,21665],{"className":21666,"style":4304},[97],[40,21668,1050],{"className":21669,"style":1242},[102,335],[40,21671,4196],{"className":21672,"style":1242},[102,335],[25,21674],{},[21676,21677,21679],"h3",{"id":21678},"references","References",[9743,21681,21682,21729,21744],{},[9746,21683,21684,21685,21688,21689,21722,21723,21726],{},"The Quantum Aviary — ",[15,21686,21687],{},"How the Cross-Resonance Gate Works"," (overview of ",[40,21690,21692,21707],{"className":21691},[43],[40,21693,21695],{"className":21694},[47],[49,21696,21697],{"xmlns":51},[53,21698,21699,21705],{},[56,21700,21701,21703],{},[154,21702,4263],{},[154,21704,4196],{},[82,21706,21402],{"encoding":84},[40,21708,21710],{"className":21709,"ariaHidden":72},[89],[40,21711,21713,21716,21719],{"className":21712},[93],[40,21714],{"className":21715,"style":4304},[97],[40,21717,4263],{"className":21718,"style":4668},[102,335],[40,21720,4196],{"className":21721,"style":1242},[102,335],", echo, and intuition)",[21724,21725],"br",{},[19,21727,21728],{"href":21728},"https:\u002F\u002Fthequantumaviary.blogspot.com\u002F2021\u002F07\u002Fhow-cross-resonance-gate-works.html",[9746,21730,21731,21732,10484,21735,21738,21739,21741],{},"Malekakhlagh, Magesan & McKay — ",[15,21733,21734],{},"First-principles analysis of cross-resonance gate operation",[35,21736,21737],{},"Phys. Rev. A"," 102, 042605 (2020)",[21724,21740],{},[19,21742,21743],{"href":21743},"https:\u002F\u002Flink.aps.org\u002Fdoi\u002F10.1103\u002FPhysRevA.102.042605",[9746,21745,21746,21747,10484,21750,21752,21753,21755],{},"Magesan & Gambetta — ",[15,21748,21749],{},"Effective Hamiltonian models of the cross-resonance gate",[35,21751,21737],{}," 101, 052308 (2020)",[21724,21754],{},[19,21756,21757],{"href":21757},"https:\u002F\u002Flink.aps.org\u002Fdoi\u002F10.1103\u002FPhysRevA.101.052308",[10082,21759,21761],{"className":10269,"code":21760,"language":10271,"meta":10090,"style":10090},"#@title Cross-Resonance (CR) gate — slices & heatmaps (PRA 2020 formulas)\n\nfrom dataclasses import dataclass\nfrom typing import Tuple\nimport math\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# -----------------------------\n# Control knobs (MHz)\n# -----------------------------\n@dataclass\nclass CRKnobs:\n    f_c: float = 5114.0        # control frequency\n    f_t: float = 4914.0        # target frequency\n    alpha_c: float = -330.0    # control anharmonicity (negative)\n    alpha_t: float = -330.0    # target anharmonicity\n    J: float = 3.8             # exchange coupling\n    Omega0: float = 30.0       # default drive for Δ-slices\n    use_energy_basis: bool = False  # toggle ν-corrections (~10–15%)\n\n    # Sweep grids\n    delta_span: Tuple[float, float] = (-500.0, 500.0)\n    delta_points: int = 1801\n    omega_span: Tuple[float, float] = (0.0, 60.0)\n    omega_points: int = 361\n\nK = CRKnobs()\n\n# -----------------------------\n# Helpers & ν-coefficients\n# -----------------------------\ndef _safe_div(a: float, b: float, eps: float = 1e-12) -> float:\n    \"\"\"Return a\u002Fb with a small guard to avoid blow-ups at poles.\"\"\"\n    if abs(b) \u003C eps:\n        return np.sign(b) * 1e12\n    return a \u002F b\n\ndef _epsilon_from_alpha_over_omega(alpha_mhz: float, omega_mhz: float) -> float:\n    \"\"\"Solve ε from (9 − 4r) ε² + 16(1 − r) ε + 64 r = 0, with r = α\u002Fω.\"\"\"\n    r = alpha_mhz \u002F omega_mhz\n    A = (9.0 - 4.0 * r)\n    B = 16.0 * (1.0 - r)\n    C = 64.0 * r\n    disc = max(B * B - 4.0 * A * C, 0.0)\n    if abs(A) \u003C 1e-15:\n        return 0.2\n    e1 = (-B + math.sqrt(disc)) \u002F (2.0 * A)\n    e2 = (-B - math.sqrt(disc)) \u002F (2.0 * A)\n    roots = [e for e in (e1, e2) if e > 0.0 and np.isfinite(e)]\n    return min(roots) if roots else 0.2\n\ndef _nu_energy_basis(alpha_mhz: float, omega_mhz: float) -> Tuple[float, float]:\n    \"\"\"Energy-basis ν_01, ν_12 up to O(ε²).\"\"\"\n    eps = _epsilon_from_alpha_over_omega(alpha_mhz, omega_mhz)\n    nu01 = 1.0 - (1.0 \u002F 8.0) * eps - (11.0 \u002F 256.0) * (eps ** 2)\n    nu12 = (1.0 - 0.25 * eps - (73.0 \u002F 512.0) * (eps ** 2)) \u002F math.sqrt(2.0)\n    return nu01, nu12\n\ndef _nu_kerr() -> Tuple[float, float]:\n    \"\"\"Kerr-limit charge matrix elements.\"\"\"\n    return 1.0, 1.0 \u002F math.sqrt(2.0)\n\nif K.use_energy_basis:\n    nu_c01, nu_c12 = _nu_energy_basis(K.alpha_c, K.f_c)\n    nu_t01, nu_t12 = _nu_energy_basis(K.alpha_t, K.f_t)\nelse:\n    nu_c01, nu_c12 = _nu_kerr()\n    nu_t01, nu_t12 = _nu_kerr()\n\n# -----------------------------\n# Leading-order rates (MHz)\n# -----------------------------\ndef omega_zx_mhz(delta_ct: float, J: float, Omega: float, alpha_c: float) -> float:\n    r\"\"\"Entangling rate ω_ZX ≈ JΩ[(Δ+α_c)^{-1} − Δ^{-1}] with small ν-correction.\"\"\"\n    kerr = J * Omega * (_safe_div(1.0, delta_ct + alpha_c) - _safe_div(1.0, delta_ct))\n    eb = 0.5 * J * Omega * nu_t01 * (\n        (nu_c12 ** 2) * _safe_div(1.0, delta_ct + alpha_c) - 2.0 * (nu_c01 ** 2) * _safe_div(1.0, delta_ct)\n    )\n    return 0.5 * (kerr + eb)\n\ndef omega_zz_mhz(delta_ct: float, J: float, alpha_c: float, alpha_t: float) -> float:\n    r\"\"\"Static ZZ ≈ J²[(Δ−α_t)^{-1} − (Δ+α_c)^{-1}] with small ν-correction.\"\"\"\n    kerr = (J ** 2) * (_safe_div(1.0, delta_ct - alpha_t) - _safe_div(1.0, delta_ct + alpha_c))\n    eb = 0.5 * (J ** 2) * (\n        (nu_c01 ** 2) * (nu_t12 ** 2) * _safe_div(1.0, delta_ct - alpha_t) -\n        (nu_t01 ** 2) * (nu_c12 ** 2) * _safe_div(1.0, delta_ct + alpha_c)\n    )\n    return 0.5 * (kerr + eb)\n\n# Region boundaries (poles) for I–V:\nREGION_LINES = (K.alpha_t, 0.0, -K.alpha_c \u002F 2.0, -K.alpha_c, -1.5 * K.alpha_c)\ndef _annotate_regions(ax):\n    for x in REGION_LINES:\n        ax.axvline(x, linestyle=\"--\", linewidth=1.0)\n\n# -----------------------------\n# Slices\n# -----------------------------\nDELTA = np.linspace(K.delta_span[0], K.delta_span[1], K.delta_points)\nOMEGA = np.linspace(K.omega_span[0], K.omega_span[1], K.omega_points)\n\nwzx_slice = np.array([omega_zx_mhz(d, K.J, K.Omega0, K.alpha_c) for d in DELTA])\nwzz_slice = np.array([omega_zz_mhz(d, K.J, K.alpha_c, K.alpha_t) for d in DELTA])\nratio_slice = np.abs(np.where(np.abs(wzz_slice) > 1e-12, wzx_slice \u002F wzz_slice, np.nan))\n\n# Plot 1: ω_ZX vs Δ_ct\nplt.figure(figsize=(8, 5), dpi=150)\nplt.plot(DELTA, wzx_slice, label=rf'$\\omega_{{ZX}}$ (Ω={K.Omega0:.1f} MHz)', color='blue')\nplt.axhline(0.0, linestyle=\":\")\n_annotate_regions(plt.gca())\nplt.xlabel(r'$\\Delta_{ct}$ (MHz)')\nplt.ylabel(r'$\\omega_{ZX}$ (MHz)')\nplt.title(r'Leading-order $\\omega_{ZX}$ vs detuning')\nplt.legend()\nplt.show()\n\n# Plot 2: ω_ZZ vs Δ_ct\nplt.figure(figsize=(8, 5), dpi=150)\nplt.plot(DELTA, wzz_slice, label=r'$\\omega_{ZZ}$ (static; lowest order)', color='blue')\nplt.axhline(0.0, linestyle=\":\")\n_annotate_regions(plt.gca())\nplt.xlabel(r'$\\Delta_{ct}$ (MHz)')\nplt.ylabel(r'$\\omega_{ZZ}$ (MHz)')\nplt.title(r'Lowest-order $\\omega_{ZZ}$ vs detuning')\nplt.legend()\nplt.show()\n\n# Heatmap 1: ω_ZX(Δ, Ω)\nwzx_map = np.empty((DELTA.size, OMEGA.size))\nfor i, d in enumerate(DELTA):\n    for j, om in enumerate(OMEGA):\n        wzx_map[i, j] = omega_zx_mhz(d, K.J, om, K.alpha_c)\n\nplt.figure(figsize=(8, 6), dpi=150)\nextent = [OMEGA.min(), OMEGA.max(), DELTA.min(), DELTA.max()]\nplt.imshow(wzx_map, origin=\"lower\", aspect=\"auto\", extent=extent, cmap=\"magma\")\nplt.colorbar(label=r'$\\omega_{ZX}$ (MHz)')\nplt.xlabel(r'$\\Omega$ (MHz)')\nplt.ylabel(r'$\\Delta_{ct}$ (MHz)')\nplt.title(r'$\\omega_{ZX}(\\Delta_{ct}, \\Omega)$ (leading-order)')\nplt.show()\n\n# Heatmap 2: |ω_ZX\u002Fω_ZZ|(Δ, Ω) (clipped for visibility)\nratio_map = np.empty_like(wzx_map)\nfor i, d in enumerate(DELTA):\n    zz = omega_zz_mhz(d, K.J, K.alpha_c, K.alpha_t)\n    for j, om in enumerate(OMEGA):\n        zx = omega_zx_mhz(d, K.J, om, K.alpha_c)\n        ratio_map[i, j] = abs(zx \u002F zz) if abs(zz) > 1e-12 else np.nan\n\nclip_max = np.nanpercentile(ratio_map[np.isfinite(ratio_map)], 99.5)\nratio_map_vis = np.clip(ratio_map, 0.0, clip_max)\n\nplt.figure(figsize=(8, 6), dpi=150)\nplt.imshow(ratio_map_vis, origin=\"lower\", aspect=\"auto\", extent=extent)\nplt.colorbar(label=r'$|\\omega_{ZX}\u002F\\omega_{ZZ}|$')\nplt.xlabel(r'$\\Omega$ (MHz)')\nplt.ylabel(r'$\\Delta_{ct}$ (MHz)')\nplt.title(r'Heuristic $|\\omega_{ZX}\u002F\\omega_{ZZ}|$ map (clipped at 99.5th pct)')\nplt.show()\n",[10088,21762,21763,21768,21772,21782,21792,21799,21809,21819,21823,21828,21833,21837,21841,21850,21865,21880,21897,21913,21928,21943,21959,21963,21968,21996,22008,22033,22045,22049,22060,22064,22068,22073,22077,22122,22127,22142,22158,22170,22174,22205,22210,22224,22246,22266,22281,22317,22333,22340,22375,22406,22447,22467,22471,22504,22509,22521,22576,22632,22639,22643,22661,22666,22688,22692,22699,22711,22723,22729,22739,22749,22753,22757,22762,22766,22815,22823,22868,22895,22946,22950,22966,22970,23016,23023,23070,23092,23131,23168,23172,23186,23190,23195,23235,23248,23262,23290,23294,23298,23303,23307,23331,23355,23359,23391,23420,23456,23460,23465,23499,23558,23580,23594,23620,23650,23671,23680,23689,23693,23698,23728,23773,23793,23803,23825,23851,23871,23879,23887,23891,23896,23920,23938,23955,23967,23971,24001,24044,24085,24115,24143,24165,24208,24216,24220,24225,24240,24256,24268,24284,24295,24328,24332,24357,24377,24381,24411,24443,24479,24505,24527,24566],{"__ignoreMap":10090},[40,21764,21765],{"class":10276,"line":10277},[40,21766,21767],{"class":10280},"#@title Cross-Resonance (CR) gate — slices & heatmaps (PRA 2020 formulas)\n",[40,21769,21770],{"class":10276,"line":10284},[40,21771,10301],{"emptyLinePlaceholder":10300},[40,21773,21774,21776,21778,21780],{"class":10276,"line":10291},[40,21775,10353],{"class":10352},[40,21777,10530],{"class":10363},[40,21779,10375],{"class":10352},[40,21781,10535],{"class":10363},[40,21783,21784,21786,21788,21790],{"class":10276,"line":10297},[40,21785,10353],{"class":10352},[40,21787,10554],{"class":10363},[40,21789,10375],{"class":10352},[40,21791,17784],{"class":10363},[40,21793,21794,21796],{"class":10276,"line":10304},[40,21795,10375],{"class":10352},[40,21797,21798],{"class":10363}," math\n",[40,21800,21801,21803,21805,21807],{"class":10276,"line":10310},[40,21802,10375],{"class":10352},[40,21804,10378],{"class":10363},[40,21806,10381],{"class":10352},[40,21808,10384],{"class":10363},[40,21810,21811,21813,21815,21817],{"class":10276,"line":10316},[40,21812,10375],{"class":10352},[40,21814,10405],{"class":10363},[40,21816,10381],{"class":10352},[40,21818,10410],{"class":10363},[40,21820,21821],{"class":10276,"line":10322},[40,21822,10301],{"emptyLinePlaceholder":10300},[40,21824,21825],{"class":10276,"line":10327},[40,21826,21827],{"class":10280},"# -----------------------------\n",[40,21829,21830],{"class":10276,"line":10333},[40,21831,21832],{"class":10280},"# Control knobs (MHz)\n",[40,21834,21835],{"class":10276,"line":10339},[40,21836,21827],{"class":10280},[40,21838,21839],{"class":10276,"line":10344},[40,21840,10627],{"class":10430},[40,21842,21843,21845,21848],{"class":10276,"line":10349},[40,21844,10632],{"class":10352},[40,21846,21847],{"class":10635}," CRKnobs",[40,21849,10639],{"class":10363},[40,21851,21852,21855,21857,21859,21862],{"class":10276,"line":10367},[40,21853,21854],{"class":10363},"    f_c: ",[40,21856,10784],{"class":10732},[40,21858,10772],{"class":10732},[40,21860,21861],{"class":10446}," 5114.0",[40,21863,21864],{"class":10280},"        # control frequency\n",[40,21866,21867,21870,21872,21874,21877],{"class":10276,"line":10372},[40,21868,21869],{"class":10363},"    f_t: ",[40,21871,10784],{"class":10732},[40,21873,10772],{"class":10732},[40,21875,21876],{"class":10446}," 4914.0",[40,21878,21879],{"class":10280},"        # target frequency\n",[40,21881,21882,21885,21887,21889,21891,21894],{"class":10276,"line":10387},[40,21883,21884],{"class":10363},"    alpha_c: ",[40,21886,10784],{"class":10732},[40,21888,10772],{"class":10732},[40,21890,11764],{"class":10732},[40,21892,21893],{"class":10446},"330.0",[40,21895,21896],{"class":10280},"    # control anharmonicity (negative)\n",[40,21898,21899,21902,21904,21906,21908,21910],{"class":10276,"line":10400},[40,21900,21901],{"class":10363},"    alpha_t: ",[40,21903,10784],{"class":10732},[40,21905,10772],{"class":10732},[40,21907,11764],{"class":10732},[40,21909,21893],{"class":10446},[40,21911,21912],{"class":10280},"    # target anharmonicity\n",[40,21914,21915,21918,21920,21922,21925],{"class":10276,"line":10413},[40,21916,21917],{"class":10363},"    J: ",[40,21919,10784],{"class":10732},[40,21921,10772],{"class":10732},[40,21923,21924],{"class":10446}," 3.8",[40,21926,21927],{"class":10280},"             # exchange coupling\n",[40,21929,21930,21933,21935,21937,21940],{"class":10276,"line":10418},[40,21931,21932],{"class":10363},"    Omega0: ",[40,21934,10784],{"class":10732},[40,21936,10772],{"class":10732},[40,21938,21939],{"class":10446}," 30.0",[40,21941,21942],{"class":10280},"       # default drive for Δ-slices\n",[40,21944,21945,21948,21951,21953,21956],{"class":10276,"line":10424},[40,21946,21947],{"class":10363},"    use_energy_basis: ",[40,21949,21950],{"class":10732},"bool",[40,21952,10772],{"class":10732},[40,21954,21955],{"class":10446}," False",[40,21957,21958],{"class":10280},"  # toggle ν-corrections (~10–15%)\n",[40,21960,21961],{"class":10276,"line":10437},[40,21962,10301],{"emptyLinePlaceholder":10300},[40,21964,21965],{"class":10276,"line":10456},[40,21966,21967],{"class":10280},"    # Sweep grids\n",[40,21969,21970,21973,21975,21977,21979,21981,21983,21985,21987,21990,21992,21994],{"class":10276,"line":10472},[40,21971,21972],{"class":10363},"    delta_span: Tuple[",[40,21974,10784],{"class":10732},[40,21976,10484],{"class":10363},[40,21978,10784],{"class":10732},[40,21980,10741],{"class":10363},[40,21982,162],{"class":10732},[40,21984,920],{"class":10363},[40,21986,11056],{"class":10732},[40,21988,21989],{"class":10446},"500.0",[40,21991,10484],{"class":10363},[40,21993,21989],{"class":10446},[40,21995,10761],{"class":10363},[40,21997,21998,22001,22003,22005],{"class":10276,"line":10496},[40,21999,22000],{"class":10363},"    delta_points: ",[40,22002,10821],{"class":10732},[40,22004,10772],{"class":10732},[40,22006,22007],{"class":10446}," 1801\n",[40,22009,22010,22013,22015,22017,22019,22021,22023,22025,22027,22029,22031],{"class":10276,"line":10510},[40,22011,22012],{"class":10363},"    omega_span: Tuple[",[40,22014,10784],{"class":10732},[40,22016,10484],{"class":10363},[40,22018,10784],{"class":10732},[40,22020,10741],{"class":10363},[40,22022,162],{"class":10732},[40,22024,920],{"class":10363},[40,22026,10801],{"class":10446},[40,22028,10484],{"class":10363},[40,22030,17601],{"class":10446},[40,22032,10761],{"class":10363},[40,22034,22035,22038,22040,22042],{"class":10276,"line":10696},[40,22036,22037],{"class":10363},"    omega_points: ",[40,22039,10821],{"class":10732},[40,22041,10772],{"class":10732},[40,22043,22044],{"class":10446}," 361\n",[40,22046,22047],{"class":10276,"line":10702},[40,22048,10301],{"emptyLinePlaceholder":10300},[40,22050,22051,22054,22056,22058],{"class":10276,"line":10708},[40,22052,22053],{"class":10363},"K ",[40,22055,162],{"class":10732},[40,22057,21847],{"class":10430},[40,22059,12174],{"class":10363},[40,22061,22062],{"class":10276,"line":10714},[40,22063,10301],{"emptyLinePlaceholder":10300},[40,22065,22066],{"class":10276,"line":10720},[40,22067,21827],{"class":10280},[40,22069,22070],{"class":10276,"line":10726},[40,22071,22072],{"class":10280},"# Helpers & ν-coefficients\n",[40,22074,22075],{"class":10276,"line":10764},[40,22076,21827],{"class":10280},[40,22078,22079,22081,22084,22086,22088,22090,22092,22094,22097,22099,22101,22103,22106,22108,22110,22112,22115,22118,22120],{"class":10276,"line":10778},[40,22080,10965],{"class":10352},[40,22082,22083],{"class":10430}," _safe_div",[40,22085,167],{"class":10363},[40,22087,19],{"class":10973},[40,22089,10443],{"class":10363},[40,22091,10784],{"class":10732},[40,22093,10484],{"class":10363},[40,22095,22096],{"class":10973},"b",[40,22098,10443],{"class":10363},[40,22100,10784],{"class":10732},[40,22102,10484],{"class":10363},[40,22104,22105],{"class":10973},"eps",[40,22107,10443],{"class":10363},[40,22109,10784],{"class":10732},[40,22111,10772],{"class":10732},[40,22113,22114],{"class":10446}," 1e-12",[40,22116,22117],{"class":10363},") -> ",[40,22119,10784],{"class":10732},[40,22121,10639],{"class":10363},[40,22123,22124],{"class":10276,"line":10815},[40,22125,22126],{"class":10287},"    \"\"\"Return a\u002Fb with a small guard to avoid blow-ups at poles.\"\"\"\n",[40,22128,22129,22131,22134,22137,22139],{"class":10276,"line":10829},[40,22130,11323],{"class":10352},[40,22132,22133],{"class":10732}," abs",[40,22135,22136],{"class":10363},"(b) ",[40,22138,10184],{"class":10732},[40,22140,22141],{"class":10363}," eps:\n",[40,22143,22144,22146,22148,22151,22153,22155],{"class":10276,"line":10842},[40,22145,11339],{"class":10352},[40,22147,11156],{"class":10363},[40,22149,22150],{"class":10430},"sign",[40,22152,22136],{"class":10363},[40,22154,11752],{"class":10732},[40,22156,22157],{"class":10446}," 1e12\n",[40,22159,22160,22162,22165,22167],{"class":10276,"line":10855},[40,22161,11201],{"class":10352},[40,22163,22164],{"class":10363}," a ",[40,22166,11243],{"class":10732},[40,22168,22169],{"class":10363}," b\n",[40,22171,22172],{"class":10276,"line":10868},[40,22173,10301],{"emptyLinePlaceholder":10300},[40,22175,22176,22178,22181,22183,22186,22188,22190,22192,22195,22197,22199,22201,22203],{"class":10276,"line":10881},[40,22177,10965],{"class":10352},[40,22179,22180],{"class":10430}," _epsilon_from_alpha_over_omega",[40,22182,167],{"class":10363},[40,22184,22185],{"class":10973},"alpha_mhz",[40,22187,10443],{"class":10363},[40,22189,10784],{"class":10732},[40,22191,10484],{"class":10363},[40,22193,22194],{"class":10973},"omega_mhz",[40,22196,10443],{"class":10363},[40,22198,10784],{"class":10732},[40,22200,22117],{"class":10363},[40,22202,10784],{"class":10732},[40,22204,10639],{"class":10363},[40,22206,22207],{"class":10276,"line":10907},[40,22208,22209],{"class":10287},"    \"\"\"Solve ε from (9 − 4r) ε² + 16(1 − r) ε + 64 r = 0, with r = α\u002Fω.\"\"\"\n",[40,22211,22212,22214,22216,22219,22221],{"class":10276,"line":10920},[40,22213,11151],{"class":10363},[40,22215,162],{"class":10732},[40,22217,22218],{"class":10363}," alpha_mhz ",[40,22220,11243],{"class":10732},[40,22222,22223],{"class":10363}," omega_mhz\n",[40,22225,22226,22229,22231,22233,22236,22238,22241,22243],{"class":10276,"line":10933},[40,22227,22228],{"class":10363},"    A ",[40,22230,162],{"class":10732},[40,22232,920],{"class":10363},[40,22234,22235],{"class":10446},"9.0",[40,22237,11764],{"class":10732},[40,22239,22240],{"class":10446}," 4.0",[40,22242,11881],{"class":10732},[40,22244,22245],{"class":10363}," r)\n",[40,22247,22248,22251,22253,22256,22258,22260,22262,22264],{"class":10276,"line":10946},[40,22249,22250],{"class":10363},"    B ",[40,22252,162],{"class":10732},[40,22254,22255],{"class":10446}," 16.0",[40,22257,11881],{"class":10732},[40,22259,920],{"class":10363},[40,22261,10810],{"class":10446},[40,22263,11764],{"class":10732},[40,22265,22245],{"class":10363},[40,22267,22268,22271,22273,22276,22278],{"class":10276,"line":10951},[40,22269,22270],{"class":10363},"    C ",[40,22272,162],{"class":10732},[40,22274,22275],{"class":10446}," 64.0",[40,22277,11881],{"class":10732},[40,22279,22280],{"class":10363}," r\n",[40,22282,22283,22286,22288,22291,22294,22296,22299,22301,22303,22305,22308,22310,22313,22315],{"class":10276,"line":10956},[40,22284,22285],{"class":10363},"    disc ",[40,22287,162],{"class":10732},[40,22289,22290],{"class":10732}," max",[40,22292,22293],{"class":10363},"(B ",[40,22295,11752],{"class":10732},[40,22297,22298],{"class":10363}," B ",[40,22300,11056],{"class":10732},[40,22302,22240],{"class":10446},[40,22304,11881],{"class":10732},[40,22306,22307],{"class":10363}," A ",[40,22309,11752],{"class":10732},[40,22311,22312],{"class":10363}," C, ",[40,22314,10801],{"class":10446},[40,22316,10761],{"class":10363},[40,22318,22319,22321,22323,22326,22328,22331],{"class":10276,"line":10962},[40,22320,11323],{"class":10352},[40,22322,22133],{"class":10732},[40,22324,22325],{"class":10363},"(A) ",[40,22327,10184],{"class":10732},[40,22329,22330],{"class":10446}," 1e-15",[40,22332,10639],{"class":10363},[40,22334,22335,22337],{"class":10276,"line":10983},[40,22336,11339],{"class":10352},[40,22338,22339],{"class":10446}," 0.2\n",[40,22341,22342,22345,22347,22349,22351,22354,22356,22359,22361,22364,22366,22368,22370,22372],{"class":10276,"line":11005},[40,22343,22344],{"class":10363},"    e1 ",[40,22346,162],{"class":10732},[40,22348,920],{"class":10363},[40,22350,11056],{"class":10732},[40,22352,22353],{"class":10363},"B ",[40,22355,1053],{"class":10732},[40,22357,22358],{"class":10363}," math.",[40,22360,2218],{"class":10430},[40,22362,22363],{"class":10363},"(disc)) ",[40,22365,11243],{"class":10732},[40,22367,920],{"class":10363},[40,22369,11878],{"class":10446},[40,22371,11881],{"class":10732},[40,22373,22374],{"class":10363}," A)\n",[40,22376,22377,22380,22382,22384,22386,22388,22390,22392,22394,22396,22398,22400,22402,22404],{"class":10276,"line":11011},[40,22378,22379],{"class":10363},"    e2 ",[40,22381,162],{"class":10732},[40,22383,920],{"class":10363},[40,22385,11056],{"class":10732},[40,22387,22353],{"class":10363},[40,22389,11056],{"class":10732},[40,22391,22358],{"class":10363},[40,22393,2218],{"class":10430},[40,22395,22363],{"class":10363},[40,22397,11243],{"class":10732},[40,22399,920],{"class":10363},[40,22401,11878],{"class":10446},[40,22403,11881],{"class":10732},[40,22405,22374],{"class":10363},[40,22407,22408,22411,22413,22416,22418,22421,22423,22426,22428,22430,22433,22436,22439,22441,22444],{"class":10276,"line":11022},[40,22409,22410],{"class":10363},"    roots ",[40,22412,162],{"class":10732},[40,22414,22415],{"class":10363}," [e ",[40,22417,13840],{"class":10352},[40,22419,22420],{"class":10363}," e ",[40,22422,13846],{"class":10352},[40,22424,22425],{"class":10363}," (e1, e2) ",[40,22427,11225],{"class":10352},[40,22429,22420],{"class":10363},[40,22431,22432],{"class":10732},">",[40,22434,22435],{"class":10446}," 0.0",[40,22437,22438],{"class":10352}," and",[40,22440,11156],{"class":10363},[40,22442,22443],{"class":10430},"isfinite",[40,22445,22446],{"class":10363},"(e)]\n",[40,22448,22449,22451,22454,22457,22459,22462,22465],{"class":10276,"line":11064},[40,22450,11201],{"class":10352},[40,22452,22453],{"class":10732}," min",[40,22455,22456],{"class":10363},"(roots) ",[40,22458,11225],{"class":10352},[40,22460,22461],{"class":10363}," roots ",[40,22463,22464],{"class":10352},"else",[40,22466,22339],{"class":10446},[40,22468,22469],{"class":10276,"line":11103},[40,22470,10301],{"emptyLinePlaceholder":10300},[40,22472,22473,22475,22478,22480,22482,22484,22486,22488,22490,22492,22494,22496,22498,22500,22502],{"class":10276,"line":11142},[40,22474,10965],{"class":10352},[40,22476,22477],{"class":10430}," _nu_energy_basis",[40,22479,167],{"class":10363},[40,22481,22185],{"class":10973},[40,22483,10443],{"class":10363},[40,22485,10784],{"class":10732},[40,22487,10484],{"class":10363},[40,22489,22194],{"class":10973},[40,22491,10443],{"class":10363},[40,22493,10784],{"class":10732},[40,22495,12138],{"class":10363},[40,22497,10784],{"class":10732},[40,22499,10484],{"class":10363},[40,22501,10784],{"class":10732},[40,22503,12152],{"class":10363},[40,22505,22506],{"class":10276,"line":11148},[40,22507,22508],{"class":10287},"    \"\"\"Energy-basis ν_01, ν_12 up to O(ε²).\"\"\"\n",[40,22510,22511,22514,22516,22518],{"class":10276,"line":11181},[40,22512,22513],{"class":10363},"    eps ",[40,22515,162],{"class":10732},[40,22517,22180],{"class":10430},[40,22519,22520],{"class":10363},"(alpha_mhz, omega_mhz)\n",[40,22522,22523,22526,22528,22531,22533,22535,22537,22540,22543,22545,22547,22550,22552,22554,22557,22559,22562,22564,22566,22569,22572,22574],{"class":10276,"line":11198},[40,22524,22525],{"class":10363},"    nu01 ",[40,22527,162],{"class":10732},[40,22529,22530],{"class":10446}," 1.0",[40,22532,11764],{"class":10732},[40,22534,920],{"class":10363},[40,22536,10810],{"class":10446},[40,22538,22539],{"class":10732}," \u002F",[40,22541,22542],{"class":10446}," 8.0",[40,22544,12055],{"class":10363},[40,22546,11752],{"class":10732},[40,22548,22549],{"class":10363}," eps ",[40,22551,11056],{"class":10732},[40,22553,920],{"class":10363},[40,22555,22556],{"class":10446},"11.0",[40,22558,22539],{"class":10732},[40,22560,22561],{"class":10446}," 256.0",[40,22563,12055],{"class":10363},[40,22565,11752],{"class":10732},[40,22567,22568],{"class":10363}," (eps ",[40,22570,22571],{"class":10732},"**",[40,22573,12960],{"class":10446},[40,22575,10761],{"class":10363},[40,22577,22578,22581,22583,22585,22587,22589,22592,22594,22596,22598,22600,22603,22605,22608,22610,22612,22614,22616,22618,22620,22622,22624,22626,22628,22630],{"class":10276,"line":11249},[40,22579,22580],{"class":10363},"    nu12 ",[40,22582,162],{"class":10732},[40,22584,920],{"class":10363},[40,22586,10810],{"class":10446},[40,22588,11764],{"class":10732},[40,22590,22591],{"class":10446}," 0.25",[40,22593,11881],{"class":10732},[40,22595,22549],{"class":10363},[40,22597,11056],{"class":10732},[40,22599,920],{"class":10363},[40,22601,22602],{"class":10446},"73.0",[40,22604,22539],{"class":10732},[40,22606,22607],{"class":10446}," 512.0",[40,22609,12055],{"class":10363},[40,22611,11752],{"class":10732},[40,22613,22568],{"class":10363},[40,22615,22571],{"class":10732},[40,22617,12960],{"class":10446},[40,22619,13376],{"class":10363},[40,22621,11243],{"class":10732},[40,22623,22358],{"class":10363},[40,22625,2218],{"class":10430},[40,22627,167],{"class":10363},[40,22629,11878],{"class":10446},[40,22631,10761],{"class":10363},[40,22633,22634,22636],{"class":10276,"line":11254},[40,22635,11201],{"class":10352},[40,22637,22638],{"class":10363}," nu01, nu12\n",[40,22640,22641],{"class":10276,"line":11259},[40,22642,10301],{"emptyLinePlaceholder":10300},[40,22644,22645,22647,22650,22653,22655,22657,22659],{"class":10276,"line":11265},[40,22646,10965],{"class":10352},[40,22648,22649],{"class":10430}," _nu_kerr",[40,22651,22652],{"class":10363},"() -> Tuple[",[40,22654,10784],{"class":10732},[40,22656,10484],{"class":10363},[40,22658,10784],{"class":10732},[40,22660,12152],{"class":10363},[40,22662,22663],{"class":10276,"line":11294},[40,22664,22665],{"class":10287},"    \"\"\"Kerr-limit charge matrix elements.\"\"\"\n",[40,22667,22668,22670,22672,22674,22676,22678,22680,22682,22684,22686],{"class":10276,"line":11300},[40,22669,11201],{"class":10352},[40,22671,22530],{"class":10446},[40,22673,10484],{"class":10363},[40,22675,10810],{"class":10446},[40,22677,22539],{"class":10732},[40,22679,22358],{"class":10363},[40,22681,2218],{"class":10430},[40,22683,167],{"class":10363},[40,22685,11878],{"class":10446},[40,22687,10761],{"class":10363},[40,22689,22690],{"class":10276,"line":11320},[40,22691,10301],{"emptyLinePlaceholder":10300},[40,22693,22694,22696],{"class":10276,"line":11336},[40,22695,11225],{"class":10352},[40,22697,22698],{"class":10363}," K.use_energy_basis:\n",[40,22700,22701,22704,22706,22708],{"class":10276,"line":11381},[40,22702,22703],{"class":10363},"    nu_c01, nu_c12 ",[40,22705,162],{"class":10732},[40,22707,22477],{"class":10430},[40,22709,22710],{"class":10363},"(K.alpha_c, K.f_c)\n",[40,22712,22713,22716,22718,22720],{"class":10276,"line":11395},[40,22714,22715],{"class":10363},"    nu_t01, nu_t12 ",[40,22717,162],{"class":10732},[40,22719,22477],{"class":10430},[40,22721,22722],{"class":10363},"(K.alpha_t, K.f_t)\n",[40,22724,22725,22727],{"class":10276,"line":11438},[40,22726,22464],{"class":10352},[40,22728,10639],{"class":10363},[40,22730,22731,22733,22735,22737],{"class":10276,"line":11452},[40,22732,22703],{"class":10363},[40,22734,162],{"class":10732},[40,22736,22649],{"class":10430},[40,22738,12174],{"class":10363},[40,22740,22741,22743,22745,22747],{"class":10276,"line":11493},[40,22742,22715],{"class":10363},[40,22744,162],{"class":10732},[40,22746,22649],{"class":10430},[40,22748,12174],{"class":10363},[40,22750,22751],{"class":10276,"line":11507},[40,22752,10301],{"emptyLinePlaceholder":10300},[40,22754,22755],{"class":10276,"line":11512},[40,22756,21827],{"class":10280},[40,22758,22759],{"class":10276,"line":11517},[40,22760,22761],{"class":10280},"# Leading-order rates (MHz)\n",[40,22763,22764],{"class":10276,"line":11545},[40,22765,21827],{"class":10280},[40,22767,22768,22770,22773,22775,22778,22780,22782,22784,22787,22789,22791,22793,22796,22798,22800,22802,22805,22807,22809,22811,22813],{"class":10276,"line":11557},[40,22769,10965],{"class":10352},[40,22771,22772],{"class":10430}," omega_zx_mhz",[40,22774,167],{"class":10363},[40,22776,22777],{"class":10973},"delta_ct",[40,22779,10443],{"class":10363},[40,22781,10784],{"class":10732},[40,22783,10484],{"class":10363},[40,22785,22786],{"class":10973},"J",[40,22788,10443],{"class":10363},[40,22790,10784],{"class":10732},[40,22792,10484],{"class":10363},[40,22794,22795],{"class":10973},"Omega",[40,22797,10443],{"class":10363},[40,22799,10784],{"class":10732},[40,22801,10484],{"class":10363},[40,22803,22804],{"class":10973},"alpha_c",[40,22806,10443],{"class":10363},[40,22808,10784],{"class":10732},[40,22810,22117],{"class":10363},[40,22812,10784],{"class":10732},[40,22814,10639],{"class":10363},[40,22816,22817,22820],{"class":10276,"line":11563},[40,22818,22819],{"class":10352},"    r",[40,22821,22822],{"class":10287},"\"\"\"Entangling rate ω_ZX ≈ JΩ[(Δ+α_c)^{-1} − Δ^{-1}] with small ν-correction.\"\"\"\n",[40,22824,22825,22828,22830,22833,22835,22838,22840,22842,22845,22847,22849,22852,22854,22857,22859,22861,22863,22865],{"class":10276,"line":11582},[40,22826,22827],{"class":10363},"    kerr ",[40,22829,162],{"class":10732},[40,22831,22832],{"class":10363}," J ",[40,22834,11752],{"class":10732},[40,22836,22837],{"class":10363}," Omega ",[40,22839,11752],{"class":10732},[40,22841,920],{"class":10363},[40,22843,22844],{"class":10430},"_safe_div",[40,22846,167],{"class":10363},[40,22848,10810],{"class":10446},[40,22850,22851],{"class":10363},", delta_ct ",[40,22853,1053],{"class":10732},[40,22855,22856],{"class":10363}," alpha_c) ",[40,22858,11056],{"class":10732},[40,22860,22083],{"class":10430},[40,22862,167],{"class":10363},[40,22864,10810],{"class":10446},[40,22866,22867],{"class":10363},", delta_ct))\n",[40,22869,22870,22873,22875,22877,22879,22881,22883,22885,22887,22890,22892],{"class":10276,"line":11596},[40,22871,22872],{"class":10363},"    eb ",[40,22874,162],{"class":10732},[40,22876,12039],{"class":10446},[40,22878,11881],{"class":10732},[40,22880,22832],{"class":10363},[40,22882,11752],{"class":10732},[40,22884,22837],{"class":10363},[40,22886,11752],{"class":10732},[40,22888,22889],{"class":10363}," nu_t01 ",[40,22891,11752],{"class":10732},[40,22893,22894],{"class":10363}," (\n",[40,22896,22897,22900,22902,22904,22906,22908,22910,22912,22914,22916,22918,22920,22922,22924,22926,22929,22931,22933,22935,22937,22939,22941,22943],{"class":10276,"line":11609},[40,22898,22899],{"class":10363},"        (nu_c12 ",[40,22901,22571],{"class":10732},[40,22903,12960],{"class":10446},[40,22905,12055],{"class":10363},[40,22907,11752],{"class":10732},[40,22909,22083],{"class":10430},[40,22911,167],{"class":10363},[40,22913,10810],{"class":10446},[40,22915,22851],{"class":10363},[40,22917,1053],{"class":10732},[40,22919,22856],{"class":10363},[40,22921,11056],{"class":10732},[40,22923,12060],{"class":10446},[40,22925,11881],{"class":10732},[40,22927,22928],{"class":10363}," (nu_c01 ",[40,22930,22571],{"class":10732},[40,22932,12960],{"class":10446},[40,22934,12055],{"class":10363},[40,22936,11752],{"class":10732},[40,22938,22083],{"class":10430},[40,22940,167],{"class":10363},[40,22942,10810],{"class":10446},[40,22944,22945],{"class":10363},", delta_ct)\n",[40,22947,22948],{"class":10276,"line":11622},[40,22949,12631],{"class":10363},[40,22951,22952,22954,22956,22958,22961,22963],{"class":10276,"line":11637},[40,22953,11201],{"class":10352},[40,22955,12039],{"class":10446},[40,22957,11881],{"class":10732},[40,22959,22960],{"class":10363}," (kerr ",[40,22962,1053],{"class":10732},[40,22964,22965],{"class":10363}," eb)\n",[40,22967,22968],{"class":10276,"line":11708},[40,22969,10301],{"emptyLinePlaceholder":10300},[40,22971,22972,22974,22977,22979,22981,22983,22985,22987,22989,22991,22993,22995,22997,22999,23001,23003,23006,23008,23010,23012,23014],{"class":10276,"line":11727},[40,22973,10965],{"class":10352},[40,22975,22976],{"class":10430}," omega_zz_mhz",[40,22978,167],{"class":10363},[40,22980,22777],{"class":10973},[40,22982,10443],{"class":10363},[40,22984,10784],{"class":10732},[40,22986,10484],{"class":10363},[40,22988,22786],{"class":10973},[40,22990,10443],{"class":10363},[40,22992,10784],{"class":10732},[40,22994,10484],{"class":10363},[40,22996,22804],{"class":10973},[40,22998,10443],{"class":10363},[40,23000,10784],{"class":10732},[40,23002,10484],{"class":10363},[40,23004,23005],{"class":10973},"alpha_t",[40,23007,10443],{"class":10363},[40,23009,10784],{"class":10732},[40,23011,22117],{"class":10363},[40,23013,10784],{"class":10732},[40,23015,10639],{"class":10363},[40,23017,23018,23020],{"class":10276,"line":11744},[40,23019,22819],{"class":10352},[40,23021,23022],{"class":10287},"\"\"\"Static ZZ ≈ J²[(Δ−α_t)^{-1} − (Δ+α_c)^{-1}] with small ν-correction.\"\"\"\n",[40,23024,23025,23027,23029,23032,23034,23036,23038,23040,23042,23044,23046,23048,23050,23052,23055,23057,23059,23061,23063,23065,23067],{"class":10276,"line":11790},[40,23026,22827],{"class":10363},[40,23028,162],{"class":10732},[40,23030,23031],{"class":10363}," (J ",[40,23033,22571],{"class":10732},[40,23035,12960],{"class":10446},[40,23037,12055],{"class":10363},[40,23039,11752],{"class":10732},[40,23041,920],{"class":10363},[40,23043,22844],{"class":10430},[40,23045,167],{"class":10363},[40,23047,10810],{"class":10446},[40,23049,22851],{"class":10363},[40,23051,11056],{"class":10732},[40,23053,23054],{"class":10363}," alpha_t) ",[40,23056,11056],{"class":10732},[40,23058,22083],{"class":10430},[40,23060,167],{"class":10363},[40,23062,10810],{"class":10446},[40,23064,22851],{"class":10363},[40,23066,1053],{"class":10732},[40,23068,23069],{"class":10363}," alpha_c))\n",[40,23071,23072,23074,23076,23078,23080,23082,23084,23086,23088,23090],{"class":10276,"line":11795},[40,23073,22872],{"class":10363},[40,23075,162],{"class":10732},[40,23077,12039],{"class":10446},[40,23079,11881],{"class":10732},[40,23081,23031],{"class":10363},[40,23083,22571],{"class":10732},[40,23085,12960],{"class":10446},[40,23087,12055],{"class":10363},[40,23089,11752],{"class":10732},[40,23091,22894],{"class":10363},[40,23093,23094,23097,23099,23101,23103,23105,23108,23110,23112,23114,23116,23118,23120,23122,23124,23126,23128],{"class":10276,"line":11800},[40,23095,23096],{"class":10363},"        (nu_c01 ",[40,23098,22571],{"class":10732},[40,23100,12960],{"class":10446},[40,23102,12055],{"class":10363},[40,23104,11752],{"class":10732},[40,23106,23107],{"class":10363}," (nu_t12 ",[40,23109,22571],{"class":10732},[40,23111,12960],{"class":10446},[40,23113,12055],{"class":10363},[40,23115,11752],{"class":10732},[40,23117,22083],{"class":10430},[40,23119,167],{"class":10363},[40,23121,10810],{"class":10446},[40,23123,22851],{"class":10363},[40,23125,11056],{"class":10732},[40,23127,23054],{"class":10363},[40,23129,23130],{"class":10732},"-\n",[40,23132,23133,23136,23138,23140,23142,23144,23147,23149,23151,23153,23155,23157,23159,23161,23163,23165],{"class":10276,"line":11806},[40,23134,23135],{"class":10363},"        (nu_t01 ",[40,23137,22571],{"class":10732},[40,23139,12960],{"class":10446},[40,23141,12055],{"class":10363},[40,23143,11752],{"class":10732},[40,23145,23146],{"class":10363}," (nu_c12 ",[40,23148,22571],{"class":10732},[40,23150,12960],{"class":10446},[40,23152,12055],{"class":10363},[40,23154,11752],{"class":10732},[40,23156,22083],{"class":10430},[40,23158,167],{"class":10363},[40,23160,10810],{"class":10446},[40,23162,22851],{"class":10363},[40,23164,1053],{"class":10732},[40,23166,23167],{"class":10363}," alpha_c)\n",[40,23169,23170],{"class":10276,"line":11834},[40,23171,12631],{"class":10363},[40,23173,23174,23176,23178,23180,23182,23184],{"class":10276,"line":11840},[40,23175,11201],{"class":10352},[40,23177,12039],{"class":10446},[40,23179,11881],{"class":10732},[40,23181,22960],{"class":10363},[40,23183,1053],{"class":10732},[40,23185,22965],{"class":10363},[40,23187,23188],{"class":10276,"line":11860},[40,23189,10301],{"emptyLinePlaceholder":10300},[40,23191,23192],{"class":10276,"line":11887},[40,23193,23194],{"class":10280},"# Region boundaries (poles) for I–V:\n",[40,23196,23197,23200,23202,23205,23207,23209,23211,23214,23216,23218,23220,23222,23225,23227,23230,23232],{"class":10276,"line":11914},[40,23198,23199],{"class":10446},"REGION_LINES",[40,23201,10772],{"class":10732},[40,23203,23204],{"class":10363}," (K.alpha_t, ",[40,23206,10801],{"class":10446},[40,23208,10484],{"class":10363},[40,23210,11056],{"class":10732},[40,23212,23213],{"class":10363},"K.alpha_c ",[40,23215,11243],{"class":10732},[40,23217,12060],{"class":10446},[40,23219,10484],{"class":10363},[40,23221,11056],{"class":10732},[40,23223,23224],{"class":10363},"K.alpha_c, ",[40,23226,11056],{"class":10732},[40,23228,23229],{"class":10446},"1.5",[40,23231,11881],{"class":10732},[40,23233,23234],{"class":10363}," K.alpha_c)\n",[40,23236,23237,23239,23242,23244,23246],{"class":10276,"line":11938},[40,23238,10965],{"class":10352},[40,23240,23241],{"class":10430}," _annotate_regions",[40,23243,167],{"class":10363},[40,23245,12429],{"class":10973},[40,23247,13353],{"class":10363},[40,23249,23250,23252,23255,23257,23260],{"class":10276,"line":11960},[40,23251,17425],{"class":10352},[40,23253,23254],{"class":10363}," x ",[40,23256,13846],{"class":10352},[40,23258,23259],{"class":10446}," REGION_LINES",[40,23261,10639],{"class":10363},[40,23263,23264,23266,23269,23272,23275,23277,23280,23282,23284,23286,23288],{"class":10276,"line":11975},[40,23265,16463],{"class":10363},[40,23267,23268],{"class":10430},"axvline",[40,23270,23271],{"class":10363},"(x, ",[40,23273,23274],{"class":11903},"linestyle",[40,23276,162],{"class":10732},[40,23278,23279],{"class":10287},"\"--\"",[40,23281,10484],{"class":10363},[40,23283,16619],{"class":11903},[40,23285,162],{"class":10732},[40,23287,10810],{"class":10446},[40,23289,10761],{"class":10363},[40,23291,23292],{"class":10276,"line":11996},[40,23293,10301],{"emptyLinePlaceholder":10300},[40,23295,23296],{"class":10276,"line":12001},[40,23297,21827],{"class":10280},[40,23299,23300],{"class":10276,"line":12006},[40,23301,23302],{"class":10280},"# Slices\n",[40,23304,23305],{"class":10276,"line":12028},[40,23306,21827],{"class":10280},[40,23308,23309,23312,23314,23316,23318,23321,23323,23326,23328],{"class":10276,"line":12034},[40,23310,23311],{"class":10446},"DELTA",[40,23313,10772],{"class":10732},[40,23315,11156],{"class":10363},[40,23317,11850],{"class":10430},[40,23319,23320],{"class":10363},"(K.delta_span[",[40,23322,66],{"class":10446},[40,23324,23325],{"class":10363},"], K.delta_span[",[40,23327,78],{"class":10446},[40,23329,23330],{"class":10363},"], K.delta_points)\n",[40,23332,23333,23336,23338,23340,23342,23345,23347,23350,23352],{"class":10276,"line":12105},[40,23334,23335],{"class":10446},"OMEGA",[40,23337,10772],{"class":10732},[40,23339,11156],{"class":10363},[40,23341,11850],{"class":10430},[40,23343,23344],{"class":10363},"(K.omega_span[",[40,23346,66],{"class":10446},[40,23348,23349],{"class":10363},"], K.omega_span[",[40,23351,78],{"class":10446},[40,23353,23354],{"class":10363},"], K.omega_points)\n",[40,23356,23357],{"class":10276,"line":12110},[40,23358,10301],{"emptyLinePlaceholder":10300},[40,23360,23361,23364,23366,23368,23370,23372,23375,23378,23380,23383,23385,23388],{"class":10276,"line":12115},[40,23362,23363],{"class":10363},"wzx_slice ",[40,23365,162],{"class":10732},[40,23367,11156],{"class":10363},[40,23369,11206],{"class":10430},[40,23371,11209],{"class":10363},[40,23373,23374],{"class":10430},"omega_zx_mhz",[40,23376,23377],{"class":10363},"(d, K.J, K.Omega0, K.alpha_c) ",[40,23379,13840],{"class":10352},[40,23381,23382],{"class":10363}," d ",[40,23384,13846],{"class":10352},[40,23386,23387],{"class":10446}," DELTA",[40,23389,23390],{"class":10363},"])\n",[40,23392,23393,23396,23398,23400,23402,23404,23407,23410,23412,23414,23416,23418],{"class":10276,"line":12121},[40,23394,23395],{"class":10363},"wzz_slice ",[40,23397,162],{"class":10732},[40,23399,11156],{"class":10363},[40,23401,11206],{"class":10430},[40,23403,11209],{"class":10363},[40,23405,23406],{"class":10430},"omega_zz_mhz",[40,23408,23409],{"class":10363},"(d, K.J, K.alpha_c, K.alpha_t) ",[40,23411,13840],{"class":10352},[40,23413,23382],{"class":10363},[40,23415,13846],{"class":10352},[40,23417,23387],{"class":10446},[40,23419,23390],{"class":10363},[40,23421,23422,23425,23427,23429,23432,23434,23437,23439,23441,23444,23446,23448,23451,23453],{"class":10276,"line":12155},[40,23423,23424],{"class":10363},"ratio_slice ",[40,23426,162],{"class":10732},[40,23428,11156],{"class":10363},[40,23430,23431],{"class":10430},"abs",[40,23433,13108],{"class":10363},[40,23435,23436],{"class":10430},"where",[40,23438,13108],{"class":10363},[40,23440,23431],{"class":10430},[40,23442,23443],{"class":10363},"(wzz_slice) ",[40,23445,22432],{"class":10732},[40,23447,22114],{"class":10446},[40,23449,23450],{"class":10363},", wzx_slice ",[40,23452,11243],{"class":10732},[40,23454,23455],{"class":10363}," wzz_slice, np.nan))\n",[40,23457,23458],{"class":10276,"line":12161},[40,23459,10301],{"emptyLinePlaceholder":10300},[40,23461,23462],{"class":10276,"line":12177},[40,23463,23464],{"class":10280},"# Plot 1: ω_ZX vs Δ_ct\n",[40,23466,23467,23470,23472,23474,23476,23478,23480,23483,23485,23487,23489,23492,23494,23497],{"class":10276,"line":12192},[40,23468,23469],{"class":10363},"plt.",[40,23471,16373],{"class":10430},[40,23473,167],{"class":10363},[40,23475,13078],{"class":11903},[40,23477,162],{"class":10732},[40,23479,167],{"class":10363},[40,23481,23482],{"class":10446},"8",[40,23484,10484],{"class":10363},[40,23486,12566],{"class":10446},[40,23488,11040],{"class":10363},[40,23490,23491],{"class":11903},"dpi",[40,23493,162],{"class":10732},[40,23495,23496],{"class":10446},"150",[40,23498,10761],{"class":10363},[40,23500,23501,23503,23506,23508,23510,23513,23515,23517,23520,23523,23526,23528,23531,23534,23536,23539,23541,23543,23546,23548,23551,23553,23556],{"class":10276,"line":12231},[40,23502,23469],{"class":10363},[40,23504,23505],{"class":10430},"plot",[40,23507,167],{"class":10363},[40,23509,23311],{"class":10446},[40,23511,23512],{"class":10363},", wzx_slice, ",[40,23514,15415],{"class":11903},[40,23516,162],{"class":10732},[40,23518,23519],{"class":10352},"rf",[40,23521,23522],{"class":10287},"'$\\omega_",[40,23524,23525],{"class":10732},"{{",[40,23527,21402],{"class":10287},[40,23529,23530],{"class":10732},"}}",[40,23532,23533],{"class":10287},"$ (Ω=",[40,23535,13268],{"class":10446},[40,23537,23538],{"class":10363},"K.Omega0",[40,23540,14652],{"class":10352},[40,23542,13280],{"class":10446},[40,23544,23545],{"class":10287}," MHz)'",[40,23547,10484],{"class":10363},[40,23549,23550],{"class":11903},"color",[40,23552,162],{"class":10732},[40,23554,23555],{"class":10287},"'blue'",[40,23557,10761],{"class":10363},[40,23559,23560,23562,23565,23567,23569,23571,23573,23575,23578],{"class":10276,"line":12245},[40,23561,23469],{"class":10363},[40,23563,23564],{"class":10430},"axhline",[40,23566,167],{"class":10363},[40,23568,10801],{"class":10446},[40,23570,10484],{"class":10363},[40,23572,23274],{"class":11903},[40,23574,162],{"class":10732},[40,23576,23577],{"class":10287},"\":\"",[40,23579,10761],{"class":10363},[40,23581,23582,23585,23588,23591],{"class":10276,"line":12280},[40,23583,23584],{"class":10430},"_annotate_regions",[40,23586,23587],{"class":10363},"(plt.",[40,23589,23590],{"class":10430},"gca",[40,23592,23593],{"class":10363},"())\n",[40,23595,23596,23598,23601,23603,23605,23608,23610,23613,23615,23618],{"class":10276,"line":12294},[40,23597,23469],{"class":10363},[40,23599,23600],{"class":10430},"xlabel",[40,23602,167],{"class":10363},[40,23604,936],{"class":10352},[40,23606,23607],{"class":10356},"'$\\Delta_{ct}$ ",[40,23609,167],{"class":10446},[40,23611,23612],{"class":10356},"MHz",[40,23614,207],{"class":10446},[40,23616,23617],{"class":10356},"'",[40,23619,10761],{"class":10363},[40,23621,23622,23624,23627,23629,23631,23634,23637,23640,23642,23644,23646,23648],{"class":10276,"line":12329},[40,23623,23469],{"class":10363},[40,23625,23626],{"class":10430},"ylabel",[40,23628,167],{"class":10363},[40,23630,936],{"class":10352},[40,23632,23633],{"class":10356},"'$",[40,23635,23636],{"class":10732},"\\o",[40,23638,23639],{"class":10356},"mega_{ZX}$ ",[40,23641,167],{"class":10446},[40,23643,23612],{"class":10356},[40,23645,207],{"class":10446},[40,23647,23617],{"class":10356},[40,23649,10761],{"class":10363},[40,23651,23652,23654,23657,23659,23661,23664,23666,23669],{"class":10276,"line":12343},[40,23653,23469],{"class":10363},[40,23655,23656],{"class":10430},"title",[40,23658,167],{"class":10363},[40,23660,936],{"class":10352},[40,23662,23663],{"class":10356},"'Leading-order $",[40,23665,23636],{"class":10732},[40,23667,23668],{"class":10356},"mega_{ZX}$ vs detuning'",[40,23670,10761],{"class":10363},[40,23672,23673,23675,23678],{"class":10276,"line":12376},[40,23674,23469],{"class":10363},[40,23676,23677],{"class":10430},"legend",[40,23679,12174],{"class":10363},[40,23681,23682,23684,23687],{"class":10276,"line":12409},[40,23683,23469],{"class":10363},[40,23685,23686],{"class":10430},"show",[40,23688,12174],{"class":10363},[40,23690,23691],{"class":10276,"line":12414},[40,23692,10301],{"emptyLinePlaceholder":10300},[40,23694,23695],{"class":10276,"line":12419},[40,23696,23697],{"class":10280},"# Plot 2: ω_ZZ vs Δ_ct\n",[40,23699,23700,23702,23704,23706,23708,23710,23712,23714,23716,23718,23720,23722,23724,23726],{"class":10276,"line":12435},[40,23701,23469],{"class":10363},[40,23703,16373],{"class":10430},[40,23705,167],{"class":10363},[40,23707,13078],{"class":11903},[40,23709,162],{"class":10732},[40,23711,167],{"class":10363},[40,23713,23482],{"class":10446},[40,23715,10484],{"class":10363},[40,23717,12566],{"class":10446},[40,23719,11040],{"class":10363},[40,23721,23491],{"class":11903},[40,23723,162],{"class":10732},[40,23725,23496],{"class":10446},[40,23727,10761],{"class":10363},[40,23729,23730,23732,23734,23736,23738,23741,23743,23745,23747,23749,23751,23754,23756,23759,23761,23763,23765,23767,23769,23771],{"class":10276,"line":12447},[40,23731,23469],{"class":10363},[40,23733,23505],{"class":10430},[40,23735,167],{"class":10363},[40,23737,23311],{"class":10446},[40,23739,23740],{"class":10363},", wzz_slice, ",[40,23742,15415],{"class":11903},[40,23744,162],{"class":10732},[40,23746,936],{"class":10352},[40,23748,23633],{"class":10356},[40,23750,23636],{"class":10732},[40,23752,23753],{"class":10356},"mega_{ZZ}$ ",[40,23755,167],{"class":10446},[40,23757,23758],{"class":10356},"static; lowest order",[40,23760,207],{"class":10446},[40,23762,23617],{"class":10356},[40,23764,10484],{"class":10363},[40,23766,23550],{"class":11903},[40,23768,162],{"class":10732},[40,23770,23555],{"class":10287},[40,23772,10761],{"class":10363},[40,23774,23775,23777,23779,23781,23783,23785,23787,23789,23791],{"class":10276,"line":12463},[40,23776,23469],{"class":10363},[40,23778,23564],{"class":10430},[40,23780,167],{"class":10363},[40,23782,10801],{"class":10446},[40,23784,10484],{"class":10363},[40,23786,23274],{"class":11903},[40,23788,162],{"class":10732},[40,23790,23577],{"class":10287},[40,23792,10761],{"class":10363},[40,23794,23795,23797,23799,23801],{"class":10276,"line":12480},[40,23796,23584],{"class":10430},[40,23798,23587],{"class":10363},[40,23800,23590],{"class":10430},[40,23802,23593],{"class":10363},[40,23804,23805,23807,23809,23811,23813,23815,23817,23819,23821,23823],{"class":10276,"line":12498},[40,23806,23469],{"class":10363},[40,23808,23600],{"class":10430},[40,23810,167],{"class":10363},[40,23812,936],{"class":10352},[40,23814,23607],{"class":10356},[40,23816,167],{"class":10446},[40,23818,23612],{"class":10356},[40,23820,207],{"class":10446},[40,23822,23617],{"class":10356},[40,23824,10761],{"class":10363},[40,23826,23827,23829,23831,23833,23835,23837,23839,23841,23843,23845,23847,23849],{"class":10276,"line":12504},[40,23828,23469],{"class":10363},[40,23830,23626],{"class":10430},[40,23832,167],{"class":10363},[40,23834,936],{"class":10352},[40,23836,23633],{"class":10356},[40,23838,23636],{"class":10732},[40,23840,23753],{"class":10356},[40,23842,167],{"class":10446},[40,23844,23612],{"class":10356},[40,23846,207],{"class":10446},[40,23848,23617],{"class":10356},[40,23850,10761],{"class":10363},[40,23852,23853,23855,23857,23859,23861,23864,23866,23869],{"class":10276,"line":12517},[40,23854,23469],{"class":10363},[40,23856,23656],{"class":10430},[40,23858,167],{"class":10363},[40,23860,936],{"class":10352},[40,23862,23863],{"class":10356},"'Lowest-order $",[40,23865,23636],{"class":10732},[40,23867,23868],{"class":10356},"mega_{ZZ}$ vs detuning'",[40,23870,10761],{"class":10363},[40,23872,23873,23875,23877],{"class":10276,"line":12530},[40,23874,23469],{"class":10363},[40,23876,23677],{"class":10430},[40,23878,12174],{"class":10363},[40,23880,23881,23883,23885],{"class":10276,"line":12542},[40,23882,23469],{"class":10363},[40,23884,23686],{"class":10430},[40,23886,12174],{"class":10363},[40,23888,23889],{"class":10276,"line":12571},[40,23890,10301],{"emptyLinePlaceholder":10300},[40,23892,23893],{"class":10276,"line":12622},[40,23894,23895],{"class":10280},"# Heatmap 1: ω_ZX(Δ, Ω)\n",[40,23897,23898,23901,23903,23905,23908,23910,23912,23915,23917],{"class":10276,"line":12628},[40,23899,23900],{"class":10363},"wzx_map ",[40,23902,162],{"class":10732},[40,23904,11156],{"class":10363},[40,23906,23907],{"class":10430},"empty",[40,23909,16469],{"class":10363},[40,23911,23311],{"class":10446},[40,23913,23914],{"class":10363},".size, ",[40,23916,23335],{"class":10446},[40,23918,23919],{"class":10363},".size))\n",[40,23921,23922,23924,23927,23929,23932,23934,23936],{"class":10276,"line":12634},[40,23923,13840],{"class":10352},[40,23925,23926],{"class":10363}," i, d ",[40,23928,13846],{"class":10352},[40,23930,23931],{"class":10732}," enumerate",[40,23933,167],{"class":10363},[40,23935,23311],{"class":10446},[40,23937,13353],{"class":10363},[40,23939,23940,23942,23945,23947,23949,23951,23953],{"class":10276,"line":12639},[40,23941,17425],{"class":10352},[40,23943,23944],{"class":10363}," j, om ",[40,23946,13846],{"class":10352},[40,23948,23931],{"class":10732},[40,23950,167],{"class":10363},[40,23952,23335],{"class":10446},[40,23954,13353],{"class":10363},[40,23956,23957,23960,23962,23964],{"class":10276,"line":12644},[40,23958,23959],{"class":10363},"        wzx_map[i, j] ",[40,23961,162],{"class":10732},[40,23963,22772],{"class":10430},[40,23965,23966],{"class":10363},"(d, K.J, om, K.alpha_c)\n",[40,23968,23969],{"class":10276,"line":12667},[40,23970,10301],{"emptyLinePlaceholder":10300},[40,23972,23973,23975,23977,23979,23981,23983,23985,23987,23989,23991,23993,23995,23997,23999],{"class":10276,"line":12673},[40,23974,23469],{"class":10363},[40,23976,16373],{"class":10430},[40,23978,167],{"class":10363},[40,23980,13078],{"class":11903},[40,23982,162],{"class":10732},[40,23984,167],{"class":10363},[40,23986,23482],{"class":10446},[40,23988,10484],{"class":10363},[40,23990,12785],{"class":10446},[40,23992,11040],{"class":10363},[40,23994,23491],{"class":11903},[40,23996,162],{"class":10732},[40,23998,23496],{"class":10446},[40,24000,10761],{"class":10363},[40,24002,24003,24006,24008,24011,24013,24015,24017,24019,24021,24023,24025,24027,24029,24031,24033,24035,24037,24039,24041],{"class":10276,"line":12684},[40,24004,24005],{"class":10363},"extent ",[40,24007,162],{"class":10732},[40,24009,24010],{"class":10363}," [",[40,24012,23335],{"class":10446},[40,24014,1123],{"class":10363},[40,24016,3067],{"class":10430},[40,24018,13575],{"class":10363},[40,24020,23335],{"class":10446},[40,24022,1123],{"class":10363},[40,24024,3032],{"class":10430},[40,24026,13575],{"class":10363},[40,24028,23311],{"class":10446},[40,24030,1123],{"class":10363},[40,24032,3067],{"class":10430},[40,24034,13575],{"class":10363},[40,24036,23311],{"class":10446},[40,24038,1123],{"class":10363},[40,24040,3032],{"class":10430},[40,24042,24043],{"class":10363},"()]\n",[40,24045,24046,24048,24050,24053,24055,24057,24059,24061,24063,24065,24067,24069,24071,24073,24076,24078,24080,24083],{"class":10276,"line":12697},[40,24047,23469],{"class":10363},[40,24049,13105],{"class":10430},[40,24051,24052],{"class":10363},"(wzx_map, ",[40,24054,13117],{"class":11903},[40,24056,162],{"class":10732},[40,24058,13122],{"class":10287},[40,24060,10484],{"class":10363},[40,24062,13153],{"class":11903},[40,24064,162],{"class":10732},[40,24066,13158],{"class":10287},[40,24068,10484],{"class":10363},[40,24070,13127],{"class":11903},[40,24072,162],{"class":10732},[40,24074,24075],{"class":10363},"extent, ",[40,24077,16338],{"class":11903},[40,24079,162],{"class":10732},[40,24081,24082],{"class":10287},"\"magma\"",[40,24084,10761],{"class":10363},[40,24086,24087,24089,24091,24093,24095,24097,24099,24101,24103,24105,24107,24109,24111,24113],{"class":10276,"line":12723},[40,24088,23469],{"class":10363},[40,24090,13173],{"class":10430},[40,24092,167],{"class":10363},[40,24094,15415],{"class":11903},[40,24096,162],{"class":10732},[40,24098,936],{"class":10352},[40,24100,23633],{"class":10356},[40,24102,23636],{"class":10732},[40,24104,23639],{"class":10356},[40,24106,167],{"class":10446},[40,24108,23612],{"class":10356},[40,24110,207],{"class":10446},[40,24112,23617],{"class":10356},[40,24114,10761],{"class":10363},[40,24116,24117,24119,24121,24123,24125,24127,24130,24133,24135,24137,24139,24141],{"class":10276,"line":12736},[40,24118,23469],{"class":10363},[40,24120,23600],{"class":10430},[40,24122,167],{"class":10363},[40,24124,936],{"class":10352},[40,24126,23633],{"class":10356},[40,24128,24129],{"class":10732},"\\O",[40,24131,24132],{"class":10356},"mega$ ",[40,24134,167],{"class":10446},[40,24136,23612],{"class":10356},[40,24138,207],{"class":10446},[40,24140,23617],{"class":10356},[40,24142,10761],{"class":10363},[40,24144,24145,24147,24149,24151,24153,24155,24157,24159,24161,24163],{"class":10276,"line":12777},[40,24146,23469],{"class":10363},[40,24148,23626],{"class":10430},[40,24150,167],{"class":10363},[40,24152,936],{"class":10352},[40,24154,23607],{"class":10356},[40,24156,167],{"class":10446},[40,24158,23612],{"class":10356},[40,24160,207],{"class":10446},[40,24162,23617],{"class":10356},[40,24164,10761],{"class":10363},[40,24166,24167,24169,24171,24173,24175,24177,24179,24182,24184,24187,24189,24192,24194,24197,24199,24202,24204,24206],{"class":10276,"line":12790},[40,24168,23469],{"class":10363},[40,24170,23656],{"class":10430},[40,24172,167],{"class":10363},[40,24174,936],{"class":10352},[40,24176,23633],{"class":10356},[40,24178,23636],{"class":10732},[40,24180,24181],{"class":10356},"mega_{ZX}",[40,24183,167],{"class":10446},[40,24185,24186],{"class":10356},"\\Delta_{ct}, ",[40,24188,24129],{"class":10732},[40,24190,24191],{"class":10356},"mega",[40,24193,207],{"class":10446},[40,24195,24196],{"class":10356},"$ ",[40,24198,167],{"class":10446},[40,24200,24201],{"class":10356},"leading-order",[40,24203,207],{"class":10446},[40,24205,23617],{"class":10356},[40,24207,10761],{"class":10363},[40,24209,24210,24212,24214],{"class":10276,"line":16980},[40,24211,23469],{"class":10363},[40,24213,23686],{"class":10430},[40,24215,12174],{"class":10363},[40,24217,24218],{"class":10276,"line":16991},[40,24219,10301],{"emptyLinePlaceholder":10300},[40,24221,24222],{"class":10276,"line":16996},[40,24223,24224],{"class":10280},"# Heatmap 2: |ω_ZX\u002Fω_ZZ|(Δ, Ω) (clipped for visibility)\n",[40,24226,24227,24230,24232,24234,24237],{"class":10276,"line":17001},[40,24228,24229],{"class":10363},"ratio_map ",[40,24231,162],{"class":10732},[40,24233,11156],{"class":10363},[40,24235,24236],{"class":10430},"empty_like",[40,24238,24239],{"class":10363},"(wzx_map)\n",[40,24241,24242,24244,24246,24248,24250,24252,24254],{"class":10276,"line":17019},[40,24243,13840],{"class":10352},[40,24245,23926],{"class":10363},[40,24247,13846],{"class":10352},[40,24249,23931],{"class":10732},[40,24251,167],{"class":10363},[40,24253,23311],{"class":10446},[40,24255,13353],{"class":10363},[40,24257,24258,24261,24263,24265],{"class":10276,"line":17047},[40,24259,24260],{"class":10363},"    zz ",[40,24262,162],{"class":10732},[40,24264,22976],{"class":10430},[40,24266,24267],{"class":10363},"(d, K.J, K.alpha_c, K.alpha_t)\n",[40,24269,24270,24272,24274,24276,24278,24280,24282],{"class":10276,"line":17064},[40,24271,17425],{"class":10352},[40,24273,23944],{"class":10363},[40,24275,13846],{"class":10352},[40,24277,23931],{"class":10732},[40,24279,167],{"class":10363},[40,24281,23335],{"class":10446},[40,24283,13353],{"class":10363},[40,24285,24286,24289,24291,24293],{"class":10276,"line":17077},[40,24287,24288],{"class":10363},"        zx ",[40,24290,162],{"class":10732},[40,24292,22772],{"class":10430},[40,24294,23966],{"class":10363},[40,24296,24297,24300,24302,24304,24307,24309,24312,24314,24316,24319,24321,24323,24325],{"class":10276,"line":17110},[40,24298,24299],{"class":10363},"        ratio_map[i, j] ",[40,24301,162],{"class":10732},[40,24303,22133],{"class":10732},[40,24305,24306],{"class":10363},"(zx ",[40,24308,11243],{"class":10732},[40,24310,24311],{"class":10363}," zz) ",[40,24313,11225],{"class":10352},[40,24315,22133],{"class":10732},[40,24317,24318],{"class":10363},"(zz) ",[40,24320,22432],{"class":10732},[40,24322,22114],{"class":10446},[40,24324,11237],{"class":10352},[40,24326,24327],{"class":10363}," np.nan\n",[40,24329,24330],{"class":10276,"line":17145},[40,24331,10301],{"emptyLinePlaceholder":10300},[40,24333,24334,24337,24339,24341,24344,24347,24349,24352,24355],{"class":10276,"line":17194},[40,24335,24336],{"class":10363},"clip_max ",[40,24338,162],{"class":10732},[40,24340,11156],{"class":10363},[40,24342,24343],{"class":10430},"nanpercentile",[40,24345,24346],{"class":10363},"(ratio_map[np.",[40,24348,22443],{"class":10430},[40,24350,24351],{"class":10363},"(ratio_map)], ",[40,24353,24354],{"class":10446},"99.5",[40,24356,10761],{"class":10363},[40,24358,24359,24362,24364,24366,24369,24372,24374],{"class":10276,"line":17202},[40,24360,24361],{"class":10363},"ratio_map_vis ",[40,24363,162],{"class":10732},[40,24365,11156],{"class":10363},[40,24367,24368],{"class":10430},"clip",[40,24370,24371],{"class":10363},"(ratio_map, ",[40,24373,10801],{"class":10446},[40,24375,24376],{"class":10363},", clip_max)\n",[40,24378,24379],{"class":10276,"line":17207},[40,24380,10301],{"emptyLinePlaceholder":10300},[40,24382,24383,24385,24387,24389,24391,24393,24395,24397,24399,24401,24403,24405,24407,24409],{"class":10276,"line":17241},[40,24384,23469],{"class":10363},[40,24386,16373],{"class":10430},[40,24388,167],{"class":10363},[40,24390,13078],{"class":11903},[40,24392,162],{"class":10732},[40,24394,167],{"class":10363},[40,24396,23482],{"class":10446},[40,24398,10484],{"class":10363},[40,24400,12785],{"class":10446},[40,24402,11040],{"class":10363},[40,24404,23491],{"class":11903},[40,24406,162],{"class":10732},[40,24408,23496],{"class":10446},[40,24410,10761],{"class":10363},[40,24412,24413,24415,24417,24420,24422,24424,24426,24428,24430,24432,24434,24436,24438,24440],{"class":10276,"line":17269},[40,24414,23469],{"class":10363},[40,24416,13105],{"class":10430},[40,24418,24419],{"class":10363},"(ratio_map_vis, ",[40,24421,13117],{"class":11903},[40,24423,162],{"class":10732},[40,24425,13122],{"class":10287},[40,24427,10484],{"class":10363},[40,24429,13153],{"class":11903},[40,24431,162],{"class":10732},[40,24433,13158],{"class":10287},[40,24435,10484],{"class":10363},[40,24437,13127],{"class":11903},[40,24439,162],{"class":10732},[40,24441,24442],{"class":10363},"extent)\n",[40,24444,24445,24447,24449,24451,24453,24455,24457,24459,24462,24464,24467,24469,24472,24474,24477],{"class":10276,"line":17295},[40,24446,23469],{"class":10363},[40,24448,13173],{"class":10430},[40,24450,167],{"class":10363},[40,24452,15415],{"class":11903},[40,24454,162],{"class":10732},[40,24456,936],{"class":10352},[40,24458,23633],{"class":10356},[40,24460,24461],{"class":10363},"|",[40,24463,23636],{"class":10732},[40,24465,24466],{"class":10356},"mega_{ZX}\u002F",[40,24468,23636],{"class":10732},[40,24470,24471],{"class":10356},"mega_{ZZ}",[40,24473,24461],{"class":10363},[40,24475,24476],{"class":10356},"$'",[40,24478,10761],{"class":10363},[40,24480,24481,24483,24485,24487,24489,24491,24493,24495,24497,24499,24501,24503],{"class":10276,"line":17324},[40,24482,23469],{"class":10363},[40,24484,23600],{"class":10430},[40,24486,167],{"class":10363},[40,24488,936],{"class":10352},[40,24490,23633],{"class":10356},[40,24492,24129],{"class":10732},[40,24494,24132],{"class":10356},[40,24496,167],{"class":10446},[40,24498,23612],{"class":10356},[40,24500,207],{"class":10446},[40,24502,23617],{"class":10356},[40,24504,10761],{"class":10363},[40,24506,24507,24509,24511,24513,24515,24517,24519,24521,24523,24525],{"class":10276,"line":17341},[40,24508,23469],{"class":10363},[40,24510,23626],{"class":10430},[40,24512,167],{"class":10363},[40,24514,936],{"class":10352},[40,24516,23607],{"class":10356},[40,24518,167],{"class":10446},[40,24520,23612],{"class":10356},[40,24522,207],{"class":10446},[40,24524,23617],{"class":10356},[40,24526,10761],{"class":10363},[40,24528,24529,24531,24533,24535,24537,24540,24542,24544,24546,24548,24550,24552,24555,24557,24560,24562,24564],{"class":10276,"line":17350},[40,24530,23469],{"class":10363},[40,24532,23656],{"class":10430},[40,24534,167],{"class":10363},[40,24536,936],{"class":10352},[40,24538,24539],{"class":10356},"'Heuristic $",[40,24541,24461],{"class":10363},[40,24543,23636],{"class":10732},[40,24545,24466],{"class":10356},[40,24547,23636],{"class":10732},[40,24549,24471],{"class":10356},[40,24551,24461],{"class":10363},[40,24553,24554],{"class":10356},"$ map ",[40,24556,167],{"class":10446},[40,24558,24559],{"class":10356},"clipped at 99.5th pct",[40,24561,207],{"class":10446},[40,24563,23617],{"class":10356},[40,24565,10761],{"class":10363},[40,24567,24568,24570,24572],{"class":10276,"line":17358},[40,24569,23469],{"class":10363},[40,24571,23686],{"class":10430},[40,24573,12174],{"class":10363},[13997,24575],{"alt":13999,"src":24576},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-01.webp",[13997,24578],{"alt":14003,"src":24579},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-02.webp",[13997,24581],{"alt":14007,"src":24582},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-03.webp",[13997,24584],{"alt":15009,"src":24585},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-04.webp",[10082,24587,24589],{"className":10269,"code":24588,"language":10271,"meta":10090,"style":10090},"#@title Echoed-CR demonstration (two-segment echo, Eqs. 28a–d in PRA 2020)\n\nimport math\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef omega_echo_coeffs(zx_mhz: float, ix_mhz: float, iz_mhz: float, zz_mhz: float, tau_ns: float):\n    \"\"\"Return (u_ii, u_iy, u_iz, u_zx) after a +Ω \u002F Xπ \u002F −Ω echo; MHz→rad\u002Fns inside.\"\"\"\n    conv = 2.0 * math.pi \u002F 1000.0\n    zx, ix, iz, zz = zx_mhz*conv, ix_mhz*conv, iz_mhz*conv, zz_mhz*conv\n    w_plus  = math.sqrt((zx + ix)**2 + (iz + zz)**2)\n    w_minus = math.sqrt((zx - ix)**2 + (iz - zz)**2)\n    c_plus, s_plus = math.cos(0.5*w_plus*tau_ns), math.sin(0.5*w_plus*tau_ns)\n    c_minus, s_minus = math.cos(0.5*w_minus*tau_ns), math.sin(0.5*w_minus*tau_ns)\n    denom = (w_plus*w_minus + 1e-30)\n    u_ii = c_plus*c_minus + ((ix**2 - iz**2 - zx**2 + zz**2)\u002Fdenom)*s_plus*s_minus\n    u_iy = 2j*(zx*zz - ix*iz)\u002Fdenom * s_plus*s_minus\n    u_iz = 1j*((zz - iz)\u002F(w_minus + 1e-30))*c_plus*s_minus - 1j*((zz + iz)\u002F(w_plus + 1e-30))*s_plus*c_minus\n    u_zx = 1j*((ix - zx)\u002F(w_minus + 1e-30))*c_plus*s_minus - 1j*((ix + zx)\u002F(w_plus + 1e-30))*s_plus*c_minus\n    return u_ii, u_iy, u_iz, u_zx\n\n# Representative point near Region III speed-up:\nf_c, f_t = 5114.0, 4914.0\nalpha_c = -330.0\nJ, Omega = 3.8, 30.0\ndelta0 = f_c - f_t\n\ndef omega_zx_mhz(delta_ct, J, Omega, alpha_c):\n    return J*Omega*(1.0\u002F(delta_ct + alpha_c) - 1.0\u002Fmax(delta_ct, 1e-12))\n\nw_zx = omega_zx_mhz(delta0, J, Omega, alpha_c)\nw_ix, w_iz, w_zz = 1.0, 10.0, 0.1   # illustrative single-qubit terms \u002F static ZZ\ntau_ns = 100.0\n\nu_ii, u_iy, u_iz, u_zx = omega_echo_coeffs(w_zx, w_ix, w_iz, w_zz, tau_ns)\n\nplt.figure(figsize=(7.5, 4.8), dpi=150)\nlabels = [r'$|u_{ZX}|$', r'$|u_{IY}|$', r'$|u_{IZ}|$']\nvals = [abs(u_zx), abs(u_iy), abs(u_iz)]\nx = np.arange(len(labels))\nplt.bar(x, vals)\nplt.xticks(x, labels)\nplt.ylabel('Coefficient magnitude (unitless)')\nplt.title('Echoed-CR: effective coefficients after two-segment echo')\nplt.show()\n",[10088,24590,24591,24596,24600,24606,24616,24626,24630,24684,24689,24708,24738,24777,24812,24853,24891,24913,24981,25026,25102,25173,25180,25184,25189,25203,25215,25229,25244,25248,25272,25313,25317,25329,25351,25361,25365,25377,25381,25412,25467,25491,25511,25521,25531,25544,25557],{"__ignoreMap":10090},[40,24592,24593],{"class":10276,"line":10277},[40,24594,24595],{"class":10280},"#@title Echoed-CR demonstration (two-segment echo, Eqs. 28a–d in PRA 2020)\n",[40,24597,24598],{"class":10276,"line":10284},[40,24599,10301],{"emptyLinePlaceholder":10300},[40,24601,24602,24604],{"class":10276,"line":10291},[40,24603,10375],{"class":10352},[40,24605,21798],{"class":10363},[40,24607,24608,24610,24612,24614],{"class":10276,"line":10297},[40,24609,10375],{"class":10352},[40,24611,10378],{"class":10363},[40,24613,10381],{"class":10352},[40,24615,10384],{"class":10363},[40,24617,24618,24620,24622,24624],{"class":10276,"line":10304},[40,24619,10375],{"class":10352},[40,24621,10405],{"class":10363},[40,24623,10381],{"class":10352},[40,24625,10410],{"class":10363},[40,24627,24628],{"class":10276,"line":10310},[40,24629,10301],{"emptyLinePlaceholder":10300},[40,24631,24632,24634,24637,24639,24642,24644,24646,24648,24651,24653,24655,24657,24660,24662,24664,24666,24669,24671,24673,24675,24678,24680,24682],{"class":10276,"line":10316},[40,24633,10965],{"class":10352},[40,24635,24636],{"class":10430}," omega_echo_coeffs",[40,24638,167],{"class":10363},[40,24640,24641],{"class":10973},"zx_mhz",[40,24643,10443],{"class":10363},[40,24645,10784],{"class":10732},[40,24647,10484],{"class":10363},[40,24649,24650],{"class":10973},"ix_mhz",[40,24652,10443],{"class":10363},[40,24654,10784],{"class":10732},[40,24656,10484],{"class":10363},[40,24658,24659],{"class":10973},"iz_mhz",[40,24661,10443],{"class":10363},[40,24663,10784],{"class":10732},[40,24665,10484],{"class":10363},[40,24667,24668],{"class":10973},"zz_mhz",[40,24670,10443],{"class":10363},[40,24672,10784],{"class":10732},[40,24674,10484],{"class":10363},[40,24676,24677],{"class":10973},"tau_ns",[40,24679,10443],{"class":10363},[40,24681,10784],{"class":10732},[40,24683,13353],{"class":10363},[40,24685,24686],{"class":10276,"line":10322},[40,24687,24688],{"class":10287},"    \"\"\"Return (u_ii, u_iy, u_iz, u_zx) after a +Ω \u002F Xπ \u002F −Ω echo; MHz→rad\u002Fns inside.\"\"\"\n",[40,24690,24691,24694,24696,24698,24700,24703,24705],{"class":10276,"line":10327},[40,24692,24693],{"class":10363},"    conv ",[40,24695,162],{"class":10732},[40,24697,12060],{"class":10446},[40,24699,11881],{"class":10732},[40,24701,24702],{"class":10363}," math.pi ",[40,24704,11243],{"class":10732},[40,24706,24707],{"class":10446}," 1000.0\n",[40,24709,24710,24713,24715,24718,24720,24723,24725,24728,24730,24733,24735],{"class":10276,"line":10333},[40,24711,24712],{"class":10363},"    zx, ix, iz, zz ",[40,24714,162],{"class":10732},[40,24716,24717],{"class":10363}," zx_mhz",[40,24719,11752],{"class":10732},[40,24721,24722],{"class":10363},"conv, ix_mhz",[40,24724,11752],{"class":10732},[40,24726,24727],{"class":10363},"conv, iz_mhz",[40,24729,11752],{"class":10732},[40,24731,24732],{"class":10363},"conv, zz_mhz",[40,24734,11752],{"class":10732},[40,24736,24737],{"class":10363},"conv\n",[40,24739,24740,24743,24745,24747,24749,24752,24754,24757,24759,24761,24763,24766,24768,24771,24773,24775],{"class":10276,"line":10339},[40,24741,24742],{"class":10363},"    w_plus  ",[40,24744,162],{"class":10732},[40,24746,22358],{"class":10363},[40,24748,2218],{"class":10430},[40,24750,24751],{"class":10363},"((zx ",[40,24753,1053],{"class":10732},[40,24755,24756],{"class":10363}," ix)",[40,24758,22571],{"class":10732},[40,24760,1044],{"class":10446},[40,24762,12042],{"class":10732},[40,24764,24765],{"class":10363}," (iz ",[40,24767,1053],{"class":10732},[40,24769,24770],{"class":10363}," zz)",[40,24772,22571],{"class":10732},[40,24774,1044],{"class":10446},[40,24776,10761],{"class":10363},[40,24778,24779,24782,24784,24786,24788,24790,24792,24794,24796,24798,24800,24802,24804,24806,24808,24810],{"class":10276,"line":10344},[40,24780,24781],{"class":10363},"    w_minus ",[40,24783,162],{"class":10732},[40,24785,22358],{"class":10363},[40,24787,2218],{"class":10430},[40,24789,24751],{"class":10363},[40,24791,11056],{"class":10732},[40,24793,24756],{"class":10363},[40,24795,22571],{"class":10732},[40,24797,1044],{"class":10446},[40,24799,12042],{"class":10732},[40,24801,24765],{"class":10363},[40,24803,11056],{"class":10732},[40,24805,24770],{"class":10363},[40,24807,22571],{"class":10732},[40,24809,1044],{"class":10446},[40,24811,10761],{"class":10363},[40,24813,24814,24817,24819,24821,24823,24825,24828,24830,24833,24835,24838,24840,24842,24844,24846,24848,24850],{"class":10276,"line":10349},[40,24815,24816],{"class":10363},"    c_plus, s_plus ",[40,24818,162],{"class":10732},[40,24820,22358],{"class":10363},[40,24822,1820],{"class":10430},[40,24824,167],{"class":10363},[40,24826,24827],{"class":10446},"0.5",[40,24829,11752],{"class":10732},[40,24831,24832],{"class":10363},"w_plus",[40,24834,11752],{"class":10732},[40,24836,24837],{"class":10363},"tau_ns), math.",[40,24839,1813],{"class":10430},[40,24841,167],{"class":10363},[40,24843,24827],{"class":10446},[40,24845,11752],{"class":10732},[40,24847,24832],{"class":10363},[40,24849,11752],{"class":10732},[40,24851,24852],{"class":10363},"tau_ns)\n",[40,24854,24855,24858,24860,24862,24864,24866,24868,24870,24873,24875,24877,24879,24881,24883,24885,24887,24889],{"class":10276,"line":10367},[40,24856,24857],{"class":10363},"    c_minus, s_minus ",[40,24859,162],{"class":10732},[40,24861,22358],{"class":10363},[40,24863,1820],{"class":10430},[40,24865,167],{"class":10363},[40,24867,24827],{"class":10446},[40,24869,11752],{"class":10732},[40,24871,24872],{"class":10363},"w_minus",[40,24874,11752],{"class":10732},[40,24876,24837],{"class":10363},[40,24878,1813],{"class":10430},[40,24880,167],{"class":10363},[40,24882,24827],{"class":10446},[40,24884,11752],{"class":10732},[40,24886,24872],{"class":10363},[40,24888,11752],{"class":10732},[40,24890,24852],{"class":10363},[40,24892,24893,24896,24898,24901,24903,24906,24908,24911],{"class":10276,"line":10372},[40,24894,24895],{"class":10363},"    denom ",[40,24897,162],{"class":10732},[40,24899,24900],{"class":10363}," (w_plus",[40,24902,11752],{"class":10732},[40,24904,24905],{"class":10363},"w_minus ",[40,24907,1053],{"class":10732},[40,24909,24910],{"class":10446}," 1e-30",[40,24912,10761],{"class":10363},[40,24914,24915,24918,24920,24923,24925,24928,24930,24933,24935,24937,24939,24942,24944,24946,24948,24951,24953,24955,24957,24960,24962,24964,24966,24968,24971,24973,24976,24978],{"class":10276,"line":10387},[40,24916,24917],{"class":10363},"    u_ii ",[40,24919,162],{"class":10732},[40,24921,24922],{"class":10363}," c_plus",[40,24924,11752],{"class":10732},[40,24926,24927],{"class":10363},"c_minus ",[40,24929,1053],{"class":10732},[40,24931,24932],{"class":10363}," ((ix",[40,24934,22571],{"class":10732},[40,24936,1044],{"class":10446},[40,24938,11764],{"class":10732},[40,24940,24941],{"class":10363}," iz",[40,24943,22571],{"class":10732},[40,24945,1044],{"class":10446},[40,24947,11764],{"class":10732},[40,24949,24950],{"class":10363}," zx",[40,24952,22571],{"class":10732},[40,24954,1044],{"class":10446},[40,24956,12042],{"class":10732},[40,24958,24959],{"class":10363}," zz",[40,24961,22571],{"class":10732},[40,24963,1044],{"class":10446},[40,24965,207],{"class":10363},[40,24967,11243],{"class":10732},[40,24969,24970],{"class":10363},"denom)",[40,24972,11752],{"class":10732},[40,24974,24975],{"class":10363},"s_plus",[40,24977,11752],{"class":10732},[40,24979,24980],{"class":10363},"s_minus\n",[40,24982,24983,24986,24988,24990,24992,24994,24997,24999,25002,25004,25007,25009,25012,25014,25017,25019,25022,25024],{"class":10276,"line":10400},[40,24984,24985],{"class":10363},"    u_iy ",[40,24987,162],{"class":10732},[40,24989,12960],{"class":10446},[40,24991,1075],{"class":10352},[40,24993,11752],{"class":10732},[40,24995,24996],{"class":10363},"(zx",[40,24998,11752],{"class":10732},[40,25000,25001],{"class":10363},"zz ",[40,25003,11056],{"class":10732},[40,25005,25006],{"class":10363}," ix",[40,25008,11752],{"class":10732},[40,25010,25011],{"class":10363},"iz)",[40,25013,11243],{"class":10732},[40,25015,25016],{"class":10363},"denom ",[40,25018,11752],{"class":10732},[40,25020,25021],{"class":10363}," s_plus",[40,25023,11752],{"class":10732},[40,25025,24980],{"class":10363},[40,25027,25028,25031,25033,25035,25037,25039,25042,25044,25047,25049,25052,25054,25056,25058,25060,25063,25065,25068,25070,25072,25074,25076,25078,25080,25082,25084,25087,25089,25091,25093,25095,25097,25099],{"class":10276,"line":10413},[40,25029,25030],{"class":10363},"    u_iz ",[40,25032,162],{"class":10732},[40,25034,13373],{"class":10446},[40,25036,1075],{"class":10352},[40,25038,11752],{"class":10732},[40,25040,25041],{"class":10363},"((zz ",[40,25043,11056],{"class":10732},[40,25045,25046],{"class":10363}," iz)",[40,25048,11243],{"class":10732},[40,25050,25051],{"class":10363},"(w_minus ",[40,25053,1053],{"class":10732},[40,25055,24910],{"class":10446},[40,25057,2407],{"class":10363},[40,25059,11752],{"class":10732},[40,25061,25062],{"class":10363},"c_plus",[40,25064,11752],{"class":10732},[40,25066,25067],{"class":10363},"s_minus ",[40,25069,11056],{"class":10732},[40,25071,13373],{"class":10446},[40,25073,1075],{"class":10352},[40,25075,11752],{"class":10732},[40,25077,25041],{"class":10363},[40,25079,1053],{"class":10732},[40,25081,25046],{"class":10363},[40,25083,11243],{"class":10732},[40,25085,25086],{"class":10363},"(w_plus ",[40,25088,1053],{"class":10732},[40,25090,24910],{"class":10446},[40,25092,2407],{"class":10363},[40,25094,11752],{"class":10732},[40,25096,24975],{"class":10363},[40,25098,11752],{"class":10732},[40,25100,25101],{"class":10363},"c_minus\n",[40,25103,25104,25107,25109,25111,25113,25115,25118,25120,25123,25125,25127,25129,25131,25133,25135,25137,25139,25141,25143,25145,25147,25149,25151,25153,25155,25157,25159,25161,25163,25165,25167,25169,25171],{"class":10276,"line":10418},[40,25105,25106],{"class":10363},"    u_zx ",[40,25108,162],{"class":10732},[40,25110,13373],{"class":10446},[40,25112,1075],{"class":10352},[40,25114,11752],{"class":10732},[40,25116,25117],{"class":10363},"((ix ",[40,25119,11056],{"class":10732},[40,25121,25122],{"class":10363}," zx)",[40,25124,11243],{"class":10732},[40,25126,25051],{"class":10363},[40,25128,1053],{"class":10732},[40,25130,24910],{"class":10446},[40,25132,2407],{"class":10363},[40,25134,11752],{"class":10732},[40,25136,25062],{"class":10363},[40,25138,11752],{"class":10732},[40,25140,25067],{"class":10363},[40,25142,11056],{"class":10732},[40,25144,13373],{"class":10446},[40,25146,1075],{"class":10352},[40,25148,11752],{"class":10732},[40,25150,25117],{"class":10363},[40,25152,1053],{"class":10732},[40,25154,25122],{"class":10363},[40,25156,11243],{"class":10732},[40,25158,25086],{"class":10363},[40,25160,1053],{"class":10732},[40,25162,24910],{"class":10446},[40,25164,2407],{"class":10363},[40,25166,11752],{"class":10732},[40,25168,24975],{"class":10363},[40,25170,11752],{"class":10732},[40,25172,25101],{"class":10363},[40,25174,25175,25177],{"class":10276,"line":10424},[40,25176,11201],{"class":10352},[40,25178,25179],{"class":10363}," u_ii, u_iy, u_iz, u_zx\n",[40,25181,25182],{"class":10276,"line":10437},[40,25183,10301],{"emptyLinePlaceholder":10300},[40,25185,25186],{"class":10276,"line":10456},[40,25187,25188],{"class":10280},"# Representative point near Region III speed-up:\n",[40,25190,25191,25194,25196,25198,25200],{"class":10276,"line":10472},[40,25192,25193],{"class":10363},"f_c, f_t ",[40,25195,162],{"class":10732},[40,25197,21861],{"class":10446},[40,25199,10484],{"class":10363},[40,25201,25202],{"class":10446},"4914.0\n",[40,25204,25205,25208,25210,25212],{"class":10276,"line":10496},[40,25206,25207],{"class":10363},"alpha_c ",[40,25209,162],{"class":10732},[40,25211,11764],{"class":10732},[40,25213,25214],{"class":10446},"330.0\n",[40,25216,25217,25220,25222,25224,25226],{"class":10276,"line":10510},[40,25218,25219],{"class":10363},"J, Omega ",[40,25221,162],{"class":10732},[40,25223,21924],{"class":10446},[40,25225,10484],{"class":10363},[40,25227,25228],{"class":10446},"30.0\n",[40,25230,25231,25234,25236,25239,25241],{"class":10276,"line":10696},[40,25232,25233],{"class":10363},"delta0 ",[40,25235,162],{"class":10732},[40,25237,25238],{"class":10363}," f_c ",[40,25240,11056],{"class":10732},[40,25242,25243],{"class":10363}," f_t\n",[40,25245,25246],{"class":10276,"line":10702},[40,25247,10301],{"emptyLinePlaceholder":10300},[40,25249,25250,25252,25254,25256,25258,25260,25262,25264,25266,25268,25270],{"class":10276,"line":10708},[40,25251,10965],{"class":10352},[40,25253,22772],{"class":10430},[40,25255,167],{"class":10363},[40,25257,22777],{"class":10973},[40,25259,10484],{"class":10363},[40,25261,22786],{"class":10973},[40,25263,10484],{"class":10363},[40,25265,22795],{"class":10973},[40,25267,10484],{"class":10363},[40,25269,22804],{"class":10973},[40,25271,13353],{"class":10363},[40,25273,25274,25276,25279,25281,25283,25285,25287,25289,25291,25294,25296,25298,25300,25302,25305,25308,25311],{"class":10276,"line":10714},[40,25275,11201],{"class":10352},[40,25277,25278],{"class":10363}," J",[40,25280,11752],{"class":10732},[40,25282,22795],{"class":10363},[40,25284,11752],{"class":10732},[40,25286,167],{"class":10363},[40,25288,10810],{"class":10446},[40,25290,11243],{"class":10732},[40,25292,25293],{"class":10363},"(delta_ct ",[40,25295,1053],{"class":10732},[40,25297,22856],{"class":10363},[40,25299,11056],{"class":10732},[40,25301,22530],{"class":10446},[40,25303,25304],{"class":10732},"\u002Fmax",[40,25306,25307],{"class":10363},"(delta_ct, ",[40,25309,25310],{"class":10446},"1e-12",[40,25312,13612],{"class":10363},[40,25314,25315],{"class":10276,"line":10720},[40,25316,10301],{"emptyLinePlaceholder":10300},[40,25318,25319,25322,25324,25326],{"class":10276,"line":10726},[40,25320,25321],{"class":10363},"w_zx ",[40,25323,162],{"class":10732},[40,25325,22772],{"class":10430},[40,25327,25328],{"class":10363},"(delta0, J, Omega, alpha_c)\n",[40,25330,25331,25334,25336,25338,25340,25343,25345,25348],{"class":10276,"line":10764},[40,25332,25333],{"class":10363},"w_ix, w_iz, w_zz ",[40,25335,162],{"class":10732},[40,25337,22530],{"class":10446},[40,25339,10484],{"class":10363},[40,25341,25342],{"class":10446},"10.0",[40,25344,10484],{"class":10363},[40,25346,25347],{"class":10446},"0.1",[40,25349,25350],{"class":10280},"   # illustrative single-qubit terms \u002F static ZZ\n",[40,25352,25353,25356,25358],{"class":10276,"line":10778},[40,25354,25355],{"class":10363},"tau_ns ",[40,25357,162],{"class":10732},[40,25359,25360],{"class":10446}," 100.0\n",[40,25362,25363],{"class":10276,"line":10815},[40,25364,10301],{"emptyLinePlaceholder":10300},[40,25366,25367,25370,25372,25374],{"class":10276,"line":10829},[40,25368,25369],{"class":10363},"u_ii, u_iy, u_iz, u_zx ",[40,25371,162],{"class":10732},[40,25373,24636],{"class":10430},[40,25375,25376],{"class":10363},"(w_zx, w_ix, w_iz, w_zz, tau_ns)\n",[40,25378,25379],{"class":10276,"line":10842},[40,25380,10301],{"emptyLinePlaceholder":10300},[40,25382,25383,25385,25387,25389,25391,25393,25395,25398,25400,25402,25404,25406,25408,25410],{"class":10276,"line":10855},[40,25384,23469],{"class":10363},[40,25386,16373],{"class":10430},[40,25388,167],{"class":10363},[40,25390,13078],{"class":11903},[40,25392,162],{"class":10732},[40,25394,167],{"class":10363},[40,25396,25397],{"class":10446},"7.5",[40,25399,10484],{"class":10363},[40,25401,15378],{"class":10446},[40,25403,11040],{"class":10363},[40,25405,23491],{"class":11903},[40,25407,162],{"class":10732},[40,25409,23496],{"class":10446},[40,25411,10761],{"class":10363},[40,25413,25414,25417,25419,25421,25423,25425,25427,25430,25432,25434,25436,25438,25440,25442,25445,25447,25449,25451,25453,25455,25457,25460,25462,25464],{"class":10276,"line":10868},[40,25415,25416],{"class":10363},"labels ",[40,25418,162],{"class":10732},[40,25420,24010],{"class":10363},[40,25422,936],{"class":10352},[40,25424,23633],{"class":10356},[40,25426,24461],{"class":10363},[40,25428,25429],{"class":10356},"u_{ZX}",[40,25431,24461],{"class":10363},[40,25433,24476],{"class":10356},[40,25435,10484],{"class":10363},[40,25437,936],{"class":10352},[40,25439,23633],{"class":10356},[40,25441,24461],{"class":10363},[40,25443,25444],{"class":10356},"u_{IY}",[40,25446,24461],{"class":10363},[40,25448,24476],{"class":10356},[40,25450,10484],{"class":10363},[40,25452,936],{"class":10352},[40,25454,23633],{"class":10356},[40,25456,24461],{"class":10363},[40,25458,25459],{"class":10356},"u_{IZ}",[40,25461,24461],{"class":10363},[40,25463,24476],{"class":10356},[40,25465,25466],{"class":10363},"]\n",[40,25468,25469,25472,25474,25476,25478,25481,25483,25486,25488],{"class":10276,"line":10881},[40,25470,25471],{"class":10363},"vals ",[40,25473,162],{"class":10732},[40,25475,24010],{"class":10363},[40,25477,23431],{"class":10732},[40,25479,25480],{"class":10363},"(u_zx), ",[40,25482,23431],{"class":10732},[40,25484,25485],{"class":10363},"(u_iy), ",[40,25487,23431],{"class":10732},[40,25489,25490],{"class":10363},"(u_iz)]\n",[40,25492,25493,25496,25498,25500,25503,25505,25508],{"class":10276,"line":10907},[40,25494,25495],{"class":10363},"x ",[40,25497,162],{"class":10732},[40,25499,11156],{"class":10363},[40,25501,25502],{"class":10430},"arange",[40,25504,167],{"class":10363},[40,25506,25507],{"class":10732},"len",[40,25509,25510],{"class":10363},"(labels))\n",[40,25512,25513,25515,25518],{"class":10276,"line":10920},[40,25514,23469],{"class":10363},[40,25516,25517],{"class":10430},"bar",[40,25519,25520],{"class":10363},"(x, vals)\n",[40,25522,25523,25525,25528],{"class":10276,"line":10933},[40,25524,23469],{"class":10363},[40,25526,25527],{"class":10430},"xticks",[40,25529,25530],{"class":10363},"(x, labels)\n",[40,25532,25533,25535,25537,25539,25542],{"class":10276,"line":10946},[40,25534,23469],{"class":10363},[40,25536,23626],{"class":10430},[40,25538,167],{"class":10363},[40,25540,25541],{"class":10287},"'Coefficient magnitude (unitless)'",[40,25543,10761],{"class":10363},[40,25545,25546,25548,25550,25552,25555],{"class":10276,"line":10951},[40,25547,23469],{"class":10363},[40,25549,23656],{"class":10430},[40,25551,167],{"class":10363},[40,25553,25554],{"class":10287},"'Echoed-CR: effective coefficients after two-segment echo'",[40,25556,10761],{"class":10363},[40,25558,25559,25561,25563],{"class":10276,"line":10956},[40,25560,23469],{"class":10363},[40,25562,23686],{"class":10430},[40,25564,12174],{"class":10363},[13997,25566],{"alt":17711,"src":25567},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-05.webp",[10082,25569,25571],{"className":10269,"code":25570,"language":10271,"meta":10090,"style":10090},"#@title Conditional target Rabi under CR effective Hamiltonian\n# we diagonalize once and propagate.\n\nfrom dataclasses import dataclass\nimport math\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# -----------------------------\n# Control knobs (MHz unless noted)\n# -----------------------------\n@dataclass\nclass RabiKnobs:\n    f_c: float = 5114.0          # control frequency\n    f_t: float = 4914.0          # target frequency\n    alpha_c: float = -330.0      # control anharmonicity\n    alpha_t: float = -330.0      # target anharmonicity\n    J: float = 3.8               # exchange coupling\n    Omega: float = 30.0          # CR drive amplitude on control\n\n    # Residuals (typical orders, tweak for your device)\n    wIX_mhz: float = 1.0         # IX (drive crosstalk on target)\n    wIZ_khz: float = 20.0        # IZ (AC Stark on target)\n    wZI_mhz: float = 10.0        # ZI (AC Stark on control)\n    wZZ_khz: float = 100.0       # ZZ (static coupling residual)\n\n    # Time axis\n    t_end_ns: float = 400.0\n    n_steps: int = 2001\n\nK = RabiKnobs()\n\n# -----------------------------\n# Helpers and literature-rate formulas\n# -----------------------------\ndef _safe_div(a: float, b: float, eps: float = 1e-12) -> float:\n    return a \u002F (b if abs(b) >= eps else np.sign(b) * eps)\n\ndef hz_to_rad_per_ns(hz: float) -> float:\n    return 2.0 * math.pi * hz * 1e-9\n\ndef mhz_to_rad_per_ns(mhz: float) -> float:\n    return hz_to_rad_per_ns(mhz * 1e6)\n\ndef khz_to_rad_per_ns(khz: float) -> float:\n    return hz_to_rad_per_ns(khz * 1e3)\n\n# Leading-order CR analytics (weak drive, effective model)\n# ω_ZX ≈ J Ω [1\u002F(Δ+α_c) − 1\u002FΔ],   ω_ZZ ≈ J^2 [1\u002F(Δ−α_t) − 1\u002F(Δ+α_c)]\ndef omega_zx_mhz(delta_ct: float, J: float, Omega: float, alpha_c: float) -> float:\n    return J * Omega * (_safe_div(1.0, delta_ct + alpha_c) - _safe_div(1.0, delta_ct))\n\ndef omega_zz_mhz(delta_ct: float, J: float, alpha_c: float, alpha_t: float) -> float:\n    return (J ** 2) * (_safe_div(1.0, delta_ct - alpha_t) - _safe_div(1.0, delta_ct + alpha_c))\n\n# -----------------------------\n# Build H_eff (4×4) in rad\u002Fns\n# -----------------------------\nI2 = np.eye(2)\nX = np.array([[0.0, 1.0],\n              [1.0, 0.0]])\nZ = np.array([[1.0, 0.0],\n              [0.0, -1.0]])\n\ndef kron2(a, b): return np.kron(a, b)\n\ndef build_heff_radns(K: RabiKnobs) -> np.ndarray:\n    Δ = K.f_c - K.f_t\n    w_zx = mhz_to_rad_per_ns(omega_zx_mhz(Δ, K.J, K.Omega, K.alpha_c))\n    w_zz = mhz_to_rad_per_ns(omega_zz_mhz(Δ, K.J, K.alpha_c, K.alpha_t)) + khz_to_rad_per_ns(K.wZZ_khz)\n    w_ix = mhz_to_rad_per_ns(K.wIX_mhz)\n    w_iz = khz_to_rad_per_ns(K.wIZ_khz)\n    w_zi = mhz_to_rad_per_ns(K.wZI_mhz)\n    return ((w_ix\u002F2.0) * kron2(I2, X) +\n            (w_iz\u002F2.0) * kron2(I2, Z) +\n            (w_zi\u002F2.0) * kron2(Z,  I2) +\n            (w_zx\u002F2.0) * kron2(Z,  X) +\n            (w_zz\u002F2.0) * kron2(Z,  Z))\n\n# -----------------------------\n# Time evolution via diagonalization\n# -----------------------------\ndef evolve_state_constH(H: np.ndarray, psi0: np.ndarray, t_ns: np.ndarray) -> np.ndarray:\n    E, V = np.linalg.eigh(H)\n    Vinv = V.conj().T\n    amps0 = Vinv @ psi0\n    out = np.empty((psi0.size, t_ns.size), dtype=complex)\n    for k, t in enumerate(t_ns):\n        out[:, k] = V @ (np.exp(-1j * E * t) * amps0)\n    return out\n\n# Basis |00>, |01>, |10>, |11>\ne00 = np.array([1,0,0,0], dtype=complex)\ne01 = np.array([0,1,0,0], dtype=complex)\ne10 = np.array([0,0,1,0], dtype=complex)\ne11 = np.array([0,0,0,1], dtype=complex)\n\n# -----------------------------\n# Conditional Rabi simulation\n# -----------------------------\nH = build_heff_radns(K)\nt_ns = np.linspace(0.0, K.t_end_ns, K.n_steps)\n\n# Prepare control in |0> (|00>) vs |1> (|10>), target initially |0>\nψ_c0 = evolve_state_constH(H, e00, t_ns)\nψ_c1 = evolve_state_constH(H, e10, t_ns)\n\n# Target |1> probability = P(|01>) + P(|11>)\np_t1_c0 = np.abs(e01.conj() @ ψ_c0)**2 + np.abs(e11.conj() @ ψ_c0)**2\np_t1_c1 = np.abs(e01.conj() @ ψ_c1)**2 + np.abs(e11.conj() @ ψ_c1)**2\n\n# -----------------------------\n# Plot\n# -----------------------------\nplt.figure(figsize=(8.2, 4.8), dpi=150)\nplt.plot(t_ns, p_t1_c0, label='Control |0⟩, init |00⟩', color='blue')\nplt.plot(t_ns, p_t1_c1, label='Control |1⟩, init |10⟩', color='orange')\nplt.xlabel('Time (ns)')\nplt.ylabel('Target |1⟩ probability')\nplt.title('Conditional target Rabi under CR effective Hamiltonian')\nplt.legend()\nplt.show()\n",[10088,25572,25573,25578,25583,25587,25597,25603,25613,25623,25627,25631,25636,25640,25644,25653,25666,25679,25694,25709,25722,25736,25740,25745,25759,25774,25789,25804,25808,25813,25825,25837,25841,25851,25855,25859,25864,25868,25908,25942,25946,25968,25988,25992,26014,26030,26034,26056,26072,26076,26081,26086,26130,26166,26170,26214,26256,26260,26264,26269,26273,26290,26311,26325,26346,26360,26364,26393,26397,26411,26426,26442,26465,26477,26489,26501,26524,26544,26564,26584,26602,26606,26610,26615,26619,26643,26658,26674,26689,26710,26724,26763,26769,26773,26778,26813,26848,26883,26918,26922,26926,26931,26935,26947,26965,26969,26974,26986,26998,27002,27007,27057,27103,27107,27111,27116,27120,27151,27177,27204,27217,27230,27243,27251],{"__ignoreMap":10090},[40,25574,25575],{"class":10276,"line":10277},[40,25576,25577],{"class":10280},"#@title Conditional target Rabi under CR effective Hamiltonian\n",[40,25579,25580],{"class":10276,"line":10284},[40,25581,25582],{"class":10280},"# we diagonalize once and propagate.\n",[40,25584,25585],{"class":10276,"line":10291},[40,25586,10301],{"emptyLinePlaceholder":10300},[40,25588,25589,25591,25593,25595],{"class":10276,"line":10297},[40,25590,10353],{"class":10352},[40,25592,10530],{"class":10363},[40,25594,10375],{"class":10352},[40,25596,10535],{"class":10363},[40,25598,25599,25601],{"class":10276,"line":10304},[40,25600,10375],{"class":10352},[40,25602,21798],{"class":10363},[40,25604,25605,25607,25609,25611],{"class":10276,"line":10310},[40,25606,10375],{"class":10352},[40,25608,10378],{"class":10363},[40,25610,10381],{"class":10352},[40,25612,10384],{"class":10363},[40,25614,25615,25617,25619,25621],{"class":10276,"line":10316},[40,25616,10375],{"class":10352},[40,25618,10405],{"class":10363},[40,25620,10381],{"class":10352},[40,25622,10410],{"class":10363},[40,25624,25625],{"class":10276,"line":10322},[40,25626,10301],{"emptyLinePlaceholder":10300},[40,25628,25629],{"class":10276,"line":10327},[40,25630,21827],{"class":10280},[40,25632,25633],{"class":10276,"line":10333},[40,25634,25635],{"class":10280},"# Control knobs (MHz unless noted)\n",[40,25637,25638],{"class":10276,"line":10339},[40,25639,21827],{"class":10280},[40,25641,25642],{"class":10276,"line":10344},[40,25643,10627],{"class":10430},[40,25645,25646,25648,25651],{"class":10276,"line":10349},[40,25647,10632],{"class":10352},[40,25649,25650],{"class":10635}," RabiKnobs",[40,25652,10639],{"class":10363},[40,25654,25655,25657,25659,25661,25663],{"class":10276,"line":10367},[40,25656,21854],{"class":10363},[40,25658,10784],{"class":10732},[40,25660,10772],{"class":10732},[40,25662,21861],{"class":10446},[40,25664,25665],{"class":10280},"          # control frequency\n",[40,25667,25668,25670,25672,25674,25676],{"class":10276,"line":10372},[40,25669,21869],{"class":10363},[40,25671,10784],{"class":10732},[40,25673,10772],{"class":10732},[40,25675,21876],{"class":10446},[40,25677,25678],{"class":10280},"          # target frequency\n",[40,25680,25681,25683,25685,25687,25689,25691],{"class":10276,"line":10387},[40,25682,21884],{"class":10363},[40,25684,10784],{"class":10732},[40,25686,10772],{"class":10732},[40,25688,11764],{"class":10732},[40,25690,21893],{"class":10446},[40,25692,25693],{"class":10280},"      # control anharmonicity\n",[40,25695,25696,25698,25700,25702,25704,25706],{"class":10276,"line":10400},[40,25697,21901],{"class":10363},[40,25699,10784],{"class":10732},[40,25701,10772],{"class":10732},[40,25703,11764],{"class":10732},[40,25705,21893],{"class":10446},[40,25707,25708],{"class":10280},"      # target anharmonicity\n",[40,25710,25711,25713,25715,25717,25719],{"class":10276,"line":10413},[40,25712,21917],{"class":10363},[40,25714,10784],{"class":10732},[40,25716,10772],{"class":10732},[40,25718,21924],{"class":10446},[40,25720,25721],{"class":10280},"               # exchange coupling\n",[40,25723,25724,25727,25729,25731,25733],{"class":10276,"line":10418},[40,25725,25726],{"class":10363},"    Omega: ",[40,25728,10784],{"class":10732},[40,25730,10772],{"class":10732},[40,25732,21939],{"class":10446},[40,25734,25735],{"class":10280},"          # CR drive amplitude on control\n",[40,25737,25738],{"class":10276,"line":10424},[40,25739,10301],{"emptyLinePlaceholder":10300},[40,25741,25742],{"class":10276,"line":10437},[40,25743,25744],{"class":10280},"    # Residuals (typical orders, tweak for your device)\n",[40,25746,25747,25750,25752,25754,25756],{"class":10276,"line":10456},[40,25748,25749],{"class":10363},"    wIX_mhz: ",[40,25751,10784],{"class":10732},[40,25753,10772],{"class":10732},[40,25755,22530],{"class":10446},[40,25757,25758],{"class":10280},"         # IX (drive crosstalk on target)\n",[40,25760,25761,25764,25766,25768,25771],{"class":10276,"line":10472},[40,25762,25763],{"class":10363},"    wIZ_khz: ",[40,25765,10784],{"class":10732},[40,25767,10772],{"class":10732},[40,25769,25770],{"class":10446}," 20.0",[40,25772,25773],{"class":10280},"        # IZ (AC Stark on target)\n",[40,25775,25776,25779,25781,25783,25786],{"class":10276,"line":10496},[40,25777,25778],{"class":10363},"    wZI_mhz: ",[40,25780,10784],{"class":10732},[40,25782,10772],{"class":10732},[40,25784,25785],{"class":10446}," 10.0",[40,25787,25788],{"class":10280},"        # ZI (AC Stark on control)\n",[40,25790,25791,25794,25796,25798,25801],{"class":10276,"line":10510},[40,25792,25793],{"class":10363},"    wZZ_khz: ",[40,25795,10784],{"class":10732},[40,25797,10772],{"class":10732},[40,25799,25800],{"class":10446}," 100.0",[40,25802,25803],{"class":10280},"       # ZZ (static coupling residual)\n",[40,25805,25806],{"class":10276,"line":10696},[40,25807,10301],{"emptyLinePlaceholder":10300},[40,25809,25810],{"class":10276,"line":10702},[40,25811,25812],{"class":10280},"    # Time axis\n",[40,25814,25815,25818,25820,25822],{"class":10276,"line":10708},[40,25816,25817],{"class":10363},"    t_end_ns: ",[40,25819,10784],{"class":10732},[40,25821,10772],{"class":10732},[40,25823,25824],{"class":10446}," 400.0\n",[40,25826,25827,25830,25832,25834],{"class":10276,"line":10714},[40,25828,25829],{"class":10363},"    n_steps: ",[40,25831,10821],{"class":10732},[40,25833,10772],{"class":10732},[40,25835,25836],{"class":10446}," 2001\n",[40,25838,25839],{"class":10276,"line":10720},[40,25840,10301],{"emptyLinePlaceholder":10300},[40,25842,25843,25845,25847,25849],{"class":10276,"line":10726},[40,25844,22053],{"class":10363},[40,25846,162],{"class":10732},[40,25848,25650],{"class":10430},[40,25850,12174],{"class":10363},[40,25852,25853],{"class":10276,"line":10764},[40,25854,10301],{"emptyLinePlaceholder":10300},[40,25856,25857],{"class":10276,"line":10778},[40,25858,21827],{"class":10280},[40,25860,25861],{"class":10276,"line":10815},[40,25862,25863],{"class":10280},"# Helpers and literature-rate formulas\n",[40,25865,25866],{"class":10276,"line":10829},[40,25867,21827],{"class":10280},[40,25869,25870,25872,25874,25876,25878,25880,25882,25884,25886,25888,25890,25892,25894,25896,25898,25900,25902,25904,25906],{"class":10276,"line":10842},[40,25871,10965],{"class":10352},[40,25873,22083],{"class":10430},[40,25875,167],{"class":10363},[40,25877,19],{"class":10973},[40,25879,10443],{"class":10363},[40,25881,10784],{"class":10732},[40,25883,10484],{"class":10363},[40,25885,22096],{"class":10973},[40,25887,10443],{"class":10363},[40,25889,10784],{"class":10732},[40,25891,10484],{"class":10363},[40,25893,22105],{"class":10973},[40,25895,10443],{"class":10363},[40,25897,10784],{"class":10732},[40,25899,10772],{"class":10732},[40,25901,22114],{"class":10446},[40,25903,22117],{"class":10363},[40,25905,10784],{"class":10732},[40,25907,10639],{"class":10363},[40,25909,25910,25912,25914,25916,25919,25921,25923,25925,25927,25929,25931,25933,25935,25937,25939],{"class":10276,"line":10855},[40,25911,11201],{"class":10352},[40,25913,22164],{"class":10363},[40,25915,11243],{"class":10732},[40,25917,25918],{"class":10363}," (b ",[40,25920,11225],{"class":10352},[40,25922,22133],{"class":10732},[40,25924,22136],{"class":10363},[40,25926,12957],{"class":10732},[40,25928,22549],{"class":10363},[40,25930,22464],{"class":10352},[40,25932,11156],{"class":10363},[40,25934,22150],{"class":10430},[40,25936,22136],{"class":10363},[40,25938,11752],{"class":10732},[40,25940,25941],{"class":10363}," eps)\n",[40,25943,25944],{"class":10276,"line":10868},[40,25945,10301],{"emptyLinePlaceholder":10300},[40,25947,25948,25950,25953,25955,25958,25960,25962,25964,25966],{"class":10276,"line":10881},[40,25949,10965],{"class":10352},[40,25951,25952],{"class":10430}," hz_to_rad_per_ns",[40,25954,167],{"class":10363},[40,25956,25957],{"class":10973},"hz",[40,25959,10443],{"class":10363},[40,25961,10784],{"class":10732},[40,25963,22117],{"class":10363},[40,25965,10784],{"class":10732},[40,25967,10639],{"class":10363},[40,25969,25970,25972,25974,25976,25978,25980,25983,25985],{"class":10276,"line":10907},[40,25971,11201],{"class":10352},[40,25973,12060],{"class":10446},[40,25975,11881],{"class":10732},[40,25977,24702],{"class":10363},[40,25979,11752],{"class":10732},[40,25981,25982],{"class":10363}," hz ",[40,25984,11752],{"class":10732},[40,25986,25987],{"class":10446}," 1e-9\n",[40,25989,25990],{"class":10276,"line":10920},[40,25991,10301],{"emptyLinePlaceholder":10300},[40,25993,25994,25996,25999,26001,26004,26006,26008,26010,26012],{"class":10276,"line":10933},[40,25995,10965],{"class":10352},[40,25997,25998],{"class":10430}," mhz_to_rad_per_ns",[40,26000,167],{"class":10363},[40,26002,26003],{"class":10973},"mhz",[40,26005,10443],{"class":10363},[40,26007,10784],{"class":10732},[40,26009,22117],{"class":10363},[40,26011,10784],{"class":10732},[40,26013,10639],{"class":10363},[40,26015,26016,26018,26020,26023,26025,26028],{"class":10276,"line":10946},[40,26017,11201],{"class":10352},[40,26019,25952],{"class":10430},[40,26021,26022],{"class":10363},"(mhz ",[40,26024,11752],{"class":10732},[40,26026,26027],{"class":10446}," 1e6",[40,26029,10761],{"class":10363},[40,26031,26032],{"class":10276,"line":10951},[40,26033,10301],{"emptyLinePlaceholder":10300},[40,26035,26036,26038,26041,26043,26046,26048,26050,26052,26054],{"class":10276,"line":10956},[40,26037,10965],{"class":10352},[40,26039,26040],{"class":10430}," khz_to_rad_per_ns",[40,26042,167],{"class":10363},[40,26044,26045],{"class":10973},"khz",[40,26047,10443],{"class":10363},[40,26049,10784],{"class":10732},[40,26051,22117],{"class":10363},[40,26053,10784],{"class":10732},[40,26055,10639],{"class":10363},[40,26057,26058,26060,26062,26065,26067,26070],{"class":10276,"line":10962},[40,26059,11201],{"class":10352},[40,26061,25952],{"class":10430},[40,26063,26064],{"class":10363},"(khz ",[40,26066,11752],{"class":10732},[40,26068,26069],{"class":10446}," 1e3",[40,26071,10761],{"class":10363},[40,26073,26074],{"class":10276,"line":10983},[40,26075,10301],{"emptyLinePlaceholder":10300},[40,26077,26078],{"class":10276,"line":11005},[40,26079,26080],{"class":10280},"# Leading-order CR analytics (weak drive, effective model)\n",[40,26082,26083],{"class":10276,"line":11011},[40,26084,26085],{"class":10280},"# ω_ZX ≈ J Ω [1\u002F(Δ+α_c) − 1\u002FΔ],   ω_ZZ ≈ J^2 [1\u002F(Δ−α_t) − 1\u002F(Δ+α_c)]\n",[40,26087,26088,26090,26092,26094,26096,26098,26100,26102,26104,26106,26108,26110,26112,26114,26116,26118,26120,26122,26124,26126,26128],{"class":10276,"line":11022},[40,26089,10965],{"class":10352},[40,26091,22772],{"class":10430},[40,26093,167],{"class":10363},[40,26095,22777],{"class":10973},[40,26097,10443],{"class":10363},[40,26099,10784],{"class":10732},[40,26101,10484],{"class":10363},[40,26103,22786],{"class":10973},[40,26105,10443],{"class":10363},[40,26107,10784],{"class":10732},[40,26109,10484],{"class":10363},[40,26111,22795],{"class":10973},[40,26113,10443],{"class":10363},[40,26115,10784],{"class":10732},[40,26117,10484],{"class":10363},[40,26119,22804],{"class":10973},[40,26121,10443],{"class":10363},[40,26123,10784],{"class":10732},[40,26125,22117],{"class":10363},[40,26127,10784],{"class":10732},[40,26129,10639],{"class":10363},[40,26131,26132,26134,26136,26138,26140,26142,26144,26146,26148,26150,26152,26154,26156,26158,26160,26162,26164],{"class":10276,"line":11064},[40,26133,11201],{"class":10352},[40,26135,22832],{"class":10363},[40,26137,11752],{"class":10732},[40,26139,22837],{"class":10363},[40,26141,11752],{"class":10732},[40,26143,920],{"class":10363},[40,26145,22844],{"class":10430},[40,26147,167],{"class":10363},[40,26149,10810],{"class":10446},[40,26151,22851],{"class":10363},[40,26153,1053],{"class":10732},[40,26155,22856],{"class":10363},[40,26157,11056],{"class":10732},[40,26159,22083],{"class":10430},[40,26161,167],{"class":10363},[40,26163,10810],{"class":10446},[40,26165,22867],{"class":10363},[40,26167,26168],{"class":10276,"line":11103},[40,26169,10301],{"emptyLinePlaceholder":10300},[40,26171,26172,26174,26176,26178,26180,26182,26184,26186,26188,26190,26192,26194,26196,26198,26200,26202,26204,26206,26208,26210,26212],{"class":10276,"line":11142},[40,26173,10965],{"class":10352},[40,26175,22976],{"class":10430},[40,26177,167],{"class":10363},[40,26179,22777],{"class":10973},[40,26181,10443],{"class":10363},[40,26183,10784],{"class":10732},[40,26185,10484],{"class":10363},[40,26187,22786],{"class":10973},[40,26189,10443],{"class":10363},[40,26191,10784],{"class":10732},[40,26193,10484],{"class":10363},[40,26195,22804],{"class":10973},[40,26197,10443],{"class":10363},[40,26199,10784],{"class":10732},[40,26201,10484],{"class":10363},[40,26203,23005],{"class":10973},[40,26205,10443],{"class":10363},[40,26207,10784],{"class":10732},[40,26209,22117],{"class":10363},[40,26211,10784],{"class":10732},[40,26213,10639],{"class":10363},[40,26215,26216,26218,26220,26222,26224,26226,26228,26230,26232,26234,26236,26238,26240,26242,26244,26246,26248,26250,26252,26254],{"class":10276,"line":11148},[40,26217,11201],{"class":10352},[40,26219,23031],{"class":10363},[40,26221,22571],{"class":10732},[40,26223,12960],{"class":10446},[40,26225,12055],{"class":10363},[40,26227,11752],{"class":10732},[40,26229,920],{"class":10363},[40,26231,22844],{"class":10430},[40,26233,167],{"class":10363},[40,26235,10810],{"class":10446},[40,26237,22851],{"class":10363},[40,26239,11056],{"class":10732},[40,26241,23054],{"class":10363},[40,26243,11056],{"class":10732},[40,26245,22083],{"class":10430},[40,26247,167],{"class":10363},[40,26249,10810],{"class":10446},[40,26251,22851],{"class":10363},[40,26253,1053],{"class":10732},[40,26255,23069],{"class":10363},[40,26257,26258],{"class":10276,"line":11181},[40,26259,10301],{"emptyLinePlaceholder":10300},[40,26261,26262],{"class":10276,"line":11198},[40,26263,21827],{"class":10280},[40,26265,26266],{"class":10276,"line":11249},[40,26267,26268],{"class":10280},"# Build H_eff (4×4) in rad\u002Fns\n",[40,26270,26271],{"class":10276,"line":11254},[40,26272,21827],{"class":10280},[40,26274,26275,26278,26280,26282,26284,26286,26288],{"class":10276,"line":11259},[40,26276,26277],{"class":10363},"I2 ",[40,26279,162],{"class":10732},[40,26281,11156],{"class":10363},[40,26283,11718],{"class":10430},[40,26285,167],{"class":10363},[40,26287,1044],{"class":10446},[40,26289,10761],{"class":10363},[40,26291,26292,26295,26297,26299,26301,26303,26305,26307,26309],{"class":10276,"line":11265},[40,26293,26294],{"class":10363},"X ",[40,26296,162],{"class":10732},[40,26298,11156],{"class":10363},[40,26300,11206],{"class":10430},[40,26302,11346],{"class":10363},[40,26304,10801],{"class":10446},[40,26306,10484],{"class":10363},[40,26308,10810],{"class":10446},[40,26310,11542],{"class":10363},[40,26312,26313,26316,26318,26320,26322],{"class":10276,"line":11294},[40,26314,26315],{"class":10363},"              [",[40,26317,10810],{"class":10446},[40,26319,10484],{"class":10363},[40,26321,10801],{"class":10446},[40,26323,26324],{"class":10363},"]])\n",[40,26326,26327,26330,26332,26334,26336,26338,26340,26342,26344],{"class":10276,"line":11300},[40,26328,26329],{"class":10363},"Z ",[40,26331,162],{"class":10732},[40,26333,11156],{"class":10363},[40,26335,11206],{"class":10430},[40,26337,11346],{"class":10363},[40,26339,10810],{"class":10446},[40,26341,10484],{"class":10363},[40,26343,10801],{"class":10446},[40,26345,11542],{"class":10363},[40,26347,26348,26350,26352,26354,26356,26358],{"class":10276,"line":11320},[40,26349,26315],{"class":10363},[40,26351,10801],{"class":10446},[40,26353,10484],{"class":10363},[40,26355,11056],{"class":10732},[40,26357,10810],{"class":10446},[40,26359,26324],{"class":10363},[40,26361,26362],{"class":10276,"line":11336},[40,26363,10301],{"emptyLinePlaceholder":10300},[40,26365,26366,26368,26371,26373,26375,26377,26379,26382,26385,26387,26390],{"class":10276,"line":11381},[40,26367,10965],{"class":10352},[40,26369,26370],{"class":10430}," kron2",[40,26372,167],{"class":10363},[40,26374,19],{"class":10973},[40,26376,10484],{"class":10363},[40,26378,22096],{"class":10973},[40,26380,26381],{"class":10363},"): ",[40,26383,26384],{"class":10352},"return",[40,26386,11156],{"class":10363},[40,26388,26389],{"class":10430},"kron",[40,26391,26392],{"class":10363},"(a, b)\n",[40,26394,26395],{"class":10276,"line":11395},[40,26396,10301],{"emptyLinePlaceholder":10300},[40,26398,26399,26401,26404,26406,26408],{"class":10276,"line":11438},[40,26400,10965],{"class":10352},[40,26402,26403],{"class":10430}," build_heff_radns",[40,26405,167],{"class":10363},[40,26407,6232],{"class":10973},[40,26409,26410],{"class":10363},": RabiKnobs) -> np.ndarray:\n",[40,26412,26413,26416,26418,26421,26423],{"class":10276,"line":11452},[40,26414,26415],{"class":10363},"    Δ ",[40,26417,162],{"class":10732},[40,26419,26420],{"class":10363}," K.f_c ",[40,26422,11056],{"class":10732},[40,26424,26425],{"class":10363}," K.f_t\n",[40,26427,26428,26431,26433,26435,26437,26439],{"class":10276,"line":11493},[40,26429,26430],{"class":10363},"    w_zx ",[40,26432,162],{"class":10732},[40,26434,25998],{"class":10430},[40,26436,167],{"class":10363},[40,26438,23374],{"class":10430},[40,26440,26441],{"class":10363},"(Δ, K.J, K.Omega, K.alpha_c))\n",[40,26443,26444,26447,26449,26451,26453,26455,26458,26460,26462],{"class":10276,"line":11507},[40,26445,26446],{"class":10363},"    w_zz ",[40,26448,162],{"class":10732},[40,26450,25998],{"class":10430},[40,26452,167],{"class":10363},[40,26454,23406],{"class":10430},[40,26456,26457],{"class":10363},"(Δ, K.J, K.alpha_c, K.alpha_t)) ",[40,26459,1053],{"class":10732},[40,26461,26040],{"class":10430},[40,26463,26464],{"class":10363},"(K.wZZ_khz)\n",[40,26466,26467,26470,26472,26474],{"class":10276,"line":11512},[40,26468,26469],{"class":10363},"    w_ix ",[40,26471,162],{"class":10732},[40,26473,25998],{"class":10430},[40,26475,26476],{"class":10363},"(K.wIX_mhz)\n",[40,26478,26479,26482,26484,26486],{"class":10276,"line":11517},[40,26480,26481],{"class":10363},"    w_iz ",[40,26483,162],{"class":10732},[40,26485,26040],{"class":10430},[40,26487,26488],{"class":10363},"(K.wIZ_khz)\n",[40,26490,26491,26494,26496,26498],{"class":10276,"line":11545},[40,26492,26493],{"class":10363},"    w_zi ",[40,26495,162],{"class":10732},[40,26497,25998],{"class":10430},[40,26499,26500],{"class":10363},"(K.wZI_mhz)\n",[40,26502,26503,26505,26508,26510,26512,26514,26516,26518,26521],{"class":10276,"line":11557},[40,26504,11201],{"class":10352},[40,26506,26507],{"class":10363}," ((w_ix",[40,26509,11243],{"class":10732},[40,26511,11878],{"class":10446},[40,26513,12055],{"class":10363},[40,26515,11752],{"class":10732},[40,26517,26370],{"class":10430},[40,26519,26520],{"class":10363},"(I2, X) ",[40,26522,26523],{"class":10732},"+\n",[40,26525,26526,26529,26531,26533,26535,26537,26539,26542],{"class":10276,"line":11563},[40,26527,26528],{"class":10363},"            (w_iz",[40,26530,11243],{"class":10732},[40,26532,11878],{"class":10446},[40,26534,12055],{"class":10363},[40,26536,11752],{"class":10732},[40,26538,26370],{"class":10430},[40,26540,26541],{"class":10363},"(I2, Z) ",[40,26543,26523],{"class":10732},[40,26545,26546,26549,26551,26553,26555,26557,26559,26562],{"class":10276,"line":11582},[40,26547,26548],{"class":10363},"            (w_zi",[40,26550,11243],{"class":10732},[40,26552,11878],{"class":10446},[40,26554,12055],{"class":10363},[40,26556,11752],{"class":10732},[40,26558,26370],{"class":10430},[40,26560,26561],{"class":10363},"(Z,  I2) ",[40,26563,26523],{"class":10732},[40,26565,26566,26569,26571,26573,26575,26577,26579,26582],{"class":10276,"line":11596},[40,26567,26568],{"class":10363},"            (w_zx",[40,26570,11243],{"class":10732},[40,26572,11878],{"class":10446},[40,26574,12055],{"class":10363},[40,26576,11752],{"class":10732},[40,26578,26370],{"class":10430},[40,26580,26581],{"class":10363},"(Z,  X) ",[40,26583,26523],{"class":10732},[40,26585,26586,26589,26591,26593,26595,26597,26599],{"class":10276,"line":11609},[40,26587,26588],{"class":10363},"            (w_zz",[40,26590,11243],{"class":10732},[40,26592,11878],{"class":10446},[40,26594,12055],{"class":10363},[40,26596,11752],{"class":10732},[40,26598,26370],{"class":10430},[40,26600,26601],{"class":10363},"(Z,  Z))\n",[40,26603,26604],{"class":10276,"line":11622},[40,26605,10301],{"emptyLinePlaceholder":10300},[40,26607,26608],{"class":10276,"line":11637},[40,26609,21827],{"class":10280},[40,26611,26612],{"class":10276,"line":11708},[40,26613,26614],{"class":10280},"# Time evolution via diagonalization\n",[40,26616,26617],{"class":10276,"line":11727},[40,26618,21827],{"class":10280},[40,26620,26621,26623,26626,26628,26631,26633,26636,26638,26641],{"class":10276,"line":11744},[40,26622,10965],{"class":10352},[40,26624,26625],{"class":10430}," evolve_state_constH",[40,26627,167],{"class":10363},[40,26629,26630],{"class":10973},"H",[40,26632,12019],{"class":10363},[40,26634,26635],{"class":10973},"psi0",[40,26637,12019],{"class":10363},[40,26639,26640],{"class":10973},"t_ns",[40,26642,12025],{"class":10363},[40,26644,26645,26648,26650,26652,26655],{"class":10276,"line":11790},[40,26646,26647],{"class":10363},"    E, V ",[40,26649,162],{"class":10732},[40,26651,11189],{"class":10363},[40,26653,26654],{"class":10430},"eigh",[40,26656,26657],{"class":10363},"(H)\n",[40,26659,26660,26663,26665,26668,26671],{"class":10276,"line":11795},[40,26661,26662],{"class":10363},"    Vinv ",[40,26664,162],{"class":10732},[40,26666,26667],{"class":10363}," V.",[40,26669,26670],{"class":10430},"conj",[40,26672,26673],{"class":10363},"().T\n",[40,26675,26676,26679,26681,26684,26686],{"class":10276,"line":11800},[40,26677,26678],{"class":10363},"    amps0 ",[40,26680,162],{"class":10732},[40,26682,26683],{"class":10363}," Vinv ",[40,26685,13404],{"class":10732},[40,26687,26688],{"class":10363}," psi0\n",[40,26690,26691,26693,26695,26697,26699,26702,26705,26708],{"class":10276,"line":11806},[40,26692,17244],{"class":10363},[40,26694,162],{"class":10732},[40,26696,11156],{"class":10363},[40,26698,23907],{"class":10430},[40,26700,26701],{"class":10363},"((psi0.size, t_ns.size), ",[40,26703,26704],{"class":11903},"dtype",[40,26706,26707],{"class":10732},"=complex",[40,26709,10761],{"class":10363},[40,26711,26712,26714,26717,26719,26721],{"class":10276,"line":11834},[40,26713,17425],{"class":10352},[40,26715,26716],{"class":10363}," k, t ",[40,26718,13846],{"class":10352},[40,26720,23931],{"class":10732},[40,26722,26723],{"class":10363},"(t_ns):\n",[40,26725,26726,26729,26731,26734,26736,26738,26740,26742,26744,26746,26748,26750,26753,26755,26758,26760],{"class":10276,"line":11840},[40,26727,26728],{"class":10363},"        out[:, k] ",[40,26730,162],{"class":10732},[40,26732,26733],{"class":10363}," V ",[40,26735,13404],{"class":10732},[40,26737,12045],{"class":10363},[40,26739,5268],{"class":10430},[40,26741,167],{"class":10363},[40,26743,11056],{"class":10732},[40,26745,78],{"class":10446},[40,26747,1075],{"class":10352},[40,26749,11881],{"class":10732},[40,26751,26752],{"class":10363}," E ",[40,26754,11752],{"class":10732},[40,26756,26757],{"class":10363}," t) ",[40,26759,11752],{"class":10732},[40,26761,26762],{"class":10363}," amps0)\n",[40,26764,26765,26767],{"class":10276,"line":11860},[40,26766,11201],{"class":10352},[40,26768,17355],{"class":10363},[40,26770,26771],{"class":10276,"line":11887},[40,26772,10301],{"emptyLinePlaceholder":10300},[40,26774,26775],{"class":10276,"line":11914},[40,26776,26777],{"class":10280},"# Basis |00>, |01>, |10>, |11>\n",[40,26779,26780,26783,26785,26787,26789,26791,26793,26795,26797,26799,26801,26803,26805,26807,26809,26811],{"class":10276,"line":11938},[40,26781,26782],{"class":10363},"e00 ",[40,26784,162],{"class":10732},[40,26786,11156],{"class":10363},[40,26788,11206],{"class":10430},[40,26790,11209],{"class":10363},[40,26792,78],{"class":10446},[40,26794,73],{"class":10363},[40,26796,66],{"class":10446},[40,26798,73],{"class":10363},[40,26800,66],{"class":10446},[40,26802,73],{"class":10363},[40,26804,66],{"class":10446},[40,26806,11672],{"class":10363},[40,26808,26704],{"class":11903},[40,26810,26707],{"class":10732},[40,26812,10761],{"class":10363},[40,26814,26815,26818,26820,26822,26824,26826,26828,26830,26832,26834,26836,26838,26840,26842,26844,26846],{"class":10276,"line":11960},[40,26816,26817],{"class":10363},"e01 ",[40,26819,162],{"class":10732},[40,26821,11156],{"class":10363},[40,26823,11206],{"class":10430},[40,26825,11209],{"class":10363},[40,26827,66],{"class":10446},[40,26829,73],{"class":10363},[40,26831,78],{"class":10446},[40,26833,73],{"class":10363},[40,26835,66],{"class":10446},[40,26837,73],{"class":10363},[40,26839,66],{"class":10446},[40,26841,11672],{"class":10363},[40,26843,26704],{"class":11903},[40,26845,26707],{"class":10732},[40,26847,10761],{"class":10363},[40,26849,26850,26853,26855,26857,26859,26861,26863,26865,26867,26869,26871,26873,26875,26877,26879,26881],{"class":10276,"line":11975},[40,26851,26852],{"class":10363},"e10 ",[40,26854,162],{"class":10732},[40,26856,11156],{"class":10363},[40,26858,11206],{"class":10430},[40,26860,11209],{"class":10363},[40,26862,66],{"class":10446},[40,26864,73],{"class":10363},[40,26866,66],{"class":10446},[40,26868,73],{"class":10363},[40,26870,78],{"class":10446},[40,26872,73],{"class":10363},[40,26874,66],{"class":10446},[40,26876,11672],{"class":10363},[40,26878,26704],{"class":11903},[40,26880,26707],{"class":10732},[40,26882,10761],{"class":10363},[40,26884,26885,26888,26890,26892,26894,26896,26898,26900,26902,26904,26906,26908,26910,26912,26914,26916],{"class":10276,"line":11996},[40,26886,26887],{"class":10363},"e11 ",[40,26889,162],{"class":10732},[40,26891,11156],{"class":10363},[40,26893,11206],{"class":10430},[40,26895,11209],{"class":10363},[40,26897,66],{"class":10446},[40,26899,73],{"class":10363},[40,26901,66],{"class":10446},[40,26903,73],{"class":10363},[40,26905,66],{"class":10446},[40,26907,73],{"class":10363},[40,26909,78],{"class":10446},[40,26911,11672],{"class":10363},[40,26913,26704],{"class":11903},[40,26915,26707],{"class":10732},[40,26917,10761],{"class":10363},[40,26919,26920],{"class":10276,"line":12001},[40,26921,10301],{"emptyLinePlaceholder":10300},[40,26923,26924],{"class":10276,"line":12006},[40,26925,21827],{"class":10280},[40,26927,26928],{"class":10276,"line":12028},[40,26929,26930],{"class":10280},"# Conditional Rabi simulation\n",[40,26932,26933],{"class":10276,"line":12034},[40,26934,21827],{"class":10280},[40,26936,26937,26940,26942,26944],{"class":10276,"line":12105},[40,26938,26939],{"class":10363},"H ",[40,26941,162],{"class":10732},[40,26943,26403],{"class":10430},[40,26945,26946],{"class":10363},"(K)\n",[40,26948,26949,26952,26954,26956,26958,26960,26962],{"class":10276,"line":12110},[40,26950,26951],{"class":10363},"t_ns ",[40,26953,162],{"class":10732},[40,26955,11156],{"class":10363},[40,26957,11850],{"class":10430},[40,26959,167],{"class":10363},[40,26961,10801],{"class":10446},[40,26963,26964],{"class":10363},", K.t_end_ns, K.n_steps)\n",[40,26966,26967],{"class":10276,"line":12115},[40,26968,10301],{"emptyLinePlaceholder":10300},[40,26970,26971],{"class":10276,"line":12121},[40,26972,26973],{"class":10280},"# Prepare control in |0> (|00>) vs |1> (|10>), target initially |0>\n",[40,26975,26976,26979,26981,26983],{"class":10276,"line":12155},[40,26977,26978],{"class":10363},"ψ_c0 ",[40,26980,162],{"class":10732},[40,26982,26625],{"class":10430},[40,26984,26985],{"class":10363},"(H, e00, t_ns)\n",[40,26987,26988,26991,26993,26995],{"class":10276,"line":12161},[40,26989,26990],{"class":10363},"ψ_c1 ",[40,26992,162],{"class":10732},[40,26994,26625],{"class":10430},[40,26996,26997],{"class":10363},"(H, e10, t_ns)\n",[40,26999,27000],{"class":10276,"line":12177},[40,27001,10301],{"emptyLinePlaceholder":10300},[40,27003,27004],{"class":10276,"line":12192},[40,27005,27006],{"class":10280},"# Target |1> probability = P(|01>) + P(|11>)\n",[40,27008,27009,27012,27014,27016,27018,27021,27023,27026,27028,27031,27033,27035,27037,27039,27041,27044,27046,27048,27050,27052,27054],{"class":10276,"line":12231},[40,27010,27011],{"class":10363},"p_t1_c0 ",[40,27013,162],{"class":10732},[40,27015,11156],{"class":10363},[40,27017,23431],{"class":10430},[40,27019,27020],{"class":10363},"(e01.",[40,27022,26670],{"class":10430},[40,27024,27025],{"class":10363},"() ",[40,27027,13404],{"class":10732},[40,27029,27030],{"class":10363}," ψ_c0)",[40,27032,22571],{"class":10732},[40,27034,1044],{"class":10446},[40,27036,12042],{"class":10732},[40,27038,11156],{"class":10363},[40,27040,23431],{"class":10430},[40,27042,27043],{"class":10363},"(e11.",[40,27045,26670],{"class":10430},[40,27047,27025],{"class":10363},[40,27049,13404],{"class":10732},[40,27051,27030],{"class":10363},[40,27053,22571],{"class":10732},[40,27055,27056],{"class":10446},"2\n",[40,27058,27059,27062,27064,27066,27068,27070,27072,27074,27076,27079,27081,27083,27085,27087,27089,27091,27093,27095,27097,27099,27101],{"class":10276,"line":12245},[40,27060,27061],{"class":10363},"p_t1_c1 ",[40,27063,162],{"class":10732},[40,27065,11156],{"class":10363},[40,27067,23431],{"class":10430},[40,27069,27020],{"class":10363},[40,27071,26670],{"class":10430},[40,27073,27025],{"class":10363},[40,27075,13404],{"class":10732},[40,27077,27078],{"class":10363}," ψ_c1)",[40,27080,22571],{"class":10732},[40,27082,1044],{"class":10446},[40,27084,12042],{"class":10732},[40,27086,11156],{"class":10363},[40,27088,23431],{"class":10430},[40,27090,27043],{"class":10363},[40,27092,26670],{"class":10430},[40,27094,27025],{"class":10363},[40,27096,13404],{"class":10732},[40,27098,27078],{"class":10363},[40,27100,22571],{"class":10732},[40,27102,27056],{"class":10446},[40,27104,27105],{"class":10276,"line":12280},[40,27106,10301],{"emptyLinePlaceholder":10300},[40,27108,27109],{"class":10276,"line":12294},[40,27110,21827],{"class":10280},[40,27112,27113],{"class":10276,"line":12329},[40,27114,27115],{"class":10280},"# Plot\n",[40,27117,27118],{"class":10276,"line":12343},[40,27119,21827],{"class":10280},[40,27121,27122,27124,27126,27128,27130,27132,27134,27137,27139,27141,27143,27145,27147,27149],{"class":10276,"line":12376},[40,27123,23469],{"class":10363},[40,27125,16373],{"class":10430},[40,27127,167],{"class":10363},[40,27129,13078],{"class":11903},[40,27131,162],{"class":10732},[40,27133,167],{"class":10363},[40,27135,27136],{"class":10446},"8.2",[40,27138,10484],{"class":10363},[40,27140,15378],{"class":10446},[40,27142,11040],{"class":10363},[40,27144,23491],{"class":11903},[40,27146,162],{"class":10732},[40,27148,23496],{"class":10446},[40,27150,10761],{"class":10363},[40,27152,27153,27155,27157,27160,27162,27164,27167,27169,27171,27173,27175],{"class":10276,"line":12409},[40,27154,23469],{"class":10363},[40,27156,23505],{"class":10430},[40,27158,27159],{"class":10363},"(t_ns, p_t1_c0, ",[40,27161,15415],{"class":11903},[40,27163,162],{"class":10732},[40,27165,27166],{"class":10287},"'Control |0⟩, init |00⟩'",[40,27168,10484],{"class":10363},[40,27170,23550],{"class":11903},[40,27172,162],{"class":10732},[40,27174,23555],{"class":10287},[40,27176,10761],{"class":10363},[40,27178,27179,27181,27183,27186,27188,27190,27193,27195,27197,27199,27202],{"class":10276,"line":12414},[40,27180,23469],{"class":10363},[40,27182,23505],{"class":10430},[40,27184,27185],{"class":10363},"(t_ns, p_t1_c1, ",[40,27187,15415],{"class":11903},[40,27189,162],{"class":10732},[40,27191,27192],{"class":10287},"'Control |1⟩, init |10⟩'",[40,27194,10484],{"class":10363},[40,27196,23550],{"class":11903},[40,27198,162],{"class":10732},[40,27200,27201],{"class":10287},"'orange'",[40,27203,10761],{"class":10363},[40,27205,27206,27208,27210,27212,27215],{"class":10276,"line":12419},[40,27207,23469],{"class":10363},[40,27209,23600],{"class":10430},[40,27211,167],{"class":10363},[40,27213,27214],{"class":10287},"'Time (ns)'",[40,27216,10761],{"class":10363},[40,27218,27219,27221,27223,27225,27228],{"class":10276,"line":12435},[40,27220,23469],{"class":10363},[40,27222,23626],{"class":10430},[40,27224,167],{"class":10363},[40,27226,27227],{"class":10287},"'Target |1⟩ probability'",[40,27229,10761],{"class":10363},[40,27231,27232,27234,27236,27238,27241],{"class":10276,"line":12447},[40,27233,23469],{"class":10363},[40,27235,23656],{"class":10430},[40,27237,167],{"class":10363},[40,27239,27240],{"class":10287},"'Conditional target Rabi under CR effective Hamiltonian'",[40,27242,10761],{"class":10363},[40,27244,27245,27247,27249],{"class":10276,"line":12463},[40,27246,23469],{"class":10363},[40,27248,23677],{"class":10430},[40,27250,12174],{"class":10363},[40,27252,27253,27255,27257],{"class":10276,"line":12480},[40,27254,23469],{"class":10363},[40,27256,23686],{"class":10430},[40,27258,12174],{"class":10363},[13997,27260],{"alt":17715,"src":27261},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-06.webp",[20069,27263,27264],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sU0A5, html code.shiki .sU0A5{--shiki-default:#E5C07B}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":10090,"searchDepth":10284,"depth":10284,"links":27266},[27267],{"id":20139,"depth":10284,"text":20140,"children":27268},[27269],{"id":21678,"depth":10291,"text":21679},[20088,20089,20090,27271],"Cross-resonance gate",[27273],{"username":7,"name":20094,"role":20095,"bio":20096,"links":27274},[27275,27276],{"label":20099,"href":20100},{"label":20102,"href":20136},{"username":7,"name":20094,"role":20104},"How the ZX entangling rate and static ZZ vary with detuning and drive amplitude, plotted from the PRA 2020 cross-resonance formulas.","Deep dive · Hardware",{},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fcross-resonance-gate-visualization","9 min read",[],{"title":20126,"description":27278},"blog\u002Fexpert-notes\u002Fcross-resonance-gate-visualization",[27288,20120],"hardware","6JUxWAxIWSaDZoHV5LLIvRE1rlXI7cOr95RKDN01afk",{"id":27291,"title":27292,"authors":27293,"body":27294,"breadcrumb":39368,"builders":39370,"byline":39375,"challenge":20105,"courseAuthor":20105,"courseLead":20105,"dek":20105,"description":39376,"draft":20107,"extension":20108,"eyebrow":27279,"featured":20107,"finish":20105,"fork":20105,"hero":20105,"heroAlt":20105,"heroCta":20105,"heroImage":20105,"homepageFeatured":20107,"kind":20110,"lessonCount":20105,"meta":39377,"navigation":10300,"newsItems":20105,"next":20105,"ogImage":39378,"order":20105,"outcomes":20105,"path":39379,"publishDate":20114,"readingTime":39380,"related":39381,"relatedProjects":20105,"seo":39382,"stem":39383,"tags":39384,"track":20105,"trackName":20105,"__hash__":39385},"blog\u002Fblog\u002Fexpert-notes\u002Fpulse-shapes-and-envelopes.md","Pulse Shapes and Envelopes, Visualized",[7],{"type":9,"value":27295,"toc":39362},[27296,27304,27307,27318,27341,27594,27597,27864,27867,28194,28197,28547,28550,28788,28791,29098,29101,29453,29456,29838,29841,29844,30436,30440,30992,30996,31618,31622,32252,32256,32259,32266,32272,32633,32843,32849,33397,33597,33603,34193,34423,34429,34646,34648,34653,37222,37808,37812,37816,38271,38275,38648,38652,39355,39359],[12,27297,27298],{},[15,27299,17,27300,27303],{},[19,27301,22],{"href":27302},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FPulse_Shapes_and_Envelopes_Visualization.ipynb",". The text, code, and figures below are Onri's, as published.",[12,27305,27306],{},"This notebook authored by Onri Jay Benally shows some plots to help visualize the process of how pulses are shaped to control a quantum chip.",[9743,27308,27309,27312,27315],{},[9746,27310,27311],{},"While Gaussian is prevalent due to its favorable spectral properties, sinusoidal or other shaped envelopes also exist and are actively used in various quantum computing architectures.",[9746,27313,27314],{},"The choice of modulation function, which resembles an envelope, is dictated by the need to balance gate fidelity, spectral leakage, robustness against noise, and suppression of higher energy transitions.",[9746,27316,27317],{},"Optimal control techniques, such as GRAPE or CRAB, can generate tailored pulse shapes that might include sinusoidal modulations.",[27319,27320,27321],"blockquote",{},[9743,27322,27323,27329,27335],{},[9746,27324,27325,27328],{},[35,27326,27327],{},"Gradient Ascent Pulse Engineering (GRAPE)",": A numerical algorithm used in quantum optimal control. It uses time discretization in order to identify the optimal control pulses for quantum systems.",[9746,27330,27331,27334],{},[35,27332,27333],{},"Chopped Random Basis (CRAB)",": A method for quantum optimal control that uses a randomly truncated basis to optimize control pulses.",[9746,27336,27337,27340],{},[35,27338,27339],{},"Derivative Removal by Adiabatic Gate (DRAG)",": Widely used in implementations of high-fidelity single-qubit gates in superconducting qubit architectures, such as those found in IBM Quantum and Google Quantum processors.",[10082,27342,27344],{"className":10269,"code":27343,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Onri prefers using rcParams to increase the quality of all plots to be higher than the default settings.\n# This only needs to be done once at beginning of every Jupyter Notebook.\nplt.rcParams['figure.dpi'] = 200\n\n# Define the parameters for the Gaussian modulation function\nmean = 0\nstd_dev = 1\ntime = np.linspace(-5, 5, 1000)  # Generate time values from -5 to 5\n\n# Calculate the Gaussian modulation function\ngaussian_modulation = np.exp(-(time - mean) ** 2 \u002F (2 * std_dev ** 2))\n\n# Plot the Gaussian modulation function with blue color\nplt.plot(time, gaussian_modulation, color='blue')\nplt.title('Gaussian Modulation Function')\nplt.xlabel('Time')\nplt.ylabel('Amplitude')\nplt.grid(False)\nplt.show()\n",[10088,27345,27346,27356,27366,27370,27375,27380,27395,27399,27404,27414,27424,27455,27459,27464,27508,27512,27517,27534,27547,27560,27573,27586],{"__ignoreMap":10090},[40,27347,27348,27350,27352,27354],{"class":10276,"line":10277},[40,27349,10375],{"class":10352},[40,27351,10378],{"class":10363},[40,27353,10381],{"class":10352},[40,27355,10384],{"class":10363},[40,27357,27358,27360,27362,27364],{"class":10276,"line":10284},[40,27359,10375],{"class":10352},[40,27361,10405],{"class":10363},[40,27363,10381],{"class":10352},[40,27365,10410],{"class":10363},[40,27367,27368],{"class":10276,"line":10291},[40,27369,10301],{"emptyLinePlaceholder":10300},[40,27371,27372],{"class":10276,"line":10297},[40,27373,27374],{"class":10280},"# Onri prefers using rcParams to increase the quality of all plots to be higher than the default settings.\n",[40,27376,27377],{"class":10276,"line":10304},[40,27378,27379],{"class":10280},"# This only needs to be done once at beginning of every Jupyter Notebook.\n",[40,27381,27382,27385,27388,27390,27392],{"class":10276,"line":10310},[40,27383,27384],{"class":10363},"plt.rcParams[",[40,27386,27387],{"class":10287},"'figure.dpi'",[40,27389,10741],{"class":10363},[40,27391,162],{"class":10732},[40,27393,27394],{"class":10446}," 200\n",[40,27396,27397],{"class":10276,"line":10316},[40,27398,10301],{"emptyLinePlaceholder":10300},[40,27400,27401],{"class":10276,"line":10322},[40,27402,27403],{"class":10280},"# Define the parameters for the Gaussian modulation function\n",[40,27405,27406,27409,27411],{"class":10276,"line":10327},[40,27407,27408],{"class":10363},"mean ",[40,27410,162],{"class":10732},[40,27412,27413],{"class":10446}," 0\n",[40,27415,27416,27419,27421],{"class":10276,"line":10333},[40,27417,27418],{"class":10363},"std_dev ",[40,27420,162],{"class":10732},[40,27422,27423],{"class":10446}," 1\n",[40,27425,27426,27429,27431,27433,27435,27437,27439,27441,27443,27445,27447,27450,27452],{"class":10276,"line":10339},[40,27427,27428],{"class":10363},"time ",[40,27430,162],{"class":10732},[40,27432,11156],{"class":10363},[40,27434,11850],{"class":10430},[40,27436,167],{"class":10363},[40,27438,11056],{"class":10732},[40,27440,12566],{"class":10446},[40,27442,10484],{"class":10363},[40,27444,12566],{"class":10446},[40,27446,10484],{"class":10363},[40,27448,27449],{"class":10446},"1000",[40,27451,17908],{"class":10363},[40,27453,27454],{"class":10280},"# Generate time values from -5 to 5\n",[40,27456,27457],{"class":10276,"line":10344},[40,27458,10301],{"emptyLinePlaceholder":10300},[40,27460,27461],{"class":10276,"line":10349},[40,27462,27463],{"class":10280},"# Calculate the Gaussian modulation function\n",[40,27465,27466,27469,27471,27473,27475,27477,27479,27482,27484,27487,27489,27491,27493,27495,27497,27499,27502,27504,27506],{"class":10276,"line":10367},[40,27467,27468],{"class":10363},"gaussian_modulation ",[40,27470,162],{"class":10732},[40,27472,11156],{"class":10363},[40,27474,5268],{"class":10430},[40,27476,167],{"class":10363},[40,27478,11056],{"class":10732},[40,27480,27481],{"class":10363},"(time ",[40,27483,11056],{"class":10732},[40,27485,27486],{"class":10363}," mean) ",[40,27488,22571],{"class":10732},[40,27490,12960],{"class":10446},[40,27492,22539],{"class":10732},[40,27494,920],{"class":10363},[40,27496,1044],{"class":10446},[40,27498,11881],{"class":10732},[40,27500,27501],{"class":10363}," std_dev ",[40,27503,22571],{"class":10732},[40,27505,12960],{"class":10446},[40,27507,13612],{"class":10363},[40,27509,27510],{"class":10276,"line":10372},[40,27511,10301],{"emptyLinePlaceholder":10300},[40,27513,27514],{"class":10276,"line":10387},[40,27515,27516],{"class":10280},"# Plot the Gaussian modulation function with blue color\n",[40,27518,27519,27521,27523,27526,27528,27530,27532],{"class":10276,"line":10400},[40,27520,23469],{"class":10363},[40,27522,23505],{"class":10430},[40,27524,27525],{"class":10363},"(time, gaussian_modulation, ",[40,27527,23550],{"class":11903},[40,27529,162],{"class":10732},[40,27531,23555],{"class":10287},[40,27533,10761],{"class":10363},[40,27535,27536,27538,27540,27542,27545],{"class":10276,"line":10413},[40,27537,23469],{"class":10363},[40,27539,23656],{"class":10430},[40,27541,167],{"class":10363},[40,27543,27544],{"class":10287},"'Gaussian Modulation Function'",[40,27546,10761],{"class":10363},[40,27548,27549,27551,27553,27555,27558],{"class":10276,"line":10418},[40,27550,23469],{"class":10363},[40,27552,23600],{"class":10430},[40,27554,167],{"class":10363},[40,27556,27557],{"class":10287},"'Time'",[40,27559,10761],{"class":10363},[40,27561,27562,27564,27566,27568,27571],{"class":10276,"line":10424},[40,27563,23469],{"class":10363},[40,27565,23626],{"class":10430},[40,27567,167],{"class":10363},[40,27569,27570],{"class":10287},"'Amplitude'",[40,27572,10761],{"class":10363},[40,27574,27575,27577,27580,27582,27584],{"class":10276,"line":10437},[40,27576,23469],{"class":10363},[40,27578,27579],{"class":10430},"grid",[40,27581,167],{"class":10363},[40,27583,10504],{"class":10446},[40,27585,10761],{"class":10363},[40,27587,27588,27590,27592],{"class":10276,"line":10456},[40,27589,23469],{"class":10363},[40,27591,23686],{"class":10430},[40,27593,12174],{"class":10363},[13997,27595],{"alt":13999,"src":27596},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-01.webp",[10082,27598,27600],{"className":10269,"code":27599,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the parameters for the Gaussian modulation function\namplitude = 1\nmean = 0\nstd_dev = 1\ntime = np.linspace(-5, 5, 1000)  # Generate time values from -5 to 5\n\n# Calculate the Gaussian modulation function\ngaussian_modulation = amplitude * np.exp(-(time - mean) ** 2 \u002F (2 * std_dev ** 2))\n\n# Plot the Gaussian modulation function\nplt.plot(time, gaussian_modulation, color='red', linestyle='dashed', label='Gaussian Modulation')\nplt.title('Gaussian Modulation Function')\nplt.xlabel('Time')\nplt.ylabel('Amplitude')\nplt.ylim(-1.0, 1.1)  # Force the y-axis to display the range from -1.0 to 1.0\n# plt.legend()\nplt.grid(False)\nplt.show()\n",[10088,27601,27602,27612,27622,27626,27630,27639,27647,27655,27683,27687,27691,27736,27740,27745,27780,27792,27804,27816,27839,27844,27856],{"__ignoreMap":10090},[40,27603,27604,27606,27608,27610],{"class":10276,"line":10277},[40,27605,10375],{"class":10352},[40,27607,10378],{"class":10363},[40,27609,10381],{"class":10352},[40,27611,10384],{"class":10363},[40,27613,27614,27616,27618,27620],{"class":10276,"line":10284},[40,27615,10375],{"class":10352},[40,27617,10405],{"class":10363},[40,27619,10381],{"class":10352},[40,27621,10410],{"class":10363},[40,27623,27624],{"class":10276,"line":10291},[40,27625,10301],{"emptyLinePlaceholder":10300},[40,27627,27628],{"class":10276,"line":10297},[40,27629,27403],{"class":10280},[40,27631,27632,27635,27637],{"class":10276,"line":10304},[40,27633,27634],{"class":10363},"amplitude ",[40,27636,162],{"class":10732},[40,27638,27423],{"class":10446},[40,27640,27641,27643,27645],{"class":10276,"line":10310},[40,27642,27408],{"class":10363},[40,27644,162],{"class":10732},[40,27646,27413],{"class":10446},[40,27648,27649,27651,27653],{"class":10276,"line":10316},[40,27650,27418],{"class":10363},[40,27652,162],{"class":10732},[40,27654,27423],{"class":10446},[40,27656,27657,27659,27661,27663,27665,27667,27669,27671,27673,27675,27677,27679,27681],{"class":10276,"line":10322},[40,27658,27428],{"class":10363},[40,27660,162],{"class":10732},[40,27662,11156],{"class":10363},[40,27664,11850],{"class":10430},[40,27666,167],{"class":10363},[40,27668,11056],{"class":10732},[40,27670,12566],{"class":10446},[40,27672,10484],{"class":10363},[40,27674,12566],{"class":10446},[40,27676,10484],{"class":10363},[40,27678,27449],{"class":10446},[40,27680,17908],{"class":10363},[40,27682,27454],{"class":10280},[40,27684,27685],{"class":10276,"line":10327},[40,27686,10301],{"emptyLinePlaceholder":10300},[40,27688,27689],{"class":10276,"line":10333},[40,27690,27463],{"class":10280},[40,27692,27693,27695,27697,27700,27702,27704,27706,27708,27710,27712,27714,27716,27718,27720,27722,27724,27726,27728,27730,27732,27734],{"class":10276,"line":10339},[40,27694,27468],{"class":10363},[40,27696,162],{"class":10732},[40,27698,27699],{"class":10363}," amplitude ",[40,27701,11752],{"class":10732},[40,27703,11156],{"class":10363},[40,27705,5268],{"class":10430},[40,27707,167],{"class":10363},[40,27709,11056],{"class":10732},[40,27711,27481],{"class":10363},[40,27713,11056],{"class":10732},[40,27715,27486],{"class":10363},[40,27717,22571],{"class":10732},[40,27719,12960],{"class":10446},[40,27721,22539],{"class":10732},[40,27723,920],{"class":10363},[40,27725,1044],{"class":10446},[40,27727,11881],{"class":10732},[40,27729,27501],{"class":10363},[40,27731,22571],{"class":10732},[40,27733,12960],{"class":10446},[40,27735,13612],{"class":10363},[40,27737,27738],{"class":10276,"line":10344},[40,27739,10301],{"emptyLinePlaceholder":10300},[40,27741,27742],{"class":10276,"line":10349},[40,27743,27744],{"class":10280},"# Plot the Gaussian modulation function\n",[40,27746,27747,27749,27751,27753,27755,27757,27760,27762,27764,27766,27769,27771,27773,27775,27778],{"class":10276,"line":10367},[40,27748,23469],{"class":10363},[40,27750,23505],{"class":10430},[40,27752,27525],{"class":10363},[40,27754,23550],{"class":11903},[40,27756,162],{"class":10732},[40,27758,27759],{"class":10287},"'red'",[40,27761,10484],{"class":10363},[40,27763,23274],{"class":11903},[40,27765,162],{"class":10732},[40,27767,27768],{"class":10287},"'dashed'",[40,27770,10484],{"class":10363},[40,27772,15415],{"class":11903},[40,27774,162],{"class":10732},[40,27776,27777],{"class":10287},"'Gaussian Modulation'",[40,27779,10761],{"class":10363},[40,27781,27782,27784,27786,27788,27790],{"class":10276,"line":10372},[40,27783,23469],{"class":10363},[40,27785,23656],{"class":10430},[40,27787,167],{"class":10363},[40,27789,27544],{"class":10287},[40,27791,10761],{"class":10363},[40,27793,27794,27796,27798,27800,27802],{"class":10276,"line":10387},[40,27795,23469],{"class":10363},[40,27797,23600],{"class":10430},[40,27799,167],{"class":10363},[40,27801,27557],{"class":10287},[40,27803,10761],{"class":10363},[40,27805,27806,27808,27810,27812,27814],{"class":10276,"line":10400},[40,27807,23469],{"class":10363},[40,27809,23626],{"class":10430},[40,27811,167],{"class":10363},[40,27813,27570],{"class":10287},[40,27815,10761],{"class":10363},[40,27817,27818,27820,27823,27825,27827,27829,27831,27834,27836],{"class":10276,"line":10413},[40,27819,23469],{"class":10363},[40,27821,27822],{"class":10430},"ylim",[40,27824,167],{"class":10363},[40,27826,11056],{"class":10732},[40,27828,10810],{"class":10446},[40,27830,10484],{"class":10363},[40,27832,27833],{"class":10446},"1.1",[40,27835,17908],{"class":10363},[40,27837,27838],{"class":10280},"# Force the y-axis to display the range from -1.0 to 1.0\n",[40,27840,27841],{"class":10276,"line":10418},[40,27842,27843],{"class":10280},"# plt.legend()\n",[40,27845,27846,27848,27850,27852,27854],{"class":10276,"line":10424},[40,27847,23469],{"class":10363},[40,27849,27579],{"class":10430},[40,27851,167],{"class":10363},[40,27853,10504],{"class":10446},[40,27855,10761],{"class":10363},[40,27857,27858,27860,27862],{"class":10276,"line":10437},[40,27859,23469],{"class":10363},[40,27861,23686],{"class":10430},[40,27863,12174],{"class":10363},[13997,27865],{"alt":14003,"src":27866},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-02.webp",[10082,27868,27870],{"className":10269,"code":27869,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the parameters for the Gaussian function\namplitude = 1\nmean = 0\nstd_dev = 1\ntime = np.linspace(-5, 5, 20)  # Reduce points for lollipop plot clarity\n\n# Increase the number of points for the lollipop plot\ntime = np.linspace(-5, 5, 40)  # Doubling the number of points\n\n# Calculate the Gaussian function\ngaussian_modulation = amplitude * np.exp(-(time - mean) ** 2 \u002F (2 * std_dev ** 2))\n\n# Create the updated lollipop plot with blue color\nplt.figure(figsize=(8, 6))\nplt.stem(time, gaussian_modulation, linefmt='b-', markerfmt='bo', basefmt='k-')\nplt.title('Lollipop Plot of Gaussian Function')\nplt.xlabel('Time')\nplt.ylabel('Amplitude')\nplt.ylim(-1.0, 1.1)  # Force the y-axis to display the range from -1.0 to 1.1\nplt.grid(False)\nplt.show()\n",[10088,27871,27872,27882,27892,27896,27901,27909,27917,27925,27954,27958,27963,27993,27997,28002,28046,28050,28055,28077,28116,28129,28141,28153,28174,28186],{"__ignoreMap":10090},[40,27873,27874,27876,27878,27880],{"class":10276,"line":10277},[40,27875,10375],{"class":10352},[40,27877,10378],{"class":10363},[40,27879,10381],{"class":10352},[40,27881,10384],{"class":10363},[40,27883,27884,27886,27888,27890],{"class":10276,"line":10284},[40,27885,10375],{"class":10352},[40,27887,10405],{"class":10363},[40,27889,10381],{"class":10352},[40,27891,10410],{"class":10363},[40,27893,27894],{"class":10276,"line":10291},[40,27895,10301],{"emptyLinePlaceholder":10300},[40,27897,27898],{"class":10276,"line":10297},[40,27899,27900],{"class":10280},"# Define the parameters for the Gaussian function\n",[40,27902,27903,27905,27907],{"class":10276,"line":10304},[40,27904,27634],{"class":10363},[40,27906,162],{"class":10732},[40,27908,27423],{"class":10446},[40,27910,27911,27913,27915],{"class":10276,"line":10310},[40,27912,27408],{"class":10363},[40,27914,162],{"class":10732},[40,27916,27413],{"class":10446},[40,27918,27919,27921,27923],{"class":10276,"line":10316},[40,27920,27418],{"class":10363},[40,27922,162],{"class":10732},[40,27924,27423],{"class":10446},[40,27926,27927,27929,27931,27933,27935,27937,27939,27941,27943,27945,27947,27949,27951],{"class":10276,"line":10322},[40,27928,27428],{"class":10363},[40,27930,162],{"class":10732},[40,27932,11156],{"class":10363},[40,27934,11850],{"class":10430},[40,27936,167],{"class":10363},[40,27938,11056],{"class":10732},[40,27940,12566],{"class":10446},[40,27942,10484],{"class":10363},[40,27944,12566],{"class":10446},[40,27946,10484],{"class":10363},[40,27948,13742],{"class":10446},[40,27950,17908],{"class":10363},[40,27952,27953],{"class":10280},"# Reduce points for lollipop plot clarity\n",[40,27955,27956],{"class":10276,"line":10327},[40,27957,10301],{"emptyLinePlaceholder":10300},[40,27959,27960],{"class":10276,"line":10333},[40,27961,27962],{"class":10280},"# Increase the number of points for the lollipop plot\n",[40,27964,27965,27967,27969,27971,27973,27975,27977,27979,27981,27983,27985,27988,27990],{"class":10276,"line":10339},[40,27966,27428],{"class":10363},[40,27968,162],{"class":10732},[40,27970,11156],{"class":10363},[40,27972,11850],{"class":10430},[40,27974,167],{"class":10363},[40,27976,11056],{"class":10732},[40,27978,12566],{"class":10446},[40,27980,10484],{"class":10363},[40,27982,12566],{"class":10446},[40,27984,10484],{"class":10363},[40,27986,27987],{"class":10446},"40",[40,27989,17908],{"class":10363},[40,27991,27992],{"class":10280},"# Doubling the number of points\n",[40,27994,27995],{"class":10276,"line":10344},[40,27996,10301],{"emptyLinePlaceholder":10300},[40,27998,27999],{"class":10276,"line":10349},[40,28000,28001],{"class":10280},"# Calculate the Gaussian function\n",[40,28003,28004,28006,28008,28010,28012,28014,28016,28018,28020,28022,28024,28026,28028,28030,28032,28034,28036,28038,28040,28042,28044],{"class":10276,"line":10367},[40,28005,27468],{"class":10363},[40,28007,162],{"class":10732},[40,28009,27699],{"class":10363},[40,28011,11752],{"class":10732},[40,28013,11156],{"class":10363},[40,28015,5268],{"class":10430},[40,28017,167],{"class":10363},[40,28019,11056],{"class":10732},[40,28021,27481],{"class":10363},[40,28023,11056],{"class":10732},[40,28025,27486],{"class":10363},[40,28027,22571],{"class":10732},[40,28029,12960],{"class":10446},[40,28031,22539],{"class":10732},[40,28033,920],{"class":10363},[40,28035,1044],{"class":10446},[40,28037,11881],{"class":10732},[40,28039,27501],{"class":10363},[40,28041,22571],{"class":10732},[40,28043,12960],{"class":10446},[40,28045,13612],{"class":10363},[40,28047,28048],{"class":10276,"line":10372},[40,28049,10301],{"emptyLinePlaceholder":10300},[40,28051,28052],{"class":10276,"line":10387},[40,28053,28054],{"class":10280},"# Create the updated lollipop plot with blue color\n",[40,28056,28057,28059,28061,28063,28065,28067,28069,28071,28073,28075],{"class":10276,"line":10400},[40,28058,23469],{"class":10363},[40,28060,16373],{"class":10430},[40,28062,167],{"class":10363},[40,28064,13078],{"class":11903},[40,28066,162],{"class":10732},[40,28068,167],{"class":10363},[40,28070,23482],{"class":10446},[40,28072,10484],{"class":10363},[40,28074,12785],{"class":10446},[40,28076,13612],{"class":10363},[40,28078,28079,28081,28084,28086,28089,28091,28094,28096,28099,28101,28104,28106,28109,28111,28114],{"class":10276,"line":10413},[40,28080,23469],{"class":10363},[40,28082,28083],{"class":10430},"stem",[40,28085,27525],{"class":10363},[40,28087,28088],{"class":11903},"linefmt",[40,28090,162],{"class":10732},[40,28092,28093],{"class":10287},"'b-'",[40,28095,10484],{"class":10363},[40,28097,28098],{"class":11903},"markerfmt",[40,28100,162],{"class":10732},[40,28102,28103],{"class":10287},"'bo'",[40,28105,10484],{"class":10363},[40,28107,28108],{"class":11903},"basefmt",[40,28110,162],{"class":10732},[40,28112,28113],{"class":10287},"'k-'",[40,28115,10761],{"class":10363},[40,28117,28118,28120,28122,28124,28127],{"class":10276,"line":10418},[40,28119,23469],{"class":10363},[40,28121,23656],{"class":10430},[40,28123,167],{"class":10363},[40,28125,28126],{"class":10287},"'Lollipop Plot of Gaussian Function'",[40,28128,10761],{"class":10363},[40,28130,28131,28133,28135,28137,28139],{"class":10276,"line":10424},[40,28132,23469],{"class":10363},[40,28134,23600],{"class":10430},[40,28136,167],{"class":10363},[40,28138,27557],{"class":10287},[40,28140,10761],{"class":10363},[40,28142,28143,28145,28147,28149,28151],{"class":10276,"line":10437},[40,28144,23469],{"class":10363},[40,28146,23626],{"class":10430},[40,28148,167],{"class":10363},[40,28150,27570],{"class":10287},[40,28152,10761],{"class":10363},[40,28154,28155,28157,28159,28161,28163,28165,28167,28169,28171],{"class":10276,"line":10456},[40,28156,23469],{"class":10363},[40,28158,27822],{"class":10430},[40,28160,167],{"class":10363},[40,28162,11056],{"class":10732},[40,28164,10810],{"class":10446},[40,28166,10484],{"class":10363},[40,28168,27833],{"class":10446},[40,28170,17908],{"class":10363},[40,28172,28173],{"class":10280},"# Force the y-axis to display the range from -1.0 to 1.1\n",[40,28175,28176,28178,28180,28182,28184],{"class":10276,"line":10472},[40,28177,23469],{"class":10363},[40,28179,27579],{"class":10430},[40,28181,167],{"class":10363},[40,28183,10504],{"class":10446},[40,28185,10761],{"class":10363},[40,28187,28188,28190,28192],{"class":10276,"line":10496},[40,28189,23469],{"class":10363},[40,28191,23686],{"class":10430},[40,28193,12174],{"class":10363},[13997,28195],{"alt":14007,"src":28196},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-03.webp",[10082,28198,28200],{"className":10269,"code":28199,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the parameters for the Gaussian-modulated sine wave\namplitude = 1\nmean = 0\nstd_dev = 1\nfrequency = 5  # Frequency of the sine wave\ntime = np.linspace(-5, 5, 1000)  # Generate time values from -5 to 5\n\n# Calculate the Gaussian function\ngaussian_modulation = np.exp(-(time - mean) ** 2 \u002F (2 * std_dev ** 2))\n\n# Calculate the sine wave\nsine_wave = amplitude * np.sin(2 * np.pi * frequency * time)\n\n# Modulate the sine wave with the Gaussian function\ngaussian_modulated_sine_wave = gaussian_modulation * sine_wave\n\n# Plot only the function of the Gaussian-modulated sine wave\nplt.plot(time, gaussian_modulation, color='red', linestyle='dashed', label='Modulation')\nplt.plot(time, -gaussian_modulation, color='red', linestyle='dashed')  # Add the negative function for completeness\nplt.title('Upper & Lower Gaussian Modulation Function')\nplt.xlabel('Time')\nplt.ylabel('Amplitude')\n# plt.legend()\nplt.grid(False)\nplt.show()\n",[10088,28201,28202,28212,28222,28226,28231,28239,28247,28255,28268,28296,28300,28304,28344,28348,28353,28387,28391,28396,28411,28415,28420,28453,28486,28499,28511,28523,28527,28539],{"__ignoreMap":10090},[40,28203,28204,28206,28208,28210],{"class":10276,"line":10277},[40,28205,10375],{"class":10352},[40,28207,10378],{"class":10363},[40,28209,10381],{"class":10352},[40,28211,10384],{"class":10363},[40,28213,28214,28216,28218,28220],{"class":10276,"line":10284},[40,28215,10375],{"class":10352},[40,28217,10405],{"class":10363},[40,28219,10381],{"class":10352},[40,28221,10410],{"class":10363},[40,28223,28224],{"class":10276,"line":10291},[40,28225,10301],{"emptyLinePlaceholder":10300},[40,28227,28228],{"class":10276,"line":10297},[40,28229,28230],{"class":10280},"# Define the parameters for the Gaussian-modulated sine wave\n",[40,28232,28233,28235,28237],{"class":10276,"line":10304},[40,28234,27634],{"class":10363},[40,28236,162],{"class":10732},[40,28238,27423],{"class":10446},[40,28240,28241,28243,28245],{"class":10276,"line":10310},[40,28242,27408],{"class":10363},[40,28244,162],{"class":10732},[40,28246,27413],{"class":10446},[40,28248,28249,28251,28253],{"class":10276,"line":10316},[40,28250,27418],{"class":10363},[40,28252,162],{"class":10732},[40,28254,27423],{"class":10446},[40,28256,28257,28260,28262,28265],{"class":10276,"line":10322},[40,28258,28259],{"class":10363},"frequency ",[40,28261,162],{"class":10732},[40,28263,28264],{"class":10446}," 5",[40,28266,28267],{"class":10280},"  # Frequency of the sine wave\n",[40,28269,28270,28272,28274,28276,28278,28280,28282,28284,28286,28288,28290,28292,28294],{"class":10276,"line":10327},[40,28271,27428],{"class":10363},[40,28273,162],{"class":10732},[40,28275,11156],{"class":10363},[40,28277,11850],{"class":10430},[40,28279,167],{"class":10363},[40,28281,11056],{"class":10732},[40,28283,12566],{"class":10446},[40,28285,10484],{"class":10363},[40,28287,12566],{"class":10446},[40,28289,10484],{"class":10363},[40,28291,27449],{"class":10446},[40,28293,17908],{"class":10363},[40,28295,27454],{"class":10280},[40,28297,28298],{"class":10276,"line":10333},[40,28299,10301],{"emptyLinePlaceholder":10300},[40,28301,28302],{"class":10276,"line":10339},[40,28303,28001],{"class":10280},[40,28305,28306,28308,28310,28312,28314,28316,28318,28320,28322,28324,28326,28328,28330,28332,28334,28336,28338,28340,28342],{"class":10276,"line":10344},[40,28307,27468],{"class":10363},[40,28309,162],{"class":10732},[40,28311,11156],{"class":10363},[40,28313,5268],{"class":10430},[40,28315,167],{"class":10363},[40,28317,11056],{"class":10732},[40,28319,27481],{"class":10363},[40,28321,11056],{"class":10732},[40,28323,27486],{"class":10363},[40,28325,22571],{"class":10732},[40,28327,12960],{"class":10446},[40,28329,22539],{"class":10732},[40,28331,920],{"class":10363},[40,28333,1044],{"class":10446},[40,28335,11881],{"class":10732},[40,28337,27501],{"class":10363},[40,28339,22571],{"class":10732},[40,28341,12960],{"class":10446},[40,28343,13612],{"class":10363},[40,28345,28346],{"class":10276,"line":10349},[40,28347,10301],{"emptyLinePlaceholder":10300},[40,28349,28350],{"class":10276,"line":10367},[40,28351,28352],{"class":10280},"# Calculate the sine wave\n",[40,28354,28355,28358,28360,28362,28364,28366,28368,28370,28372,28374,28377,28379,28382,28384],{"class":10276,"line":10372},[40,28356,28357],{"class":10363},"sine_wave ",[40,28359,162],{"class":10732},[40,28361,27699],{"class":10363},[40,28363,11752],{"class":10732},[40,28365,11156],{"class":10363},[40,28367,1813],{"class":10430},[40,28369,167],{"class":10363},[40,28371,1044],{"class":10446},[40,28373,11881],{"class":10732},[40,28375,28376],{"class":10363}," np.pi ",[40,28378,11752],{"class":10732},[40,28380,28381],{"class":10363}," frequency ",[40,28383,11752],{"class":10732},[40,28385,28386],{"class":10363}," time)\n",[40,28388,28389],{"class":10276,"line":10387},[40,28390,10301],{"emptyLinePlaceholder":10300},[40,28392,28393],{"class":10276,"line":10400},[40,28394,28395],{"class":10280},"# Modulate the sine wave with the Gaussian function\n",[40,28397,28398,28401,28403,28406,28408],{"class":10276,"line":10413},[40,28399,28400],{"class":10363},"gaussian_modulated_sine_wave ",[40,28402,162],{"class":10732},[40,28404,28405],{"class":10363}," gaussian_modulation ",[40,28407,11752],{"class":10732},[40,28409,28410],{"class":10363}," sine_wave\n",[40,28412,28413],{"class":10276,"line":10418},[40,28414,10301],{"emptyLinePlaceholder":10300},[40,28416,28417],{"class":10276,"line":10424},[40,28418,28419],{"class":10280},"# Plot only the function of the Gaussian-modulated sine wave\n",[40,28421,28422,28424,28426,28428,28430,28432,28434,28436,28438,28440,28442,28444,28446,28448,28451],{"class":10276,"line":10437},[40,28423,23469],{"class":10363},[40,28425,23505],{"class":10430},[40,28427,27525],{"class":10363},[40,28429,23550],{"class":11903},[40,28431,162],{"class":10732},[40,28433,27759],{"class":10287},[40,28435,10484],{"class":10363},[40,28437,23274],{"class":11903},[40,28439,162],{"class":10732},[40,28441,27768],{"class":10287},[40,28443,10484],{"class":10363},[40,28445,15415],{"class":11903},[40,28447,162],{"class":10732},[40,28449,28450],{"class":10287},"'Modulation'",[40,28452,10761],{"class":10363},[40,28454,28455,28457,28459,28462,28464,28467,28469,28471,28473,28475,28477,28479,28481,28483],{"class":10276,"line":10456},[40,28456,23469],{"class":10363},[40,28458,23505],{"class":10430},[40,28460,28461],{"class":10363},"(time, ",[40,28463,11056],{"class":10732},[40,28465,28466],{"class":10363},"gaussian_modulation, ",[40,28468,23550],{"class":11903},[40,28470,162],{"class":10732},[40,28472,27759],{"class":10287},[40,28474,10484],{"class":10363},[40,28476,23274],{"class":11903},[40,28478,162],{"class":10732},[40,28480,27768],{"class":10287},[40,28482,17908],{"class":10363},[40,28484,28485],{"class":10280},"# Add the negative function for completeness\n",[40,28487,28488,28490,28492,28494,28497],{"class":10276,"line":10472},[40,28489,23469],{"class":10363},[40,28491,23656],{"class":10430},[40,28493,167],{"class":10363},[40,28495,28496],{"class":10287},"'Upper & Lower Gaussian Modulation Function'",[40,28498,10761],{"class":10363},[40,28500,28501,28503,28505,28507,28509],{"class":10276,"line":10496},[40,28502,23469],{"class":10363},[40,28504,23600],{"class":10430},[40,28506,167],{"class":10363},[40,28508,27557],{"class":10287},[40,28510,10761],{"class":10363},[40,28512,28513,28515,28517,28519,28521],{"class":10276,"line":10510},[40,28514,23469],{"class":10363},[40,28516,23626],{"class":10430},[40,28518,167],{"class":10363},[40,28520,27570],{"class":10287},[40,28522,10761],{"class":10363},[40,28524,28525],{"class":10276,"line":10696},[40,28526,27843],{"class":10280},[40,28528,28529,28531,28533,28535,28537],{"class":10276,"line":10702},[40,28530,23469],{"class":10363},[40,28532,27579],{"class":10430},[40,28534,167],{"class":10363},[40,28536,10504],{"class":10446},[40,28538,10761],{"class":10363},[40,28540,28541,28543,28545],{"class":10276,"line":10708},[40,28542,23469],{"class":10363},[40,28544,23686],{"class":10430},[40,28546,12174],{"class":10363},[13997,28548],{"alt":15009,"src":28549},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-04.webp",[10082,28551,28553],{"className":10269,"code":28552,"language":10271,"meta":10090,"style":10090},"'''\nThis plot represents a sine wave at a chosen frequency, commonly used to match the resonance frequency\nof a quantum device on a chip. In quantum computing, such signals serve as carrier waves for qubit\ncontrol pulses before modulation is applied.\n'''\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# Define sine wave parameters\namplitude = 1  # Peak amplitude of the wave\nfrequency = 5  # Frequency of the sine wave (in arbitrary units, could be GHz for quantum systems)\ntime = np.linspace(-5, 5, 1000)  # Generate 1000 evenly spaced time values from -5 to 5\n\n# Compute the sine wave, representing an unmodulated carrier signal\nsine_wave = amplitude * np.sin(2 * np.pi * frequency * time)\n\n# Plot the sine wave with blue color\nplt.plot(time, sine_wave, color='blue')\nplt.title('Unmodulated Sine Wave (Carrier Signal)')\nplt.xlabel('Time')\nplt.ylabel('Amplitude')\nplt.grid(False)  # No grid for a cleaner look\nplt.show()\n",[10088,28554,28555,28560,28565,28570,28575,28579,28583,28593,28603,28607,28612,28623,28634,28663,28667,28672,28702,28706,28711,28728,28741,28753,28765,28780],{"__ignoreMap":10090},[40,28556,28557],{"class":10276,"line":10277},[40,28558,28559],{"class":10287},"'''\n",[40,28561,28562],{"class":10276,"line":10284},[40,28563,28564],{"class":10287},"This plot represents a sine wave at a chosen frequency, commonly used to match the resonance frequency\n",[40,28566,28567],{"class":10276,"line":10291},[40,28568,28569],{"class":10287},"of a quantum device on a chip. In quantum computing, such signals serve as carrier waves for qubit\n",[40,28571,28572],{"class":10276,"line":10297},[40,28573,28574],{"class":10287},"control pulses before modulation is applied.\n",[40,28576,28577],{"class":10276,"line":10304},[40,28578,28559],{"class":10287},[40,28580,28581],{"class":10276,"line":10310},[40,28582,10301],{"emptyLinePlaceholder":10300},[40,28584,28585,28587,28589,28591],{"class":10276,"line":10316},[40,28586,10375],{"class":10352},[40,28588,10378],{"class":10363},[40,28590,10381],{"class":10352},[40,28592,10384],{"class":10363},[40,28594,28595,28597,28599,28601],{"class":10276,"line":10322},[40,28596,10375],{"class":10352},[40,28598,10405],{"class":10363},[40,28600,10381],{"class":10352},[40,28602,10410],{"class":10363},[40,28604,28605],{"class":10276,"line":10327},[40,28606,10301],{"emptyLinePlaceholder":10300},[40,28608,28609],{"class":10276,"line":10333},[40,28610,28611],{"class":10280},"# Define sine wave parameters\n",[40,28613,28614,28616,28618,28620],{"class":10276,"line":10339},[40,28615,27634],{"class":10363},[40,28617,162],{"class":10732},[40,28619,13373],{"class":10446},[40,28621,28622],{"class":10280},"  # Peak amplitude of the wave\n",[40,28624,28625,28627,28629,28631],{"class":10276,"line":10344},[40,28626,28259],{"class":10363},[40,28628,162],{"class":10732},[40,28630,28264],{"class":10446},[40,28632,28633],{"class":10280},"  # Frequency of the sine wave (in arbitrary units, could be GHz for quantum systems)\n",[40,28635,28636,28638,28640,28642,28644,28646,28648,28650,28652,28654,28656,28658,28660],{"class":10276,"line":10349},[40,28637,27428],{"class":10363},[40,28639,162],{"class":10732},[40,28641,11156],{"class":10363},[40,28643,11850],{"class":10430},[40,28645,167],{"class":10363},[40,28647,11056],{"class":10732},[40,28649,12566],{"class":10446},[40,28651,10484],{"class":10363},[40,28653,12566],{"class":10446},[40,28655,10484],{"class":10363},[40,28657,27449],{"class":10446},[40,28659,17908],{"class":10363},[40,28661,28662],{"class":10280},"# Generate 1000 evenly spaced time values from -5 to 5\n",[40,28664,28665],{"class":10276,"line":10367},[40,28666,10301],{"emptyLinePlaceholder":10300},[40,28668,28669],{"class":10276,"line":10372},[40,28670,28671],{"class":10280},"# Compute the sine wave, representing an unmodulated carrier signal\n",[40,28673,28674,28676,28678,28680,28682,28684,28686,28688,28690,28692,28694,28696,28698,28700],{"class":10276,"line":10387},[40,28675,28357],{"class":10363},[40,28677,162],{"class":10732},[40,28679,27699],{"class":10363},[40,28681,11752],{"class":10732},[40,28683,11156],{"class":10363},[40,28685,1813],{"class":10430},[40,28687,167],{"class":10363},[40,28689,1044],{"class":10446},[40,28691,11881],{"class":10732},[40,28693,28376],{"class":10363},[40,28695,11752],{"class":10732},[40,28697,28381],{"class":10363},[40,28699,11752],{"class":10732},[40,28701,28386],{"class":10363},[40,28703,28704],{"class":10276,"line":10400},[40,28705,10301],{"emptyLinePlaceholder":10300},[40,28707,28708],{"class":10276,"line":10413},[40,28709,28710],{"class":10280},"# Plot the sine wave with blue color\n",[40,28712,28713,28715,28717,28720,28722,28724,28726],{"class":10276,"line":10418},[40,28714,23469],{"class":10363},[40,28716,23505],{"class":10430},[40,28718,28719],{"class":10363},"(time, sine_wave, ",[40,28721,23550],{"class":11903},[40,28723,162],{"class":10732},[40,28725,23555],{"class":10287},[40,28727,10761],{"class":10363},[40,28729,28730,28732,28734,28736,28739],{"class":10276,"line":10424},[40,28731,23469],{"class":10363},[40,28733,23656],{"class":10430},[40,28735,167],{"class":10363},[40,28737,28738],{"class":10287},"'Unmodulated Sine Wave (Carrier Signal)'",[40,28740,10761],{"class":10363},[40,28742,28743,28745,28747,28749,28751],{"class":10276,"line":10437},[40,28744,23469],{"class":10363},[40,28746,23600],{"class":10430},[40,28748,167],{"class":10363},[40,28750,27557],{"class":10287},[40,28752,10761],{"class":10363},[40,28754,28755,28757,28759,28761,28763],{"class":10276,"line":10456},[40,28756,23469],{"class":10363},[40,28758,23626],{"class":10430},[40,28760,167],{"class":10363},[40,28762,27570],{"class":10287},[40,28764,10761],{"class":10363},[40,28766,28767,28769,28771,28773,28775,28777],{"class":10276,"line":10472},[40,28768,23469],{"class":10363},[40,28770,27579],{"class":10430},[40,28772,167],{"class":10363},[40,28774,10504],{"class":10446},[40,28776,17908],{"class":10363},[40,28778,28779],{"class":10280},"# No grid for a cleaner look\n",[40,28781,28782,28784,28786],{"class":10276,"line":10496},[40,28783,23469],{"class":10363},[40,28785,23686],{"class":10430},[40,28787,12174],{"class":10363},[13997,28789],{"alt":17711,"src":28790},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-05.webp",[10082,28792,28794],{"className":10269,"code":28793,"language":10271,"meta":10090,"style":10090},"'''\nWhen a modulating pulse shape is applied to an unmodulated sine wave tone, then the result will be a\nmodulated sine wave output as shown in the plot below. The example shown is modulation by a\nGaussian wave.\n'''\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the parameters for the Gaussian-modulated sine wave\namplitude = 1\nmean = 0\nstd_dev = 1\nfrequency = 5  # Frequency of the sine wave\ntime = np.linspace(-5, 5, 1000)  # Generate time values from -5 to 5\n\n# Calculate the Gaussian function\ngaussian_modulation = np.exp(-(time - mean) ** 2 \u002F (2 * std_dev ** 2))\n\n# Calculate the sine wave\nsine_wave = amplitude * np.sin(2 * np.pi * frequency * time)\n\n# Modulate the sine wave with the Gaussian function\ngaussian_modulated_sine_wave = gaussian_modulation * sine_wave\n\n# Plot the Gaussian-modulated sine wave with blue color\nplt.plot(time, gaussian_modulated_sine_wave, color='blue')\nplt.title('Gaussian-Modulated Sine Wave Pulse')\nplt.xlabel('Time')\nplt.ylabel('Amplitude')\nplt.grid(False)\nplt.show()\n",[10088,28795,28796,28800,28805,28810,28815,28819,28829,28839,28843,28847,28855,28863,28871,28881,28909,28913,28917,28957,28961,28965,28995,28999,29003,29015,29019,29024,29041,29054,29066,29078,29090],{"__ignoreMap":10090},[40,28797,28798],{"class":10276,"line":10277},[40,28799,28559],{"class":10287},[40,28801,28802],{"class":10276,"line":10284},[40,28803,28804],{"class":10287},"When a modulating pulse shape is applied to an unmodulated sine wave tone, then the result will be a\n",[40,28806,28807],{"class":10276,"line":10291},[40,28808,28809],{"class":10287},"modulated sine wave output as shown in the plot below. The example shown is modulation by a\n",[40,28811,28812],{"class":10276,"line":10297},[40,28813,28814],{"class":10287},"Gaussian wave.\n",[40,28816,28817],{"class":10276,"line":10304},[40,28818,28559],{"class":10287},[40,28820,28821,28823,28825,28827],{"class":10276,"line":10310},[40,28822,10375],{"class":10352},[40,28824,10378],{"class":10363},[40,28826,10381],{"class":10352},[40,28828,10384],{"class":10363},[40,28830,28831,28833,28835,28837],{"class":10276,"line":10316},[40,28832,10375],{"class":10352},[40,28834,10405],{"class":10363},[40,28836,10381],{"class":10352},[40,28838,10410],{"class":10363},[40,28840,28841],{"class":10276,"line":10322},[40,28842,10301],{"emptyLinePlaceholder":10300},[40,28844,28845],{"class":10276,"line":10327},[40,28846,28230],{"class":10280},[40,28848,28849,28851,28853],{"class":10276,"line":10333},[40,28850,27634],{"class":10363},[40,28852,162],{"class":10732},[40,28854,27423],{"class":10446},[40,28856,28857,28859,28861],{"class":10276,"line":10339},[40,28858,27408],{"class":10363},[40,28860,162],{"class":10732},[40,28862,27413],{"class":10446},[40,28864,28865,28867,28869],{"class":10276,"line":10344},[40,28866,27418],{"class":10363},[40,28868,162],{"class":10732},[40,28870,27423],{"class":10446},[40,28872,28873,28875,28877,28879],{"class":10276,"line":10349},[40,28874,28259],{"class":10363},[40,28876,162],{"class":10732},[40,28878,28264],{"class":10446},[40,28880,28267],{"class":10280},[40,28882,28883,28885,28887,28889,28891,28893,28895,28897,28899,28901,28903,28905,28907],{"class":10276,"line":10367},[40,28884,27428],{"class":10363},[40,28886,162],{"class":10732},[40,28888,11156],{"class":10363},[40,28890,11850],{"class":10430},[40,28892,167],{"class":10363},[40,28894,11056],{"class":10732},[40,28896,12566],{"class":10446},[40,28898,10484],{"class":10363},[40,28900,12566],{"class":10446},[40,28902,10484],{"class":10363},[40,28904,27449],{"class":10446},[40,28906,17908],{"class":10363},[40,28908,27454],{"class":10280},[40,28910,28911],{"class":10276,"line":10372},[40,28912,10301],{"emptyLinePlaceholder":10300},[40,28914,28915],{"class":10276,"line":10387},[40,28916,28001],{"class":10280},[40,28918,28919,28921,28923,28925,28927,28929,28931,28933,28935,28937,28939,28941,28943,28945,28947,28949,28951,28953,28955],{"class":10276,"line":10400},[40,28920,27468],{"class":10363},[40,28922,162],{"class":10732},[40,28924,11156],{"class":10363},[40,28926,5268],{"class":10430},[40,28928,167],{"class":10363},[40,28930,11056],{"class":10732},[40,28932,27481],{"class":10363},[40,28934,11056],{"class":10732},[40,28936,27486],{"class":10363},[40,28938,22571],{"class":10732},[40,28940,12960],{"class":10446},[40,28942,22539],{"class":10732},[40,28944,920],{"class":10363},[40,28946,1044],{"class":10446},[40,28948,11881],{"class":10732},[40,28950,27501],{"class":10363},[40,28952,22571],{"class":10732},[40,28954,12960],{"class":10446},[40,28956,13612],{"class":10363},[40,28958,28959],{"class":10276,"line":10413},[40,28960,10301],{"emptyLinePlaceholder":10300},[40,28962,28963],{"class":10276,"line":10418},[40,28964,28352],{"class":10280},[40,28966,28967,28969,28971,28973,28975,28977,28979,28981,28983,28985,28987,28989,28991,28993],{"class":10276,"line":10424},[40,28968,28357],{"class":10363},[40,28970,162],{"class":10732},[40,28972,27699],{"class":10363},[40,28974,11752],{"class":10732},[40,28976,11156],{"class":10363},[40,28978,1813],{"class":10430},[40,28980,167],{"class":10363},[40,28982,1044],{"class":10446},[40,28984,11881],{"class":10732},[40,28986,28376],{"class":10363},[40,28988,11752],{"class":10732},[40,28990,28381],{"class":10363},[40,28992,11752],{"class":10732},[40,28994,28386],{"class":10363},[40,28996,28997],{"class":10276,"line":10437},[40,28998,10301],{"emptyLinePlaceholder":10300},[40,29000,29001],{"class":10276,"line":10456},[40,29002,28395],{"class":10280},[40,29004,29005,29007,29009,29011,29013],{"class":10276,"line":10472},[40,29006,28400],{"class":10363},[40,29008,162],{"class":10732},[40,29010,28405],{"class":10363},[40,29012,11752],{"class":10732},[40,29014,28410],{"class":10363},[40,29016,29017],{"class":10276,"line":10496},[40,29018,10301],{"emptyLinePlaceholder":10300},[40,29020,29021],{"class":10276,"line":10510},[40,29022,29023],{"class":10280},"# Plot the Gaussian-modulated sine wave with blue color\n",[40,29025,29026,29028,29030,29033,29035,29037,29039],{"class":10276,"line":10696},[40,29027,23469],{"class":10363},[40,29029,23505],{"class":10430},[40,29031,29032],{"class":10363},"(time, gaussian_modulated_sine_wave, ",[40,29034,23550],{"class":11903},[40,29036,162],{"class":10732},[40,29038,23555],{"class":10287},[40,29040,10761],{"class":10363},[40,29042,29043,29045,29047,29049,29052],{"class":10276,"line":10702},[40,29044,23469],{"class":10363},[40,29046,23656],{"class":10430},[40,29048,167],{"class":10363},[40,29050,29051],{"class":10287},"'Gaussian-Modulated Sine Wave Pulse'",[40,29053,10761],{"class":10363},[40,29055,29056,29058,29060,29062,29064],{"class":10276,"line":10708},[40,29057,23469],{"class":10363},[40,29059,23600],{"class":10430},[40,29061,167],{"class":10363},[40,29063,27557],{"class":10287},[40,29065,10761],{"class":10363},[40,29067,29068,29070,29072,29074,29076],{"class":10276,"line":10714},[40,29069,23469],{"class":10363},[40,29071,23626],{"class":10430},[40,29073,167],{"class":10363},[40,29075,27570],{"class":10287},[40,29077,10761],{"class":10363},[40,29079,29080,29082,29084,29086,29088],{"class":10276,"line":10720},[40,29081,23469],{"class":10363},[40,29083,27579],{"class":10430},[40,29085,167],{"class":10363},[40,29087,10504],{"class":10446},[40,29089,10761],{"class":10363},[40,29091,29092,29094,29096],{"class":10276,"line":10726},[40,29093,23469],{"class":10363},[40,29095,23686],{"class":10430},[40,29097,12174],{"class":10363},[13997,29099],{"alt":17715,"src":29100},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-06.webp",[10082,29102,29104],{"className":10269,"code":29103,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the parameters for the Gaussian-modulated sine wave\namplitude = 1\nmean = 0\nstd_dev = 1\nfrequency = 5  # Frequency of the sine wave\ntime = np.linspace(-5, 5, 1000)  # Generate time values from -5 to 5\n\n# Calculate the Gaussian function\ngaussian_modulation = np.exp(-(time - mean) ** 2 \u002F (2 * std_dev ** 2))\n\n# Calculate the sine wave\nsine_wave = amplitude * np.sin(2 * np.pi * frequency * time)\n\n# Modulate the sine wave with the Gaussian function\ngaussian_modulated_sine_wave = gaussian_modulation * sine_wave\n\n# Calculate the upper envelope of the modulated sine wave\nupper_modulation = np.abs(gaussian_modulation)\n\n# Plot the Gaussian-modulated sine wave and its upper envelope\nplt.plot(time, gaussian_modulated_sine_wave, color='blue', label='Gaussian-Modulated Sine Wave Pulse')\nplt.plot(time, upper_modulation, color='red', linestyle='dashed', label='Upper Modulation Function')\nplt.title('Gaussian-Modulated Sine Wave with Upper Modulation Function')\nplt.xlabel('Time')\nplt.ylabel('Amplitude')\n# plt.legend()\nplt.grid(False)\nplt.show()\n",[10088,29105,29106,29116,29126,29130,29134,29142,29150,29158,29168,29196,29200,29204,29244,29248,29252,29282,29286,29290,29302,29306,29311,29325,29329,29334,29358,29392,29405,29417,29429,29433,29445],{"__ignoreMap":10090},[40,29107,29108,29110,29112,29114],{"class":10276,"line":10277},[40,29109,10375],{"class":10352},[40,29111,10378],{"class":10363},[40,29113,10381],{"class":10352},[40,29115,10384],{"class":10363},[40,29117,29118,29120,29122,29124],{"class":10276,"line":10284},[40,29119,10375],{"class":10352},[40,29121,10405],{"class":10363},[40,29123,10381],{"class":10352},[40,29125,10410],{"class":10363},[40,29127,29128],{"class":10276,"line":10291},[40,29129,10301],{"emptyLinePlaceholder":10300},[40,29131,29132],{"class":10276,"line":10297},[40,29133,28230],{"class":10280},[40,29135,29136,29138,29140],{"class":10276,"line":10304},[40,29137,27634],{"class":10363},[40,29139,162],{"class":10732},[40,29141,27423],{"class":10446},[40,29143,29144,29146,29148],{"class":10276,"line":10310},[40,29145,27408],{"class":10363},[40,29147,162],{"class":10732},[40,29149,27413],{"class":10446},[40,29151,29152,29154,29156],{"class":10276,"line":10316},[40,29153,27418],{"class":10363},[40,29155,162],{"class":10732},[40,29157,27423],{"class":10446},[40,29159,29160,29162,29164,29166],{"class":10276,"line":10322},[40,29161,28259],{"class":10363},[40,29163,162],{"class":10732},[40,29165,28264],{"class":10446},[40,29167,28267],{"class":10280},[40,29169,29170,29172,29174,29176,29178,29180,29182,29184,29186,29188,29190,29192,29194],{"class":10276,"line":10327},[40,29171,27428],{"class":10363},[40,29173,162],{"class":10732},[40,29175,11156],{"class":10363},[40,29177,11850],{"class":10430},[40,29179,167],{"class":10363},[40,29181,11056],{"class":10732},[40,29183,12566],{"class":10446},[40,29185,10484],{"class":10363},[40,29187,12566],{"class":10446},[40,29189,10484],{"class":10363},[40,29191,27449],{"class":10446},[40,29193,17908],{"class":10363},[40,29195,27454],{"class":10280},[40,29197,29198],{"class":10276,"line":10333},[40,29199,10301],{"emptyLinePlaceholder":10300},[40,29201,29202],{"class":10276,"line":10339},[40,29203,28001],{"class":10280},[40,29205,29206,29208,29210,29212,29214,29216,29218,29220,29222,29224,29226,29228,29230,29232,29234,29236,29238,29240,29242],{"class":10276,"line":10344},[40,29207,27468],{"class":10363},[40,29209,162],{"class":10732},[40,29211,11156],{"class":10363},[40,29213,5268],{"class":10430},[40,29215,167],{"class":10363},[40,29217,11056],{"class":10732},[40,29219,27481],{"class":10363},[40,29221,11056],{"class":10732},[40,29223,27486],{"class":10363},[40,29225,22571],{"class":10732},[40,29227,12960],{"class":10446},[40,29229,22539],{"class":10732},[40,29231,920],{"class":10363},[40,29233,1044],{"class":10446},[40,29235,11881],{"class":10732},[40,29237,27501],{"class":10363},[40,29239,22571],{"class":10732},[40,29241,12960],{"class":10446},[40,29243,13612],{"class":10363},[40,29245,29246],{"class":10276,"line":10349},[40,29247,10301],{"emptyLinePlaceholder":10300},[40,29249,29250],{"class":10276,"line":10367},[40,29251,28352],{"class":10280},[40,29253,29254,29256,29258,29260,29262,29264,29266,29268,29270,29272,29274,29276,29278,29280],{"class":10276,"line":10372},[40,29255,28357],{"class":10363},[40,29257,162],{"class":10732},[40,29259,27699],{"class":10363},[40,29261,11752],{"class":10732},[40,29263,11156],{"class":10363},[40,29265,1813],{"class":10430},[40,29267,167],{"class":10363},[40,29269,1044],{"class":10446},[40,29271,11881],{"class":10732},[40,29273,28376],{"class":10363},[40,29275,11752],{"class":10732},[40,29277,28381],{"class":10363},[40,29279,11752],{"class":10732},[40,29281,28386],{"class":10363},[40,29283,29284],{"class":10276,"line":10387},[40,29285,10301],{"emptyLinePlaceholder":10300},[40,29287,29288],{"class":10276,"line":10400},[40,29289,28395],{"class":10280},[40,29291,29292,29294,29296,29298,29300],{"class":10276,"line":10413},[40,29293,28400],{"class":10363},[40,29295,162],{"class":10732},[40,29297,28405],{"class":10363},[40,29299,11752],{"class":10732},[40,29301,28410],{"class":10363},[40,29303,29304],{"class":10276,"line":10418},[40,29305,10301],{"emptyLinePlaceholder":10300},[40,29307,29308],{"class":10276,"line":10424},[40,29309,29310],{"class":10280},"# Calculate the upper envelope of the modulated sine wave\n",[40,29312,29313,29316,29318,29320,29322],{"class":10276,"line":10437},[40,29314,29315],{"class":10363},"upper_modulation ",[40,29317,162],{"class":10732},[40,29319,11156],{"class":10363},[40,29321,23431],{"class":10430},[40,29323,29324],{"class":10363},"(gaussian_modulation)\n",[40,29326,29327],{"class":10276,"line":10456},[40,29328,10301],{"emptyLinePlaceholder":10300},[40,29330,29331],{"class":10276,"line":10472},[40,29332,29333],{"class":10280},"# Plot the Gaussian-modulated sine wave and its upper envelope\n",[40,29335,29336,29338,29340,29342,29344,29346,29348,29350,29352,29354,29356],{"class":10276,"line":10496},[40,29337,23469],{"class":10363},[40,29339,23505],{"class":10430},[40,29341,29032],{"class":10363},[40,29343,23550],{"class":11903},[40,29345,162],{"class":10732},[40,29347,23555],{"class":10287},[40,29349,10484],{"class":10363},[40,29351,15415],{"class":11903},[40,29353,162],{"class":10732},[40,29355,29051],{"class":10287},[40,29357,10761],{"class":10363},[40,29359,29360,29362,29364,29367,29369,29371,29373,29375,29377,29379,29381,29383,29385,29387,29390],{"class":10276,"line":10510},[40,29361,23469],{"class":10363},[40,29363,23505],{"class":10430},[40,29365,29366],{"class":10363},"(time, upper_modulation, ",[40,29368,23550],{"class":11903},[40,29370,162],{"class":10732},[40,29372,27759],{"class":10287},[40,29374,10484],{"class":10363},[40,29376,23274],{"class":11903},[40,29378,162],{"class":10732},[40,29380,27768],{"class":10287},[40,29382,10484],{"class":10363},[40,29384,15415],{"class":11903},[40,29386,162],{"class":10732},[40,29388,29389],{"class":10287},"'Upper Modulation Function'",[40,29391,10761],{"class":10363},[40,29393,29394,29396,29398,29400,29403],{"class":10276,"line":10696},[40,29395,23469],{"class":10363},[40,29397,23656],{"class":10430},[40,29399,167],{"class":10363},[40,29401,29402],{"class":10287},"'Gaussian-Modulated Sine Wave with Upper Modulation Function'",[40,29404,10761],{"class":10363},[40,29406,29407,29409,29411,29413,29415],{"class":10276,"line":10702},[40,29408,23469],{"class":10363},[40,29410,23600],{"class":10430},[40,29412,167],{"class":10363},[40,29414,27557],{"class":10287},[40,29416,10761],{"class":10363},[40,29418,29419,29421,29423,29425,29427],{"class":10276,"line":10708},[40,29420,23469],{"class":10363},[40,29422,23626],{"class":10430},[40,29424,167],{"class":10363},[40,29426,27570],{"class":10287},[40,29428,10761],{"class":10363},[40,29430,29431],{"class":10276,"line":10714},[40,29432,27843],{"class":10280},[40,29434,29435,29437,29439,29441,29443],{"class":10276,"line":10720},[40,29436,23469],{"class":10363},[40,29438,27579],{"class":10430},[40,29440,167],{"class":10363},[40,29442,10504],{"class":10446},[40,29444,10761],{"class":10363},[40,29446,29447,29449,29451],{"class":10276,"line":10726},[40,29448,23469],{"class":10363},[40,29450,23686],{"class":10430},[40,29452,12174],{"class":10363},[13997,29454],{"alt":17719,"src":29455},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-07.webp",[10082,29457,29459],{"className":10269,"code":29458,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define the parameters for the Gaussian-modulated sine wave\namplitude = 1\nmean = 0\nstd_dev = 1\nfrequency = 5  # Frequency of the sine wave\ntime = np.linspace(-5, 5, 1000)  # Generate time values from -5 to 5\n\n# Calculate the Gaussian function\ngaussian_modulation = np.exp(-(time - mean) ** 2 \u002F (2 * std_dev ** 2))\n\n# Calculate the sine wave\nsine_wave = amplitude * np.sin(2 * np.pi * frequency * time)\n\n# Modulate the sine wave with the Gaussian function\ngaussian_modulated_sine_wave = gaussian_modulation * sine_wave\n\n# Calculate the function of the modulated sine wave\nmodulation = np.abs(gaussian_modulation)\n\n# Plot the Gaussian-modulated sine wave and its modulation function\nplt.plot(time, gaussian_modulated_sine_wave, color='blue', label='Gaussian-Modulated Sine Wave Pulse')\nplt.plot(time, modulation, color='red', linestyle='dashed', label='Modulation')\nplt.plot(time, -modulation, color='red', linestyle='dashed')  # Add the negative modulation function for completeness\nplt.title('Gaussian-Modulated Sine Wave with Modulation Function')\nplt.xlabel('Time')\nplt.ylabel('Amplitude')\n# plt.legend()\nplt.grid(False)\nplt.show()\n",[10088,29460,29461,29471,29481,29485,29489,29497,29505,29513,29523,29551,29555,29559,29599,29603,29607,29637,29641,29645,29657,29661,29666,29679,29683,29688,29712,29745,29777,29790,29802,29814,29818,29830],{"__ignoreMap":10090},[40,29462,29463,29465,29467,29469],{"class":10276,"line":10277},[40,29464,10375],{"class":10352},[40,29466,10378],{"class":10363},[40,29468,10381],{"class":10352},[40,29470,10384],{"class":10363},[40,29472,29473,29475,29477,29479],{"class":10276,"line":10284},[40,29474,10375],{"class":10352},[40,29476,10405],{"class":10363},[40,29478,10381],{"class":10352},[40,29480,10410],{"class":10363},[40,29482,29483],{"class":10276,"line":10291},[40,29484,10301],{"emptyLinePlaceholder":10300},[40,29486,29487],{"class":10276,"line":10297},[40,29488,28230],{"class":10280},[40,29490,29491,29493,29495],{"class":10276,"line":10304},[40,29492,27634],{"class":10363},[40,29494,162],{"class":10732},[40,29496,27423],{"class":10446},[40,29498,29499,29501,29503],{"class":10276,"line":10310},[40,29500,27408],{"class":10363},[40,29502,162],{"class":10732},[40,29504,27413],{"class":10446},[40,29506,29507,29509,29511],{"class":10276,"line":10316},[40,29508,27418],{"class":10363},[40,29510,162],{"class":10732},[40,29512,27423],{"class":10446},[40,29514,29515,29517,29519,29521],{"class":10276,"line":10322},[40,29516,28259],{"class":10363},[40,29518,162],{"class":10732},[40,29520,28264],{"class":10446},[40,29522,28267],{"class":10280},[40,29524,29525,29527,29529,29531,29533,29535,29537,29539,29541,29543,29545,29547,29549],{"class":10276,"line":10327},[40,29526,27428],{"class":10363},[40,29528,162],{"class":10732},[40,29530,11156],{"class":10363},[40,29532,11850],{"class":10430},[40,29534,167],{"class":10363},[40,29536,11056],{"class":10732},[40,29538,12566],{"class":10446},[40,29540,10484],{"class":10363},[40,29542,12566],{"class":10446},[40,29544,10484],{"class":10363},[40,29546,27449],{"class":10446},[40,29548,17908],{"class":10363},[40,29550,27454],{"class":10280},[40,29552,29553],{"class":10276,"line":10333},[40,29554,10301],{"emptyLinePlaceholder":10300},[40,29556,29557],{"class":10276,"line":10339},[40,29558,28001],{"class":10280},[40,29560,29561,29563,29565,29567,29569,29571,29573,29575,29577,29579,29581,29583,29585,29587,29589,29591,29593,29595,29597],{"class":10276,"line":10344},[40,29562,27468],{"class":10363},[40,29564,162],{"class":10732},[40,29566,11156],{"class":10363},[40,29568,5268],{"class":10430},[40,29570,167],{"class":10363},[40,29572,11056],{"class":10732},[40,29574,27481],{"class":10363},[40,29576,11056],{"class":10732},[40,29578,27486],{"class":10363},[40,29580,22571],{"class":10732},[40,29582,12960],{"class":10446},[40,29584,22539],{"class":10732},[40,29586,920],{"class":10363},[40,29588,1044],{"class":10446},[40,29590,11881],{"class":10732},[40,29592,27501],{"class":10363},[40,29594,22571],{"class":10732},[40,29596,12960],{"class":10446},[40,29598,13612],{"class":10363},[40,29600,29601],{"class":10276,"line":10349},[40,29602,10301],{"emptyLinePlaceholder":10300},[40,29604,29605],{"class":10276,"line":10367},[40,29606,28352],{"class":10280},[40,29608,29609,29611,29613,29615,29617,29619,29621,29623,29625,29627,29629,29631,29633,29635],{"class":10276,"line":10372},[40,29610,28357],{"class":10363},[40,29612,162],{"class":10732},[40,29614,27699],{"class":10363},[40,29616,11752],{"class":10732},[40,29618,11156],{"class":10363},[40,29620,1813],{"class":10430},[40,29622,167],{"class":10363},[40,29624,1044],{"class":10446},[40,29626,11881],{"class":10732},[40,29628,28376],{"class":10363},[40,29630,11752],{"class":10732},[40,29632,28381],{"class":10363},[40,29634,11752],{"class":10732},[40,29636,28386],{"class":10363},[40,29638,29639],{"class":10276,"line":10387},[40,29640,10301],{"emptyLinePlaceholder":10300},[40,29642,29643],{"class":10276,"line":10400},[40,29644,28395],{"class":10280},[40,29646,29647,29649,29651,29653,29655],{"class":10276,"line":10413},[40,29648,28400],{"class":10363},[40,29650,162],{"class":10732},[40,29652,28405],{"class":10363},[40,29654,11752],{"class":10732},[40,29656,28410],{"class":10363},[40,29658,29659],{"class":10276,"line":10418},[40,29660,10301],{"emptyLinePlaceholder":10300},[40,29662,29663],{"class":10276,"line":10424},[40,29664,29665],{"class":10280},"# Calculate the function of the modulated sine wave\n",[40,29667,29668,29671,29673,29675,29677],{"class":10276,"line":10437},[40,29669,29670],{"class":10363},"modulation ",[40,29672,162],{"class":10732},[40,29674,11156],{"class":10363},[40,29676,23431],{"class":10430},[40,29678,29324],{"class":10363},[40,29680,29681],{"class":10276,"line":10456},[40,29682,10301],{"emptyLinePlaceholder":10300},[40,29684,29685],{"class":10276,"line":10472},[40,29686,29687],{"class":10280},"# Plot the Gaussian-modulated sine wave and its modulation function\n",[40,29689,29690,29692,29694,29696,29698,29700,29702,29704,29706,29708,29710],{"class":10276,"line":10496},[40,29691,23469],{"class":10363},[40,29693,23505],{"class":10430},[40,29695,29032],{"class":10363},[40,29697,23550],{"class":11903},[40,29699,162],{"class":10732},[40,29701,23555],{"class":10287},[40,29703,10484],{"class":10363},[40,29705,15415],{"class":11903},[40,29707,162],{"class":10732},[40,29709,29051],{"class":10287},[40,29711,10761],{"class":10363},[40,29713,29714,29716,29718,29721,29723,29725,29727,29729,29731,29733,29735,29737,29739,29741,29743],{"class":10276,"line":10510},[40,29715,23469],{"class":10363},[40,29717,23505],{"class":10430},[40,29719,29720],{"class":10363},"(time, modulation, ",[40,29722,23550],{"class":11903},[40,29724,162],{"class":10732},[40,29726,27759],{"class":10287},[40,29728,10484],{"class":10363},[40,29730,23274],{"class":11903},[40,29732,162],{"class":10732},[40,29734,27768],{"class":10287},[40,29736,10484],{"class":10363},[40,29738,15415],{"class":11903},[40,29740,162],{"class":10732},[40,29742,28450],{"class":10287},[40,29744,10761],{"class":10363},[40,29746,29747,29749,29751,29753,29755,29758,29760,29762,29764,29766,29768,29770,29772,29774],{"class":10276,"line":10696},[40,29748,23469],{"class":10363},[40,29750,23505],{"class":10430},[40,29752,28461],{"class":10363},[40,29754,11056],{"class":10732},[40,29756,29757],{"class":10363},"modulation, ",[40,29759,23550],{"class":11903},[40,29761,162],{"class":10732},[40,29763,27759],{"class":10287},[40,29765,10484],{"class":10363},[40,29767,23274],{"class":11903},[40,29769,162],{"class":10732},[40,29771,27768],{"class":10287},[40,29773,17908],{"class":10363},[40,29775,29776],{"class":10280},"# Add the negative modulation function for completeness\n",[40,29778,29779,29781,29783,29785,29788],{"class":10276,"line":10702},[40,29780,23469],{"class":10363},[40,29782,23656],{"class":10430},[40,29784,167],{"class":10363},[40,29786,29787],{"class":10287},"'Gaussian-Modulated Sine Wave with Modulation Function'",[40,29789,10761],{"class":10363},[40,29791,29792,29794,29796,29798,29800],{"class":10276,"line":10708},[40,29793,23469],{"class":10363},[40,29795,23600],{"class":10430},[40,29797,167],{"class":10363},[40,29799,27557],{"class":10287},[40,29801,10761],{"class":10363},[40,29803,29804,29806,29808,29810,29812],{"class":10276,"line":10714},[40,29805,23469],{"class":10363},[40,29807,23626],{"class":10430},[40,29809,167],{"class":10363},[40,29811,27570],{"class":10287},[40,29813,10761],{"class":10363},[40,29815,29816],{"class":10276,"line":10720},[40,29817,27843],{"class":10280},[40,29819,29820,29822,29824,29826,29828],{"class":10276,"line":10726},[40,29821,23469],{"class":10363},[40,29823,27579],{"class":10430},[40,29825,167],{"class":10363},[40,29827,10504],{"class":10446},[40,29829,10761],{"class":10363},[40,29831,29832,29834,29836],{"class":10276,"line":10764},[40,29833,23469],{"class":10363},[40,29835,23686],{"class":10430},[40,29837,12174],{"class":10363},[13997,29839],{"alt":20066,"src":29840},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-08.webp",[12,29842,29843],{},"Below are some common pulse envelope shapes used to modulate a sine wave tone.",[10082,29845,29847],{"className":10269,"code":29846,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.signal.windows import blackmanharris, tukey\n\n# Define time vector\nt = np.linspace(-2, 2, 1000)\n\n# Define different pulse envelopes\ngaussian = np.exp(-t**2)\nsin2 = np.sin(np.pi * (t - t.min()) \u002F (t.max() - t.min()))**2\ncos2 = np.cos(np.pi * (t - t.min()) \u002F (t.max() - t.min()))**2\nrectangular = np.ones_like(t)\nrectangular[np.abs(t) > 1] = 0\nblackman_harris = blackmanharris(len(t))\ntukey_envelope = tukey(len(t), alpha=0.5)\nsech = 1 \u002F np.cosh(t)\n\n# List of envelopes and titles\nenvelopes = [(gaussian, \"Gaussian Modulation Function\"),\n             (sin2, \"$Sin^{2}$ Modulation Function\"),\n             (cos2, \"$Cos^{2}$ Modulation Function\"),\n             (rectangular, \"Rectangular Modulation Function\"),\n             (blackman_harris, \"Blackman-Harris Modulation Function\"),\n             (tukey_envelope, \"Tukey Modulation Function\"),\n             (sech, \"Sech Modulation Function\")]\n\n# Create subplots\nfig, axes = plt.subplots(3, 3, figsize=(12, 9))\naxes = axes.flatten()\n\nfor i, (envelope, title) in enumerate(envelopes):\n    axes[i].plot(t, envelope, 'b')\n    axes[i].set_title(title)\n    axes[i].set_xlabel(\"Time\")\n    axes[i].set_ylabel(\"Amplitude\")\n    # axes[i].legend([title])\n\n# Hide unused subplots\nfor j in range(i + 1, len(axes)):\n    fig.delaxes(axes[j])\n\nplt.tight_layout()\nplt.show()\n",[10088,29848,29849,29859,29869,29881,29885,29890,29917,29921,29926,29949,30000,30045,30060,30080,30097,30122,30140,30144,30149,30164,30180,30194,30204,30214,30224,30235,30239,30244,30280,30295,30299,30313,30328,30338,30351,30364,30369,30373,30378,30404,30415,30419,30428],{"__ignoreMap":10090},[40,29850,29851,29853,29855,29857],{"class":10276,"line":10277},[40,29852,10375],{"class":10352},[40,29854,10378],{"class":10363},[40,29856,10381],{"class":10352},[40,29858,10384],{"class":10363},[40,29860,29861,29863,29865,29867],{"class":10276,"line":10284},[40,29862,10375],{"class":10352},[40,29864,10405],{"class":10363},[40,29866,10381],{"class":10352},[40,29868,10410],{"class":10363},[40,29870,29871,29873,29876,29878],{"class":10276,"line":10291},[40,29872,10353],{"class":10352},[40,29874,29875],{"class":10363}," scipy.signal.windows ",[40,29877,10375],{"class":10352},[40,29879,29880],{"class":10363}," blackmanharris, tukey\n",[40,29882,29883],{"class":10276,"line":10297},[40,29884,10301],{"emptyLinePlaceholder":10300},[40,29886,29887],{"class":10276,"line":10304},[40,29888,29889],{"class":10280},"# Define time vector\n",[40,29891,29892,29895,29897,29899,29901,29903,29905,29907,29909,29911,29913,29915],{"class":10276,"line":10310},[40,29893,29894],{"class":10363},"t ",[40,29896,162],{"class":10732},[40,29898,11156],{"class":10363},[40,29900,11850],{"class":10430},[40,29902,167],{"class":10363},[40,29904,11056],{"class":10732},[40,29906,1044],{"class":10446},[40,29908,10484],{"class":10363},[40,29910,1044],{"class":10446},[40,29912,10484],{"class":10363},[40,29914,27449],{"class":10446},[40,29916,10761],{"class":10363},[40,29918,29919],{"class":10276,"line":10316},[40,29920,10301],{"emptyLinePlaceholder":10300},[40,29922,29923],{"class":10276,"line":10322},[40,29924,29925],{"class":10280},"# Define different pulse envelopes\n",[40,29927,29928,29931,29933,29935,29937,29939,29941,29943,29945,29947],{"class":10276,"line":10327},[40,29929,29930],{"class":10363},"gaussian ",[40,29932,162],{"class":10732},[40,29934,11156],{"class":10363},[40,29936,5268],{"class":10430},[40,29938,167],{"class":10363},[40,29940,11056],{"class":10732},[40,29942,20277],{"class":10363},[40,29944,22571],{"class":10732},[40,29946,1044],{"class":10446},[40,29948,10761],{"class":10363},[40,29950,29951,29954,29956,29958,29960,29963,29965,29968,29970,29973,29975,29978,29980,29983,29985,29987,29989,29991,29993,29996,29998],{"class":10276,"line":10333},[40,29952,29953],{"class":10363},"sin2 ",[40,29955,162],{"class":10732},[40,29957,11156],{"class":10363},[40,29959,1813],{"class":10430},[40,29961,29962],{"class":10363},"(np.pi ",[40,29964,11752],{"class":10732},[40,29966,29967],{"class":10363}," (t ",[40,29969,11056],{"class":10732},[40,29971,29972],{"class":10363}," t.",[40,29974,3067],{"class":10430},[40,29976,29977],{"class":10363},"()) ",[40,29979,11243],{"class":10732},[40,29981,29982],{"class":10363}," (t.",[40,29984,3032],{"class":10430},[40,29986,27025],{"class":10363},[40,29988,11056],{"class":10732},[40,29990,29972],{"class":10363},[40,29992,3067],{"class":10430},[40,29994,29995],{"class":10363},"()))",[40,29997,22571],{"class":10732},[40,29999,27056],{"class":10446},[40,30001,30002,30005,30007,30009,30011,30013,30015,30017,30019,30021,30023,30025,30027,30029,30031,30033,30035,30037,30039,30041,30043],{"class":10276,"line":10339},[40,30003,30004],{"class":10363},"cos2 ",[40,30006,162],{"class":10732},[40,30008,11156],{"class":10363},[40,30010,1820],{"class":10430},[40,30012,29962],{"class":10363},[40,30014,11752],{"class":10732},[40,30016,29967],{"class":10363},[40,30018,11056],{"class":10732},[40,30020,29972],{"class":10363},[40,30022,3067],{"class":10430},[40,30024,29977],{"class":10363},[40,30026,11243],{"class":10732},[40,30028,29982],{"class":10363},[40,30030,3032],{"class":10430},[40,30032,27025],{"class":10363},[40,30034,11056],{"class":10732},[40,30036,29972],{"class":10363},[40,30038,3067],{"class":10430},[40,30040,29995],{"class":10363},[40,30042,22571],{"class":10732},[40,30044,27056],{"class":10446},[40,30046,30047,30050,30052,30054,30057],{"class":10276,"line":10344},[40,30048,30049],{"class":10363},"rectangular ",[40,30051,162],{"class":10732},[40,30053,11156],{"class":10363},[40,30055,30056],{"class":10430},"ones_like",[40,30058,30059],{"class":10363},"(t)\n",[40,30061,30062,30065,30067,30070,30072,30074,30076,30078],{"class":10276,"line":10349},[40,30063,30064],{"class":10363},"rectangular[np.",[40,30066,23431],{"class":10430},[40,30068,30069],{"class":10363},"(t) ",[40,30071,22432],{"class":10732},[40,30073,13373],{"class":10446},[40,30075,10741],{"class":10363},[40,30077,162],{"class":10732},[40,30079,27413],{"class":10446},[40,30081,30082,30085,30087,30090,30092,30094],{"class":10276,"line":10367},[40,30083,30084],{"class":10363},"blackman_harris ",[40,30086,162],{"class":10732},[40,30088,30089],{"class":10430}," blackmanharris",[40,30091,167],{"class":10363},[40,30093,25507],{"class":10732},[40,30095,30096],{"class":10363},"(t))\n",[40,30098,30099,30102,30104,30107,30109,30111,30114,30116,30118,30120],{"class":10276,"line":10372},[40,30100,30101],{"class":10363},"tukey_envelope ",[40,30103,162],{"class":10732},[40,30105,30106],{"class":10430}," tukey",[40,30108,167],{"class":10363},[40,30110,25507],{"class":10732},[40,30112,30113],{"class":10363},"(t), ",[40,30115,12602],{"class":11903},[40,30117,162],{"class":10732},[40,30119,24827],{"class":10446},[40,30121,10761],{"class":10363},[40,30123,30124,30127,30129,30131,30133,30135,30138],{"class":10276,"line":10387},[40,30125,30126],{"class":10363},"sech ",[40,30128,162],{"class":10732},[40,30130,13373],{"class":10446},[40,30132,22539],{"class":10732},[40,30134,11156],{"class":10363},[40,30136,30137],{"class":10430},"cosh",[40,30139,30059],{"class":10363},[40,30141,30142],{"class":10276,"line":10400},[40,30143,10301],{"emptyLinePlaceholder":10300},[40,30145,30146],{"class":10276,"line":10413},[40,30147,30148],{"class":10280},"# List of envelopes and titles\n",[40,30150,30151,30154,30156,30159,30162],{"class":10276,"line":10418},[40,30152,30153],{"class":10363},"envelopes ",[40,30155,162],{"class":10732},[40,30157,30158],{"class":10363}," [(gaussian, ",[40,30160,30161],{"class":10287},"\"Gaussian Modulation Function\"",[40,30163,11061],{"class":10363},[40,30165,30166,30169,30172,30175,30178],{"class":10276,"line":10424},[40,30167,30168],{"class":10363},"             (sin2, ",[40,30170,30171],{"class":10287},"\"$Sin^",[40,30173,30174],{"class":10446},"{2}",[40,30176,30177],{"class":10287},"$ Modulation Function\"",[40,30179,11061],{"class":10363},[40,30181,30182,30185,30188,30190,30192],{"class":10276,"line":10437},[40,30183,30184],{"class":10363},"             (cos2, ",[40,30186,30187],{"class":10287},"\"$Cos^",[40,30189,30174],{"class":10446},[40,30191,30177],{"class":10287},[40,30193,11061],{"class":10363},[40,30195,30196,30199,30202],{"class":10276,"line":10456},[40,30197,30198],{"class":10363},"             (rectangular, ",[40,30200,30201],{"class":10287},"\"Rectangular Modulation Function\"",[40,30203,11061],{"class":10363},[40,30205,30206,30209,30212],{"class":10276,"line":10472},[40,30207,30208],{"class":10363},"             (blackman_harris, ",[40,30210,30211],{"class":10287},"\"Blackman-Harris Modulation Function\"",[40,30213,11061],{"class":10363},[40,30215,30216,30219,30222],{"class":10276,"line":10496},[40,30217,30218],{"class":10363},"             (tukey_envelope, ",[40,30220,30221],{"class":10287},"\"Tukey Modulation Function\"",[40,30223,11061],{"class":10363},[40,30225,30226,30229,30232],{"class":10276,"line":10510},[40,30227,30228],{"class":10363},"             (sech, ",[40,30230,30231],{"class":10287},"\"Sech Modulation Function\"",[40,30233,30234],{"class":10363},")]\n",[40,30236,30237],{"class":10276,"line":10696},[40,30238,10301],{"emptyLinePlaceholder":10300},[40,30240,30241],{"class":10276,"line":10702},[40,30242,30243],{"class":10280},"# Create subplots\n",[40,30245,30246,30249,30251,30253,30255,30257,30259,30261,30263,30265,30267,30269,30271,30274,30276,30278],{"class":10276,"line":10708},[40,30247,30248],{"class":10363},"fig, axes ",[40,30250,162],{"class":10732},[40,30252,13070],{"class":10363},[40,30254,13073],{"class":10430},[40,30256,167],{"class":10363},[40,30258,949],{"class":10446},[40,30260,10484],{"class":10363},[40,30262,949],{"class":10446},[40,30264,10484],{"class":10363},[40,30266,13078],{"class":11903},[40,30268,162],{"class":10732},[40,30270,167],{"class":10363},[40,30272,30273],{"class":10446},"12",[40,30275,10484],{"class":10363},[40,30277,13793],{"class":10446},[40,30279,13612],{"class":10363},[40,30281,30282,30285,30287,30290,30293],{"class":10276,"line":10714},[40,30283,30284],{"class":10363},"axes ",[40,30286,162],{"class":10732},[40,30288,30289],{"class":10363}," axes.",[40,30291,30292],{"class":10430},"flatten",[40,30294,12174],{"class":10363},[40,30296,30297],{"class":10276,"line":10720},[40,30298,10301],{"emptyLinePlaceholder":10300},[40,30300,30301,30303,30306,30308,30310],{"class":10276,"line":10726},[40,30302,13840],{"class":10352},[40,30304,30305],{"class":10363}," i, (envelope, title) ",[40,30307,13846],{"class":10352},[40,30309,23931],{"class":10732},[40,30311,30312],{"class":10363},"(envelopes):\n",[40,30314,30315,30318,30320,30323,30326],{"class":10276,"line":10764},[40,30316,30317],{"class":10363},"    axes[i].",[40,30319,23505],{"class":10430},[40,30321,30322],{"class":10363},"(t, envelope, ",[40,30324,30325],{"class":10287},"'b'",[40,30327,10761],{"class":10363},[40,30329,30330,30332,30335],{"class":10276,"line":10778},[40,30331,30317],{"class":10363},[40,30333,30334],{"class":10430},"set_title",[40,30336,30337],{"class":10363},"(title)\n",[40,30339,30340,30342,30344,30346,30349],{"class":10276,"line":10815},[40,30341,30317],{"class":10363},[40,30343,13230],{"class":10430},[40,30345,167],{"class":10363},[40,30347,30348],{"class":10287},"\"Time\"",[40,30350,10761],{"class":10363},[40,30352,30353,30355,30357,30359,30362],{"class":10276,"line":10829},[40,30354,30317],{"class":10363},[40,30356,13244],{"class":10430},[40,30358,167],{"class":10363},[40,30360,30361],{"class":10287},"\"Amplitude\"",[40,30363,10761],{"class":10363},[40,30365,30366],{"class":10276,"line":10842},[40,30367,30368],{"class":10280},"    # axes[i].legend([title])\n",[40,30370,30371],{"class":10276,"line":10855},[40,30372,10301],{"emptyLinePlaceholder":10300},[40,30374,30375],{"class":10276,"line":10868},[40,30376,30377],{"class":10280},"# Hide unused subplots\n",[40,30379,30380,30382,30385,30387,30390,30393,30395,30397,30399,30401],{"class":10276,"line":10881},[40,30381,13840],{"class":10352},[40,30383,30384],{"class":10363}," j ",[40,30386,13846],{"class":10352},[40,30388,30389],{"class":10732}," range",[40,30391,30392],{"class":10363},"(i ",[40,30394,1053],{"class":10732},[40,30396,13373],{"class":10446},[40,30398,10484],{"class":10363},[40,30400,25507],{"class":10732},[40,30402,30403],{"class":10363},"(axes)):\n",[40,30405,30406,30409,30412],{"class":10276,"line":10907},[40,30407,30408],{"class":10363},"    fig.",[40,30410,30411],{"class":10430},"delaxes",[40,30413,30414],{"class":10363},"(axes[j])\n",[40,30416,30417],{"class":10276,"line":10920},[40,30418,10301],{"emptyLinePlaceholder":10300},[40,30420,30421,30423,30426],{"class":10276,"line":10933},[40,30422,23469],{"class":10363},[40,30424,30425],{"class":10430},"tight_layout",[40,30427,12174],{"class":10363},[40,30429,30430,30432,30434],{"class":10276,"line":10946},[40,30431,23469],{"class":10363},[40,30433,23686],{"class":10430},[40,30435,12174],{"class":10363},[13997,30437],{"alt":30438,"src":30439},"Output 9 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-09.webp",[10082,30441,30443],{"className":10269,"code":30442,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.signal.windows import blackmanharris, tukey\n\n# Define time vector with reduced points for lollipop plot clarity\nt = np.linspace(-2, 2, 40)\n\n# Define different modulation functions\ngaussian_modulation = np.exp(-t**2)\nsin2_modulation = np.sin(np.pi * (t - t.min()) \u002F (t.max() - t.min()))**2\ncos2_modulation = np.cos(np.pi * (t - t.min()) \u002F (t.max() - t.min()))**2\nrectangular_modulation = np.ones_like(t)\nrectangular_modulation[np.abs(t) > 1] = 0\nblackman_harris_modulation = blackmanharris(len(t))\ntukey_modulation = tukey(len(t), alpha=0.5)\nsech_modulation = 1 \u002F np.cosh(t)\n\n# List of modulation functions and titles\nmodulations = [(gaussian_modulation, \"Gaussian Modulation Function\"),\n               (sin2_modulation, \"Sin² Modulation Function\"),\n               (cos2_modulation, \"Cos² Modulation Function\"),\n               (rectangular_modulation, \"Rectangular Modulation Function\"),\n               (blackman_harris_modulation, \"Blackman-Harris Modulation Function\"),\n               (tukey_modulation, \"Tukey Modulation Function\"),\n               (sech_modulation, \"Sech Modulation Function\")]\n\n# Create subplots\nfig, axes = plt.subplots(3, 3, figsize=(12, 9))\naxes = axes.flatten()\n\nfor i, (modulation, title) in enumerate(modulations):\n    axes[i].stem(t, modulation, linefmt='b-', markerfmt='bo', basefmt='k-')\n    axes[i].set_title(title)\n    axes[i].set_xlabel(\"Time\")\n    axes[i].set_ylabel(\"Amplitude\")\n\n# Hide unused subplots\nfor j in range(i + 1, len(axes)):\n    fig.delaxes(axes[j])\n\nplt.tight_layout()\nplt.show()\n",[10088,30444,30445,30455,30465,30475,30479,30484,30510,30514,30519,30541,30586,30631,30644,30663,30678,30701,30718,30722,30727,30741,30751,30761,30770,30779,30788,30797,30801,30805,30839,30851,30855,30869,30902,30910,30922,30934,30938,30942,30964,30972,30976,30984],{"__ignoreMap":10090},[40,30446,30447,30449,30451,30453],{"class":10276,"line":10277},[40,30448,10375],{"class":10352},[40,30450,10378],{"class":10363},[40,30452,10381],{"class":10352},[40,30454,10384],{"class":10363},[40,30456,30457,30459,30461,30463],{"class":10276,"line":10284},[40,30458,10375],{"class":10352},[40,30460,10405],{"class":10363},[40,30462,10381],{"class":10352},[40,30464,10410],{"class":10363},[40,30466,30467,30469,30471,30473],{"class":10276,"line":10291},[40,30468,10353],{"class":10352},[40,30470,29875],{"class":10363},[40,30472,10375],{"class":10352},[40,30474,29880],{"class":10363},[40,30476,30477],{"class":10276,"line":10297},[40,30478,10301],{"emptyLinePlaceholder":10300},[40,30480,30481],{"class":10276,"line":10304},[40,30482,30483],{"class":10280},"# Define time vector with reduced points for lollipop plot clarity\n",[40,30485,30486,30488,30490,30492,30494,30496,30498,30500,30502,30504,30506,30508],{"class":10276,"line":10310},[40,30487,29894],{"class":10363},[40,30489,162],{"class":10732},[40,30491,11156],{"class":10363},[40,30493,11850],{"class":10430},[40,30495,167],{"class":10363},[40,30497,11056],{"class":10732},[40,30499,1044],{"class":10446},[40,30501,10484],{"class":10363},[40,30503,1044],{"class":10446},[40,30505,10484],{"class":10363},[40,30507,27987],{"class":10446},[40,30509,10761],{"class":10363},[40,30511,30512],{"class":10276,"line":10316},[40,30513,10301],{"emptyLinePlaceholder":10300},[40,30515,30516],{"class":10276,"line":10322},[40,30517,30518],{"class":10280},"# Define different modulation functions\n",[40,30520,30521,30523,30525,30527,30529,30531,30533,30535,30537,30539],{"class":10276,"line":10327},[40,30522,27468],{"class":10363},[40,30524,162],{"class":10732},[40,30526,11156],{"class":10363},[40,30528,5268],{"class":10430},[40,30530,167],{"class":10363},[40,30532,11056],{"class":10732},[40,30534,20277],{"class":10363},[40,30536,22571],{"class":10732},[40,30538,1044],{"class":10446},[40,30540,10761],{"class":10363},[40,30542,30543,30546,30548,30550,30552,30554,30556,30558,30560,30562,30564,30566,30568,30570,30572,30574,30576,30578,30580,30582,30584],{"class":10276,"line":10333},[40,30544,30545],{"class":10363},"sin2_modulation ",[40,30547,162],{"class":10732},[40,30549,11156],{"class":10363},[40,30551,1813],{"class":10430},[40,30553,29962],{"class":10363},[40,30555,11752],{"class":10732},[40,30557,29967],{"class":10363},[40,30559,11056],{"class":10732},[40,30561,29972],{"class":10363},[40,30563,3067],{"class":10430},[40,30565,29977],{"class":10363},[40,30567,11243],{"class":10732},[40,30569,29982],{"class":10363},[40,30571,3032],{"class":10430},[40,30573,27025],{"class":10363},[40,30575,11056],{"class":10732},[40,30577,29972],{"class":10363},[40,30579,3067],{"class":10430},[40,30581,29995],{"class":10363},[40,30583,22571],{"class":10732},[40,30585,27056],{"class":10446},[40,30587,30588,30591,30593,30595,30597,30599,30601,30603,30605,30607,30609,30611,30613,30615,30617,30619,30621,30623,30625,30627,30629],{"class":10276,"line":10339},[40,30589,30590],{"class":10363},"cos2_modulation ",[40,30592,162],{"class":10732},[40,30594,11156],{"class":10363},[40,30596,1820],{"class":10430},[40,30598,29962],{"class":10363},[40,30600,11752],{"class":10732},[40,30602,29967],{"class":10363},[40,30604,11056],{"class":10732},[40,30606,29972],{"class":10363},[40,30608,3067],{"class":10430},[40,30610,29977],{"class":10363},[40,30612,11243],{"class":10732},[40,30614,29982],{"class":10363},[40,30616,3032],{"class":10430},[40,30618,27025],{"class":10363},[40,30620,11056],{"class":10732},[40,30622,29972],{"class":10363},[40,30624,3067],{"class":10430},[40,30626,29995],{"class":10363},[40,30628,22571],{"class":10732},[40,30630,27056],{"class":10446},[40,30632,30633,30636,30638,30640,30642],{"class":10276,"line":10344},[40,30634,30635],{"class":10363},"rectangular_modulation ",[40,30637,162],{"class":10732},[40,30639,11156],{"class":10363},[40,30641,30056],{"class":10430},[40,30643,30059],{"class":10363},[40,30645,30646,30649,30651,30653,30655,30657,30659,30661],{"class":10276,"line":10349},[40,30647,30648],{"class":10363},"rectangular_modulation[np.",[40,30650,23431],{"class":10430},[40,30652,30069],{"class":10363},[40,30654,22432],{"class":10732},[40,30656,13373],{"class":10446},[40,30658,10741],{"class":10363},[40,30660,162],{"class":10732},[40,30662,27413],{"class":10446},[40,30664,30665,30668,30670,30672,30674,30676],{"class":10276,"line":10367},[40,30666,30667],{"class":10363},"blackman_harris_modulation ",[40,30669,162],{"class":10732},[40,30671,30089],{"class":10430},[40,30673,167],{"class":10363},[40,30675,25507],{"class":10732},[40,30677,30096],{"class":10363},[40,30679,30680,30683,30685,30687,30689,30691,30693,30695,30697,30699],{"class":10276,"line":10372},[40,30681,30682],{"class":10363},"tukey_modulation ",[40,30684,162],{"class":10732},[40,30686,30106],{"class":10430},[40,30688,167],{"class":10363},[40,30690,25507],{"class":10732},[40,30692,30113],{"class":10363},[40,30694,12602],{"class":11903},[40,30696,162],{"class":10732},[40,30698,24827],{"class":10446},[40,30700,10761],{"class":10363},[40,30702,30703,30706,30708,30710,30712,30714,30716],{"class":10276,"line":10387},[40,30704,30705],{"class":10363},"sech_modulation ",[40,30707,162],{"class":10732},[40,30709,13373],{"class":10446},[40,30711,22539],{"class":10732},[40,30713,11156],{"class":10363},[40,30715,30137],{"class":10430},[40,30717,30059],{"class":10363},[40,30719,30720],{"class":10276,"line":10400},[40,30721,10301],{"emptyLinePlaceholder":10300},[40,30723,30724],{"class":10276,"line":10413},[40,30725,30726],{"class":10280},"# List of modulation functions and titles\n",[40,30728,30729,30732,30734,30737,30739],{"class":10276,"line":10418},[40,30730,30731],{"class":10363},"modulations ",[40,30733,162],{"class":10732},[40,30735,30736],{"class":10363}," [(gaussian_modulation, ",[40,30738,30161],{"class":10287},[40,30740,11061],{"class":10363},[40,30742,30743,30746,30749],{"class":10276,"line":10424},[40,30744,30745],{"class":10363},"               (sin2_modulation, ",[40,30747,30748],{"class":10287},"\"Sin² Modulation Function\"",[40,30750,11061],{"class":10363},[40,30752,30753,30756,30759],{"class":10276,"line":10437},[40,30754,30755],{"class":10363},"               (cos2_modulation, ",[40,30757,30758],{"class":10287},"\"Cos² Modulation Function\"",[40,30760,11061],{"class":10363},[40,30762,30763,30766,30768],{"class":10276,"line":10456},[40,30764,30765],{"class":10363},"               (rectangular_modulation, ",[40,30767,30201],{"class":10287},[40,30769,11061],{"class":10363},[40,30771,30772,30775,30777],{"class":10276,"line":10472},[40,30773,30774],{"class":10363},"               (blackman_harris_modulation, ",[40,30776,30211],{"class":10287},[40,30778,11061],{"class":10363},[40,30780,30781,30784,30786],{"class":10276,"line":10496},[40,30782,30783],{"class":10363},"               (tukey_modulation, ",[40,30785,30221],{"class":10287},[40,30787,11061],{"class":10363},[40,30789,30790,30793,30795],{"class":10276,"line":10510},[40,30791,30792],{"class":10363},"               (sech_modulation, ",[40,30794,30231],{"class":10287},[40,30796,30234],{"class":10363},[40,30798,30799],{"class":10276,"line":10696},[40,30800,10301],{"emptyLinePlaceholder":10300},[40,30802,30803],{"class":10276,"line":10702},[40,30804,30243],{"class":10280},[40,30806,30807,30809,30811,30813,30815,30817,30819,30821,30823,30825,30827,30829,30831,30833,30835,30837],{"class":10276,"line":10708},[40,30808,30248],{"class":10363},[40,30810,162],{"class":10732},[40,30812,13070],{"class":10363},[40,30814,13073],{"class":10430},[40,30816,167],{"class":10363},[40,30818,949],{"class":10446},[40,30820,10484],{"class":10363},[40,30822,949],{"class":10446},[40,30824,10484],{"class":10363},[40,30826,13078],{"class":11903},[40,30828,162],{"class":10732},[40,30830,167],{"class":10363},[40,30832,30273],{"class":10446},[40,30834,10484],{"class":10363},[40,30836,13793],{"class":10446},[40,30838,13612],{"class":10363},[40,30840,30841,30843,30845,30847,30849],{"class":10276,"line":10714},[40,30842,30284],{"class":10363},[40,30844,162],{"class":10732},[40,30846,30289],{"class":10363},[40,30848,30292],{"class":10430},[40,30850,12174],{"class":10363},[40,30852,30853],{"class":10276,"line":10720},[40,30854,10301],{"emptyLinePlaceholder":10300},[40,30856,30857,30859,30862,30864,30866],{"class":10276,"line":10726},[40,30858,13840],{"class":10352},[40,30860,30861],{"class":10363}," i, (modulation, title) ",[40,30863,13846],{"class":10352},[40,30865,23931],{"class":10732},[40,30867,30868],{"class":10363},"(modulations):\n",[40,30870,30871,30873,30875,30878,30880,30882,30884,30886,30888,30890,30892,30894,30896,30898,30900],{"class":10276,"line":10764},[40,30872,30317],{"class":10363},[40,30874,28083],{"class":10430},[40,30876,30877],{"class":10363},"(t, modulation, ",[40,30879,28088],{"class":11903},[40,30881,162],{"class":10732},[40,30883,28093],{"class":10287},[40,30885,10484],{"class":10363},[40,30887,28098],{"class":11903},[40,30889,162],{"class":10732},[40,30891,28103],{"class":10287},[40,30893,10484],{"class":10363},[40,30895,28108],{"class":11903},[40,30897,162],{"class":10732},[40,30899,28113],{"class":10287},[40,30901,10761],{"class":10363},[40,30903,30904,30906,30908],{"class":10276,"line":10778},[40,30905,30317],{"class":10363},[40,30907,30334],{"class":10430},[40,30909,30337],{"class":10363},[40,30911,30912,30914,30916,30918,30920],{"class":10276,"line":10815},[40,30913,30317],{"class":10363},[40,30915,13230],{"class":10430},[40,30917,167],{"class":10363},[40,30919,30348],{"class":10287},[40,30921,10761],{"class":10363},[40,30923,30924,30926,30928,30930,30932],{"class":10276,"line":10829},[40,30925,30317],{"class":10363},[40,30927,13244],{"class":10430},[40,30929,167],{"class":10363},[40,30931,30361],{"class":10287},[40,30933,10761],{"class":10363},[40,30935,30936],{"class":10276,"line":10842},[40,30937,10301],{"emptyLinePlaceholder":10300},[40,30939,30940],{"class":10276,"line":10855},[40,30941,30377],{"class":10280},[40,30943,30944,30946,30948,30950,30952,30954,30956,30958,30960,30962],{"class":10276,"line":10868},[40,30945,13840],{"class":10352},[40,30947,30384],{"class":10363},[40,30949,13846],{"class":10352},[40,30951,30389],{"class":10732},[40,30953,30392],{"class":10363},[40,30955,1053],{"class":10732},[40,30957,13373],{"class":10446},[40,30959,10484],{"class":10363},[40,30961,25507],{"class":10732},[40,30963,30403],{"class":10363},[40,30965,30966,30968,30970],{"class":10276,"line":10881},[40,30967,30408],{"class":10363},[40,30969,30411],{"class":10430},[40,30971,30414],{"class":10363},[40,30973,30974],{"class":10276,"line":10907},[40,30975,10301],{"emptyLinePlaceholder":10300},[40,30977,30978,30980,30982],{"class":10276,"line":10920},[40,30979,23469],{"class":10363},[40,30981,30425],{"class":10430},[40,30983,12174],{"class":10363},[40,30985,30986,30988,30990],{"class":10276,"line":10933},[40,30987,23469],{"class":10363},[40,30989,23686],{"class":10430},[40,30991,12174],{"class":10363},[13997,30993],{"alt":30994,"src":30995},"Output 10 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-10.webp",[10082,30997,30999],{"className":10269,"code":30998,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.signal.windows import blackmanharris, tukey\n\n# Define time vector\ntime = np.linspace(-5, 5, 1000)\n\n# Define different modulated sine waves\ngaussian_modulated_sine = np.exp(-time**2)\nsin2_modulated_sine = np.sin(np.pi * (time - time.min()) \u002F (time.max() - time.min()))**2\ncos2_modulated_sine = np.cos(np.pi * (time - time.min()) \u002F (time.max() - time.min()))**2\nrectangular_modulated_sine = np.ones_like(time)\nrectangular_modulated_sine[np.abs(time) > 2] = 0\nblackman_harris_modulated_sine = blackmanharris(len(time))\ntukey_modulated_sine = tukey(len(time), alpha=0.5)\nsech_modulated_sine = 1 \u002F np.cosh(time)\n\n# Define sine wave parameters\namplitude = 1\nfrequency = 5  # Frequency of the sine wave\nsine_wave = amplitude * np.sin(2 * np.pi * frequency * time)\n\n# List of modulated sine waves and titles\nmodulated_sines = [(gaussian_modulated_sine, \"Gaussian Modulated Sine Wave\"),\n                    (sin2_modulated_sine, \"$Sin^{2}$ Modulated Sine Wave\"),\n                    (cos2_modulated_sine, \"$Cos^{2}$ Modulated Sine Wave\"),\n                    (rectangular_modulated_sine, \"Rectangular Modulated Sine Wave\"),\n                    (blackman_harris_modulated_sine, \"Blackman-Harris Modulated Sine Wave\"),\n                    (tukey_modulated_sine, \"Tukey Modulated Sine Wave\"),\n                    (sech_modulated_sine, \"Sech Modulated Sine Wave\")]\n\n# Create subplots\nfig, axes = plt.subplots(3, 3, figsize=(12, 9))\naxes = axes.flatten()\n\nfor i, (modulated_sine, title) in enumerate(modulated_sines):\n    modulated_wave = modulated_sine * sine_wave\n    axes[i].plot(time, modulated_wave, 'b')\n    axes[i].set_title(title)\n    axes[i].set_xlabel(\"Time\")\n    axes[i].set_ylabel(\"Amplitude\")\n\n# Hide unused subplots\nfor j in range(i + 1, len(axes)):\n    fig.delaxes(axes[j])\n\nplt.tight_layout()\nplt.show()\n",[10088,31000,31001,31011,31021,31031,31035,31039,31065,31069,31074,31098,31146,31191,31205,31225,31241,31265,31282,31286,31290,31298,31308,31338,31342,31347,31362,31376,31389,31399,31409,31419,31429,31433,31437,31471,31483,31487,31501,31515,31528,31536,31548,31560,31564,31568,31590,31598,31602,31610],{"__ignoreMap":10090},[40,31002,31003,31005,31007,31009],{"class":10276,"line":10277},[40,31004,10375],{"class":10352},[40,31006,10378],{"class":10363},[40,31008,10381],{"class":10352},[40,31010,10384],{"class":10363},[40,31012,31013,31015,31017,31019],{"class":10276,"line":10284},[40,31014,10375],{"class":10352},[40,31016,10405],{"class":10363},[40,31018,10381],{"class":10352},[40,31020,10410],{"class":10363},[40,31022,31023,31025,31027,31029],{"class":10276,"line":10291},[40,31024,10353],{"class":10352},[40,31026,29875],{"class":10363},[40,31028,10375],{"class":10352},[40,31030,29880],{"class":10363},[40,31032,31033],{"class":10276,"line":10297},[40,31034,10301],{"emptyLinePlaceholder":10300},[40,31036,31037],{"class":10276,"line":10304},[40,31038,29889],{"class":10280},[40,31040,31041,31043,31045,31047,31049,31051,31053,31055,31057,31059,31061,31063],{"class":10276,"line":10310},[40,31042,27428],{"class":10363},[40,31044,162],{"class":10732},[40,31046,11156],{"class":10363},[40,31048,11850],{"class":10430},[40,31050,167],{"class":10363},[40,31052,11056],{"class":10732},[40,31054,12566],{"class":10446},[40,31056,10484],{"class":10363},[40,31058,12566],{"class":10446},[40,31060,10484],{"class":10363},[40,31062,27449],{"class":10446},[40,31064,10761],{"class":10363},[40,31066,31067],{"class":10276,"line":10316},[40,31068,10301],{"emptyLinePlaceholder":10300},[40,31070,31071],{"class":10276,"line":10322},[40,31072,31073],{"class":10280},"# Define different modulated sine waves\n",[40,31075,31076,31079,31081,31083,31085,31087,31089,31092,31094,31096],{"class":10276,"line":10327},[40,31077,31078],{"class":10363},"gaussian_modulated_sine ",[40,31080,162],{"class":10732},[40,31082,11156],{"class":10363},[40,31084,5268],{"class":10430},[40,31086,167],{"class":10363},[40,31088,11056],{"class":10732},[40,31090,31091],{"class":10363},"time",[40,31093,22571],{"class":10732},[40,31095,1044],{"class":10446},[40,31097,10761],{"class":10363},[40,31099,31100,31103,31105,31107,31109,31111,31113,31116,31118,31121,31123,31125,31127,31130,31132,31134,31136,31138,31140,31142,31144],{"class":10276,"line":10333},[40,31101,31102],{"class":10363},"sin2_modulated_sine ",[40,31104,162],{"class":10732},[40,31106,11156],{"class":10363},[40,31108,1813],{"class":10430},[40,31110,29962],{"class":10363},[40,31112,11752],{"class":10732},[40,31114,31115],{"class":10363}," (time ",[40,31117,11056],{"class":10732},[40,31119,31120],{"class":10363}," time.",[40,31122,3067],{"class":10430},[40,31124,29977],{"class":10363},[40,31126,11243],{"class":10732},[40,31128,31129],{"class":10363}," (time.",[40,31131,3032],{"class":10430},[40,31133,27025],{"class":10363},[40,31135,11056],{"class":10732},[40,31137,31120],{"class":10363},[40,31139,3067],{"class":10430},[40,31141,29995],{"class":10363},[40,31143,22571],{"class":10732},[40,31145,27056],{"class":10446},[40,31147,31148,31151,31153,31155,31157,31159,31161,31163,31165,31167,31169,31171,31173,31175,31177,31179,31181,31183,31185,31187,31189],{"class":10276,"line":10339},[40,31149,31150],{"class":10363},"cos2_modulated_sine ",[40,31152,162],{"class":10732},[40,31154,11156],{"class":10363},[40,31156,1820],{"class":10430},[40,31158,29962],{"class":10363},[40,31160,11752],{"class":10732},[40,31162,31115],{"class":10363},[40,31164,11056],{"class":10732},[40,31166,31120],{"class":10363},[40,31168,3067],{"class":10430},[40,31170,29977],{"class":10363},[40,31172,11243],{"class":10732},[40,31174,31129],{"class":10363},[40,31176,3032],{"class":10430},[40,31178,27025],{"class":10363},[40,31180,11056],{"class":10732},[40,31182,31120],{"class":10363},[40,31184,3067],{"class":10430},[40,31186,29995],{"class":10363},[40,31188,22571],{"class":10732},[40,31190,27056],{"class":10446},[40,31192,31193,31196,31198,31200,31202],{"class":10276,"line":10344},[40,31194,31195],{"class":10363},"rectangular_modulated_sine ",[40,31197,162],{"class":10732},[40,31199,11156],{"class":10363},[40,31201,30056],{"class":10430},[40,31203,31204],{"class":10363},"(time)\n",[40,31206,31207,31210,31212,31215,31217,31219,31221,31223],{"class":10276,"line":10349},[40,31208,31209],{"class":10363},"rectangular_modulated_sine[np.",[40,31211,23431],{"class":10430},[40,31213,31214],{"class":10363},"(time) ",[40,31216,22432],{"class":10732},[40,31218,12960],{"class":10446},[40,31220,10741],{"class":10363},[40,31222,162],{"class":10732},[40,31224,27413],{"class":10446},[40,31226,31227,31230,31232,31234,31236,31238],{"class":10276,"line":10367},[40,31228,31229],{"class":10363},"blackman_harris_modulated_sine ",[40,31231,162],{"class":10732},[40,31233,30089],{"class":10430},[40,31235,167],{"class":10363},[40,31237,25507],{"class":10732},[40,31239,31240],{"class":10363},"(time))\n",[40,31242,31243,31246,31248,31250,31252,31254,31257,31259,31261,31263],{"class":10276,"line":10372},[40,31244,31245],{"class":10363},"tukey_modulated_sine ",[40,31247,162],{"class":10732},[40,31249,30106],{"class":10430},[40,31251,167],{"class":10363},[40,31253,25507],{"class":10732},[40,31255,31256],{"class":10363},"(time), ",[40,31258,12602],{"class":11903},[40,31260,162],{"class":10732},[40,31262,24827],{"class":10446},[40,31264,10761],{"class":10363},[40,31266,31267,31270,31272,31274,31276,31278,31280],{"class":10276,"line":10387},[40,31268,31269],{"class":10363},"sech_modulated_sine ",[40,31271,162],{"class":10732},[40,31273,13373],{"class":10446},[40,31275,22539],{"class":10732},[40,31277,11156],{"class":10363},[40,31279,30137],{"class":10430},[40,31281,31204],{"class":10363},[40,31283,31284],{"class":10276,"line":10400},[40,31285,10301],{"emptyLinePlaceholder":10300},[40,31287,31288],{"class":10276,"line":10413},[40,31289,28611],{"class":10280},[40,31291,31292,31294,31296],{"class":10276,"line":10418},[40,31293,27634],{"class":10363},[40,31295,162],{"class":10732},[40,31297,27423],{"class":10446},[40,31299,31300,31302,31304,31306],{"class":10276,"line":10424},[40,31301,28259],{"class":10363},[40,31303,162],{"class":10732},[40,31305,28264],{"class":10446},[40,31307,28267],{"class":10280},[40,31309,31310,31312,31314,31316,31318,31320,31322,31324,31326,31328,31330,31332,31334,31336],{"class":10276,"line":10437},[40,31311,28357],{"class":10363},[40,31313,162],{"class":10732},[40,31315,27699],{"class":10363},[40,31317,11752],{"class":10732},[40,31319,11156],{"class":10363},[40,31321,1813],{"class":10430},[40,31323,167],{"class":10363},[40,31325,1044],{"class":10446},[40,31327,11881],{"class":10732},[40,31329,28376],{"class":10363},[40,31331,11752],{"class":10732},[40,31333,28381],{"class":10363},[40,31335,11752],{"class":10732},[40,31337,28386],{"class":10363},[40,31339,31340],{"class":10276,"line":10456},[40,31341,10301],{"emptyLinePlaceholder":10300},[40,31343,31344],{"class":10276,"line":10472},[40,31345,31346],{"class":10280},"# List of modulated sine waves and titles\n",[40,31348,31349,31352,31354,31357,31360],{"class":10276,"line":10496},[40,31350,31351],{"class":10363},"modulated_sines ",[40,31353,162],{"class":10732},[40,31355,31356],{"class":10363}," [(gaussian_modulated_sine, ",[40,31358,31359],{"class":10287},"\"Gaussian Modulated Sine Wave\"",[40,31361,11061],{"class":10363},[40,31363,31364,31367,31369,31371,31374],{"class":10276,"line":10510},[40,31365,31366],{"class":10363},"                    (sin2_modulated_sine, ",[40,31368,30171],{"class":10287},[40,31370,30174],{"class":10446},[40,31372,31373],{"class":10287},"$ Modulated Sine Wave\"",[40,31375,11061],{"class":10363},[40,31377,31378,31381,31383,31385,31387],{"class":10276,"line":10696},[40,31379,31380],{"class":10363},"                    (cos2_modulated_sine, ",[40,31382,30187],{"class":10287},[40,31384,30174],{"class":10446},[40,31386,31373],{"class":10287},[40,31388,11061],{"class":10363},[40,31390,31391,31394,31397],{"class":10276,"line":10702},[40,31392,31393],{"class":10363},"                    (rectangular_modulated_sine, ",[40,31395,31396],{"class":10287},"\"Rectangular Modulated Sine Wave\"",[40,31398,11061],{"class":10363},[40,31400,31401,31404,31407],{"class":10276,"line":10708},[40,31402,31403],{"class":10363},"                    (blackman_harris_modulated_sine, ",[40,31405,31406],{"class":10287},"\"Blackman-Harris Modulated Sine Wave\"",[40,31408,11061],{"class":10363},[40,31410,31411,31414,31417],{"class":10276,"line":10714},[40,31412,31413],{"class":10363},"                    (tukey_modulated_sine, ",[40,31415,31416],{"class":10287},"\"Tukey Modulated Sine Wave\"",[40,31418,11061],{"class":10363},[40,31420,31421,31424,31427],{"class":10276,"line":10720},[40,31422,31423],{"class":10363},"                    (sech_modulated_sine, ",[40,31425,31426],{"class":10287},"\"Sech Modulated Sine Wave\"",[40,31428,30234],{"class":10363},[40,31430,31431],{"class":10276,"line":10726},[40,31432,10301],{"emptyLinePlaceholder":10300},[40,31434,31435],{"class":10276,"line":10764},[40,31436,30243],{"class":10280},[40,31438,31439,31441,31443,31445,31447,31449,31451,31453,31455,31457,31459,31461,31463,31465,31467,31469],{"class":10276,"line":10778},[40,31440,30248],{"class":10363},[40,31442,162],{"class":10732},[40,31444,13070],{"class":10363},[40,31446,13073],{"class":10430},[40,31448,167],{"class":10363},[40,31450,949],{"class":10446},[40,31452,10484],{"class":10363},[40,31454,949],{"class":10446},[40,31456,10484],{"class":10363},[40,31458,13078],{"class":11903},[40,31460,162],{"class":10732},[40,31462,167],{"class":10363},[40,31464,30273],{"class":10446},[40,31466,10484],{"class":10363},[40,31468,13793],{"class":10446},[40,31470,13612],{"class":10363},[40,31472,31473,31475,31477,31479,31481],{"class":10276,"line":10815},[40,31474,30284],{"class":10363},[40,31476,162],{"class":10732},[40,31478,30289],{"class":10363},[40,31480,30292],{"class":10430},[40,31482,12174],{"class":10363},[40,31484,31485],{"class":10276,"line":10829},[40,31486,10301],{"emptyLinePlaceholder":10300},[40,31488,31489,31491,31494,31496,31498],{"class":10276,"line":10842},[40,31490,13840],{"class":10352},[40,31492,31493],{"class":10363}," i, (modulated_sine, title) ",[40,31495,13846],{"class":10352},[40,31497,23931],{"class":10732},[40,31499,31500],{"class":10363},"(modulated_sines):\n",[40,31502,31503,31506,31508,31511,31513],{"class":10276,"line":10855},[40,31504,31505],{"class":10363},"    modulated_wave ",[40,31507,162],{"class":10732},[40,31509,31510],{"class":10363}," modulated_sine ",[40,31512,11752],{"class":10732},[40,31514,28410],{"class":10363},[40,31516,31517,31519,31521,31524,31526],{"class":10276,"line":10868},[40,31518,30317],{"class":10363},[40,31520,23505],{"class":10430},[40,31522,31523],{"class":10363},"(time, modulated_wave, ",[40,31525,30325],{"class":10287},[40,31527,10761],{"class":10363},[40,31529,31530,31532,31534],{"class":10276,"line":10881},[40,31531,30317],{"class":10363},[40,31533,30334],{"class":10430},[40,31535,30337],{"class":10363},[40,31537,31538,31540,31542,31544,31546],{"class":10276,"line":10907},[40,31539,30317],{"class":10363},[40,31541,13230],{"class":10430},[40,31543,167],{"class":10363},[40,31545,30348],{"class":10287},[40,31547,10761],{"class":10363},[40,31549,31550,31552,31554,31556,31558],{"class":10276,"line":10920},[40,31551,30317],{"class":10363},[40,31553,13244],{"class":10430},[40,31555,167],{"class":10363},[40,31557,30361],{"class":10287},[40,31559,10761],{"class":10363},[40,31561,31562],{"class":10276,"line":10933},[40,31563,10301],{"emptyLinePlaceholder":10300},[40,31565,31566],{"class":10276,"line":10946},[40,31567,30377],{"class":10280},[40,31569,31570,31572,31574,31576,31578,31580,31582,31584,31586,31588],{"class":10276,"line":10951},[40,31571,13840],{"class":10352},[40,31573,30384],{"class":10363},[40,31575,13846],{"class":10352},[40,31577,30389],{"class":10732},[40,31579,30392],{"class":10363},[40,31581,1053],{"class":10732},[40,31583,13373],{"class":10446},[40,31585,10484],{"class":10363},[40,31587,25507],{"class":10732},[40,31589,30403],{"class":10363},[40,31591,31592,31594,31596],{"class":10276,"line":10956},[40,31593,30408],{"class":10363},[40,31595,30411],{"class":10430},[40,31597,30414],{"class":10363},[40,31599,31600],{"class":10276,"line":10962},[40,31601,10301],{"emptyLinePlaceholder":10300},[40,31603,31604,31606,31608],{"class":10276,"line":10983},[40,31605,23469],{"class":10363},[40,31607,30425],{"class":10430},[40,31609,12174],{"class":10363},[40,31611,31612,31614,31616],{"class":10276,"line":11005},[40,31613,23469],{"class":10363},[40,31615,23686],{"class":10430},[40,31617,12174],{"class":10363},[13997,31619],{"alt":31620,"src":31621},"Output 11 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-11.webp",[10082,31623,31625],{"className":10269,"code":31624,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.signal.windows import blackmanharris, tukey\n\n# Define time vector\ntime = np.linspace(-5, 5, 1000)\n\n# Define different modulated sine waves\ngaussian_modulated_sine = np.exp(-time**2)\nsin2_modulated_sine = np.sin(np.pi * (time - time.min()) \u002F (time.max() - time.min()))**2\ncos2_modulated_sine = np.cos(np.pi * (time - time.min()) \u002F (time.max() - time.min()))**2\nrectangular_modulated_sine = np.ones_like(time)\nrectangular_modulated_sine[np.abs(time) > 2] = 0\nblackman_harris_modulated_sine = blackmanharris(len(time))\ntukey_modulated_sine = tukey(len(time), alpha=0.5)\nsech_modulated_sine = 1 \u002F np.cosh(time)\n\n# Define sine wave parameters\namplitude = 1\nfrequency = 5  # Frequency of the sine wave\nsine_wave = amplitude * np.sin(2 * np.pi * frequency * time)\n\n# List of modulated sine waves and titles\nmodulated_sines = [(gaussian_modulated_sine, \"Gaussian Modulated Sine Wave\"),\n                   (sin2_modulated_sine, \"$Sin^{2}$ Modulated Sine Wave\"),\n                   (cos2_modulated_sine, \"$Cos^{2}$ Modulated Sine Wave\"),\n                   (rectangular_modulated_sine, \"Rectangular Modulated Sine Wave\"),\n                   (blackman_harris_modulated_sine, \"Blackman-Harris Modulated Sine Wave\"),\n                   (tukey_modulated_sine, \"Tukey Modulated Sine Wave\"),\n                   (sech_modulated_sine, \"Sech Modulated Sine Wave\")]\n\n# Create subplots\nfig, axes = plt.subplots(3, 3, figsize=(12, 9))\naxes = axes.flatten()\n\nfor i, (modulated_sine, title) in enumerate(modulated_sines):\n    modulated_wave = modulated_sine * sine_wave\n    axes[i].plot(time, modulated_wave, 'b')\n    axes[i].plot(time, modulated_sine, 'r', linestyle='--', label='Upper Modulation Function')\n    axes[i].set_title(title)\n    axes[i].set_xlabel(\"Time\")\n    axes[i].set_ylabel(\"Amplitude\")\n    axes[i].legend()\n\n# Hide unused subplots\nfor j in range(i + 1, len(axes)):\n    fig.delaxes(axes[j])\n\nplt.tight_layout()\nplt.show()\n",[10088,31626,31627,31637,31647,31657,31661,31665,31691,31695,31699,31721,31765,31809,31821,31839,31853,31875,31891,31895,31899,31907,31917,31947,31951,31955,31967,31980,31993,32002,32011,32020,32029,32033,32037,32071,32083,32087,32099,32111,32123,32154,32162,32174,32186,32194,32198,32202,32224,32232,32236,32244],{"__ignoreMap":10090},[40,31628,31629,31631,31633,31635],{"class":10276,"line":10277},[40,31630,10375],{"class":10352},[40,31632,10378],{"class":10363},[40,31634,10381],{"class":10352},[40,31636,10384],{"class":10363},[40,31638,31639,31641,31643,31645],{"class":10276,"line":10284},[40,31640,10375],{"class":10352},[40,31642,10405],{"class":10363},[40,31644,10381],{"class":10352},[40,31646,10410],{"class":10363},[40,31648,31649,31651,31653,31655],{"class":10276,"line":10291},[40,31650,10353],{"class":10352},[40,31652,29875],{"class":10363},[40,31654,10375],{"class":10352},[40,31656,29880],{"class":10363},[40,31658,31659],{"class":10276,"line":10297},[40,31660,10301],{"emptyLinePlaceholder":10300},[40,31662,31663],{"class":10276,"line":10304},[40,31664,29889],{"class":10280},[40,31666,31667,31669,31671,31673,31675,31677,31679,31681,31683,31685,31687,31689],{"class":10276,"line":10310},[40,31668,27428],{"class":10363},[40,31670,162],{"class":10732},[40,31672,11156],{"class":10363},[40,31674,11850],{"class":10430},[40,31676,167],{"class":10363},[40,31678,11056],{"class":10732},[40,31680,12566],{"class":10446},[40,31682,10484],{"class":10363},[40,31684,12566],{"class":10446},[40,31686,10484],{"class":10363},[40,31688,27449],{"class":10446},[40,31690,10761],{"class":10363},[40,31692,31693],{"class":10276,"line":10316},[40,31694,10301],{"emptyLinePlaceholder":10300},[40,31696,31697],{"class":10276,"line":10322},[40,31698,31073],{"class":10280},[40,31700,31701,31703,31705,31707,31709,31711,31713,31715,31717,31719],{"class":10276,"line":10327},[40,31702,31078],{"class":10363},[40,31704,162],{"class":10732},[40,31706,11156],{"class":10363},[40,31708,5268],{"class":10430},[40,31710,167],{"class":10363},[40,31712,11056],{"class":10732},[40,31714,31091],{"class":10363},[40,31716,22571],{"class":10732},[40,31718,1044],{"class":10446},[40,31720,10761],{"class":10363},[40,31722,31723,31725,31727,31729,31731,31733,31735,31737,31739,31741,31743,31745,31747,31749,31751,31753,31755,31757,31759,31761,31763],{"class":10276,"line":10333},[40,31724,31102],{"class":10363},[40,31726,162],{"class":10732},[40,31728,11156],{"class":10363},[40,31730,1813],{"class":10430},[40,31732,29962],{"class":10363},[40,31734,11752],{"class":10732},[40,31736,31115],{"class":10363},[40,31738,11056],{"class":10732},[40,31740,31120],{"class":10363},[40,31742,3067],{"class":10430},[40,31744,29977],{"class":10363},[40,31746,11243],{"class":10732},[40,31748,31129],{"class":10363},[40,31750,3032],{"class":10430},[40,31752,27025],{"class":10363},[40,31754,11056],{"class":10732},[40,31756,31120],{"class":10363},[40,31758,3067],{"class":10430},[40,31760,29995],{"class":10363},[40,31762,22571],{"class":10732},[40,31764,27056],{"class":10446},[40,31766,31767,31769,31771,31773,31775,31777,31779,31781,31783,31785,31787,31789,31791,31793,31795,31797,31799,31801,31803,31805,31807],{"class":10276,"line":10339},[40,31768,31150],{"class":10363},[40,31770,162],{"class":10732},[40,31772,11156],{"class":10363},[40,31774,1820],{"class":10430},[40,31776,29962],{"class":10363},[40,31778,11752],{"class":10732},[40,31780,31115],{"class":10363},[40,31782,11056],{"class":10732},[40,31784,31120],{"class":10363},[40,31786,3067],{"class":10430},[40,31788,29977],{"class":10363},[40,31790,11243],{"class":10732},[40,31792,31129],{"class":10363},[40,31794,3032],{"class":10430},[40,31796,27025],{"class":10363},[40,31798,11056],{"class":10732},[40,31800,31120],{"class":10363},[40,31802,3067],{"class":10430},[40,31804,29995],{"class":10363},[40,31806,22571],{"class":10732},[40,31808,27056],{"class":10446},[40,31810,31811,31813,31815,31817,31819],{"class":10276,"line":10344},[40,31812,31195],{"class":10363},[40,31814,162],{"class":10732},[40,31816,11156],{"class":10363},[40,31818,30056],{"class":10430},[40,31820,31204],{"class":10363},[40,31822,31823,31825,31827,31829,31831,31833,31835,31837],{"class":10276,"line":10349},[40,31824,31209],{"class":10363},[40,31826,23431],{"class":10430},[40,31828,31214],{"class":10363},[40,31830,22432],{"class":10732},[40,31832,12960],{"class":10446},[40,31834,10741],{"class":10363},[40,31836,162],{"class":10732},[40,31838,27413],{"class":10446},[40,31840,31841,31843,31845,31847,31849,31851],{"class":10276,"line":10367},[40,31842,31229],{"class":10363},[40,31844,162],{"class":10732},[40,31846,30089],{"class":10430},[40,31848,167],{"class":10363},[40,31850,25507],{"class":10732},[40,31852,31240],{"class":10363},[40,31854,31855,31857,31859,31861,31863,31865,31867,31869,31871,31873],{"class":10276,"line":10372},[40,31856,31245],{"class":10363},[40,31858,162],{"class":10732},[40,31860,30106],{"class":10430},[40,31862,167],{"class":10363},[40,31864,25507],{"class":10732},[40,31866,31256],{"class":10363},[40,31868,12602],{"class":11903},[40,31870,162],{"class":10732},[40,31872,24827],{"class":10446},[40,31874,10761],{"class":10363},[40,31876,31877,31879,31881,31883,31885,31887,31889],{"class":10276,"line":10387},[40,31878,31269],{"class":10363},[40,31880,162],{"class":10732},[40,31882,13373],{"class":10446},[40,31884,22539],{"class":10732},[40,31886,11156],{"class":10363},[40,31888,30137],{"class":10430},[40,31890,31204],{"class":10363},[40,31892,31893],{"class":10276,"line":10400},[40,31894,10301],{"emptyLinePlaceholder":10300},[40,31896,31897],{"class":10276,"line":10413},[40,31898,28611],{"class":10280},[40,31900,31901,31903,31905],{"class":10276,"line":10418},[40,31902,27634],{"class":10363},[40,31904,162],{"class":10732},[40,31906,27423],{"class":10446},[40,31908,31909,31911,31913,31915],{"class":10276,"line":10424},[40,31910,28259],{"class":10363},[40,31912,162],{"class":10732},[40,31914,28264],{"class":10446},[40,31916,28267],{"class":10280},[40,31918,31919,31921,31923,31925,31927,31929,31931,31933,31935,31937,31939,31941,31943,31945],{"class":10276,"line":10437},[40,31920,28357],{"class":10363},[40,31922,162],{"class":10732},[40,31924,27699],{"class":10363},[40,31926,11752],{"class":10732},[40,31928,11156],{"class":10363},[40,31930,1813],{"class":10430},[40,31932,167],{"class":10363},[40,31934,1044],{"class":10446},[40,31936,11881],{"class":10732},[40,31938,28376],{"class":10363},[40,31940,11752],{"class":10732},[40,31942,28381],{"class":10363},[40,31944,11752],{"class":10732},[40,31946,28386],{"class":10363},[40,31948,31949],{"class":10276,"line":10456},[40,31950,10301],{"emptyLinePlaceholder":10300},[40,31952,31953],{"class":10276,"line":10472},[40,31954,31346],{"class":10280},[40,31956,31957,31959,31961,31963,31965],{"class":10276,"line":10496},[40,31958,31351],{"class":10363},[40,31960,162],{"class":10732},[40,31962,31356],{"class":10363},[40,31964,31359],{"class":10287},[40,31966,11061],{"class":10363},[40,31968,31969,31972,31974,31976,31978],{"class":10276,"line":10510},[40,31970,31971],{"class":10363},"                   (sin2_modulated_sine, ",[40,31973,30171],{"class":10287},[40,31975,30174],{"class":10446},[40,31977,31373],{"class":10287},[40,31979,11061],{"class":10363},[40,31981,31982,31985,31987,31989,31991],{"class":10276,"line":10696},[40,31983,31984],{"class":10363},"                   (cos2_modulated_sine, ",[40,31986,30187],{"class":10287},[40,31988,30174],{"class":10446},[40,31990,31373],{"class":10287},[40,31992,11061],{"class":10363},[40,31994,31995,31998,32000],{"class":10276,"line":10702},[40,31996,31997],{"class":10363},"                   (rectangular_modulated_sine, ",[40,31999,31396],{"class":10287},[40,32001,11061],{"class":10363},[40,32003,32004,32007,32009],{"class":10276,"line":10708},[40,32005,32006],{"class":10363},"                   (blackman_harris_modulated_sine, ",[40,32008,31406],{"class":10287},[40,32010,11061],{"class":10363},[40,32012,32013,32016,32018],{"class":10276,"line":10714},[40,32014,32015],{"class":10363},"                   (tukey_modulated_sine, ",[40,32017,31416],{"class":10287},[40,32019,11061],{"class":10363},[40,32021,32022,32025,32027],{"class":10276,"line":10720},[40,32023,32024],{"class":10363},"                   (sech_modulated_sine, ",[40,32026,31426],{"class":10287},[40,32028,30234],{"class":10363},[40,32030,32031],{"class":10276,"line":10726},[40,32032,10301],{"emptyLinePlaceholder":10300},[40,32034,32035],{"class":10276,"line":10764},[40,32036,30243],{"class":10280},[40,32038,32039,32041,32043,32045,32047,32049,32051,32053,32055,32057,32059,32061,32063,32065,32067,32069],{"class":10276,"line":10778},[40,32040,30248],{"class":10363},[40,32042,162],{"class":10732},[40,32044,13070],{"class":10363},[40,32046,13073],{"class":10430},[40,32048,167],{"class":10363},[40,32050,949],{"class":10446},[40,32052,10484],{"class":10363},[40,32054,949],{"class":10446},[40,32056,10484],{"class":10363},[40,32058,13078],{"class":11903},[40,32060,162],{"class":10732},[40,32062,167],{"class":10363},[40,32064,30273],{"class":10446},[40,32066,10484],{"class":10363},[40,32068,13793],{"class":10446},[40,32070,13612],{"class":10363},[40,32072,32073,32075,32077,32079,32081],{"class":10276,"line":10815},[40,32074,30284],{"class":10363},[40,32076,162],{"class":10732},[40,32078,30289],{"class":10363},[40,32080,30292],{"class":10430},[40,32082,12174],{"class":10363},[40,32084,32085],{"class":10276,"line":10829},[40,32086,10301],{"emptyLinePlaceholder":10300},[40,32088,32089,32091,32093,32095,32097],{"class":10276,"line":10842},[40,32090,13840],{"class":10352},[40,32092,31493],{"class":10363},[40,32094,13846],{"class":10352},[40,32096,23931],{"class":10732},[40,32098,31500],{"class":10363},[40,32100,32101,32103,32105,32107,32109],{"class":10276,"line":10855},[40,32102,31505],{"class":10363},[40,32104,162],{"class":10732},[40,32106,31510],{"class":10363},[40,32108,11752],{"class":10732},[40,32110,28410],{"class":10363},[40,32112,32113,32115,32117,32119,32121],{"class":10276,"line":10868},[40,32114,30317],{"class":10363},[40,32116,23505],{"class":10430},[40,32118,31523],{"class":10363},[40,32120,30325],{"class":10287},[40,32122,10761],{"class":10363},[40,32124,32125,32127,32129,32132,32135,32137,32139,32141,32144,32146,32148,32150,32152],{"class":10276,"line":10881},[40,32126,30317],{"class":10363},[40,32128,23505],{"class":10430},[40,32130,32131],{"class":10363},"(time, modulated_sine, ",[40,32133,32134],{"class":10287},"'r'",[40,32136,10484],{"class":10363},[40,32138,23274],{"class":11903},[40,32140,162],{"class":10732},[40,32142,32143],{"class":10287},"'--'",[40,32145,10484],{"class":10363},[40,32147,15415],{"class":11903},[40,32149,162],{"class":10732},[40,32151,29389],{"class":10287},[40,32153,10761],{"class":10363},[40,32155,32156,32158,32160],{"class":10276,"line":10907},[40,32157,30317],{"class":10363},[40,32159,30334],{"class":10430},[40,32161,30337],{"class":10363},[40,32163,32164,32166,32168,32170,32172],{"class":10276,"line":10920},[40,32165,30317],{"class":10363},[40,32167,13230],{"class":10430},[40,32169,167],{"class":10363},[40,32171,30348],{"class":10287},[40,32173,10761],{"class":10363},[40,32175,32176,32178,32180,32182,32184],{"class":10276,"line":10933},[40,32177,30317],{"class":10363},[40,32179,13244],{"class":10430},[40,32181,167],{"class":10363},[40,32183,30361],{"class":10287},[40,32185,10761],{"class":10363},[40,32187,32188,32190,32192],{"class":10276,"line":10946},[40,32189,30317],{"class":10363},[40,32191,23677],{"class":10430},[40,32193,12174],{"class":10363},[40,32195,32196],{"class":10276,"line":10951},[40,32197,10301],{"emptyLinePlaceholder":10300},[40,32199,32200],{"class":10276,"line":10956},[40,32201,30377],{"class":10280},[40,32203,32204,32206,32208,32210,32212,32214,32216,32218,32220,32222],{"class":10276,"line":10962},[40,32205,13840],{"class":10352},[40,32207,30384],{"class":10363},[40,32209,13846],{"class":10352},[40,32211,30389],{"class":10732},[40,32213,30392],{"class":10363},[40,32215,1053],{"class":10732},[40,32217,13373],{"class":10446},[40,32219,10484],{"class":10363},[40,32221,25507],{"class":10732},[40,32223,30403],{"class":10363},[40,32225,32226,32228,32230],{"class":10276,"line":10983},[40,32227,30408],{"class":10363},[40,32229,30411],{"class":10430},[40,32231,30414],{"class":10363},[40,32233,32234],{"class":10276,"line":11005},[40,32235,10301],{"emptyLinePlaceholder":10300},[40,32237,32238,32240,32242],{"class":10276,"line":11011},[40,32239,23469],{"class":10363},[40,32241,30425],{"class":10430},[40,32243,12174],{"class":10363},[40,32245,32246,32248,32250],{"class":10276,"line":11022},[40,32247,23469],{"class":10363},[40,32249,23686],{"class":10430},[40,32251,12174],{"class":10363},[13997,32253],{"alt":32254,"src":32255},"Output 12 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-12.webp",[12,32257,32258],{},"Below are some applied examples in the context of common quantum gates.",[12,32260,32261,32262,32265],{},"Relevant equations for the ",[35,32263,32264],{},"modulated Gaussian pulses"," used in the plots for some common quantum gates:",[21676,32267,32269],{"id":32268},"_1-gaussian-envelope",[35,32270,32271],{},"1. Gaussian Envelope",[12,32273,32274,32275,32632],{},"The Gaussian envelope is used to smoothly shape the pulse to minimize spectral leakage:\n",[40,32276,32278,32350],{"className":32277},[43],[40,32279,32281],{"className":32280},[47],[49,32282,32283],{"xmlns":51},[53,32284,32285,32347],{},[56,32286,32287,32289,32291,32293,32295,32297,32303,32305,32307],{},[154,32288,8234],{},[59,32290,167],{"stretchy":61},[154,32292,20277],{},[59,32294,207],{"stretchy":61},[59,32296,162],{},[151,32298,32299,32301],{},[154,32300,8234],{},[64,32302,66],{},[154,32304,5268],{},[59,32306,1084],{},[56,32308,32309,32311,32313,32345],{},[59,32310,167],{"fence":72},[59,32312,243],{},[1038,32314,32315,32335],{},[56,32316,32317,32319,32321,32323,32329],{},[59,32318,167],{"stretchy":61},[154,32320,20277],{},[59,32322,243],{},[151,32324,32325,32327],{},[154,32326,20277],{},[64,32328,66],{},[941,32330,32331,32333],{},[59,32332,207],{"stretchy":61},[64,32334,1044],{},[56,32336,32337,32339],{},[64,32338,1044],{},[941,32340,32341,32343],{},[154,32342,156],{},[64,32344,1044],{},[59,32346,207],{"fence":72},[82,32348,32349],{"encoding":84},"A(t) = A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right)",[40,32351,32353,32380],{"className":32352,"ariaHidden":72},[89],[40,32354,32356,32359,32362,32365,32368,32371,32374,32377],{"className":32355},[93],[40,32357],{"className":32358,"style":98},[97],[40,32360,8234],{"className":32361},[102,335],[40,32363,167],{"className":32364},[106],[40,32366,20277],{"className":32367},[102,335],[40,32369,207],{"className":32370},[113],[40,32372],{"className":32373,"style":390},[121],[40,32375,162],{"className":32376},[394],[40,32378],{"className":32379,"style":390},[121],[40,32381,32383,32386,32426,32429,32432,32435],{"className":32382},[93],[40,32384],{"className":32385,"style":2120},[97],[40,32387,32389,32392],{"className":32388},[102],[40,32390,8234],{"className":32391},[102,335],[40,32393,32395],{"className":32394},[340],[40,32396,32398,32418],{"className":32397},[344,345],[40,32399,32401,32415],{"className":32400},[349],[40,32402,32404],{"className":32403,"style":7201},[353],[40,32405,32406,32409],{"style":6098},[40,32407],{"className":32408,"style":362},[361],[40,32410,32412],{"className":32411},[366,367,368,369],[40,32413,66],{"className":32414},[102,369],[40,32416,377],{"className":32417},[376],[40,32419,32421],{"className":32420},[349],[40,32422,32424],{"className":32423,"style":384},[353],[40,32425],{},[40,32427],{"className":32428,"style":122},[121],[40,32430,5268],{"className":32431},[1369],[40,32433],{"className":32434,"style":122},[121],[40,32436,32438,32444,32447,32626],{"className":32437},[408],[40,32439,32441],{"className":32440,"style":413},[106,412],[40,32442,167],{"className":32443},[417,2195],[40,32445,243],{"className":32446},[102],[40,32448,32450,32453,32623],{"className":32449},[102],[40,32451],{"className":32452},[106,1162],[40,32454,32456],{"className":32455},[1038],[40,32457,32459,32615],{"className":32458},[344,345],[40,32460,32462,32612],{"className":32461},[349],[40,32463,32466,32510,32518],{"className":32464,"style":32465},[353],"height:1.1089em;",[40,32467,32468,32471],{"style":1178},[40,32469],{"className":32470,"style":444},[361],[40,32472,32474],{"className":32473},[366,367,368,369],[40,32475,32477,32480],{"className":32476},[102,369],[40,32478,1044],{"className":32479},[102,369],[40,32481,32483,32486],{"className":32482},[102,369],[40,32484,156],{"className":32485,"style":336},[102,335,369],[40,32487,32489],{"className":32488},[340],[40,32490,32492],{"className":32491},[344],[40,32493,32495],{"className":32494},[349],[40,32496,32499],{"className":32497,"style":32498},[353],"height:0.7463em;",[40,32500,32501,32504],{"style":3827},[40,32502],{"className":32503,"style":3831},[361],[40,32505,32507],{"className":32506},[366,3835,1238,369],[40,32508,1044],{"className":32509},[102,369],[40,32511,32512,32515],{"style":1193},[40,32513],{"className":32514,"style":444},[361],[40,32516],{"className":32517,"style":1201},[1200],[40,32519,32521,32524],{"style":32520},"top:-3.485em;",[40,32522],{"className":32523,"style":444},[361],[40,32525,32527],{"className":32526},[366,367,368,369],[40,32528,32530,32533,32536,32539,32581],{"className":32529},[102,369],[40,32531,167],{"className":32532},[106,369],[40,32534,20277],{"className":32535},[102,335,369],[40,32537,243],{"className":32538},[1250,369],[40,32540,32542,32545],{"className":32541},[102,369],[40,32543,20277],{"className":32544},[102,335,369],[40,32546,32548],{"className":32547},[340],[40,32549,32551,32573],{"className":32550},[344,345],[40,32552,32554,32570],{"className":32553},[349],[40,32555,32558],{"className":32556,"style":32557},[353],"height:0.3173em;",[40,32559,32561,32564],{"style":32560},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[40,32562],{"className":32563,"style":3831},[361],[40,32565,32567],{"className":32566},[366,3835,1238,369],[40,32568,66],{"className":32569},[102,369],[40,32571,377],{"className":32572},[376],[40,32574,32576],{"className":32575},[349],[40,32577,32579],{"className":32578,"style":4479},[353],[40,32580],{},[40,32582,32584,32587],{"className":32583},[113,369],[40,32585,207],{"className":32586},[113,369],[40,32588,32590],{"className":32589},[340],[40,32591,32593],{"className":32592},[344],[40,32594,32596],{"className":32595},[349],[40,32597,32600],{"className":32598,"style":32599},[353],"height:0.8913em;",[40,32601,32603,32606],{"style":32602},"top:-2.931em;margin-right:0.0714em;",[40,32604],{"className":32605,"style":3831},[361],[40,32607,32609],{"className":32608},[366,3835,1238,369],[40,32610,1044],{"className":32611},[102,369],[40,32613,377],{"className":32614},[376],[40,32616,32618],{"className":32617},[349],[40,32619,32621],{"className":32620,"style":1226},[353],[40,32622],{},[40,32624],{"className":32625},[113,1162],[40,32627,32629],{"className":32628,"style":413},[113,412],[40,32630,207],{"className":32631},[417,2195],"\nwhere:",[9743,32634,32635,32718,32801],{},[9746,32636,32637,32717],{},[40,32638,32640,32662],{"className":32639},[43],[40,32641,32643],{"className":32642},[47],[49,32644,32645],{"xmlns":51},[53,32646,32647,32659],{},[56,32648,32649,32651,32657],{},[59,32650,167],{"stretchy":61},[151,32652,32653,32655],{},[154,32654,8234],{},[64,32656,66],{},[59,32658,207],{"stretchy":61},[82,32660,32661],{"encoding":84},"( A_0 )",[40,32663,32665],{"className":32664,"ariaHidden":72},[89],[40,32666,32668,32671,32674,32714],{"className":32667},[93],[40,32669],{"className":32670,"style":98},[97],[40,32672,167],{"className":32673},[106],[40,32675,32677,32680],{"className":32676},[102],[40,32678,8234],{"className":32679},[102,335],[40,32681,32683],{"className":32682},[340],[40,32684,32686,32706],{"className":32685},[344,345],[40,32687,32689,32703],{"className":32688},[349],[40,32690,32692],{"className":32691,"style":7201},[353],[40,32693,32694,32697],{"style":6098},[40,32695],{"className":32696,"style":362},[361],[40,32698,32700],{"className":32699},[366,367,368,369],[40,32701,66],{"className":32702},[102,369],[40,32704,377],{"className":32705},[376],[40,32707,32709],{"className":32708},[349],[40,32710,32712],{"className":32711,"style":384},[353],[40,32713],{},[40,32715,207],{"className":32716},[113]," is the peak amplitude,",[9746,32719,32720,32800],{},[40,32721,32723,32745],{"className":32722},[43],[40,32724,32726],{"className":32725},[47],[49,32727,32728],{"xmlns":51},[53,32729,32730,32742],{},[56,32731,32732,32734,32740],{},[59,32733,167],{"stretchy":61},[151,32735,32736,32738],{},[154,32737,20277],{},[64,32739,66],{},[59,32741,207],{"stretchy":61},[82,32743,32744],{"encoding":84},"( t_0 )",[40,32746,32748],{"className":32747,"ariaHidden":72},[89],[40,32749,32751,32754,32757,32797],{"className":32750},[93],[40,32752],{"className":32753,"style":98},[97],[40,32755,167],{"className":32756},[106],[40,32758,32760,32763],{"className":32759},[102],[40,32761,20277],{"className":32762},[102,335],[40,32764,32766],{"className":32765},[340],[40,32767,32769,32789],{"className":32768},[344,345],[40,32770,32772,32786],{"className":32771},[349],[40,32773,32775],{"className":32774,"style":7201},[353],[40,32776,32777,32780],{"style":6098},[40,32778],{"className":32779,"style":362},[361],[40,32781,32783],{"className":32782},[366,367,368,369],[40,32784,66],{"className":32785},[102,369],[40,32787,377],{"className":32788},[376],[40,32790,32792],{"className":32791},[349],[40,32793,32795],{"className":32794,"style":384},[353],[40,32796],{},[40,32798,207],{"className":32799},[113]," is the pulse center,",[9746,32802,32803,32842],{},[40,32804,32806,32824],{"className":32805},[43],[40,32807,32809],{"className":32808},[47],[49,32810,32811],{"xmlns":51},[53,32812,32813,32821],{},[56,32814,32815,32817,32819],{},[59,32816,167],{"stretchy":61},[154,32818,156],{},[59,32820,207],{"stretchy":61},[82,32822,32823],{"encoding":84},"( \\sigma )",[40,32825,32827],{"className":32826,"ariaHidden":72},[89],[40,32828,32830,32833,32836,32839],{"className":32829},[93],[40,32831],{"className":32832,"style":98},[97],[40,32834,167],{"className":32835},[106],[40,32837,156],{"className":32838,"style":336},[102,335],[40,32840,207],{"className":32841},[113]," determines the width of the pulse.",[21676,32844,32846],{"id":32845},"_2-modulated-pulse-for-pauli-x-gate",[35,32847,32848],{},"2. Modulated Pulse for Pauli-X Gate",[12,32850,8707,32851,32854,32855,10639,32858,32632],{},[35,32852,32853],{},"Pauli-X gate"," is implemented with a ",[35,32856,32857],{},"zero-phase cosine modulation",[40,32859,32861,32966],{"className":32860},[43],[40,32862,32864],{"className":32863},[47],[49,32865,32866],{"xmlns":51},[53,32867,32868,32963],{},[56,32869,32870,32877,32879,32881,32883,32885,32891,32893,32895,32935,32937,32939,32941,32943,32945,32951,32953,32955,32961],{},[151,32871,32872,32875],{},[154,32873,32874],{},"E",[154,32876,4196],{},[59,32878,167],{"stretchy":61},[154,32880,20277],{},[59,32882,207],{"stretchy":61},[59,32884,162],{},[151,32886,32887,32889],{},[154,32888,8234],{},[64,32890,66],{},[154,32892,5268],{},[59,32894,1084],{},[56,32896,32897,32899,32901,32933],{},[59,32898,167],{"fence":72},[59,32900,243],{},[1038,32902,32903,32923],{},[56,32904,32905,32907,32909,32911,32917],{},[59,32906,167],{"stretchy":61},[154,32908,20277],{},[59,32910,243],{},[151,32912,32913,32915],{},[154,32914,20277],{},[64,32916,66],{},[941,32918,32919,32921],{},[59,32920,207],{"stretchy":61},[64,32922,1044],{},[56,32924,32925,32927],{},[64,32926,1044],{},[941,32928,32929,32931],{},[154,32930,156],{},[64,32932,1044],{},[59,32934,207],{"fence":72},[154,32936,1820],{},[59,32938,1084],{},[59,32940,167],{"stretchy":61},[64,32942,1044],{},[154,32944,1661],{},[151,32946,32947,32949],{},[154,32948,13262],{},[154,32950,20274],{},[154,32952,20277],{},[59,32954,1053],{},[151,32956,32957,32959],{},[154,32958,1671],{},[154,32960,4196],{},[59,32962,207],{"stretchy":61},[82,32964,32965],{"encoding":84},"E_X(t) = A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right) \\cos(2\\pi f_c t + \\phi_X)",[40,32967,32969,33034,33348],{"className":32968,"ariaHidden":72},[89],[40,32970,32972,32975,33016,33019,33022,33025,33028,33031],{"className":32971},[93],[40,32973],{"className":32974,"style":98},[97],[40,32976,32978,32981],{"className":32977},[102],[40,32979,32874],{"className":32980,"style":2383},[102,335],[40,32982,32984],{"className":32983},[340],[40,32985,32987,33008],{"className":32986},[344,345],[40,32988,32990,33005],{"className":32989},[349],[40,32991,32993],{"className":32992,"style":20224},[353],[40,32994,32996,32999],{"style":32995},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[40,32997],{"className":32998,"style":362},[361],[40,33000,33002],{"className":33001},[366,367,368,369],[40,33003,4196],{"className":33004,"style":1242},[102,335,369],[40,33006,377],{"className":33007},[376],[40,33009,33011],{"className":33010},[349],[40,33012,33014],{"className":33013,"style":384},[353],[40,33015],{},[40,33017,167],{"className":33018},[106],[40,33020,20277],{"className":33021},[102,335],[40,33023,207],{"className":33024},[113],[40,33026],{"className":33027,"style":390},[121],[40,33029,162],{"className":33030},[394],[40,33032],{"className":33033,"style":390},[121],[40,33035,33037,33040,33080,33083,33086,33089,33279,33282,33285,33288,33291,33294,33336,33339,33342,33345],{"className":33036},[93],[40,33038],{"className":33039,"style":2120},[97],[40,33041,33043,33046],{"className":33042},[102],[40,33044,8234],{"className":33045},[102,335],[40,33047,33049],{"className":33048},[340],[40,33050,33052,33072],{"className":33051},[344,345],[40,33053,33055,33069],{"className":33054},[349],[40,33056,33058],{"className":33057,"style":7201},[353],[40,33059,33060,33063],{"style":6098},[40,33061],{"className":33062,"style":362},[361],[40,33064,33066],{"className":33065},[366,367,368,369],[40,33067,66],{"className":33068},[102,369],[40,33070,377],{"className":33071},[376],[40,33073,33075],{"className":33074},[349],[40,33076,33078],{"className":33077,"style":384},[353],[40,33079],{},[40,33081],{"className":33082,"style":122},[121],[40,33084,5268],{"className":33085},[1369],[40,33087],{"className":33088,"style":122},[121],[40,33090,33092,33098,33101,33273],{"className":33091},[408],[40,33093,33095],{"className":33094,"style":413},[106,412],[40,33096,167],{"className":33097},[417,2195],[40,33099,243],{"className":33100},[102],[40,33102,33104,33107,33270],{"className":33103},[102],[40,33105],{"className":33106},[106,1162],[40,33108,33110],{"className":33109},[1038],[40,33111,33113,33262],{"className":33112},[344,345],[40,33114,33116,33259],{"className":33115},[349],[40,33117,33119,33162,33170],{"className":33118,"style":32465},[353],[40,33120,33121,33124],{"style":1178},[40,33122],{"className":33123,"style":444},[361],[40,33125,33127],{"className":33126},[366,367,368,369],[40,33128,33130,33133],{"className":33129},[102,369],[40,33131,1044],{"className":33132},[102,369],[40,33134,33136,33139],{"className":33135},[102,369],[40,33137,156],{"className":33138,"style":336},[102,335,369],[40,33140,33142],{"className":33141},[340],[40,33143,33145],{"className":33144},[344],[40,33146,33148],{"className":33147},[349],[40,33149,33151],{"className":33150,"style":32498},[353],[40,33152,33153,33156],{"style":3827},[40,33154],{"className":33155,"style":3831},[361],[40,33157,33159],{"className":33158},[366,3835,1238,369],[40,33160,1044],{"className":33161},[102,369],[40,33163,33164,33167],{"style":1193},[40,33165],{"className":33166,"style":444},[361],[40,33168],{"className":33169,"style":1201},[1200],[40,33171,33172,33175],{"style":32520},[40,33173],{"className":33174,"style":444},[361],[40,33176,33178],{"className":33177},[366,367,368,369],[40,33179,33181,33184,33187,33190,33230],{"className":33180},[102,369],[40,33182,167],{"className":33183},[106,369],[40,33185,20277],{"className":33186},[102,335,369],[40,33188,243],{"className":33189},[1250,369],[40,33191,33193,33196],{"className":33192},[102,369],[40,33194,20277],{"className":33195},[102,335,369],[40,33197,33199],{"className":33198},[340],[40,33200,33202,33222],{"className":33201},[344,345],[40,33203,33205,33219],{"className":33204},[349],[40,33206,33208],{"className":33207,"style":32557},[353],[40,33209,33210,33213],{"style":32560},[40,33211],{"className":33212,"style":3831},[361],[40,33214,33216],{"className":33215},[366,3835,1238,369],[40,33217,66],{"className":33218},[102,369],[40,33220,377],{"className":33221},[376],[40,33223,33225],{"className":33224},[349],[40,33226,33228],{"className":33227,"style":4479},[353],[40,33229],{},[40,33231,33233,33236],{"className":33232},[113,369],[40,33234,207],{"className":33235},[113,369],[40,33237,33239],{"className":33238},[340],[40,33240,33242],{"className":33241},[344],[40,33243,33245],{"className":33244},[349],[40,33246,33248],{"className":33247,"style":32599},[353],[40,33249,33250,33253],{"style":32602},[40,33251],{"className":33252,"style":3831},[361],[40,33254,33256],{"className":33255},[366,3835,1238,369],[40,33257,1044],{"className":33258},[102,369],[40,33260,377],{"className":33261},[376],[40,33263,33265],{"className":33264},[349],[40,33266,33268],{"className":33267,"style":1226},[353],[40,33269],{},[40,33271],{"className":33272},[113,1162],[40,33274,33276],{"className":33275,"style":413},[113,412],[40,33277,207],{"className":33278},[417,2195],[40,33280],{"className":33281,"style":122},[121],[40,33283,1820],{"className":33284},[1369],[40,33286,167],{"className":33287},[106],[40,33289,1044],{"className":33290},[102],[40,33292,1661],{"className":33293,"style":336},[102,335],[40,33295,33297,33301],{"className":33296},[102],[40,33298,13262],{"className":33299,"style":33300},[102,335],"margin-right:0.1076em;",[40,33302,33304],{"className":33303},[340],[40,33305,33307,33328],{"className":33306},[344,345],[40,33308,33310,33325],{"className":33309},[349],[40,33311,33313],{"className":33312,"style":354},[353],[40,33314,33316,33319],{"style":33315},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[40,33317],{"className":33318,"style":362},[361],[40,33320,33322],{"className":33321},[366,367,368,369],[40,33323,20274],{"className":33324},[102,335,369],[40,33326,377],{"className":33327},[376],[40,33329,33331],{"className":33330},[349],[40,33332,33334],{"className":33333,"style":384},[353],[40,33335],{},[40,33337,20277],{"className":33338},[102,335],[40,33340],{"className":33341,"style":1246},[121],[40,33343,1053],{"className":33344},[1250],[40,33346],{"className":33347,"style":1246},[121],[40,33349,33351,33354,33394],{"className":33350},[93],[40,33352],{"className":33353,"style":98},[97],[40,33355,33357,33360],{"className":33356},[102],[40,33358,1671],{"className":33359},[102,335],[40,33361,33363],{"className":33362},[340],[40,33364,33366,33386],{"className":33365},[344,345],[40,33367,33369,33383],{"className":33368},[349],[40,33370,33372],{"className":33371,"style":20224},[353],[40,33373,33374,33377],{"style":6098},[40,33375],{"className":33376,"style":362},[361],[40,33378,33380],{"className":33379},[366,367,368,369],[40,33381,4196],{"className":33382,"style":1242},[102,335,369],[40,33384,377],{"className":33385},[376],[40,33387,33389],{"className":33388},[349],[40,33390,33392],{"className":33391,"style":384},[353],[40,33393],{},[40,33395,207],{"className":33396},[113],[9743,33398,33399,33482,33590],{},[9746,33400,33401,33481],{},[40,33402,33404,33426],{"className":33403},[43],[40,33405,33407],{"className":33406},[47],[49,33408,33409],{"xmlns":51},[53,33410,33411,33423],{},[56,33412,33413,33415,33421],{},[59,33414,167],{"stretchy":61},[151,33416,33417,33419],{},[154,33418,13262],{},[154,33420,20274],{},[59,33422,207],{"stretchy":61},[82,33424,33425],{"encoding":84},"( f_c )",[40,33427,33429],{"className":33428,"ariaHidden":72},[89],[40,33430,33432,33435,33438,33478],{"className":33431},[93],[40,33433],{"className":33434,"style":98},[97],[40,33436,167],{"className":33437},[106],[40,33439,33441,33444],{"className":33440},[102],[40,33442,13262],{"className":33443,"style":33300},[102,335],[40,33445,33447],{"className":33446},[340],[40,33448,33450,33470],{"className":33449},[344,345],[40,33451,33453,33467],{"className":33452},[349],[40,33454,33456],{"className":33455,"style":354},[353],[40,33457,33458,33461],{"style":33315},[40,33459],{"className":33460,"style":362},[361],[40,33462,33464],{"className":33463},[366,367,368,369],[40,33465,20274],{"className":33466},[102,335,369],[40,33468,377],{"className":33469},[376],[40,33471,33473],{"className":33472},[349],[40,33474,33476],{"className":33475,"style":384},[353],[40,33477],{},[40,33479,207],{"className":33480},[113]," is the carrier frequency (typically GHz range),",[9746,33483,33484,33586,33587,1123],{},[40,33485,33487,33513],{"className":33486},[43],[40,33488,33490],{"className":33489},[47],[49,33491,33492],{"xmlns":51},[53,33493,33494,33510],{},[56,33495,33496,33498,33504,33506,33508],{},[59,33497,167],{"stretchy":61},[151,33499,33500,33502],{},[154,33501,1671],{},[154,33503,4196],{},[59,33505,162],{},[64,33507,66],{},[59,33509,207],{"stretchy":61},[82,33511,33512],{"encoding":84},"( \\phi_X = 0 )",[40,33514,33516,33574],{"className":33515,"ariaHidden":72},[89],[40,33517,33519,33522,33525,33565,33568,33571],{"className":33518},[93],[40,33520],{"className":33521,"style":98},[97],[40,33523,167],{"className":33524},[106],[40,33526,33528,33531],{"className":33527},[102],[40,33529,1671],{"className":33530},[102,335],[40,33532,33534],{"className":33533},[340],[40,33535,33537,33557],{"className":33536},[344,345],[40,33538,33540,33554],{"className":33539},[349],[40,33541,33543],{"className":33542,"style":20224},[353],[40,33544,33545,33548],{"style":6098},[40,33546],{"className":33547,"style":362},[361],[40,33549,33551],{"className":33550},[366,367,368,369],[40,33552,4196],{"className":33553,"style":1242},[102,335,369],[40,33555,377],{"className":33556},[376],[40,33558,33560],{"className":33559},[349],[40,33561,33563],{"className":33562,"style":384},[353],[40,33564],{},[40,33566],{"className":33567,"style":390},[121],[40,33569,162],{"className":33570},[394],[40,33572],{"className":33573,"style":390},[121],[40,33575,33577,33580,33583],{"className":33576},[93],[40,33578],{"className":33579,"style":98},[97],[40,33581,66],{"className":33582},[102],[40,33584,207],{"className":33585},[113]," for an ",[35,33588,33589],{},"X gate",[9746,33591,33592,33593,33596],{},"The modulation is centered at the qubit frequency with ",[35,33594,33595],{},"zero phase shift",", leading to a cosine-like oscillation.",[21676,33598,33600],{"id":33599},"_3-modulated-pulse-for-pauli-y-gate",[35,33601,33602],{},"3. Modulated Pulse for Pauli-Y Gate",[12,33604,8707,33605,32854,33608,10639,33658,32632],{},[35,33606,33607],{},"Pauli-Y gate",[35,33609,33610,33657],{},[40,33611,33613,33635],{"className":33612},[43],[40,33614,33616],{"className":33615},[47],[49,33617,33618],{"xmlns":51},[53,33619,33620,33632],{},[56,33621,33622,33624,33626,33628,33630],{},[59,33623,167],{"stretchy":61},[154,33625,1661],{},[154,33627,11243],{"mathvariant":1080},[64,33629,1044],{},[59,33631,207],{"stretchy":61},[82,33633,33634],{"encoding":84},"(\\pi\u002F2)",[40,33636,33638],{"className":33637,"ariaHidden":72},[89],[40,33639,33641,33644,33647,33650,33654],{"className":33640},[93],[40,33642],{"className":33643,"style":98},[97],[40,33645,167],{"className":33646},[106],[40,33648,1661],{"className":33649,"style":336},[102,335],[40,33651,33653],{"className":33652},[102],"\u002F2",[40,33655,207],{"className":33656},[113]," phase-shifted cosine modulation",[40,33659,33661,33765],{"className":33660},[43],[40,33662,33664],{"className":33663},[47],[49,33665,33666],{"xmlns":51},[53,33667,33668,33762],{},[56,33669,33670,33676,33678,33680,33682,33684,33690,33692,33694,33734,33736,33738,33740,33742,33744,33750,33752,33754,33760],{},[151,33671,33672,33674],{},[154,33673,32874],{},[154,33675,4230],{},[59,33677,167],{"stretchy":61},[154,33679,20277],{},[59,33681,207],{"stretchy":61},[59,33683,162],{},[151,33685,33686,33688],{},[154,33687,8234],{},[64,33689,66],{},[154,33691,5268],{},[59,33693,1084],{},[56,33695,33696,33698,33700,33732],{},[59,33697,167],{"fence":72},[59,33699,243],{},[1038,33701,33702,33722],{},[56,33703,33704,33706,33708,33710,33716],{},[59,33705,167],{"stretchy":61},[154,33707,20277],{},[59,33709,243],{},[151,33711,33712,33714],{},[154,33713,20277],{},[64,33715,66],{},[941,33717,33718,33720],{},[59,33719,207],{"stretchy":61},[64,33721,1044],{},[56,33723,33724,33726],{},[64,33725,1044],{},[941,33727,33728,33730],{},[154,33729,156],{},[64,33731,1044],{},[59,33733,207],{"fence":72},[154,33735,1820],{},[59,33737,1084],{},[59,33739,167],{"stretchy":61},[64,33741,1044],{},[154,33743,1661],{},[151,33745,33746,33748],{},[154,33747,13262],{},[154,33749,20274],{},[154,33751,20277],{},[59,33753,1053],{},[151,33755,33756,33758],{},[154,33757,1671],{},[154,33759,4230],{},[59,33761,207],{"stretchy":61},[82,33763,33764],{"encoding":84},"E_Y(t) = A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right) \\cos(2\\pi f_c t + \\phi_Y)",[40,33766,33768,33832,34144],{"className":33767,"ariaHidden":72},[89],[40,33769,33771,33774,33814,33817,33820,33823,33826,33829],{"className":33770},[93],[40,33772],{"className":33773,"style":98},[97],[40,33775,33777,33780],{"className":33776},[102],[40,33778,32874],{"className":33779,"style":2383},[102,335],[40,33781,33783],{"className":33782},[340],[40,33784,33786,33806],{"className":33785},[344,345],[40,33787,33789,33803],{"className":33788},[349],[40,33790,33792],{"className":33791,"style":20224},[353],[40,33793,33794,33797],{"style":32995},[40,33795],{"className":33796,"style":362},[361],[40,33798,33800],{"className":33799},[366,367,368,369],[40,33801,4230],{"className":33802,"style":1246},[102,335,369],[40,33804,377],{"className":33805},[376],[40,33807,33809],{"className":33808},[349],[40,33810,33812],{"className":33811,"style":384},[353],[40,33813],{},[40,33815,167],{"className":33816},[106],[40,33818,20277],{"className":33819},[102,335],[40,33821,207],{"className":33822},[113],[40,33824],{"className":33825,"style":390},[121],[40,33827,162],{"className":33828},[394],[40,33830],{"className":33831,"style":390},[121],[40,33833,33835,33838,33878,33881,33884,33887,34077,34080,34083,34086,34089,34092,34132,34135,34138,34141],{"className":33834},[93],[40,33836],{"className":33837,"style":2120},[97],[40,33839,33841,33844],{"className":33840},[102],[40,33842,8234],{"className":33843},[102,335],[40,33845,33847],{"className":33846},[340],[40,33848,33850,33870],{"className":33849},[344,345],[40,33851,33853,33867],{"className":33852},[349],[40,33854,33856],{"className":33855,"style":7201},[353],[40,33857,33858,33861],{"style":6098},[40,33859],{"className":33860,"style":362},[361],[40,33862,33864],{"className":33863},[366,367,368,369],[40,33865,66],{"className":33866},[102,369],[40,33868,377],{"className":33869},[376],[40,33871,33873],{"className":33872},[349],[40,33874,33876],{"className":33875,"style":384},[353],[40,33877],{},[40,33879],{"className":33880,"style":122},[121],[40,33882,5268],{"className":33883},[1369],[40,33885],{"className":33886,"style":122},[121],[40,33888,33890,33896,33899,34071],{"className":33889},[408],[40,33891,33893],{"className":33892,"style":413},[106,412],[40,33894,167],{"className":33895},[417,2195],[40,33897,243],{"className":33898},[102],[40,33900,33902,33905,34068],{"className":33901},[102],[40,33903],{"className":33904},[106,1162],[40,33906,33908],{"className":33907},[1038],[40,33909,33911,34060],{"className":33910},[344,345],[40,33912,33914,34057],{"className":33913},[349],[40,33915,33917,33960,33968],{"className":33916,"style":32465},[353],[40,33918,33919,33922],{"style":1178},[40,33920],{"className":33921,"style":444},[361],[40,33923,33925],{"className":33924},[366,367,368,369],[40,33926,33928,33931],{"className":33927},[102,369],[40,33929,1044],{"className":33930},[102,369],[40,33932,33934,33937],{"className":33933},[102,369],[40,33935,156],{"className":33936,"style":336},[102,335,369],[40,33938,33940],{"className":33939},[340],[40,33941,33943],{"className":33942},[344],[40,33944,33946],{"className":33945},[349],[40,33947,33949],{"className":33948,"style":32498},[353],[40,33950,33951,33954],{"style":3827},[40,33952],{"className":33953,"style":3831},[361],[40,33955,33957],{"className":33956},[366,3835,1238,369],[40,33958,1044],{"className":33959},[102,369],[40,33961,33962,33965],{"style":1193},[40,33963],{"className":33964,"style":444},[361],[40,33966],{"className":33967,"style":1201},[1200],[40,33969,33970,33973],{"style":32520},[40,33971],{"className":33972,"style":444},[361],[40,33974,33976],{"className":33975},[366,367,368,369],[40,33977,33979,33982,33985,33988,34028],{"className":33978},[102,369],[40,33980,167],{"className":33981},[106,369],[40,33983,20277],{"className":33984},[102,335,369],[40,33986,243],{"className":33987},[1250,369],[40,33989,33991,33994],{"className":33990},[102,369],[40,33992,20277],{"className":33993},[102,335,369],[40,33995,33997],{"className":33996},[340],[40,33998,34000,34020],{"className":33999},[344,345],[40,34001,34003,34017],{"className":34002},[349],[40,34004,34006],{"className":34005,"style":32557},[353],[40,34007,34008,34011],{"style":32560},[40,34009],{"className":34010,"style":3831},[361],[40,34012,34014],{"className":34013},[366,3835,1238,369],[40,34015,66],{"className":34016},[102,369],[40,34018,377],{"className":34019},[376],[40,34021,34023],{"className":34022},[349],[40,34024,34026],{"className":34025,"style":4479},[353],[40,34027],{},[40,34029,34031,34034],{"className":34030},[113,369],[40,34032,207],{"className":34033},[113,369],[40,34035,34037],{"className":34036},[340],[40,34038,34040],{"className":34039},[344],[40,34041,34043],{"className":34042},[349],[40,34044,34046],{"className":34045,"style":32599},[353],[40,34047,34048,34051],{"style":32602},[40,34049],{"className":34050,"style":3831},[361],[40,34052,34054],{"className":34053},[366,3835,1238,369],[40,34055,1044],{"className":34056},[102,369],[40,34058,377],{"className":34059},[376],[40,34061,34063],{"className":34062},[349],[40,34064,34066],{"className":34065,"style":1226},[353],[40,34067],{},[40,34069],{"className":34070},[113,1162],[40,34072,34074],{"className":34073,"style":413},[113,412],[40,34075,207],{"className":34076},[417,2195],[40,34078],{"className":34079,"style":122},[121],[40,34081,1820],{"className":34082},[1369],[40,34084,167],{"className":34085},[106],[40,34087,1044],{"className":34088},[102],[40,34090,1661],{"className":34091,"style":336},[102,335],[40,34093,34095,34098],{"className":34094},[102],[40,34096,13262],{"className":34097,"style":33300},[102,335],[40,34099,34101],{"className":34100},[340],[40,34102,34104,34124],{"className":34103},[344,345],[40,34105,34107,34121],{"className":34106},[349],[40,34108,34110],{"className":34109,"style":354},[353],[40,34111,34112,34115],{"style":33315},[40,34113],{"className":34114,"style":362},[361],[40,34116,34118],{"className":34117},[366,367,368,369],[40,34119,20274],{"className":34120},[102,335,369],[40,34122,377],{"className":34123},[376],[40,34125,34127],{"className":34126},[349],[40,34128,34130],{"className":34129,"style":384},[353],[40,34131],{},[40,34133,20277],{"className":34134},[102,335],[40,34136],{"className":34137,"style":1246},[121],[40,34139,1053],{"className":34140},[1250],[40,34142],{"className":34143,"style":1246},[121],[40,34145,34147,34150,34190],{"className":34146},[93],[40,34148],{"className":34149,"style":98},[97],[40,34151,34153,34156],{"className":34152},[102],[40,34154,1671],{"className":34155},[102,335],[40,34157,34159],{"className":34158},[340],[40,34160,34162,34182],{"className":34161},[344,345],[40,34163,34165,34179],{"className":34164},[349],[40,34166,34168],{"className":34167,"style":20224},[353],[40,34169,34170,34173],{"style":6098},[40,34171],{"className":34172,"style":362},[361],[40,34174,34176],{"className":34175},[366,367,368,369],[40,34177,4230],{"className":34178,"style":1246},[102,335,369],[40,34180,377],{"className":34181},[376],[40,34183,34185],{"className":34184},[349],[40,34186,34188],{"className":34187,"style":384},[353],[40,34189],{},[40,34191,207],{"className":34192},[113],[9743,34194,34195,34370],{},[9746,34196,34197,34369],{},[40,34198,34200,34230],{"className":34199},[43],[40,34201,34203],{"className":34202},[47],[49,34204,34205],{"xmlns":51},[53,34206,34207,34227],{},[56,34208,34209,34211,34217,34219,34225],{},[59,34210,167],{"stretchy":61},[151,34212,34213,34215],{},[154,34214,1671],{},[154,34216,4230],{},[59,34218,162],{},[1038,34220,34221,34223],{},[154,34222,1661],{},[64,34224,1044],{},[59,34226,207],{"stretchy":61},[82,34228,34229],{"encoding":84},"( \\phi_Y = \\frac{\\pi}{2} )",[40,34231,34233,34291],{"className":34232,"ariaHidden":72},[89],[40,34234,34236,34239,34242,34282,34285,34288],{"className":34235},[93],[40,34237],{"className":34238,"style":98},[97],[40,34240,167],{"className":34241},[106],[40,34243,34245,34248],{"className":34244},[102],[40,34246,1671],{"className":34247},[102,335],[40,34249,34251],{"className":34250},[340],[40,34252,34254,34274],{"className":34253},[344,345],[40,34255,34257,34271],{"className":34256},[349],[40,34258,34260],{"className":34259,"style":20224},[353],[40,34261,34262,34265],{"style":6098},[40,34263],{"className":34264,"style":362},[361],[40,34266,34268],{"className":34267},[366,367,368,369],[40,34269,4230],{"className":34270,"style":1246},[102,335,369],[40,34272,377],{"className":34273},[376],[40,34275,34277],{"className":34276},[349],[40,34278,34280],{"className":34279,"style":384},[353],[40,34281],{},[40,34283],{"className":34284,"style":390},[121],[40,34286,162],{"className":34287},[394],[40,34289],{"className":34290,"style":390},[121],[40,34292,34294,34298,34366],{"className":34293},[93],[40,34295],{"className":34296,"style":34297},[97],"height:1.095em;vertical-align:-0.345em;",[40,34299,34301,34304,34363],{"className":34300},[102],[40,34302],{"className":34303},[106,1162],[40,34305,34307],{"className":34306},[1038],[40,34308,34310,34355],{"className":34309},[344,345],[40,34311,34313,34352],{"className":34312},[349],[40,34314,34316,34330,34338],{"className":34315,"style":5436},[353],[40,34317,34318,34321],{"style":1178},[40,34319],{"className":34320,"style":444},[361],[40,34322,34324],{"className":34323},[366,367,368,369],[40,34325,34327],{"className":34326},[102,369],[40,34328,1044],{"className":34329},[102,369],[40,34331,34332,34335],{"style":1193},[40,34333],{"className":34334,"style":444},[361],[40,34336],{"className":34337,"style":1201},[1200],[40,34339,34340,34343],{"style":1204},[40,34341],{"className":34342,"style":444},[361],[40,34344,34346],{"className":34345},[366,367,368,369],[40,34347,34349],{"className":34348},[102,369],[40,34350,1661],{"className":34351,"style":336},[102,335,369],[40,34353,377],{"className":34354},[376],[40,34356,34358],{"className":34357},[349],[40,34359,34361],{"className":34360,"style":1226},[353],[40,34362],{},[40,34364],{"className":34365},[113,1162],[40,34367,207],{"className":34368},[113]," shifts the phase by 90° (producing a sine-like waveform).",[9746,34371,34372,34373,34422],{},"The modulation has a ",[35,34374,34375,34421],{},[40,34376,34378,34400],{"className":34377},[43],[40,34379,34381],{"className":34380},[47],[49,34382,34383],{"xmlns":51},[53,34384,34385,34397],{},[56,34386,34387,34389,34391,34393,34395],{},[59,34388,167],{"stretchy":61},[154,34390,1661],{},[154,34392,11243],{"mathvariant":1080},[64,34394,1044],{},[59,34396,207],{"stretchy":61},[82,34398,34399],{"encoding":84},"( \\pi\u002F2 )",[40,34401,34403],{"className":34402,"ariaHidden":72},[89],[40,34404,34406,34409,34412,34415,34418],{"className":34405},[93],[40,34407],{"className":34408,"style":98},[97],[40,34410,167],{"className":34411},[106],[40,34413,1661],{"className":34414,"style":336},[102,335],[40,34416,33653],{"className":34417},[102],[40,34419,207],{"className":34420},[113]," phase shift",", which shifts the waveform by a quarter cycle, making it a sine-like oscillation.",[21676,34424,34426],{"id":34425},"_4-why-no-pulse-for-pauli-z-gate",[35,34427,34428],{},"4. Why No Pulse for Pauli-Z Gate?",[12,34430,34431,34432,34474,34475,10639,34478,34645],{},"For ",[35,34433,34434,34435],{},"Pauli-Z ",[40,34436,34438,34456],{"className":34437},[43],[40,34439,34441],{"className":34440},[47],[49,34442,34443],{"xmlns":51},[53,34444,34445,34453],{},[56,34446,34447,34449,34451],{},[59,34448,167],{"stretchy":61},[154,34450,4263],{},[59,34452,207],{"stretchy":61},[82,34454,34455],{"encoding":84},"( Z )",[40,34457,34459],{"className":34458,"ariaHidden":72},[89],[40,34460,34462,34465,34468,34471],{"className":34461},[93],[40,34463],{"className":34464,"style":98},[97],[40,34466,167],{"className":34467},[106],[40,34469,4263],{"className":34470,"style":4668},[102,335],[40,34472,207],{"className":34473},[113],", no physical pulse is needed because it is implemented as a ",[35,34476,34477],{},"virtual phase shift",[40,34479,34481,34535],{"className":34480},[43],[40,34482,34484],{"className":34483},[47],[49,34485,34486],{"xmlns":51},[53,34487,34488,34532],{},[56,34489,34490,34492,34494,34496,34498,34500,34502,34504,34506,34508,34510,34512,34514,34516,34526,34528,34530],{},[154,34491,62],{"mathvariant":1080},[64,34493,66],{},[59,34495,69],{"stretchy":61},[59,34497,5620],{},[154,34499,62],{"mathvariant":1080},[64,34501,66],{},[59,34503,69],{"stretchy":61},[59,34505,73],{"separator":72},[121,34507],{"width":173},[154,34509,62],{"mathvariant":1080},[64,34511,78],{},[59,34513,69],{"stretchy":61},[59,34515,5620],{},[941,34517,34518,34520],{},[154,34519,4203],{},[56,34521,34522,34524],{},[154,34523,246],{},[154,34525,1649],{},[154,34527,62],{"mathvariant":1080},[64,34529,78],{},[59,34531,69],{"stretchy":61},[82,34533,34534],{"encoding":84},"|0\\rangle \\rightarrow |0\\rangle, \\quad |1\\rangle \\rightarrow e^{i\\theta}|1\\rangle",[40,34536,34538,34560,34597],{"className":34537,"ariaHidden":72},[89],[40,34539,34541,34544,34548,34551,34554,34557],{"className":34540},[93],[40,34542],{"className":34543,"style":98},[97],[40,34545,34547],{"className":34546},[102],"∣0",[40,34549,69],{"className":34550},[113],[40,34552],{"className":34553,"style":390},[121],[40,34555,5620],{"className":34556},[394],[40,34558],{"className":34559,"style":390},[121],[40,34561,34563,34566,34569,34572,34575,34578,34581,34585,34588,34591,34594],{"className":34562},[93],[40,34564],{"className":34565,"style":98},[97],[40,34567,34547],{"className":34568},[102],[40,34570,69],{"className":34571},[113],[40,34573,73],{"className":34574},[117],[40,34576],{"className":34577,"style":543},[121],[40,34579],{"className":34580,"style":122},[121],[40,34582,34584],{"className":34583},[102],"∣1",[40,34586,69],{"className":34587},[113],[40,34589],{"className":34590,"style":390},[121],[40,34592,5620],{"className":34593},[394],[40,34595],{"className":34596,"style":390},[121],[40,34598,34600,34604,34639,34642],{"className":34599},[93],[40,34601],{"className":34602,"style":34603},[97],"height:1.0991em;vertical-align:-0.25em;",[40,34605,34607,34610],{"className":34606},[102],[40,34608,4203],{"className":34609},[102,335],[40,34611,34613],{"className":34612},[340],[40,34614,34616],{"className":34615},[344],[40,34617,34619],{"className":34618},[349],[40,34620,34622],{"className":34621,"style":9999},[353],[40,34623,34624,34627],{"style":1004},[40,34625],{"className":34626,"style":362},[361],[40,34628,34630],{"className":34629},[366,367,368,369],[40,34631,34633,34636],{"className":34632},[102,369],[40,34634,246],{"className":34635},[102,335,369],[40,34637,1649],{"className":34638,"style":1312},[102,335,369],[40,34640,34584],{"className":34641},[102],[40,34643,69],{"className":34644},[113],"\nThis shift is typically done in software rather than by applying a physical pulse.",[25,34647],{},[12,34649,34650,34651,32265],{},"Relevant equations in table form, for the ",[35,34652,32264],{},[20142,34654,34655,34665],{},[20145,34656,34657],{},[20148,34658,34659,34662],{},[20151,34660,34661],{},"Gate",[20151,34663,34664],{},"Equation",[20162,34666,34667,35025,35568,36111,36656],{},[20148,34668,34669,34674],{},[20167,34670,34671],{},[35,34672,34673],{},"Gaussian Envelope",[20167,34675,34676],{},[40,34677,34679,34750],{"className":34678},[43],[40,34680,34682],{"className":34681},[47],[49,34683,34684],{"xmlns":51},[53,34685,34686,34748],{},[56,34687,34688,34690,34692,34694,34696,34698,34704,34706,34708],{},[154,34689,8234],{},[59,34691,167],{"stretchy":61},[154,34693,20277],{},[59,34695,207],{"stretchy":61},[59,34697,162],{},[151,34699,34700,34702],{},[154,34701,8234],{},[64,34703,66],{},[154,34705,5268],{},[59,34707,1084],{},[56,34709,34710,34712,34714,34746],{},[59,34711,167],{"fence":72},[59,34713,243],{},[1038,34715,34716,34736],{},[56,34717,34718,34720,34722,34724,34730],{},[59,34719,167],{"stretchy":61},[154,34721,20277],{},[59,34723,243],{},[151,34725,34726,34728],{},[154,34727,20277],{},[64,34729,66],{},[941,34731,34732,34734],{},[59,34733,207],{"stretchy":61},[64,34735,1044],{},[56,34737,34738,34740],{},[64,34739,1044],{},[941,34741,34742,34744],{},[154,34743,156],{},[64,34745,1044],{},[59,34747,207],{"fence":72},[82,34749,32349],{"encoding":84},[40,34751,34753,34780],{"className":34752,"ariaHidden":72},[89],[40,34754,34756,34759,34762,34765,34768,34771,34774,34777],{"className":34755},[93],[40,34757],{"className":34758,"style":98},[97],[40,34760,8234],{"className":34761},[102,335],[40,34763,167],{"className":34764},[106],[40,34766,20277],{"className":34767},[102,335],[40,34769,207],{"className":34770},[113],[40,34772],{"className":34773,"style":390},[121],[40,34775,162],{"className":34776},[394],[40,34778],{"className":34779,"style":390},[121],[40,34781,34783,34786,34826,34829,34832,34835],{"className":34782},[93],[40,34784],{"className":34785,"style":2120},[97],[40,34787,34789,34792],{"className":34788},[102],[40,34790,8234],{"className":34791},[102,335],[40,34793,34795],{"className":34794},[340],[40,34796,34798,34818],{"className":34797},[344,345],[40,34799,34801,34815],{"className":34800},[349],[40,34802,34804],{"className":34803,"style":7201},[353],[40,34805,34806,34809],{"style":6098},[40,34807],{"className":34808,"style":362},[361],[40,34810,34812],{"className":34811},[366,367,368,369],[40,34813,66],{"className":34814},[102,369],[40,34816,377],{"className":34817},[376],[40,34819,34821],{"className":34820},[349],[40,34822,34824],{"className":34823,"style":384},[353],[40,34825],{},[40,34827],{"className":34828,"style":122},[121],[40,34830,5268],{"className":34831},[1369],[40,34833],{"className":34834,"style":122},[121],[40,34836,34838,34844,34847,35019],{"className":34837},[408],[40,34839,34841],{"className":34840,"style":413},[106,412],[40,34842,167],{"className":34843},[417,2195],[40,34845,243],{"className":34846},[102],[40,34848,34850,34853,35016],{"className":34849},[102],[40,34851],{"className":34852},[106,1162],[40,34854,34856],{"className":34855},[1038],[40,34857,34859,35008],{"className":34858},[344,345],[40,34860,34862,35005],{"className":34861},[349],[40,34863,34865,34908,34916],{"className":34864,"style":32465},[353],[40,34866,34867,34870],{"style":1178},[40,34868],{"className":34869,"style":444},[361],[40,34871,34873],{"className":34872},[366,367,368,369],[40,34874,34876,34879],{"className":34875},[102,369],[40,34877,1044],{"className":34878},[102,369],[40,34880,34882,34885],{"className":34881},[102,369],[40,34883,156],{"className":34884,"style":336},[102,335,369],[40,34886,34888],{"className":34887},[340],[40,34889,34891],{"className":34890},[344],[40,34892,34894],{"className":34893},[349],[40,34895,34897],{"className":34896,"style":32498},[353],[40,34898,34899,34902],{"style":3827},[40,34900],{"className":34901,"style":3831},[361],[40,34903,34905],{"className":34904},[366,3835,1238,369],[40,34906,1044],{"className":34907},[102,369],[40,34909,34910,34913],{"style":1193},[40,34911],{"className":34912,"style":444},[361],[40,34914],{"className":34915,"style":1201},[1200],[40,34917,34918,34921],{"style":32520},[40,34919],{"className":34920,"style":444},[361],[40,34922,34924],{"className":34923},[366,367,368,369],[40,34925,34927,34930,34933,34936,34976],{"className":34926},[102,369],[40,34928,167],{"className":34929},[106,369],[40,34931,20277],{"className":34932},[102,335,369],[40,34934,243],{"className":34935},[1250,369],[40,34937,34939,34942],{"className":34938},[102,369],[40,34940,20277],{"className":34941},[102,335,369],[40,34943,34945],{"className":34944},[340],[40,34946,34948,34968],{"className":34947},[344,345],[40,34949,34951,34965],{"className":34950},[349],[40,34952,34954],{"className":34953,"style":32557},[353],[40,34955,34956,34959],{"style":32560},[40,34957],{"className":34958,"style":3831},[361],[40,34960,34962],{"className":34961},[366,3835,1238,369],[40,34963,66],{"className":34964},[102,369],[40,34966,377],{"className":34967},[376],[40,34969,34971],{"className":34970},[349],[40,34972,34974],{"className":34973,"style":4479},[353],[40,34975],{},[40,34977,34979,34982],{"className":34978},[113,369],[40,34980,207],{"className":34981},[113,369],[40,34983,34985],{"className":34984},[340],[40,34986,34988],{"className":34987},[344],[40,34989,34991],{"className":34990},[349],[40,34992,34994],{"className":34993,"style":32599},[353],[40,34995,34996,34999],{"style":32602},[40,34997],{"className":34998,"style":3831},[361],[40,35000,35002],{"className":35001},[366,3835,1238,369],[40,35003,1044],{"className":35004},[102,369],[40,35006,377],{"className":35007},[376],[40,35009,35011],{"className":35010},[349],[40,35012,35014],{"className":35013,"style":1226},[353],[40,35015],{},[40,35017],{"className":35018},[113,1162],[40,35020,35022],{"className":35021,"style":413},[113,412],[40,35023,207],{"className":35024},[417,2195],[20148,35026,35027,35032],{},[20167,35028,35029],{},[35,35030,35031],{},"Pauli-X Gate",[20167,35033,35034],{},[40,35035,35037,35140],{"className":35036},[43],[40,35038,35040],{"className":35039},[47],[49,35041,35042],{"xmlns":51},[53,35043,35044,35138],{},[56,35045,35046,35052,35054,35056,35058,35060,35066,35068,35070,35110,35112,35114,35116,35118,35120,35126,35128,35130,35136],{},[151,35047,35048,35050],{},[154,35049,32874],{},[154,35051,4196],{},[59,35053,167],{"stretchy":61},[154,35055,20277],{},[59,35057,207],{"stretchy":61},[59,35059,162],{},[151,35061,35062,35064],{},[154,35063,8234],{},[64,35065,66],{},[154,35067,5268],{},[59,35069,1084],{},[56,35071,35072,35074,35076,35108],{},[59,35073,167],{"fence":72},[59,35075,243],{},[1038,35077,35078,35098],{},[56,35079,35080,35082,35084,35086,35092],{},[59,35081,167],{"stretchy":61},[154,35083,20277],{},[59,35085,243],{},[151,35087,35088,35090],{},[154,35089,20277],{},[64,35091,66],{},[941,35093,35094,35096],{},[59,35095,207],{"stretchy":61},[64,35097,1044],{},[56,35099,35100,35102],{},[64,35101,1044],{},[941,35103,35104,35106],{},[154,35105,156],{},[64,35107,1044],{},[59,35109,207],{"fence":72},[154,35111,1820],{},[59,35113,1084],{},[59,35115,167],{"stretchy":61},[64,35117,1044],{},[154,35119,1661],{},[151,35121,35122,35124],{},[154,35123,13262],{},[154,35125,20274],{},[154,35127,20277],{},[59,35129,1053],{},[151,35131,35132,35134],{},[154,35133,1671],{},[154,35135,4196],{},[59,35137,207],{"stretchy":61},[82,35139,32965],{"encoding":84},[40,35141,35143,35207,35519],{"className":35142,"ariaHidden":72},[89],[40,35144,35146,35149,35189,35192,35195,35198,35201,35204],{"className":35145},[93],[40,35147],{"className":35148,"style":98},[97],[40,35150,35152,35155],{"className":35151},[102],[40,35153,32874],{"className":35154,"style":2383},[102,335],[40,35156,35158],{"className":35157},[340],[40,35159,35161,35181],{"className":35160},[344,345],[40,35162,35164,35178],{"className":35163},[349],[40,35165,35167],{"className":35166,"style":20224},[353],[40,35168,35169,35172],{"style":32995},[40,35170],{"className":35171,"style":362},[361],[40,35173,35175],{"className":35174},[366,367,368,369],[40,35176,4196],{"className":35177,"style":1242},[102,335,369],[40,35179,377],{"className":35180},[376],[40,35182,35184],{"className":35183},[349],[40,35185,35187],{"className":35186,"style":384},[353],[40,35188],{},[40,35190,167],{"className":35191},[106],[40,35193,20277],{"className":35194},[102,335],[40,35196,207],{"className":35197},[113],[40,35199],{"className":35200,"style":390},[121],[40,35202,162],{"className":35203},[394],[40,35205],{"className":35206,"style":390},[121],[40,35208,35210,35213,35253,35256,35259,35262,35452,35455,35458,35461,35464,35467,35507,35510,35513,35516],{"className":35209},[93],[40,35211],{"className":35212,"style":2120},[97],[40,35214,35216,35219],{"className":35215},[102],[40,35217,8234],{"className":35218},[102,335],[40,35220,35222],{"className":35221},[340],[40,35223,35225,35245],{"className":35224},[344,345],[40,35226,35228,35242],{"className":35227},[349],[40,35229,35231],{"className":35230,"style":7201},[353],[40,35232,35233,35236],{"style":6098},[40,35234],{"className":35235,"style":362},[361],[40,35237,35239],{"className":35238},[366,367,368,369],[40,35240,66],{"className":35241},[102,369],[40,35243,377],{"className":35244},[376],[40,35246,35248],{"className":35247},[349],[40,35249,35251],{"className":35250,"style":384},[353],[40,35252],{},[40,35254],{"className":35255,"style":122},[121],[40,35257,5268],{"className":35258},[1369],[40,35260],{"className":35261,"style":122},[121],[40,35263,35265,35271,35274,35446],{"className":35264},[408],[40,35266,35268],{"className":35267,"style":413},[106,412],[40,35269,167],{"className":35270},[417,2195],[40,35272,243],{"className":35273},[102],[40,35275,35277,35280,35443],{"className":35276},[102],[40,35278],{"className":35279},[106,1162],[40,35281,35283],{"className":35282},[1038],[40,35284,35286,35435],{"className":35285},[344,345],[40,35287,35289,35432],{"className":35288},[349],[40,35290,35292,35335,35343],{"className":35291,"style":32465},[353],[40,35293,35294,35297],{"style":1178},[40,35295],{"className":35296,"style":444},[361],[40,35298,35300],{"className":35299},[366,367,368,369],[40,35301,35303,35306],{"className":35302},[102,369],[40,35304,1044],{"className":35305},[102,369],[40,35307,35309,35312],{"className":35308},[102,369],[40,35310,156],{"className":35311,"style":336},[102,335,369],[40,35313,35315],{"className":35314},[340],[40,35316,35318],{"className":35317},[344],[40,35319,35321],{"className":35320},[349],[40,35322,35324],{"className":35323,"style":32498},[353],[40,35325,35326,35329],{"style":3827},[40,35327],{"className":35328,"style":3831},[361],[40,35330,35332],{"className":35331},[366,3835,1238,369],[40,35333,1044],{"className":35334},[102,369],[40,35336,35337,35340],{"style":1193},[40,35338],{"className":35339,"style":444},[361],[40,35341],{"className":35342,"style":1201},[1200],[40,35344,35345,35348],{"style":32520},[40,35346],{"className":35347,"style":444},[361],[40,35349,35351],{"className":35350},[366,367,368,369],[40,35352,35354,35357,35360,35363,35403],{"className":35353},[102,369],[40,35355,167],{"className":35356},[106,369],[40,35358,20277],{"className":35359},[102,335,369],[40,35361,243],{"className":35362},[1250,369],[40,35364,35366,35369],{"className":35365},[102,369],[40,35367,20277],{"className":35368},[102,335,369],[40,35370,35372],{"className":35371},[340],[40,35373,35375,35395],{"className":35374},[344,345],[40,35376,35378,35392],{"className":35377},[349],[40,35379,35381],{"className":35380,"style":32557},[353],[40,35382,35383,35386],{"style":32560},[40,35384],{"className":35385,"style":3831},[361],[40,35387,35389],{"className":35388},[366,3835,1238,369],[40,35390,66],{"className":35391},[102,369],[40,35393,377],{"className":35394},[376],[40,35396,35398],{"className":35397},[349],[40,35399,35401],{"className":35400,"style":4479},[353],[40,35402],{},[40,35404,35406,35409],{"className":35405},[113,369],[40,35407,207],{"className":35408},[113,369],[40,35410,35412],{"className":35411},[340],[40,35413,35415],{"className":35414},[344],[40,35416,35418],{"className":35417},[349],[40,35419,35421],{"className":35420,"style":32599},[353],[40,35422,35423,35426],{"style":32602},[40,35424],{"className":35425,"style":3831},[361],[40,35427,35429],{"className":35428},[366,3835,1238,369],[40,35430,1044],{"className":35431},[102,369],[40,35433,377],{"className":35434},[376],[40,35436,35438],{"className":35437},[349],[40,35439,35441],{"className":35440,"style":1226},[353],[40,35442],{},[40,35444],{"className":35445},[113,1162],[40,35447,35449],{"className":35448,"style":413},[113,412],[40,35450,207],{"className":35451},[417,2195],[40,35453],{"className":35454,"style":122},[121],[40,35456,1820],{"className":35457},[1369],[40,35459,167],{"className":35460},[106],[40,35462,1044],{"className":35463},[102],[40,35465,1661],{"className":35466,"style":336},[102,335],[40,35468,35470,35473],{"className":35469},[102],[40,35471,13262],{"className":35472,"style":33300},[102,335],[40,35474,35476],{"className":35475},[340],[40,35477,35479,35499],{"className":35478},[344,345],[40,35480,35482,35496],{"className":35481},[349],[40,35483,35485],{"className":35484,"style":354},[353],[40,35486,35487,35490],{"style":33315},[40,35488],{"className":35489,"style":362},[361],[40,35491,35493],{"className":35492},[366,367,368,369],[40,35494,20274],{"className":35495},[102,335,369],[40,35497,377],{"className":35498},[376],[40,35500,35502],{"className":35501},[349],[40,35503,35505],{"className":35504,"style":384},[353],[40,35506],{},[40,35508,20277],{"className":35509},[102,335],[40,35511],{"className":35512,"style":1246},[121],[40,35514,1053],{"className":35515},[1250],[40,35517],{"className":35518,"style":1246},[121],[40,35520,35522,35525,35565],{"className":35521},[93],[40,35523],{"className":35524,"style":98},[97],[40,35526,35528,35531],{"className":35527},[102],[40,35529,1671],{"className":35530},[102,335],[40,35532,35534],{"className":35533},[340],[40,35535,35537,35557],{"className":35536},[344,345],[40,35538,35540,35554],{"className":35539},[349],[40,35541,35543],{"className":35542,"style":20224},[353],[40,35544,35545,35548],{"style":6098},[40,35546],{"className":35547,"style":362},[361],[40,35549,35551],{"className":35550},[366,367,368,369],[40,35552,4196],{"className":35553,"style":1242},[102,335,369],[40,35555,377],{"className":35556},[376],[40,35558,35560],{"className":35559},[349],[40,35561,35563],{"className":35562,"style":384},[353],[40,35564],{},[40,35566,207],{"className":35567},[113],[20148,35569,35570,35575],{},[20167,35571,35572],{},[35,35573,35574],{},"Pauli-Y Gate",[20167,35576,35577],{},[40,35578,35580,35683],{"className":35579},[43],[40,35581,35583],{"className":35582},[47],[49,35584,35585],{"xmlns":51},[53,35586,35587,35681],{},[56,35588,35589,35595,35597,35599,35601,35603,35609,35611,35613,35653,35655,35657,35659,35661,35663,35669,35671,35673,35679],{},[151,35590,35591,35593],{},[154,35592,32874],{},[154,35594,4230],{},[59,35596,167],{"stretchy":61},[154,35598,20277],{},[59,35600,207],{"stretchy":61},[59,35602,162],{},[151,35604,35605,35607],{},[154,35606,8234],{},[64,35608,66],{},[154,35610,5268],{},[59,35612,1084],{},[56,35614,35615,35617,35619,35651],{},[59,35616,167],{"fence":72},[59,35618,243],{},[1038,35620,35621,35641],{},[56,35622,35623,35625,35627,35629,35635],{},[59,35624,167],{"stretchy":61},[154,35626,20277],{},[59,35628,243],{},[151,35630,35631,35633],{},[154,35632,20277],{},[64,35634,66],{},[941,35636,35637,35639],{},[59,35638,207],{"stretchy":61},[64,35640,1044],{},[56,35642,35643,35645],{},[64,35644,1044],{},[941,35646,35647,35649],{},[154,35648,156],{},[64,35650,1044],{},[59,35652,207],{"fence":72},[154,35654,1820],{},[59,35656,1084],{},[59,35658,167],{"stretchy":61},[64,35660,1044],{},[154,35662,1661],{},[151,35664,35665,35667],{},[154,35666,13262],{},[154,35668,20274],{},[154,35670,20277],{},[59,35672,1053],{},[151,35674,35675,35677],{},[154,35676,1671],{},[154,35678,4230],{},[59,35680,207],{"stretchy":61},[82,35682,33764],{"encoding":84},[40,35684,35686,35750,36062],{"className":35685,"ariaHidden":72},[89],[40,35687,35689,35692,35732,35735,35738,35741,35744,35747],{"className":35688},[93],[40,35690],{"className":35691,"style":98},[97],[40,35693,35695,35698],{"className":35694},[102],[40,35696,32874],{"className":35697,"style":2383},[102,335],[40,35699,35701],{"className":35700},[340],[40,35702,35704,35724],{"className":35703},[344,345],[40,35705,35707,35721],{"className":35706},[349],[40,35708,35710],{"className":35709,"style":20224},[353],[40,35711,35712,35715],{"style":32995},[40,35713],{"className":35714,"style":362},[361],[40,35716,35718],{"className":35717},[366,367,368,369],[40,35719,4230],{"className":35720,"style":1246},[102,335,369],[40,35722,377],{"className":35723},[376],[40,35725,35727],{"className":35726},[349],[40,35728,35730],{"className":35729,"style":384},[353],[40,35731],{},[40,35733,167],{"className":35734},[106],[40,35736,20277],{"className":35737},[102,335],[40,35739,207],{"className":35740},[113],[40,35742],{"className":35743,"style":390},[121],[40,35745,162],{"className":35746},[394],[40,35748],{"className":35749,"style":390},[121],[40,35751,35753,35756,35796,35799,35802,35805,35995,35998,36001,36004,36007,36010,36050,36053,36056,36059],{"className":35752},[93],[40,35754],{"className":35755,"style":2120},[97],[40,35757,35759,35762],{"className":35758},[102],[40,35760,8234],{"className":35761},[102,335],[40,35763,35765],{"className":35764},[340],[40,35766,35768,35788],{"className":35767},[344,345],[40,35769,35771,35785],{"className":35770},[349],[40,35772,35774],{"className":35773,"style":7201},[353],[40,35775,35776,35779],{"style":6098},[40,35777],{"className":35778,"style":362},[361],[40,35780,35782],{"className":35781},[366,367,368,369],[40,35783,66],{"className":35784},[102,369],[40,35786,377],{"className":35787},[376],[40,35789,35791],{"className":35790},[349],[40,35792,35794],{"className":35793,"style":384},[353],[40,35795],{},[40,35797],{"className":35798,"style":122},[121],[40,35800,5268],{"className":35801},[1369],[40,35803],{"className":35804,"style":122},[121],[40,35806,35808,35814,35817,35989],{"className":35807},[408],[40,35809,35811],{"className":35810,"style":413},[106,412],[40,35812,167],{"className":35813},[417,2195],[40,35815,243],{"className":35816},[102],[40,35818,35820,35823,35986],{"className":35819},[102],[40,35821],{"className":35822},[106,1162],[40,35824,35826],{"className":35825},[1038],[40,35827,35829,35978],{"className":35828},[344,345],[40,35830,35832,35975],{"className":35831},[349],[40,35833,35835,35878,35886],{"className":35834,"style":32465},[353],[40,35836,35837,35840],{"style":1178},[40,35838],{"className":35839,"style":444},[361],[40,35841,35843],{"className":35842},[366,367,368,369],[40,35844,35846,35849],{"className":35845},[102,369],[40,35847,1044],{"className":35848},[102,369],[40,35850,35852,35855],{"className":35851},[102,369],[40,35853,156],{"className":35854,"style":336},[102,335,369],[40,35856,35858],{"className":35857},[340],[40,35859,35861],{"className":35860},[344],[40,35862,35864],{"className":35863},[349],[40,35865,35867],{"className":35866,"style":32498},[353],[40,35868,35869,35872],{"style":3827},[40,35870],{"className":35871,"style":3831},[361],[40,35873,35875],{"className":35874},[366,3835,1238,369],[40,35876,1044],{"className":35877},[102,369],[40,35879,35880,35883],{"style":1193},[40,35881],{"className":35882,"style":444},[361],[40,35884],{"className":35885,"style":1201},[1200],[40,35887,35888,35891],{"style":32520},[40,35889],{"className":35890,"style":444},[361],[40,35892,35894],{"className":35893},[366,367,368,369],[40,35895,35897,35900,35903,35906,35946],{"className":35896},[102,369],[40,35898,167],{"className":35899},[106,369],[40,35901,20277],{"className":35902},[102,335,369],[40,35904,243],{"className":35905},[1250,369],[40,35907,35909,35912],{"className":35908},[102,369],[40,35910,20277],{"className":35911},[102,335,369],[40,35913,35915],{"className":35914},[340],[40,35916,35918,35938],{"className":35917},[344,345],[40,35919,35921,35935],{"className":35920},[349],[40,35922,35924],{"className":35923,"style":32557},[353],[40,35925,35926,35929],{"style":32560},[40,35927],{"className":35928,"style":3831},[361],[40,35930,35932],{"className":35931},[366,3835,1238,369],[40,35933,66],{"className":35934},[102,369],[40,35936,377],{"className":35937},[376],[40,35939,35941],{"className":35940},[349],[40,35942,35944],{"className":35943,"style":4479},[353],[40,35945],{},[40,35947,35949,35952],{"className":35948},[113,369],[40,35950,207],{"className":35951},[113,369],[40,35953,35955],{"className":35954},[340],[40,35956,35958],{"className":35957},[344],[40,35959,35961],{"className":35960},[349],[40,35962,35964],{"className":35963,"style":32599},[353],[40,35965,35966,35969],{"style":32602},[40,35967],{"className":35968,"style":3831},[361],[40,35970,35972],{"className":35971},[366,3835,1238,369],[40,35973,1044],{"className":35974},[102,369],[40,35976,377],{"className":35977},[376],[40,35979,35981],{"className":35980},[349],[40,35982,35984],{"className":35983,"style":1226},[353],[40,35985],{},[40,35987],{"className":35988},[113,1162],[40,35990,35992],{"className":35991,"style":413},[113,412],[40,35993,207],{"className":35994},[417,2195],[40,35996],{"className":35997,"style":122},[121],[40,35999,1820],{"className":36000},[1369],[40,36002,167],{"className":36003},[106],[40,36005,1044],{"className":36006},[102],[40,36008,1661],{"className":36009,"style":336},[102,335],[40,36011,36013,36016],{"className":36012},[102],[40,36014,13262],{"className":36015,"style":33300},[102,335],[40,36017,36019],{"className":36018},[340],[40,36020,36022,36042],{"className":36021},[344,345],[40,36023,36025,36039],{"className":36024},[349],[40,36026,36028],{"className":36027,"style":354},[353],[40,36029,36030,36033],{"style":33315},[40,36031],{"className":36032,"style":362},[361],[40,36034,36036],{"className":36035},[366,367,368,369],[40,36037,20274],{"className":36038},[102,335,369],[40,36040,377],{"className":36041},[376],[40,36043,36045],{"className":36044},[349],[40,36046,36048],{"className":36047,"style":384},[353],[40,36049],{},[40,36051,20277],{"className":36052},[102,335],[40,36054],{"className":36055,"style":1246},[121],[40,36057,1053],{"className":36058},[1250],[40,36060],{"className":36061,"style":1246},[121],[40,36063,36065,36068,36108],{"className":36064},[93],[40,36066],{"className":36067,"style":98},[97],[40,36069,36071,36074],{"className":36070},[102],[40,36072,1671],{"className":36073},[102,335],[40,36075,36077],{"className":36076},[340],[40,36078,36080,36100],{"className":36079},[344,345],[40,36081,36083,36097],{"className":36082},[349],[40,36084,36086],{"className":36085,"style":20224},[353],[40,36087,36088,36091],{"style":6098},[40,36089],{"className":36090,"style":362},[361],[40,36092,36094],{"className":36093},[366,367,368,369],[40,36095,4230],{"className":36096,"style":1246},[102,335,369],[40,36098,377],{"className":36099},[376],[40,36101,36103],{"className":36102},[349],[40,36104,36106],{"className":36105,"style":384},[353],[40,36107],{},[40,36109,207],{"className":36110},[113],[20148,36112,36113,36118],{},[20167,36114,36115],{},[35,36116,36117],{},"Hadamard Gate",[20167,36119,36120],{},[40,36121,36123,36227],{"className":36122},[43],[40,36124,36126],{"className":36125},[47],[49,36127,36128],{"xmlns":51},[53,36129,36130,36224],{},[56,36131,36132,36138,36140,36142,36144,36146,36152,36154,36156,36196,36198,36200,36202,36204,36206,36212,36214,36216,36222],{},[151,36133,36134,36136],{},[154,36135,32874],{},[154,36137,26630],{},[59,36139,167],{"stretchy":61},[154,36141,20277],{},[59,36143,207],{"stretchy":61},[59,36145,162],{},[151,36147,36148,36150],{},[154,36149,8234],{},[64,36151,66],{},[154,36153,5268],{},[59,36155,1084],{},[56,36157,36158,36160,36162,36194],{},[59,36159,167],{"fence":72},[59,36161,243],{},[1038,36163,36164,36184],{},[56,36165,36166,36168,36170,36172,36178],{},[59,36167,167],{"stretchy":61},[154,36169,20277],{},[59,36171,243],{},[151,36173,36174,36176],{},[154,36175,20277],{},[64,36177,66],{},[941,36179,36180,36182],{},[59,36181,207],{"stretchy":61},[64,36183,1044],{},[56,36185,36186,36188],{},[64,36187,1044],{},[941,36189,36190,36192],{},[154,36191,156],{},[64,36193,1044],{},[59,36195,207],{"fence":72},[154,36197,1820],{},[59,36199,1084],{},[59,36201,167],{"stretchy":61},[64,36203,1044],{},[154,36205,1661],{},[151,36207,36208,36210],{},[154,36209,13262],{},[154,36211,20274],{},[154,36213,20277],{},[59,36215,1053],{},[151,36217,36218,36220],{},[154,36219,1671],{},[154,36221,26630],{},[59,36223,207],{"stretchy":61},[82,36225,36226],{"encoding":84},"E_H(t) = A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right) \\cos(2\\pi f_c t + \\phi_H)",[40,36228,36230,36295,36607],{"className":36229,"ariaHidden":72},[89],[40,36231,36233,36236,36277,36280,36283,36286,36289,36292],{"className":36232},[93],[40,36234],{"className":36235,"style":98},[97],[40,36237,36239,36242],{"className":36238},[102],[40,36240,32874],{"className":36241,"style":2383},[102,335],[40,36243,36245],{"className":36244},[340],[40,36246,36248,36269],{"className":36247},[344,345],[40,36249,36251,36266],{"className":36250},[349],[40,36252,36254],{"className":36253,"style":20224},[353],[40,36255,36256,36259],{"style":32995},[40,36257],{"className":36258,"style":362},[361],[40,36260,36262],{"className":36261},[366,367,368,369],[40,36263,26630],{"className":36264,"style":36265},[102,335,369],"margin-right:0.0813em;",[40,36267,377],{"className":36268},[376],[40,36270,36272],{"className":36271},[349],[40,36273,36275],{"className":36274,"style":384},[353],[40,36276],{},[40,36278,167],{"className":36279},[106],[40,36281,20277],{"className":36282},[102,335],[40,36284,207],{"className":36285},[113],[40,36287],{"className":36288,"style":390},[121],[40,36290,162],{"className":36291},[394],[40,36293],{"className":36294,"style":390},[121],[40,36296,36298,36301,36341,36344,36347,36350,36540,36543,36546,36549,36552,36555,36595,36598,36601,36604],{"className":36297},[93],[40,36299],{"className":36300,"style":2120},[97],[40,36302,36304,36307],{"className":36303},[102],[40,36305,8234],{"className":36306},[102,335],[40,36308,36310],{"className":36309},[340],[40,36311,36313,36333],{"className":36312},[344,345],[40,36314,36316,36330],{"className":36315},[349],[40,36317,36319],{"className":36318,"style":7201},[353],[40,36320,36321,36324],{"style":6098},[40,36322],{"className":36323,"style":362},[361],[40,36325,36327],{"className":36326},[366,367,368,369],[40,36328,66],{"className":36329},[102,369],[40,36331,377],{"className":36332},[376],[40,36334,36336],{"className":36335},[349],[40,36337,36339],{"className":36338,"style":384},[353],[40,36340],{},[40,36342],{"className":36343,"style":122},[121],[40,36345,5268],{"className":36346},[1369],[40,36348],{"className":36349,"style":122},[121],[40,36351,36353,36359,36362,36534],{"className":36352},[408],[40,36354,36356],{"className":36355,"style":413},[106,412],[40,36357,167],{"className":36358},[417,2195],[40,36360,243],{"className":36361},[102],[40,36363,36365,36368,36531],{"className":36364},[102],[40,36366],{"className":36367},[106,1162],[40,36369,36371],{"className":36370},[1038],[40,36372,36374,36523],{"className":36373},[344,345],[40,36375,36377,36520],{"className":36376},[349],[40,36378,36380,36423,36431],{"className":36379,"style":32465},[353],[40,36381,36382,36385],{"style":1178},[40,36383],{"className":36384,"style":444},[361],[40,36386,36388],{"className":36387},[366,367,368,369],[40,36389,36391,36394],{"className":36390},[102,369],[40,36392,1044],{"className":36393},[102,369],[40,36395,36397,36400],{"className":36396},[102,369],[40,36398,156],{"className":36399,"style":336},[102,335,369],[40,36401,36403],{"className":36402},[340],[40,36404,36406],{"className":36405},[344],[40,36407,36409],{"className":36408},[349],[40,36410,36412],{"className":36411,"style":32498},[353],[40,36413,36414,36417],{"style":3827},[40,36415],{"className":36416,"style":3831},[361],[40,36418,36420],{"className":36419},[366,3835,1238,369],[40,36421,1044],{"className":36422},[102,369],[40,36424,36425,36428],{"style":1193},[40,36426],{"className":36427,"style":444},[361],[40,36429],{"className":36430,"style":1201},[1200],[40,36432,36433,36436],{"style":32520},[40,36434],{"className":36435,"style":444},[361],[40,36437,36439],{"className":36438},[366,367,368,369],[40,36440,36442,36445,36448,36451,36491],{"className":36441},[102,369],[40,36443,167],{"className":36444},[106,369],[40,36446,20277],{"className":36447},[102,335,369],[40,36449,243],{"className":36450},[1250,369],[40,36452,36454,36457],{"className":36453},[102,369],[40,36455,20277],{"className":36456},[102,335,369],[40,36458,36460],{"className":36459},[340],[40,36461,36463,36483],{"className":36462},[344,345],[40,36464,36466,36480],{"className":36465},[349],[40,36467,36469],{"className":36468,"style":32557},[353],[40,36470,36471,36474],{"style":32560},[40,36472],{"className":36473,"style":3831},[361],[40,36475,36477],{"className":36476},[366,3835,1238,369],[40,36478,66],{"className":36479},[102,369],[40,36481,377],{"className":36482},[376],[40,36484,36486],{"className":36485},[349],[40,36487,36489],{"className":36488,"style":4479},[353],[40,36490],{},[40,36492,36494,36497],{"className":36493},[113,369],[40,36495,207],{"className":36496},[113,369],[40,36498,36500],{"className":36499},[340],[40,36501,36503],{"className":36502},[344],[40,36504,36506],{"className":36505},[349],[40,36507,36509],{"className":36508,"style":32599},[353],[40,36510,36511,36514],{"style":32602},[40,36512],{"className":36513,"style":3831},[361],[40,36515,36517],{"className":36516},[366,3835,1238,369],[40,36518,1044],{"className":36519},[102,369],[40,36521,377],{"className":36522},[376],[40,36524,36526],{"className":36525},[349],[40,36527,36529],{"className":36528,"style":1226},[353],[40,36530],{},[40,36532],{"className":36533},[113,1162],[40,36535,36537],{"className":36536,"style":413},[113,412],[40,36538,207],{"className":36539},[417,2195],[40,36541],{"className":36542,"style":122},[121],[40,36544,1820],{"className":36545},[1369],[40,36547,167],{"className":36548},[106],[40,36550,1044],{"className":36551},[102],[40,36553,1661],{"className":36554,"style":336},[102,335],[40,36556,36558,36561],{"className":36557},[102],[40,36559,13262],{"className":36560,"style":33300},[102,335],[40,36562,36564],{"className":36563},[340],[40,36565,36567,36587],{"className":36566},[344,345],[40,36568,36570,36584],{"className":36569},[349],[40,36571,36573],{"className":36572,"style":354},[353],[40,36574,36575,36578],{"style":33315},[40,36576],{"className":36577,"style":362},[361],[40,36579,36581],{"className":36580},[366,367,368,369],[40,36582,20274],{"className":36583},[102,335,369],[40,36585,377],{"className":36586},[376],[40,36588,36590],{"className":36589},[349],[40,36591,36593],{"className":36592,"style":384},[353],[40,36594],{},[40,36596,20277],{"className":36597},[102,335],[40,36599],{"className":36600,"style":1246},[121],[40,36602,1053],{"className":36603},[1250],[40,36605],{"className":36606,"style":1246},[121],[40,36608,36610,36613,36653],{"className":36609},[93],[40,36611],{"className":36612,"style":98},[97],[40,36614,36616,36619],{"className":36615},[102],[40,36617,1671],{"className":36618},[102,335],[40,36620,36622],{"className":36621},[340],[40,36623,36625,36645],{"className":36624},[344,345],[40,36626,36628,36642],{"className":36627},[349],[40,36629,36631],{"className":36630,"style":20224},[353],[40,36632,36633,36636],{"style":6098},[40,36634],{"className":36635,"style":362},[361],[40,36637,36639],{"className":36638},[366,367,368,369],[40,36640,26630],{"className":36641,"style":36265},[102,335,369],[40,36643,377],{"className":36644},[376],[40,36646,36648],{"className":36647},[349],[40,36649,36651],{"className":36650,"style":384},[353],[40,36652],{},[40,36654,207],{"className":36655},[113],[20148,36657,36658,36663],{},[20167,36659,36660],{},[35,36661,36662],{},"CNOT Target Qubit Pulse",[20167,36664,36665],{},[40,36666,36668,36782],{"className":36667},[43],[40,36669,36671],{"className":36670},[47],[49,36672,36673],{"xmlns":51},[53,36674,36675,36779],{},[56,36676,36677,36693,36695,36697,36699,36701,36707,36709,36711,36751,36753,36755,36757,36759,36761,36767,36769,36771,36777],{},[151,36678,36679,36681],{},[154,36680,32874],{},[56,36682,36683,36686,36689,36691],{},[154,36684,36685],{},"C",[154,36687,36688],{},"N",[154,36690,4055],{},[154,36692,7059],{},[59,36694,167],{"stretchy":61},[154,36696,20277],{},[59,36698,207],{"stretchy":61},[59,36700,162],{},[151,36702,36703,36705],{},[154,36704,8234],{},[64,36706,66],{},[154,36708,5268],{},[59,36710,1084],{},[56,36712,36713,36715,36717,36749],{},[59,36714,167],{"fence":72},[59,36716,243],{},[1038,36718,36719,36739],{},[56,36720,36721,36723,36725,36727,36733],{},[59,36722,167],{"stretchy":61},[154,36724,20277],{},[59,36726,243],{},[151,36728,36729,36731],{},[154,36730,20277],{},[64,36732,66],{},[941,36734,36735,36737],{},[59,36736,207],{"stretchy":61},[64,36738,1044],{},[56,36740,36741,36743],{},[64,36742,1044],{},[941,36744,36745,36747],{},[154,36746,156],{},[64,36748,1044],{},[59,36750,207],{"fence":72},[154,36752,1820],{},[59,36754,1084],{},[59,36756,167],{"stretchy":61},[64,36758,1044],{},[154,36760,1661],{},[151,36762,36763,36765],{},[154,36764,13262],{},[154,36766,20274],{},[154,36768,20277],{},[59,36770,1053],{},[151,36772,36773,36775],{},[154,36774,1671],{},[154,36776,4196],{},[59,36778,207],{"stretchy":61},[82,36780,36781],{"encoding":84},"E_{CNOT}(t) = A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right) \\cos(2\\pi f_c t + \\phi_X)",[40,36783,36785,36861,37173],{"className":36784,"ariaHidden":72},[89],[40,36786,36788,36791,36843,36846,36849,36852,36855,36858],{"className":36787},[93],[40,36789],{"className":36790,"style":98},[97],[40,36792,36794,36797],{"className":36793},[102],[40,36795,32874],{"className":36796,"style":2383},[102,335],[40,36798,36800],{"className":36799},[340],[40,36801,36803,36835],{"className":36802},[344,345],[40,36804,36806,36832],{"className":36805},[349],[40,36807,36809],{"className":36808,"style":20224},[353],[40,36810,36811,36814],{"style":32995},[40,36812],{"className":36813,"style":362},[361],[40,36815,36817],{"className":36816},[366,367,368,369],[40,36818,36820,36823,36826,36829],{"className":36819},[102,369],[40,36821,36685],{"className":36822,"style":4668},[102,335,369],[40,36824,36688],{"className":36825,"style":3802},[102,335,369],[40,36827,4055],{"className":36828,"style":1312},[102,335,369],[40,36830,7059],{"className":36831,"style":2541},[102,335,369],[40,36833,377],{"className":36834},[376],[40,36836,36838],{"className":36837},[349],[40,36839,36841],{"className":36840,"style":384},[353],[40,36842],{},[40,36844,167],{"className":36845},[106],[40,36847,20277],{"className":36848},[102,335],[40,36850,207],{"className":36851},[113],[40,36853],{"className":36854,"style":390},[121],[40,36856,162],{"className":36857},[394],[40,36859],{"className":36860,"style":390},[121],[40,36862,36864,36867,36907,36910,36913,36916,37106,37109,37112,37115,37118,37121,37161,37164,37167,37170],{"className":36863},[93],[40,36865],{"className":36866,"style":2120},[97],[40,36868,36870,36873],{"className":36869},[102],[40,36871,8234],{"className":36872},[102,335],[40,36874,36876],{"className":36875},[340],[40,36877,36879,36899],{"className":36878},[344,345],[40,36880,36882,36896],{"className":36881},[349],[40,36883,36885],{"className":36884,"style":7201},[353],[40,36886,36887,36890],{"style":6098},[40,36888],{"className":36889,"style":362},[361],[40,36891,36893],{"className":36892},[366,367,368,369],[40,36894,66],{"className":36895},[102,369],[40,36897,377],{"className":36898},[376],[40,36900,36902],{"className":36901},[349],[40,36903,36905],{"className":36904,"style":384},[353],[40,36906],{},[40,36908],{"className":36909,"style":122},[121],[40,36911,5268],{"className":36912},[1369],[40,36914],{"className":36915,"style":122},[121],[40,36917,36919,36925,36928,37100],{"className":36918},[408],[40,36920,36922],{"className":36921,"style":413},[106,412],[40,36923,167],{"className":36924},[417,2195],[40,36926,243],{"className":36927},[102],[40,36929,36931,36934,37097],{"className":36930},[102],[40,36932],{"className":36933},[106,1162],[40,36935,36937],{"className":36936},[1038],[40,36938,36940,37089],{"className":36939},[344,345],[40,36941,36943,37086],{"className":36942},[349],[40,36944,36946,36989,36997],{"className":36945,"style":32465},[353],[40,36947,36948,36951],{"style":1178},[40,36949],{"className":36950,"style":444},[361],[40,36952,36954],{"className":36953},[366,367,368,369],[40,36955,36957,36960],{"className":36956},[102,369],[40,36958,1044],{"className":36959},[102,369],[40,36961,36963,36966],{"className":36962},[102,369],[40,36964,156],{"className":36965,"style":336},[102,335,369],[40,36967,36969],{"className":36968},[340],[40,36970,36972],{"className":36971},[344],[40,36973,36975],{"className":36974},[349],[40,36976,36978],{"className":36977,"style":32498},[353],[40,36979,36980,36983],{"style":3827},[40,36981],{"className":36982,"style":3831},[361],[40,36984,36986],{"className":36985},[366,3835,1238,369],[40,36987,1044],{"className":36988},[102,369],[40,36990,36991,36994],{"style":1193},[40,36992],{"className":36993,"style":444},[361],[40,36995],{"className":36996,"style":1201},[1200],[40,36998,36999,37002],{"style":32520},[40,37000],{"className":37001,"style":444},[361],[40,37003,37005],{"className":37004},[366,367,368,369],[40,37006,37008,37011,37014,37017,37057],{"className":37007},[102,369],[40,37009,167],{"className":37010},[106,369],[40,37012,20277],{"className":37013},[102,335,369],[40,37015,243],{"className":37016},[1250,369],[40,37018,37020,37023],{"className":37019},[102,369],[40,37021,20277],{"className":37022},[102,335,369],[40,37024,37026],{"className":37025},[340],[40,37027,37029,37049],{"className":37028},[344,345],[40,37030,37032,37046],{"className":37031},[349],[40,37033,37035],{"className":37034,"style":32557},[353],[40,37036,37037,37040],{"style":32560},[40,37038],{"className":37039,"style":3831},[361],[40,37041,37043],{"className":37042},[366,3835,1238,369],[40,37044,66],{"className":37045},[102,369],[40,37047,377],{"className":37048},[376],[40,37050,37052],{"className":37051},[349],[40,37053,37055],{"className":37054,"style":4479},[353],[40,37056],{},[40,37058,37060,37063],{"className":37059},[113,369],[40,37061,207],{"className":37062},[113,369],[40,37064,37066],{"className":37065},[340],[40,37067,37069],{"className":37068},[344],[40,37070,37072],{"className":37071},[349],[40,37073,37075],{"className":37074,"style":32599},[353],[40,37076,37077,37080],{"style":32602},[40,37078],{"className":37079,"style":3831},[361],[40,37081,37083],{"className":37082},[366,3835,1238,369],[40,37084,1044],{"className":37085},[102,369],[40,37087,377],{"className":37088},[376],[40,37090,37092],{"className":37091},[349],[40,37093,37095],{"className":37094,"style":1226},[353],[40,37096],{},[40,37098],{"className":37099},[113,1162],[40,37101,37103],{"className":37102,"style":413},[113,412],[40,37104,207],{"className":37105},[417,2195],[40,37107],{"className":37108,"style":122},[121],[40,37110,1820],{"className":37111},[1369],[40,37113,167],{"className":37114},[106],[40,37116,1044],{"className":37117},[102],[40,37119,1661],{"className":37120,"style":336},[102,335],[40,37122,37124,37127],{"className":37123},[102],[40,37125,13262],{"className":37126,"style":33300},[102,335],[40,37128,37130],{"className":37129},[340],[40,37131,37133,37153],{"className":37132},[344,345],[40,37134,37136,37150],{"className":37135},[349],[40,37137,37139],{"className":37138,"style":354},[353],[40,37140,37141,37144],{"style":33315},[40,37142],{"className":37143,"style":362},[361],[40,37145,37147],{"className":37146},[366,367,368,369],[40,37148,20274],{"className":37149},[102,335,369],[40,37151,377],{"className":37152},[376],[40,37154,37156],{"className":37155},[349],[40,37157,37159],{"className":37158,"style":384},[353],[40,37160],{},[40,37162,20277],{"className":37163},[102,335],[40,37165],{"className":37166,"style":1246},[121],[40,37168,1053],{"className":37169},[1250],[40,37171],{"className":37172,"style":1246},[121],[40,37174,37176,37179,37219],{"className":37175},[93],[40,37177],{"className":37178,"style":98},[97],[40,37180,37182,37185],{"className":37181},[102],[40,37183,1671],{"className":37184},[102,335],[40,37186,37188],{"className":37187},[340],[40,37189,37191,37211],{"className":37190},[344,345],[40,37192,37194,37208],{"className":37193},[349],[40,37195,37197],{"className":37196,"style":20224},[353],[40,37198,37199,37202],{"style":6098},[40,37200],{"className":37201,"style":362},[361],[40,37203,37205],{"className":37204},[366,367,368,369],[40,37206,4196],{"className":37207,"style":1242},[102,335,369],[40,37209,377],{"className":37210},[376],[40,37212,37214],{"className":37213},[349],[40,37215,37217],{"className":37216,"style":384},[353],[40,37218],{},[40,37220,207],{"className":37221},[113],[10082,37223,37225],{"className":10269,"code":37224,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define parameters\nt0 = 0      # Pulse center\nsigma = 5   # Standard deviation of Gaussian modulation function (ns)\nA0 = 1      # Peak amplitude\n\n# Increase the number of points by 1.5x for denser lollipop visualization\nnum_points = int(40 * 1.5)\nt = np.linspace(-20, 20, num_points)  # Adjusted time axis\n\n# Compute Gaussian modulation function\ngaussian_modulation = A0 * np.exp(-(t - t0)**2 \u002F (2 * sigma**2))\n\n# Store modulation functions and titles in an array\nmodulations = [\n    (gaussian_modulation, \"Gaussian Modulation Function for Pauli-X Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for Pauli-Y Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for Hadamard Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for CNOT Gate Target Qubit\")\n]\n\n# Create a figure with constrained layout\nfig, axs = plt.subplots(2, 2, figsize=(12, 8), constrained_layout=True)\naxs = axs.flatten()\n\n# Lollipop plot setup\nfor ax, (modulation, title) in zip(axs, modulations):\n    ax.stem(t, modulation, linefmt='b-', markerfmt='bo', basefmt='k-')\n    ax.set_title(title)\n    ax.set_xlabel(\"Time (ns)\")\n    ax.set_ylabel(\"Amplitude\")\n\n# Memory Addressing with Lollipop Representation\nfig_combined, ax_combined = plt.subplots(figsize=(12, 6), constrained_layout=True)\n\n# Memory addresses as an index range\nmemory_addresses = np.arange(len(t))\n\n# Lollipop-style plot for memory addressing\nax_combined.stem(memory_addresses, gaussian_modulation, linefmt='b-', markerfmt='bo', basefmt='k-')\n\n# Labels and title\nax_combined.set_title(\"Memory Addressing with Lollipop Representation for Quantum Gate Control\")\nax_combined.set_xlabel(\"Memory Address Index\")\nax_combined.set_ylabel(\"Amplitude\")\nax_combined.grid(True)\n\n# Display plots\nplt.show()\n",[10088,37226,37227,37237,37247,37251,37256,37268,37280,37292,37296,37301,37322,37348,37352,37357,37405,37409,37414,37423,37433,37442,37451,37460,37464,37468,37473,37516,37530,37534,37539,37554,37586,37594,37607,37619,37623,37628,37663,37667,37672,37689,37693,37698,37732,37736,37741,37754,37767,37779,37791,37795,37800],{"__ignoreMap":10090},[40,37228,37229,37231,37233,37235],{"class":10276,"line":10277},[40,37230,10375],{"class":10352},[40,37232,10378],{"class":10363},[40,37234,10381],{"class":10352},[40,37236,10384],{"class":10363},[40,37238,37239,37241,37243,37245],{"class":10276,"line":10284},[40,37240,10375],{"class":10352},[40,37242,10405],{"class":10363},[40,37244,10381],{"class":10352},[40,37246,10410],{"class":10363},[40,37248,37249],{"class":10276,"line":10291},[40,37250,10301],{"emptyLinePlaceholder":10300},[40,37252,37253],{"class":10276,"line":10297},[40,37254,37255],{"class":10280},"# Define parameters\n",[40,37257,37258,37261,37263,37265],{"class":10276,"line":10304},[40,37259,37260],{"class":10363},"t0 ",[40,37262,162],{"class":10732},[40,37264,11234],{"class":10446},[40,37266,37267],{"class":10280},"      # Pulse center\n",[40,37269,37270,37273,37275,37277],{"class":10276,"line":10310},[40,37271,37272],{"class":10363},"sigma ",[40,37274,162],{"class":10732},[40,37276,28264],{"class":10446},[40,37278,37279],{"class":10280},"   # Standard deviation of Gaussian modulation function (ns)\n",[40,37281,37282,37285,37287,37289],{"class":10276,"line":10316},[40,37283,37284],{"class":10363},"A0 ",[40,37286,162],{"class":10732},[40,37288,13373],{"class":10446},[40,37290,37291],{"class":10280},"      # Peak amplitude\n",[40,37293,37294],{"class":10276,"line":10322},[40,37295,10301],{"emptyLinePlaceholder":10300},[40,37297,37298],{"class":10276,"line":10327},[40,37299,37300],{"class":10280},"# Increase the number of points by 1.5x for denser lollipop visualization\n",[40,37302,37303,37306,37308,37311,37313,37315,37317,37320],{"class":10276,"line":10333},[40,37304,37305],{"class":10363},"num_points ",[40,37307,162],{"class":10732},[40,37309,37310],{"class":10732}," int",[40,37312,167],{"class":10363},[40,37314,27987],{"class":10446},[40,37316,11881],{"class":10732},[40,37318,37319],{"class":10446}," 1.5",[40,37321,10761],{"class":10363},[40,37323,37324,37326,37328,37330,37332,37334,37336,37338,37340,37342,37345],{"class":10276,"line":10339},[40,37325,29894],{"class":10363},[40,37327,162],{"class":10732},[40,37329,11156],{"class":10363},[40,37331,11850],{"class":10430},[40,37333,167],{"class":10363},[40,37335,11056],{"class":10732},[40,37337,13742],{"class":10446},[40,37339,10484],{"class":10363},[40,37341,13742],{"class":10446},[40,37343,37344],{"class":10363},", num_points)  ",[40,37346,37347],{"class":10280},"# Adjusted time axis\n",[40,37349,37350],{"class":10276,"line":10344},[40,37351,10301],{"emptyLinePlaceholder":10300},[40,37353,37354],{"class":10276,"line":10349},[40,37355,37356],{"class":10280},"# Compute Gaussian modulation function\n",[40,37358,37359,37361,37363,37366,37368,37370,37372,37374,37376,37379,37381,37384,37386,37388,37390,37392,37394,37396,37399,37401,37403],{"class":10276,"line":10367},[40,37360,27468],{"class":10363},[40,37362,162],{"class":10732},[40,37364,37365],{"class":10363}," A0 ",[40,37367,11752],{"class":10732},[40,37369,11156],{"class":10363},[40,37371,5268],{"class":10430},[40,37373,167],{"class":10363},[40,37375,11056],{"class":10732},[40,37377,37378],{"class":10363},"(t ",[40,37380,11056],{"class":10732},[40,37382,37383],{"class":10363}," t0)",[40,37385,22571],{"class":10732},[40,37387,1044],{"class":10446},[40,37389,22539],{"class":10732},[40,37391,920],{"class":10363},[40,37393,1044],{"class":10446},[40,37395,11881],{"class":10732},[40,37397,37398],{"class":10363}," sigma",[40,37400,22571],{"class":10732},[40,37402,1044],{"class":10446},[40,37404,13612],{"class":10363},[40,37406,37407],{"class":10276,"line":10372},[40,37408,10301],{"emptyLinePlaceholder":10300},[40,37410,37411],{"class":10276,"line":10387},[40,37412,37413],{"class":10280},"# Store modulation functions and titles in an array\n",[40,37415,37416,37418,37420],{"class":10276,"line":10400},[40,37417,30731],{"class":10363},[40,37419,162],{"class":10732},[40,37421,37422],{"class":10363}," [\n",[40,37424,37425,37428,37431],{"class":10276,"line":10413},[40,37426,37427],{"class":10363},"    (gaussian_modulation, ",[40,37429,37430],{"class":10287},"\"Gaussian Modulation Function for Pauli-X Gate\"",[40,37432,11061],{"class":10363},[40,37434,37435,37437,37440],{"class":10276,"line":10418},[40,37436,37427],{"class":10363},[40,37438,37439],{"class":10287},"\"Gaussian Modulation Function for Pauli-Y Gate\"",[40,37441,11061],{"class":10363},[40,37443,37444,37446,37449],{"class":10276,"line":10424},[40,37445,37427],{"class":10363},[40,37447,37448],{"class":10287},"\"Gaussian Modulation Function for Hadamard Gate\"",[40,37450,11061],{"class":10363},[40,37452,37453,37455,37458],{"class":10276,"line":10437},[40,37454,37427],{"class":10363},[40,37456,37457],{"class":10287},"\"Gaussian Modulation Function for CNOT Gate Target Qubit\"",[40,37459,10761],{"class":10363},[40,37461,37462],{"class":10276,"line":10456},[40,37463,25466],{"class":10363},[40,37465,37466],{"class":10276,"line":10472},[40,37467,10301],{"emptyLinePlaceholder":10300},[40,37469,37470],{"class":10276,"line":10496},[40,37471,37472],{"class":10280},"# Create a figure with constrained layout\n",[40,37474,37475,37478,37480,37482,37484,37486,37488,37490,37492,37494,37496,37498,37500,37502,37504,37506,37508,37510,37512,37514],{"class":10276,"line":10510},[40,37476,37477],{"class":10363},"fig, axs ",[40,37479,162],{"class":10732},[40,37481,13070],{"class":10363},[40,37483,13073],{"class":10430},[40,37485,167],{"class":10363},[40,37487,1044],{"class":10446},[40,37489,10484],{"class":10363},[40,37491,1044],{"class":10446},[40,37493,10484],{"class":10363},[40,37495,13078],{"class":11903},[40,37497,162],{"class":10732},[40,37499,167],{"class":10363},[40,37501,30273],{"class":10446},[40,37503,10484],{"class":10363},[40,37505,23482],{"class":10446},[40,37507,11040],{"class":10363},[40,37509,13086],{"class":11903},[40,37511,162],{"class":10732},[40,37513,13091],{"class":10446},[40,37515,10761],{"class":10363},[40,37517,37518,37521,37523,37526,37528],{"class":10276,"line":10696},[40,37519,37520],{"class":10363},"axs ",[40,37522,162],{"class":10732},[40,37524,37525],{"class":10363}," axs.",[40,37527,30292],{"class":10430},[40,37529,12174],{"class":10363},[40,37531,37532],{"class":10276,"line":10702},[40,37533,10301],{"emptyLinePlaceholder":10300},[40,37535,37536],{"class":10276,"line":10708},[40,37537,37538],{"class":10280},"# Lollipop plot setup\n",[40,37540,37541,37543,37546,37548,37551],{"class":10276,"line":10714},[40,37542,13840],{"class":10352},[40,37544,37545],{"class":10363}," ax, (modulation, title) ",[40,37547,13846],{"class":10352},[40,37549,37550],{"class":10732}," zip",[40,37552,37553],{"class":10363},"(axs, modulations):\n",[40,37555,37556,37558,37560,37562,37564,37566,37568,37570,37572,37574,37576,37578,37580,37582,37584],{"class":10276,"line":10720},[40,37557,13227],{"class":10363},[40,37559,28083],{"class":10430},[40,37561,30877],{"class":10363},[40,37563,28088],{"class":11903},[40,37565,162],{"class":10732},[40,37567,28093],{"class":10287},[40,37569,10484],{"class":10363},[40,37571,28098],{"class":11903},[40,37573,162],{"class":10732},[40,37575,28103],{"class":10287},[40,37577,10484],{"class":10363},[40,37579,28108],{"class":11903},[40,37581,162],{"class":10732},[40,37583,28113],{"class":10287},[40,37585,10761],{"class":10363},[40,37587,37588,37590,37592],{"class":10276,"line":10726},[40,37589,13227],{"class":10363},[40,37591,30334],{"class":10430},[40,37593,30337],{"class":10363},[40,37595,37596,37598,37600,37602,37605],{"class":10276,"line":10764},[40,37597,13227],{"class":10363},[40,37599,13230],{"class":10430},[40,37601,167],{"class":10363},[40,37603,37604],{"class":10287},"\"Time (ns)\"",[40,37606,10761],{"class":10363},[40,37608,37609,37611,37613,37615,37617],{"class":10276,"line":10778},[40,37610,13227],{"class":10363},[40,37612,13244],{"class":10430},[40,37614,167],{"class":10363},[40,37616,30361],{"class":10287},[40,37618,10761],{"class":10363},[40,37620,37621],{"class":10276,"line":10815},[40,37622,10301],{"emptyLinePlaceholder":10300},[40,37624,37625],{"class":10276,"line":10829},[40,37626,37627],{"class":10280},"# Memory Addressing with Lollipop Representation\n",[40,37629,37630,37633,37635,37637,37639,37641,37643,37645,37647,37649,37651,37653,37655,37657,37659,37661],{"class":10276,"line":10842},[40,37631,37632],{"class":10363},"fig_combined, ax_combined ",[40,37634,162],{"class":10732},[40,37636,13070],{"class":10363},[40,37638,13073],{"class":10430},[40,37640,167],{"class":10363},[40,37642,13078],{"class":11903},[40,37644,162],{"class":10732},[40,37646,167],{"class":10363},[40,37648,30273],{"class":10446},[40,37650,10484],{"class":10363},[40,37652,12785],{"class":10446},[40,37654,11040],{"class":10363},[40,37656,13086],{"class":11903},[40,37658,162],{"class":10732},[40,37660,13091],{"class":10446},[40,37662,10761],{"class":10363},[40,37664,37665],{"class":10276,"line":10855},[40,37666,10301],{"emptyLinePlaceholder":10300},[40,37668,37669],{"class":10276,"line":10868},[40,37670,37671],{"class":10280},"# Memory addresses as an index range\n",[40,37673,37674,37677,37679,37681,37683,37685,37687],{"class":10276,"line":10881},[40,37675,37676],{"class":10363},"memory_addresses ",[40,37678,162],{"class":10732},[40,37680,11156],{"class":10363},[40,37682,25502],{"class":10430},[40,37684,167],{"class":10363},[40,37686,25507],{"class":10732},[40,37688,30096],{"class":10363},[40,37690,37691],{"class":10276,"line":10907},[40,37692,10301],{"emptyLinePlaceholder":10300},[40,37694,37695],{"class":10276,"line":10920},[40,37696,37697],{"class":10280},"# Lollipop-style plot for memory addressing\n",[40,37699,37700,37703,37705,37708,37710,37712,37714,37716,37718,37720,37722,37724,37726,37728,37730],{"class":10276,"line":10933},[40,37701,37702],{"class":10363},"ax_combined.",[40,37704,28083],{"class":10430},[40,37706,37707],{"class":10363},"(memory_addresses, gaussian_modulation, ",[40,37709,28088],{"class":11903},[40,37711,162],{"class":10732},[40,37713,28093],{"class":10287},[40,37715,10484],{"class":10363},[40,37717,28098],{"class":11903},[40,37719,162],{"class":10732},[40,37721,28103],{"class":10287},[40,37723,10484],{"class":10363},[40,37725,28108],{"class":11903},[40,37727,162],{"class":10732},[40,37729,28113],{"class":10287},[40,37731,10761],{"class":10363},[40,37733,37734],{"class":10276,"line":10946},[40,37735,10301],{"emptyLinePlaceholder":10300},[40,37737,37738],{"class":10276,"line":10951},[40,37739,37740],{"class":10280},"# Labels and title\n",[40,37742,37743,37745,37747,37749,37752],{"class":10276,"line":10956},[40,37744,37702],{"class":10363},[40,37746,30334],{"class":10430},[40,37748,167],{"class":10363},[40,37750,37751],{"class":10287},"\"Memory Addressing with Lollipop Representation for Quantum Gate Control\"",[40,37753,10761],{"class":10363},[40,37755,37756,37758,37760,37762,37765],{"class":10276,"line":10962},[40,37757,37702],{"class":10363},[40,37759,13230],{"class":10430},[40,37761,167],{"class":10363},[40,37763,37764],{"class":10287},"\"Memory Address Index\"",[40,37766,10761],{"class":10363},[40,37768,37769,37771,37773,37775,37777],{"class":10276,"line":10983},[40,37770,37702],{"class":10363},[40,37772,13244],{"class":10430},[40,37774,167],{"class":10363},[40,37776,30361],{"class":10287},[40,37778,10761],{"class":10363},[40,37780,37781,37783,37785,37787,37789],{"class":10276,"line":11005},[40,37782,37702],{"class":10363},[40,37784,27579],{"class":10430},[40,37786,167],{"class":10363},[40,37788,13091],{"class":10446},[40,37790,10761],{"class":10363},[40,37792,37793],{"class":10276,"line":11011},[40,37794,10301],{"emptyLinePlaceholder":10300},[40,37796,37797],{"class":10276,"line":11022},[40,37798,37799],{"class":10280},"# Display plots\n",[40,37801,37802,37804,37806],{"class":10276,"line":11064},[40,37803,23469],{"class":10363},[40,37805,23686],{"class":10430},[40,37807,12174],{"class":10363},[13997,37809],{"alt":37810,"src":37811},"Output 13 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-13.webp",[13997,37813],{"alt":37814,"src":37815},"Output 14 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-14.webp",[10082,37817,37819],{"className":10269,"code":37818,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define parameters\nt0 = 0      # Pulse center\nsigma = 5   # Standard deviation of Gaussian modulation function (ns)\nA0 = 1      # Peak amplitude\nfrequency = 0.2  # Frequency of the carrier wave (normalized)\n\n# Increase the number of points by 1.5x for denser visualization\nnum_points = int(40 * 1.5)\nt = np.linspace(-20, 20, num_points)  # Adjusted time axis\n\n# Compute Gaussian modulation function\ngaussian_modulation = A0 * np.exp(-(t - t0)**2 \u002F (2 * sigma**2))\n\n# Generate In-phase (I) and Quadrature (Q) components\nI = gaussian_modulation * np.cos(2 * np.pi * frequency * t)\nQ = gaussian_modulation * np.sin(2 * np.pi * frequency * t)\n\n# Plot IQ diagram\nfig, ax = plt.subplots(figsize=(6, 6), constrained_layout=True)\nax.plot(I, Q, 'bo-', alpha=0.7, label=\"IQ Trajectory\")\nax.set_xlabel(\"In-phase (I)\")\nax.set_ylabel(\"Quadrature (Q)\")\nax.set_title(\"IQ Diagram for Gaussian Modulated Pulse\")\nax.axhline(0, color='k', linewidth=0.5)\nax.axvline(0, color='k', linewidth=0.5)\nax.grid(True)\nax.legend()\n\nplt.show()\n",[10088,37820,37821,37831,37841,37845,37849,37859,37869,37879,37891,37895,37900,37918,37942,37946,37950,37994,37998,38003,38035,38066,38070,38075,38110,38143,38156,38169,38182,38211,38239,38251,38259,38263],{"__ignoreMap":10090},[40,37822,37823,37825,37827,37829],{"class":10276,"line":10277},[40,37824,10375],{"class":10352},[40,37826,10378],{"class":10363},[40,37828,10381],{"class":10352},[40,37830,10384],{"class":10363},[40,37832,37833,37835,37837,37839],{"class":10276,"line":10284},[40,37834,10375],{"class":10352},[40,37836,10405],{"class":10363},[40,37838,10381],{"class":10352},[40,37840,10410],{"class":10363},[40,37842,37843],{"class":10276,"line":10291},[40,37844,10301],{"emptyLinePlaceholder":10300},[40,37846,37847],{"class":10276,"line":10297},[40,37848,37255],{"class":10280},[40,37850,37851,37853,37855,37857],{"class":10276,"line":10304},[40,37852,37260],{"class":10363},[40,37854,162],{"class":10732},[40,37856,11234],{"class":10446},[40,37858,37267],{"class":10280},[40,37860,37861,37863,37865,37867],{"class":10276,"line":10310},[40,37862,37272],{"class":10363},[40,37864,162],{"class":10732},[40,37866,28264],{"class":10446},[40,37868,37279],{"class":10280},[40,37870,37871,37873,37875,37877],{"class":10276,"line":10316},[40,37872,37284],{"class":10363},[40,37874,162],{"class":10732},[40,37876,13373],{"class":10446},[40,37878,37291],{"class":10280},[40,37880,37881,37883,37885,37888],{"class":10276,"line":10322},[40,37882,28259],{"class":10363},[40,37884,162],{"class":10732},[40,37886,37887],{"class":10446}," 0.2",[40,37889,37890],{"class":10280},"  # Frequency of the carrier wave (normalized)\n",[40,37892,37893],{"class":10276,"line":10327},[40,37894,10301],{"emptyLinePlaceholder":10300},[40,37896,37897],{"class":10276,"line":10333},[40,37898,37899],{"class":10280},"# Increase the number of points by 1.5x for denser visualization\n",[40,37901,37902,37904,37906,37908,37910,37912,37914,37916],{"class":10276,"line":10339},[40,37903,37305],{"class":10363},[40,37905,162],{"class":10732},[40,37907,37310],{"class":10732},[40,37909,167],{"class":10363},[40,37911,27987],{"class":10446},[40,37913,11881],{"class":10732},[40,37915,37319],{"class":10446},[40,37917,10761],{"class":10363},[40,37919,37920,37922,37924,37926,37928,37930,37932,37934,37936,37938,37940],{"class":10276,"line":10344},[40,37921,29894],{"class":10363},[40,37923,162],{"class":10732},[40,37925,11156],{"class":10363},[40,37927,11850],{"class":10430},[40,37929,167],{"class":10363},[40,37931,11056],{"class":10732},[40,37933,13742],{"class":10446},[40,37935,10484],{"class":10363},[40,37937,13742],{"class":10446},[40,37939,37344],{"class":10363},[40,37941,37347],{"class":10280},[40,37943,37944],{"class":10276,"line":10349},[40,37945,10301],{"emptyLinePlaceholder":10300},[40,37947,37948],{"class":10276,"line":10367},[40,37949,37356],{"class":10280},[40,37951,37952,37954,37956,37958,37960,37962,37964,37966,37968,37970,37972,37974,37976,37978,37980,37982,37984,37986,37988,37990,37992],{"class":10276,"line":10372},[40,37953,27468],{"class":10363},[40,37955,162],{"class":10732},[40,37957,37365],{"class":10363},[40,37959,11752],{"class":10732},[40,37961,11156],{"class":10363},[40,37963,5268],{"class":10430},[40,37965,167],{"class":10363},[40,37967,11056],{"class":10732},[40,37969,37378],{"class":10363},[40,37971,11056],{"class":10732},[40,37973,37383],{"class":10363},[40,37975,22571],{"class":10732},[40,37977,1044],{"class":10446},[40,37979,22539],{"class":10732},[40,37981,920],{"class":10363},[40,37983,1044],{"class":10446},[40,37985,11881],{"class":10732},[40,37987,37398],{"class":10363},[40,37989,22571],{"class":10732},[40,37991,1044],{"class":10446},[40,37993,13612],{"class":10363},[40,37995,37996],{"class":10276,"line":10387},[40,37997,10301],{"emptyLinePlaceholder":10300},[40,37999,38000],{"class":10276,"line":10400},[40,38001,38002],{"class":10280},"# Generate In-phase (I) and Quadrature (Q) components\n",[40,38004,38005,38008,38010,38012,38014,38016,38018,38020,38022,38024,38026,38028,38030,38032],{"class":10276,"line":10413},[40,38006,38007],{"class":10363},"I ",[40,38009,162],{"class":10732},[40,38011,28405],{"class":10363},[40,38013,11752],{"class":10732},[40,38015,11156],{"class":10363},[40,38017,1820],{"class":10430},[40,38019,167],{"class":10363},[40,38021,1044],{"class":10446},[40,38023,11881],{"class":10732},[40,38025,28376],{"class":10363},[40,38027,11752],{"class":10732},[40,38029,28381],{"class":10363},[40,38031,11752],{"class":10732},[40,38033,38034],{"class":10363}," t)\n",[40,38036,38037,38040,38042,38044,38046,38048,38050,38052,38054,38056,38058,38060,38062,38064],{"class":10276,"line":10418},[40,38038,38039],{"class":10363},"Q ",[40,38041,162],{"class":10732},[40,38043,28405],{"class":10363},[40,38045,11752],{"class":10732},[40,38047,11156],{"class":10363},[40,38049,1813],{"class":10430},[40,38051,167],{"class":10363},[40,38053,1044],{"class":10446},[40,38055,11881],{"class":10732},[40,38057,28376],{"class":10363},[40,38059,11752],{"class":10732},[40,38061,28381],{"class":10363},[40,38063,11752],{"class":10732},[40,38065,38034],{"class":10363},[40,38067,38068],{"class":10276,"line":10424},[40,38069,10301],{"emptyLinePlaceholder":10300},[40,38071,38072],{"class":10276,"line":10437},[40,38073,38074],{"class":10280},"# Plot IQ diagram\n",[40,38076,38077,38080,38082,38084,38086,38088,38090,38092,38094,38096,38098,38100,38102,38104,38106,38108],{"class":10276,"line":10456},[40,38078,38079],{"class":10363},"fig, ax ",[40,38081,162],{"class":10732},[40,38083,13070],{"class":10363},[40,38085,13073],{"class":10430},[40,38087,167],{"class":10363},[40,38089,13078],{"class":11903},[40,38091,162],{"class":10732},[40,38093,167],{"class":10363},[40,38095,12785],{"class":10446},[40,38097,10484],{"class":10363},[40,38099,12785],{"class":10446},[40,38101,11040],{"class":10363},[40,38103,13086],{"class":11903},[40,38105,162],{"class":10732},[40,38107,13091],{"class":10446},[40,38109,10761],{"class":10363},[40,38111,38112,38115,38117,38120,38123,38125,38127,38129,38132,38134,38136,38138,38141],{"class":10276,"line":10472},[40,38113,38114],{"class":10363},"ax.",[40,38116,23505],{"class":10430},[40,38118,38119],{"class":10363},"(I, Q, ",[40,38121,38122],{"class":10287},"'bo-'",[40,38124,10484],{"class":10363},[40,38126,12602],{"class":11903},[40,38128,162],{"class":10732},[40,38130,38131],{"class":10446},"0.7",[40,38133,10484],{"class":10363},[40,38135,15415],{"class":11903},[40,38137,162],{"class":10732},[40,38139,38140],{"class":10287},"\"IQ Trajectory\"",[40,38142,10761],{"class":10363},[40,38144,38145,38147,38149,38151,38154],{"class":10276,"line":10496},[40,38146,38114],{"class":10363},[40,38148,13230],{"class":10430},[40,38150,167],{"class":10363},[40,38152,38153],{"class":10287},"\"In-phase (I)\"",[40,38155,10761],{"class":10363},[40,38157,38158,38160,38162,38164,38167],{"class":10276,"line":10510},[40,38159,38114],{"class":10363},[40,38161,13244],{"class":10430},[40,38163,167],{"class":10363},[40,38165,38166],{"class":10287},"\"Quadrature (Q)\"",[40,38168,10761],{"class":10363},[40,38170,38171,38173,38175,38177,38180],{"class":10276,"line":10696},[40,38172,38114],{"class":10363},[40,38174,30334],{"class":10430},[40,38176,167],{"class":10363},[40,38178,38179],{"class":10287},"\"IQ Diagram for Gaussian Modulated Pulse\"",[40,38181,10761],{"class":10363},[40,38183,38184,38186,38188,38190,38192,38194,38196,38198,38201,38203,38205,38207,38209],{"class":10276,"line":10702},[40,38185,38114],{"class":10363},[40,38187,23564],{"class":10430},[40,38189,167],{"class":10363},[40,38191,66],{"class":10446},[40,38193,10484],{"class":10363},[40,38195,23550],{"class":11903},[40,38197,162],{"class":10732},[40,38199,38200],{"class":10287},"'k'",[40,38202,10484],{"class":10363},[40,38204,16619],{"class":11903},[40,38206,162],{"class":10732},[40,38208,24827],{"class":10446},[40,38210,10761],{"class":10363},[40,38212,38213,38215,38217,38219,38221,38223,38225,38227,38229,38231,38233,38235,38237],{"class":10276,"line":10708},[40,38214,38114],{"class":10363},[40,38216,23268],{"class":10430},[40,38218,167],{"class":10363},[40,38220,66],{"class":10446},[40,38222,10484],{"class":10363},[40,38224,23550],{"class":11903},[40,38226,162],{"class":10732},[40,38228,38200],{"class":10287},[40,38230,10484],{"class":10363},[40,38232,16619],{"class":11903},[40,38234,162],{"class":10732},[40,38236,24827],{"class":10446},[40,38238,10761],{"class":10363},[40,38240,38241,38243,38245,38247,38249],{"class":10276,"line":10714},[40,38242,38114],{"class":10363},[40,38244,27579],{"class":10430},[40,38246,167],{"class":10363},[40,38248,13091],{"class":10446},[40,38250,10761],{"class":10363},[40,38252,38253,38255,38257],{"class":10276,"line":10720},[40,38254,38114],{"class":10363},[40,38256,23677],{"class":10430},[40,38258,12174],{"class":10363},[40,38260,38261],{"class":10276,"line":10726},[40,38262,10301],{"emptyLinePlaceholder":10300},[40,38264,38265,38267,38269],{"class":10276,"line":10764},[40,38266,23469],{"class":10363},[40,38268,23686],{"class":10430},[40,38270,12174],{"class":10363},[13997,38272],{"alt":38273,"src":38274},"Output 15 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-15.webp",[10082,38276,38278],{"className":10269,"code":38277,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define parameters\nt0 = 0      # Pulse center\nsigma = 5   # Standard deviation of Gaussian modulation function (ns)\nA0 = 1      # Peak amplitude\n\n# Increase the number of points by 1.5x for denser lollipop visualization\nnum_points = int(40 * 1.5)\nt = np.linspace(-20, 20, num_points)  # Adjusted time axis\n\n# Compute Gaussian modulation function\ngaussian_modulation = A0 * np.exp(-(t - t0)**2 \u002F (2 * sigma**2))\n\n# Store modulation functions and titles in an array\nmodulations = [\n    (gaussian_modulation, \"Gaussian Modulation Function for Pauli-X Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for Pauli-Y Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for Hadamard Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for CNOT Gate Target Qubit\")\n]\n\n# Create lollipop plots\nfig, axs = plt.subplots(2, 2, figsize=(12, 8), constrained_layout=True)\naxs = axs.flatten()\n\nfor ax, (modulation, title) in zip(axs, modulations):\n    ax.stem(t, gaussian_modulation, linefmt='b-', markerfmt='bo', basefmt='k-')\n    ax.set_title(title)\n    ax.set_xlabel(\"Time (ns)\")\n    ax.set_ylabel(\"Amplitude\")\n\nplt.show()\n",[10088,38279,38280,38290,38300,38304,38308,38318,38328,38338,38342,38346,38364,38388,38392,38396,38440,38444,38448,38456,38464,38472,38480,38488,38492,38496,38501,38543,38555,38559,38571,38604,38612,38624,38636,38640],{"__ignoreMap":10090},[40,38281,38282,38284,38286,38288],{"class":10276,"line":10277},[40,38283,10375],{"class":10352},[40,38285,10378],{"class":10363},[40,38287,10381],{"class":10352},[40,38289,10384],{"class":10363},[40,38291,38292,38294,38296,38298],{"class":10276,"line":10284},[40,38293,10375],{"class":10352},[40,38295,10405],{"class":10363},[40,38297,10381],{"class":10352},[40,38299,10410],{"class":10363},[40,38301,38302],{"class":10276,"line":10291},[40,38303,10301],{"emptyLinePlaceholder":10300},[40,38305,38306],{"class":10276,"line":10297},[40,38307,37255],{"class":10280},[40,38309,38310,38312,38314,38316],{"class":10276,"line":10304},[40,38311,37260],{"class":10363},[40,38313,162],{"class":10732},[40,38315,11234],{"class":10446},[40,38317,37267],{"class":10280},[40,38319,38320,38322,38324,38326],{"class":10276,"line":10310},[40,38321,37272],{"class":10363},[40,38323,162],{"class":10732},[40,38325,28264],{"class":10446},[40,38327,37279],{"class":10280},[40,38329,38330,38332,38334,38336],{"class":10276,"line":10316},[40,38331,37284],{"class":10363},[40,38333,162],{"class":10732},[40,38335,13373],{"class":10446},[40,38337,37291],{"class":10280},[40,38339,38340],{"class":10276,"line":10322},[40,38341,10301],{"emptyLinePlaceholder":10300},[40,38343,38344],{"class":10276,"line":10327},[40,38345,37300],{"class":10280},[40,38347,38348,38350,38352,38354,38356,38358,38360,38362],{"class":10276,"line":10333},[40,38349,37305],{"class":10363},[40,38351,162],{"class":10732},[40,38353,37310],{"class":10732},[40,38355,167],{"class":10363},[40,38357,27987],{"class":10446},[40,38359,11881],{"class":10732},[40,38361,37319],{"class":10446},[40,38363,10761],{"class":10363},[40,38365,38366,38368,38370,38372,38374,38376,38378,38380,38382,38384,38386],{"class":10276,"line":10339},[40,38367,29894],{"class":10363},[40,38369,162],{"class":10732},[40,38371,11156],{"class":10363},[40,38373,11850],{"class":10430},[40,38375,167],{"class":10363},[40,38377,11056],{"class":10732},[40,38379,13742],{"class":10446},[40,38381,10484],{"class":10363},[40,38383,13742],{"class":10446},[40,38385,37344],{"class":10363},[40,38387,37347],{"class":10280},[40,38389,38390],{"class":10276,"line":10344},[40,38391,10301],{"emptyLinePlaceholder":10300},[40,38393,38394],{"class":10276,"line":10349},[40,38395,37356],{"class":10280},[40,38397,38398,38400,38402,38404,38406,38408,38410,38412,38414,38416,38418,38420,38422,38424,38426,38428,38430,38432,38434,38436,38438],{"class":10276,"line":10367},[40,38399,27468],{"class":10363},[40,38401,162],{"class":10732},[40,38403,37365],{"class":10363},[40,38405,11752],{"class":10732},[40,38407,11156],{"class":10363},[40,38409,5268],{"class":10430},[40,38411,167],{"class":10363},[40,38413,11056],{"class":10732},[40,38415,37378],{"class":10363},[40,38417,11056],{"class":10732},[40,38419,37383],{"class":10363},[40,38421,22571],{"class":10732},[40,38423,1044],{"class":10446},[40,38425,22539],{"class":10732},[40,38427,920],{"class":10363},[40,38429,1044],{"class":10446},[40,38431,11881],{"class":10732},[40,38433,37398],{"class":10363},[40,38435,22571],{"class":10732},[40,38437,1044],{"class":10446},[40,38439,13612],{"class":10363},[40,38441,38442],{"class":10276,"line":10372},[40,38443,10301],{"emptyLinePlaceholder":10300},[40,38445,38446],{"class":10276,"line":10387},[40,38447,37413],{"class":10280},[40,38449,38450,38452,38454],{"class":10276,"line":10400},[40,38451,30731],{"class":10363},[40,38453,162],{"class":10732},[40,38455,37422],{"class":10363},[40,38457,38458,38460,38462],{"class":10276,"line":10413},[40,38459,37427],{"class":10363},[40,38461,37430],{"class":10287},[40,38463,11061],{"class":10363},[40,38465,38466,38468,38470],{"class":10276,"line":10418},[40,38467,37427],{"class":10363},[40,38469,37439],{"class":10287},[40,38471,11061],{"class":10363},[40,38473,38474,38476,38478],{"class":10276,"line":10424},[40,38475,37427],{"class":10363},[40,38477,37448],{"class":10287},[40,38479,11061],{"class":10363},[40,38481,38482,38484,38486],{"class":10276,"line":10437},[40,38483,37427],{"class":10363},[40,38485,37457],{"class":10287},[40,38487,10761],{"class":10363},[40,38489,38490],{"class":10276,"line":10456},[40,38491,25466],{"class":10363},[40,38493,38494],{"class":10276,"line":10472},[40,38495,10301],{"emptyLinePlaceholder":10300},[40,38497,38498],{"class":10276,"line":10496},[40,38499,38500],{"class":10280},"# Create lollipop plots\n",[40,38502,38503,38505,38507,38509,38511,38513,38515,38517,38519,38521,38523,38525,38527,38529,38531,38533,38535,38537,38539,38541],{"class":10276,"line":10510},[40,38504,37477],{"class":10363},[40,38506,162],{"class":10732},[40,38508,13070],{"class":10363},[40,38510,13073],{"class":10430},[40,38512,167],{"class":10363},[40,38514,1044],{"class":10446},[40,38516,10484],{"class":10363},[40,38518,1044],{"class":10446},[40,38520,10484],{"class":10363},[40,38522,13078],{"class":11903},[40,38524,162],{"class":10732},[40,38526,167],{"class":10363},[40,38528,30273],{"class":10446},[40,38530,10484],{"class":10363},[40,38532,23482],{"class":10446},[40,38534,11040],{"class":10363},[40,38536,13086],{"class":11903},[40,38538,162],{"class":10732},[40,38540,13091],{"class":10446},[40,38542,10761],{"class":10363},[40,38544,38545,38547,38549,38551,38553],{"class":10276,"line":10696},[40,38546,37520],{"class":10363},[40,38548,162],{"class":10732},[40,38550,37525],{"class":10363},[40,38552,30292],{"class":10430},[40,38554,12174],{"class":10363},[40,38556,38557],{"class":10276,"line":10702},[40,38558,10301],{"emptyLinePlaceholder":10300},[40,38560,38561,38563,38565,38567,38569],{"class":10276,"line":10708},[40,38562,13840],{"class":10352},[40,38564,37545],{"class":10363},[40,38566,13846],{"class":10352},[40,38568,37550],{"class":10732},[40,38570,37553],{"class":10363},[40,38572,38573,38575,38577,38580,38582,38584,38586,38588,38590,38592,38594,38596,38598,38600,38602],{"class":10276,"line":10714},[40,38574,13227],{"class":10363},[40,38576,28083],{"class":10430},[40,38578,38579],{"class":10363},"(t, gaussian_modulation, ",[40,38581,28088],{"class":11903},[40,38583,162],{"class":10732},[40,38585,28093],{"class":10287},[40,38587,10484],{"class":10363},[40,38589,28098],{"class":11903},[40,38591,162],{"class":10732},[40,38593,28103],{"class":10287},[40,38595,10484],{"class":10363},[40,38597,28108],{"class":11903},[40,38599,162],{"class":10732},[40,38601,28113],{"class":10287},[40,38603,10761],{"class":10363},[40,38605,38606,38608,38610],{"class":10276,"line":10720},[40,38607,13227],{"class":10363},[40,38609,30334],{"class":10430},[40,38611,30337],{"class":10363},[40,38613,38614,38616,38618,38620,38622],{"class":10276,"line":10726},[40,38615,13227],{"class":10363},[40,38617,13230],{"class":10430},[40,38619,167],{"class":10363},[40,38621,37604],{"class":10287},[40,38623,10761],{"class":10363},[40,38625,38626,38628,38630,38632,38634],{"class":10276,"line":10764},[40,38627,13227],{"class":10363},[40,38629,13244],{"class":10430},[40,38631,167],{"class":10363},[40,38633,30361],{"class":10287},[40,38635,10761],{"class":10363},[40,38637,38638],{"class":10276,"line":10778},[40,38639,10301],{"emptyLinePlaceholder":10300},[40,38641,38642,38644,38646],{"class":10276,"line":10815},[40,38643,23469],{"class":10363},[40,38645,23686],{"class":10430},[40,38647,12174],{"class":10363},[13997,38649],{"alt":38650,"src":38651},"Output 16 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-16.webp",[10082,38653,38655],{"className":10269,"code":38654,"language":10271,"meta":10090,"style":10090},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define parameters\nt0 = 0      # Pulse center\nsigma = 5   # Standard deviation of Gaussian modulation (ns)\nA0 = 1      # Peak amplitude\nfc = 5e9    # Carrier frequency (5 GHz, typical for superconducting qubits)\nphi_X = 0           # X gate (Pauli-X) - $\\pi$ rotation around X-axis\nphi_Y = np.pi \u002F 2   # Y gate (Pauli-Y) - $\\pi\u002F2$ rotation around Y-axis\nphi_H = np.pi \u002F 2   # Hadamard gate (H) - $\\pi\u002F2$ rotation around Y-axis\nt = np.linspace(-20, 20, 1000)  # Time axis in nanoseconds\n\n# Compute Gaussian modulation function\ngaussian_envelope = A0 * np.exp(-(t - t0)**2 \u002F (2 * sigma**2))\n\n# Compute modulated pulses for X, Y, H, and CNOT gates\nmodulated_pulse_X = gaussian_envelope * np.cos(2 * np.pi * fc * t * 1e-9 + phi_X)\nmodulated_pulse_Y = gaussian_envelope * np.cos(2 * np.pi * fc * t * 1e-9 + phi_Y)\nmodulated_pulse_H = gaussian_envelope * np.cos(2 * np.pi * fc * t * 1e-9 + phi_H)\nmodulated_pulse_CNOT = gaussian_envelope * np.cos(2 * np.pi * fc * t * 1e-9 + phi_X)  # Similar to X gate\n\n# Store pulses and titles in an array\npulses = np.array([\n    (modulated_pulse_X, \"Pauli-X Gate\"),\n    (modulated_pulse_Y, \"Pauli-Y Gate\"),\n    (modulated_pulse_H, \"Hadamard Gate\"),\n    (modulated_pulse_CNOT, \"CNOT Gate Target Qubit\")\n], dtype=object)\n\n# Create plots with blue pulse lines\nfig, axs = plt.subplots(2, 2, figsize=(12, 8), constrained_layout=True)\naxs = axs.flatten()\n\nfor ax, (pulse, title) in zip(axs, pulses):\n    ax.plot(t, pulse, 'b', linewidth=0.5, label=f\"{title} Modulated Pulse\")  # Blue line with reduced thickness\n    ax.plot(t, gaussian_envelope, 'r--', linewidth=1.0, label=\"Gaussian Envelope\")\n    ax.plot(t, -gaussian_envelope, 'r--', linewidth=1.0)\n    ax.set_title(f\"Modulated Pulse for {title}\")\n    ax.set_xlabel(\"Time (ns)\")\n    ax.set_ylabel(\"Amplitude\")\n    # ax.legend()\n    # ax.grid()\n\nplt.show()\n",[10088,38656,38657,38667,38677,38681,38685,38695,38706,38716,38729,38741,38757,38773,38802,38806,38810,38855,38859,38864,38908,38948,38988,39031,39035,39040,39054,39064,39074,39084,39094,39105,39109,39114,39156,39168,39172,39186,39229,39260,39286,39309,39321,39333,39338,39343,39347],{"__ignoreMap":10090},[40,38658,38659,38661,38663,38665],{"class":10276,"line":10277},[40,38660,10375],{"class":10352},[40,38662,10378],{"class":10363},[40,38664,10381],{"class":10352},[40,38666,10384],{"class":10363},[40,38668,38669,38671,38673,38675],{"class":10276,"line":10284},[40,38670,10375],{"class":10352},[40,38672,10405],{"class":10363},[40,38674,10381],{"class":10352},[40,38676,10410],{"class":10363},[40,38678,38679],{"class":10276,"line":10291},[40,38680,10301],{"emptyLinePlaceholder":10300},[40,38682,38683],{"class":10276,"line":10297},[40,38684,37255],{"class":10280},[40,38686,38687,38689,38691,38693],{"class":10276,"line":10304},[40,38688,37260],{"class":10363},[40,38690,162],{"class":10732},[40,38692,11234],{"class":10446},[40,38694,37267],{"class":10280},[40,38696,38697,38699,38701,38703],{"class":10276,"line":10310},[40,38698,37272],{"class":10363},[40,38700,162],{"class":10732},[40,38702,28264],{"class":10446},[40,38704,38705],{"class":10280},"   # Standard deviation of Gaussian modulation (ns)\n",[40,38707,38708,38710,38712,38714],{"class":10276,"line":10316},[40,38709,37284],{"class":10363},[40,38711,162],{"class":10732},[40,38713,13373],{"class":10446},[40,38715,37291],{"class":10280},[40,38717,38718,38721,38723,38726],{"class":10276,"line":10322},[40,38719,38720],{"class":10363},"fc ",[40,38722,162],{"class":10732},[40,38724,38725],{"class":10446}," 5e9",[40,38727,38728],{"class":10280},"    # Carrier frequency (5 GHz, typical for superconducting qubits)\n",[40,38730,38731,38734,38736,38738],{"class":10276,"line":10327},[40,38732,38733],{"class":10363},"phi_X ",[40,38735,162],{"class":10732},[40,38737,11234],{"class":10446},[40,38739,38740],{"class":10280},"           # X gate (Pauli-X) - $\\pi$ rotation around X-axis\n",[40,38742,38743,38746,38748,38750,38752,38754],{"class":10276,"line":10333},[40,38744,38745],{"class":10363},"phi_Y ",[40,38747,162],{"class":10732},[40,38749,28376],{"class":10363},[40,38751,11243],{"class":10732},[40,38753,12960],{"class":10446},[40,38755,38756],{"class":10280},"   # Y gate (Pauli-Y) - $\\pi\u002F2$ rotation around Y-axis\n",[40,38758,38759,38762,38764,38766,38768,38770],{"class":10276,"line":10339},[40,38760,38761],{"class":10363},"phi_H ",[40,38763,162],{"class":10732},[40,38765,28376],{"class":10363},[40,38767,11243],{"class":10732},[40,38769,12960],{"class":10446},[40,38771,38772],{"class":10280},"   # Hadamard gate (H) - $\\pi\u002F2$ rotation around Y-axis\n",[40,38774,38775,38777,38779,38781,38783,38785,38787,38789,38791,38793,38795,38797,38799],{"class":10276,"line":10344},[40,38776,29894],{"class":10363},[40,38778,162],{"class":10732},[40,38780,11156],{"class":10363},[40,38782,11850],{"class":10430},[40,38784,167],{"class":10363},[40,38786,11056],{"class":10732},[40,38788,13742],{"class":10446},[40,38790,10484],{"class":10363},[40,38792,13742],{"class":10446},[40,38794,10484],{"class":10363},[40,38796,27449],{"class":10446},[40,38798,17908],{"class":10363},[40,38800,38801],{"class":10280},"# Time axis in nanoseconds\n",[40,38803,38804],{"class":10276,"line":10349},[40,38805,10301],{"emptyLinePlaceholder":10300},[40,38807,38808],{"class":10276,"line":10367},[40,38809,37356],{"class":10280},[40,38811,38812,38815,38817,38819,38821,38823,38825,38827,38829,38831,38833,38835,38837,38839,38841,38843,38845,38847,38849,38851,38853],{"class":10276,"line":10372},[40,38813,38814],{"class":10363},"gaussian_envelope ",[40,38816,162],{"class":10732},[40,38818,37365],{"class":10363},[40,38820,11752],{"class":10732},[40,38822,11156],{"class":10363},[40,38824,5268],{"class":10430},[40,38826,167],{"class":10363},[40,38828,11056],{"class":10732},[40,38830,37378],{"class":10363},[40,38832,11056],{"class":10732},[40,38834,37383],{"class":10363},[40,38836,22571],{"class":10732},[40,38838,1044],{"class":10446},[40,38840,22539],{"class":10732},[40,38842,920],{"class":10363},[40,38844,1044],{"class":10446},[40,38846,11881],{"class":10732},[40,38848,37398],{"class":10363},[40,38850,22571],{"class":10732},[40,38852,1044],{"class":10446},[40,38854,13612],{"class":10363},[40,38856,38857],{"class":10276,"line":10387},[40,38858,10301],{"emptyLinePlaceholder":10300},[40,38860,38861],{"class":10276,"line":10400},[40,38862,38863],{"class":10280},"# Compute modulated pulses for X, Y, H, and CNOT gates\n",[40,38865,38866,38869,38871,38874,38876,38878,38880,38882,38884,38886,38888,38890,38893,38895,38898,38900,38903,38905],{"class":10276,"line":10413},[40,38867,38868],{"class":10363},"modulated_pulse_X ",[40,38870,162],{"class":10732},[40,38872,38873],{"class":10363}," gaussian_envelope ",[40,38875,11752],{"class":10732},[40,38877,11156],{"class":10363},[40,38879,1820],{"class":10430},[40,38881,167],{"class":10363},[40,38883,1044],{"class":10446},[40,38885,11881],{"class":10732},[40,38887,28376],{"class":10363},[40,38889,11752],{"class":10732},[40,38891,38892],{"class":10363}," fc ",[40,38894,11752],{"class":10732},[40,38896,38897],{"class":10363}," t ",[40,38899,11752],{"class":10732},[40,38901,38902],{"class":10446}," 1e-9",[40,38904,12042],{"class":10732},[40,38906,38907],{"class":10363}," phi_X)\n",[40,38909,38910,38913,38915,38917,38919,38921,38923,38925,38927,38929,38931,38933,38935,38937,38939,38941,38943,38945],{"class":10276,"line":10418},[40,38911,38912],{"class":10363},"modulated_pulse_Y ",[40,38914,162],{"class":10732},[40,38916,38873],{"class":10363},[40,38918,11752],{"class":10732},[40,38920,11156],{"class":10363},[40,38922,1820],{"class":10430},[40,38924,167],{"class":10363},[40,38926,1044],{"class":10446},[40,38928,11881],{"class":10732},[40,38930,28376],{"class":10363},[40,38932,11752],{"class":10732},[40,38934,38892],{"class":10363},[40,38936,11752],{"class":10732},[40,38938,38897],{"class":10363},[40,38940,11752],{"class":10732},[40,38942,38902],{"class":10446},[40,38944,12042],{"class":10732},[40,38946,38947],{"class":10363}," phi_Y)\n",[40,38949,38950,38953,38955,38957,38959,38961,38963,38965,38967,38969,38971,38973,38975,38977,38979,38981,38983,38985],{"class":10276,"line":10424},[40,38951,38952],{"class":10363},"modulated_pulse_H ",[40,38954,162],{"class":10732},[40,38956,38873],{"class":10363},[40,38958,11752],{"class":10732},[40,38960,11156],{"class":10363},[40,38962,1820],{"class":10430},[40,38964,167],{"class":10363},[40,38966,1044],{"class":10446},[40,38968,11881],{"class":10732},[40,38970,28376],{"class":10363},[40,38972,11752],{"class":10732},[40,38974,38892],{"class":10363},[40,38976,11752],{"class":10732},[40,38978,38897],{"class":10363},[40,38980,11752],{"class":10732},[40,38982,38902],{"class":10446},[40,38984,12042],{"class":10732},[40,38986,38987],{"class":10363}," phi_H)\n",[40,38989,38990,38993,38995,38997,38999,39001,39003,39005,39007,39009,39011,39013,39015,39017,39019,39021,39023,39025,39028],{"class":10276,"line":10437},[40,38991,38992],{"class":10363},"modulated_pulse_CNOT ",[40,38994,162],{"class":10732},[40,38996,38873],{"class":10363},[40,38998,11752],{"class":10732},[40,39000,11156],{"class":10363},[40,39002,1820],{"class":10430},[40,39004,167],{"class":10363},[40,39006,1044],{"class":10446},[40,39008,11881],{"class":10732},[40,39010,28376],{"class":10363},[40,39012,11752],{"class":10732},[40,39014,38892],{"class":10363},[40,39016,11752],{"class":10732},[40,39018,38897],{"class":10363},[40,39020,11752],{"class":10732},[40,39022,38902],{"class":10446},[40,39024,12042],{"class":10732},[40,39026,39027],{"class":10363}," phi_X)  ",[40,39029,39030],{"class":10280},"# Similar to X gate\n",[40,39032,39033],{"class":10276,"line":10456},[40,39034,10301],{"emptyLinePlaceholder":10300},[40,39036,39037],{"class":10276,"line":10472},[40,39038,39039],{"class":10280},"# Store pulses and titles in an array\n",[40,39041,39042,39045,39047,39049,39051],{"class":10276,"line":10496},[40,39043,39044],{"class":10363},"pulses ",[40,39046,162],{"class":10732},[40,39048,11156],{"class":10363},[40,39050,11206],{"class":10430},[40,39052,39053],{"class":10363},"([\n",[40,39055,39056,39059,39062],{"class":10276,"line":10510},[40,39057,39058],{"class":10363},"    (modulated_pulse_X, ",[40,39060,39061],{"class":10287},"\"Pauli-X Gate\"",[40,39063,11061],{"class":10363},[40,39065,39066,39069,39072],{"class":10276,"line":10696},[40,39067,39068],{"class":10363},"    (modulated_pulse_Y, ",[40,39070,39071],{"class":10287},"\"Pauli-Y Gate\"",[40,39073,11061],{"class":10363},[40,39075,39076,39079,39082],{"class":10276,"line":10702},[40,39077,39078],{"class":10363},"    (modulated_pulse_H, ",[40,39080,39081],{"class":10287},"\"Hadamard Gate\"",[40,39083,11061],{"class":10363},[40,39085,39086,39089,39092],{"class":10276,"line":10708},[40,39087,39088],{"class":10363},"    (modulated_pulse_CNOT, ",[40,39090,39091],{"class":10287},"\"CNOT Gate Target Qubit\"",[40,39093,10761],{"class":10363},[40,39095,39096,39098,39100,39103],{"class":10276,"line":10714},[40,39097,11672],{"class":10363},[40,39099,26704],{"class":11903},[40,39101,39102],{"class":10732},"=object",[40,39104,10761],{"class":10363},[40,39106,39107],{"class":10276,"line":10720},[40,39108,10301],{"emptyLinePlaceholder":10300},[40,39110,39111],{"class":10276,"line":10726},[40,39112,39113],{"class":10280},"# Create plots with blue pulse lines\n",[40,39115,39116,39118,39120,39122,39124,39126,39128,39130,39132,39134,39136,39138,39140,39142,39144,39146,39148,39150,39152,39154],{"class":10276,"line":10764},[40,39117,37477],{"class":10363},[40,39119,162],{"class":10732},[40,39121,13070],{"class":10363},[40,39123,13073],{"class":10430},[40,39125,167],{"class":10363},[40,39127,1044],{"class":10446},[40,39129,10484],{"class":10363},[40,39131,1044],{"class":10446},[40,39133,10484],{"class":10363},[40,39135,13078],{"class":11903},[40,39137,162],{"class":10732},[40,39139,167],{"class":10363},[40,39141,30273],{"class":10446},[40,39143,10484],{"class":10363},[40,39145,23482],{"class":10446},[40,39147,11040],{"class":10363},[40,39149,13086],{"class":11903},[40,39151,162],{"class":10732},[40,39153,13091],{"class":10446},[40,39155,10761],{"class":10363},[40,39157,39158,39160,39162,39164,39166],{"class":10276,"line":10778},[40,39159,37520],{"class":10363},[40,39161,162],{"class":10732},[40,39163,37525],{"class":10363},[40,39165,30292],{"class":10430},[40,39167,12174],{"class":10363},[40,39169,39170],{"class":10276,"line":10815},[40,39171,10301],{"emptyLinePlaceholder":10300},[40,39173,39174,39176,39179,39181,39183],{"class":10276,"line":10829},[40,39175,13840],{"class":10352},[40,39177,39178],{"class":10363}," ax, (pulse, title) ",[40,39180,13846],{"class":10352},[40,39182,37550],{"class":10732},[40,39184,39185],{"class":10363},"(axs, pulses):\n",[40,39187,39188,39190,39192,39195,39197,39199,39201,39203,39205,39207,39209,39211,39213,39215,39217,39219,39221,39224,39226],{"class":10276,"line":10842},[40,39189,13227],{"class":10363},[40,39191,23505],{"class":10430},[40,39193,39194],{"class":10363},"(t, pulse, ",[40,39196,30325],{"class":10287},[40,39198,10484],{"class":10363},[40,39200,16619],{"class":11903},[40,39202,162],{"class":10732},[40,39204,24827],{"class":10446},[40,39206,10484],{"class":10363},[40,39208,15415],{"class":11903},[40,39210,162],{"class":10732},[40,39212,13262],{"class":10352},[40,39214,13478],{"class":10287},[40,39216,13268],{"class":10446},[40,39218,23656],{"class":10363},[40,39220,13280],{"class":10446},[40,39222,39223],{"class":10287}," Modulated Pulse\"",[40,39225,17908],{"class":10363},[40,39227,39228],{"class":10280},"# Blue line with reduced thickness\n",[40,39230,39231,39233,39235,39238,39241,39243,39245,39247,39249,39251,39253,39255,39258],{"class":10276,"line":10855},[40,39232,13227],{"class":10363},[40,39234,23505],{"class":10430},[40,39236,39237],{"class":10363},"(t, gaussian_envelope, ",[40,39239,39240],{"class":10287},"'r--'",[40,39242,10484],{"class":10363},[40,39244,16619],{"class":11903},[40,39246,162],{"class":10732},[40,39248,10810],{"class":10446},[40,39250,10484],{"class":10363},[40,39252,15415],{"class":11903},[40,39254,162],{"class":10732},[40,39256,39257],{"class":10287},"\"Gaussian Envelope\"",[40,39259,10761],{"class":10363},[40,39261,39262,39264,39266,39269,39271,39274,39276,39278,39280,39282,39284],{"class":10276,"line":10868},[40,39263,13227],{"class":10363},[40,39265,23505],{"class":10430},[40,39267,39268],{"class":10363},"(t, ",[40,39270,11056],{"class":10732},[40,39272,39273],{"class":10363},"gaussian_envelope, ",[40,39275,39240],{"class":10287},[40,39277,10484],{"class":10363},[40,39279,16619],{"class":11903},[40,39281,162],{"class":10732},[40,39283,10810],{"class":10446},[40,39285,10761],{"class":10363},[40,39287,39288,39290,39292,39294,39296,39299,39301,39303,39305,39307],{"class":10276,"line":10881},[40,39289,13227],{"class":10363},[40,39291,30334],{"class":10430},[40,39293,167],{"class":10363},[40,39295,13262],{"class":10352},[40,39297,39298],{"class":10287},"\"Modulated Pulse for ",[40,39300,13268],{"class":10446},[40,39302,23656],{"class":10363},[40,39304,13280],{"class":10446},[40,39306,13478],{"class":10287},[40,39308,10761],{"class":10363},[40,39310,39311,39313,39315,39317,39319],{"class":10276,"line":10907},[40,39312,13227],{"class":10363},[40,39314,13230],{"class":10430},[40,39316,167],{"class":10363},[40,39318,37604],{"class":10287},[40,39320,10761],{"class":10363},[40,39322,39323,39325,39327,39329,39331],{"class":10276,"line":10920},[40,39324,13227],{"class":10363},[40,39326,13244],{"class":10430},[40,39328,167],{"class":10363},[40,39330,30361],{"class":10287},[40,39332,10761],{"class":10363},[40,39334,39335],{"class":10276,"line":10933},[40,39336,39337],{"class":10280},"    # ax.legend()\n",[40,39339,39340],{"class":10276,"line":10946},[40,39341,39342],{"class":10280},"    # ax.grid()\n",[40,39344,39345],{"class":10276,"line":10951},[40,39346,10301],{"emptyLinePlaceholder":10300},[40,39348,39349,39351,39353],{"class":10276,"line":10956},[40,39350,23469],{"class":10363},[40,39352,23686],{"class":10430},[40,39354,12174],{"class":10363},[13997,39356],{"alt":39357,"src":39358},"Output 17 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-17.webp",[20069,39360,39361],{},"html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":10090,"searchDepth":10284,"depth":10284,"links":39363},[39364,39365,39366,39367],{"id":32268,"depth":10291,"text":32271},{"id":32845,"depth":10291,"text":32848},{"id":33599,"depth":10291,"text":33602},{"id":34425,"depth":10291,"text":34428},[20088,20089,20090,39369],"Pulse shapes",[39371],{"username":7,"name":20094,"role":20095,"bio":20096,"links":39372},[39373,39374],{"label":20099,"href":20100},{"label":20102,"href":27302},{"username":7,"name":20094,"role":20104},"How microwave pulses are shaped to control a quantum chip: Gaussian and sinusoidal envelopes, and the modulated pulses behind common gates.",{},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fpulse-shapes-and-envelopes","16 min read",[],{"title":27292,"description":39376},"blog\u002Fexpert-notes\u002Fpulse-shapes-and-envelopes",[27288,20120],"lUWuH8SAXjG9DJfLvm20n28sMwmAzXp3sg2F30H98eQ",{"id":4,"title":5,"authors":39387,"body":39388,"breadcrumb":57950,"builders":57951,"byline":57956,"challenge":20105,"courseAuthor":20105,"courseLead":20105,"dek":20105,"description":20106,"draft":20107,"extension":20108,"eyebrow":20109,"featured":20107,"finish":20105,"fork":20105,"hero":20105,"heroAlt":20105,"heroCta":20105,"heroImage":20105,"homepageFeatured":20107,"kind":20110,"lessonCount":20105,"meta":57957,"navigation":10300,"newsItems":20105,"next":20105,"ogImage":20112,"order":20105,"outcomes":20105,"path":20113,"publishDate":20114,"readingTime":20115,"related":57958,"relatedProjects":20105,"seo":57959,"stem":20118,"tags":57960,"track":20105,"trackName":20105,"__hash__":20122},[7],{"type":9,"value":39389,"toc":57938},[39390,39396,39398,39400,39468,40188,40274,40700,40837,40977,41184,41186,41188,41190,41505,41567,42086,42148,42776,42780,43133,43207,43209,43239,43269,43902,44186,44188,44190,44280,44600,44726,44957,44959,44989,45232,45975,45977,46473,46573,47067,47069,47121,47270,47669,47673,47945,47947,47949,47951,48013,48685,48687,48689,48691,48995,48997,48999,49001,49006,49008,49010,49170,49336,51216,52198,52203,52208,52213,52215,52217,52219,53155,53160,53162,53164,55614,55619,55624,55629,55631,55633,55635,57929,57934,57936],[12,39391,39392],{},[15,39393,17,39394,23],{},[19,39395,22],{"href":21},[25,39397],{},[28,39399,31],{"id":30},[12,39401,39402,38,39404,132],{},[35,39403,37],{},[40,39405,39407,39432],{"className":39406},[43],[40,39408,39410],{"className":39409},[47],[49,39411,39412],{"xmlns":51},[53,39413,39414,39430],{},[56,39415,39416,39418,39420,39422,39424,39426,39428],{},[59,39417,62],{"stretchy":61},[64,39419,66],{},[59,39421,69],{"stretchy":61},[59,39423,73],{"separator":72},[59,39425,62],{"stretchy":61},[64,39427,78],{},[59,39429,69],{"stretchy":61},[82,39431,85],{"encoding":84},[40,39433,39435],{"className":39434,"ariaHidden":72},[89],[40,39436,39438,39441],{"className":39437},[93],[40,39439],{"className":39440,"style":98},[97],[40,39442,39444,39447,39450,39453,39456,39459,39462,39465],{"className":39443},[102],[40,39445,62],{"className":39446},[106],[40,39448,66],{"className":39449},[102],[40,39451,69],{"className":39452},[113],[40,39454,73],{"className":39455},[117],[40,39457],{"className":39458,"style":122},[121],[40,39460,62],{"className":39461},[106],[40,39463,78],{"className":39464},[102],[40,39466,69],{"className":39467},[113],[40,39469,39471],{"className":39470},[136],[40,39472,39474,39633],{"className":39473},[43],[40,39475,39477],{"className":39476},[47],[49,39478,39479],{"xmlns":51,"display":145},[53,39480,39481,39631],{},[56,39482,39483,39489,39491,39527,39529,39531,39537,39539,39579,39581,39583,39589,39591],{},[151,39484,39485,39487],{},[154,39486,156],{},[154,39488,159],{},[59,39490,162],{},[56,39492,39493,39495,39525],{},[59,39494,167],{"fence":72},[169,39496,39497,39511],{"rowspacing":171,"columnalign":172,"columnspacing":173},[175,39498,39499,39505],{},[178,39500,39501],{},[181,39502,39503],{"scriptlevel":66,"displaystyle":61},[64,39504,66],{},[178,39506,39507],{},[181,39508,39509],{"scriptlevel":66,"displaystyle":61},[64,39510,78],{},[175,39512,39513,39519],{},[178,39514,39515],{},[181,39516,39517],{"scriptlevel":66,"displaystyle":61},[64,39518,78],{},[178,39520,39521],{},[181,39522,39523],{"scriptlevel":66,"displaystyle":61},[64,39524,66],{},[59,39526,207],{"fence":72},[59,39528,73],{"separator":72},[121,39530],{"width":173},[151,39532,39533,39535],{},[154,39534,156],{},[154,39536,218],{},[59,39538,162],{},[56,39540,39541,39543,39577],{},[59,39542,167],{"fence":72},[169,39544,39545,39563],{"rowspacing":171,"columnalign":172,"columnspacing":173},[175,39546,39547,39553],{},[178,39548,39549],{},[181,39550,39551],{"scriptlevel":66,"displaystyle":61},[64,39552,66],{},[178,39554,39555],{},[181,39556,39557],{"scriptlevel":66,"displaystyle":61},[56,39558,39559,39561],{},[59,39560,243],{},[154,39562,246],{},[175,39564,39565,39571],{},[178,39566,39567],{},[181,39568,39569],{"scriptlevel":66,"displaystyle":61},[154,39570,246],{},[178,39572,39573],{},[181,39574,39575],{"scriptlevel":66,"displaystyle":61},[64,39576,66],{},[59,39578,207],{"fence":72},[59,39580,73],{"separator":72},[121,39582],{"width":173},[151,39584,39585,39587],{},[154,39586,156],{},[154,39588,273],{},[59,39590,162],{},[56,39592,39593,39595,39629],{},[59,39594,167],{"fence":72},[169,39596,39597,39611],{"rowspacing":171,"columnalign":172,"columnspacing":173},[175,39598,39599,39605],{},[178,39600,39601],{},[181,39602,39603],{"scriptlevel":66,"displaystyle":61},[64,39604,78],{},[178,39606,39607],{},[181,39608,39609],{"scriptlevel":66,"displaystyle":61},[64,39610,66],{},[175,39612,39613,39619],{},[178,39614,39615],{},[181,39616,39617],{"scriptlevel":66,"displaystyle":61},[64,39618,66],{},[178,39620,39621],{},[181,39622,39623],{"scriptlevel":66,"displaystyle":61},[56,39624,39625,39627],{},[59,39626,243],{},[64,39628,78],{},[59,39630,207],{"fence":72},[82,39632,318],{"encoding":84},[40,39634,39636,39691,39875,40062],{"className":39635,"ariaHidden":72},[89],[40,39637,39639,39642,39682,39685,39688],{"className":39638},[93],[40,39640],{"className":39641,"style":328},[97],[40,39643,39645,39648],{"className":39644},[102],[40,39646,156],{"className":39647,"style":336},[102,335],[40,39649,39651],{"className":39650},[340],[40,39652,39654,39674],{"className":39653},[344,345],[40,39655,39657,39671],{"className":39656},[349],[40,39658,39660],{"className":39659,"style":354},[353],[40,39661,39662,39665],{"style":357},[40,39663],{"className":39664,"style":362},[361],[40,39666,39668],{"className":39667},[366,367,368,369],[40,39669,159],{"className":39670},[102,335,369],[40,39672,377],{"className":39673},[376],[40,39675,39677],{"className":39676},[349],[40,39678,39680],{"className":39679,"style":384},[353],[40,39681],{},[40,39683],{"className":39684,"style":390},[121],[40,39686,162],{"className":39687},[394],[40,39689],{"className":39690,"style":390},[121],[40,39692,39694,39697,39814,39817,39820,39823,39826,39866,39869,39872],{"className":39693},[93],[40,39695],{"className":39696,"style":404},[97],[40,39698,39700,39706,39808],{"className":39699},[408],[40,39701,39703],{"className":39702,"style":413},[106,412],[40,39704,167],{"className":39705},[417,368],[40,39707,39709],{"className":39708},[102],[40,39710,39712,39757,39760,39763],{"className":39711},[169],[40,39713,39715],{"className":39714},[427],[40,39716,39718,39749],{"className":39717},[344,345],[40,39719,39721,39746],{"className":39720},[349],[40,39722,39724,39735],{"className":39723,"style":437},[353],[40,39725,39726,39729],{"style":440},[40,39727],{"className":39728,"style":444},[361],[40,39730,39732],{"className":39731},[102],[40,39733,66],{"className":39734},[102],[40,39736,39737,39740],{"style":453},[40,39738],{"className":39739,"style":444},[361],[40,39741,39743],{"className":39742},[102],[40,39744,78],{"className":39745},[102],[40,39747,377],{"className":39748},[376],[40,39750,39752],{"className":39751},[349],[40,39753,39755],{"className":39754,"style":472},[353],[40,39756],{},[40,39758],{"className":39759,"style":479},[478],[40,39761],{"className":39762,"style":479},[478],[40,39764,39766],{"className":39765},[427],[40,39767,39769,39800],{"className":39768},[344,345],[40,39770,39772,39797],{"className":39771},[349],[40,39773,39775,39786],{"className":39774,"style":437},[353],[40,39776,39777,39780],{"style":440},[40,39778],{"className":39779,"style":444},[361],[40,39781,39783],{"className":39782},[102],[40,39784,78],{"className":39785},[102],[40,39787,39788,39791],{"style":453},[40,39789],{"className":39790,"style":444},[361],[40,39792,39794],{"className":39793},[102],[40,39795,66],{"className":39796},[102],[40,39798,377],{"className":39799},[376],[40,39801,39803],{"className":39802},[349],[40,39804,39806],{"className":39805,"style":472},[353],[40,39807],{},[40,39809,39811],{"className":39810,"style":413},[113,412],[40,39812,207],{"className":39813},[417,368],[40,39815],{"className":39816,"style":122},[121],[40,39818,73],{"className":39819},[117],[40,39821],{"className":39822,"style":543},[121],[40,39824],{"className":39825,"style":122},[121],[40,39827,39829,39832],{"className":39828},[102],[40,39830,156],{"className":39831,"style":336},[102,335],[40,39833,39835],{"className":39834},[340],[40,39836,39838,39858],{"className":39837},[344,345],[40,39839,39841,39855],{"className":39840},[349],[40,39842,39844],{"className":39843,"style":354},[353],[40,39845,39846,39849],{"style":357},[40,39847],{"className":39848,"style":362},[361],[40,39850,39852],{"className":39851},[366,367,368,369],[40,39853,218],{"className":39854,"style":336},[102,335,369],[40,39856,377],{"className":39857},[376],[40,39859,39861],{"className":39860},[349],[40,39862,39864],{"className":39863,"style":585},[353],[40,39865],{},[40,39867],{"className":39868,"style":390},[121],[40,39870,162],{"className":39871},[394],[40,39873],{"className":39874,"style":390},[121],[40,39876,39878,39881,40001,40004,40007,40010,40013,40053,40056,40059],{"className":39877},[93],[40,39879],{"className":39880,"style":404},[97],[40,39882,39884,39890,39995],{"className":39883},[408],[40,39885,39887],{"className":39886,"style":413},[106,412],[40,39888,167],{"className":39889},[417,368],[40,39891,39893],{"className":39892},[102],[40,39894,39896,39941,39944,39947],{"className":39895},[169],[40,39897,39899],{"className":39898},[427],[40,39900,39902,39933],{"className":39901},[344,345],[40,39903,39905,39930],{"className":39904},[349],[40,39906,39908,39919],{"className":39907,"style":437},[353],[40,39909,39910,39913],{"style":440},[40,39911],{"className":39912,"style":444},[361],[40,39914,39916],{"className":39915},[102],[40,39917,66],{"className":39918},[102],[40,39920,39921,39924],{"style":453},[40,39922],{"className":39923,"style":444},[361],[40,39925,39927],{"className":39926},[102],[40,39928,246],{"className":39929},[102,335],[40,39931,377],{"className":39932},[376],[40,39934,39936],{"className":39935},[349],[40,39937,39939],{"className":39938,"style":472},[353],[40,39940],{},[40,39942],{"className":39943,"style":479},[478],[40,39945],{"className":39946,"style":479},[478],[40,39948,39950],{"className":39949},[427],[40,39951,39953,39987],{"className":39952},[344,345],[40,39954,39956,39984],{"className":39955},[349],[40,39957,39959,39973],{"className":39958,"style":437},[353],[40,39960,39961,39964],{"style":440},[40,39962],{"className":39963,"style":444},[361],[40,39965,39967,39970],{"className":39966},[102],[40,39968,243],{"className":39969},[102],[40,39971,246],{"className":39972},[102,335],[40,39974,39975,39978],{"style":453},[40,39976],{"className":39977,"style":444},[361],[40,39979,39981],{"className":39980},[102],[40,39982,66],{"className":39983},[102],[40,39985,377],{"className":39986},[376],[40,39988,39990],{"className":39989},[349],[40,39991,39993],{"className":39992,"style":472},[353],[40,39994],{},[40,39996,39998],{"className":39997,"style":413},[113,412],[40,39999,207],{"className":40000},[417,368],[40,40002],{"className":40003,"style":122},[121],[40,40005,73],{"className":40006},[117],[40,40008],{"className":40009,"style":543},[121],[40,40011],{"className":40012,"style":122},[121],[40,40014,40016,40019],{"className":40015},[102],[40,40017,156],{"className":40018,"style":336},[102,335],[40,40020,40022],{"className":40021},[340],[40,40023,40025,40045],{"className":40024},[344,345],[40,40026,40028,40042],{"className":40027},[349],[40,40029,40031],{"className":40030,"style":354},[353],[40,40032,40033,40036],{"style":357},[40,40034],{"className":40035,"style":362},[361],[40,40037,40039],{"className":40038},[366,367,368,369],[40,40040,273],{"className":40041,"style":764},[102,335,369],[40,40043,377],{"className":40044},[376],[40,40046,40048],{"className":40047},[349],[40,40049,40051],{"className":40050,"style":384},[353],[40,40052],{},[40,40054],{"className":40055,"style":390},[121],[40,40057,162],{"className":40058},[394],[40,40060],{"className":40061,"style":390},[121],[40,40063,40065,40068],{"className":40064},[93],[40,40066],{"className":40067,"style":404},[97],[40,40069,40071,40077,40182],{"className":40070},[408],[40,40072,40074],{"className":40073,"style":413},[106,412],[40,40075,167],{"className":40076},[417,368],[40,40078,40080],{"className":40079},[102],[40,40081,40083,40128,40131,40134],{"className":40082},[169],[40,40084,40086],{"className":40085},[427],[40,40087,40089,40120],{"className":40088},[344,345],[40,40090,40092,40117],{"className":40091},[349],[40,40093,40095,40106],{"className":40094,"style":437},[353],[40,40096,40097,40100],{"style":440},[40,40098],{"className":40099,"style":444},[361],[40,40101,40103],{"className":40102},[102],[40,40104,78],{"className":40105},[102],[40,40107,40108,40111],{"style":453},[40,40109],{"className":40110,"style":444},[361],[40,40112,40114],{"className":40113},[102],[40,40115,66],{"className":40116},[102],[40,40118,377],{"className":40119},[376],[40,40121,40123],{"className":40122},[349],[40,40124,40126],{"className":40125,"style":472},[353],[40,40127],{},[40,40129],{"className":40130,"style":479},[478],[40,40132],{"className":40133,"style":479},[478],[40,40135,40137],{"className":40136},[427],[40,40138,40140,40174],{"className":40139},[344,345],[40,40141,40143,40171],{"className":40142},[349],[40,40144,40146,40157],{"className":40145,"style":437},[353],[40,40147,40148,40151],{"style":440},[40,40149],{"className":40150,"style":444},[361],[40,40152,40154],{"className":40153},[102],[40,40155,66],{"className":40156},[102],[40,40158,40159,40162],{"style":453},[40,40160],{"className":40161,"style":444},[361],[40,40163,40165,40168],{"className":40164},[102],[40,40166,243],{"className":40167},[102],[40,40169,78],{"className":40170},[102],[40,40172,377],{"className":40173},[376],[40,40175,40177],{"className":40176},[349],[40,40178,40180],{"className":40179,"style":472},[353],[40,40181],{},[40,40183,40185],{"className":40184,"style":413},[113,412],[40,40186,207],{"className":40187},[417,368],[12,40189,40190,916,40192,920,40194,1014],{},[35,40191,915],{},[35,40193,919],{},[40,40195,40197,40218],{"className":40196},[43],[40,40198,40200],{"className":40199},[47],[49,40201,40202],{"xmlns":51},[53,40203,40204,40216],{},[56,40205,40206,40208,40210],{},[154,40207,936],{"mathvariant":935},[59,40209,939],{},[941,40211,40212,40214],{},[154,40213,946],{"mathvariant":945},[64,40215,949],{},[82,40217,952],{"encoding":84},[40,40219,40221,40239],{"className":40220,"ariaHidden":72},[89],[40,40222,40224,40227,40230,40233,40236],{"className":40223},[93],[40,40225],{"className":40226,"style":962},[97],[40,40228,936],{"className":40229},[102,966],[40,40231],{"className":40232,"style":390},[121],[40,40234,939],{"className":40235},[394],[40,40237],{"className":40238,"style":390},[121],[40,40240,40242,40245],{"className":40241},[93],[40,40243],{"className":40244,"style":982},[97],[40,40246,40248,40251],{"className":40247},[102],[40,40249,946],{"className":40250},[102,989],[40,40252,40254],{"className":40253},[340],[40,40255,40257],{"className":40256},[344],[40,40258,40260],{"className":40259},[349],[40,40261,40263],{"className":40262,"style":982},[353],[40,40264,40265,40268],{"style":1004},[40,40266],{"className":40267,"style":362},[361],[40,40269,40271],{"className":40270},[366,367,368,369],[40,40272,949],{"className":40273},[102,369],[40,40275,40277],{"className":40276},[136],[40,40278,40280,40369],{"className":40279},[43],[40,40281,40283],{"className":40282},[47],[49,40284,40285],{"xmlns":51,"display":145},[53,40286,40287,40367],{},[56,40288,40289,40291,40293,40301,40303,40305,40307,40309,40311,40313,40315,40317,40319,40325,40327,40329,40331,40333,40335,40337,40343,40345,40347,40349,40351,40353,40365],{},[154,40290,1032],{},[59,40292,162],{},[181,40294,40295],{"scriptlevel":66,"displaystyle":61},[1038,40296,40297,40299],{},[64,40298,78],{},[64,40300,1044],{},[59,40302,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,40304,1050],{},[59,40306,1053],{},[154,40308,936],{"mathvariant":935},[59,40310,1058],{},[154,40312,156],{"mathvariant":1061},[59,40314,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,40316,73],{"separator":72},[121,40318],{"width":1068},[151,40320,40321,40323],{},[154,40322,936],{},[154,40324,1075],{},[59,40326,162],{},[154,40328,1081],{"mathvariant":1080},[59,40330,1084],{},[59,40332,167],{"stretchy":61},[154,40334,1032],{},[59,40336,73],{"separator":72},[151,40338,40339,40341],{},[154,40340,156],{},[154,40342,1075],{},[59,40344,207],{"stretchy":61},[59,40346,73],{"separator":72},[1102,40348,1104],{},[154,40350,1075],{},[59,40352,939],{},[56,40354,40355,40357,40359,40361,40363],{},[154,40356,159],{},[59,40358,73],{"separator":72},[154,40360,218],{},[59,40362,73],{"separator":72},[154,40364,273],{},[154,40366,1123],{"mathvariant":1080},[82,40368,1126],{"encoding":84},[40,40370,40372,40390,40482,40500,40579,40667],{"className":40371,"ariaHidden":72},[89],[40,40373,40375,40378,40381,40384,40387],{"className":40374},[93],[40,40376],{"className":40377,"style":1136},[97],[40,40379,1032],{"className":40380},[102,335],[40,40382],{"className":40383,"style":390},[121],[40,40385,162],{"className":40386},[394],[40,40388],{"className":40389,"style":390},[121],[40,40391,40393,40396,40464,40470,40473,40476,40479],{"className":40392},[93],[40,40394],{"className":40395,"style":1155},[97],[40,40397,40399,40402,40461],{"className":40398},[102],[40,40400],{"className":40401},[106,1162],[40,40403,40405],{"className":40404},[1038],[40,40406,40408,40453],{"className":40407},[344,345],[40,40409,40411,40450],{"className":40410},[349],[40,40412,40414,40428,40436],{"className":40413,"style":1175},[353],[40,40415,40416,40419],{"style":1178},[40,40417],{"className":40418,"style":444},[361],[40,40420,40422],{"className":40421},[366,367,368,369],[40,40423,40425],{"className":40424},[102,369],[40,40426,1044],{"className":40427},[102,369],[40,40429,40430,40433],{"style":1193},[40,40431],{"className":40432,"style":444},[361],[40,40434],{"className":40435,"style":1201},[1200],[40,40437,40438,40441],{"style":1204},[40,40439],{"className":40440,"style":444},[361],[40,40442,40444],{"className":40443},[366,367,368,369],[40,40445,40447],{"className":40446},[102,369],[40,40448,78],{"className":40449},[102,369],[40,40451,377],{"className":40452},[376],[40,40454,40456],{"className":40455},[349],[40,40457,40459],{"className":40458,"style":1226},[353],[40,40460],{},[40,40462],{"className":40463},[113,1162],[40,40465,40467],{"className":40466},[102],[40,40468,167],{"className":40469},[417,1238],[40,40471,1050],{"className":40472,"style":1242},[102,335],[40,40474],{"className":40475,"style":1246},[121],[40,40477,1053],{"className":40478},[1250],[40,40480],{"className":40481,"style":1246},[121],[40,40483,40485,40488,40491,40494,40497],{"className":40484},[93],[40,40486],{"className":40487,"style":1260},[97],[40,40489,936],{"className":40490},[102,966],[40,40492],{"className":40493,"style":1246},[121],[40,40495,1058],{"className":40496},[1250],[40,40498],{"className":40499,"style":1246},[121],[40,40501,40503,40506,40515,40521,40524,40527,40530,40570,40573,40576],{"className":40502},[93],[40,40504],{"className":40505,"style":1155},[97],[40,40507,40509],{"className":40508},[102],[40,40510,40512],{"className":40511},[102],[40,40513,156],{"className":40514,"style":1289},[102,1288],[40,40516,40518],{"className":40517},[102],[40,40519,207],{"className":40520},[417,1238],[40,40522,73],{"className":40523},[117],[40,40525],{"className":40526,"style":1302},[121],[40,40528],{"className":40529,"style":122},[121],[40,40531,40533,40536],{"className":40532},[102],[40,40534,936],{"className":40535,"style":1312},[102,335],[40,40537,40539],{"className":40538},[340],[40,40540,40542,40562],{"className":40541},[344,345],[40,40543,40545,40559],{"className":40544},[349],[40,40546,40548],{"className":40547,"style":1325},[353],[40,40549,40550,40553],{"style":1328},[40,40551],{"className":40552,"style":362},[361],[40,40554,40556],{"className":40555},[366,367,368,369],[40,40557,1075],{"className":40558,"style":1338},[102,335,369],[40,40560,377],{"className":40561},[376],[40,40563,40565],{"className":40564},[349],[40,40566,40568],{"className":40567,"style":585},[353],[40,40569],{},[40,40571],{"className":40572,"style":390},[121],[40,40574,162],{"className":40575},[394],[40,40577],{"className":40578,"style":390},[121],[40,40580,40582,40585,40591,40594,40597,40600,40603,40643,40646,40649,40652,40655,40658,40661,40664],{"className":40581},[93],[40,40583],{"className":40584,"style":1365},[97],[40,40586,40588],{"className":40587},[1369],[40,40589,1081],{"className":40590},[102,1373],[40,40592,167],{"className":40593},[106],[40,40595,1032],{"className":40596},[102,335],[40,40598,73],{"className":40599},[117],[40,40601],{"className":40602,"style":122},[121],[40,40604,40606,40609],{"className":40605},[102],[40,40607,156],{"className":40608,"style":336},[102,335],[40,40610,40612],{"className":40611},[340],[40,40613,40615,40635],{"className":40614},[344,345],[40,40616,40618,40632],{"className":40617},[349],[40,40619,40621],{"className":40620,"style":1325},[353],[40,40622,40623,40626],{"style":357},[40,40624],{"className":40625,"style":362},[361],[40,40627,40629],{"className":40628},[366,367,368,369],[40,40630,1075],{"className":40631,"style":1338},[102,335,369],[40,40633,377],{"className":40634},[376],[40,40636,40638],{"className":40637},[349],[40,40639,40641],{"className":40640,"style":585},[353],[40,40642],{},[40,40644,207],{"className":40645},[113],[40,40647,73],{"className":40648},[117],[40,40650,1104],{"className":40651},[121],[40,40653],{"className":40654,"style":122},[121],[40,40656,1075],{"className":40657,"style":1338},[102,335],[40,40659],{"className":40660,"style":390},[121],[40,40662,939],{"className":40663},[394],[40,40665],{"className":40666,"style":390},[121],[40,40668,40670,40673,40697],{"className":40669},[93],[40,40671],{"className":40672,"style":1136},[97],[40,40674,40676,40679,40682,40685,40688,40691,40694],{"className":40675},[102],[40,40677,159],{"className":40678},[102,335],[40,40680,73],{"className":40681},[117],[40,40683],{"className":40684,"style":122},[121],[40,40686,218],{"className":40687,"style":336},[102,335],[40,40689,73],{"className":40690},[117],[40,40692],{"className":40693,"style":122},[121],[40,40695,273],{"className":40696,"style":764},[102,335],[40,40698,1123],{"className":40699},[102],[12,40701,1485,40702,1564,40777,1628],{},[40,40703,40705,40732],{"className":40704},[43],[40,40706,40708],{"className":40707},[47],[49,40709,40710],{"xmlns":51},[53,40711,40712,40730],{},[56,40713,40714,40716,40718,40720,40722,40724,40726,40728],{},[154,40715,1032],{},[59,40717,162],{},[59,40719,62],{"stretchy":61},[154,40721,1506],{},[59,40723,69],{"stretchy":61},[59,40725,1511],{"stretchy":61},[154,40727,1506],{},[59,40729,62],{"stretchy":61},[82,40731,1518],{"encoding":84},[40,40733,40735,40753],{"className":40734,"ariaHidden":72},[89],[40,40736,40738,40741,40744,40747,40750],{"className":40737},[93],[40,40739],{"className":40740,"style":1136},[97],[40,40742,1032],{"className":40743},[102,335],[40,40745],{"className":40746,"style":390},[121],[40,40748,162],{"className":40749},[394],[40,40751],{"className":40752,"style":390},[121],[40,40754,40756,40759,40762,40765,40768,40771,40774],{"className":40755},[93],[40,40757],{"className":40758,"style":98},[97],[40,40760,62],{"className":40761},[106],[40,40763,1506],{"className":40764,"style":336},[102,335],[40,40766,69],{"className":40767},[113],[40,40769,1511],{"className":40770},[106],[40,40772,1506],{"className":40773,"style":336},[102,335],[40,40775,62],{"className":40776},[113],[40,40778,40780,40801],{"className":40779},[43],[40,40781,40783],{"className":40782},[47],[49,40784,40785],{"xmlns":51},[53,40786,40787,40799],{},[56,40788,40789,40791,40793,40795,40797],{},[59,40790,1579],{"stretchy":61},[154,40792,936],{"mathvariant":935},[59,40794,1579],{"stretchy":61},[59,40796,162],{},[64,40798,78],{},[82,40800,1590],{"encoding":84},[40,40802,40804,40828],{"className":40803,"ariaHidden":72},[89],[40,40805,40807,40810,40813,40816,40819,40822,40825],{"className":40806},[93],[40,40808],{"className":40809,"style":98},[97],[40,40811,1579],{"className":40812},[106],[40,40814,936],{"className":40815},[102,966],[40,40817,1579],{"className":40818},[113],[40,40820],{"className":40821,"style":390},[121],[40,40823,162],{"className":40824},[394],[40,40826],{"className":40827,"style":390},[121],[40,40829,40831,40834],{"className":40830},[93],[40,40832],{"className":40833,"style":1624},[97],[40,40835,78],{"className":40836},[102],[12,40838,40839,1634,40841,1778],{},[35,40840,1633],{},[40,40842,40844,40887],{"className":40843},[43],[40,40845,40847],{"className":40846},[47],[49,40848,40849],{"xmlns":51},[53,40850,40851,40885],{},[56,40852,40853,40855,40857,40859,40861,40863,40865,40867,40869,40871,40873,40875,40877,40879,40881,40883],{},[154,40854,1649],{},[59,40856,939],{},[59,40858,1654],{"stretchy":61},[64,40860,66],{},[59,40862,73],{"separator":72},[154,40864,1661],{},[59,40866,1664],{"stretchy":61},[59,40868,73],{"separator":72},[1102,40870,1104],{},[154,40872,1671],{},[59,40874,939],{},[59,40876,1654],{"stretchy":61},[64,40878,66],{},[59,40880,73],{"separator":72},[64,40882,1044],{},[154,40884,1661],{},[82,40886,1686],{"encoding":84},[40,40888,40890,40908,40953],{"className":40889,"ariaHidden":72},[89],[40,40891,40893,40896,40899,40902,40905],{"className":40892},[93],[40,40894],{"className":40895,"style":1696},[97],[40,40897,1649],{"className":40898,"style":1312},[102,335],[40,40900],{"className":40901,"style":390},[121],[40,40903,939],{"className":40904},[394],[40,40906],{"className":40907,"style":390},[121],[40,40909,40911,40914,40917,40920,40923,40926,40929,40932,40935,40938,40941,40944,40947,40950],{"className":40910},[93],[40,40912],{"className":40913,"style":98},[97],[40,40915,1654],{"className":40916},[106],[40,40918,66],{"className":40919},[102],[40,40921,73],{"className":40922},[117],[40,40924],{"className":40925,"style":122},[121],[40,40927,1661],{"className":40928,"style":336},[102,335],[40,40930,1664],{"className":40931},[113],[40,40933,73],{"className":40934},[117],[40,40936,1104],{"className":40937},[121],[40,40939],{"className":40940,"style":122},[121],[40,40942,1671],{"className":40943},[102,335],[40,40945],{"className":40946,"style":390},[121],[40,40948,939],{"className":40949},[394],[40,40951],{"className":40952,"style":390},[121],[40,40954,40956,40959,40962,40965,40968,40971,40974],{"className":40955},[93],[40,40957],{"className":40958,"style":98},[97],[40,40960,1654],{"className":40961},[106],[40,40963,66],{"className":40964},[102],[40,40966,73],{"className":40967},[117],[40,40969],{"className":40970,"style":122},[121],[40,40972,1044],{"className":40973},[102],[40,40975,1661],{"className":40976,"style":336},[102,335],[40,40978,40980],{"className":40979},[136],[40,40981,40983,41052],{"className":40982},[43],[40,40984,40986],{"className":40985},[47],[49,40987,40988],{"xmlns":51,"display":145},[53,40989,40990,41050],{},[56,40991,40992,40994,40996,40998,41000,41002,41004,41006,41008,41010,41012,41014,41016,41018,41020,41022,41024,41026,41028,41030,41032,41034,41036,41038,41040,41042,41044,41046,41048],{},[154,40993,1796],{"mathvariant":935},[59,40995,167],{"stretchy":61},[154,40997,1649],{},[59,40999,73],{"separator":72},[154,41001,1671],{},[59,41003,207],{"stretchy":61},[59,41005,162],{},[59,41007,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,41009,1813],{},[59,41011,1084],{},[154,41013,1649],{},[154,41015,1820],{},[59,41017,1084],{},[154,41019,1671],{},[59,41021,73],{"separator":72},[1102,41023,1104],{},[154,41025,1813],{},[59,41027,1084],{},[154,41029,1649],{},[154,41031,1813],{},[59,41033,1084],{},[154,41035,1671],{},[59,41037,73],{"separator":72},[1102,41039,1104],{},[154,41041,1820],{},[59,41043,1084],{},[154,41045,1649],{},[59,41047,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,41049,1123],{"mathvariant":1080},[82,41051,1857],{"encoding":84},[40,41053,41055,41091],{"className":41054,"ariaHidden":72},[89],[40,41056,41058,41061,41064,41067,41070,41073,41076,41079,41082,41085,41088],{"className":41057},[93],[40,41059],{"className":41060,"style":98},[97],[40,41062,1796],{"className":41063},[102,966],[40,41065,167],{"className":41066},[106],[40,41068,1649],{"className":41069,"style":1312},[102,335],[40,41071,73],{"className":41072},[117],[40,41074],{"className":41075,"style":122},[121],[40,41077,1671],{"className":41078},[102,335],[40,41080,207],{"className":41081},[113],[40,41083],{"className":41084,"style":390},[121],[40,41086,162],{"className":41087},[394],[40,41089],{"className":41090,"style":390},[121],[40,41092,41094,41097,41103,41106,41109,41112,41115,41118,41121,41124,41127,41130,41133,41136,41139,41142,41145,41148,41151,41154,41157,41160,41163,41166,41169,41172,41175,41181],{"className":41093},[93],[40,41095],{"className":41096,"style":1155},[97],[40,41098,41100],{"className":41099},[102],[40,41101,167],{"className":41102},[417,1238],[40,41104],{"className":41105,"style":122},[121],[40,41107,1813],{"className":41108},[1369],[40,41110],{"className":41111,"style":122},[121],[40,41113,1649],{"className":41114,"style":1312},[102,335],[40,41116],{"className":41117,"style":122},[121],[40,41119,1820],{"className":41120},[1369],[40,41122],{"className":41123,"style":122},[121],[40,41125,1671],{"className":41126},[102,335],[40,41128,73],{"className":41129},[117],[40,41131,1104],{"className":41132},[121],[40,41134],{"className":41135,"style":122},[121],[40,41137,1813],{"className":41138},[1369],[40,41140],{"className":41141,"style":122},[121],[40,41143,1649],{"className":41144,"style":1312},[102,335],[40,41146],{"className":41147,"style":122},[121],[40,41149,1813],{"className":41150},[1369],[40,41152],{"className":41153,"style":122},[121],[40,41155,1671],{"className":41156},[102,335],[40,41158,73],{"className":41159},[117],[40,41161,1104],{"className":41162},[121],[40,41164],{"className":41165,"style":122},[121],[40,41167,1820],{"className":41168},[1369],[40,41170],{"className":41171,"style":122},[121],[40,41173,1649],{"className":41174,"style":1312},[102,335],[40,41176,41178],{"className":41177},[102],[40,41179,207],{"className":41180},[417,1238],[40,41182,1123],{"className":41183},[102],[25,41185],{},[28,41187,1995],{"id":1994},[12,41189,1998],{},[40,41191,41193],{"className":41192},[136],[40,41194,41196,41261],{"className":41195},[43],[40,41197,41199],{"className":41198},[47],[49,41200,41201],{"xmlns":51,"display":145},[53,41202,41203,41259],{},[56,41204,41205,41207,41209,41211,41213,41215,41217,41219,41227,41229,41231,41233,41237,41239,41241,41243,41245,41247,41249,41251,41253,41255,41257],{},[154,41206,2016],{"mathvariant":1080},[59,41208,167],{"stretchy":61},[154,41210,1649],{},[59,41212,73],{"separator":72},[154,41214,1671],{},[59,41216,207],{"stretchy":61},[59,41218,162],{},[181,41220,41221],{"scriptlevel":66,"displaystyle":61},[1038,41222,41223,41225],{},[64,41224,78],{},[64,41226,1044],{},[59,41228,167],{"fence":61,"stretchy":72,"minsize":2039,"maxsize":2039},[154,41230,1050],{},[59,41232,1053],{},[2045,41234,41235],{},[64,41236,949],{},[59,41238,2051],{"separator":72},[154,41240,1796],{"mathvariant":935},[59,41242,167],{"stretchy":61},[154,41244,1649],{},[59,41246,73],{"separator":72},[154,41248,1671],{},[59,41250,207],{"stretchy":61},[59,41252,1058],{},[154,41254,156],{"mathvariant":1061},[59,41256,207],{"fence":61,"stretchy":72,"minsize":2039,"maxsize":2039},[59,41258,73],{"separator":72},[82,41260,2074],{"encoding":84},[40,41262,41264,41300,41392,41481],{"className":41263,"ariaHidden":72},[89],[40,41265,41267,41270,41273,41276,41279,41282,41285,41288,41291,41294,41297],{"className":41266},[93],[40,41268],{"className":41269,"style":98},[97],[40,41271,2016],{"className":41272},[102],[40,41274,167],{"className":41275},[106],[40,41277,1649],{"className":41278,"style":1312},[102,335],[40,41280,73],{"className":41281},[117],[40,41283],{"className":41284,"style":122},[121],[40,41286,1671],{"className":41287},[102,335],[40,41289,207],{"className":41290},[113],[40,41292],{"className":41293,"style":390},[121],[40,41295,162],{"className":41296},[394],[40,41298],{"className":41299,"style":390},[121],[40,41301,41303,41306,41374,41380,41383,41386,41389],{"className":41302},[93],[40,41304],{"className":41305,"style":2120},[97],[40,41307,41309,41312,41371],{"className":41308},[102],[40,41310],{"className":41311},[106,1162],[40,41313,41315],{"className":41314},[1038],[40,41316,41318,41363],{"className":41317},[344,345],[40,41319,41321,41360],{"className":41320},[349],[40,41322,41324,41338,41346],{"className":41323,"style":1175},[353],[40,41325,41326,41329],{"style":1178},[40,41327],{"className":41328,"style":444},[361],[40,41330,41332],{"className":41331},[366,367,368,369],[40,41333,41335],{"className":41334},[102,369],[40,41336,1044],{"className":41337},[102,369],[40,41339,41340,41343],{"style":1193},[40,41341],{"className":41342,"style":444},[361],[40,41344],{"className":41345,"style":1201},[1200],[40,41347,41348,41351],{"style":1204},[40,41349],{"className":41350,"style":444},[361],[40,41352,41354],{"className":41353},[366,367,368,369],[40,41355,41357],{"className":41356},[102,369],[40,41358,78],{"className":41359},[102,369],[40,41361,377],{"className":41362},[376],[40,41364,41366],{"className":41365},[349],[40,41367,41369],{"className":41368,"style":1226},[353],[40,41370],{},[40,41372],{"className":41373},[113,1162],[40,41375,41377],{"className":41376},[102],[40,41378,167],{"className":41379},[417,2195],[40,41381,1050],{"className":41382,"style":1242},[102,335],[40,41384],{"className":41385,"style":1246},[121],[40,41387,1053],{"className":41388},[1250],[40,41390],{"className":41391,"style":1246},[121],[40,41393,41395,41398,41445,41448,41451,41454,41457,41460,41463,41466,41469,41472,41475,41478],{"className":41394},[93],[40,41396],{"className":41397,"style":2214},[97],[40,41399,41401],{"className":41400},[102,2218],[40,41402,41404,41437],{"className":41403},[344,345],[40,41405,41407,41434],{"className":41406},[349],[40,41408,41410,41422],{"className":41409,"style":2228},[353],[40,41411,41413,41416],{"className":41412,"style":2233},[2232],[40,41414],{"className":41415,"style":444},[361],[40,41417,41419],{"className":41418,"style":2240},[102],[40,41420,949],{"className":41421},[102],[40,41423,41424,41427],{"style":2246},[40,41425],{"className":41426,"style":444},[361],[40,41428,41430],{"className":41429,"style":2254},[2253],[2256,41431,41432],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,41433],{"d":2266},[40,41435,377],{"className":41436},[376],[40,41438,41440],{"className":41439},[349],[40,41441,41443],{"className":41442,"style":2276},[353],[40,41444],{},[40,41446,2051],{"className":41447},[117],[40,41449],{"className":41450,"style":122},[121],[40,41452,1796],{"className":41453},[102,966],[40,41455,167],{"className":41456},[106],[40,41458,1649],{"className":41459,"style":1312},[102,335],[40,41461,73],{"className":41462},[117],[40,41464],{"className":41465,"style":122},[121],[40,41467,1671],{"className":41468},[102,335],[40,41470,207],{"className":41471},[113],[40,41473],{"className":41474,"style":1246},[121],[40,41476,1058],{"className":41477},[1250],[40,41479],{"className":41480,"style":1246},[121],[40,41482,41484,41487,41496,41502],{"className":41483},[93],[40,41485],{"className":41486,"style":2120},[97],[40,41488,41490],{"className":41489},[102],[40,41491,41493],{"className":41492},[102],[40,41494,156],{"className":41495,"style":1289},[102,1288],[40,41497,41499],{"className":41498},[102],[40,41500,207],{"className":41501},[417,2195],[40,41503,73],{"className":41504},[117],[12,41506,2341,41507,2345,41509,2407],{},[35,41508,2344],{},[40,41510,41512,41529],{"className":41511},[43],[40,41513,41515],{"className":41514},[47],[49,41516,41517],{"xmlns":51},[53,41518,41519,41527],{},[56,41520,41521],{},[941,41522,41523,41525],{},[154,41524,2362],{},[64,41526,1044],{},[82,41528,2367],{"encoding":84},[40,41530,41532],{"className":41531,"ariaHidden":72},[89],[40,41533,41535,41538],{"className":41534},[93],[40,41536],{"className":41537,"style":982},[97],[40,41539,41541,41544],{"className":41540},[102],[40,41542,2362],{"className":41543,"style":2383},[102,335],[40,41545,41547],{"className":41546},[340],[40,41548,41550],{"className":41549},[344],[40,41551,41553],{"className":41552},[349],[40,41554,41556],{"className":41555,"style":982},[353],[40,41557,41558,41561],{"style":1004},[40,41559],{"className":41560,"style":362},[361],[40,41562,41564],{"className":41563},[366,367,368,369],[40,41565,1044],{"className":41566},[102,369],[40,41568,41570],{"className":41569},[136],[40,41571,41573,41680],{"className":41572},[43],[40,41574,41576],{"className":41575},[47],[49,41577,41578],{"xmlns":51,"display":145},[53,41579,41580,41678],{},[169,41581,41582],{"width":2423},[175,41583,41584,41586,41672,41674],{},[178,41585],{"width":2428},[178,41587,41588],{},[56,41589,41590,41596,41598,41600,41602,41604,41606,41608,41610,41612,41614,41616,41618,41620,41622,41624,41626,41628,41630,41632,41634,41642,41644,41652,41654,41656,41658,41660,41662,41664,41666,41668,41670],{},[151,41591,41592,41594],{},[154,41593,2437],{},[154,41595,1032],{},[59,41597,167],{"stretchy":61},[154,41599,1649],{},[59,41601,73],{"separator":72},[154,41603,1671],{},[59,41605,207],{"stretchy":61},[59,41607,162],{},[154,41609,1081],{"mathvariant":1080},[59,41611,1084],{},[59,41613,1654],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,41615,1032],{},[59,41617,73],{"separator":72},[154,41619,2016],{"mathvariant":1080},[59,41621,167],{"stretchy":61},[154,41623,1649],{},[59,41625,73],{"separator":72},[154,41627,1671],{},[59,41629,207],{"stretchy":61},[59,41631,1664],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,41633,162],{},[181,41635,41636],{"scriptlevel":66,"displaystyle":61},[1038,41637,41638,41640],{},[64,41639,78],{},[64,41641,1044],{},[59,41643,1053],{},[1038,41645,41646,41650],{},[2045,41647,41648],{},[64,41649,949],{},[64,41651,1044],{},[59,41653,2051],{"separator":72},[154,41655,936],{"mathvariant":935},[59,41657,1058],{},[154,41659,1796],{"mathvariant":935},[59,41661,167],{"stretchy":61},[154,41663,1649],{},[59,41665,73],{"separator":72},[154,41667,1671],{},[59,41669,207],{"stretchy":61},[154,41671,1123],{"mathvariant":1080},[178,41673],{"width":2428},[178,41675,41676],{},[1102,41677,2522],{},[82,41679,2525],{"encoding":84},[40,41681,41683,41756,41822,41905,42035,42065],{"className":41682,"ariaHidden":72},[89],[40,41684,41686,41689,41729,41732,41735,41738,41741,41744,41747,41750,41753],{"className":41685},[93],[40,41687],{"className":41688,"style":1365},[97],[40,41690,41692,41695],{"className":41691},[102],[40,41693,2437],{"className":41694,"style":2541},[102,335],[40,41696,41698],{"className":41697},[340],[40,41699,41701,41721],{"className":41700},[344,345],[40,41702,41704,41718],{"className":41703},[349],[40,41705,41707],{"className":41706,"style":354},[353],[40,41708,41709,41712],{"style":2556},[40,41710],{"className":41711,"style":362},[361],[40,41713,41715],{"className":41714},[366,367,368,369],[40,41716,1032],{"className":41717},[102,335,369],[40,41719,377],{"className":41720},[376],[40,41722,41724],{"className":41723},[349],[40,41725,41727],{"className":41726,"style":585},[353],[40,41728],{},[40,41730,167],{"className":41731},[106],[40,41733,1649],{"className":41734,"style":1312},[102,335],[40,41736,73],{"className":41737},[117],[40,41739],{"className":41740,"style":122},[121],[40,41742,1671],{"className":41743},[102,335],[40,41745,207],{"className":41746},[113],[40,41748],{"className":41749,"style":390},[121],[40,41751,162],{"className":41752},[394],[40,41754],{"className":41755,"style":390},[121],[40,41757,41759,41762,41768,41771,41777,41780,41783,41786,41789,41792,41795,41798,41801,41804,41807,41813,41816,41819],{"className":41758},[93],[40,41760],{"className":41761,"style":1155},[97],[40,41763,41765],{"className":41764},[1369],[40,41766,1081],{"className":41767},[102,1373],[40,41769],{"className":41770,"style":122},[121],[40,41772,41774],{"className":41773},[102],[40,41775,1654],{"className":41776},[417,1238],[40,41778,1032],{"className":41779},[102,335],[40,41781,73],{"className":41782},[117],[40,41784],{"className":41785,"style":122},[121],[40,41787,2016],{"className":41788},[102],[40,41790,167],{"className":41791},[106],[40,41793,1649],{"className":41794,"style":1312},[102,335],[40,41796,73],{"className":41797},[117],[40,41799],{"className":41800,"style":122},[121],[40,41802,1671],{"className":41803},[102,335],[40,41805,207],{"className":41806},[113],[40,41808,41810],{"className":41809},[102],[40,41811,1664],{"className":41812},[417,1238],[40,41814],{"className":41815,"style":390},[121],[40,41817,162],{"className":41818},[394],[40,41820],{"className":41821,"style":390},[121],[40,41823,41825,41828,41896,41899,41902],{"className":41824},[93],[40,41826],{"className":41827,"style":2676},[97],[40,41829,41831,41834,41893],{"className":41830},[102],[40,41832],{"className":41833},[106,1162],[40,41835,41837],{"className":41836},[1038],[40,41838,41840,41885],{"className":41839},[344,345],[40,41841,41843,41882],{"className":41842},[349],[40,41844,41846,41860,41868],{"className":41845,"style":1175},[353],[40,41847,41848,41851],{"style":1178},[40,41849],{"className":41850,"style":444},[361],[40,41852,41854],{"className":41853},[366,367,368,369],[40,41855,41857],{"className":41856},[102,369],[40,41858,1044],{"className":41859},[102,369],[40,41861,41862,41865],{"style":1193},[40,41863],{"className":41864,"style":444},[361],[40,41866],{"className":41867,"style":1201},[1200],[40,41869,41870,41873],{"style":1204},[40,41871],{"className":41872,"style":444},[361],[40,41874,41876],{"className":41875},[366,367,368,369],[40,41877,41879],{"className":41878},[102,369],[40,41880,78],{"className":41881},[102,369],[40,41883,377],{"className":41884},[376],[40,41886,41888],{"className":41887},[349],[40,41889,41891],{"className":41890,"style":1226},[353],[40,41892],{},[40,41894],{"className":41895},[113,1162],[40,41897],{"className":41898,"style":1246},[121],[40,41900,1053],{"className":41901},[1250],[40,41903],{"className":41904,"style":1246},[121],[40,41906,41908,41911,42017,42020,42023,42026,42029,42032],{"className":41907},[93],[40,41909],{"className":41910,"style":2760},[97],[40,41912,41914,41917,42014],{"className":41913},[102],[40,41915],{"className":41916},[106,1162],[40,41918,41920],{"className":41919},[1038],[40,41921,41923,42006],{"className":41922},[344,345],[40,41924,41926,42003],{"className":41925},[349],[40,41927,41929,41940,41948],{"className":41928,"style":2779},[353],[40,41930,41931,41934],{"style":2782},[40,41932],{"className":41933,"style":444},[361],[40,41935,41937],{"className":41936},[102],[40,41938,1044],{"className":41939},[102],[40,41941,41942,41945],{"style":1193},[40,41943],{"className":41944,"style":444},[361],[40,41946],{"className":41947,"style":1201},[1200],[40,41949,41950,41953],{"style":2802},[40,41951],{"className":41952,"style":444},[361],[40,41954,41956],{"className":41955},[102],[40,41957,41959],{"className":41958},[102,2218],[40,41960,41962,41995],{"className":41961},[344,345],[40,41963,41965,41992],{"className":41964},[349],[40,41966,41968,41980],{"className":41967,"style":2821},[353],[40,41969,41971,41974],{"className":41970,"style":2233},[2232],[40,41972],{"className":41973,"style":444},[361],[40,41975,41977],{"className":41976,"style":2240},[102],[40,41978,949],{"className":41979},[102],[40,41981,41982,41985],{"style":2836},[40,41983],{"className":41984,"style":444},[361],[40,41986,41988],{"className":41987,"style":2254},[2253],[2256,41989,41990],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,41991],{"d":2266},[40,41993,377],{"className":41994},[376],[40,41996,41998],{"className":41997},[349],[40,41999,42001],{"className":42000,"style":2856},[353],[40,42002],{},[40,42004,377],{"className":42005},[376],[40,42007,42009],{"className":42008},[349],[40,42010,42012],{"className":42011,"style":2868},[353],[40,42013],{},[40,42015],{"className":42016},[113,1162],[40,42018,2051],{"className":42019},[117],[40,42021],{"className":42022,"style":122},[121],[40,42024,936],{"className":42025},[102,966],[40,42027],{"className":42028,"style":1246},[121],[40,42030,1058],{"className":42031},[1250],[40,42033],{"className":42034,"style":1246},[121],[40,42036,42038,42041,42044,42047,42050,42053,42056,42059,42062],{"className":42037},[93],[40,42039],{"className":42040,"style":98},[97],[40,42042,1796],{"className":42043},[102,966],[40,42045,167],{"className":42046},[106],[40,42048,1649],{"className":42049,"style":1312},[102,335],[40,42051,73],{"className":42052},[117],[40,42054],{"className":42055,"style":122},[121],[40,42057,1671],{"className":42058},[102,335],[40,42060,207],{"className":42061},[113],[40,42063,1123],{"className":42064},[102],[40,42066,42068,42071],{"className":42067},[2925],[40,42069],{"className":42070,"style":2760},[97],[40,42072,42074,42077,42083],{"className":42073},[102,2932],[40,42075,167],{"className":42076},[102],[40,42078,42080],{"className":42079},[102],[40,42081,2942],{"className":42082},[102],[40,42084,207],{"className":42085},[102],[12,42087,2948,42088,3010],{},[40,42089,42091,42112],{"className":42090},[43],[40,42092,42094],{"className":42093},[47],[49,42095,42096],{"xmlns":51},[53,42097,42098,42110],{},[56,42099,42100,42102,42104,42106,42108],{},[59,42101,1579],{"stretchy":61},[154,42103,936],{"mathvariant":935},[59,42105,1579],{"stretchy":61},[59,42107,162],{},[64,42109,78],{},[82,42111,2973],{"encoding":84},[40,42113,42115,42139],{"className":42114,"ariaHidden":72},[89],[40,42116,42118,42121,42124,42127,42130,42133,42136],{"className":42117},[93],[40,42119],{"className":42120,"style":98},[97],[40,42122,1579],{"className":42123},[106],[40,42125,936],{"className":42126},[102,966],[40,42128,1579],{"className":42129},[113],[40,42131],{"className":42132,"style":390},[121],[40,42134,162],{"className":42135},[394],[40,42137],{"className":42138,"style":390},[121],[40,42140,42142,42145],{"className":42141},[93],[40,42143],{"className":42144,"style":1624},[97],[40,42146,78],{"className":42147},[102],[40,42149,42151],{"className":42150},[136],[40,42152,42154,42231],{"className":42153},[43],[40,42155,42157],{"className":42156},[47],[49,42158,42159],{"xmlns":51,"display":145},[53,42160,42161,42229],{},[56,42162,42163,42171,42173,42181,42183,42193,42195,42197,42205,42207,42215,42217,42227],{},[151,42164,42165,42167,42169],{},[154,42166,2437],{},[154,42168,3032],{},[59,42170,1084],{},[59,42172,162],{},[181,42174,42175],{"scriptlevel":66,"displaystyle":61},[1038,42176,42177,42179],{},[64,42178,78],{},[64,42180,1044],{},[59,42182,1053],{},[181,42184,42185],{"scriptlevel":66,"displaystyle":61},[1038,42186,42187,42191],{},[2045,42188,42189],{},[64,42190,949],{},[64,42192,1044],{},[59,42194,73],{"separator":72},[121,42196],{"width":1068},[151,42198,42199,42201,42203],{},[154,42200,2437],{},[154,42202,3067],{},[59,42204,1084],{},[59,42206,162],{},[181,42208,42209],{"scriptlevel":66,"displaystyle":61},[1038,42210,42211,42213],{},[64,42212,78],{},[64,42214,1044],{},[59,42216,243],{},[181,42218,42219],{"scriptlevel":66,"displaystyle":61},[1038,42220,42221,42225],{},[2045,42222,42223],{},[64,42224,949],{},[64,42226,1044],{},[154,42228,1123],{"mathvariant":1080},[82,42230,3096],{"encoding":84},[40,42232,42234,42301,42384,42572,42655],{"className":42233,"ariaHidden":72},[89],[40,42235,42237,42240,42292,42295,42298],{"className":42236},[93],[40,42238],{"className":42239,"style":3106},[97],[40,42241,42243,42246],{"className":42242},[102],[40,42244,2437],{"className":42245,"style":2541},[102,335],[40,42247,42249],{"className":42248},[340],[40,42250,42252,42284],{"className":42251},[344,345],[40,42253,42255,42281],{"className":42254},[349],[40,42256,42258],{"className":42257,"style":354},[353],[40,42259,42260,42263],{"style":2556},[40,42261],{"className":42262,"style":362},[361],[40,42264,42266],{"className":42265},[366,367,368,369],[40,42267,42269],{"className":42268},[102,369],[40,42270,42272,42275,42278],{"className":42271},[1369,369],[40,42273,3142],{"className":42274},[369],[40,42276,19],{"className":42277},[369],[40,42279,159],{"className":42280},[369],[40,42282,377],{"className":42283},[376],[40,42285,42287],{"className":42286},[349],[40,42288,42290],{"className":42289,"style":384},[353],[40,42291],{},[40,42293],{"className":42294,"style":390},[121],[40,42296,162],{"className":42297},[394],[40,42299],{"className":42300,"style":390},[121],[40,42302,42304,42307,42375,42378,42381],{"className":42303},[93],[40,42305],{"className":42306,"style":2676},[97],[40,42308,42310,42313,42372],{"className":42309},[102],[40,42311],{"className":42312},[106,1162],[40,42314,42316],{"className":42315},[1038],[40,42317,42319,42364],{"className":42318},[344,345],[40,42320,42322,42361],{"className":42321},[349],[40,42323,42325,42339,42347],{"className":42324,"style":1175},[353],[40,42326,42327,42330],{"style":1178},[40,42328],{"className":42329,"style":444},[361],[40,42331,42333],{"className":42332},[366,367,368,369],[40,42334,42336],{"className":42335},[102,369],[40,42337,1044],{"className":42338},[102,369],[40,42340,42341,42344],{"style":1193},[40,42342],{"className":42343,"style":444},[361],[40,42345],{"className":42346,"style":1201},[1200],[40,42348,42349,42352],{"style":1204},[40,42350],{"className":42351,"style":444},[361],[40,42353,42355],{"className":42354},[366,367,368,369],[40,42356,42358],{"className":42357},[102,369],[40,42359,78],{"className":42360},[102,369],[40,42362,377],{"className":42363},[376],[40,42365,42367],{"className":42366},[349],[40,42368,42370],{"className":42369,"style":1226},[353],[40,42371],{},[40,42373],{"className":42374},[113,1162],[40,42376],{"className":42377,"style":1246},[121],[40,42379,1053],{"className":42380},[1250],[40,42382],{"className":42383,"style":1246},[121],[40,42385,42387,42390,42502,42505,42508,42511,42563,42566,42569],{"className":42386},[93],[40,42388],{"className":42389,"style":3258},[97],[40,42391,42393,42396,42499],{"className":42392},[102],[40,42394],{"className":42395},[106,1162],[40,42397,42399],{"className":42398},[1038],[40,42400,42402,42491],{"className":42401},[344,345],[40,42403,42405,42488],{"className":42404},[349],[40,42406,42408,42422,42430],{"className":42407,"style":3277},[353],[40,42409,42410,42413],{"style":1178},[40,42411],{"className":42412,"style":444},[361],[40,42414,42416],{"className":42415},[366,367,368,369],[40,42417,42419],{"className":42418},[102,369],[40,42420,1044],{"className":42421},[102,369],[40,42423,42424,42427],{"style":1193},[40,42425],{"className":42426,"style":444},[361],[40,42428],{"className":42429,"style":1201},[1200],[40,42431,42432,42435],{"style":3302},[40,42433],{"className":42434,"style":444},[361],[40,42436,42438],{"className":42437},[366,367,368,369],[40,42439,42441],{"className":42440},[102,369],[40,42442,42444],{"className":42443},[102,2218,369],[40,42445,42447,42480],{"className":42446},[344,345],[40,42448,42450,42477],{"className":42449},[349],[40,42451,42453,42465],{"className":42452,"style":3324},[353],[40,42454,42456,42459],{"className":42455,"style":2233},[2232],[40,42457],{"className":42458,"style":444},[361],[40,42460,42462],{"className":42461,"style":2240},[102,369],[40,42463,949],{"className":42464},[102,369],[40,42466,42467,42470],{"style":3339},[40,42468],{"className":42469,"style":444},[361],[40,42471,42473],{"className":42472,"style":2254},[2253,369],[2256,42474,42475],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,42476],{"d":2266},[40,42478,377],{"className":42479},[376],[40,42481,42483],{"className":42482},[349],[40,42484,42486],{"className":42485,"style":3359},[353],[40,42487],{},[40,42489,377],{"className":42490},[376],[40,42492,42494],{"className":42493},[349],[40,42495,42497],{"className":42496,"style":1226},[353],[40,42498],{},[40,42500],{"className":42501},[113,1162],[40,42503,73],{"className":42504},[117],[40,42506],{"className":42507,"style":1302},[121],[40,42509],{"className":42510,"style":122},[121],[40,42512,42514,42517],{"className":42513},[102],[40,42515,2437],{"className":42516,"style":2541},[102,335],[40,42518,42520],{"className":42519},[340],[40,42521,42523,42555],{"className":42522},[344,345],[40,42524,42526,42552],{"className":42525},[349],[40,42527,42529],{"className":42528,"style":3403},[353],[40,42530,42531,42534],{"style":2556},[40,42532],{"className":42533,"style":362},[361],[40,42535,42537],{"className":42536},[366,367,368,369],[40,42538,42540],{"className":42539},[102,369],[40,42541,42543,42546,42549],{"className":42542},[1369,369],[40,42544,3142],{"className":42545},[369],[40,42547,246],{"className":42548},[369],[40,42550,1796],{"className":42551},[369],[40,42553,377],{"className":42554},[376],[40,42556,42558],{"className":42557},[349],[40,42559,42561],{"className":42560,"style":384},[353],[40,42562],{},[40,42564],{"className":42565,"style":390},[121],[40,42567,162],{"className":42568},[394],[40,42570],{"className":42571,"style":390},[121],[40,42573,42575,42578,42646,42649,42652],{"className":42574},[93],[40,42576],{"className":42577,"style":2676},[97],[40,42579,42581,42584,42643],{"className":42580},[102],[40,42582],{"className":42583},[106,1162],[40,42585,42587],{"className":42586},[1038],[40,42588,42590,42635],{"className":42589},[344,345],[40,42591,42593,42632],{"className":42592},[349],[40,42594,42596,42610,42618],{"className":42595,"style":1175},[353],[40,42597,42598,42601],{"style":1178},[40,42599],{"className":42600,"style":444},[361],[40,42602,42604],{"className":42603},[366,367,368,369],[40,42605,42607],{"className":42606},[102,369],[40,42608,1044],{"className":42609},[102,369],[40,42611,42612,42615],{"style":1193},[40,42613],{"className":42614,"style":444},[361],[40,42616],{"className":42617,"style":1201},[1200],[40,42619,42620,42623],{"style":1204},[40,42621],{"className":42622,"style":444},[361],[40,42624,42626],{"className":42625},[366,367,368,369],[40,42627,42629],{"className":42628},[102,369],[40,42630,78],{"className":42631},[102,369],[40,42633,377],{"className":42634},[376],[40,42636,42638],{"className":42637},[349],[40,42639,42641],{"className":42640,"style":1226},[353],[40,42642],{},[40,42644],{"className":42645},[113,1162],[40,42647],{"className":42648,"style":1246},[121],[40,42650,243],{"className":42651},[1250],[40,42653],{"className":42654,"style":1246},[121],[40,42656,42658,42661,42773],{"className":42657},[93],[40,42659],{"className":42660,"style":3258},[97],[40,42662,42664,42667,42770],{"className":42663},[102],[40,42665],{"className":42666},[106,1162],[40,42668,42670],{"className":42669},[1038],[40,42671,42673,42762],{"className":42672},[344,345],[40,42674,42676,42759],{"className":42675},[349],[40,42677,42679,42693,42701],{"className":42678,"style":3277},[353],[40,42680,42681,42684],{"style":1178},[40,42682],{"className":42683,"style":444},[361],[40,42685,42687],{"className":42686},[366,367,368,369],[40,42688,42690],{"className":42689},[102,369],[40,42691,1044],{"className":42692},[102,369],[40,42694,42695,42698],{"style":1193},[40,42696],{"className":42697,"style":444},[361],[40,42699],{"className":42700,"style":1201},[1200],[40,42702,42703,42706],{"style":3302},[40,42704],{"className":42705,"style":444},[361],[40,42707,42709],{"className":42708},[366,367,368,369],[40,42710,42712],{"className":42711},[102,369],[40,42713,42715],{"className":42714},[102,2218,369],[40,42716,42718,42751],{"className":42717},[344,345],[40,42719,42721,42748],{"className":42720},[349],[40,42722,42724,42736],{"className":42723,"style":3324},[353],[40,42725,42727,42730],{"className":42726,"style":2233},[2232],[40,42728],{"className":42729,"style":444},[361],[40,42731,42733],{"className":42732,"style":2240},[102,369],[40,42734,949],{"className":42735},[102,369],[40,42737,42738,42741],{"style":3339},[40,42739],{"className":42740,"style":444},[361],[40,42742,42744],{"className":42743,"style":2254},[2253,369],[2256,42745,42746],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,42747],{"d":2266},[40,42749,377],{"className":42750},[376],[40,42752,42754],{"className":42753},[349],[40,42755,42757],{"className":42756,"style":3359},[353],[40,42758],{},[40,42760,377],{"className":42761},[376],[40,42763,42765],{"className":42764},[349],[40,42766,42768],{"className":42767,"style":1226},[353],[40,42769],{},[40,42771],{"className":42772},[113,1162],[40,42774,1123],{"className":42775},[102],[12,42777,42778,3656],{},[35,42779,3655],{},[40,42781,42783],{"className":42782},[136],[40,42784,42786,42875],{"className":42785},[43],[40,42787,42789],{"className":42788},[47],[49,42790,42791],{"xmlns":51,"display":145},[53,42792,42793,42873],{},[169,42794,42795],{"width":2423},[175,42796,42797,42799,42867,42869],{},[178,42798],{"width":2428},[178,42800,42801],{},[56,42802,42803,42819,42821,42823,42825,42827,42829,42831,42837,42839,42841,42851,42853,42855,42857,42859,42861,42863,42865],{},[151,42804,42805,42807],{},[154,42806,2437],{},[56,42808,42809,42811,42813],{},[154,42810,3688],{},[154,42812,1032],{},[941,42814,42815,42817],{},[154,42816,3688],{},[59,42818,3698],{"lspace":3697,"rspace":3697},[59,42820,167],{"stretchy":61},[154,42822,1649],{},[59,42824,73],{"separator":72},[154,42826,1671],{},[59,42828,207],{"stretchy":61},[59,42830,162],{},[151,42832,42833,42835],{},[154,42834,2437],{},[154,42836,1032],{},[59,42838,3719],{"stretchy":61},[59,42840,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[941,42842,42843,42845],{},[154,42844,946],{},[56,42846,42847,42849],{},[59,42848,243],{},[64,42850,78],{},[154,42852,1796],{"mathvariant":935},[59,42854,167],{"stretchy":61},[154,42856,1649],{},[59,42858,73],{"separator":72},[154,42860,1671],{},[59,42862,207],{"stretchy":61},[59,42864,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,42866,73],{"separator":72},[178,42868],{"width":2428},[178,42870,42871],{},[1102,42872,3754],{},[82,42874,3757],{"encoding":84},[40,42876,42878,42989,43112],{"className":42877,"ariaHidden":72},[89],[40,42879,42881,42884,42962,42965,42968,42971,42974,42977,42980,42983,42986],{"className":42880},[93],[40,42882],{"className":42883,"style":3767},[97],[40,42885,42887,42890],{"className":42886},[102],[40,42888,2437],{"className":42889,"style":2541},[102,335],[40,42891,42893],{"className":42892},[340],[40,42894,42896,42954],{"className":42895},[344,345],[40,42897,42899,42951],{"className":42898},[349],[40,42900,42902],{"className":42901,"style":3786},[353],[40,42903,42904,42907],{"style":3789},[40,42905],{"className":42906,"style":362},[361],[40,42908,42910],{"className":42909},[366,367,368,369],[40,42911,42913,42916,42919],{"className":42912},[102,369],[40,42914,3688],{"className":42915,"style":3802},[102,335,369],[40,42917,1032],{"className":42918},[102,335,369],[40,42920,42922,42925],{"className":42921},[102,369],[40,42923,3688],{"className":42924,"style":3802},[102,335,369],[40,42926,42928],{"className":42927},[340],[40,42929,42931],{"className":42930},[344],[40,42932,42934],{"className":42933},[349],[40,42935,42937],{"className":42936,"style":3824},[353],[40,42938,42939,42942],{"style":3827},[40,42940],{"className":42941,"style":3831},[361],[40,42943,42945],{"className":42944},[366,3835,1238,369],[40,42946,42948],{"className":42947},[102,369],[40,42949,3698],{"className":42950},[102,369],[40,42952,377],{"className":42953},[376],[40,42955,42957],{"className":42956},[349],[40,42958,42960],{"className":42959,"style":3851},[353],[40,42961],{},[40,42963,167],{"className":42964},[106],[40,42966,1649],{"className":42967,"style":1312},[102,335],[40,42969,73],{"className":42970},[117],[40,42972],{"className":42973,"style":122},[121],[40,42975,1671],{"className":42976},[102,335],[40,42978,207],{"className":42979},[113],[40,42981],{"className":42982,"style":390},[121],[40,42984,162],{"className":42985},[394],[40,42987],{"className":42988,"style":390},[121],[40,42990,42992,42995,43038,43041,43047,43082,43085,43088,43091,43094,43097,43100,43103,43109],{"className":42991},[93],[40,42993],{"className":42994,"style":3887},[97],[40,42996,42998,43001],{"className":42997},[102],[40,42999,2437],{"className":43000,"style":2541},[102,335],[40,43002,43004],{"className":43003},[340],[40,43005,43007,43030],{"className":43006},[344,345],[40,43008,43010,43027],{"className":43009},[349],[40,43011,43013],{"className":43012,"style":354},[353],[40,43014,43015,43018],{"style":2556},[40,43016],{"className":43017,"style":362},[361],[40,43019,43021],{"className":43020},[366,367,368,369],[40,43022,43024],{"className":43023},[102,369],[40,43025,1032],{"className":43026},[102,335,369],[40,43028,377],{"className":43029},[376],[40,43031,43033],{"className":43032},[349],[40,43034,43036],{"className":43035,"style":585},[353],[40,43037],{},[40,43039,3719],{"className":43040},[113],[40,43042,43044],{"className":43043},[102],[40,43045,167],{"className":43046},[417,1238],[40,43048,43050,43053],{"className":43049},[102],[40,43051,946],{"className":43052,"style":3946},[102,335],[40,43054,43056],{"className":43055},[340],[40,43057,43059],{"className":43058},[344],[40,43060,43062],{"className":43061},[349],[40,43063,43065],{"className":43064,"style":3959},[353],[40,43066,43067,43070],{"style":3962},[40,43068],{"className":43069,"style":362},[361],[40,43071,43073],{"className":43072},[366,367,368,369],[40,43074,43076,43079],{"className":43075},[102,369],[40,43077,243],{"className":43078},[102,369],[40,43080,78],{"className":43081},[102,369],[40,43083,1796],{"className":43084},[102,966],[40,43086,167],{"className":43087},[106],[40,43089,1649],{"className":43090,"style":1312},[102,335],[40,43092,73],{"className":43093},[117],[40,43095],{"className":43096,"style":122},[121],[40,43098,1671],{"className":43099},[102,335],[40,43101,207],{"className":43102},[113],[40,43104,43106],{"className":43105},[102],[40,43107,207],{"className":43108},[417,1238],[40,43110,73],{"className":43111},[117],[40,43113,43115,43118],{"className":43114},[2925],[40,43116],{"className":43117,"style":3887},[97],[40,43119,43121,43124,43130],{"className":43120},[102,2932],[40,43122,167],{"className":43123},[102],[40,43125,43127],{"className":43126},[102],[40,43128,4026],{"className":43129},[102],[40,43131,207],{"className":43132},[102],[12,43134,4032,43135,4109],{},[40,43136,43138,43165],{"className":43137},[43],[40,43139,43141],{"className":43140},[47],[49,43142,43143],{"xmlns":51},[53,43144,43145,43163],{},[56,43146,43147,43149,43151,43157,43159,43161],{},[154,43148,946],{},[59,43150,939],{},[56,43152,43153,43155],{},[154,43154,2362],{"mathvariant":1080},[154,43156,4055],{"mathvariant":1080},[59,43158,167],{"stretchy":61},[64,43160,949],{},[59,43162,207],{"stretchy":61},[82,43164,4064],{"encoding":84},[40,43166,43168,43186],{"className":43167,"ariaHidden":72},[89],[40,43169,43171,43174,43177,43180,43183],{"className":43170},[93],[40,43172],{"className":43173,"style":4074},[97],[40,43175,946],{"className":43176,"style":3946},[102,335],[40,43178],{"className":43179,"style":390},[121],[40,43181,939],{"className":43182},[394],[40,43184],{"className":43185,"style":390},[121],[40,43187,43189,43192,43198,43201,43204],{"className":43188},[93],[40,43190],{"className":43191,"style":98},[97],[40,43193,43195],{"className":43194},[102],[40,43196,4099],{"className":43197},[102,1373],[40,43199,167],{"className":43200},[106],[40,43202,949],{"className":43203},[102],[40,43205,207],{"className":43206},[113],[25,43208],{},[28,43210,4115,43211,4146],{"id":4114},[40,43212,43214,43227],{"className":43213},[43],[40,43215,43217],{"className":43216},[47],[49,43218,43219],{"xmlns":51},[53,43220,43221,43225],{},[56,43222,43223],{},[154,43224,1661],{},[82,43226,4132],{"encoding":84},[40,43228,43230],{"className":43229,"ariaHidden":72},[89],[40,43231,43233,43236],{"className":43232},[93],[40,43234],{"className":43235,"style":4142},[97],[40,43237,1661],{"className":43238,"style":336},[102,335],[12,43240,4149,43241,4178],{},[40,43242,43244,43257],{"className":43243},[43],[40,43245,43247],{"className":43246},[47],[49,43248,43249],{"xmlns":51},[53,43250,43251,43255],{},[56,43252,43253],{},[154,43254,1661],{},[82,43256,4132],{"encoding":84},[40,43258,43260],{"className":43259,"ariaHidden":72},[89],[40,43261,43263,43266],{"className":43262},[93],[40,43264],{"className":43265,"style":4142},[97],[40,43267,1661],{"className":43268,"style":336},[102,335],[40,43270,43272],{"className":43271},[136],[40,43273,43275,43380],{"className":43274},[43],[40,43276,43278],{"className":43277},[47],[49,43279,43280],{"xmlns":51,"display":145},[53,43281,43282,43378],{},[56,43283,43284,43286,43288,43312,43314,43316,43318,43320,43344,43346,43348,43350,43352,43376],{},[154,43285,4196],{},[59,43287,162],{},[941,43289,43290,43292],{},[154,43291,4203],{},[56,43293,43294,43296,43298,43304,43306],{},[59,43295,243],{},[154,43297,246],{},[1038,43299,43300,43302],{},[154,43301,1661],{},[64,43303,1044],{},[59,43305,73],{"separator":72},[151,43307,43308,43310],{},[154,43309,156],{},[154,43311,159],{},[59,43313,73],{"separator":72},[121,43315],{"width":1068},[154,43317,4230],{},[59,43319,162],{},[941,43321,43322,43324],{},[154,43323,4203],{},[56,43325,43326,43328,43330,43336,43338],{},[59,43327,243],{},[154,43329,246],{},[1038,43331,43332,43334],{},[154,43333,1661],{},[64,43335,1044],{},[59,43337,73],{"separator":72},[151,43339,43340,43342],{},[154,43341,156],{},[154,43343,218],{},[59,43345,73],{"separator":72},[121,43347],{"width":1068},[154,43349,4263],{},[59,43351,162],{},[941,43353,43354,43356],{},[154,43355,4203],{},[56,43357,43358,43360,43362,43368,43370],{},[59,43359,243],{},[154,43361,246],{},[1038,43363,43364,43366],{},[154,43365,1661],{},[64,43367,1044],{},[59,43369,73],{"separator":72},[151,43371,43372,43374],{},[154,43373,156],{},[154,43375,273],{},[154,43377,1123],{"mathvariant":1080},[82,43379,4294],{"encoding":84},[40,43381,43383,43401,43574,43747],{"className":43382,"ariaHidden":72},[89],[40,43384,43386,43389,43392,43395,43398],{"className":43385},[93],[40,43387],{"className":43388,"style":4304},[97],[40,43390,4196],{"className":43391,"style":1242},[102,335],[40,43393],{"className":43394,"style":390},[121],[40,43396,162],{"className":43397},[394],[40,43399],{"className":43400,"style":390},[121],[40,43402,43404,43407,43553,43556,43559,43562,43565,43568,43571],{"className":43403},[93],[40,43405],{"className":43406,"style":4323},[97],[40,43408,43410,43413],{"className":43409},[102],[40,43411,4203],{"className":43412},[102,335],[40,43414,43416],{"className":43415},[340],[40,43417,43419],{"className":43418},[344],[40,43420,43422],{"className":43421},[349],[40,43423,43425],{"className":43424,"style":4342},[353],[40,43426,43427,43430],{"style":4345},[40,43428],{"className":43429,"style":444},[361],[40,43431,43433],{"className":43432},[366,367,368,369],[40,43434,43436,43439,43442,43510,43513],{"className":43435},[102,369],[40,43437,243],{"className":43438},[102,369],[40,43440,246],{"className":43441},[102,335,369],[40,43443,43445,43448,43507],{"className":43444},[102,369],[40,43446],{"className":43447},[106,1162,366,3835,4367],[40,43449,43451],{"className":43450},[1038],[40,43452,43454,43499],{"className":43453},[344,345],[40,43455,43457,43496],{"className":43456},[349],[40,43458,43460,43474,43482],{"className":43459,"style":4380},[353],[40,43461,43462,43465],{"style":4383},[40,43463],{"className":43464,"style":444},[361],[40,43466,43468],{"className":43467},[366,3835,1238,369],[40,43469,43471],{"className":43470},[102,369],[40,43472,1044],{"className":43473},[102,369],[40,43475,43476,43479],{"style":4398},[40,43477],{"className":43478,"style":444},[361],[40,43480],{"className":43481,"style":4405},[1200,369],[40,43483,43484,43487],{"style":4408},[40,43485],{"className":43486,"style":444},[361],[40,43488,43490],{"className":43489},[366,3835,1238,369],[40,43491,43493],{"className":43492},[102,369],[40,43494,1661],{"className":43495,"style":336},[102,335,369],[40,43497,377],{"className":43498},[376],[40,43500,43502],{"className":43501},[349],[40,43503,43505],{"className":43504,"style":4430},[353],[40,43506],{},[40,43508],{"className":43509},[113,1162,366,3835,4367],[40,43511,73],{"className":43512},[117,369],[40,43514,43516,43519],{"className":43515},[102,369],[40,43517,156],{"className":43518,"style":336},[102,335,369],[40,43520,43522],{"className":43521},[340],[40,43523,43525,43545],{"className":43524},[344,345],[40,43526,43528,43542],{"className":43527},[349],[40,43529,43531],{"className":43530,"style":4457},[353],[40,43532,43533,43536],{"style":4460},[40,43534],{"className":43535,"style":3831},[361],[40,43537,43539],{"className":43538},[366,3835,1238,369],[40,43540,159],{"className":43541},[102,335,369],[40,43543,377],{"className":43544},[376],[40,43546,43548],{"className":43547},[349],[40,43549,43551],{"className":43550,"style":4479},[353],[40,43552],{},[40,43554,73],{"className":43555},[117],[40,43557],{"className":43558,"style":1302},[121],[40,43560],{"className":43561,"style":122},[121],[40,43563,4230],{"className":43564,"style":1246},[102,335],[40,43566],{"className":43567,"style":390},[121],[40,43569,162],{"className":43570},[394],[40,43572],{"className":43573,"style":390},[121],[40,43575,43577,43580,43726,43729,43732,43735,43738,43741,43744],{"className":43576},[93],[40,43578],{"className":43579,"style":4323},[97],[40,43581,43583,43586],{"className":43582},[102],[40,43584,4203],{"className":43585},[102,335],[40,43587,43589],{"className":43588},[340],[40,43590,43592],{"className":43591},[344],[40,43593,43595],{"className":43594},[349],[40,43596,43598],{"className":43597,"style":4342},[353],[40,43599,43600,43603],{"style":4345},[40,43601],{"className":43602,"style":444},[361],[40,43604,43606],{"className":43605},[366,367,368,369],[40,43607,43609,43612,43615,43683,43686],{"className":43608},[102,369],[40,43610,243],{"className":43611},[102,369],[40,43613,246],{"className":43614},[102,335,369],[40,43616,43618,43621,43680],{"className":43617},[102,369],[40,43619],{"className":43620},[106,1162,366,3835,4367],[40,43622,43624],{"className":43623},[1038],[40,43625,43627,43672],{"className":43626},[344,345],[40,43628,43630,43669],{"className":43629},[349],[40,43631,43633,43647,43655],{"className":43632,"style":4380},[353],[40,43634,43635,43638],{"style":4383},[40,43636],{"className":43637,"style":444},[361],[40,43639,43641],{"className":43640},[366,3835,1238,369],[40,43642,43644],{"className":43643},[102,369],[40,43645,1044],{"className":43646},[102,369],[40,43648,43649,43652],{"style":4398},[40,43650],{"className":43651,"style":444},[361],[40,43653],{"className":43654,"style":4405},[1200,369],[40,43656,43657,43660],{"style":4408},[40,43658],{"className":43659,"style":444},[361],[40,43661,43663],{"className":43662},[366,3835,1238,369],[40,43664,43666],{"className":43665},[102,369],[40,43667,1661],{"className":43668,"style":336},[102,335,369],[40,43670,377],{"className":43671},[376],[40,43673,43675],{"className":43674},[349],[40,43676,43678],{"className":43677,"style":4430},[353],[40,43679],{},[40,43681],{"className":43682},[113,1162,366,3835,4367],[40,43684,73],{"className":43685},[117,369],[40,43687,43689,43692],{"className":43688},[102,369],[40,43690,156],{"className":43691,"style":336},[102,335,369],[40,43693,43695],{"className":43694},[340],[40,43696,43698,43718],{"className":43697},[344,345],[40,43699,43701,43715],{"className":43700},[349],[40,43702,43704],{"className":43703,"style":4457},[353],[40,43705,43706,43709],{"style":4460},[40,43707],{"className":43708,"style":3831},[361],[40,43710,43712],{"className":43711},[366,3835,1238,369],[40,43713,218],{"className":43714,"style":336},[102,335,369],[40,43716,377],{"className":43717},[376],[40,43719,43721],{"className":43720},[349],[40,43722,43724],{"className":43723,"style":4653},[353],[40,43725],{},[40,43727,73],{"className":43728},[117],[40,43730],{"className":43731,"style":1302},[121],[40,43733],{"className":43734,"style":122},[121],[40,43736,4263],{"className":43737,"style":4668},[102,335],[40,43739],{"className":43740,"style":390},[121],[40,43742,162],{"className":43743},[394],[40,43745],{"className":43746,"style":390},[121],[40,43748,43750,43753,43899],{"className":43749},[93],[40,43751],{"className":43752,"style":4342},[97],[40,43754,43756,43759],{"className":43755},[102],[40,43757,4203],{"className":43758},[102,335],[40,43760,43762],{"className":43761},[340],[40,43763,43765],{"className":43764},[344],[40,43766,43768],{"className":43767},[349],[40,43769,43771],{"className":43770,"style":4342},[353],[40,43772,43773,43776],{"style":4345},[40,43774],{"className":43775,"style":444},[361],[40,43777,43779],{"className":43778},[366,367,368,369],[40,43780,43782,43785,43788,43856,43859],{"className":43781},[102,369],[40,43783,243],{"className":43784},[102,369],[40,43786,246],{"className":43787},[102,335,369],[40,43789,43791,43794,43853],{"className":43790},[102,369],[40,43792],{"className":43793},[106,1162,366,3835,4367],[40,43795,43797],{"className":43796},[1038],[40,43798,43800,43845],{"className":43799},[344,345],[40,43801,43803,43842],{"className":43802},[349],[40,43804,43806,43820,43828],{"className":43805,"style":4380},[353],[40,43807,43808,43811],{"style":4383},[40,43809],{"className":43810,"style":444},[361],[40,43812,43814],{"className":43813},[366,3835,1238,369],[40,43815,43817],{"className":43816},[102,369],[40,43818,1044],{"className":43819},[102,369],[40,43821,43822,43825],{"style":4398},[40,43823],{"className":43824,"style":444},[361],[40,43826],{"className":43827,"style":4405},[1200,369],[40,43829,43830,43833],{"style":4408},[40,43831],{"className":43832,"style":444},[361],[40,43834,43836],{"className":43835},[366,3835,1238,369],[40,43837,43839],{"className":43838},[102,369],[40,43840,1661],{"className":43841,"style":336},[102,335,369],[40,43843,377],{"className":43844},[376],[40,43846,43848],{"className":43847},[349],[40,43849,43851],{"className":43850,"style":4430},[353],[40,43852],{},[40,43854],{"className":43855},[113,1162,366,3835,4367],[40,43857,73],{"className":43858},[117,369],[40,43860,43862,43865],{"className":43861},[102,369],[40,43863,156],{"className":43864,"style":336},[102,335,369],[40,43866,43868],{"className":43867},[340],[40,43869,43871,43891],{"className":43870},[344,345],[40,43872,43874,43888],{"className":43873},[349],[40,43875,43877],{"className":43876,"style":4457},[353],[40,43878,43879,43882],{"style":4460},[40,43880],{"className":43881,"style":3831},[361],[40,43883,43885],{"className":43884},[366,3835,1238,369],[40,43886,273],{"className":43887,"style":764},[102,335,369],[40,43889,377],{"className":43890},[376],[40,43892,43894],{"className":43893},[349],[40,43895,43897],{"className":43896,"style":4479},[353],[40,43898],{},[40,43900,1123],{"className":43901},[102],[12,43903,4835,43904,5060,44126,5063,44128,1628],{},[40,43905,43907,43958],{"className":43906},[43],[40,43908,43910],{"className":43909},[47],[49,43911,43912],{"xmlns":51},[53,43913,43914,43956],{},[56,43915,43916,43922,43924,43926,43928,43930,43936,43938,43940,43942,43944,43950,43952,43954],{},[151,43917,43918,43920],{},[154,43919,946],{},[154,43921,159],{},[59,43923,167],{"stretchy":61},[154,43925,1661],{},[59,43927,207],{"stretchy":61},[59,43929,73],{"separator":72},[151,43931,43932,43934],{},[154,43933,946],{},[154,43935,218],{},[59,43937,167],{"stretchy":61},[154,43939,1661],{},[59,43941,207],{"stretchy":61},[59,43943,73],{"separator":72},[151,43945,43946,43948],{},[154,43947,946],{},[154,43949,273],{},[59,43951,167],{"stretchy":61},[154,43953,1661],{},[59,43955,207],{"stretchy":61},[82,43957,4890],{"encoding":84},[40,43959,43961],{"className":43960,"ariaHidden":72},[89],[40,43962,43964,43967,44007,44010,44013,44016,44019,44022,44062,44065,44068,44071,44074,44077,44117,44120,44123],{"className":43963},[93],[40,43965],{"className":43966,"style":1365},[97],[40,43968,43970,43973],{"className":43969},[102],[40,43971,946],{"className":43972,"style":3946},[102,335],[40,43974,43976],{"className":43975},[340],[40,43977,43979,43999],{"className":43978},[344,345],[40,43980,43982,43996],{"className":43981},[349],[40,43983,43985],{"className":43984,"style":354},[353],[40,43986,43987,43990],{"style":4920},[40,43988],{"className":43989,"style":362},[361],[40,43991,43993],{"className":43992},[366,367,368,369],[40,43994,159],{"className":43995},[102,335,369],[40,43997,377],{"className":43998},[376],[40,44000,44002],{"className":44001},[349],[40,44003,44005],{"className":44004,"style":384},[353],[40,44006],{},[40,44008,167],{"className":44009},[106],[40,44011,1661],{"className":44012,"style":336},[102,335],[40,44014,207],{"className":44015},[113],[40,44017,73],{"className":44018},[117],[40,44020],{"className":44021,"style":122},[121],[40,44023,44025,44028],{"className":44024},[102],[40,44026,946],{"className":44027,"style":3946},[102,335],[40,44029,44031],{"className":44030},[340],[40,44032,44034,44054],{"className":44033},[344,345],[40,44035,44037,44051],{"className":44036},[349],[40,44038,44040],{"className":44039,"style":354},[353],[40,44041,44042,44045],{"style":4920},[40,44043],{"className":44044,"style":362},[361],[40,44046,44048],{"className":44047},[366,367,368,369],[40,44049,218],{"className":44050,"style":336},[102,335,369],[40,44052,377],{"className":44053},[376],[40,44055,44057],{"className":44056},[349],[40,44058,44060],{"className":44059,"style":585},[353],[40,44061],{},[40,44063,167],{"className":44064},[106],[40,44066,1661],{"className":44067,"style":336},[102,335],[40,44069,207],{"className":44070},[113],[40,44072,73],{"className":44073},[117],[40,44075],{"className":44076,"style":122},[121],[40,44078,44080,44083],{"className":44079},[102],[40,44081,946],{"className":44082,"style":3946},[102,335],[40,44084,44086],{"className":44085},[340],[40,44087,44089,44109],{"className":44088},[344,345],[40,44090,44092,44106],{"className":44091},[349],[40,44093,44095],{"className":44094,"style":354},[353],[40,44096,44097,44100],{"style":4920},[40,44098],{"className":44099,"style":362},[361],[40,44101,44103],{"className":44102},[366,367,368,369],[40,44104,273],{"className":44105,"style":764},[102,335,369],[40,44107,377],{"className":44108},[376],[40,44110,44112],{"className":44111},[349],[40,44113,44115],{"className":44114,"style":384},[353],[40,44116],{},[40,44118,167],{"className":44119},[106],[40,44121,1661],{"className":44122,"style":336},[102,335],[40,44124,207],{"className":44125},[113],[35,44127,3754],{},[40,44129,44131,44148],{"className":44130},[43],[40,44132,44134],{"className":44133},[47],[49,44135,44136],{"xmlns":51},[53,44137,44138,44146],{},[56,44139,44140],{},[941,44141,44142,44144],{},[154,44143,2362],{},[64,44145,1044],{},[82,44147,2367],{"encoding":84},[40,44149,44151],{"className":44150,"ariaHidden":72},[89],[40,44152,44154,44157],{"className":44153},[93],[40,44155],{"className":44156,"style":982},[97],[40,44158,44160,44163],{"className":44159},[102],[40,44161,2362],{"className":44162,"style":2383},[102,335],[40,44164,44166],{"className":44165},[340],[40,44167,44169],{"className":44168},[344],[40,44170,44172],{"className":44171},[349],[40,44173,44175],{"className":44174,"style":982},[353],[40,44176,44177,44180],{"style":1004},[40,44178],{"className":44179,"style":362},[361],[40,44181,44183],{"className":44182},[366,367,368,369],[40,44184,1044],{"className":44185},[102,369],[25,44187],{},[28,44189,5127],{"id":5126},[12,44191,5130,44192,5200,44252,5232],{},[40,44193,44195,44212],{"className":44194},[43],[40,44196,44198],{"className":44197},[47],[49,44199,44200],{"xmlns":51},[53,44201,44202,44210],{},[56,44203,44204],{},[5144,44205,44206,44208],{"accent":72},[154,44207,5148],{"mathvariant":935},[59,44209,5151],{},[82,44211,5154],{"encoding":84},[40,44213,44215],{"className":44214,"ariaHidden":72},[89],[40,44216,44218,44221],{"className":44217},[93],[40,44219],{"className":44220,"style":5164},[97],[40,44222,44224],{"className":44223},[102,5168],[40,44225,44227],{"className":44226},[344],[40,44228,44230],{"className":44229},[349],[40,44231,44233,44241],{"className":44232,"style":5164},[353],[40,44234,44235,44238],{"style":2233},[40,44236],{"className":44237,"style":444},[361],[40,44239,5148],{"className":44240},[102,966],[40,44242,44243,44246],{"style":5188},[40,44244],{"className":44245,"style":444},[361],[40,44247,44249],{"className":44248,"style":5196},[5195],[40,44250,5151],{"className":44251},[102],[40,44253,44255,44268],{"className":44254},[43],[40,44256,44258],{"className":44257},[47],[49,44259,44260],{"xmlns":51},[53,44261,44262,44266],{},[56,44263,44264],{},[154,44265,5215],{},[82,44267,5218],{"encoding":84},[40,44269,44271],{"className":44270,"ariaHidden":72},[89],[40,44272,44274,44277],{"className":44273},[93],[40,44275],{"className":44276,"style":4142},[97],[40,44278,5215],{"className":44279,"style":5231},[102,335],[40,44281,44283],{"className":44282},[136],[40,44284,44286,44353],{"className":44285},[43],[40,44287,44289],{"className":44288},[47],[49,44290,44291],{"xmlns":51,"display":145},[53,44292,44293,44351],{},[56,44294,44295,44297,44299,44301,44303,44309,44311,44313,44315,44317,44319,44321,44323,44325,44327,44335,44337,44343,44345,44347,44349],{},[154,44296,3688],{},[59,44298,167],{"stretchy":61},[154,44300,5215],{},[59,44302,73],{"separator":72},[5144,44304,44305,44307],{"accent":72},[154,44306,5148],{"mathvariant":935},[59,44308,5151],{},[59,44310,207],{"stretchy":61},[59,44312,162],{},[154,44314,5268],{},[59,44316,1084],{},[59,44318,3719],{"stretchy":61},[59,44320,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,44322,243],{},[154,44324,246],{},[59,44326,73],{"separator":72},[181,44328,44329],{"scriptlevel":66,"displaystyle":61},[1038,44330,44331,44333],{},[154,44332,5215],{},[64,44334,1044],{},[59,44336,73],{"separator":72},[5144,44338,44339,44341],{"accent":72},[154,44340,5148],{"mathvariant":935},[59,44342,5151],{},[59,44344,1058],{},[154,44346,156],{"mathvariant":1061},[59,44348,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,44350,1123],{"mathvariant":1080},[82,44352,5307],{"encoding":84},[40,44354,44356,44420,44447,44576],{"className":44355,"ariaHidden":72},[89],[40,44357,44359,44362,44365,44368,44371,44374,44377,44408,44411,44414,44417],{"className":44358},[93],[40,44360],{"className":44361,"style":98},[97],[40,44363,3688],{"className":44364,"style":3802},[102,335],[40,44366,167],{"className":44367},[106],[40,44369,5215],{"className":44370,"style":5231},[102,335],[40,44372,73],{"className":44373},[117],[40,44375],{"className":44376,"style":122},[121],[40,44378,44380],{"className":44379},[102,5168],[40,44381,44383],{"className":44382},[344],[40,44384,44386],{"className":44385},[349],[40,44387,44389,44397],{"className":44388,"style":5164},[353],[40,44390,44391,44394],{"style":2233},[40,44392],{"className":44393,"style":444},[361],[40,44395,5148],{"className":44396},[102,966],[40,44398,44399,44402],{"style":5188},[40,44400],{"className":44401,"style":444},[361],[40,44403,44405],{"className":44404,"style":5196},[5195],[40,44406,5151],{"className":44407},[102],[40,44409,207],{"className":44410},[113],[40,44412],{"className":44413,"style":390},[121],[40,44415,162],{"className":44416},[394],[40,44418],{"className":44419,"style":390},[121],[40,44421,44423,44426,44429,44432,44438,44441,44444],{"className":44422},[93],[40,44424],{"className":44425,"style":1155},[97],[40,44427,5268],{"className":44428},[1369],[40,44430,3719],{"className":44431},[113],[40,44433,44435],{"className":44434},[102],[40,44436,167],{"className":44437},[417,1238],[40,44439],{"className":44440,"style":1246},[121],[40,44442,243],{"className":44443},[1250],[40,44445],{"className":44446,"style":1246},[121],[40,44448,44450,44453,44456,44459,44462,44530,44533,44536,44567,44570,44573],{"className":44449},[93],[40,44451],{"className":44452,"style":5408},[97],[40,44454,246],{"className":44455},[102,335],[40,44457,73],{"className":44458},[117],[40,44460],{"className":44461,"style":122},[121],[40,44463,44465,44468,44527],{"className":44464},[102],[40,44466],{"className":44467},[106,1162],[40,44469,44471],{"className":44470},[1038],[40,44472,44474,44519],{"className":44473},[344,345],[40,44475,44477,44516],{"className":44476},[349],[40,44478,44480,44494,44502],{"className":44479,"style":5436},[353],[40,44481,44482,44485],{"style":1178},[40,44483],{"className":44484,"style":444},[361],[40,44486,44488],{"className":44487},[366,367,368,369],[40,44489,44491],{"className":44490},[102,369],[40,44492,1044],{"className":44493},[102,369],[40,44495,44496,44499],{"style":1193},[40,44497],{"className":44498,"style":444},[361],[40,44500],{"className":44501,"style":1201},[1200],[40,44503,44504,44507],{"style":1204},[40,44505],{"className":44506,"style":444},[361],[40,44508,44510],{"className":44509},[366,367,368,369],[40,44511,44513],{"className":44512},[102,369],[40,44514,5215],{"className":44515,"style":5231},[102,335,369],[40,44517,377],{"className":44518},[376],[40,44520,44522],{"className":44521},[349],[40,44523,44525],{"className":44524,"style":1226},[353],[40,44526],{},[40,44528],{"className":44529},[113,1162],[40,44531,73],{"className":44532},[117],[40,44534],{"className":44535,"style":122},[121],[40,44537,44539],{"className":44538},[102,5168],[40,44540,44542],{"className":44541},[344],[40,44543,44545],{"className":44544},[349],[40,44546,44548,44556],{"className":44547,"style":5164},[353],[40,44549,44550,44553],{"style":2233},[40,44551],{"className":44552,"style":444},[361],[40,44554,5148],{"className":44555},[102,966],[40,44557,44558,44561],{"style":5188},[40,44559],{"className":44560,"style":444},[361],[40,44562,44564],{"className":44563,"style":5196},[5195],[40,44565,5151],{"className":44566},[102],[40,44568],{"className":44569,"style":1246},[121],[40,44571,1058],{"className":44572},[1250],[40,44574],{"className":44575,"style":1246},[121],[40,44577,44579,44582,44591,44597],{"className":44578},[93],[40,44580],{"className":44581,"style":1155},[97],[40,44583,44585],{"className":44584},[102],[40,44586,44588],{"className":44587},[102],[40,44589,156],{"className":44590,"style":1289},[102,1288],[40,44592,44594],{"className":44593},[102],[40,44595,207],{"className":44596},[417,1238],[40,44598,1123],{"className":44599},[102],[12,44601,5559,44602,5590,44630,920,44632,1014],{},[40,44603,44605,44618],{"className":44604},[43],[40,44606,44608],{"className":44607},[47],[49,44609,44610],{"xmlns":51},[53,44611,44612,44616],{},[56,44613,44614],{},[154,44615,936],{"mathvariant":935},[82,44617,5576],{"encoding":84},[40,44619,44621],{"className":44620,"ariaHidden":72},[89],[40,44622,44624,44627],{"className":44623},[93],[40,44625],{"className":44626,"style":5586},[97],[40,44628,936],{"className":44629},[102,966],[35,44631,5593],{},[40,44633,44635,44672],{"className":44634},[43],[40,44636,44638],{"className":44637},[47],[49,44639,44640],{"xmlns":51},[53,44641,44642,44670],{},[56,44643,44644,44650,44652,44654,44656,44658,44664,44666,44668],{},[56,44645,44646,44648],{},[154,44647,2362],{"mathvariant":1080},[154,44649,3688],{"mathvariant":1080},[59,44651,167],{"stretchy":61},[64,44653,1044],{},[59,44655,207],{"stretchy":61},[59,44657,5620],{},[56,44659,44660,44662],{},[154,44661,2362],{"mathvariant":1080},[154,44663,4055],{"mathvariant":1080},[59,44665,167],{"stretchy":61},[64,44667,949],{},[59,44669,207],{"stretchy":61},[82,44671,5635],{"encoding":84},[40,44673,44675,44705],{"className":44674,"ariaHidden":72},[89],[40,44676,44678,44681,44687,44690,44693,44696,44699,44702],{"className":44677},[93],[40,44679],{"className":44680,"style":98},[97],[40,44682,44684],{"className":44683},[102],[40,44685,5651],{"className":44686},[102,1373],[40,44688,167],{"className":44689},[106],[40,44691,1044],{"className":44692},[102],[40,44694,207],{"className":44695},[113],[40,44697],{"className":44698,"style":390},[121],[40,44700,5620],{"className":44701},[394],[40,44703],{"className":44704,"style":390},[121],[40,44706,44708,44711,44717,44720,44723],{"className":44707},[93],[40,44709],{"className":44710,"style":98},[97],[40,44712,44714],{"className":44713},[102],[40,44715,4099],{"className":44716},[102,1373],[40,44718,167],{"className":44719},[106],[40,44721,949],{"className":44722},[102],[40,44724,207],{"className":44725},[113],[40,44727,44729],{"className":44728},[136],[40,44730,44732,44795],{"className":44731},[43],[40,44733,44735],{"className":44734},[47],[49,44736,44737],{"xmlns":51,"display":145},[53,44738,44739,44793],{},[169,44740,44741],{"width":2423},[175,44742,44743,44745,44787,44789],{},[178,44744],{"width":2428},[178,44746,44747],{},[56,44748,44749,44751,44753,44755,44757,44763,44765,44767,44769,44771,44773,44779,44781,44783,44785],{},[154,44750,936],{"mathvariant":935},[59,44752,2051],{"separator":72},[59,44754,5720],{},[59,44756,2051],{"separator":72},[941,44758,44759,44761],{},[154,44760,936],{"mathvariant":935},[59,44762,5729],{"mathvariant":1080,"lspace":3697,"rspace":3697},[59,44764,162],{},[154,44766,946],{},[59,44768,167],{"stretchy":61},[154,44770,5215],{},[59,44772,73],{"separator":72},[5144,44774,44775,44777],{"accent":72},[154,44776,5148],{"mathvariant":935},[59,44778,5151],{},[59,44780,207],{"stretchy":61},[59,44782,73],{"separator":72},[154,44784,936],{"mathvariant":935},[154,44786,1123],{"mathvariant":1080},[178,44788],{"width":2428},[178,44790,44791],{},[1102,44792,5760],{},[82,44794,5763],{"encoding":84},[40,44796,44798,44816,44869,44936],{"className":44797,"ariaHidden":72},[89],[40,44799,44801,44804,44807,44810,44813],{"className":44800},[93],[40,44802],{"className":44803,"style":5773},[97],[40,44805,936],{"className":44806},[102,966],[40,44808,2051],{"className":44809},[117],[40,44811],{"className":44812,"style":390},[121],[40,44814,5720],{"className":44815},[394],[40,44817,44819,44822,44825,44828,44860,44863,44866],{"className":44818},[93],[40,44820],{"className":44821,"style":5792},[97],[40,44823,2051],{"className":44824},[117],[40,44826],{"className":44827,"style":122},[121],[40,44829,44831,44834],{"className":44830},[102],[40,44832,936],{"className":44833},[102,966],[40,44835,44837],{"className":44836},[340],[40,44838,44840],{"className":44839},[344],[40,44841,44843],{"className":44842},[349],[40,44844,44846],{"className":44845,"style":5817},[353],[40,44847,44848,44851],{"style":3962},[40,44849],{"className":44850,"style":362},[361],[40,44852,44854],{"className":44853},[366,367,368,369],[40,44855,44857],{"className":44856},[102,369],[40,44858,5729],{"className":44859},[102,369],[40,44861],{"className":44862,"style":390},[121],[40,44864,162],{"className":44865},[394],[40,44867],{"className":44868,"style":390},[121],[40,44870,44872,44875,44878,44881,44884,44887,44890,44921,44924,44927,44930,44933],{"className":44871},[93],[40,44873],{"className":44874,"style":98},[97],[40,44876,946],{"className":44877,"style":3946},[102,335],[40,44879,167],{"className":44880},[106],[40,44882,5215],{"className":44883,"style":5231},[102,335],[40,44885,73],{"className":44886},[117],[40,44888],{"className":44889,"style":122},[121],[40,44891,44893],{"className":44892},[102,5168],[40,44894,44896],{"className":44895},[344],[40,44897,44899],{"className":44898},[349],[40,44900,44902,44910],{"className":44901,"style":5164},[353],[40,44903,44904,44907],{"style":2233},[40,44905],{"className":44906,"style":444},[361],[40,44908,5148],{"className":44909},[102,966],[40,44911,44912,44915],{"style":5188},[40,44913],{"className":44914,"style":444},[361],[40,44916,44918],{"className":44917,"style":5196},[5195],[40,44919,5151],{"className":44920},[102],[40,44922,207],{"className":44923},[113],[40,44925,73],{"className":44926},[117],[40,44928],{"className":44929,"style":122},[121],[40,44931,936],{"className":44932},[102,966],[40,44934,1123],{"className":44935},[102],[40,44937,44939,44942],{"className":44938},[2925],[40,44940],{"className":44941,"style":5914},[97],[40,44943,44945,44948,44954],{"className":44944},[102,2932],[40,44946,167],{"className":44947},[102],[40,44949,44951],{"className":44950},[102],[40,44952,5927],{"className":44953},[102],[40,44955,207],{"className":44956},[102],[25,44958],{},[28,44960,5936,44961,5967],{"id":5935},[40,44962,44964,44977],{"className":44963},[43],[40,44965,44967],{"className":44966},[47],[49,44968,44969],{"xmlns":51},[53,44970,44971,44975],{},[56,44972,44973],{},[59,44974,5620],{},[82,44976,5953],{"encoding":84},[40,44978,44980],{"className":44979,"ariaHidden":72},[89],[40,44981,44983,44986],{"className":44982},[93],[40,44984],{"className":44985,"style":5963},[97],[40,44987,5620],{"className":44988},[394],[12,44990,5970,44991,6214],{},[40,44992,44994,45039],{"className":44993},[43],[40,44995,44997],{"className":44996},[47],[49,44998,44999],{"xmlns":51},[53,45000,45001,45037],{},[56,45002,45003,45009,45011,45013,45019,45021,45027,45029,45035],{},[5144,45004,45005,45007],{"accent":72},[154,45006,5148],{"mathvariant":935},[59,45008,5151],{},[59,45010,162],{},[59,45012,167],{"stretchy":61},[151,45014,45015,45017],{},[154,45016,5148],{},[154,45018,159],{},[59,45020,73],{"separator":72},[151,45022,45023,45025],{},[154,45024,5148],{},[154,45026,218],{},[59,45028,73],{"separator":72},[151,45030,45031,45033],{},[154,45032,5148],{},[154,45034,273],{},[59,45036,207],{"stretchy":61},[82,45038,6019],{"encoding":84},[40,45040,45042,45088],{"className":45041,"ariaHidden":72},[89],[40,45043,45045,45048,45079,45082,45085],{"className":45044},[93],[40,45046],{"className":45047,"style":5164},[97],[40,45049,45051],{"className":45050},[102,5168],[40,45052,45054],{"className":45053},[344],[40,45055,45057],{"className":45056},[349],[40,45058,45060,45068],{"className":45059,"style":5164},[353],[40,45061,45062,45065],{"style":2233},[40,45063],{"className":45064,"style":444},[361],[40,45066,5148],{"className":45067},[102,966],[40,45069,45070,45073],{"style":5188},[40,45071],{"className":45072,"style":444},[361],[40,45074,45076],{"className":45075,"style":5196},[5195],[40,45077,5151],{"className":45078},[102],[40,45080],{"className":45081,"style":390},[121],[40,45083,162],{"className":45084},[394],[40,45086],{"className":45087,"style":390},[121],[40,45089,45091,45094,45097,45137,45140,45143,45183,45186,45189,45229],{"className":45090},[93],[40,45092],{"className":45093,"style":1365},[97],[40,45095,167],{"className":45096},[106],[40,45098,45100,45103],{"className":45099},[102],[40,45101,5148],{"className":45102},[102,335],[40,45104,45106],{"className":45105},[340],[40,45107,45109,45129],{"className":45108},[344,345],[40,45110,45112,45126],{"className":45111},[349],[40,45113,45115],{"className":45114,"style":354},[353],[40,45116,45117,45120],{"style":6098},[40,45118],{"className":45119,"style":362},[361],[40,45121,45123],{"className":45122},[366,367,368,369],[40,45124,159],{"className":45125},[102,335,369],[40,45127,377],{"className":45128},[376],[40,45130,45132],{"className":45131},[349],[40,45133,45135],{"className":45134,"style":384},[353],[40,45136],{},[40,45138,73],{"className":45139},[117],[40,45141],{"className":45142,"style":122},[121],[40,45144,45146,45149],{"className":45145},[102],[40,45147,5148],{"className":45148},[102,335],[40,45150,45152],{"className":45151},[340],[40,45153,45155,45175],{"className":45154},[344,345],[40,45156,45158,45172],{"className":45157},[349],[40,45159,45161],{"className":45160,"style":354},[353],[40,45162,45163,45166],{"style":6098},[40,45164],{"className":45165,"style":362},[361],[40,45167,45169],{"className":45168},[366,367,368,369],[40,45170,218],{"className":45171,"style":336},[102,335,369],[40,45173,377],{"className":45174},[376],[40,45176,45178],{"className":45177},[349],[40,45179,45181],{"className":45180,"style":585},[353],[40,45182],{},[40,45184,73],{"className":45185},[117],[40,45187],{"className":45188,"style":122},[121],[40,45190,45192,45195],{"className":45191},[102],[40,45193,5148],{"className":45194},[102,335],[40,45196,45198],{"className":45197},[340],[40,45199,45201,45221],{"className":45200},[344,345],[40,45202,45204,45218],{"className":45203},[349],[40,45205,45207],{"className":45206,"style":354},[353],[40,45208,45209,45212],{"style":6098},[40,45210],{"className":45211,"style":362},[361],[40,45213,45215],{"className":45214},[366,367,368,369],[40,45216,273],{"className":45217,"style":764},[102,335,369],[40,45219,377],{"className":45220},[376],[40,45222,45224],{"className":45223},[349],[40,45225,45227],{"className":45226,"style":384},[353],[40,45228],{},[40,45230,207],{"className":45231},[113],[40,45233,45235],{"className":45234},[136],[40,45236,45238,45361],{"className":45237},[43],[40,45239,45241],{"className":45240},[47],[49,45242,45243],{"xmlns":51,"display":145},[53,45244,45245,45359],{},[56,45246,45247,45249,45251,45257,45259,45261,45357],{},[154,45248,6232],{},[59,45250,167],{"stretchy":61},[5144,45252,45253,45255],{"accent":72},[154,45254,5148],{"mathvariant":935},[59,45256,5151],{},[59,45258,207],{"stretchy":61},[59,45260,162],{},[56,45262,45263,45265,45355],{},[59,45264,167],{"fence":72},[169,45266,45267],{"rowspacing":171,"columnalign":6251,"columnspacing":173},[175,45268,45269,45275,45289,45309,45315,45339,45349],{},[178,45270,45271],{},[181,45272,45273],{"scriptlevel":66,"displaystyle":61},[64,45274,66],{},[178,45276,45277],{},[181,45278,45279],{"scriptlevel":66,"displaystyle":61},[56,45280,45281,45283],{},[59,45282,243],{},[151,45284,45285,45287],{},[154,45286,5148],{},[154,45288,273],{},[178,45290,45291],{},[181,45292,45293],{"scriptlevel":66,"displaystyle":61},[56,45294,45295,45301,45303],{},[151,45296,45297,45299],{},[154,45298,5148],{},[154,45300,218],{},[1102,45302,1104],{},[151,45304,45305,45307],{},[154,45306,5148],{},[154,45308,273],{},[178,45310,45311],{},[181,45312,45313],{"scriptlevel":66,"displaystyle":61},[64,45314,66],{},[178,45316,45317],{},[181,45318,45319],{"scriptlevel":66,"displaystyle":61},[56,45320,45321,45323,45329,45331,45333],{},[59,45322,243],{},[151,45324,45325,45327],{},[154,45326,5148],{},[154,45328,159],{},[1102,45330,1104],{},[59,45332,243],{},[151,45334,45335,45337],{},[154,45336,5148],{},[154,45338,218],{},[178,45340,45341],{},[181,45342,45343],{"scriptlevel":66,"displaystyle":61},[151,45344,45345,45347],{},[154,45346,5148],{},[154,45348,159],{},[178,45350,45351],{},[181,45352,45353],{"scriptlevel":66,"displaystyle":61},[64,45354,66],{},[59,45356,207],{"fence":72},[154,45358,1123],{"mathvariant":1080},[82,45360,6346],{"encoding":84},[40,45362,45364,45419],{"className":45363,"ariaHidden":72},[89],[40,45365,45367,45370,45373,45376,45407,45410,45413,45416],{"className":45366},[93],[40,45368],{"className":45369,"style":98},[97],[40,45371,6232],{"className":45372,"style":4668},[102,335],[40,45374,167],{"className":45375},[106],[40,45377,45379],{"className":45378},[102,5168],[40,45380,45382],{"className":45381},[344],[40,45383,45385],{"className":45384},[349],[40,45386,45388,45396],{"className":45387,"style":5164},[353],[40,45389,45390,45393],{"style":2233},[40,45391],{"className":45392,"style":444},[361],[40,45394,5148],{"className":45395},[102,966],[40,45397,45398,45401],{"style":5188},[40,45399],{"className":45400,"style":444},[361],[40,45402,45404],{"className":45403,"style":5196},[5195],[40,45405,5151],{"className":45406},[102],[40,45408,207],{"className":45409},[113],[40,45411],{"className":45412,"style":390},[121],[40,45414,162],{"className":45415},[394],[40,45417],{"className":45418,"style":390},[121],[40,45420,45422,45425,45969,45972],{"className":45421},[93],[40,45423],{"className":45424,"style":1155},[97],[40,45426,45428,45434,45963],{"className":45427},[408],[40,45429,45431],{"className":45430,"style":413},[106,412],[40,45432,167],{"className":45433},[417,1238],[40,45435,45437],{"className":45436},[102],[40,45438,45440,45474,45477,45480,45554,45557,45560,45674,45677,45680,45714,45717,45720,45846,45849,45852,45923,45926,45929],{"className":45439},[169],[40,45441,45443],{"className":45442},[427],[40,45444,45446,45466],{"className":45445},[344,345],[40,45447,45449,45463],{"className":45448},[349],[40,45450,45452],{"className":45451,"style":6438},[353],[40,45453,45454,45457],{"style":6441},[40,45455],{"className":45456,"style":444},[361],[40,45458,45460],{"className":45459},[102],[40,45461,66],{"className":45462},[102],[40,45464,377],{"className":45465},[376],[40,45467,45469],{"className":45468},[349],[40,45470,45472],{"className":45471,"style":6460},[353],[40,45473],{},[40,45475],{"className":45476,"style":479},[478],[40,45478],{"className":45479,"style":479},[478],[40,45481,45483],{"className":45482},[427],[40,45484,45486,45546],{"className":45485},[344,345],[40,45487,45489,45543],{"className":45488},[349],[40,45490,45492],{"className":45491,"style":6438},[353],[40,45493,45494,45497],{"style":6441},[40,45495],{"className":45496,"style":444},[361],[40,45498,45500,45503],{"className":45499},[102],[40,45501,243],{"className":45502},[102],[40,45504,45506,45509],{"className":45505},[102],[40,45507,5148],{"className":45508},[102,335],[40,45510,45512],{"className":45511},[340],[40,45513,45515,45535],{"className":45514},[344,345],[40,45516,45518,45532],{"className":45517},[349],[40,45519,45521],{"className":45520,"style":354},[353],[40,45522,45523,45526],{"style":6098},[40,45524],{"className":45525,"style":362},[361],[40,45527,45529],{"className":45528},[366,367,368,369],[40,45530,273],{"className":45531,"style":764},[102,335,369],[40,45533,377],{"className":45534},[376],[40,45536,45538],{"className":45537},[349],[40,45539,45541],{"className":45540,"style":384},[353],[40,45542],{},[40,45544,377],{"className":45545},[376],[40,45547,45549],{"className":45548},[349],[40,45550,45552],{"className":45551,"style":6460},[353],[40,45553],{},[40,45555],{"className":45556,"style":479},[478],[40,45558],{"className":45559,"style":479},[478],[40,45561,45563],{"className":45562},[427],[40,45564,45566,45666],{"className":45565},[344,345],[40,45567,45569,45663],{"className":45568},[349],[40,45570,45572],{"className":45571,"style":6438},[353],[40,45573,45574,45577],{"style":6441},[40,45575],{"className":45576,"style":444},[361],[40,45578,45580,45620,45623],{"className":45579},[102],[40,45581,45583,45586],{"className":45582},[102],[40,45584,5148],{"className":45585},[102,335],[40,45587,45589],{"className":45588},[340],[40,45590,45592,45612],{"className":45591},[344,345],[40,45593,45595,45609],{"className":45594},[349],[40,45596,45598],{"className":45597,"style":354},[353],[40,45599,45600,45603],{"style":6098},[40,45601],{"className":45602,"style":362},[361],[40,45604,45606],{"className":45605},[366,367,368,369],[40,45607,218],{"className":45608,"style":336},[102,335,369],[40,45610,377],{"className":45611},[376],[40,45613,45615],{"className":45614},[349],[40,45616,45618],{"className":45617,"style":585},[353],[40,45619],{},[40,45621,1104],{"className":45622},[121],[40,45624,45626,45629],{"className":45625},[102],[40,45627,5148],{"className":45628},[102,335],[40,45630,45632],{"className":45631},[340],[40,45633,45635,45655],{"className":45634},[344,345],[40,45636,45638,45652],{"className":45637},[349],[40,45639,45641],{"className":45640,"style":354},[353],[40,45642,45643,45646],{"style":6098},[40,45644],{"className":45645,"style":362},[361],[40,45647,45649],{"className":45648},[366,367,368,369],[40,45650,273],{"className":45651,"style":764},[102,335,369],[40,45653,377],{"className":45654},[376],[40,45656,45658],{"className":45657},[349],[40,45659,45661],{"className":45660,"style":384},[353],[40,45662],{},[40,45664,377],{"className":45665},[376],[40,45667,45669],{"className":45668},[349],[40,45670,45672],{"className":45671,"style":6460},[353],[40,45673],{},[40,45675],{"className":45676,"style":479},[478],[40,45678],{"className":45679,"style":479},[478],[40,45681,45683],{"className":45682},[427],[40,45684,45686,45706],{"className":45685},[344,345],[40,45687,45689,45703],{"className":45688},[349],[40,45690,45692],{"className":45691,"style":6438},[353],[40,45693,45694,45697],{"style":6441},[40,45695],{"className":45696,"style":444},[361],[40,45698,45700],{"className":45699},[102],[40,45701,66],{"className":45702},[102],[40,45704,377],{"className":45705},[376],[40,45707,45709],{"className":45708},[349],[40,45710,45712],{"className":45711,"style":6460},[353],[40,45713],{},[40,45715],{"className":45716,"style":479},[478],[40,45718],{"className":45719,"style":479},[478],[40,45721,45723],{"className":45722},[427],[40,45724,45726,45838],{"className":45725},[344,345],[40,45727,45729,45835],{"className":45728},[349],[40,45730,45732],{"className":45731,"style":6438},[353],[40,45733,45734,45737],{"style":6441},[40,45735],{"className":45736,"style":444},[361],[40,45738,45740,45743,45783,45786,45789,45792,45795],{"className":45739},[102],[40,45741,243],{"className":45742},[102],[40,45744,45746,45749],{"className":45745},[102],[40,45747,5148],{"className":45748},[102,335],[40,45750,45752],{"className":45751},[340],[40,45753,45755,45775],{"className":45754},[344,345],[40,45756,45758,45772],{"className":45757},[349],[40,45759,45761],{"className":45760,"style":354},[353],[40,45762,45763,45766],{"style":6098},[40,45764],{"className":45765,"style":362},[361],[40,45767,45769],{"className":45768},[366,367,368,369],[40,45770,159],{"className":45771},[102,335,369],[40,45773,377],{"className":45774},[376],[40,45776,45778],{"className":45777},[349],[40,45779,45781],{"className":45780,"style":384},[353],[40,45782],{},[40,45784,1104],{"className":45785},[121],[40,45787],{"className":45788,"style":1246},[121],[40,45790,243],{"className":45791},[1250],[40,45793],{"className":45794,"style":1246},[121],[40,45796,45798,45801],{"className":45797},[102],[40,45799,5148],{"className":45800},[102,335],[40,45802,45804],{"className":45803},[340],[40,45805,45807,45827],{"className":45806},[344,345],[40,45808,45810,45824],{"className":45809},[349],[40,45811,45813],{"className":45812,"style":354},[353],[40,45814,45815,45818],{"style":6098},[40,45816],{"className":45817,"style":362},[361],[40,45819,45821],{"className":45820},[366,367,368,369],[40,45822,218],{"className":45823,"style":336},[102,335,369],[40,45825,377],{"className":45826},[376],[40,45828,45830],{"className":45829},[349],[40,45831,45833],{"className":45832,"style":585},[353],[40,45834],{},[40,45836,377],{"className":45837},[376],[40,45839,45841],{"className":45840},[349],[40,45842,45844],{"className":45843,"style":6460},[353],[40,45845],{},[40,45847],{"className":45848,"style":479},[478],[40,45850],{"className":45851,"style":479},[478],[40,45853,45855],{"className":45854},[427],[40,45856,45858,45915],{"className":45857},[344,345],[40,45859,45861,45912],{"className":45860},[349],[40,45862,45864],{"className":45863,"style":6438},[353],[40,45865,45866,45869],{"style":6441},[40,45867],{"className":45868,"style":444},[361],[40,45870,45872],{"className":45871},[102],[40,45873,45875,45878],{"className":45874},[102],[40,45876,5148],{"className":45877},[102,335],[40,45879,45881],{"className":45880},[340],[40,45882,45884,45904],{"className":45883},[344,345],[40,45885,45887,45901],{"className":45886},[349],[40,45888,45890],{"className":45889,"style":354},[353],[40,45891,45892,45895],{"style":6098},[40,45893],{"className":45894,"style":362},[361],[40,45896,45898],{"className":45897},[366,367,368,369],[40,45899,159],{"className":45900},[102,335,369],[40,45902,377],{"className":45903},[376],[40,45905,45907],{"className":45906},[349],[40,45908,45910],{"className":45909,"style":384},[353],[40,45911],{},[40,45913,377],{"className":45914},[376],[40,45916,45918],{"className":45917},[349],[40,45919,45921],{"className":45920,"style":6460},[353],[40,45922],{},[40,45924],{"className":45925,"style":479},[478],[40,45927],{"className":45928,"style":479},[478],[40,45930,45932],{"className":45931},[427],[40,45933,45935,45955],{"className":45934},[344,345],[40,45936,45938,45952],{"className":45937},[349],[40,45939,45941],{"className":45940,"style":6438},[353],[40,45942,45943,45946],{"style":6441},[40,45944],{"className":45945,"style":444},[361],[40,45947,45949],{"className":45948},[102],[40,45950,66],{"className":45951},[102],[40,45953,377],{"className":45954},[376],[40,45956,45958],{"className":45957},[349],[40,45959,45961],{"className":45960,"style":6460},[353],[40,45962],{},[40,45964,45966],{"className":45965,"style":413},[113,412],[40,45967,207],{"className":45968},[417,1238],[40,45970],{"className":45971,"style":122},[121],[40,45973,1123],{"className":45974},[102],[12,45976,6966],{},[40,45978,45980],{"className":45979},[136],[40,45981,45983,46100],{"className":45982},[43],[40,45984,45986],{"className":45985},[47],[49,45987,45988],{"xmlns":51,"display":145},[53,45989,45990,46098],{},[169,45991,45992],{"width":2423},[175,45993,45994,45996,46092,46094],{},[178,45995],{"width":2428},[178,45997,45998],{},[56,45999,46000,46002,46004,46006,46008,46014,46016,46018,46020,46022,46024,46026,46032,46034,46036,46038,46040,46042,46044,46046,46048,46050,46056,46058,46068,46070,46072,46074,46076,46078,46080,46082,46088,46090],{},[154,46001,946],{},[59,46003,167],{"stretchy":61},[154,46005,5215],{},[59,46007,73],{"separator":72},[5144,46009,46010,46012],{"accent":72},[154,46011,5148],{"mathvariant":935},[59,46013,5151],{},[59,46015,207],{"stretchy":61},[59,46017,162],{},[154,46019,1820],{},[59,46021,1084],{},[154,46023,5215],{},[59,46025,73],{"separator":72},[151,46027,46028,46030],{},[154,46029,1050],{},[64,46031,949],{},[59,46033,1053],{},[59,46035,167],{"stretchy":61},[64,46037,78],{},[59,46039,243],{},[154,46041,1820],{},[59,46043,1084],{},[154,46045,5215],{},[59,46047,207],{"stretchy":61},[59,46049,73],{"separator":72},[5144,46051,46052,46054],{"accent":72},[154,46053,5148],{"mathvariant":935},[59,46055,5151],{},[59,46057,73],{"separator":72},[941,46059,46060,46066],{},[5144,46061,46062,46064],{"accent":72},[154,46063,5148],{"mathvariant":935},[59,46065,5151],{},[154,46067,7059],{"mathvariant":7058},[59,46069,1053],{},[154,46071,1813],{},[59,46073,1084],{},[154,46075,5215],{},[59,46077,73],{"separator":72},[154,46079,6232],{},[59,46081,167],{"stretchy":61},[5144,46083,46084,46086],{"accent":72},[154,46085,5148],{"mathvariant":935},[59,46087,5151],{},[59,46089,207],{"stretchy":61},[154,46091,1123],{"mathvariant":1080},[178,46093],{"width":2428},[178,46095,46096],{},[1102,46097,7090],{},[82,46099,7093],{"encoding":84},[40,46101,46103,46167,46237,46258,46388,46452],{"className":46102,"ariaHidden":72},[89],[40,46104,46106,46109,46112,46115,46118,46121,46124,46155,46158,46161,46164],{"className":46105},[93],[40,46107],{"className":46108,"style":98},[97],[40,46110,946],{"className":46111,"style":3946},[102,335],[40,46113,167],{"className":46114},[106],[40,46116,5215],{"className":46117,"style":5231},[102,335],[40,46119,73],{"className":46120},[117],[40,46122],{"className":46123,"style":122},[121],[40,46125,46127],{"className":46126},[102,5168],[40,46128,46130],{"className":46129},[344],[40,46131,46133],{"className":46132},[349],[40,46134,46136,46144],{"className":46135,"style":5164},[353],[40,46137,46138,46141],{"style":2233},[40,46139],{"className":46140,"style":444},[361],[40,46142,5148],{"className":46143},[102,966],[40,46145,46146,46149],{"style":5188},[40,46147],{"className":46148,"style":444},[361],[40,46150,46152],{"className":46151,"style":5196},[5195],[40,46153,5151],{"className":46154},[102],[40,46156,207],{"className":46157},[113],[40,46159],{"className":46160,"style":390},[121],[40,46162,162],{"className":46163},[394],[40,46165],{"className":46166,"style":390},[121],[40,46168,46170,46173,46176,46179,46182,46185,46188,46228,46231,46234],{"className":46169},[93],[40,46171],{"className":46172,"style":7167},[97],[40,46174,1820],{"className":46175},[1369],[40,46177],{"className":46178,"style":122},[121],[40,46180,5215],{"className":46181,"style":5231},[102,335],[40,46183,73],{"className":46184},[117],[40,46186],{"className":46187,"style":122},[121],[40,46189,46191,46194],{"className":46190},[102],[40,46192,1050],{"className":46193,"style":1242},[102,335],[40,46195,46197],{"className":46196},[340],[40,46198,46200,46220],{"className":46199},[344,345],[40,46201,46203,46217],{"className":46202},[349],[40,46204,46206],{"className":46205,"style":7201},[353],[40,46207,46208,46211],{"style":7204},[40,46209],{"className":46210,"style":362},[361],[40,46212,46214],{"className":46213},[366,367,368,369],[40,46215,949],{"className":46216},[102,369],[40,46218,377],{"className":46219},[376],[40,46221,46223],{"className":46222},[349],[40,46224,46226],{"className":46225,"style":384},[353],[40,46227],{},[40,46229],{"className":46230,"style":1246},[121],[40,46232,1053],{"className":46233},[1250],[40,46235],{"className":46236,"style":1246},[121],[40,46238,46240,46243,46246,46249,46252,46255],{"className":46239},[93],[40,46241],{"className":46242,"style":98},[97],[40,46244,167],{"className":46245},[106],[40,46247,78],{"className":46248},[102],[40,46250],{"className":46251,"style":1246},[121],[40,46253,243],{"className":46254},[1250],[40,46256],{"className":46257,"style":1246},[121],[40,46259,46261,46264,46267,46270,46273,46276,46279,46282,46313,46316,46319,46379,46382,46385],{"className":46260},[93],[40,46262],{"className":46263,"style":7261},[97],[40,46265,1820],{"className":46266},[1369],[40,46268],{"className":46269,"style":122},[121],[40,46271,5215],{"className":46272,"style":5231},[102,335],[40,46274,207],{"className":46275},[113],[40,46277,73],{"className":46278},[117],[40,46280],{"className":46281,"style":122},[121],[40,46283,46285],{"className":46284},[102,5168],[40,46286,46288],{"className":46287},[344],[40,46289,46291],{"className":46290},[349],[40,46292,46294,46302],{"className":46293,"style":5164},[353],[40,46295,46296,46299],{"style":2233},[40,46297],{"className":46298,"style":444},[361],[40,46300,5148],{"className":46301},[102,966],[40,46303,46304,46307],{"style":5188},[40,46305],{"className":46306,"style":444},[361],[40,46308,46310],{"className":46309,"style":5196},[5195],[40,46311,5151],{"className":46312},[102],[40,46314,73],{"className":46315},[117],[40,46317],{"className":46318,"style":122},[121],[40,46320,46322,46353],{"className":46321},[102],[40,46323,46325],{"className":46324},[102,5168],[40,46326,46328],{"className":46327},[344],[40,46329,46331],{"className":46330},[349],[40,46332,46334,46342],{"className":46333,"style":5164},[353],[40,46335,46336,46339],{"style":2233},[40,46337],{"className":46338,"style":444},[361],[40,46340,5148],{"className":46341},[102,966],[40,46343,46344,46347],{"style":5188},[40,46345],{"className":46346,"style":444},[361],[40,46348,46350],{"className":46349,"style":5196},[5195],[40,46351,5151],{"className":46352},[102],[40,46354,46356],{"className":46355},[340],[40,46357,46359],{"className":46358},[344],[40,46360,46362],{"className":46361},[349],[40,46363,46365],{"className":46364,"style":7363},[353],[40,46366,46367,46370],{"style":3962},[40,46368],{"className":46369,"style":362},[361],[40,46371,46373],{"className":46372},[366,367,368,369],[40,46374,46376],{"className":46375},[102,369],[40,46377,7059],{"className":46378},[102,7378,369],[40,46380],{"className":46381,"style":1246},[121],[40,46383,1053],{"className":46384},[1250],[40,46386],{"className":46387,"style":1246},[121],[40,46389,46391,46394,46397,46400,46403,46406,46409,46412,46415,46446,46449],{"className":46390},[93],[40,46392],{"className":46393,"style":98},[97],[40,46395,1813],{"className":46396},[1369],[40,46398],{"className":46399,"style":122},[121],[40,46401,5215],{"className":46402,"style":5231},[102,335],[40,46404,73],{"className":46405},[117],[40,46407],{"className":46408,"style":122},[121],[40,46410,6232],{"className":46411,"style":4668},[102,335],[40,46413,167],{"className":46414},[106],[40,46416,46418],{"className":46417},[102,5168],[40,46419,46421],{"className":46420},[344],[40,46422,46424],{"className":46423},[349],[40,46425,46427,46435],{"className":46426,"style":5164},[353],[40,46428,46429,46432],{"style":2233},[40,46430],{"className":46431,"style":444},[361],[40,46433,5148],{"className":46434},[102,966],[40,46436,46437,46440],{"style":5188},[40,46438],{"className":46439,"style":444},[361],[40,46441,46443],{"className":46442,"style":5196},[5195],[40,46444,5151],{"className":46445},[102],[40,46447,207],{"className":46448},[113],[40,46450,1123],{"className":46451},[102],[40,46453,46455,46458],{"className":46454},[2925],[40,46456],{"className":46457,"style":7261},[97],[40,46459,46461,46464,46470],{"className":46460},[102,2932],[40,46462,167],{"className":46463},[102],[40,46465,46467],{"className":46466},[102],[40,46468,7470],{"className":46469},[102],[40,46471,207],{"className":46472},[102],[12,46474,7476,46475,7575,46571,7578],{},[40,46476,46478,46503],{"className":46477},[43],[40,46479,46481],{"className":46480},[47],[49,46482,46483],{"xmlns":51},[53,46484,46485,46501],{},[56,46486,46487,46493,46495,46497,46499],{},[941,46488,46489,46491],{},[154,46490,936],{"mathvariant":935},[59,46492,5729],{"mathvariant":1080,"lspace":3697,"rspace":3697},[59,46494,162],{},[154,46496,946],{},[59,46498,73],{"separator":72},[154,46500,936],{"mathvariant":935},[82,46502,7505],{"encoding":84},[40,46504,46506,46553],{"className":46505,"ariaHidden":72},[89],[40,46507,46509,46512,46544,46547,46550],{"className":46508},[93],[40,46510],{"className":46511,"style":7515},[97],[40,46513,46515,46518],{"className":46514},[102],[40,46516,936],{"className":46517},[102,966],[40,46519,46521],{"className":46520},[340],[40,46522,46524],{"className":46523},[344],[40,46525,46527],{"className":46526},[349],[40,46528,46530],{"className":46529,"style":7515},[353],[40,46531,46532,46535],{"style":1004},[40,46533],{"className":46534,"style":362},[361],[40,46536,46538],{"className":46537},[366,367,368,369],[40,46539,46541],{"className":46540},[102,369],[40,46542,5729],{"className":46543},[102,369],[40,46545],{"className":46546,"style":390},[121],[40,46548,162],{"className":46549},[394],[40,46551],{"className":46552,"style":390},[121],[40,46554,46556,46559,46562,46565,46568],{"className":46555},[93],[40,46557],{"className":46558,"style":7167},[97],[40,46560,946],{"className":46561,"style":3946},[102,335],[40,46563,73],{"className":46564},[117],[40,46566],{"className":46567,"style":122},[121],[40,46569,936],{"className":46570},[102,966],[35,46572,3754],{},[40,46574,46576],{"className":46575},[136],[40,46577,46579,46672],{"className":46578},[43],[40,46580,46582],{"className":46581},[47],[49,46583,46584],{"xmlns":51,"display":145},[53,46585,46586,46670],{},[56,46587,46588,46594,46596,46598,46600,46602,46604,46606,46608,46610,46618,46620,46628,46630,46632,46634,46636,46638,46640,46646,46648,46650,46652,46654,46656,46658,46660,46662,46664,46666,46668],{},[151,46589,46590,46592],{},[154,46591,2437],{},[154,46593,1032],{},[59,46595,167],{"stretchy":61},[154,46597,1649],{},[59,46599,73],{"separator":72},[154,46601,1671],{},[59,46603,2051],{"separator":72},[154,46605,5215],{},[59,46607,207],{"stretchy":61},[59,46609,162],{},[181,46611,46612],{"scriptlevel":66,"displaystyle":61},[1038,46613,46614,46616],{},[64,46615,78],{},[64,46617,1044],{},[59,46619,1053],{},[1038,46621,46622,46626],{},[2045,46623,46624],{},[64,46625,949],{},[64,46627,1044],{},[59,46629,2051],{"separator":72},[59,46631,167],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[154,46633,946],{},[59,46635,167],{"stretchy":61},[154,46637,5215],{},[59,46639,73],{"separator":72},[5144,46641,46642,46644],{"accent":72},[154,46643,5148],{"mathvariant":935},[59,46645,5151],{},[59,46647,207],{"stretchy":61},[59,46649,73],{"separator":72},[154,46651,936],{"mathvariant":935},[59,46653,207],{"fence":61,"stretchy":72,"minsize":1047,"maxsize":1047},[59,46655,1058],{},[154,46657,1796],{"mathvariant":935},[59,46659,167],{"stretchy":61},[154,46661,1649],{},[59,46663,73],{"separator":72},[154,46665,1671],{},[59,46667,207],{"stretchy":61},[154,46669,1123],{"mathvariant":1080},[82,46671,7678],{"encoding":84},[40,46673,46675,46757,46840,47037],{"className":46674,"ariaHidden":72},[89],[40,46676,46678,46681,46721,46724,46727,46730,46733,46736,46739,46742,46745,46748,46751,46754],{"className":46677},[93],[40,46679],{"className":46680,"style":1365},[97],[40,46682,46684,46687],{"className":46683},[102],[40,46685,2437],{"className":46686,"style":2541},[102,335],[40,46688,46690],{"className":46689},[340],[40,46691,46693,46713],{"className":46692},[344,345],[40,46694,46696,46710],{"className":46695},[349],[40,46697,46699],{"className":46698,"style":354},[353],[40,46700,46701,46704],{"style":2556},[40,46702],{"className":46703,"style":362},[361],[40,46705,46707],{"className":46706},[366,367,368,369],[40,46708,1032],{"className":46709},[102,335,369],[40,46711,377],{"className":46712},[376],[40,46714,46716],{"className":46715},[349],[40,46717,46719],{"className":46718,"style":585},[353],[40,46720],{},[40,46722,167],{"className":46723},[106],[40,46725,1649],{"className":46726,"style":1312},[102,335],[40,46728,73],{"className":46729},[117],[40,46731],{"className":46732,"style":122},[121],[40,46734,1671],{"className":46735},[102,335],[40,46737,2051],{"className":46738},[117],[40,46740],{"className":46741,"style":122},[121],[40,46743,5215],{"className":46744,"style":5231},[102,335],[40,46746,207],{"className":46747},[113],[40,46749],{"className":46750,"style":390},[121],[40,46752,162],{"className":46753},[394],[40,46755],{"className":46756,"style":390},[121],[40,46758,46760,46763,46831,46834,46837],{"className":46759},[93],[40,46761],{"className":46762,"style":2676},[97],[40,46764,46766,46769,46828],{"className":46765},[102],[40,46767],{"className":46768},[106,1162],[40,46770,46772],{"className":46771},[1038],[40,46773,46775,46820],{"className":46774},[344,345],[40,46776,46778,46817],{"className":46777},[349],[40,46779,46781,46795,46803],{"className":46780,"style":1175},[353],[40,46782,46783,46786],{"style":1178},[40,46784],{"className":46785,"style":444},[361],[40,46787,46789],{"className":46788},[366,367,368,369],[40,46790,46792],{"className":46791},[102,369],[40,46793,1044],{"className":46794},[102,369],[40,46796,46797,46800],{"style":1193},[40,46798],{"className":46799,"style":444},[361],[40,46801],{"className":46802,"style":1201},[1200],[40,46804,46805,46808],{"style":1204},[40,46806],{"className":46807,"style":444},[361],[40,46809,46811],{"className":46810},[366,367,368,369],[40,46812,46814],{"className":46813},[102,369],[40,46815,78],{"className":46816},[102,369],[40,46818,377],{"className":46819},[376],[40,46821,46823],{"className":46822},[349],[40,46824,46826],{"className":46825,"style":1226},[353],[40,46827],{},[40,46829],{"className":46830},[113,1162],[40,46832],{"className":46833,"style":1246},[121],[40,46835,1053],{"className":46836},[1250],[40,46838],{"className":46839,"style":1246},[121],[40,46841,46843,46846,46952,46955,46958,46964,46967,46970,46973,46976,46979,47010,47013,47016,47019,47022,47028,47031,47034],{"className":46842},[93],[40,46844],{"className":46845,"style":2760},[97],[40,46847,46849,46852,46949],{"className":46848},[102],[40,46850],{"className":46851},[106,1162],[40,46853,46855],{"className":46854},[1038],[40,46856,46858,46941],{"className":46857},[344,345],[40,46859,46861,46938],{"className":46860},[349],[40,46862,46864,46875,46883],{"className":46863,"style":2779},[353],[40,46865,46866,46869],{"style":2782},[40,46867],{"className":46868,"style":444},[361],[40,46870,46872],{"className":46871},[102],[40,46873,1044],{"className":46874},[102],[40,46876,46877,46880],{"style":1193},[40,46878],{"className":46879,"style":444},[361],[40,46881],{"className":46882,"style":1201},[1200],[40,46884,46885,46888],{"style":2802},[40,46886],{"className":46887,"style":444},[361],[40,46889,46891],{"className":46890},[102],[40,46892,46894],{"className":46893},[102,2218],[40,46895,46897,46930],{"className":46896},[344,345],[40,46898,46900,46927],{"className":46899},[349],[40,46901,46903,46915],{"className":46902,"style":2821},[353],[40,46904,46906,46909],{"className":46905,"style":2233},[2232],[40,46907],{"className":46908,"style":444},[361],[40,46910,46912],{"className":46911,"style":2240},[102],[40,46913,949],{"className":46914},[102],[40,46916,46917,46920],{"style":2836},[40,46918],{"className":46919,"style":444},[361],[40,46921,46923],{"className":46922,"style":2254},[2253],[2256,46924,46925],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,46926],{"d":2266},[40,46928,377],{"className":46929},[376],[40,46931,46933],{"className":46932},[349],[40,46934,46936],{"className":46935,"style":2856},[353],[40,46937],{},[40,46939,377],{"className":46940},[376],[40,46942,46944],{"className":46943},[349],[40,46945,46947],{"className":46946,"style":2868},[353],[40,46948],{},[40,46950],{"className":46951},[113,1162],[40,46953,2051],{"className":46954},[117],[40,46956],{"className":46957,"style":122},[121],[40,46959,46961],{"className":46960},[102],[40,46962,167],{"className":46963},[417,1238],[40,46965,946],{"className":46966,"style":3946},[102,335],[40,46968,167],{"className":46969},[106],[40,46971,5215],{"className":46972,"style":5231},[102,335],[40,46974,73],{"className":46975},[117],[40,46977],{"className":46978,"style":122},[121],[40,46980,46982],{"className":46981},[102,5168],[40,46983,46985],{"className":46984},[344],[40,46986,46988],{"className":46987},[349],[40,46989,46991,46999],{"className":46990,"style":5164},[353],[40,46992,46993,46996],{"style":2233},[40,46994],{"className":46995,"style":444},[361],[40,46997,5148],{"className":46998},[102,966],[40,47000,47001,47004],{"style":5188},[40,47002],{"className":47003,"style":444},[361],[40,47005,47007],{"className":47006,"style":5196},[5195],[40,47008,5151],{"className":47009},[102],[40,47011,207],{"className":47012},[113],[40,47014,73],{"className":47015},[117],[40,47017],{"className":47018,"style":122},[121],[40,47020,936],{"className":47021},[102,966],[40,47023,47025],{"className":47024},[102],[40,47026,207],{"className":47027},[417,1238],[40,47029],{"className":47030,"style":1246},[121],[40,47032,1058],{"className":47033},[1250],[40,47035],{"className":47036,"style":1246},[121],[40,47038,47040,47043,47046,47049,47052,47055,47058,47061,47064],{"className":47039},[93],[40,47041],{"className":47042,"style":98},[97],[40,47044,1796],{"className":47045},[102,966],[40,47047,167],{"className":47048},[106],[40,47050,1649],{"className":47051,"style":1312},[102,335],[40,47053,73],{"className":47054},[117],[40,47056],{"className":47057,"style":122},[121],[40,47059,1671],{"className":47060},[102,335],[40,47062,207],{"className":47063},[113],[40,47065,1123],{"className":47066},[102],[25,47068],{},[28,47070,8079,47071,8133],{"id":8078},[40,47072,47074,47091],{"className":47073},[43],[40,47075,47077],{"className":47076},[47],[49,47078,47079],{"xmlns":51},[53,47080,47081,47089],{},[56,47082,47083,47085,47087],{},[64,47084,1044],{},[59,47086,8096],{},[64,47088,1044],{},[82,47090,8101],{"encoding":84},[40,47092,47094,47112],{"className":47093,"ariaHidden":72},[89],[40,47095,47097,47100,47103,47106,47109],{"className":47096},[93],[40,47098],{"className":47099,"style":8111},[97],[40,47101,1044],{"className":47102},[102],[40,47104],{"className":47105,"style":1246},[121],[40,47107,8096],{"className":47108},[1250],[40,47110],{"className":47111,"style":1246},[121],[40,47113,47115,47118],{"className":47114},[93],[40,47116],{"className":47117,"style":1624},[97],[40,47119,1044],{"className":47120},[102],[12,47122,8136,47123,8217,47201,8289],{},[40,47124,47126,47153],{"className":47125},[43],[40,47127,47129],{"className":47128},[47],[49,47130,47131],{"xmlns":51},[53,47132,47133,47151],{},[56,47134,47135,47137,47139],{},[154,47136,5215],{},[59,47138,939],{},[56,47140,47141,47143,47145,47147,47149],{},[59,47142,167],{"stretchy":61},[154,47144,8159],{},[59,47146,73],{"separator":72},[154,47148,12],{},[59,47150,207],{"stretchy":61},[82,47152,8168],{"encoding":84},[40,47154,47156,47174],{"className":47155,"ariaHidden":72},[89],[40,47157,47159,47162,47165,47168,47171],{"className":47158},[93],[40,47160],{"className":47161,"style":962},[97],[40,47163,5215],{"className":47164,"style":5231},[102,335],[40,47166],{"className":47167,"style":390},[121],[40,47169,939],{"className":47170},[394],[40,47172],{"className":47173,"style":390},[121],[40,47175,47177,47180],{"className":47176},[93],[40,47178],{"className":47179,"style":98},[97],[40,47181,47183,47186,47189,47192,47195,47198],{"className":47182},[102],[40,47184,167],{"className":47185},[106],[40,47187,8159],{"className":47188,"style":336},[102,335],[40,47190,73],{"className":47191},[117],[40,47193],{"className":47194,"style":122},[121],[40,47196,12],{"className":47197},[102,335],[40,47199,207],{"className":47200},[113],[40,47202,47204,47221],{"className":47203},[43],[40,47205,47207],{"className":47206},[47],[49,47208,47209],{"xmlns":51},[53,47210,47211,47219],{},[56,47212,47213],{},[151,47214,47215,47217],{},[154,47216,8234],{},[154,47218,5215],{},[82,47220,8239],{"encoding":84},[40,47222,47224],{"className":47223,"ariaHidden":72},[89],[40,47225,47227,47230],{"className":47226},[93],[40,47228],{"className":47229,"style":3106},[97],[40,47231,47233,47236],{"className":47232},[102],[40,47234,8234],{"className":47235},[102,335],[40,47237,47239],{"className":47238},[340],[40,47240,47242,47262],{"className":47241},[344,345],[40,47243,47245,47259],{"className":47244},[349],[40,47246,47248],{"className":47247,"style":354},[353],[40,47249,47250,47253],{"style":6098},[40,47251],{"className":47252,"style":362},[361],[40,47254,47256],{"className":47255},[366,367,368,369],[40,47257,5215],{"className":47258,"style":5231},[102,335,369],[40,47260,377],{"className":47261},[376],[40,47263,47265],{"className":47264},[349],[40,47266,47268],{"className":47267,"style":384},[353],[40,47269],{},[40,47271,47273],{"className":47272},[136],[40,47274,47276,47349],{"className":47275},[43],[40,47277,47279],{"className":47278},[47],[49,47280,47281],{"xmlns":51,"display":145},[53,47282,47283,47347],{},[56,47284,47285,47291,47297,47299,47301,47303,47305,47307,47309,47311,47313,47319,47321,47323,47325,47327,47329,47335,47337,47345],{},[8306,47286,47287,47289],{},[59,47288,8310],{},[154,47290,5215],{},[151,47292,47293,47295],{},[154,47294,8234],{},[154,47296,5215],{},[59,47298,162],{},[64,47300,1044],{},[154,47302,1050],{},[59,47304,73],{"separator":72},[121,47306],{"width":173},[154,47308,1081],{"mathvariant":1080},[59,47310,1084],{},[59,47312,167],{"stretchy":61},[151,47314,47315,47317],{},[154,47316,8234],{},[154,47318,5215],{},[59,47320,207],{"stretchy":61},[59,47322,162],{},[64,47324,78],{},[59,47326,73],{"separator":72},[121,47328],{"width":173},[151,47330,47331,47333],{},[154,47332,8234],{},[154,47334,5215],{},[59,47336,162],{},[8360,47338,47339,47341,47343],{},[154,47340,8234],{},[154,47342,5215],{},[59,47344,3698],{"lspace":3697,"rspace":3697},[154,47346,1123],{"mathvariant":1080},[82,47348,8372],{"encoding":84},[40,47350,47352,47457,47539,47606],{"className":47351,"ariaHidden":72},[89],[40,47353,47355,47358,47405,47408,47448,47451,47454],{"className":47354},[93],[40,47356],{"className":47357,"style":8382},[97],[40,47359,47361],{"className":47360},[1369,8386],[40,47362,47364,47397],{"className":47363},[344,345],[40,47365,47367,47394],{"className":47366},[349],[40,47368,47370,47384],{"className":47369,"style":8396},[353],[40,47371,47372,47375],{"style":8399},[40,47373],{"className":47374,"style":8403},[361],[40,47376,47378],{"className":47377},[366,367,368,369],[40,47379,47381],{"className":47380},[102,369],[40,47382,5215],{"className":47383,"style":5231},[102,335,369],[40,47385,47386,47389],{"style":8415},[40,47387],{"className":47388,"style":8403},[361],[40,47390,47391],{},[40,47392,8310],{"className":47393},[1369,8424,8425],[40,47395,377],{"className":47396},[376],[40,47398,47400],{"className":47399},[349],[40,47401,47403],{"className":47402,"style":8435},[353],[40,47404],{},[40,47406],{"className":47407,"style":122},[121],[40,47409,47411,47414],{"className":47410},[102],[40,47412,8234],{"className":47413},[102,335],[40,47415,47417],{"className":47416},[340],[40,47418,47420,47440],{"className":47419},[344,345],[40,47421,47423,47437],{"className":47422},[349],[40,47424,47426],{"className":47425,"style":354},[353],[40,47427,47428,47431],{"style":6098},[40,47429],{"className":47430,"style":362},[361],[40,47432,47434],{"className":47433},[366,367,368,369],[40,47435,5215],{"className":47436,"style":5231},[102,335,369],[40,47438,377],{"className":47439},[376],[40,47441,47443],{"className":47442},[349],[40,47444,47446],{"className":47445,"style":384},[353],[40,47447],{},[40,47449],{"className":47450,"style":390},[121],[40,47452,162],{"className":47453},[394],[40,47455],{"className":47456,"style":390},[121],[40,47458,47460,47463,47466,47469,47472,47475,47478,47484,47487,47527,47530,47533,47536],{"className":47459},[93],[40,47461],{"className":47462,"style":98},[97],[40,47464,1044],{"className":47465},[102],[40,47467,1050],{"className":47468,"style":1242},[102,335],[40,47470,73],{"className":47471},[117],[40,47473],{"className":47474,"style":543},[121],[40,47476],{"className":47477,"style":122},[121],[40,47479,47481],{"className":47480},[1369],[40,47482,1081],{"className":47483},[102,1373],[40,47485,167],{"className":47486},[106],[40,47488,47490,47493],{"className":47489},[102],[40,47491,8234],{"className":47492},[102,335],[40,47494,47496],{"className":47495},[340],[40,47497,47499,47519],{"className":47498},[344,345],[40,47500,47502,47516],{"className":47501},[349],[40,47503,47505],{"className":47504,"style":354},[353],[40,47506,47507,47510],{"style":6098},[40,47508],{"className":47509,"style":362},[361],[40,47511,47513],{"className":47512},[366,367,368,369],[40,47514,5215],{"className":47515,"style":5231},[102,335,369],[40,47517,377],{"className":47518},[376],[40,47520,47522],{"className":47521},[349],[40,47523,47525],{"className":47524,"style":384},[353],[40,47526],{},[40,47528,207],{"className":47529},[113],[40,47531],{"className":47532,"style":390},[121],[40,47534,162],{"className":47535},[394],[40,47537],{"className":47538,"style":390},[121],[40,47540,47542,47545,47548,47551,47554,47557,47597,47600,47603],{"className":47541},[93],[40,47543],{"className":47544,"style":7167},[97],[40,47546,78],{"className":47547},[102],[40,47549,73],{"className":47550},[117],[40,47552],{"className":47553,"style":543},[121],[40,47555],{"className":47556,"style":122},[121],[40,47558,47560,47563],{"className":47559},[102],[40,47561,8234],{"className":47562},[102,335],[40,47564,47566],{"className":47565},[340],[40,47567,47569,47589],{"className":47568},[344,345],[40,47570,47572,47586],{"className":47571},[349],[40,47573,47575],{"className":47574,"style":354},[353],[40,47576,47577,47580],{"style":6098},[40,47578],{"className":47579,"style":362},[361],[40,47581,47583],{"className":47582},[366,367,368,369],[40,47584,5215],{"className":47585,"style":5231},[102,335,369],[40,47587,377],{"className":47588},[376],[40,47590,47592],{"className":47591},[349],[40,47593,47595],{"className":47594,"style":384},[353],[40,47596],{},[40,47598],{"className":47599,"style":390},[121],[40,47601,162],{"className":47602},[394],[40,47604],{"className":47605,"style":390},[121],[40,47607,47609,47612,47666],{"className":47608},[93],[40,47610],{"className":47611,"style":8645},[97],[40,47613,47615,47618],{"className":47614},[102],[40,47616,8234],{"className":47617},[102,335],[40,47619,47621],{"className":47620},[340],[40,47622,47624,47658],{"className":47623},[344,345],[40,47625,47627,47655],{"className":47626},[349],[40,47628,47630,47641],{"className":47629,"style":7363},[353],[40,47631,47632,47635],{"style":8666},[40,47633],{"className":47634,"style":362},[361],[40,47636,47638],{"className":47637},[366,367,368,369],[40,47639,5215],{"className":47640,"style":5231},[102,335,369],[40,47642,47643,47646],{"style":3962},[40,47644],{"className":47645,"style":362},[361],[40,47647,47649],{"className":47648},[366,367,368,369],[40,47650,47652],{"className":47651},[102,369],[40,47653,3698],{"className":47654},[102,369],[40,47656,377],{"className":47657},[376],[40,47659,47661],{"className":47660},[349],[40,47662,47664],{"className":47663,"style":8699},[353],[40,47665],{},[40,47667,1123],{"className":47668},[102],[12,47670,8707,47671,8711],{},[35,47672,8710],{},[40,47674,47676],{"className":47675},[136],[40,47677,47679,47734],{"className":47678},[43],[40,47680,47682],{"className":47681},[47],[49,47683,47684],{"xmlns":51,"display":145},[53,47685,47686,47732],{},[56,47687,47688,47694,47696,47698,47700,47702,47710,47712,47714,47716,47718,47720,47722,47728,47730],{},[151,47689,47690,47692],{},[154,47691,2437],{},[154,47693,1032],{},[59,47695,167],{"stretchy":61},[154,47697,5215],{},[59,47699,207],{"stretchy":61},[59,47701,162],{},[181,47703,47704],{"scriptlevel":66,"displaystyle":61},[1038,47705,47706,47708],{},[64,47707,78],{},[64,47709,1044],{},[59,47711,73],{"separator":72},[154,47713,1081],{"mathvariant":1080},[59,47715,1084],{},[59,47717,167],{"stretchy":61},[154,47719,1032],{},[59,47721,73],{"separator":72},[151,47723,47724,47726],{},[154,47725,8234],{},[154,47727,5215],{},[59,47729,207],{"stretchy":61},[59,47731,73],{"separator":72},[82,47733,8773],{"encoding":84},[40,47735,47737,47801],{"className":47736,"ariaHidden":72},[89],[40,47738,47740,47743,47783,47786,47789,47792,47795,47798],{"className":47739},[93],[40,47741],{"className":47742,"style":1365},[97],[40,47744,47746,47749],{"className":47745},[102],[40,47747,2437],{"className":47748,"style":2541},[102,335],[40,47750,47752],{"className":47751},[340],[40,47753,47755,47775],{"className":47754},[344,345],[40,47756,47758,47772],{"className":47757},[349],[40,47759,47761],{"className":47760,"style":354},[353],[40,47762,47763,47766],{"style":2556},[40,47764],{"className":47765,"style":362},[361],[40,47767,47769],{"className":47768},[366,367,368,369],[40,47770,1032],{"className":47771},[102,335,369],[40,47773,377],{"className":47774},[376],[40,47776,47778],{"className":47777},[349],[40,47779,47781],{"className":47780,"style":585},[353],[40,47782],{},[40,47784,167],{"className":47785},[106],[40,47787,5215],{"className":47788,"style":5231},[102,335],[40,47790,207],{"className":47791},[113],[40,47793],{"className":47794,"style":390},[121],[40,47796,162],{"className":47797},[394],[40,47799],{"className":47800,"style":390},[121],[40,47802,47804,47807,47875,47878,47881,47887,47890,47893,47896,47899,47939,47942],{"className":47803},[93],[40,47805],{"className":47806,"style":2676},[97],[40,47808,47810,47813,47872],{"className":47809},[102],[40,47811],{"className":47812},[106,1162],[40,47814,47816],{"className":47815},[1038],[40,47817,47819,47864],{"className":47818},[344,345],[40,47820,47822,47861],{"className":47821},[349],[40,47823,47825,47839,47847],{"className":47824,"style":1175},[353],[40,47826,47827,47830],{"style":1178},[40,47828],{"className":47829,"style":444},[361],[40,47831,47833],{"className":47832},[366,367,368,369],[40,47834,47836],{"className":47835},[102,369],[40,47837,1044],{"className":47838},[102,369],[40,47840,47841,47844],{"style":1193},[40,47842],{"className":47843,"style":444},[361],[40,47845],{"className":47846,"style":1201},[1200],[40,47848,47849,47852],{"style":1204},[40,47850],{"className":47851,"style":444},[361],[40,47853,47855],{"className":47854},[366,367,368,369],[40,47856,47858],{"className":47857},[102,369],[40,47859,78],{"className":47860},[102,369],[40,47862,377],{"className":47863},[376],[40,47865,47867],{"className":47866},[349],[40,47868,47870],{"className":47869,"style":1226},[353],[40,47871],{},[40,47873],{"className":47874},[113,1162],[40,47876,73],{"className":47877},[117],[40,47879],{"className":47880,"style":122},[121],[40,47882,47884],{"className":47883},[1369],[40,47885,1081],{"className":47886},[102,1373],[40,47888,167],{"className":47889},[106],[40,47891,1032],{"className":47892},[102,335],[40,47894,73],{"className":47895},[117],[40,47897],{"className":47898,"style":122},[121],[40,47900,47902,47905],{"className":47901},[102],[40,47903,8234],{"className":47904},[102,335],[40,47906,47908],{"className":47907},[340],[40,47909,47911,47931],{"className":47910},[344,345],[40,47912,47914,47928],{"className":47913},[349],[40,47915,47917],{"className":47916,"style":354},[353],[40,47918,47919,47922],{"style":6098},[40,47920],{"className":47921,"style":362},[361],[40,47923,47925],{"className":47924},[366,367,368,369],[40,47926,5215],{"className":47927,"style":5231},[102,335,369],[40,47929,377],{"className":47930},[376],[40,47932,47934],{"className":47933},[349],[40,47935,47937],{"className":47936,"style":384},[353],[40,47938],{},[40,47940,207],{"className":47941},[113],[40,47943,73],{"className":47944},[117],[12,47946,8987],{},[25,47948],{},[28,47950,8993],{"id":8992},[12,47952,8996,47953,1014],{},[40,47954,47956,47977],{"className":47955},[43],[40,47957,47959],{"className":47958},[47],[49,47960,47961],{"xmlns":51},[53,47962,47963,47975],{},[56,47964,47965,47967,47969,47971,47973],{},[59,47966,1579],{"stretchy":61},[154,47968,936],{"mathvariant":935},[59,47970,1579],{"stretchy":61},[59,47972,162],{},[64,47974,78],{},[82,47976,2973],{"encoding":84},[40,47978,47980,48004],{"className":47979,"ariaHidden":72},[89],[40,47981,47983,47986,47989,47992,47995,47998,48001],{"className":47982},[93],[40,47984],{"className":47985,"style":98},[97],[40,47987,1579],{"className":47988},[106],[40,47990,936],{"className":47991},[102,966],[40,47993,1579],{"className":47994},[113],[40,47996],{"className":47997,"style":390},[121],[40,47999,162],{"className":48000},[394],[40,48002],{"className":48003,"style":390},[121],[40,48005,48007,48010],{"className":48006},[93],[40,48008],{"className":48009,"style":1624},[97],[40,48011,78],{"className":48012},[102],[40,48014,48016],{"className":48015},[136],[40,48017,48019,48104],{"className":48018},[43],[40,48020,48022],{"className":48021},[47],[49,48023,48024],{"xmlns":51,"display":145},[53,48025,48026,48102],{},[56,48027,48028,48036,48038,48046,48048,48058,48060,48062,48064,48066,48068,48076,48078,48086,48088,48098,48100],{},[151,48029,48030,48032,48034],{},[154,48031,2437],{},[154,48033,3067],{},[59,48035,1084],{},[59,48037,162],{},[181,48039,48040],{"scriptlevel":66,"displaystyle":61},[1038,48041,48042,48044],{},[64,48043,78],{},[64,48045,1044],{},[59,48047,243],{},[181,48049,48050],{"scriptlevel":66,"displaystyle":61},[1038,48051,48052,48056],{},[2045,48053,48054],{},[64,48055,949],{},[64,48057,1044],{},[59,48059,9104],{},[59,48061,243],{},[64,48063,9109],{},[59,48065,73],{"separator":72},[121,48067],{"width":1068},[151,48069,48070,48072,48074],{},[154,48071,2437],{},[154,48073,3032],{},[59,48075,1084],{},[59,48077,162],{},[181,48079,48080],{"scriptlevel":66,"displaystyle":61},[1038,48081,48082,48084],{},[64,48083,78],{},[64,48085,1044],{},[59,48087,1053],{},[181,48089,48090],{"scriptlevel":66,"displaystyle":61},[1038,48091,48092,48096],{},[2045,48093,48094],{},[64,48095,949],{},[64,48097,1044],{},[59,48099,9104],{},[64,48101,9148],{},[82,48103,9151],{"encoding":84},[40,48105,48107,48174,48257,48384,48466,48549,48676],{"className":48106,"ariaHidden":72},[89],[40,48108,48110,48113,48165,48168,48171],{"className":48109},[93],[40,48111],{"className":48112,"style":3106},[97],[40,48114,48116,48119],{"className":48115},[102],[40,48117,2437],{"className":48118,"style":2541},[102,335],[40,48120,48122],{"className":48121},[340],[40,48123,48125,48157],{"className":48124},[344,345],[40,48126,48128,48154],{"className":48127},[349],[40,48129,48131],{"className":48130,"style":3403},[353],[40,48132,48133,48136],{"style":2556},[40,48134],{"className":48135,"style":362},[361],[40,48137,48139],{"className":48138},[366,367,368,369],[40,48140,48142],{"className":48141},[102,369],[40,48143,48145,48148,48151],{"className":48144},[1369,369],[40,48146,3142],{"className":48147},[369],[40,48149,246],{"className":48150},[369],[40,48152,1796],{"className":48153},[369],[40,48155,377],{"className":48156},[376],[40,48158,48160],{"className":48159},[349],[40,48161,48163],{"className":48162,"style":384},[353],[40,48164],{},[40,48166],{"className":48167,"style":390},[121],[40,48169,162],{"className":48170},[394],[40,48172],{"className":48173,"style":390},[121],[40,48175,48177,48180,48248,48251,48254],{"className":48176},[93],[40,48178],{"className":48179,"style":2676},[97],[40,48181,48183,48186,48245],{"className":48182},[102],[40,48184],{"className":48185},[106,1162],[40,48187,48189],{"className":48188},[1038],[40,48190,48192,48237],{"className":48191},[344,345],[40,48193,48195,48234],{"className":48194},[349],[40,48196,48198,48212,48220],{"className":48197,"style":1175},[353],[40,48199,48200,48203],{"style":1178},[40,48201],{"className":48202,"style":444},[361],[40,48204,48206],{"className":48205},[366,367,368,369],[40,48207,48209],{"className":48208},[102,369],[40,48210,1044],{"className":48211},[102,369],[40,48213,48214,48217],{"style":1193},[40,48215],{"className":48216,"style":444},[361],[40,48218],{"className":48219,"style":1201},[1200],[40,48221,48222,48225],{"style":1204},[40,48223],{"className":48224,"style":444},[361],[40,48226,48228],{"className":48227},[366,367,368,369],[40,48229,48231],{"className":48230},[102,369],[40,48232,78],{"className":48233},[102,369],[40,48235,377],{"className":48236},[376],[40,48238,48240],{"className":48239},[349],[40,48241,48243],{"className":48242,"style":1226},[353],[40,48244],{},[40,48246],{"className":48247},[113,1162],[40,48249],{"className":48250,"style":1246},[121],[40,48252,243],{"className":48253},[1250],[40,48255],{"className":48256,"style":1246},[121],[40,48258,48260,48263,48375,48378,48381],{"className":48259},[93],[40,48261],{"className":48262,"style":3258},[97],[40,48264,48266,48269,48372],{"className":48265},[102],[40,48267],{"className":48268},[106,1162],[40,48270,48272],{"className":48271},[1038],[40,48273,48275,48364],{"className":48274},[344,345],[40,48276,48278,48361],{"className":48277},[349],[40,48279,48281,48295,48303],{"className":48280,"style":3277},[353],[40,48282,48283,48286],{"style":1178},[40,48284],{"className":48285,"style":444},[361],[40,48287,48289],{"className":48288},[366,367,368,369],[40,48290,48292],{"className":48291},[102,369],[40,48293,1044],{"className":48294},[102,369],[40,48296,48297,48300],{"style":1193},[40,48298],{"className":48299,"style":444},[361],[40,48301],{"className":48302,"style":1201},[1200],[40,48304,48305,48308],{"style":3302},[40,48306],{"className":48307,"style":444},[361],[40,48309,48311],{"className":48310},[366,367,368,369],[40,48312,48314],{"className":48313},[102,369],[40,48315,48317],{"className":48316},[102,2218,369],[40,48318,48320,48353],{"className":48319},[344,345],[40,48321,48323,48350],{"className":48322},[349],[40,48324,48326,48338],{"className":48325,"style":3324},[353],[40,48327,48329,48332],{"className":48328,"style":2233},[2232],[40,48330],{"className":48331,"style":444},[361],[40,48333,48335],{"className":48334,"style":2240},[102,369],[40,48336,949],{"className":48337},[102,369],[40,48339,48340,48343],{"style":3339},[40,48341],{"className":48342,"style":444},[361],[40,48344,48346],{"className":48345,"style":2254},[2253,369],[2256,48347,48348],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,48349],{"d":2266},[40,48351,377],{"className":48352},[376],[40,48354,48356],{"className":48355},[349],[40,48357,48359],{"className":48358,"style":3359},[353],[40,48360],{},[40,48362,377],{"className":48363},[376],[40,48365,48367],{"className":48366},[349],[40,48368,48370],{"className":48369,"style":1226},[353],[40,48371],{},[40,48373],{"className":48374},[113,1162],[40,48376],{"className":48377,"style":390},[121],[40,48379,9104],{"className":48380},[394],[40,48382],{"className":48383,"style":390},[121],[40,48385,48387,48390,48393,48396,48399,48402,48405,48457,48460,48463],{"className":48386},[93],[40,48388],{"className":48389,"style":7167},[97],[40,48391,243],{"className":48392},[102],[40,48394,9109],{"className":48395},[102],[40,48397,73],{"className":48398},[117],[40,48400],{"className":48401,"style":1302},[121],[40,48403],{"className":48404,"style":122},[121],[40,48406,48408,48411],{"className":48407},[102],[40,48409,2437],{"className":48410,"style":2541},[102,335],[40,48412,48414],{"className":48413},[340],[40,48415,48417,48449],{"className":48416},[344,345],[40,48418,48420,48446],{"className":48419},[349],[40,48421,48423],{"className":48422,"style":354},[353],[40,48424,48425,48428],{"style":2556},[40,48426],{"className":48427,"style":362},[361],[40,48429,48431],{"className":48430},[366,367,368,369],[40,48432,48434],{"className":48433},[102,369],[40,48435,48437,48440,48443],{"className":48436},[1369,369],[40,48438,3142],{"className":48439},[369],[40,48441,19],{"className":48442},[369],[40,48444,159],{"className":48445},[369],[40,48447,377],{"className":48448},[376],[40,48450,48452],{"className":48451},[349],[40,48453,48455],{"className":48454,"style":384},[353],[40,48456],{},[40,48458],{"className":48459,"style":390},[121],[40,48461,162],{"className":48462},[394],[40,48464],{"className":48465,"style":390},[121],[40,48467,48469,48472,48540,48543,48546],{"className":48468},[93],[40,48470],{"className":48471,"style":2676},[97],[40,48473,48475,48478,48537],{"className":48474},[102],[40,48476],{"className":48477},[106,1162],[40,48479,48481],{"className":48480},[1038],[40,48482,48484,48529],{"className":48483},[344,345],[40,48485,48487,48526],{"className":48486},[349],[40,48488,48490,48504,48512],{"className":48489,"style":1175},[353],[40,48491,48492,48495],{"style":1178},[40,48493],{"className":48494,"style":444},[361],[40,48496,48498],{"className":48497},[366,367,368,369],[40,48499,48501],{"className":48500},[102,369],[40,48502,1044],{"className":48503},[102,369],[40,48505,48506,48509],{"style":1193},[40,48507],{"className":48508,"style":444},[361],[40,48510],{"className":48511,"style":1201},[1200],[40,48513,48514,48517],{"style":1204},[40,48515],{"className":48516,"style":444},[361],[40,48518,48520],{"className":48519},[366,367,368,369],[40,48521,48523],{"className":48522},[102,369],[40,48524,78],{"className":48525},[102,369],[40,48527,377],{"className":48528},[376],[40,48530,48532],{"className":48531},[349],[40,48533,48535],{"className":48534,"style":1226},[353],[40,48536],{},[40,48538],{"className":48539},[113,1162],[40,48541],{"className":48542,"style":1246},[121],[40,48544,1053],{"className":48545},[1250],[40,48547],{"className":48548,"style":1246},[121],[40,48550,48552,48555,48667,48670,48673],{"className":48551},[93],[40,48553],{"className":48554,"style":3258},[97],[40,48556,48558,48561,48664],{"className":48557},[102],[40,48559],{"className":48560},[106,1162],[40,48562,48564],{"className":48563},[1038],[40,48565,48567,48656],{"className":48566},[344,345],[40,48568,48570,48653],{"className":48569},[349],[40,48571,48573,48587,48595],{"className":48572,"style":3277},[353],[40,48574,48575,48578],{"style":1178},[40,48576],{"className":48577,"style":444},[361],[40,48579,48581],{"className":48580},[366,367,368,369],[40,48582,48584],{"className":48583},[102,369],[40,48585,1044],{"className":48586},[102,369],[40,48588,48589,48592],{"style":1193},[40,48590],{"className":48591,"style":444},[361],[40,48593],{"className":48594,"style":1201},[1200],[40,48596,48597,48600],{"style":3302},[40,48598],{"className":48599,"style":444},[361],[40,48601,48603],{"className":48602},[366,367,368,369],[40,48604,48606],{"className":48605},[102,369],[40,48607,48609],{"className":48608},[102,2218,369],[40,48610,48612,48645],{"className":48611},[344,345],[40,48613,48615,48642],{"className":48614},[349],[40,48616,48618,48630],{"className":48617,"style":3324},[353],[40,48619,48621,48624],{"className":48620,"style":2233},[2232],[40,48622],{"className":48623,"style":444},[361],[40,48625,48627],{"className":48626,"style":2240},[102,369],[40,48628,949],{"className":48629},[102,369],[40,48631,48632,48635],{"style":3339},[40,48633],{"className":48634,"style":444},[361],[40,48636,48638],{"className":48637,"style":2254},[2253,369],[2256,48639,48640],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,48641],{"d":2266},[40,48643,377],{"className":48644},[376],[40,48646,48648],{"className":48647},[349],[40,48649,48651],{"className":48650,"style":3359},[353],[40,48652],{},[40,48654,377],{"className":48655},[376],[40,48657,48659],{"className":48658},[349],[40,48660,48662],{"className":48661,"style":1226},[353],[40,48663],{},[40,48665],{"className":48666},[113,1162],[40,48668],{"className":48669,"style":390},[121],[40,48671,9104],{"className":48672},[394],[40,48674],{"className":48675,"style":390},[121],[40,48677,48679,48682],{"className":48678},[93],[40,48680],{"className":48681,"style":1624},[97],[40,48683,9148],{"className":48684},[102],[12,48686,9735],{},[25,48688],{},[28,48690,9741],{"id":9740},[9743,48692,48693,48697,48701,48705,48709,48821],{},[9746,48694,48695,9751],{},[35,48696,9750],{},[9746,48698,48699,9757],{},[35,48700,9756],{},[9746,48702,48703,9763],{},[35,48704,9762],{},[9746,48706,48707,9769],{},[35,48708,9768],{},[9746,48710,48711,9813,48751,9855,48791,9887],{},[35,48712,167,48713,207],{},[40,48714,48716,48733],{"className":48715},[43],[40,48717,48719],{"className":48718},[47],[49,48720,48721],{"xmlns":51},[53,48722,48723,48731],{},[56,48724,48725,48727,48729],{},[59,48726,62],{"stretchy":61},[154,48728,1506],{},[59,48730,69],{"stretchy":61},[82,48732,9794],{"encoding":84},[40,48734,48736],{"className":48735,"ariaHidden":72},[89],[40,48737,48739,48742,48745,48748],{"className":48738},[93],[40,48740],{"className":48741,"style":98},[97],[40,48743,62],{"className":48744},[106],[40,48746,1506],{"className":48747,"style":336},[102,335],[40,48749,69],{"className":48750},[113],[35,48752,167,48753,207],{},[40,48754,48756,48773],{"className":48755},[43],[40,48757,48759],{"className":48758},[47],[49,48760,48761],{"xmlns":51},[53,48762,48763,48771],{},[56,48764,48765,48767,48769],{},[59,48766,1511],{"stretchy":61},[154,48768,1506],{},[59,48770,62],{"stretchy":61},[82,48772,9836],{"encoding":84},[40,48774,48776],{"className":48775,"ariaHidden":72},[89],[40,48777,48779,48782,48785,48788],{"className":48778},[93],[40,48780],{"className":48781,"style":98},[97],[40,48783,1511],{"className":48784},[106],[40,48786,1506],{"className":48787,"style":336},[102,335],[40,48789,62],{"className":48790},[113],[35,48792,167,48793,207],{},[40,48794,48796,48809],{"className":48795},[43],[40,48797,48799],{"className":48798},[47],[49,48800,48801],{"xmlns":51},[53,48802,48803,48807],{},[56,48804,48805],{},[154,48806,1032],{},[82,48808,9874],{"encoding":84},[40,48810,48812],{"className":48811,"ariaHidden":72},[89],[40,48813,48815,48818],{"className":48814},[93],[40,48816],{"className":48817,"style":1136},[97],[40,48819,1032],{"className":48820},[102,335],[9746,48822,48823,9893,48825,10014,48940,1628],{},[35,48824,9892],{},[40,48826,48828,48857],{"className":48827},[43],[40,48829,48831],{"className":48830},[47],[49,48832,48833],{"xmlns":51},[53,48834,48835,48855],{},[56,48836,48837,48839,48841,48843,48845,48847,48849],{},[154,48838,3688],{},[59,48840,9910],{},[154,48842,1032],{},[59,48844,9915],{},[154,48846,3688],{},[154,48848,1032],{},[941,48850,48851,48853],{},[154,48852,3688],{},[59,48854,3698],{"lspace":3697,"rspace":3697},[82,48856,9928],{"encoding":84},[40,48858,48860,48878,48896],{"className":48859,"ariaHidden":72},[89],[40,48861,48863,48866,48869,48872,48875],{"className":48862},[93],[40,48864],{"className":48865,"style":4304},[97],[40,48867,3688],{"className":48868,"style":3802},[102,335],[40,48870],{"className":48871,"style":390},[121],[40,48873,9910],{"className":48874},[394],[40,48876],{"className":48877,"style":390},[121],[40,48879,48881,48884,48887,48890,48893],{"className":48880},[93],[40,48882],{"className":48883,"style":5773},[97],[40,48885,1032],{"className":48886},[102,335],[40,48888],{"className":48889,"style":390},[121],[40,48891,9915],{"className":48892},[394],[40,48894],{"className":48895,"style":390},[121],[40,48897,48899,48902,48905,48908],{"className":48898},[93],[40,48900],{"className":48901,"style":9974},[97],[40,48903,3688],{"className":48904,"style":3802},[102,335],[40,48906,1032],{"className":48907},[102,335],[40,48909,48911,48914],{"className":48910},[102],[40,48912,3688],{"className":48913,"style":3802},[102,335],[40,48915,48917],{"className":48916},[340],[40,48918,48920],{"className":48919},[344],[40,48921,48923],{"className":48922},[349],[40,48924,48926],{"className":48925,"style":9999},[353],[40,48927,48928,48931],{"style":1004},[40,48929],{"className":48930,"style":362},[361],[40,48932,48934],{"className":48933},[366,367,368,369],[40,48935,48937],{"className":48936},[102,369],[40,48938,3698],{"className":48939},[102,369],[40,48941,48943,48962],{"className":48942},[43],[40,48944,48946],{"className":48945},[47],[49,48947,48948],{"xmlns":51},[53,48949,48950,48960],{},[56,48951,48952,48954,48956,48958],{},[154,48953,936],{"mathvariant":935},[59,48955,9915],{},[154,48957,946],{},[154,48959,936],{"mathvariant":935},[82,48961,10037],{"encoding":84},[40,48963,48965,48983],{"className":48964,"ariaHidden":72},[89],[40,48966,48968,48971,48974,48977,48980],{"className":48967},[93],[40,48969],{"className":48970,"style":10047},[97],[40,48972,936],{"className":48973},[102,966],[40,48975],{"className":48976,"style":390},[121],[40,48978,9915],{"className":48979},[394],[40,48981],{"className":48982,"style":390},[121],[40,48984,48986,48989,48992],{"className":48985},[93],[40,48987],{"className":48988,"style":4304},[97],[40,48990,946],{"className":48991,"style":3946},[102,335],[40,48993,936],{"className":48994},[102,966],[25,48996],{},[28,48998,10077],{"id":10076},[12,49000,10080],{},[10082,49002,49004],{"className":49003,"code":10086,"language":2932},[10085],[10088,49005,10086],{"__ignoreMap":10090},[25,49007],{},[28,49009,10096],{"id":10095},[9743,49011,49012,49070],{},[9746,49013,10101,49014,10131,49042,10160],{},[40,49015,49017,49030],{"className":49016},[43],[40,49018,49020],{"className":49019},[47],[49,49021,49022],{"xmlns":51},[53,49023,49024,49028],{},[56,49025,49026],{},[154,49027,2016],{"mathvariant":1080},[82,49029,10118],{"encoding":84},[40,49031,49033],{"className":49032,"ariaHidden":72},[89],[40,49034,49036,49039],{"className":49035},[93],[40,49037],{"className":49038,"style":4304},[97],[40,49040,2016],{"className":49041},[102],[40,49043,49045,49058],{"className":49044},[43],[40,49046,49048],{"className":49047},[47],[49,49049,49050],{"xmlns":51},[53,49051,49052,49056],{},[56,49053,49054],{},[154,49055,2437],{},[82,49057,2437],{"encoding":84},[40,49059,49061],{"className":49060,"ariaHidden":72},[89],[40,49062,49064,49067],{"className":49063},[93],[40,49065],{"className":49066,"style":4304},[97],[40,49068,2437],{"className":49069,"style":2541},[102,335],[9746,49071,10163,49072,10226,49132,10266],{},[40,49073,49075,49096],{"className":49074},[43],[40,49076,49078],{"className":49077},[47],[49,49079,49080],{"xmlns":51},[53,49081,49082,49094],{},[56,49083,49084,49086,49088,49090,49092],{},[59,49085,1579],{"stretchy":61},[154,49087,936],{"mathvariant":935},[59,49089,1579],{"stretchy":61},[59,49091,10184],{},[64,49093,78],{},[82,49095,10189],{"encoding":84},[40,49097,49099,49123],{"className":49098,"ariaHidden":72},[89],[40,49100,49102,49105,49108,49111,49114,49117,49120],{"className":49101},[93],[40,49103],{"className":49104,"style":98},[97],[40,49106,1579],{"className":49107},[106],[40,49109,936],{"className":49110},[102,966],[40,49112,1579],{"className":49113},[113],[40,49115],{"className":49116,"style":390},[121],[40,49118,10184],{"className":49119},[394],[40,49121],{"className":49122,"style":390},[121],[40,49124,49126,49129],{"className":49125},[93],[40,49127],{"className":49128,"style":1624},[97],[40,49130,78],{"className":49131},[102],[40,49133,49135,49152],{"className":49134},[43],[40,49136,49138],{"className":49137},[47],[49,49139,49140],{"xmlns":51},[53,49141,49142,49150],{},[56,49143,49144,49146,49148],{},[59,49145,1579],{"stretchy":61},[154,49147,936],{"mathvariant":935},[59,49149,1579],{"stretchy":61},[82,49151,10247],{"encoding":84},[40,49153,49155],{"className":49154,"ariaHidden":72},[89],[40,49156,49158,49161,49164,49167],{"className":49157},[93],[40,49159],{"className":49160,"style":98},[97],[40,49162,1579],{"className":49163},[106],[40,49165,936],{"className":49166},[102,966],[40,49168,1579],{"className":49169},[113],[10082,49171,49172],{"className":10269,"code":10270,"language":10271,"meta":10090,"style":10090},[10088,49173,49174,49178,49182,49186,49190,49194,49198,49202,49206,49210,49214,49218,49222,49232,49236,49246,49256,49266,49270,49274,49282,49294,49306,49322,49332],{"__ignoreMap":10090},[40,49175,49176],{"class":10276,"line":10277},[40,49177,10281],{"class":10280},[40,49179,49180],{"class":10276,"line":10284},[40,49181,10288],{"class":10287},[40,49183,49184],{"class":10276,"line":10291},[40,49185,10294],{"class":10287},[40,49187,49188],{"class":10276,"line":10297},[40,49189,10301],{"emptyLinePlaceholder":10300},[40,49191,49192],{"class":10276,"line":10304},[40,49193,10307],{"class":10287},[40,49195,49196],{"class":10276,"line":10310},[40,49197,10313],{"class":10287},[40,49199,49200],{"class":10276,"line":10316},[40,49201,10319],{"class":10287},[40,49203,49204],{"class":10276,"line":10322},[40,49205,10301],{"emptyLinePlaceholder":10300},[40,49207,49208],{"class":10276,"line":10327},[40,49209,10330],{"class":10287},[40,49211,49212],{"class":10276,"line":10333},[40,49213,10336],{"class":10287},[40,49215,49216],{"class":10276,"line":10339},[40,49217,10288],{"class":10287},[40,49219,49220],{"class":10276,"line":10344},[40,49221,10301],{"emptyLinePlaceholder":10300},[40,49223,49224,49226,49228,49230],{"class":10276,"line":10349},[40,49225,10353],{"class":10352},[40,49227,10357],{"class":10356},[40,49229,10360],{"class":10352},[40,49231,10364],{"class":10363},[40,49233,49234],{"class":10276,"line":10367},[40,49235,10301],{"emptyLinePlaceholder":10300},[40,49237,49238,49240,49242,49244],{"class":10276,"line":10372},[40,49239,10375],{"class":10352},[40,49241,10378],{"class":10363},[40,49243,10381],{"class":10352},[40,49245,10384],{"class":10363},[40,49247,49248,49250,49252,49254],{"class":10276,"line":10387},[40,49249,10375],{"class":10352},[40,49251,10392],{"class":10363},[40,49253,10381],{"class":10352},[40,49255,10397],{"class":10363},[40,49257,49258,49260,49262,49264],{"class":10276,"line":10400},[40,49259,10375],{"class":10352},[40,49261,10405],{"class":10363},[40,49263,10381],{"class":10352},[40,49265,10410],{"class":10363},[40,49267,49268],{"class":10276,"line":10413},[40,49269,10301],{"emptyLinePlaceholder":10300},[40,49271,49272],{"class":10276,"line":10418},[40,49273,10421],{"class":10280},[40,49275,49276,49278,49280],{"class":10276,"line":10424},[40,49277,10427],{"class":10363},[40,49279,10431],{"class":10430},[40,49281,10434],{"class":10363},[40,49283,49284,49286,49288,49290,49292],{"class":10276,"line":10437},[40,49285,10440],{"class":10287},[40,49287,10443],{"class":10363},[40,49289,10447],{"class":10446},[40,49291,10450],{"class":10363},[40,49293,10453],{"class":10280},[40,49295,49296,49298,49300,49302,49304],{"class":10276,"line":10456},[40,49297,10459],{"class":10287},[40,49299,10443],{"class":10363},[40,49301,10447],{"class":10446},[40,49303,10466],{"class":10363},[40,49305,10469],{"class":10280},[40,49307,49308,49310,49312,49314,49316,49318,49320],{"class":10276,"line":10472},[40,49309,10475],{"class":10287},[40,49311,10478],{"class":10363},[40,49313,10481],{"class":10446},[40,49315,10484],{"class":10363},[40,49317,10487],{"class":10446},[40,49319,10490],{"class":10363},[40,49321,10493],{"class":10280},[40,49323,49324,49326,49328,49330],{"class":10276,"line":10496},[40,49325,10499],{"class":10287},[40,49327,10443],{"class":10363},[40,49329,10504],{"class":10446},[40,49331,10507],{"class":10363},[40,49333,49334],{"class":10276,"line":10510},[40,49335,10513],{"class":10363},[10082,49337,49338],{"className":10269,"code":10516,"language":10271,"meta":10090,"style":10090},[10088,49339,49340,49344,49354,49364,49374,49380,49384,49394,49404,49414,49424,49428,49432,49436,49444,49448,49452,49456,49460,49464,49468,49472,49476,49480,49484,49488,49492,49496,49500,49504,49508,49536,49546,49578,49588,49598,49608,49618,49628,49652,49662,49672,49682,49686,49690,49694,49710,49728,49732,49740,49776,49812,49848,49852,49876,49888,49926,49930,49934,49938,49962,49966,49982,49994,50032,50044,50082,50094,50132,50142,50146,50150,50174,50184,50188,50204,50216,50228,50238,50250,50314,50330,50346,50382,50386,50390,50394,50418,50422,50438,50460,50480,50500,50520,50532,50550,50554,50558,50574,50578,50638,50642,50646,50650,50678,50682,50694,50706,50738,50750,50782,50794,50826,50838,50870,50902,50906,50910,50922,50932,50946,50960,50974,50978,50988,50998,51006,51028,51068,51072,51076,51080,51084,51104,51108,51118,51128,51152,51162,51202,51212],{"__ignoreMap":10090},[40,49341,49342],{"class":10276,"line":10277},[40,49343,10523],{"class":10280},[40,49345,49346,49348,49350,49352],{"class":10276,"line":10284},[40,49347,10353],{"class":10352},[40,49349,10530],{"class":10363},[40,49351,10375],{"class":10352},[40,49353,10535],{"class":10363},[40,49355,49356,49358,49360,49362],{"class":10276,"line":10291},[40,49357,10353],{"class":10352},[40,49359,10542],{"class":10363},[40,49361,10375],{"class":10352},[40,49363,10547],{"class":10363},[40,49365,49366,49368,49370,49372],{"class":10276,"line":10297},[40,49367,10353],{"class":10352},[40,49369,10554],{"class":10363},[40,49371,10375],{"class":10352},[40,49373,10559],{"class":10363},[40,49375,49376,49378],{"class":10276,"line":10304},[40,49377,10375],{"class":10352},[40,49379,10566],{"class":10363},[40,49381,49382],{"class":10276,"line":10310},[40,49383,10301],{"emptyLinePlaceholder":10300},[40,49385,49386,49388,49390,49392],{"class":10276,"line":10316},[40,49387,10375],{"class":10352},[40,49389,10378],{"class":10363},[40,49391,10381],{"class":10352},[40,49393,10384],{"class":10363},[40,49395,49396,49398,49400,49402],{"class":10276,"line":10322},[40,49397,10375],{"class":10352},[40,49399,10405],{"class":10363},[40,49401,10381],{"class":10352},[40,49403,10410],{"class":10363},[40,49405,49406,49408,49410,49412],{"class":10276,"line":10327},[40,49407,10353],{"class":10352},[40,49409,10597],{"class":10363},[40,49411,10375],{"class":10352},[40,49413,10602],{"class":10363},[40,49415,49416,49418,49420,49422],{"class":10276,"line":10333},[40,49417,10353],{"class":10352},[40,49419,10609],{"class":10363},[40,49421,10375],{"class":10352},[40,49423,10614],{"class":10363},[40,49425,49426],{"class":10276,"line":10339},[40,49427,10301],{"emptyLinePlaceholder":10300},[40,49429,49430],{"class":10276,"line":10344},[40,49431,10301],{"emptyLinePlaceholder":10300},[40,49433,49434],{"class":10276,"line":10349},[40,49435,10627],{"class":10430},[40,49437,49438,49440,49442],{"class":10276,"line":10367},[40,49439,10632],{"class":10352},[40,49441,10636],{"class":10635},[40,49443,10639],{"class":10363},[40,49445,49446],{"class":10276,"line":10372},[40,49447,10644],{"class":10287},[40,49449,49450],{"class":10276,"line":10387},[40,49451,10301],{"emptyLinePlaceholder":10300},[40,49453,49454],{"class":10276,"line":10400},[40,49455,10653],{"class":10287},[40,49457,49458],{"class":10276,"line":10413},[40,49459,10658],{"class":10287},[40,49461,49462],{"class":10276,"line":10418},[40,49463,10663],{"class":10287},[40,49465,49466],{"class":10276,"line":10424},[40,49467,10668],{"class":10287},[40,49469,49470],{"class":10276,"line":10437},[40,49471,10673],{"class":10287},[40,49473,49474],{"class":10276,"line":10456},[40,49475,10678],{"class":10287},[40,49477,49478],{"class":10276,"line":10472},[40,49479,10683],{"class":10287},[40,49481,49482],{"class":10276,"line":10496},[40,49483,10688],{"class":10287},[40,49485,49486],{"class":10276,"line":10510},[40,49487,10693],{"class":10287},[40,49489,49490],{"class":10276,"line":10696},[40,49491,10699],{"class":10287},[40,49493,49494],{"class":10276,"line":10702},[40,49495,10705],{"class":10287},[40,49497,49498],{"class":10276,"line":10708},[40,49499,10711],{"class":10287},[40,49501,49502],{"class":10276,"line":10714},[40,49503,10717],{"class":10287},[40,49505,49506],{"class":10276,"line":10720},[40,49507,10723],{"class":10287},[40,49509,49510,49512,49514,49516,49518,49520,49522,49524,49526,49528,49530,49532,49534],{"class":10276,"line":10726},[40,49511,10729],{"class":10363},[40,49513,10733],{"class":10732},[40,49515,10484],{"class":10363},[40,49517,10738],{"class":10446},[40,49519,10741],{"class":10363},[40,49521,162],{"class":10732},[40,49523,920],{"class":10363},[40,49525,10748],{"class":10287},[40,49527,10484],{"class":10363},[40,49529,10753],{"class":10287},[40,49531,10484],{"class":10363},[40,49533,10758],{"class":10287},[40,49535,10761],{"class":10363},[40,49537,49538,49540,49542,49544],{"class":10276,"line":10764},[40,49539,10767],{"class":10363},[40,49541,10733],{"class":10732},[40,49543,10772],{"class":10732},[40,49545,10775],{"class":10287},[40,49547,49548,49550,49552,49554,49556,49558,49560,49562,49564,49566,49568,49570,49572,49574,49576],{"class":10276,"line":10778},[40,49549,10781],{"class":10363},[40,49551,10784],{"class":10732},[40,49553,10484],{"class":10363},[40,49555,10784],{"class":10732},[40,49557,10484],{"class":10363},[40,49559,10784],{"class":10732},[40,49561,10741],{"class":10363},[40,49563,162],{"class":10732},[40,49565,920],{"class":10363},[40,49567,10801],{"class":10446},[40,49569,10484],{"class":10363},[40,49571,10801],{"class":10446},[40,49573,10484],{"class":10363},[40,49575,10810],{"class":10446},[40,49577,10761],{"class":10363},[40,49579,49580,49582,49584,49586],{"class":10276,"line":10815},[40,49581,10818],{"class":10363},[40,49583,10821],{"class":10732},[40,49585,10772],{"class":10732},[40,49587,10826],{"class":10446},[40,49589,49590,49592,49594,49596],{"class":10276,"line":10829},[40,49591,10832],{"class":10363},[40,49593,10821],{"class":10732},[40,49595,10772],{"class":10732},[40,49597,10839],{"class":10446},[40,49599,49600,49602,49604,49606],{"class":10276,"line":10842},[40,49601,10845],{"class":10363},[40,49603,10821],{"class":10732},[40,49605,10772],{"class":10732},[40,49607,10852],{"class":10446},[40,49609,49610,49612,49614,49616],{"class":10276,"line":10855},[40,49611,10858],{"class":10363},[40,49613,10821],{"class":10732},[40,49615,10772],{"class":10732},[40,49617,10865],{"class":10446},[40,49619,49620,49622,49624,49626],{"class":10276,"line":10868},[40,49621,10871],{"class":10363},[40,49623,10733],{"class":10732},[40,49625,10772],{"class":10732},[40,49627,10878],{"class":10287},[40,49629,49630,49632,49634,49636,49638,49640,49642,49644,49646,49648,49650],{"class":10276,"line":10881},[40,49631,10884],{"class":10363},[40,49633,10784],{"class":10732},[40,49635,10484],{"class":10363},[40,49637,10784],{"class":10732},[40,49639,10741],{"class":10363},[40,49641,162],{"class":10732},[40,49643,920],{"class":10363},[40,49645,10481],{"class":10446},[40,49647,10484],{"class":10363},[40,49649,10487],{"class":10446},[40,49651,10761],{"class":10363},[40,49653,49654,49656,49658,49660],{"class":10276,"line":10907},[40,49655,10910],{"class":10363},[40,49657,10733],{"class":10732},[40,49659,10772],{"class":10732},[40,49661,10917],{"class":10287},[40,49663,49664,49666,49668,49670],{"class":10276,"line":10920},[40,49665,10923],{"class":10363},[40,49667,10821],{"class":10732},[40,49669,10772],{"class":10732},[40,49671,10930],{"class":10446},[40,49673,49674,49676,49678,49680],{"class":10276,"line":10933},[40,49675,10936],{"class":10363},[40,49677,10821],{"class":10732},[40,49679,10772],{"class":10732},[40,49681,10943],{"class":10446},[40,49683,49684],{"class":10276,"line":10946},[40,49685,10301],{"emptyLinePlaceholder":10300},[40,49687,49688],{"class":10276,"line":10951},[40,49689,10301],{"emptyLinePlaceholder":10300},[40,49691,49692],{"class":10276,"line":10956},[40,49693,10959],{"class":10280},[40,49695,49696,49698,49700,49702,49704,49706,49708],{"class":10276,"line":10962},[40,49697,10965],{"class":10352},[40,49699,10968],{"class":10430},[40,49701,167],{"class":10363},[40,49703,10974],{"class":10973},[40,49705,10443],{"class":10363},[40,49707,10733],{"class":10732},[40,49709,10507],{"class":10363},[40,49711,49712,49714,49716,49718,49720,49722,49724,49726],{"class":10276,"line":10983},[40,49713,10986],{"class":10973},[40,49715,10989],{"class":10363},[40,49717,10784],{"class":10732},[40,49719,10484],{"class":10363},[40,49721,10784],{"class":10732},[40,49723,10484],{"class":10363},[40,49725,10784],{"class":10732},[40,49727,11002],{"class":10363},[40,49729,49730],{"class":10276,"line":11005},[40,49731,11008],{"class":10287},[40,49733,49734,49736,49738],{"class":10276,"line":11011},[40,49735,11014],{"class":10363},[40,49737,162],{"class":10732},[40,49739,11019],{"class":10363},[40,49741,49742,49744,49746,49748,49750,49752,49754,49756,49758,49760,49762,49764,49766,49768,49770,49772,49774],{"class":10276,"line":11022},[40,49743,11025],{"class":10287},[40,49745,10478],{"class":10363},[40,49747,10801],{"class":10446},[40,49749,10484],{"class":10363},[40,49751,10801],{"class":10446},[40,49753,10484],{"class":10363},[40,49755,10810],{"class":10446},[40,49757,11040],{"class":10363},[40,49759,11043],{"class":10287},[40,49761,10478],{"class":10363},[40,49763,10801],{"class":10446},[40,49765,10484],{"class":10363},[40,49767,10801],{"class":10446},[40,49769,10484],{"class":10363},[40,49771,11056],{"class":10732},[40,49773,10810],{"class":10446},[40,49775,11061],{"class":10363},[40,49777,49778,49780,49782,49784,49786,49788,49790,49792,49794,49796,49798,49800,49802,49804,49806,49808,49810],{"class":10276,"line":11064},[40,49779,11067],{"class":10287},[40,49781,10478],{"class":10363},[40,49783,10810],{"class":10446},[40,49785,10484],{"class":10363},[40,49787,10801],{"class":10446},[40,49789,10484],{"class":10363},[40,49791,10801],{"class":10446},[40,49793,11040],{"class":10363},[40,49795,11084],{"class":10287},[40,49797,10478],{"class":10363},[40,49799,11056],{"class":10732},[40,49801,10810],{"class":10446},[40,49803,10484],{"class":10363},[40,49805,10801],{"class":10446},[40,49807,10484],{"class":10363},[40,49809,10801],{"class":10446},[40,49811,11061],{"class":10363},[40,49813,49814,49816,49818,49820,49822,49824,49826,49828,49830,49832,49834,49836,49838,49840,49842,49844,49846],{"class":10276,"line":11103},[40,49815,11106],{"class":10287},[40,49817,10478],{"class":10363},[40,49819,10801],{"class":10446},[40,49821,10484],{"class":10363},[40,49823,10810],{"class":10446},[40,49825,10484],{"class":10363},[40,49827,10801],{"class":10446},[40,49829,11040],{"class":10363},[40,49831,11123],{"class":10287},[40,49833,10478],{"class":10363},[40,49835,10801],{"class":10446},[40,49837,10484],{"class":10363},[40,49839,11056],{"class":10732},[40,49841,10810],{"class":10446},[40,49843,10484],{"class":10363},[40,49845,10801],{"class":10446},[40,49847,11061],{"class":10363},[40,49849,49850],{"class":10276,"line":11142},[40,49851,11145],{"class":10363},[40,49853,49854,49856,49858,49860,49862,49864,49866,49868,49870,49872,49874],{"class":10276,"line":11148},[40,49855,11151],{"class":10363},[40,49857,162],{"class":10732},[40,49859,11156],{"class":10363},[40,49861,11159],{"class":10430},[40,49863,11162],{"class":10363},[40,49865,11165],{"class":10430},[40,49867,11168],{"class":10363},[40,49869,11171],{"class":10430},[40,49871,11174],{"class":10363},[40,49873,10784],{"class":10732},[40,49875,10761],{"class":10363},[40,49877,49878,49880,49882,49884,49886],{"class":10276,"line":11181},[40,49879,11184],{"class":10363},[40,49881,162],{"class":10732},[40,49883,11189],{"class":10363},[40,49885,11192],{"class":10430},[40,49887,11195],{"class":10363},[40,49889,49890,49892,49894,49896,49898,49900,49902,49904,49906,49908,49910,49912,49914,49916,49918,49920,49922,49924],{"class":10276,"line":11198},[40,49891,11201],{"class":10352},[40,49893,11156],{"class":10363},[40,49895,11206],{"class":10430},[40,49897,11209],{"class":10363},[40,49899,10801],{"class":10446},[40,49901,10484],{"class":10363},[40,49903,10801],{"class":10446},[40,49905,10484],{"class":10363},[40,49907,10810],{"class":10446},[40,49909,11222],{"class":10363},[40,49911,11225],{"class":10352},[40,49913,11228],{"class":10363},[40,49915,11231],{"class":10732},[40,49917,11234],{"class":10446},[40,49919,11237],{"class":10352},[40,49921,11240],{"class":10363},[40,49923,11243],{"class":10732},[40,49925,11246],{"class":10363},[40,49927,49928],{"class":10276,"line":11249},[40,49929,10301],{"emptyLinePlaceholder":10300},[40,49931,49932],{"class":10276,"line":11254},[40,49933,10301],{"emptyLinePlaceholder":10300},[40,49935,49936],{"class":10276,"line":11259},[40,49937,11262],{"class":10280},[40,49939,49940,49942,49944,49946,49948,49950,49952,49954,49956,49958,49960],{"class":10276,"line":11265},[40,49941,10965],{"class":10352},[40,49943,11270],{"class":10430},[40,49945,167],{"class":10363},[40,49947,11275],{"class":10973},[40,49949,10443],{"class":10363},[40,49951,10733],{"class":10732},[40,49953,10484],{"class":10363},[40,49955,11284],{"class":10973},[40,49957,10443],{"class":10363},[40,49959,10784],{"class":10732},[40,49961,11291],{"class":10363},[40,49963,49964],{"class":10276,"line":11294},[40,49965,11297],{"class":10287},[40,49967,49968,49970,49972,49974,49976,49978,49980],{"class":10276,"line":11300},[40,49969,11303],{"class":10363},[40,49971,162],{"class":10732},[40,49973,11156],{"class":10363},[40,49975,1820],{"class":10430},[40,49977,11312],{"class":10363},[40,49979,1813],{"class":10430},[40,49981,11317],{"class":10363},[40,49983,49984,49986,49988,49990,49992],{"class":10276,"line":11320},[40,49985,11323],{"class":10352},[40,49987,11326],{"class":10363},[40,49989,11231],{"class":10732},[40,49991,11331],{"class":10287},[40,49993,10639],{"class":10363},[40,49995,49996,49998,50000,50002,50004,50006,50008,50010,50012,50014,50016,50018,50020,50022,50024,50026,50028,50030],{"class":10276,"line":11336},[40,49997,11339],{"class":10352},[40,49999,11156],{"class":10363},[40,50001,11206],{"class":10430},[40,50003,11346],{"class":10363},[40,50005,78],{"class":10446},[40,50007,10484],{"class":10363},[40,50009,66],{"class":10446},[40,50011,10484],{"class":10363},[40,50013,66],{"class":10446},[40,50015,11359],{"class":10363},[40,50017,66],{"class":10446},[40,50019,11364],{"class":10363},[40,50021,11056],{"class":10732},[40,50023,11369],{"class":10363},[40,50025,66],{"class":10446},[40,50027,11374],{"class":10363},[40,50029,10784],{"class":10732},[40,50031,10761],{"class":10363},[40,50033,50034,50036,50038,50040,50042],{"class":10276,"line":11381},[40,50035,11323],{"class":10352},[40,50037,11326],{"class":10363},[40,50039,11231],{"class":10732},[40,50041,11390],{"class":10287},[40,50043,10639],{"class":10363},[40,50045,50046,50048,50050,50052,50054,50056,50058,50060,50062,50064,50066,50068,50070,50072,50074,50076,50078,50080],{"class":10276,"line":11395},[40,50047,11339],{"class":10352},[40,50049,11156],{"class":10363},[40,50051,11206],{"class":10430},[40,50053,11404],{"class":10363},[40,50055,66],{"class":10446},[40,50057,11409],{"class":10363},[40,50059,66],{"class":10446},[40,50061,10484],{"class":10363},[40,50063,78],{"class":10446},[40,50065,10484],{"class":10363},[40,50067,66],{"class":10446},[40,50069,11359],{"class":10363},[40,50071,11056],{"class":10732},[40,50073,11426],{"class":10363},[40,50075,66],{"class":10446},[40,50077,11431],{"class":10363},[40,50079,10784],{"class":10732},[40,50081,10761],{"class":10363},[40,50083,50084,50086,50088,50090,50092],{"class":10276,"line":11438},[40,50085,11323],{"class":10352},[40,50087,11326],{"class":10363},[40,50089,11231],{"class":10732},[40,50091,11447],{"class":10287},[40,50093,10639],{"class":10363},[40,50095,50096,50098,50100,50102,50104,50106,50108,50110,50112,50114,50116,50118,50120,50122,50124,50126,50128,50130],{"class":10276,"line":11452},[40,50097,11339],{"class":10352},[40,50099,11156],{"class":10363},[40,50101,11206],{"class":10430},[40,50103,11404],{"class":10363},[40,50105,11056],{"class":10732},[40,50107,11426],{"class":10363},[40,50109,66],{"class":10446},[40,50111,11469],{"class":10363},[40,50113,66],{"class":10446},[40,50115,11359],{"class":10363},[40,50117,66],{"class":10446},[40,50119,10484],{"class":10363},[40,50121,66],{"class":10446},[40,50123,10484],{"class":10363},[40,50125,78],{"class":10446},[40,50127,11486],{"class":10363},[40,50129,10784],{"class":10732},[40,50131,10761],{"class":10363},[40,50133,50134,50136,50138,50140],{"class":10276,"line":11493},[40,50135,11496],{"class":10352},[40,50137,11499],{"class":10363},[40,50139,11502],{"class":10287},[40,50141,10761],{"class":10363},[40,50143,50144],{"class":10276,"line":11507},[40,50145,10301],{"emptyLinePlaceholder":10300},[40,50147,50148],{"class":10276,"line":11512},[40,50149,10301],{"emptyLinePlaceholder":10300},[40,50151,50152,50154,50156,50158,50160,50162,50164,50166,50168,50170,50172],{"class":10276,"line":11517},[40,50153,10965],{"class":10352},[40,50155,11522],{"class":10430},[40,50157,167],{"class":10363},[40,50159,11527],{"class":10973},[40,50161,10989],{"class":10363},[40,50163,10784],{"class":10732},[40,50165,10484],{"class":10363},[40,50167,10784],{"class":10732},[40,50169,10484],{"class":10363},[40,50171,10784],{"class":10732},[40,50173,11542],{"class":10363},[40,50175,50176,50178,50180,50182],{"class":10276,"line":11545},[40,50177,11548],{"class":10973},[40,50179,10443],{"class":10363},[40,50181,10784],{"class":10732},[40,50183,11291],{"class":10363},[40,50185,50186],{"class":10276,"line":11557},[40,50187,11560],{"class":10287},[40,50189,50190,50192,50194,50196,50198,50200,50202],{"class":10276,"line":11563},[40,50191,11566],{"class":10363},[40,50193,162],{"class":10732},[40,50195,11156],{"class":10363},[40,50197,11159],{"class":10430},[40,50199,11575],{"class":10363},[40,50201,10784],{"class":10732},[40,50203,10761],{"class":10363},[40,50205,50206,50208,50210,50212,50214],{"class":10276,"line":11582},[40,50207,11184],{"class":10363},[40,50209,162],{"class":10732},[40,50211,11189],{"class":10363},[40,50213,11192],{"class":10430},[40,50215,11593],{"class":10363},[40,50217,50218,50220,50222,50224,50226],{"class":10276,"line":11596},[40,50219,11323],{"class":10352},[40,50221,11228],{"class":10363},[40,50223,11231],{"class":10732},[40,50225,11234],{"class":10446},[40,50227,10639],{"class":10363},[40,50229,50230,50232,50234,50236],{"class":10276,"line":11609},[40,50231,11612],{"class":10352},[40,50233,11499],{"class":10363},[40,50235,11617],{"class":10287},[40,50237,10761],{"class":10363},[40,50239,50240,50242,50244,50246,50248],{"class":10276,"line":11622},[40,50241,11625],{"class":10363},[40,50243,162],{"class":10732},[40,50245,11630],{"class":10363},[40,50247,11243],{"class":10732},[40,50249,11246],{"class":10363},[40,50251,50252,50254,50256,50258,50260,50262,50264,50266,50268,50270,50272,50274,50276,50278,50280,50282,50284,50286,50288,50290,50292,50294,50296,50298,50300,50302,50304,50306,50308,50310,50312],{"class":10276,"line":11637},[40,50253,11640],{"class":10363},[40,50255,162],{"class":10732},[40,50257,11156],{"class":10363},[40,50259,11206],{"class":10430},[40,50261,11346],{"class":10363},[40,50263,10801],{"class":10446},[40,50265,10484],{"class":10363},[40,50267,11056],{"class":10732},[40,50269,11657],{"class":10363},[40,50271,1044],{"class":10446},[40,50273,11662],{"class":10363},[40,50275,78],{"class":10446},[40,50277,11667],{"class":10363},[40,50279,1044],{"class":10446},[40,50281,11672],{"class":10363},[40,50283,10801],{"class":10446},[40,50285,10484],{"class":10363},[40,50287,11056],{"class":10732},[40,50289,11657],{"class":10363},[40,50291,66],{"class":10446},[40,50293,11685],{"class":10363},[40,50295,11056],{"class":10732},[40,50297,11657],{"class":10363},[40,50299,78],{"class":10446},[40,50301,11662],{"class":10363},[40,50303,66],{"class":10446},[40,50305,11672],{"class":10363},[40,50307,10801],{"class":10446},[40,50309,11486],{"class":10363},[40,50311,10784],{"class":10732},[40,50313,10761],{"class":10363},[40,50315,50316,50318,50320,50322,50324,50326,50328],{"class":10276,"line":11708},[40,50317,11711],{"class":10363},[40,50319,162],{"class":10732},[40,50321,11156],{"class":10363},[40,50323,11718],{"class":10430},[40,50325,167],{"class":10363},[40,50327,949],{"class":10446},[40,50329,10761],{"class":10363},[40,50331,50332,50334,50336,50338,50340,50342,50344],{"class":10276,"line":11727},[40,50333,11303],{"class":10363},[40,50335,162],{"class":10732},[40,50337,11156],{"class":10363},[40,50339,1820],{"class":10430},[40,50341,11312],{"class":10363},[40,50343,1813],{"class":10430},[40,50345,11317],{"class":10363},[40,50347,50348,50350,50352,50354,50356,50358,50360,50362,50364,50366,50368,50370,50372,50374,50376,50378,50380],{"class":10276,"line":11744},[40,50349,11201],{"class":10352},[40,50351,11749],{"class":10363},[40,50353,11752],{"class":10732},[40,50355,11755],{"class":10363},[40,50357,1053],{"class":10732},[40,50359,920],{"class":10363},[40,50361,78],{"class":10446},[40,50363,11764],{"class":10732},[40,50365,11767],{"class":10363},[40,50367,11752],{"class":10732},[40,50369,11156],{"class":10363},[40,50371,11774],{"class":10430},[40,50373,11777],{"class":10363},[40,50375,1053],{"class":10732},[40,50377,11782],{"class":10363},[40,50379,11752],{"class":10732},[40,50381,11787],{"class":10363},[40,50383,50384],{"class":10276,"line":11790},[40,50385,10301],{"emptyLinePlaceholder":10300},[40,50387,50388],{"class":10276,"line":11795},[40,50389,10301],{"emptyLinePlaceholder":10300},[40,50391,50392],{"class":10276,"line":11800},[40,50393,11803],{"class":10280},[40,50395,50396,50398,50400,50402,50404,50406,50408,50410,50412,50414,50416],{"class":10276,"line":11806},[40,50397,10965],{"class":10352},[40,50399,11811],{"class":10430},[40,50401,167],{"class":10363},[40,50403,11816],{"class":10973},[40,50405,10443],{"class":10363},[40,50407,10821],{"class":10732},[40,50409,10484],{"class":10363},[40,50411,11825],{"class":10973},[40,50413,10443],{"class":10363},[40,50415,10821],{"class":10732},[40,50417,11291],{"class":10363},[40,50419,50420],{"class":10276,"line":11834},[40,50421,11837],{"class":10287},[40,50423,50424,50426,50428,50430,50432,50434,50436],{"class":10276,"line":11840},[40,50425,11843],{"class":10363},[40,50427,162],{"class":10732},[40,50429,11156],{"class":10363},[40,50431,11850],{"class":10430},[40,50433,167],{"class":10363},[40,50435,10801],{"class":10446},[40,50437,11857],{"class":10363},[40,50439,50440,50442,50444,50446,50448,50450,50452,50454,50456,50458],{"class":10276,"line":11860},[40,50441,11863],{"class":10363},[40,50443,162],{"class":10732},[40,50445,11156],{"class":10363},[40,50447,11850],{"class":10430},[40,50449,167],{"class":10363},[40,50451,10801],{"class":10446},[40,50453,10484],{"class":10363},[40,50455,11878],{"class":10446},[40,50457,11881],{"class":10732},[40,50459,11884],{"class":10363},[40,50461,50462,50464,50466,50468,50470,50472,50474,50476,50478],{"class":10276,"line":11887},[40,50463,11890],{"class":10363},[40,50465,162],{"class":10732},[40,50467,11156],{"class":10363},[40,50469,11897],{"class":10430},[40,50471,11900],{"class":10363},[40,50473,11904],{"class":11903},[40,50475,162],{"class":10732},[40,50477,11909],{"class":10287},[40,50479,10761],{"class":10363},[40,50481,50482,50484,50486,50488,50490,50492,50494,50496,50498],{"class":10276,"line":11914},[40,50483,11917],{"class":10363},[40,50485,162],{"class":10732},[40,50487,11156],{"class":10363},[40,50489,1813],{"class":10430},[40,50491,11926],{"class":10363},[40,50493,11752],{"class":10732},[40,50495,11156],{"class":10363},[40,50497,1820],{"class":10430},[40,50499,11935],{"class":10363},[40,50501,50502,50504,50506,50508,50510,50512,50514,50516,50518],{"class":10276,"line":11938},[40,50503,11941],{"class":10363},[40,50505,162],{"class":10732},[40,50507,11156],{"class":10363},[40,50509,1813],{"class":10430},[40,50511,11926],{"class":10363},[40,50513,11752],{"class":10732},[40,50515,11156],{"class":10363},[40,50517,1813],{"class":10430},[40,50519,11935],{"class":10363},[40,50521,50522,50524,50526,50528,50530],{"class":10276,"line":11960},[40,50523,11963],{"class":10363},[40,50525,162],{"class":10732},[40,50527,11156],{"class":10363},[40,50529,1820],{"class":10430},[40,50531,11972],{"class":10363},[40,50533,50534,50536,50538,50540,50542,50544,50546,50548],{"class":10276,"line":11975},[40,50535,11201],{"class":10352},[40,50537,11156],{"class":10363},[40,50539,11982],{"class":10430},[40,50541,11985],{"class":10363},[40,50543,11275],{"class":11903},[40,50545,162],{"class":10732},[40,50547,66],{"class":10446},[40,50549,10761],{"class":10363},[40,50551,50552],{"class":10276,"line":11996},[40,50553,10301],{"emptyLinePlaceholder":10300},[40,50555,50556],{"class":10276,"line":12001},[40,50557,10301],{"emptyLinePlaceholder":10300},[40,50559,50560,50562,50564,50566,50568,50570,50572],{"class":10276,"line":12006},[40,50561,10965],{"class":10352},[40,50563,12011],{"class":10430},[40,50565,167],{"class":10363},[40,50567,12016],{"class":10973},[40,50569,12019],{"class":10363},[40,50571,12022],{"class":10973},[40,50573,12025],{"class":10363},[40,50575,50576],{"class":10276,"line":12028},[40,50577,12031],{"class":10287},[40,50579,50580,50582,50584,50586,50588,50590,50592,50594,50596,50598,50600,50602,50604,50606,50608,50610,50612,50614,50616,50618,50620,50622,50624,50626,50628,50630,50632,50634,50636],{"class":10276,"line":12034},[40,50581,11201],{"class":10352},[40,50583,12039],{"class":10446},[40,50585,12042],{"class":10732},[40,50587,12045],{"class":10363},[40,50589,2218],{"class":10430},[40,50591,167],{"class":10363},[40,50593,12052],{"class":10446},[40,50595,12055],{"class":10363},[40,50597,11243],{"class":10732},[40,50599,12060],{"class":10446},[40,50601,12055],{"class":10363},[40,50603,11752],{"class":10732},[40,50605,12067],{"class":10363},[40,50607,12070],{"class":10430},[40,50609,167],{"class":10363},[40,50611,949],{"class":10446},[40,50613,10484],{"class":10363},[40,50615,78],{"class":10446},[40,50617,10484],{"class":10363},[40,50619,78],{"class":10446},[40,50621,12055],{"class":10363},[40,50623,11752],{"class":10732},[40,50625,12089],{"class":10363},[40,50627,12092],{"class":10430},[40,50629,167],{"class":10363},[40,50631,11275],{"class":11903},[40,50633,162],{"class":10732},[40,50635,66],{"class":10446},[40,50637,10761],{"class":10363},[40,50639,50640],{"class":10276,"line":12105},[40,50641,10301],{"emptyLinePlaceholder":10300},[40,50643,50644],{"class":10276,"line":12110},[40,50645,10301],{"emptyLinePlaceholder":10300},[40,50647,50648],{"class":10276,"line":12115},[40,50649,12118],{"class":10280},[40,50651,50652,50654,50656,50658,50660,50662,50664,50666,50668,50670,50672,50674,50676],{"class":10276,"line":12121},[40,50653,10965],{"class":10352},[40,50655,12126],{"class":10430},[40,50657,167],{"class":10363},[40,50659,12131],{"class":10973},[40,50661,10443],{"class":10363},[40,50663,10733],{"class":10732},[40,50665,12138],{"class":10363},[40,50667,10784],{"class":10732},[40,50669,10484],{"class":10363},[40,50671,10784],{"class":10732},[40,50673,10484],{"class":10363},[40,50675,12149],{"class":10732},[40,50677,12152],{"class":10363},[40,50679,50680],{"class":10276,"line":12155},[40,50681,12158],{"class":10287},[40,50683,50684,50686,50688,50690,50692],{"class":10276,"line":12161},[40,50685,12164],{"class":10363},[40,50687,162],{"class":10732},[40,50689,12169],{"class":10363},[40,50691,11171],{"class":10430},[40,50693,12174],{"class":10363},[40,50695,50696,50698,50700,50702,50704],{"class":10276,"line":12177},[40,50697,11323],{"class":10352},[40,50699,12182],{"class":10363},[40,50701,11231],{"class":10732},[40,50703,12187],{"class":10287},[40,50705,10639],{"class":10363},[40,50707,50708,50710,50712,50714,50716,50718,50720,50722,50724,50726,50728,50730,50732,50734,50736],{"class":10276,"line":12192},[40,50709,11339],{"class":10352},[40,50711,12197],{"class":10446},[40,50713,10484],{"class":10363},[40,50715,12202],{"class":10446},[40,50717,10484],{"class":10363},[40,50719,12149],{"class":10732},[40,50721,167],{"class":10363},[40,50723,12211],{"class":11903},[40,50725,162],{"class":10732},[40,50727,12216],{"class":10287},[40,50729,10484],{"class":10363},[40,50731,12221],{"class":11903},[40,50733,162],{"class":10732},[40,50735,12226],{"class":10287},[40,50737,10761],{"class":10363},[40,50739,50740,50742,50744,50746,50748],{"class":10276,"line":12231},[40,50741,11323],{"class":10352},[40,50743,12182],{"class":10363},[40,50745,11231],{"class":10732},[40,50747,12240],{"class":10287},[40,50749,10639],{"class":10363},[40,50751,50752,50754,50756,50758,50760,50762,50764,50766,50768,50770,50772,50774,50776,50778,50780],{"class":10276,"line":12245},[40,50753,11339],{"class":10352},[40,50755,12250],{"class":10446},[40,50757,10484],{"class":10363},[40,50759,12202],{"class":10446},[40,50761,10484],{"class":10363},[40,50763,12149],{"class":10732},[40,50765,167],{"class":10363},[40,50767,12211],{"class":11903},[40,50769,162],{"class":10732},[40,50771,12267],{"class":10287},[40,50773,10484],{"class":10363},[40,50775,12221],{"class":11903},[40,50777,162],{"class":10732},[40,50779,12226],{"class":10287},[40,50781,10761],{"class":10363},[40,50783,50784,50786,50788,50790,50792],{"class":10276,"line":12280},[40,50785,11323],{"class":10352},[40,50787,12182],{"class":10363},[40,50789,11231],{"class":10732},[40,50791,12289],{"class":10287},[40,50793,10639],{"class":10363},[40,50795,50796,50798,50800,50802,50804,50806,50808,50810,50812,50814,50816,50818,50820,50822,50824],{"class":10276,"line":12294},[40,50797,11339],{"class":10352},[40,50799,12197],{"class":10446},[40,50801,10484],{"class":10363},[40,50803,12303],{"class":10446},[40,50805,10484],{"class":10363},[40,50807,12149],{"class":10732},[40,50809,167],{"class":10363},[40,50811,12211],{"class":11903},[40,50813,162],{"class":10732},[40,50815,12216],{"class":10287},[40,50817,10484],{"class":10363},[40,50819,12221],{"class":11903},[40,50821,162],{"class":10732},[40,50823,12324],{"class":10287},[40,50825,10761],{"class":10363},[40,50827,50828,50830,50832,50834,50836],{"class":10276,"line":12329},[40,50829,11323],{"class":10352},[40,50831,12182],{"class":10363},[40,50833,11231],{"class":10732},[40,50835,12338],{"class":10287},[40,50837,10639],{"class":10363},[40,50839,50840,50842,50844,50846,50848,50850,50852,50854,50856,50858,50860,50862,50864,50866,50868],{"class":10276,"line":12343},[40,50841,11339],{"class":10352},[40,50843,12250],{"class":10446},[40,50845,10484],{"class":10363},[40,50847,12303],{"class":10446},[40,50849,10484],{"class":10363},[40,50851,12149],{"class":10732},[40,50853,167],{"class":10363},[40,50855,12211],{"class":11903},[40,50857,162],{"class":10732},[40,50859,12267],{"class":10287},[40,50861,10484],{"class":10363},[40,50863,12221],{"class":11903},[40,50865,162],{"class":10732},[40,50867,12324],{"class":10287},[40,50869,10761],{"class":10363},[40,50871,50872,50874,50876,50878,50880,50882,50884,50886,50888,50890,50892,50894,50896,50898,50900],{"class":10276,"line":12376},[40,50873,11201],{"class":10352},[40,50875,12197],{"class":10446},[40,50877,10484],{"class":10363},[40,50879,12202],{"class":10446},[40,50881,10484],{"class":10363},[40,50883,12149],{"class":10732},[40,50885,167],{"class":10363},[40,50887,12211],{"class":11903},[40,50889,162],{"class":10732},[40,50891,12216],{"class":10287},[40,50893,10484],{"class":10363},[40,50895,12221],{"class":11903},[40,50897,162],{"class":10732},[40,50899,12226],{"class":10287},[40,50901,10761],{"class":10363},[40,50903,50904],{"class":10276,"line":12409},[40,50905,10301],{"emptyLinePlaceholder":10300},[40,50907,50908],{"class":10276,"line":12414},[40,50909,10301],{"emptyLinePlaceholder":10300},[40,50911,50912,50914,50916,50918,50920],{"class":10276,"line":12419},[40,50913,10965],{"class":10352},[40,50915,12424],{"class":10430},[40,50917,167],{"class":10363},[40,50919,12429],{"class":10973},[40,50921,12432],{"class":10363},[40,50923,50924,50926,50928,50930],{"class":10276,"line":12435},[40,50925,12438],{"class":10973},[40,50927,10443],{"class":10363},[40,50929,10733],{"class":10732},[40,50931,10507],{"class":10363},[40,50933,50934,50936,50938,50940,50942,50944],{"class":10276,"line":12447},[40,50935,12450],{"class":10973},[40,50937,10443],{"class":10363},[40,50939,10733],{"class":10732},[40,50941,10772],{"class":10732},[40,50943,12187],{"class":10287},[40,50945,10507],{"class":10363},[40,50947,50948,50950,50952,50954,50956,50958],{"class":10276,"line":12463},[40,50949,12466],{"class":10973},[40,50951,10443],{"class":10363},[40,50953,10821],{"class":10732},[40,50955,10772],{"class":10732},[40,50957,12475],{"class":10446},[40,50959,10507],{"class":10363},[40,50961,50962,50964,50966,50968,50970,50972],{"class":10276,"line":12480},[40,50963,12483],{"class":10973},[40,50965,10443],{"class":10363},[40,50967,10821],{"class":10732},[40,50969,10772],{"class":10732},[40,50971,12492],{"class":10446},[40,50973,12495],{"class":10363},[40,50975,50976],{"class":10276,"line":12498},[40,50977,12501],{"class":10287},[40,50979,50980,50982,50984,50986],{"class":10276,"line":12504},[40,50981,12507],{"class":10363},[40,50983,162],{"class":10732},[40,50985,12126],{"class":10430},[40,50987,12514],{"class":10363},[40,50989,50990,50992,50994,50996],{"class":10276,"line":12517},[40,50991,11201],{"class":10352},[40,50993,12522],{"class":10363},[40,50995,2932],{"class":10430},[40,50997,12527],{"class":10363},[40,50999,51000,51002,51004],{"class":10276,"line":12530},[40,51001,12533],{"class":10363},[40,51003,12536],{"class":10430},[40,51005,12539],{"class":10363},[40,51007,51008,51010,51012,51014,51016,51018,51020,51022,51024,51026],{"class":10276,"line":12542},[40,51009,12545],{"class":11903},[40,51011,162],{"class":10732},[40,51013,12550],{"class":10363},[40,51015,12553],{"class":11903},[40,51017,162],{"class":10732},[40,51019,12558],{"class":10363},[40,51021,12561],{"class":11903},[40,51023,162],{"class":10732},[40,51025,12566],{"class":10446},[40,51027,10507],{"class":10363},[40,51029,51030,51032,51034,51036,51038,51040,51042,51044,51046,51048,51050,51052,51054,51056,51058,51060,51062,51064,51066],{"class":10276,"line":12571},[40,51031,12574],{"class":11903},[40,51033,12577],{"class":10732},[40,51035,167],{"class":10363},[40,51037,12582],{"class":11903},[40,51039,162],{"class":10732},[40,51041,12587],{"class":10287},[40,51043,10484],{"class":10363},[40,51045,12592],{"class":11903},[40,51047,162],{"class":10732},[40,51049,12597],{"class":10287},[40,51051,10484],{"class":10363},[40,51053,12602],{"class":11903},[40,51055,162],{"class":10732},[40,51057,12607],{"class":10446},[40,51059,10484],{"class":10363},[40,51061,12612],{"class":11903},[40,51063,162],{"class":10732},[40,51065,12617],{"class":10287},[40,51067,11061],{"class":10363},[40,51069,51070],{"class":10276,"line":12622},[40,51071,12625],{"class":10363},[40,51073,51074],{"class":10276,"line":12628},[40,51075,12631],{"class":10363},[40,51077,51078],{"class":10276,"line":12634},[40,51079,10301],{"emptyLinePlaceholder":10300},[40,51081,51082],{"class":10276,"line":12639},[40,51083,10301],{"emptyLinePlaceholder":10300},[40,51085,51086,51088,51090,51092,51094,51096,51098,51100,51102],{"class":10276,"line":12644},[40,51087,10965],{"class":10352},[40,51089,12649],{"class":10430},[40,51091,167],{"class":10363},[40,51093,12429],{"class":10973},[40,51095,12656],{"class":10363},[40,51097,2932],{"class":10973},[40,51099,10443],{"class":10363},[40,51101,10733],{"class":10732},[40,51103,12495],{"class":10363},[40,51105,51106],{"class":10276,"line":12667},[40,51107,12670],{"class":10287},[40,51109,51110,51112,51114,51116],{"class":10276,"line":12673},[40,51111,11201],{"class":10352},[40,51113,12522],{"class":10363},[40,51115,2932],{"class":10430},[40,51117,12527],{"class":10363},[40,51119,51120,51122,51124,51126],{"class":10276,"line":12684},[40,51121,12687],{"class":10446},[40,51123,10484],{"class":10363},[40,51125,12202],{"class":10446},[40,51127,12694],{"class":10363},[40,51129,51130,51132,51134,51136,51138,51140,51142,51144,51146,51148,51150],{"class":10276,"line":12697},[40,51131,12545],{"class":11903},[40,51133,162],{"class":10732},[40,51135,12550],{"class":10363},[40,51137,12211],{"class":11903},[40,51139,162],{"class":10732},[40,51141,12710],{"class":10287},[40,51143,10484],{"class":10363},[40,51145,12221],{"class":11903},[40,51147,162],{"class":10732},[40,51149,12226],{"class":10287},[40,51151,10507],{"class":10363},[40,51153,51154,51156,51158,51160],{"class":10276,"line":12723},[40,51155,12726],{"class":11903},[40,51157,162],{"class":10732},[40,51159,12731],{"class":10446},[40,51161,10507],{"class":10363},[40,51163,51164,51166,51168,51170,51172,51174,51176,51178,51180,51182,51184,51186,51188,51190,51192,51194,51196,51198,51200],{"class":10276,"line":12736},[40,51165,12574],{"class":11903},[40,51167,12577],{"class":10732},[40,51169,167],{"class":10363},[40,51171,12582],{"class":11903},[40,51173,162],{"class":10732},[40,51175,12587],{"class":10287},[40,51177,10484],{"class":10363},[40,51179,12592],{"class":11903},[40,51181,162],{"class":10732},[40,51183,12597],{"class":10287},[40,51185,10484],{"class":10363},[40,51187,12602],{"class":11903},[40,51189,162],{"class":10732},[40,51191,12607],{"class":10446},[40,51193,10484],{"class":10363},[40,51195,12612],{"class":11903},[40,51197,162],{"class":10732},[40,51199,12617],{"class":10287},[40,51201,11061],{"class":10363},[40,51203,51204,51206,51208,51210],{"class":10276,"line":12777},[40,51205,12780],{"class":11903},[40,51207,162],{"class":10732},[40,51209,12785],{"class":10446},[40,51211,10507],{"class":10363},[40,51213,51214],{"class":10276,"line":12790},[40,51215,12631],{"class":10363},[10082,51217,51218],{"className":10269,"code":12795,"language":10271,"meta":10090,"style":10090},[10088,51219,51220,51224,51234,51238,51254,51260,51270,51280,51290,51300,51314,51324,51334,51344,51350,51354,51368,51378,51404,51430,51452,51456,51482,51512,51534,51568,51580,51584,51596,51608,51632,51636,51662,51670,51674,51690,51714,51732,51744,51790,51796,51800,51832,51856,51882,51898,51906,51912,51916,51920,51924,51936,51956,51966,51978,51998,52008,52018,52034,52060,52064,52068,52078,52086,52096,52120,52128,52134,52138,52142,52146,52150,52154,52158,52162,52172,52188],{"__ignoreMap":10090},[40,51221,51222],{"class":10276,"line":10277},[40,51223,12802],{"class":10280},[40,51225,51226,51228,51230,51232],{"class":10276,"line":10284},[40,51227,10353],{"class":10352},[40,51229,10542],{"class":10363},[40,51231,10375],{"class":10352},[40,51233,10547],{"class":10363},[40,51235,51236],{"class":10276,"line":10291},[40,51237,10301],{"emptyLinePlaceholder":10300},[40,51239,51240,51242,51244,51246,51248,51250,51252],{"class":10276,"line":10297},[40,51241,10965],{"class":10352},[40,51243,12823],{"class":10430},[40,51245,167],{"class":10363},[40,51247,11275],{"class":10973},[40,51249,10443],{"class":10363},[40,51251,10733],{"class":10732},[40,51253,10507],{"class":10363},[40,51255,51256,51258],{"class":10276,"line":10304},[40,51257,12838],{"class":10973},[40,51259,12841],{"class":10363},[40,51261,51262,51264,51266,51268],{"class":10276,"line":10310},[40,51263,12846],{"class":10973},[40,51265,10443],{"class":10363},[40,51267,10821],{"class":10732},[40,51269,10507],{"class":10363},[40,51271,51272,51274,51276,51278],{"class":10276,"line":10316},[40,51273,12857],{"class":10973},[40,51275,10443],{"class":10363},[40,51277,10821],{"class":10732},[40,51279,10507],{"class":10363},[40,51281,51282,51284,51286,51288],{"class":10276,"line":10322},[40,51283,12868],{"class":10973},[40,51285,10443],{"class":10363},[40,51287,10821],{"class":10732},[40,51289,10507],{"class":10363},[40,51291,51292,51294,51296,51298],{"class":10276,"line":10327},[40,51293,12879],{"class":10973},[40,51295,10443],{"class":10363},[40,51297,10821],{"class":10732},[40,51299,10507],{"class":10363},[40,51301,51302,51304,51306,51308,51310,51312],{"class":10276,"line":10333},[40,51303,12890],{"class":10973},[40,51305,10989],{"class":10363},[40,51307,10784],{"class":10732},[40,51309,10484],{"class":10363},[40,51311,10784],{"class":10732},[40,51313,11542],{"class":10363},[40,51315,51316,51318,51320,51322],{"class":10276,"line":10339},[40,51317,12905],{"class":10973},[40,51319,10443],{"class":10363},[40,51321,10733],{"class":10732},[40,51323,10507],{"class":10363},[40,51325,51326,51328,51330,51332],{"class":10276,"line":10344},[40,51327,12916],{"class":10973},[40,51329,10443],{"class":10363},[40,51331,10821],{"class":10732},[40,51333,10507],{"class":10363},[40,51335,51336,51338,51340,51342],{"class":10276,"line":10349},[40,51337,12927],{"class":10973},[40,51339,10443],{"class":10363},[40,51341,10821],{"class":10732},[40,51343,10507],{"class":10363},[40,51345,51346,51348],{"class":10276,"line":10367},[40,51347,12938],{"class":10973},[40,51349,12941],{"class":10363},[40,51351,51352],{"class":10276,"line":10372},[40,51353,12946],{"class":10287},[40,51355,51356,51358,51360,51362,51364,51366],{"class":10276,"line":10387},[40,51357,12951],{"class":10352},[40,51359,12954],{"class":10363},[40,51361,12957],{"class":10732},[40,51363,12960],{"class":10446},[40,51365,10484],{"class":10363},[40,51367,12965],{"class":10287},[40,51369,51370,51372,51374,51376],{"class":10276,"line":10400},[40,51371,12970],{"class":10363},[40,51373,162],{"class":10732},[40,51375,11811],{"class":10430},[40,51377,12977],{"class":10363},[40,51379,51380,51382,51384,51386,51388,51390,51392,51394,51396,51398,51400,51402],{"class":10276,"line":10413},[40,51381,12982],{"class":10363},[40,51383,162],{"class":10732},[40,51385,12039],{"class":10446},[40,51387,11764],{"class":10732},[40,51389,12045],{"class":10363},[40,51391,2218],{"class":10430},[40,51393,167],{"class":10363},[40,51395,12052],{"class":10446},[40,51397,12055],{"class":10363},[40,51399,11243],{"class":10732},[40,51401,12060],{"class":10446},[40,51403,10761],{"class":10363},[40,51405,51406,51408,51410,51412,51414,51416,51418,51420,51422,51424,51426,51428],{"class":10276,"line":10418},[40,51407,13009],{"class":10363},[40,51409,162],{"class":10732},[40,51411,12039],{"class":10446},[40,51413,12042],{"class":10732},[40,51415,12045],{"class":10363},[40,51417,2218],{"class":10430},[40,51419,167],{"class":10363},[40,51421,12052],{"class":10446},[40,51423,12055],{"class":10363},[40,51425,11243],{"class":10732},[40,51427,12060],{"class":10446},[40,51429,10761],{"class":10363},[40,51431,51432,51434,51436,51438,51440,51442,51444,51446,51448,51450],{"class":10276,"line":10424},[40,51433,13036],{"class":10363},[40,51435,162],{"class":10732},[40,51437,920],{"class":10363},[40,51439,10801],{"class":10446},[40,51441,10484],{"class":10363},[40,51443,11878],{"class":10446},[40,51445,11881],{"class":10732},[40,51447,13051],{"class":10363},[40,51449,10801],{"class":10446},[40,51451,13056],{"class":10363},[40,51453,51454],{"class":10276,"line":10437},[40,51455,10301],{"emptyLinePlaceholder":10300},[40,51457,51458,51460,51462,51464,51466,51468,51470,51472,51474,51476,51478,51480],{"class":10276,"line":10456},[40,51459,13065],{"class":10363},[40,51461,162],{"class":10732},[40,51463,13070],{"class":10363},[40,51465,13073],{"class":10430},[40,51467,167],{"class":10363},[40,51469,13078],{"class":11903},[40,51471,162],{"class":10732},[40,51473,13083],{"class":10363},[40,51475,13086],{"class":11903},[40,51477,162],{"class":10732},[40,51479,13091],{"class":10446},[40,51481,10761],{"class":10363},[40,51483,51484,51486,51488,51490,51492,51494,51496,51498,51500,51502,51504,51506,51508,51510],{"class":10276,"line":10472},[40,51485,13098],{"class":10363},[40,51487,162],{"class":10732},[40,51489,12522],{"class":10363},[40,51491,13105],{"class":10430},[40,51493,13108],{"class":10363},[40,51495,13111],{"class":10430},[40,51497,13114],{"class":10363},[40,51499,13117],{"class":11903},[40,51501,162],{"class":10732},[40,51503,13122],{"class":10287},[40,51505,10484],{"class":10363},[40,51507,13127],{"class":11903},[40,51509,162],{"class":10732},[40,51511,13132],{"class":10363},[40,51513,51514,51516,51518,51520,51522,51524,51526,51528,51530,51532],{"class":10276,"line":10496},[40,51515,13137],{"class":11903},[40,51517,162],{"class":10732},[40,51519,13142],{"class":10363},[40,51521,13145],{"class":11903},[40,51523,162],{"class":10732},[40,51525,13150],{"class":10363},[40,51527,13153],{"class":11903},[40,51529,162],{"class":10732},[40,51531,13158],{"class":10287},[40,51533,10761],{"class":10363},[40,51535,51536,51538,51540,51542,51544,51546,51548,51550,51552,51554,51556,51558,51560,51562,51564,51566],{"class":10276,"line":10510},[40,51537,13165],{"class":10363},[40,51539,162],{"class":10732},[40,51541,13170],{"class":10363},[40,51543,13173],{"class":10430},[40,51545,13176],{"class":10363},[40,51547,12429],{"class":11903},[40,51549,162],{"class":10732},[40,51551,13183],{"class":10363},[40,51553,13186],{"class":11903},[40,51555,162],{"class":10732},[40,51557,13191],{"class":10446},[40,51559,10484],{"class":10363},[40,51561,13196],{"class":11903},[40,51563,162],{"class":10732},[40,51565,13201],{"class":10446},[40,51567,10761],{"class":10363},[40,51569,51570,51572,51574,51576,51578],{"class":10276,"line":10696},[40,51571,13208],{"class":10363},[40,51573,13211],{"class":10430},[40,51575,167],{"class":10363},[40,51577,13216],{"class":10287},[40,51579,10761],{"class":10363},[40,51581,51582],{"class":10276,"line":10702},[40,51583,10301],{"emptyLinePlaceholder":10300},[40,51585,51586,51588,51590,51592,51594],{"class":10276,"line":10708},[40,51587,13227],{"class":10363},[40,51589,13230],{"class":10430},[40,51591,167],{"class":10363},[40,51593,13235],{"class":10287},[40,51595,10761],{"class":10363},[40,51597,51598,51600,51602,51604,51606],{"class":10276,"line":10714},[40,51599,13227],{"class":10363},[40,51601,13244],{"class":10430},[40,51603,167],{"class":10363},[40,51605,13249],{"class":10287},[40,51607,10761],{"class":10363},[40,51609,51610,51612,51614,51616,51618,51620,51622,51624,51626,51628,51630],{"class":10276,"line":10720},[40,51611,13256],{"class":10430},[40,51613,13259],{"class":10363},[40,51615,13262],{"class":10352},[40,51617,13265],{"class":10287},[40,51619,13268],{"class":10446},[40,51621,13271],{"class":10363},[40,51623,13274],{"class":10430},[40,51625,13277],{"class":10363},[40,51627,13280],{"class":10446},[40,51629,13283],{"class":10287},[40,51631,10761],{"class":10363},[40,51633,51634],{"class":10276,"line":10726},[40,51635,10301],{"emptyLinePlaceholder":10300},[40,51637,51638,51640,51642,51644,51646,51648,51650,51652,51654,51656,51658,51660],{"class":10276,"line":10764},[40,51639,13294],{"class":10363},[40,51641,162],{"class":10732},[40,51643,12424],{"class":10430},[40,51645,13259],{"class":10363},[40,51647,13303],{"class":10287},[40,51649,10484],{"class":10363},[40,51651,12131],{"class":11903},[40,51653,162],{"class":10732},[40,51655,13312],{"class":10363},[40,51657,12553],{"class":11903},[40,51659,162],{"class":10732},[40,51661,13319],{"class":10363},[40,51663,51664,51666,51668],{"class":10276,"line":10778},[40,51665,13324],{"class":11903},[40,51667,162],{"class":10732},[40,51669,13329],{"class":10363},[40,51671,51672],{"class":10276,"line":10815},[40,51673,10301],{"emptyLinePlaceholder":10300},[40,51675,51676,51678,51680,51682,51684,51686,51688],{"class":10276,"line":10829},[40,51677,13338],{"class":10352},[40,51679,13341],{"class":10430},[40,51681,167],{"class":10363},[40,51683,13346],{"class":10973},[40,51685,10443],{"class":10363},[40,51687,10821],{"class":10732},[40,51689,13353],{"class":10363},[40,51691,51692,51694,51696,51698,51700,51702,51704,51706,51708,51710,51712],{"class":10276,"line":10842},[40,51693,13358],{"class":10363},[40,51695,162],{"class":10732},[40,51697,13363],{"class":10363},[40,51699,11243],{"class":10732},[40,51701,13368],{"class":10363},[40,51703,11056],{"class":10732},[40,51705,13373],{"class":10446},[40,51707,13376],{"class":10363},[40,51709,11752],{"class":10732},[40,51711,13381],{"class":10363},[40,51713,13384],{"class":10280},[40,51715,51716,51718,51720,51722,51724,51726,51728,51730],{"class":10276,"line":10855},[40,51717,13389],{"class":10363},[40,51719,162],{"class":10732},[40,51721,11270],{"class":10430},[40,51723,13396],{"class":10363},[40,51725,11171],{"class":10430},[40,51727,13401],{"class":10363},[40,51729,13404],{"class":10732},[40,51731,13407],{"class":10363},[40,51733,51734,51736,51738,51740,51742],{"class":10276,"line":10868},[40,51735,13412],{"class":10363},[40,51737,13415],{"class":10430},[40,51739,167],{"class":10363},[40,51741,13420],{"class":10430},[40,51743,13423],{"class":10363},[40,51745,51746,51748,51750,51752,51754,51756,51758,51760,51762,51764,51766,51768,51770,51772,51774,51776,51778,51780,51782,51784,51786,51788],{"class":10276,"line":10881},[40,51747,13428],{"class":10363},[40,51749,13431],{"class":10430},[40,51751,167],{"class":10363},[40,51753,13262],{"class":10352},[40,51755,13438],{"class":10287},[40,51757,13268],{"class":10446},[40,51759,13443],{"class":10363},[40,51761,13446],{"class":10352},[40,51763,13280],{"class":10446},[40,51765,13451],{"class":10287},[40,51767,13454],{"class":10732},[40,51769,13457],{"class":10287},[40,51771,13268],{"class":10446},[40,51773,13346],{"class":10363},[40,51775,1053],{"class":10732},[40,51777,13466],{"class":10446},[40,51779,11243],{"class":10287},[40,51781,13268],{"class":10446},[40,51783,13473],{"class":10363},[40,51785,13280],{"class":10446},[40,51787,13478],{"class":10287},[40,51789,10761],{"class":10363},[40,51791,51792,51794],{"class":10276,"line":10907},[40,51793,11339],{"class":10352},[40,51795,13487],{"class":10363},[40,51797,51798],{"class":10276,"line":10920},[40,51799,10301],{"emptyLinePlaceholder":10300},[40,51801,51802,51804,51806,51808,51810,51812,51814,51816,51818,51820,51822,51824,51826,51828,51830],{"class":10276,"line":10933},[40,51803,13496],{"class":10363},[40,51805,162],{"class":10732},[40,51807,13501],{"class":10430},[40,51809,13504],{"class":10363},[40,51811,13473],{"class":11903},[40,51813,162],{"class":10732},[40,51815,13511],{"class":10363},[40,51817,13514],{"class":11903},[40,51819,162],{"class":10732},[40,51821,13519],{"class":10446},[40,51823,10484],{"class":10363},[40,51825,13524],{"class":11903},[40,51827,162],{"class":10732},[40,51829,10504],{"class":10446},[40,51831,10761],{"class":10363},[40,51833,51834,51836,51838,51840,51842,51844,51846,51848,51850,51852,51854],{"class":10276,"line":10946},[40,51835,13535],{"class":10363},[40,51837,13538],{"class":10430},[40,51839,167],{"class":10363},[40,51841,13543],{"class":11903},[40,51843,162],{"class":10732},[40,51845,13091],{"class":10446},[40,51847,10484],{"class":10363},[40,51849,13552],{"class":11903},[40,51851,162],{"class":10732},[40,51853,13091],{"class":10446},[40,51855,10761],{"class":10363},[40,51857,51858,51860,51862,51864,51866,51868,51870,51872,51874,51876,51878,51880],{"class":10276,"line":10951},[40,51859,13563],{"class":10363},[40,51861,13566],{"class":10430},[40,51863,13569],{"class":10363},[40,51865,13572],{"class":10430},[40,51867,13575],{"class":10363},[40,51869,13578],{"class":11903},[40,51871,162],{"class":10732},[40,51873,13583],{"class":10430},[40,51875,167],{"class":10363},[40,51877,13588],{"class":11903},[40,51879,162],{"class":10732},[40,51881,13593],{"class":10363},[40,51883,51884,51886,51888,51890,51892,51894,51896],{"class":10276,"line":10956},[40,51885,13598],{"class":11903},[40,51887,12577],{"class":10732},[40,51889,167],{"class":10363},[40,51891,13605],{"class":11903},[40,51893,162],{"class":10732},[40,51895,12597],{"class":10287},[40,51897,13612],{"class":10363},[40,51899,51900,51902,51904],{"class":10276,"line":10962},[40,51901,13617],{"class":10363},[40,51903,13620],{"class":10430},[40,51905,13623],{"class":10363},[40,51907,51908,51910],{"class":10276,"line":10983},[40,51909,11201],{"class":10352},[40,51911,13630],{"class":10363},[40,51913,51914],{"class":10276,"line":11005},[40,51915,10301],{"emptyLinePlaceholder":10300},[40,51917,51918],{"class":10276,"line":11011},[40,51919,10301],{"emptyLinePlaceholder":10300},[40,51921,51922],{"class":10276,"line":11022},[40,51923,13643],{"class":10280},[40,51925,51926,51928,51930,51932,51934],{"class":10276,"line":11064},[40,51927,13648],{"class":10363},[40,51929,162],{"class":10732},[40,51931,10636],{"class":10430},[40,51933,13655],{"class":10363},[40,51935,13658],{"class":10280},[40,51937,51938,51940,51942,51944,51946,51948,51950,51952,51954],{"class":10276,"line":11103},[40,51939,13663],{"class":11903},[40,51941,162],{"class":10732},[40,51943,167],{"class":10363},[40,51945,10748],{"class":10287},[40,51947,10484],{"class":10363},[40,51949,10753],{"class":10287},[40,51951,10484],{"class":10363},[40,51953,10758],{"class":10287},[40,51955,11061],{"class":10363},[40,51957,51958,51960,51962,51964],{"class":10276,"line":11142},[40,51959,13684],{"class":11903},[40,51961,162],{"class":10732},[40,51963,13689],{"class":10287},[40,51965,10507],{"class":10363},[40,51967,51968,51970,51972,51974,51976],{"class":10276,"line":11148},[40,51969,13696],{"class":11903},[40,51971,162],{"class":10732},[40,51973,13701],{"class":10446},[40,51975,13704],{"class":10363},[40,51977,13707],{"class":10280},[40,51979,51980,51982,51984,51986,51988,51990,51992,51994,51996],{"class":10276,"line":11181},[40,51981,13712],{"class":11903},[40,51983,162],{"class":10732},[40,51985,13717],{"class":10446},[40,51987,10484],{"class":10363},[40,51989,11825],{"class":11903},[40,51991,162],{"class":10732},[40,51993,13726],{"class":10446},[40,51995,13729],{"class":10363},[40,51997,13732],{"class":10280},[40,51999,52000,52002,52004,52006],{"class":10276,"line":11198},[40,52001,13737],{"class":11903},[40,52003,162],{"class":10732},[40,52005,13742],{"class":10446},[40,52007,10507],{"class":10363},[40,52009,52010,52012,52014,52016],{"class":10276,"line":11249},[40,52011,13749],{"class":11903},[40,52013,162],{"class":10732},[40,52015,13754],{"class":10287},[40,52017,10507],{"class":10363},[40,52019,52020,52022,52024,52026,52028,52030,52032],{"class":10276,"line":11254},[40,52021,13761],{"class":11903},[40,52023,162],{"class":10732},[40,52025,167],{"class":10363},[40,52027,10481],{"class":10446},[40,52029,10484],{"class":10363},[40,52031,10487],{"class":10446},[40,52033,11061],{"class":10363},[40,52035,52036,52038,52040,52042,52044,52046,52048,52050,52052,52054,52056,52058],{"class":10276,"line":11259},[40,52037,13778],{"class":11903},[40,52039,162],{"class":10732},[40,52041,13783],{"class":10287},[40,52043,10484],{"class":10363},[40,52045,13788],{"class":11903},[40,52047,162],{"class":10732},[40,52049,13793],{"class":10446},[40,52051,10484],{"class":10363},[40,52053,13798],{"class":11903},[40,52055,162],{"class":10732},[40,52057,13803],{"class":10446},[40,52059,10507],{"class":10363},[40,52061,52062],{"class":10276,"line":11265},[40,52063,10761],{"class":10363},[40,52065,52066],{"class":10276,"line":11294},[40,52067,10301],{"emptyLinePlaceholder":10300},[40,52069,52070,52072,52074,52076],{"class":10276,"line":11300},[40,52071,13818],{"class":10363},[40,52073,162],{"class":10732},[40,52075,10968],{"class":10430},[40,52077,13825],{"class":10363},[40,52079,52080,52082,52084],{"class":10276,"line":11320},[40,52081,13830],{"class":10363},[40,52083,162],{"class":10732},[40,52085,13835],{"class":10363},[40,52087,52088,52090,52092,52094],{"class":10276,"line":11336},[40,52089,13840],{"class":10352},[40,52091,13843],{"class":10363},[40,52093,13846],{"class":10352},[40,52095,13849],{"class":10363},[40,52097,52098,52100,52102,52104,52106,52108,52110,52112,52114,52116,52118],{"class":10276,"line":11381},[40,52099,13854],{"class":10363},[40,52101,162],{"class":10732},[40,52103,13859],{"class":10430},[40,52105,13862],{"class":10363},[40,52107,11243],{"class":10732},[40,52109,13867],{"class":10352},[40,52111,13870],{"class":10287},[40,52113,13268],{"class":10446},[40,52115,13875],{"class":10363},[40,52117,13280],{"class":10446},[40,52119,13880],{"class":10287},[40,52121,52122,52124,52126],{"class":10276,"line":11395},[40,52123,13885],{"class":10363},[40,52125,13888],{"class":10430},[40,52127,12527],{"class":10363},[40,52129,52130,52132],{"class":10276,"line":11438},[40,52131,13895],{"class":10430},[40,52133,13898],{"class":10363},[40,52135,52136],{"class":10276,"line":11452},[40,52137,13903],{"class":10363},[40,52139,52140],{"class":10276,"line":11493},[40,52141,13908],{"class":10363},[40,52143,52144],{"class":10276,"line":11507},[40,52145,13913],{"class":10363},[40,52147,52148],{"class":10276,"line":11512},[40,52149,13918],{"class":10363},[40,52151,52152],{"class":10276,"line":11517},[40,52153,12631],{"class":10363},[40,52155,52156],{"class":10276,"line":11545},[40,52157,10301],{"emptyLinePlaceholder":10300},[40,52159,52160],{"class":10276,"line":11557},[40,52161,13931],{"class":10280},[40,52163,52164,52166,52168,52170],{"class":10276,"line":11563},[40,52165,13840],{"class":10352},[40,52167,13938],{"class":10363},[40,52169,13846],{"class":10352},[40,52171,13943],{"class":10363},[40,52173,52174,52176,52178,52180,52182,52184,52186],{"class":10276,"line":11582},[40,52175,13948],{"class":10430},[40,52177,167],{"class":10363},[40,52179,13953],{"class":10430},[40,52181,167],{"class":10363},[40,52183,13958],{"class":11903},[40,52185,13961],{"class":10732},[40,52187,13964],{"class":10363},[40,52189,52190,52192,52194,52196],{"class":10276,"line":11596},[40,52191,13969],{"class":10732},[40,52193,167],{"class":10363},[40,52195,13974],{"class":10287},[40,52197,13977],{"class":10363},[10082,52199,52201],{"className":52200,"code":13981,"language":2932,"meta":10090},[10085],[10088,52202,13981],{"__ignoreMap":10090},[10082,52204,52206],{"className":52205,"code":13987,"language":2932,"meta":10090},[10085],[10088,52207,13987],{"__ignoreMap":10090},[10082,52209,52211],{"className":52210,"code":13993,"language":2932,"meta":10090},[10085],[10088,52212,13993],{"__ignoreMap":10090},[13997,52214],{"alt":13999,"src":14000},[13997,52216],{"alt":14003,"src":14004},[13997,52218],{"alt":14007,"src":14008},[10082,52220,52221],{"className":10269,"code":14011,"language":10271,"meta":10090,"style":10090},[10088,52222,52223,52227,52237,52241,52265,52275,52281,52291,52301,52311,52321,52335,52345,52355,52365,52371,52375,52389,52399,52425,52451,52473,52477,52503,52533,52555,52589,52601,52605,52617,52629,52639,52643,52669,52677,52681,52697,52723,52733,52745,52757,52765,52817,52847,52851,52857,52861,52893,52917,52943,52959,52967,52973,52977,52981,52985,52999,53005,53025,53035,53043,53055,53073,53083,53091,53105,53113,53121,53125,53129,53145],{"__ignoreMap":10090},[40,52224,52225],{"class":10276,"line":10277},[40,52226,14018],{"class":10280},[40,52228,52229,52231,52233,52235],{"class":10276,"line":10284},[40,52230,10353],{"class":10352},[40,52232,10542],{"class":10363},[40,52234,10375],{"class":10352},[40,52236,10547],{"class":10363},[40,52238,52239],{"class":10276,"line":10291},[40,52240,10301],{"emptyLinePlaceholder":10300},[40,52242,52243,52245,52247,52249,52251,52253,52255,52257,52259,52261,52263],{"class":10276,"line":10297},[40,52244,10965],{"class":10352},[40,52246,14039],{"class":10430},[40,52248,167],{"class":10363},[40,52250,11527],{"class":10973},[40,52252,10989],{"class":10363},[40,52254,10784],{"class":10732},[40,52256,10484],{"class":10363},[40,52258,10784],{"class":10732},[40,52260,10484],{"class":10363},[40,52262,10784],{"class":10732},[40,52264,11542],{"class":10363},[40,52266,52267,52269,52271,52273],{"class":10276,"line":10304},[40,52268,14062],{"class":10973},[40,52270,10443],{"class":10363},[40,52272,10784],{"class":10732},[40,52274,10507],{"class":10363},[40,52276,52277,52279],{"class":10276,"line":10310},[40,52278,14073],{"class":10973},[40,52280,12841],{"class":10363},[40,52282,52283,52285,52287,52289],{"class":10276,"line":10316},[40,52284,14080],{"class":10973},[40,52286,10443],{"class":10363},[40,52288,10821],{"class":10732},[40,52290,10507],{"class":10363},[40,52292,52293,52295,52297,52299],{"class":10276,"line":10322},[40,52294,14091],{"class":10973},[40,52296,10443],{"class":10363},[40,52298,10821],{"class":10732},[40,52300,10507],{"class":10363},[40,52302,52303,52305,52307,52309],{"class":10276,"line":10327},[40,52304,14102],{"class":10973},[40,52306,10443],{"class":10363},[40,52308,10821],{"class":10732},[40,52310,10507],{"class":10363},[40,52312,52313,52315,52317,52319],{"class":10276,"line":10333},[40,52314,14113],{"class":10973},[40,52316,10443],{"class":10363},[40,52318,10821],{"class":10732},[40,52320,10507],{"class":10363},[40,52322,52323,52325,52327,52329,52331,52333],{"class":10276,"line":10339},[40,52324,14124],{"class":10973},[40,52326,10989],{"class":10363},[40,52328,10784],{"class":10732},[40,52330,10484],{"class":10363},[40,52332,10784],{"class":10732},[40,52334,11542],{"class":10363},[40,52336,52337,52339,52341,52343],{"class":10276,"line":10344},[40,52338,14139],{"class":10973},[40,52340,10443],{"class":10363},[40,52342,10733],{"class":10732},[40,52344,10507],{"class":10363},[40,52346,52347,52349,52351,52353],{"class":10276,"line":10349},[40,52348,14150],{"class":10973},[40,52350,10443],{"class":10363},[40,52352,10821],{"class":10732},[40,52354,10507],{"class":10363},[40,52356,52357,52359,52361,52363],{"class":10276,"line":10367},[40,52358,14161],{"class":10973},[40,52360,10443],{"class":10363},[40,52362,10821],{"class":10732},[40,52364,10507],{"class":10363},[40,52366,52367,52369],{"class":10276,"line":10372},[40,52368,14172],{"class":10973},[40,52370,12941],{"class":10363},[40,52372,52373],{"class":10276,"line":10387},[40,52374,14179],{"class":10287},[40,52376,52377,52379,52381,52383,52385,52387],{"class":10276,"line":10400},[40,52378,12951],{"class":10352},[40,52380,12954],{"class":10363},[40,52382,12957],{"class":10732},[40,52384,12960],{"class":10446},[40,52386,10484],{"class":10363},[40,52388,12965],{"class":10287},[40,52390,52391,52393,52395,52397],{"class":10276,"line":10413},[40,52392,12970],{"class":10363},[40,52394,162],{"class":10732},[40,52396,11811],{"class":10430},[40,52398,12977],{"class":10363},[40,52400,52401,52403,52405,52407,52409,52411,52413,52415,52417,52419,52421,52423],{"class":10276,"line":10418},[40,52402,12982],{"class":10363},[40,52404,162],{"class":10732},[40,52406,12039],{"class":10446},[40,52408,11764],{"class":10732},[40,52410,12045],{"class":10363},[40,52412,2218],{"class":10430},[40,52414,167],{"class":10363},[40,52416,12052],{"class":10446},[40,52418,12055],{"class":10363},[40,52420,11243],{"class":10732},[40,52422,12060],{"class":10446},[40,52424,10761],{"class":10363},[40,52426,52427,52429,52431,52433,52435,52437,52439,52441,52443,52445,52447,52449],{"class":10276,"line":10424},[40,52428,13009],{"class":10363},[40,52430,162],{"class":10732},[40,52432,12039],{"class":10446},[40,52434,12042],{"class":10732},[40,52436,12045],{"class":10363},[40,52438,2218],{"class":10430},[40,52440,167],{"class":10363},[40,52442,12052],{"class":10446},[40,52444,12055],{"class":10363},[40,52446,11243],{"class":10732},[40,52448,12060],{"class":10446},[40,52450,10761],{"class":10363},[40,52452,52453,52455,52457,52459,52461,52463,52465,52467,52469,52471],{"class":10276,"line":10437},[40,52454,13036],{"class":10363},[40,52456,162],{"class":10732},[40,52458,920],{"class":10363},[40,52460,10801],{"class":10446},[40,52462,10484],{"class":10363},[40,52464,11878],{"class":10446},[40,52466,11881],{"class":10732},[40,52468,13051],{"class":10363},[40,52470,10801],{"class":10446},[40,52472,13056],{"class":10363},[40,52474,52475],{"class":10276,"line":10456},[40,52476,10301],{"emptyLinePlaceholder":10300},[40,52478,52479,52481,52483,52485,52487,52489,52491,52493,52495,52497,52499,52501],{"class":10276,"line":10472},[40,52480,13065],{"class":10363},[40,52482,162],{"class":10732},[40,52484,13070],{"class":10363},[40,52486,13073],{"class":10430},[40,52488,167],{"class":10363},[40,52490,13078],{"class":11903},[40,52492,162],{"class":10732},[40,52494,13083],{"class":10363},[40,52496,13086],{"class":11903},[40,52498,162],{"class":10732},[40,52500,13091],{"class":10446},[40,52502,10761],{"class":10363},[40,52504,52505,52507,52509,52511,52513,52515,52517,52519,52521,52523,52525,52527,52529,52531],{"class":10276,"line":10496},[40,52506,13098],{"class":10363},[40,52508,162],{"class":10732},[40,52510,12522],{"class":10363},[40,52512,13105],{"class":10430},[40,52514,13108],{"class":10363},[40,52516,13111],{"class":10430},[40,52518,13114],{"class":10363},[40,52520,13117],{"class":11903},[40,52522,162],{"class":10732},[40,52524,13122],{"class":10287},[40,52526,10484],{"class":10363},[40,52528,13127],{"class":11903},[40,52530,162],{"class":10732},[40,52532,13132],{"class":10363},[40,52534,52535,52537,52539,52541,52543,52545,52547,52549,52551,52553],{"class":10276,"line":10510},[40,52536,13137],{"class":11903},[40,52538,162],{"class":10732},[40,52540,13142],{"class":10363},[40,52542,13145],{"class":11903},[40,52544,162],{"class":10732},[40,52546,13150],{"class":10363},[40,52548,13153],{"class":11903},[40,52550,162],{"class":10732},[40,52552,13158],{"class":10287},[40,52554,10761],{"class":10363},[40,52556,52557,52559,52561,52563,52565,52567,52569,52571,52573,52575,52577,52579,52581,52583,52585,52587],{"class":10276,"line":10696},[40,52558,13165],{"class":10363},[40,52560,162],{"class":10732},[40,52562,13170],{"class":10363},[40,52564,13173],{"class":10430},[40,52566,13176],{"class":10363},[40,52568,12429],{"class":11903},[40,52570,162],{"class":10732},[40,52572,13183],{"class":10363},[40,52574,13186],{"class":11903},[40,52576,162],{"class":10732},[40,52578,13191],{"class":10446},[40,52580,10484],{"class":10363},[40,52582,13196],{"class":11903},[40,52584,162],{"class":10732},[40,52586,13201],{"class":10446},[40,52588,10761],{"class":10363},[40,52590,52591,52593,52595,52597,52599],{"class":10276,"line":10702},[40,52592,13208],{"class":10363},[40,52594,13211],{"class":10430},[40,52596,167],{"class":10363},[40,52598,13216],{"class":10287},[40,52600,10761],{"class":10363},[40,52602,52603],{"class":10276,"line":10708},[40,52604,10301],{"emptyLinePlaceholder":10300},[40,52606,52607,52609,52611,52613,52615],{"class":10276,"line":10714},[40,52608,13227],{"class":10363},[40,52610,13230],{"class":10430},[40,52612,167],{"class":10363},[40,52614,13235],{"class":10287},[40,52616,10761],{"class":10363},[40,52618,52619,52621,52623,52625,52627],{"class":10276,"line":10720},[40,52620,13227],{"class":10363},[40,52622,13244],{"class":10430},[40,52624,167],{"class":10363},[40,52626,13249],{"class":10287},[40,52628,10761],{"class":10363},[40,52630,52631,52633,52635,52637],{"class":10276,"line":10726},[40,52632,13256],{"class":10430},[40,52634,13259],{"class":10363},[40,52636,14442],{"class":10287},[40,52638,10761],{"class":10363},[40,52640,52641],{"class":10276,"line":10764},[40,52642,10301],{"emptyLinePlaceholder":10300},[40,52644,52645,52647,52649,52651,52653,52655,52657,52659,52661,52663,52665,52667],{"class":10276,"line":10778},[40,52646,13294],{"class":10363},[40,52648,162],{"class":10732},[40,52650,12424],{"class":10430},[40,52652,13259],{"class":10363},[40,52654,13303],{"class":10287},[40,52656,10484],{"class":10363},[40,52658,12131],{"class":11903},[40,52660,162],{"class":10732},[40,52662,13312],{"class":10363},[40,52664,12553],{"class":11903},[40,52666,162],{"class":10732},[40,52668,13319],{"class":10363},[40,52670,52671,52673,52675],{"class":10276,"line":10815},[40,52672,13324],{"class":11903},[40,52674,162],{"class":10732},[40,52676,13329],{"class":10363},[40,52678,52679],{"class":10276,"line":10829},[40,52680,10301],{"emptyLinePlaceholder":10300},[40,52682,52683,52685,52687,52689,52691,52693,52695],{"class":10276,"line":10842},[40,52684,13338],{"class":10352},[40,52686,13341],{"class":10430},[40,52688,167],{"class":10363},[40,52690,13346],{"class":10973},[40,52692,10443],{"class":10363},[40,52694,10821],{"class":10732},[40,52696,13353],{"class":10363},[40,52698,52699,52701,52703,52705,52707,52709,52711,52713,52715,52717,52719,52721],{"class":10276,"line":10855},[40,52700,13358],{"class":10363},[40,52702,162],{"class":10732},[40,52704,13363],{"class":10363},[40,52706,11243],{"class":10732},[40,52708,13368],{"class":10363},[40,52710,11056],{"class":10732},[40,52712,13373],{"class":10446},[40,52714,13376],{"class":10363},[40,52716,11752],{"class":10732},[40,52718,11156],{"class":10363},[40,52720,14527],{"class":10430},[40,52722,14530],{"class":10363},[40,52724,52725,52727,52729,52731],{"class":10276,"line":10868},[40,52726,14535],{"class":10363},[40,52728,162],{"class":10732},[40,52730,11522],{"class":10430},[40,52732,14542],{"class":10363},[40,52734,52735,52737,52739,52741,52743],{"class":10276,"line":10881},[40,52736,13389],{"class":10363},[40,52738,162],{"class":10732},[40,52740,14551],{"class":10363},[40,52742,13404],{"class":10732},[40,52744,13407],{"class":10363},[40,52746,52747,52749,52751,52753,52755],{"class":10276,"line":10907},[40,52748,13412],{"class":10363},[40,52750,13415],{"class":10430},[40,52752,167],{"class":10363},[40,52754,13420],{"class":10430},[40,52756,13423],{"class":10363},[40,52758,52759,52761,52763],{"class":10276,"line":10920},[40,52760,13428],{"class":10363},[40,52762,13431],{"class":10430},[40,52764,12527],{"class":10363},[40,52766,52767,52769,52771,52773,52775,52777,52779,52781,52783,52785,52787,52789,52791,52793,52795,52797,52799,52801,52803,52805,52807,52809,52811,52813,52815],{"class":10276,"line":10933},[40,52768,14580],{"class":10352},[40,52770,14583],{"class":10287},[40,52772,13268],{"class":10446},[40,52774,14588],{"class":10363},[40,52776,66],{"class":10446},[40,52778,1664],{"class":10363},[40,52780,14595],{"class":10352},[40,52782,13280],{"class":10446},[40,52784,10484],{"class":10287},[40,52786,13268],{"class":10446},[40,52788,14588],{"class":10363},[40,52790,78],{"class":10446},[40,52792,1664],{"class":10363},[40,52794,14595],{"class":10352},[40,52796,13280],{"class":10446},[40,52798,10484],{"class":10287},[40,52800,13268],{"class":10446},[40,52802,14588],{"class":10363},[40,52804,1044],{"class":10446},[40,52806,1664],{"class":10363},[40,52808,14595],{"class":10352},[40,52810,13280],{"class":10446},[40,52812,207],{"class":10287},[40,52814,13454],{"class":10732},[40,52816,14632],{"class":10287},[40,52818,52819,52821,52823,52825,52827,52829,52831,52833,52835,52837,52839,52841,52843,52845],{"class":10276,"line":10946},[40,52820,14580],{"class":10352},[40,52822,13438],{"class":10287},[40,52824,13268],{"class":10446},[40,52826,14643],{"class":10363},[40,52828,14646],{"class":10430},[40,52830,14649],{"class":10363},[40,52832,14652],{"class":10352},[40,52834,13280],{"class":10446},[40,52836,14657],{"class":10287},[40,52838,13268],{"class":10446},[40,52840,14662],{"class":10363},[40,52842,14652],{"class":10352},[40,52844,13280],{"class":10446},[40,52846,14669],{"class":10287},[40,52848,52849],{"class":10276,"line":10951},[40,52850,14674],{"class":10363},[40,52852,52853,52855],{"class":10276,"line":10956},[40,52854,11339],{"class":10352},[40,52856,13487],{"class":10363},[40,52858,52859],{"class":10276,"line":10962},[40,52860,10301],{"emptyLinePlaceholder":10300},[40,52862,52863,52865,52867,52869,52871,52873,52875,52877,52879,52881,52883,52885,52887,52889,52891],{"class":10276,"line":10983},[40,52864,13496],{"class":10363},[40,52866,162],{"class":10732},[40,52868,13501],{"class":10430},[40,52870,13504],{"class":10363},[40,52872,13473],{"class":11903},[40,52874,162],{"class":10732},[40,52876,13511],{"class":10363},[40,52878,13514],{"class":11903},[40,52880,162],{"class":10732},[40,52882,13519],{"class":10446},[40,52884,10484],{"class":10363},[40,52886,13524],{"class":11903},[40,52888,162],{"class":10732},[40,52890,10504],{"class":10446},[40,52892,10761],{"class":10363},[40,52894,52895,52897,52899,52901,52903,52905,52907,52909,52911,52913,52915],{"class":10276,"line":11005},[40,52896,13535],{"class":10363},[40,52898,13538],{"class":10430},[40,52900,167],{"class":10363},[40,52902,13543],{"class":11903},[40,52904,162],{"class":10732},[40,52906,13091],{"class":10446},[40,52908,10484],{"class":10363},[40,52910,13552],{"class":11903},[40,52912,162],{"class":10732},[40,52914,13091],{"class":10446},[40,52916,10761],{"class":10363},[40,52918,52919,52921,52923,52925,52927,52929,52931,52933,52935,52937,52939,52941],{"class":10276,"line":11011},[40,52920,13563],{"class":10363},[40,52922,13566],{"class":10430},[40,52924,13569],{"class":10363},[40,52926,13572],{"class":10430},[40,52928,13575],{"class":10363},[40,52930,13578],{"class":11903},[40,52932,162],{"class":10732},[40,52934,13583],{"class":10430},[40,52936,167],{"class":10363},[40,52938,13588],{"class":11903},[40,52940,162],{"class":10732},[40,52942,13593],{"class":10363},[40,52944,52945,52947,52949,52951,52953,52955,52957],{"class":10276,"line":11022},[40,52946,13598],{"class":11903},[40,52948,12577],{"class":10732},[40,52950,167],{"class":10363},[40,52952,13605],{"class":11903},[40,52954,162],{"class":10732},[40,52956,12597],{"class":10287},[40,52958,13612],{"class":10363},[40,52960,52961,52963,52965],{"class":10276,"line":11064},[40,52962,13617],{"class":10363},[40,52964,13620],{"class":10430},[40,52966,13623],{"class":10363},[40,52968,52969,52971],{"class":10276,"line":11103},[40,52970,11201],{"class":10352},[40,52972,13630],{"class":10363},[40,52974,52975],{"class":10276,"line":11142},[40,52976,10301],{"emptyLinePlaceholder":10300},[40,52978,52979],{"class":10276,"line":11148},[40,52980,10301],{"emptyLinePlaceholder":10300},[40,52982,52983],{"class":10276,"line":11181},[40,52984,14809],{"class":10280},[40,52986,52987,52989,52991,52993,52995,52997],{"class":10276,"line":11198},[40,52988,14814],{"class":10363},[40,52990,162],{"class":10732},[40,52992,13859],{"class":10430},[40,52994,13862],{"class":10363},[40,52996,11243],{"class":10732},[40,52998,14825],{"class":10287},[40,53000,53001,53003],{"class":10276,"line":11249},[40,53002,14830],{"class":10430},[40,53004,12527],{"class":10363},[40,53006,53007,53009,53011,53013,53015,53017,53019,53021,53023],{"class":10276,"line":11254},[40,53008,14837],{"class":11903},[40,53010,162],{"class":10732},[40,53012,167],{"class":10363},[40,53014,10801],{"class":10446},[40,53016,10484],{"class":10363},[40,53018,10810],{"class":10446},[40,53020,10484],{"class":10363},[40,53022,10801],{"class":10446},[40,53024,11061],{"class":10363},[40,53026,53027,53029,53031,53033],{"class":10276,"line":11259},[40,53028,14858],{"class":11903},[40,53030,162],{"class":10732},[40,53032,14863],{"class":10446},[40,53034,10507],{"class":10363},[40,53036,53037,53039,53041],{"class":10276,"line":11265},[40,53038,14870],{"class":11903},[40,53040,162],{"class":10732},[40,53042,14875],{"class":10363},[40,53044,53045,53047,53049,53051,53053],{"class":10276,"line":11294},[40,53046,13696],{"class":11903},[40,53048,162],{"class":10732},[40,53050,13701],{"class":10446},[40,53052,14886],{"class":10363},[40,53054,14889],{"class":10280},[40,53056,53057,53059,53061,53063,53065,53067,53069,53071],{"class":10276,"line":11300},[40,53058,13712],{"class":11903},[40,53060,162],{"class":10732},[40,53062,13717],{"class":10446},[40,53064,10484],{"class":10363},[40,53066,11825],{"class":11903},[40,53068,162],{"class":10732},[40,53070,13726],{"class":10446},[40,53072,10507],{"class":10363},[40,53074,53075,53077,53079,53081],{"class":10276,"line":11320},[40,53076,13737],{"class":11903},[40,53078,162],{"class":10732},[40,53080,13742],{"class":10446},[40,53082,10507],{"class":10363},[40,53084,53085,53087,53089],{"class":10276,"line":11336},[40,53086,13761],{"class":11903},[40,53088,162],{"class":10732},[40,53090,14926],{"class":10363},[40,53092,53093,53095,53097,53099,53101,53103],{"class":10276,"line":11381},[40,53094,13778],{"class":11903},[40,53096,162],{"class":10732},[40,53098,14935],{"class":10363},[40,53100,13788],{"class":11903},[40,53102,162],{"class":10732},[40,53104,14942],{"class":10363},[40,53106,53107,53109,53111],{"class":10276,"line":11395},[40,53108,14947],{"class":11903},[40,53110,162],{"class":10732},[40,53112,14952],{"class":10363},[40,53114,53115,53117,53119],{"class":10276,"line":11438},[40,53116,14957],{"class":11903},[40,53118,162],{"class":10732},[40,53120,14962],{"class":10363},[40,53122,53123],{"class":10276,"line":11452},[40,53124,10761],{"class":10363},[40,53126,53127],{"class":10276,"line":11493},[40,53128,10301],{"emptyLinePlaceholder":10300},[40,53130,53131,53133,53135,53137,53139,53141,53143],{"class":10276,"line":11507},[40,53132,14975],{"class":10430},[40,53134,167],{"class":10363},[40,53136,13953],{"class":10430},[40,53138,167],{"class":10363},[40,53140,13958],{"class":11903},[40,53142,13961],{"class":10732},[40,53144,14988],{"class":10363},[40,53146,53147,53149,53151,53153],{"class":10276,"line":11512},[40,53148,14993],{"class":10732},[40,53150,167],{"class":10363},[40,53152,13974],{"class":10287},[40,53154,15000],{"class":10363},[10082,53156,53158],{"className":53157,"code":15004,"language":2932,"meta":10090},[10085],[10088,53159,15004],{"__ignoreMap":10090},[13997,53161],{"alt":15009,"src":15010},[12,53163,15013],{},[10082,53165,53166],{"className":10269,"code":15016,"language":10271,"meta":10090,"style":10090},[10088,53167,53168,53172,53176,53180,53190,53194,53204,53214,53224,53228,53238,53248,53258,53268,53278,53288,53292,53296,53300,53308,53312,53316,53320,53324,53328,53332,53336,53340,53344,53348,53352,53356,53360,53364,53368,53378,53410,53420,53430,53440,53450,53460,53472,53482,53506,53516,53520,53524,53528,53560,53568,53604,53640,53676,53680,53704,53720,53730,53734,53738,53762,53778,53790,53828,53840,53878,53890,53928,53938,53942,53946,53970,53974,53990,54012,54032,54052,54072,54084,54104,54110,54114,54118,54134,54138,54198,54202,54206,54226,54230,54240,54250,54254,54258,54278,54298,54310,54314,54318,54344,54370,54392,54404,54426,54436,54440,54466,54490,54510,54522,54542,54546,54550,54572,54600,54604,54616,54620,54630,54648,54666,54678,54682,54686,54690,54724,54736,54740,54744,54756,54768,54780,54792,54820,54852,54892,54902,54906,54918,54954,54962,55002,55012,55016,55020,55036,55060,55074,55084,55112,55140,55186,55192,55196,55228,55252,55276,55302,55318,55326,55332,55336,55340,55352,55382,55386,55396,55410,55426,55446,55452,55456,55460,55464,55476,55486,55520,55538,55548,55564,55574,55578,55582],{"__ignoreMap":10090},[40,53169,53170],{"class":10276,"line":10277},[40,53171,15023],{"class":10280},[40,53173,53174],{"class":10276,"line":10284},[40,53175,15028],{"class":10280},[40,53177,53178],{"class":10276,"line":10291},[40,53179,10301],{"emptyLinePlaceholder":10300},[40,53181,53182,53184,53186,53188],{"class":10276,"line":10297},[40,53183,10353],{"class":10352},[40,53185,10357],{"class":10356},[40,53187,10360],{"class":10352},[40,53189,10364],{"class":10363},[40,53191,53192],{"class":10276,"line":10304},[40,53193,10301],{"emptyLinePlaceholder":10300},[40,53195,53196,53198,53200,53202],{"class":10276,"line":10310},[40,53197,10353],{"class":10352},[40,53199,10530],{"class":10363},[40,53201,10375],{"class":10352},[40,53203,10535],{"class":10363},[40,53205,53206,53208,53210,53212],{"class":10276,"line":10316},[40,53207,10353],{"class":10352},[40,53209,10542],{"class":10363},[40,53211,10375],{"class":10352},[40,53213,10547],{"class":10363},[40,53215,53216,53218,53220,53222],{"class":10276,"line":10322},[40,53217,10353],{"class":10352},[40,53219,10554],{"class":10363},[40,53221,10375],{"class":10352},[40,53223,10559],{"class":10363},[40,53225,53226],{"class":10276,"line":10327},[40,53227,10301],{"emptyLinePlaceholder":10300},[40,53229,53230,53232,53234,53236],{"class":10276,"line":10333},[40,53231,10375],{"class":10352},[40,53233,10378],{"class":10363},[40,53235,10381],{"class":10352},[40,53237,10384],{"class":10363},[40,53239,53240,53242,53244,53246],{"class":10276,"line":10339},[40,53241,10375],{"class":10352},[40,53243,10405],{"class":10363},[40,53245,10381],{"class":10352},[40,53247,10410],{"class":10363},[40,53249,53250,53252,53254,53256],{"class":10276,"line":10344},[40,53251,10353],{"class":10352},[40,53253,10597],{"class":10363},[40,53255,10375],{"class":10352},[40,53257,10602],{"class":10363},[40,53259,53260,53262,53264,53266],{"class":10276,"line":10349},[40,53261,10353],{"class":10352},[40,53263,15117],{"class":10363},[40,53265,10375],{"class":10352},[40,53267,15122],{"class":10363},[40,53269,53270,53272,53274,53276],{"class":10276,"line":10367},[40,53271,10353],{"class":10352},[40,53273,15129],{"class":10363},[40,53275,10375],{"class":10352},[40,53277,15134],{"class":10363},[40,53279,53280,53282,53284,53286],{"class":10276,"line":10372},[40,53281,10353],{"class":10352},[40,53283,10609],{"class":10363},[40,53285,10375],{"class":10352},[40,53287,10614],{"class":10363},[40,53289,53290],{"class":10276,"line":10387},[40,53291,10301],{"emptyLinePlaceholder":10300},[40,53293,53294],{"class":10276,"line":10400},[40,53295,10301],{"emptyLinePlaceholder":10300},[40,53297,53298],{"class":10276,"line":10413},[40,53299,10627],{"class":10430},[40,53301,53302,53304,53306],{"class":10276,"line":10418},[40,53303,10632],{"class":10352},[40,53305,15163],{"class":10635},[40,53307,10639],{"class":10363},[40,53309,53310],{"class":10276,"line":10424},[40,53311,15170],{"class":10287},[40,53313,53314],{"class":10276,"line":10437},[40,53315,10301],{"emptyLinePlaceholder":10300},[40,53317,53318],{"class":10276,"line":10456},[40,53319,10653],{"class":10287},[40,53321,53322],{"class":10276,"line":10472},[40,53323,15183],{"class":10287},[40,53325,53326],{"class":10276,"line":10496},[40,53327,10668],{"class":10287},[40,53329,53330],{"class":10276,"line":10510},[40,53331,15192],{"class":10287},[40,53333,53334],{"class":10276,"line":10696},[40,53335,15197],{"class":10287},[40,53337,53338],{"class":10276,"line":10702},[40,53339,15202],{"class":10287},[40,53341,53342],{"class":10276,"line":10708},[40,53343,10688],{"class":10287},[40,53345,53346],{"class":10276,"line":10714},[40,53347,15211],{"class":10287},[40,53349,53350],{"class":10276,"line":10720},[40,53351,15216],{"class":10287},[40,53353,53354],{"class":10276,"line":10726},[40,53355,15221],{"class":10287},[40,53357,53358],{"class":10276,"line":10764},[40,53359,15226],{"class":10287},[40,53361,53362],{"class":10276,"line":10778},[40,53363,15231],{"class":10287},[40,53365,53366],{"class":10276,"line":10815},[40,53367,10723],{"class":10287},[40,53369,53370,53372,53374,53376],{"class":10276,"line":10829},[40,53371,10767],{"class":10363},[40,53373,10733],{"class":10732},[40,53375,10772],{"class":10732},[40,53377,10775],{"class":10287},[40,53379,53380,53382,53384,53386,53388,53390,53392,53394,53396,53398,53400,53402,53404,53406,53408],{"class":10276,"line":10842},[40,53381,10781],{"class":10363},[40,53383,10784],{"class":10732},[40,53385,10484],{"class":10363},[40,53387,10784],{"class":10732},[40,53389,10484],{"class":10363},[40,53391,10784],{"class":10732},[40,53393,10741],{"class":10363},[40,53395,162],{"class":10732},[40,53397,920],{"class":10363},[40,53399,10801],{"class":10446},[40,53401,10484],{"class":10363},[40,53403,10801],{"class":10446},[40,53405,10484],{"class":10363},[40,53407,10810],{"class":10446},[40,53409,10761],{"class":10363},[40,53411,53412,53414,53416,53418],{"class":10276,"line":10855},[40,53413,10818],{"class":10363},[40,53415,10821],{"class":10732},[40,53417,10772],{"class":10732},[40,53419,10826],{"class":10446},[40,53421,53422,53424,53426,53428],{"class":10276,"line":10868},[40,53423,10832],{"class":10363},[40,53425,10821],{"class":10732},[40,53427,10772],{"class":10732},[40,53429,15298],{"class":10446},[40,53431,53432,53434,53436,53438],{"class":10276,"line":10881},[40,53433,10845],{"class":10363},[40,53435,10821],{"class":10732},[40,53437,10772],{"class":10732},[40,53439,15309],{"class":10446},[40,53441,53442,53444,53446,53448],{"class":10276,"line":10907},[40,53443,10858],{"class":10363},[40,53445,10821],{"class":10732},[40,53447,10772],{"class":10732},[40,53449,10865],{"class":10446},[40,53451,53452,53454,53456,53458],{"class":10276,"line":10920},[40,53453,15324],{"class":10363},[40,53455,10784],{"class":10732},[40,53457,10772],{"class":10732},[40,53459,15331],{"class":10446},[40,53461,53462,53464,53466,53468,53470],{"class":10276,"line":10933},[40,53463,15336],{"class":10363},[40,53465,10784],{"class":10732},[40,53467,10772],{"class":10732},[40,53469,11764],{"class":10732},[40,53471,15345],{"class":10446},[40,53473,53474,53476,53478,53480],{"class":10276,"line":10946},[40,53475,10871],{"class":10363},[40,53477,10733],{"class":10732},[40,53479,10772],{"class":10732},[40,53481,10878],{"class":10287},[40,53483,53484,53486,53488,53490,53492,53494,53496,53498,53500,53502,53504],{"class":10276,"line":10951},[40,53485,10884],{"class":10363},[40,53487,10784],{"class":10732},[40,53489,10484],{"class":10363},[40,53491,10784],{"class":10732},[40,53493,10741],{"class":10363},[40,53495,162],{"class":10732},[40,53497,920],{"class":10363},[40,53499,10481],{"class":10446},[40,53501,10484],{"class":10363},[40,53503,15378],{"class":10446},[40,53505,10761],{"class":10363},[40,53507,53508,53510,53512,53514],{"class":10276,"line":10956},[40,53509,10923],{"class":10363},[40,53511,10821],{"class":10732},[40,53513,10772],{"class":10732},[40,53515,10930],{"class":10446},[40,53517,53518],{"class":10276,"line":10962},[40,53519,10301],{"emptyLinePlaceholder":10300},[40,53521,53522],{"class":10276,"line":10983},[40,53523,10301],{"emptyLinePlaceholder":10300},[40,53525,53526],{"class":10276,"line":11005},[40,53527,15403],{"class":10280},[40,53529,53530,53532,53534,53536,53538,53540,53542,53544,53546,53548,53550,53552,53554,53556,53558],{"class":10276,"line":11011},[40,53531,10965],{"class":10352},[40,53533,15410],{"class":10430},[40,53535,167],{"class":10363},[40,53537,15415],{"class":10973},[40,53539,10443],{"class":10363},[40,53541,10733],{"class":10732},[40,53543,10484],{"class":10363},[40,53545,15424],{"class":10973},[40,53547,15427],{"class":10363},[40,53549,10801],{"class":10446},[40,53551,10484],{"class":10363},[40,53553,10801],{"class":10446},[40,53555,10484],{"class":10363},[40,53557,10810],{"class":10446},[40,53559,15440],{"class":10363},[40,53561,53562,53564,53566],{"class":10276,"line":11022},[40,53563,11014],{"class":10363},[40,53565,162],{"class":10732},[40,53567,11019],{"class":10363},[40,53569,53570,53572,53574,53576,53578,53580,53582,53584,53586,53588,53590,53592,53594,53596,53598,53600,53602],{"class":10276,"line":11064},[40,53571,11025],{"class":10287},[40,53573,10478],{"class":10363},[40,53575,10801],{"class":10446},[40,53577,10484],{"class":10363},[40,53579,10801],{"class":10446},[40,53581,10484],{"class":10363},[40,53583,10810],{"class":10446},[40,53585,11040],{"class":10363},[40,53587,11043],{"class":10287},[40,53589,10478],{"class":10363},[40,53591,10801],{"class":10446},[40,53593,10484],{"class":10363},[40,53595,10801],{"class":10446},[40,53597,10484],{"class":10363},[40,53599,11056],{"class":10732},[40,53601,10810],{"class":10446},[40,53603,11061],{"class":10363},[40,53605,53606,53608,53610,53612,53614,53616,53618,53620,53622,53624,53626,53628,53630,53632,53634,53636,53638],{"class":10276,"line":11103},[40,53607,11067],{"class":10287},[40,53609,10478],{"class":10363},[40,53611,10810],{"class":10446},[40,53613,10484],{"class":10363},[40,53615,10801],{"class":10446},[40,53617,10484],{"class":10363},[40,53619,10801],{"class":10446},[40,53621,11040],{"class":10363},[40,53623,11084],{"class":10287},[40,53625,10478],{"class":10363},[40,53627,11056],{"class":10732},[40,53629,10810],{"class":10446},[40,53631,10484],{"class":10363},[40,53633,10801],{"class":10446},[40,53635,10484],{"class":10363},[40,53637,10801],{"class":10446},[40,53639,11061],{"class":10363},[40,53641,53642,53644,53646,53648,53650,53652,53654,53656,53658,53660,53662,53664,53666,53668,53670,53672,53674],{"class":10276,"line":11142},[40,53643,11106],{"class":10287},[40,53645,10478],{"class":10363},[40,53647,10801],{"class":10446},[40,53649,10484],{"class":10363},[40,53651,10810],{"class":10446},[40,53653,10484],{"class":10363},[40,53655,10801],{"class":10446},[40,53657,11040],{"class":10363},[40,53659,11123],{"class":10287},[40,53661,10478],{"class":10363},[40,53663,10801],{"class":10446},[40,53665,10484],{"class":10363},[40,53667,11056],{"class":10732},[40,53669,10810],{"class":10446},[40,53671,10484],{"class":10363},[40,53673,10801],{"class":10446},[40,53675,11061],{"class":10363},[40,53677,53678],{"class":10276,"line":11148},[40,53679,11145],{"class":10363},[40,53681,53682,53684,53686,53688,53690,53692,53694,53696,53698,53700,53702],{"class":10276,"line":11181},[40,53683,11151],{"class":10363},[40,53685,162],{"class":10732},[40,53687,11156],{"class":10363},[40,53689,11159],{"class":10430},[40,53691,11162],{"class":10363},[40,53693,11165],{"class":10430},[40,53695,15577],{"class":10363},[40,53697,11171],{"class":10430},[40,53699,15582],{"class":10363},[40,53701,10784],{"class":10732},[40,53703,10761],{"class":10363},[40,53705,53706,53708,53710,53712,53714,53716,53718],{"class":10276,"line":11198},[40,53707,11184],{"class":10363},[40,53709,162],{"class":10732},[40,53711,11189],{"class":10363},[40,53713,11192],{"class":10430},[40,53715,15599],{"class":10363},[40,53717,15602],{"class":10352},[40,53719,15605],{"class":10446},[40,53721,53722,53724,53726,53728],{"class":10276,"line":11249},[40,53723,11201],{"class":10352},[40,53725,11240],{"class":10363},[40,53727,11243],{"class":10732},[40,53729,11246],{"class":10363},[40,53731,53732],{"class":10276,"line":11254},[40,53733,10301],{"emptyLinePlaceholder":10300},[40,53735,53736],{"class":10276,"line":11259},[40,53737,10301],{"emptyLinePlaceholder":10300},[40,53739,53740,53742,53744,53746,53748,53750,53752,53754,53756,53758,53760],{"class":10276,"line":11265},[40,53741,10965],{"class":10352},[40,53743,15630],{"class":10430},[40,53745,167],{"class":10363},[40,53747,11275],{"class":10973},[40,53749,10443],{"class":10363},[40,53751,10733],{"class":10732},[40,53753,10484],{"class":10363},[40,53755,11284],{"class":10973},[40,53757,10443],{"class":10363},[40,53759,10784],{"class":10732},[40,53761,11291],{"class":10363},[40,53763,53764,53766,53768,53770,53772,53774,53776],{"class":10276,"line":11294},[40,53765,11303],{"class":10363},[40,53767,162],{"class":10732},[40,53769,11156],{"class":10363},[40,53771,1820],{"class":10430},[40,53773,11312],{"class":10363},[40,53775,1813],{"class":10430},[40,53777,11317],{"class":10363},[40,53779,53780,53782,53784,53786,53788],{"class":10276,"line":11300},[40,53781,11323],{"class":10352},[40,53783,11326],{"class":10363},[40,53785,11231],{"class":10732},[40,53787,11331],{"class":10287},[40,53789,10639],{"class":10363},[40,53791,53792,53794,53796,53798,53800,53802,53804,53806,53808,53810,53812,53814,53816,53818,53820,53822,53824,53826],{"class":10276,"line":11320},[40,53793,11339],{"class":10352},[40,53795,11156],{"class":10363},[40,53797,11206],{"class":10430},[40,53799,11346],{"class":10363},[40,53801,78],{"class":10446},[40,53803,10484],{"class":10363},[40,53805,66],{"class":10446},[40,53807,10484],{"class":10363},[40,53809,66],{"class":10446},[40,53811,11359],{"class":10363},[40,53813,66],{"class":10446},[40,53815,11364],{"class":10363},[40,53817,11056],{"class":10732},[40,53819,11369],{"class":10363},[40,53821,66],{"class":10446},[40,53823,11374],{"class":10363},[40,53825,10784],{"class":10732},[40,53827,10761],{"class":10363},[40,53829,53830,53832,53834,53836,53838],{"class":10276,"line":11336},[40,53831,11323],{"class":10352},[40,53833,11326],{"class":10363},[40,53835,11231],{"class":10732},[40,53837,11390],{"class":10287},[40,53839,10639],{"class":10363},[40,53841,53842,53844,53846,53848,53850,53852,53854,53856,53858,53860,53862,53864,53866,53868,53870,53872,53874,53876],{"class":10276,"line":11381},[40,53843,11339],{"class":10352},[40,53845,11156],{"class":10363},[40,53847,11206],{"class":10430},[40,53849,11404],{"class":10363},[40,53851,66],{"class":10446},[40,53853,11409],{"class":10363},[40,53855,66],{"class":10446},[40,53857,10484],{"class":10363},[40,53859,78],{"class":10446},[40,53861,10484],{"class":10363},[40,53863,66],{"class":10446},[40,53865,11359],{"class":10363},[40,53867,11056],{"class":10732},[40,53869,11426],{"class":10363},[40,53871,66],{"class":10446},[40,53873,11431],{"class":10363},[40,53875,10784],{"class":10732},[40,53877,10761],{"class":10363},[40,53879,53880,53882,53884,53886,53888],{"class":10276,"line":11395},[40,53881,11323],{"class":10352},[40,53883,11326],{"class":10363},[40,53885,11231],{"class":10732},[40,53887,11447],{"class":10287},[40,53889,10639],{"class":10363},[40,53891,53892,53894,53896,53898,53900,53902,53904,53906,53908,53910,53912,53914,53916,53918,53920,53922,53924,53926],{"class":10276,"line":11438},[40,53893,11339],{"class":10352},[40,53895,11156],{"class":10363},[40,53897,11206],{"class":10430},[40,53899,11404],{"class":10363},[40,53901,11056],{"class":10732},[40,53903,11426],{"class":10363},[40,53905,66],{"class":10446},[40,53907,11469],{"class":10363},[40,53909,66],{"class":10446},[40,53911,11359],{"class":10363},[40,53913,66],{"class":10446},[40,53915,10484],{"class":10363},[40,53917,66],{"class":10446},[40,53919,10484],{"class":10363},[40,53921,78],{"class":10446},[40,53923,11486],{"class":10363},[40,53925,10784],{"class":10732},[40,53927,10761],{"class":10363},[40,53929,53930,53932,53934,53936],{"class":10276,"line":11452},[40,53931,11496],{"class":10352},[40,53933,11499],{"class":10363},[40,53935,15823],{"class":10287},[40,53937,10761],{"class":10363},[40,53939,53940],{"class":10276,"line":11493},[40,53941,10301],{"emptyLinePlaceholder":10300},[40,53943,53944],{"class":10276,"line":11507},[40,53945,10301],{"emptyLinePlaceholder":10300},[40,53947,53948,53950,53952,53954,53956,53958,53960,53962,53964,53966,53968],{"class":10276,"line":11512},[40,53949,10965],{"class":10352},[40,53951,15840],{"class":10430},[40,53953,167],{"class":10363},[40,53955,11816],{"class":10973},[40,53957,10443],{"class":10363},[40,53959,10821],{"class":10732},[40,53961,10484],{"class":10363},[40,53963,11825],{"class":10973},[40,53965,10443],{"class":10363},[40,53967,10821],{"class":10732},[40,53969,13353],{"class":10363},[40,53971,53972],{"class":10276,"line":11517},[40,53973,15863],{"class":10287},[40,53975,53976,53978,53980,53982,53984,53986,53988],{"class":10276,"line":11545},[40,53977,11843],{"class":10363},[40,53979,162],{"class":10732},[40,53981,11156],{"class":10363},[40,53983,11850],{"class":10430},[40,53985,167],{"class":10363},[40,53987,10801],{"class":10446},[40,53989,11857],{"class":10363},[40,53991,53992,53994,53996,53998,54000,54002,54004,54006,54008,54010],{"class":10276,"line":11557},[40,53993,11863],{"class":10363},[40,53995,162],{"class":10732},[40,53997,11156],{"class":10363},[40,53999,11850],{"class":10430},[40,54001,167],{"class":10363},[40,54003,10801],{"class":10446},[40,54005,10484],{"class":10363},[40,54007,11878],{"class":10446},[40,54009,11881],{"class":10732},[40,54011,11884],{"class":10363},[40,54013,54014,54016,54018,54020,54022,54024,54026,54028,54030],{"class":10276,"line":11563},[40,54015,11890],{"class":10363},[40,54017,162],{"class":10732},[40,54019,11156],{"class":10363},[40,54021,11897],{"class":10430},[40,54023,11900],{"class":10363},[40,54025,11904],{"class":11903},[40,54027,162],{"class":10732},[40,54029,11909],{"class":10287},[40,54031,10761],{"class":10363},[40,54033,54034,54036,54038,54040,54042,54044,54046,54048,54050],{"class":10276,"line":11582},[40,54035,11917],{"class":10363},[40,54037,162],{"class":10732},[40,54039,11156],{"class":10363},[40,54041,1813],{"class":10430},[40,54043,11926],{"class":10363},[40,54045,11752],{"class":10732},[40,54047,11156],{"class":10363},[40,54049,1820],{"class":10430},[40,54051,11935],{"class":10363},[40,54053,54054,54056,54058,54060,54062,54064,54066,54068,54070],{"class":10276,"line":11596},[40,54055,11941],{"class":10363},[40,54057,162],{"class":10732},[40,54059,11156],{"class":10363},[40,54061,1813],{"class":10430},[40,54063,11926],{"class":10363},[40,54065,11752],{"class":10732},[40,54067,11156],{"class":10363},[40,54069,1813],{"class":10430},[40,54071,11935],{"class":10363},[40,54073,54074,54076,54078,54080,54082],{"class":10276,"line":11609},[40,54075,11963],{"class":10363},[40,54077,162],{"class":10732},[40,54079,11156],{"class":10363},[40,54081,1820],{"class":10430},[40,54083,11972],{"class":10363},[40,54085,54086,54088,54090,54092,54094,54096,54098,54100,54102],{"class":10276,"line":11622},[40,54087,15978],{"class":10363},[40,54089,162],{"class":10732},[40,54091,11156],{"class":10363},[40,54093,11982],{"class":10430},[40,54095,11985],{"class":10363},[40,54097,11275],{"class":11903},[40,54099,162],{"class":10732},[40,54101,66],{"class":10446},[40,54103,10761],{"class":10363},[40,54105,54106,54108],{"class":10276,"line":11637},[40,54107,11201],{"class":10352},[40,54109,16001],{"class":10363},[40,54111,54112],{"class":10276,"line":11708},[40,54113,10301],{"emptyLinePlaceholder":10300},[40,54115,54116],{"class":10276,"line":11727},[40,54117,10301],{"emptyLinePlaceholder":10300},[40,54119,54120,54122,54124,54126,54128,54130,54132],{"class":10276,"line":11744},[40,54121,10965],{"class":10352},[40,54123,16016],{"class":10430},[40,54125,167],{"class":10363},[40,54127,12016],{"class":10973},[40,54129,12019],{"class":10363},[40,54131,1796],{"class":10973},[40,54133,12025],{"class":10363},[40,54135,54136],{"class":10276,"line":11790},[40,54137,16031],{"class":10287},[40,54139,54140,54142,54144,54146,54148,54150,54152,54154,54156,54158,54160,54162,54164,54166,54168,54170,54172,54174,54176,54178,54180,54182,54184,54186,54188,54190,54192,54194,54196],{"class":10276,"line":11795},[40,54141,11201],{"class":10352},[40,54143,12039],{"class":10446},[40,54145,12042],{"class":10732},[40,54147,12045],{"class":10363},[40,54149,2218],{"class":10430},[40,54151,167],{"class":10363},[40,54153,12052],{"class":10446},[40,54155,12055],{"class":10363},[40,54157,11243],{"class":10732},[40,54159,12060],{"class":10446},[40,54161,12055],{"class":10363},[40,54163,11752],{"class":10732},[40,54165,12067],{"class":10363},[40,54167,12070],{"class":10430},[40,54169,167],{"class":10363},[40,54171,949],{"class":10446},[40,54173,10484],{"class":10363},[40,54175,78],{"class":10446},[40,54177,10484],{"class":10363},[40,54179,78],{"class":10446},[40,54181,12055],{"class":10363},[40,54183,11752],{"class":10732},[40,54185,16080],{"class":10363},[40,54187,12092],{"class":10430},[40,54189,167],{"class":10363},[40,54191,11275],{"class":11903},[40,54193,162],{"class":10732},[40,54195,66],{"class":10446},[40,54197,10761],{"class":10363},[40,54199,54200],{"class":10276,"line":11800},[40,54201,10301],{"emptyLinePlaceholder":10300},[40,54203,54204],{"class":10276,"line":11806},[40,54205,10301],{"emptyLinePlaceholder":10300},[40,54207,54208,54210,54212,54214,54216,54218,54220,54222,54224],{"class":10276,"line":11834},[40,54209,10965],{"class":10352},[40,54211,16107],{"class":10430},[40,54213,167],{"class":10363},[40,54215,11275],{"class":10973},[40,54217,10443],{"class":10363},[40,54219,10733],{"class":10732},[40,54221,10484],{"class":10363},[40,54223,13346],{"class":10973},[40,54225,16122],{"class":10363},[40,54227,54228],{"class":10276,"line":11840},[40,54229,16127],{"class":10287},[40,54231,54232,54234,54236,54238],{"class":10276,"line":11860},[40,54233,16132],{"class":10363},[40,54235,162],{"class":10732},[40,54237,15410],{"class":10430},[40,54239,16139],{"class":10363},[40,54241,54242,54244,54246,54248],{"class":10276,"line":11887},[40,54243,16144],{"class":10363},[40,54245,162],{"class":10732},[40,54247,15840],{"class":10430},[40,54249,16151],{"class":10363},[40,54251,54252],{"class":10276,"line":11914},[40,54253,10301],{"emptyLinePlaceholder":10300},[40,54255,54256],{"class":10276,"line":11938},[40,54257,16160],{"class":10280},[40,54259,54260,54262,54264,54266,54268,54270,54272,54274,54276],{"class":10276,"line":11960},[40,54261,16165],{"class":10363},[40,54263,162],{"class":10732},[40,54265,11156],{"class":10363},[40,54267,1813],{"class":10430},[40,54269,11926],{"class":10363},[40,54271,11752],{"class":10732},[40,54273,11156],{"class":10363},[40,54275,1820],{"class":10430},[40,54277,11935],{"class":10363},[40,54279,54280,54282,54284,54286,54288,54290,54292,54294,54296],{"class":10276,"line":11975},[40,54281,16186],{"class":10363},[40,54283,162],{"class":10732},[40,54285,11156],{"class":10363},[40,54287,1813],{"class":10430},[40,54289,11926],{"class":10363},[40,54291,11752],{"class":10732},[40,54293,11156],{"class":10363},[40,54295,1813],{"class":10430},[40,54297,11935],{"class":10363},[40,54299,54300,54302,54304,54306,54308],{"class":10276,"line":11996},[40,54301,16207],{"class":10363},[40,54303,162],{"class":10732},[40,54305,11156],{"class":10363},[40,54307,1820],{"class":10430},[40,54309,11972],{"class":10363},[40,54311,54312],{"class":10276,"line":12001},[40,54313,10301],{"emptyLinePlaceholder":10300},[40,54315,54316],{"class":10276,"line":12006},[40,54317,16224],{"class":10280},[40,54319,54320,54322,54324,54326,54328,54330,54332,54334,54336,54338,54340,54342],{"class":10276,"line":12028},[40,54321,12982],{"class":10363},[40,54323,162],{"class":10732},[40,54325,12039],{"class":10446},[40,54327,11764],{"class":10732},[40,54329,12045],{"class":10363},[40,54331,2218],{"class":10430},[40,54333,167],{"class":10363},[40,54335,12052],{"class":10446},[40,54337,12055],{"class":10363},[40,54339,11243],{"class":10732},[40,54341,12060],{"class":10446},[40,54343,10761],{"class":10363},[40,54345,54346,54348,54350,54352,54354,54356,54358,54360,54362,54364,54366,54368],{"class":10276,"line":12034},[40,54347,13009],{"class":10363},[40,54349,162],{"class":10732},[40,54351,12039],{"class":10446},[40,54353,12042],{"class":10732},[40,54355,12045],{"class":10363},[40,54357,2218],{"class":10430},[40,54359,167],{"class":10363},[40,54361,12052],{"class":10446},[40,54363,12055],{"class":10363},[40,54365,11243],{"class":10732},[40,54367,12060],{"class":10446},[40,54369,10761],{"class":10363},[40,54371,54372,54374,54376,54378,54380,54382,54384,54386,54388,54390],{"class":10276,"line":12105},[40,54373,16281],{"class":10363},[40,54375,162],{"class":10732},[40,54377,16286],{"class":10430},[40,54379,167],{"class":10363},[40,54381,16291],{"class":11903},[40,54383,162],{"class":10732},[40,54385,13142],{"class":10363},[40,54387,13145],{"class":11903},[40,54389,162],{"class":10732},[40,54391,16302],{"class":10363},[40,54393,54394,54396,54398,54400,54402],{"class":10276,"line":12110},[40,54395,16307],{"class":10363},[40,54397,162],{"class":10732},[40,54399,13070],{"class":10363},[40,54401,16314],{"class":10430},[40,54403,12174],{"class":10363},[40,54405,54406,54408,54410,54412,54414,54416,54418,54420,54422,54424],{"class":10276,"line":12115},[40,54407,16321],{"class":10363},[40,54409,162],{"class":10732},[40,54411,16326],{"class":10430},[40,54413,167],{"class":10363},[40,54415,11192],{"class":11903},[40,54417,162],{"class":10732},[40,54419,16335],{"class":10363},[40,54421,16338],{"class":11903},[40,54423,162],{"class":10732},[40,54425,16343],{"class":10363},[40,54427,54428,54430,54432,54434],{"class":10276,"line":12121},[40,54429,16348],{"class":10363},[40,54431,16351],{"class":10430},[40,54433,16354],{"class":10363},[40,54435,16357],{"class":10280},[40,54437,54438],{"class":10276,"line":12155},[40,54439,10301],{"emptyLinePlaceholder":10300},[40,54441,54442,54444,54446,54448,54450,54452,54454,54456,54458,54460,54462,54464],{"class":10276,"line":12161},[40,54443,16366],{"class":10363},[40,54445,162],{"class":10732},[40,54447,13070],{"class":10363},[40,54449,16373],{"class":10430},[40,54451,167],{"class":10363},[40,54453,13078],{"class":11903},[40,54455,162],{"class":10732},[40,54457,16382],{"class":10363},[40,54459,13086],{"class":11903},[40,54461,162],{"class":10732},[40,54463,13091],{"class":10446},[40,54465,10761],{"class":10363},[40,54467,54468,54470,54472,54474,54476,54478,54480,54482,54484,54486,54488],{"class":10276,"line":12177},[40,54469,16395],{"class":10363},[40,54471,162],{"class":10732},[40,54473,13170],{"class":10363},[40,54475,16402],{"class":10430},[40,54477,167],{"class":10363},[40,54479,16407],{"class":10446},[40,54481,10484],{"class":10363},[40,54483,16412],{"class":11903},[40,54485,162],{"class":10732},[40,54487,16417],{"class":10287},[40,54489,10761],{"class":10363},[40,54491,54492,54494,54496,54498,54500,54502,54504,54506,54508],{"class":10276,"line":12192},[40,54493,13227],{"class":10363},[40,54495,16426],{"class":10430},[40,54497,167],{"class":10363},[40,54499,16431],{"class":11903},[40,54501,162],{"class":10732},[40,54503,16436],{"class":10363},[40,54505,16439],{"class":11903},[40,54507,162],{"class":10732},[40,54509,16444],{"class":10363},[40,54511,54512,54514,54516,54518,54520],{"class":10276,"line":12231},[40,54513,11323],{"class":10352},[40,54515,16451],{"class":10732},[40,54517,13259],{"class":10363},[40,54519,16456],{"class":10287},[40,54521,13353],{"class":10363},[40,54523,54524,54526,54528,54530,54532,54534,54536,54538,54540],{"class":10276,"line":12245},[40,54525,16463],{"class":10363},[40,54527,16466],{"class":10430},[40,54529,16469],{"class":10363},[40,54531,78],{"class":10446},[40,54533,10484],{"class":10363},[40,54535,78],{"class":10446},[40,54537,10484],{"class":10363},[40,54539,78],{"class":10446},[40,54541,13612],{"class":10363},[40,54543,54544],{"class":10276,"line":12280},[40,54545,10301],{"emptyLinePlaceholder":10300},[40,54547,54548],{"class":10276,"line":12294},[40,54549,16490],{"class":10280},[40,54551,54552,54554,54556,54558,54560,54562,54564,54566,54568,54570],{"class":10276,"line":12329},[40,54553,16495],{"class":10363},[40,54555,162],{"class":10732},[40,54557,16016],{"class":10430},[40,54559,167],{"class":10363},[40,54561,16504],{"class":10430},[40,54563,16507],{"class":10363},[40,54565,10801],{"class":10446},[40,54567,12055],{"class":10363},[40,54569,13404],{"class":10732},[40,54571,16516],{"class":10363},[40,54573,54574,54576,54578,54580,54582,54584,54586,54588,54590,54592,54594,54596,54598],{"class":10276,"line":12343},[40,54575,16521],{"class":10446},[40,54577,10772],{"class":10732},[40,54579,16526],{"class":10430},[40,54581,167],{"class":10363},[40,54583,11192],{"class":10430},[40,54585,16533],{"class":10363},[40,54587,11056],{"class":10732},[40,54589,78],{"class":10446},[40,54591,16540],{"class":10363},[40,54593,11056],{"class":10732},[40,54595,78],{"class":10446},[40,54597,16547],{"class":10363},[40,54599,16550],{"class":10280},[40,54601,54602],{"class":10276,"line":12376},[40,54603,10301],{"emptyLinePlaceholder":10300},[40,54605,54606,54608,54610,54612,54614],{"class":10276,"line":12409},[40,54607,16559],{"class":10363},[40,54609,162],{"class":10732},[40,54611,12522],{"class":10363},[40,54613,16566],{"class":10430},[40,54615,12527],{"class":10363},[40,54617,54618],{"class":10276,"line":12414},[40,54619,16573],{"class":10363},[40,54621,54622,54624,54626,54628],{"class":10276,"line":12419},[40,54623,16578],{"class":11903},[40,54625,162],{"class":10732},[40,54627,16583],{"class":10446},[40,54629,10507],{"class":10363},[40,54631,54632,54634,54636,54638,54640,54642,54644,54646],{"class":10276,"line":12435},[40,54633,16590],{"class":11903},[40,54635,162],{"class":10732},[40,54637,78],{"class":10446},[40,54639,10484],{"class":10363},[40,54641,16599],{"class":11903},[40,54643,162],{"class":10732},[40,54645,78],{"class":10446},[40,54647,10507],{"class":10363},[40,54649,54650,54652,54654,54656,54658,54660,54662,54664],{"class":10276,"line":12447},[40,54651,16610],{"class":11903},[40,54653,162],{"class":10732},[40,54655,10504],{"class":10446},[40,54657,10484],{"class":10363},[40,54659,16619],{"class":11903},[40,54661,162],{"class":10732},[40,54663,66],{"class":10446},[40,54665,10507],{"class":10363},[40,54667,54668,54670,54672,54674,54676],{"class":10276,"line":12463},[40,54669,16630],{"class":11903},[40,54671,162],{"class":10732},[40,54673,10504],{"class":10446},[40,54675,16637],{"class":10363},[40,54677,16640],{"class":10280},[40,54679,54680],{"class":10276,"line":12480},[40,54681,12631],{"class":10363},[40,54683,54684],{"class":10276,"line":12498},[40,54685,10301],{"emptyLinePlaceholder":10300},[40,54687,54688],{"class":10276,"line":12504},[40,54689,16653],{"class":10280},[40,54691,54692,54694,54696,54698,54700,54702,54704,54706,54708,54710,54712,54714,54716,54718,54720,54722],{"class":10276,"line":12517},[40,54693,13165],{"class":10363},[40,54695,162],{"class":10732},[40,54697,13170],{"class":10363},[40,54699,13173],{"class":10430},[40,54701,16666],{"class":10363},[40,54703,12429],{"class":11903},[40,54705,162],{"class":10732},[40,54707,13183],{"class":10363},[40,54709,13186],{"class":11903},[40,54711,162],{"class":10732},[40,54713,13191],{"class":10446},[40,54715,10484],{"class":10363},[40,54717,13196],{"class":11903},[40,54719,162],{"class":10732},[40,54721,13201],{"class":10446},[40,54723,10761],{"class":10363},[40,54725,54726,54728,54730,54732,54734],{"class":10276,"line":12530},[40,54727,13208],{"class":10363},[40,54729,13211],{"class":10430},[40,54731,167],{"class":10363},[40,54733,13216],{"class":10287},[40,54735,10761],{"class":10363},[40,54737,54738],{"class":10276,"line":12542},[40,54739,10301],{"emptyLinePlaceholder":10300},[40,54741,54742],{"class":10276,"line":12571},[40,54743,16709],{"class":10280},[40,54745,54746,54748,54750,54752,54754],{"class":10276,"line":12622},[40,54747,13227],{"class":10363},[40,54749,13230],{"class":10430},[40,54751,167],{"class":10363},[40,54753,10748],{"class":10287},[40,54755,10761],{"class":10363},[40,54757,54758,54760,54762,54764,54766],{"class":10276,"line":12628},[40,54759,13227],{"class":10363},[40,54761,13244],{"class":10430},[40,54763,167],{"class":10363},[40,54765,10753],{"class":10287},[40,54767,10761],{"class":10363},[40,54769,54770,54772,54774,54776,54778],{"class":10276,"line":12634},[40,54771,13227],{"class":10363},[40,54773,16740],{"class":10430},[40,54775,167],{"class":10363},[40,54777,10758],{"class":10287},[40,54779,10761],{"class":10363},[40,54781,54782,54784,54786,54788,54790],{"class":10276,"line":12639},[40,54783,16751],{"class":10363},[40,54785,162],{"class":10732},[40,54787,12522],{"class":10363},[40,54789,16758],{"class":10430},[40,54791,12527],{"class":10363},[40,54793,54794,54796,54798,54800,54802,54804,54806,54808,54810,54812,54814,54816,54818],{"class":10276,"line":12644},[40,54795,12687],{"class":10446},[40,54797,10484],{"class":10363},[40,54799,16769],{"class":10446},[40,54801,10484],{"class":10363},[40,54803,13262],{"class":10352},[40,54805,13265],{"class":10287},[40,54807,13268],{"class":10446},[40,54809,13271],{"class":10363},[40,54811,13274],{"class":10430},[40,54813,13277],{"class":10363},[40,54815,13280],{"class":10446},[40,54817,16788],{"class":10287},[40,54819,10507],{"class":10363},[40,54821,54822,54824,54826,54828,54830,54832,54834,54836,54838,54840,54842,54844,54846,54848,54850],{"class":10276,"line":12667},[40,54823,12545],{"class":11903},[40,54825,162],{"class":10732},[40,54827,12550],{"class":10363},[40,54829,12211],{"class":11903},[40,54831,162],{"class":10732},[40,54833,12710],{"class":10287},[40,54835,10484],{"class":10363},[40,54837,12221],{"class":11903},[40,54839,162],{"class":10732},[40,54841,12226],{"class":10287},[40,54843,10484],{"class":10363},[40,54845,12553],{"class":11903},[40,54847,162],{"class":10732},[40,54849,12731],{"class":10446},[40,54851,10507],{"class":10363},[40,54853,54854,54856,54858,54860,54862,54864,54866,54868,54870,54872,54874,54876,54878,54880,54882,54884,54886,54888,54890],{"class":10276,"line":12673},[40,54855,12574],{"class":11903},[40,54857,12577],{"class":10732},[40,54859,167],{"class":10363},[40,54861,12582],{"class":11903},[40,54863,162],{"class":10732},[40,54865,12587],{"class":10287},[40,54867,10484],{"class":10363},[40,54869,12592],{"class":11903},[40,54871,162],{"class":10732},[40,54873,12597],{"class":10287},[40,54875,10484],{"class":10363},[40,54877,12602],{"class":11903},[40,54879,162],{"class":10732},[40,54881,12607],{"class":10446},[40,54883,10484],{"class":10363},[40,54885,12612],{"class":11903},[40,54887,162],{"class":10732},[40,54889,12617],{"class":10287},[40,54891,11061],{"class":10363},[40,54893,54894,54896,54898,54900],{"class":10276,"line":12684},[40,54895,12780],{"class":11903},[40,54897,162],{"class":10732},[40,54899,12785],{"class":10446},[40,54901,10507],{"class":10363},[40,54903,54904],{"class":10276,"line":12697},[40,54905,12631],{"class":10363},[40,54907,54908,54910,54912,54914,54916],{"class":10276,"line":12723},[40,54909,16881],{"class":10363},[40,54911,162],{"class":10732},[40,54913,12522],{"class":10363},[40,54915,16758],{"class":10430},[40,54917,12527],{"class":10363},[40,54919,54920,54922,54924,54926,54928,54930,54932,54934,54936,54938,54940,54942,54944,54946,54948,54950,54952],{"class":10276,"line":12736},[40,54921,16894],{"class":10446},[40,54923,10484],{"class":10363},[40,54925,16899],{"class":10446},[40,54927,10484],{"class":10363},[40,54929,13303],{"class":10287},[40,54931,10484],{"class":10363},[40,54933,16908],{"class":11903},[40,54935,162],{"class":10732},[40,54937,12550],{"class":10363},[40,54939,12211],{"class":11903},[40,54941,162],{"class":10732},[40,54943,12216],{"class":10287},[40,54945,10484],{"class":10363},[40,54947,12221],{"class":11903},[40,54949,162],{"class":10732},[40,54951,12226],{"class":10287},[40,54953,10507],{"class":10363},[40,54955,54956,54958,54960],{"class":10276,"line":12777},[40,54957,12726],{"class":11903},[40,54959,162],{"class":10732},[40,54961,16937],{"class":10363},[40,54963,54964,54966,54968,54970,54972,54974,54976,54978,54980,54982,54984,54986,54988,54990,54992,54994,54996,54998,55000],{"class":10276,"line":12790},[40,54965,12574],{"class":11903},[40,54967,12577],{"class":10732},[40,54969,167],{"class":10363},[40,54971,12582],{"class":11903},[40,54973,162],{"class":10732},[40,54975,12587],{"class":10287},[40,54977,10484],{"class":10363},[40,54979,12592],{"class":11903},[40,54981,162],{"class":10732},[40,54983,12597],{"class":10287},[40,54985,10484],{"class":10363},[40,54987,12602],{"class":11903},[40,54989,162],{"class":10732},[40,54991,12607],{"class":10446},[40,54993,10484],{"class":10363},[40,54995,12612],{"class":11903},[40,54997,162],{"class":10732},[40,54999,12617],{"class":10287},[40,55001,11061],{"class":10363},[40,55003,55004,55006,55008,55010],{"class":10276,"line":16980},[40,55005,12780],{"class":11903},[40,55007,162],{"class":10732},[40,55009,12785],{"class":10446},[40,55011,10507],{"class":10363},[40,55013,55014],{"class":10276,"line":16991},[40,55015,12631],{"class":10363},[40,55017,55018],{"class":10276,"line":16996},[40,55019,10301],{"emptyLinePlaceholder":10300},[40,55021,55022,55024,55026,55028,55030,55032,55034],{"class":10276,"line":17001},[40,55023,13338],{"class":10352},[40,55025,13341],{"class":10430},[40,55027,167],{"class":10363},[40,55029,17010],{"class":10973},[40,55031,10443],{"class":10363},[40,55033,10821],{"class":10732},[40,55035,13353],{"class":10363},[40,55037,55038,55040,55042,55044,55046,55048,55050,55052,55054,55056,55058],{"class":10276,"line":17019},[40,55039,17022],{"class":10363},[40,55041,162],{"class":10732},[40,55043,17027],{"class":10363},[40,55045,11243],{"class":10732},[40,55047,17032],{"class":10363},[40,55049,11056],{"class":10732},[40,55051,13373],{"class":10446},[40,55053,13376],{"class":10363},[40,55055,11752],{"class":10732},[40,55057,13381],{"class":10363},[40,55059,13384],{"class":10280},[40,55061,55062,55064,55066,55068,55070,55072],{"class":10276,"line":17047},[40,55063,17050],{"class":10363},[40,55065,162],{"class":10732},[40,55067,15630],{"class":10430},[40,55069,17057],{"class":10363},[40,55071,13404],{"class":10732},[40,55073,13407],{"class":10363},[40,55075,55076,55078,55080,55082],{"class":10276,"line":17064},[40,55077,17067],{"class":10363},[40,55079,162],{"class":10732},[40,55081,16016],{"class":10430},[40,55083,17074],{"class":10363},[40,55085,55086,55088,55090,55092,55094,55096,55098,55100,55102,55104,55106,55108,55110],{"class":10276,"line":17077},[40,55087,17080],{"class":10446},[40,55089,10772],{"class":10732},[40,55091,16526],{"class":10430},[40,55093,167],{"class":10363},[40,55095,11192],{"class":10430},[40,55097,17091],{"class":10363},[40,55099,11056],{"class":10732},[40,55101,78],{"class":10446},[40,55103,16540],{"class":10363},[40,55105,11056],{"class":10732},[40,55107,78],{"class":10446},[40,55109,17104],{"class":10363},[40,55111,17107],{"class":10280},[40,55113,55114,55116,55118,55120,55122,55124,55126,55128,55130,55132,55134,55136,55138],{"class":10276,"line":17110},[40,55115,17113],{"class":10363},[40,55117,17116],{"class":10430},[40,55119,167],{"class":10363},[40,55121,17121],{"class":10446},[40,55123,1123],{"class":10363},[40,55125,12070],{"class":10430},[40,55127,167],{"class":10363},[40,55129,11056],{"class":10732},[40,55131,78],{"class":10446},[40,55133,10484],{"class":10363},[40,55135,17136],{"class":10446},[40,55137,17139],{"class":10363},[40,55139,17142],{"class":10280},[40,55141,55142,55144,55146,55148,55150,55152,55154,55156,55158,55160,55162,55164,55166,55168,55170,55172,55174,55176,55178,55180,55182,55184],{"class":10276,"line":17145},[40,55143,17148],{"class":10363},[40,55145,13431],{"class":10430},[40,55147,167],{"class":10363},[40,55149,13262],{"class":10352},[40,55151,13438],{"class":10287},[40,55153,13268],{"class":10446},[40,55155,11284],{"class":10363},[40,55157,13446],{"class":10352},[40,55159,13280],{"class":10446},[40,55161,13451],{"class":10287},[40,55163,13454],{"class":10732},[40,55165,13457],{"class":10287},[40,55167,13268],{"class":10446},[40,55169,17010],{"class":10363},[40,55171,1053],{"class":10732},[40,55173,13466],{"class":10446},[40,55175,11243],{"class":10287},[40,55177,13268],{"class":10446},[40,55179,17185],{"class":10363},[40,55181,13280],{"class":10446},[40,55183,13478],{"class":10287},[40,55185,10761],{"class":10363},[40,55187,55188,55190],{"class":10276,"line":17194},[40,55189,11339],{"class":10352},[40,55191,17199],{"class":10363},[40,55193,55194],{"class":10276,"line":17202},[40,55195,10301],{"emptyLinePlaceholder":10300},[40,55197,55198,55200,55202,55204,55206,55208,55210,55212,55214,55216,55218,55220,55222,55224,55226],{"class":10276,"line":17207},[40,55199,13496],{"class":10363},[40,55201,162],{"class":10732},[40,55203,13501],{"class":10430},[40,55205,13504],{"class":10363},[40,55207,13473],{"class":11903},[40,55209,162],{"class":10732},[40,55211,17222],{"class":10363},[40,55213,13514],{"class":11903},[40,55215,162],{"class":10732},[40,55217,13519],{"class":10446},[40,55219,10484],{"class":10363},[40,55221,13524],{"class":11903},[40,55223,162],{"class":10732},[40,55225,10504],{"class":10446},[40,55227,10761],{"class":10363},[40,55229,55230,55232,55234,55236,55238,55240,55242,55244,55246,55248,55250],{"class":10276,"line":17241},[40,55231,17244],{"class":10363},[40,55233,162],{"class":10732},[40,55235,13859],{"class":10430},[40,55237,17251],{"class":10363},[40,55239,11243],{"class":10732},[40,55241,13867],{"class":10352},[40,55243,17258],{"class":10287},[40,55245,13268],{"class":10446},[40,55247,11275],{"class":10363},[40,55249,13280],{"class":10446},[40,55251,13880],{"class":10287},[40,55253,55254,55256,55258,55260,55262,55264,55266,55268,55270,55272,55274],{"class":10276,"line":17269},[40,55255,17272],{"class":10363},[40,55257,13538],{"class":10430},[40,55259,167],{"class":10363},[40,55261,13543],{"class":11903},[40,55263,162],{"class":10732},[40,55265,13091],{"class":10446},[40,55267,10484],{"class":10363},[40,55269,13552],{"class":11903},[40,55271,162],{"class":10732},[40,55273,13091],{"class":10446},[40,55275,10761],{"class":10363},[40,55277,55278,55280,55282,55284,55286,55288,55290,55292,55294,55296,55298,55300],{"class":10276,"line":17295},[40,55279,13563],{"class":10363},[40,55281,13566],{"class":10430},[40,55283,17302],{"class":10363},[40,55285,13572],{"class":10430},[40,55287,13575],{"class":10363},[40,55289,13578],{"class":11903},[40,55291,162],{"class":10732},[40,55293,13583],{"class":10430},[40,55295,167],{"class":10363},[40,55297,13588],{"class":11903},[40,55299,162],{"class":10732},[40,55301,17321],{"class":10363},[40,55303,55304,55306,55308,55310,55312,55314,55316],{"class":10276,"line":17324},[40,55305,13598],{"class":11903},[40,55307,12577],{"class":10732},[40,55309,167],{"class":10363},[40,55311,13605],{"class":11903},[40,55313,162],{"class":10732},[40,55315,12597],{"class":10287},[40,55317,13612],{"class":10363},[40,55319,55320,55322,55324],{"class":10276,"line":17341},[40,55321,13617],{"class":10363},[40,55323,13620],{"class":10430},[40,55325,13623],{"class":10363},[40,55327,55328,55330],{"class":10276,"line":17350},[40,55329,11201],{"class":10352},[40,55331,17355],{"class":10363},[40,55333,55334],{"class":10276,"line":17358},[40,55335,10301],{"emptyLinePlaceholder":10300},[40,55337,55338],{"class":10276,"line":17363},[40,55339,10301],{"emptyLinePlaceholder":10300},[40,55341,55342,55344,55346,55348,55350],{"class":10276,"line":17368},[40,55343,10965],{"class":10352},[40,55345,17373],{"class":10430},[40,55347,167],{"class":10363},[40,55349,13346],{"class":10973},[40,55351,17380],{"class":10363},[40,55353,55354,55356,55358,55360,55362,55364,55366,55368,55370,55372,55374,55376,55378,55380],{"class":10276,"line":17383},[40,55355,17386],{"class":10973},[40,55357,10989],{"class":10363},[40,55359,10733],{"class":10732},[40,55361,10484],{"class":10363},[40,55363,10738],{"class":10446},[40,55365,10741],{"class":10363},[40,55367,162],{"class":10732},[40,55369,920],{"class":10363},[40,55371,10748],{"class":10287},[40,55373,10484],{"class":10363},[40,55375,10753],{"class":10287},[40,55377,10484],{"class":10363},[40,55379,10758],{"class":10287},[40,55381,17413],{"class":10363},[40,55383,55384],{"class":10276,"line":17416},[40,55385,17419],{"class":10287},[40,55387,55388,55390,55392,55394],{"class":10276,"line":17422},[40,55389,17425],{"class":10352},[40,55391,13843],{"class":10363},[40,55393,13846],{"class":10352},[40,55395,17432],{"class":10363},[40,55397,55398,55400,55402,55404,55406,55408],{"class":10276,"line":17435},[40,55399,17438],{"class":10363},[40,55401,162],{"class":10732},[40,55403,16107],{"class":10430},[40,55405,17445],{"class":10363},[40,55407,11171],{"class":10430},[40,55409,17450],{"class":10363},[40,55411,55412,55414,55416,55418,55420,55422,55424],{"class":10276,"line":17453},[40,55413,17456],{"class":10430},[40,55415,167],{"class":10363},[40,55417,13953],{"class":10430},[40,55419,167],{"class":10363},[40,55421,13958],{"class":11903},[40,55423,13961],{"class":10732},[40,55425,17469],{"class":10363},[40,55427,55428,55430,55432,55434,55436,55438,55440,55442,55444],{"class":10276,"line":17472},[40,55429,17475],{"class":10732},[40,55431,167],{"class":10363},[40,55433,13262],{"class":10352},[40,55435,17482],{"class":10287},[40,55437,13268],{"class":10446},[40,55439,2264],{"class":10363},[40,55441,13280],{"class":10446},[40,55443,13478],{"class":10287},[40,55445,10761],{"class":10363},[40,55447,55448,55450],{"class":10276,"line":17495},[40,55449,17498],{"class":10352},[40,55451,17501],{"class":10363},[40,55453,55454],{"class":10276,"line":17504},[40,55455,10301],{"emptyLinePlaceholder":10300},[40,55457,55458],{"class":10276,"line":17509},[40,55459,10301],{"emptyLinePlaceholder":10300},[40,55461,55462],{"class":10276,"line":17514},[40,55463,17517],{"class":10280},[40,55465,55466,55468,55470,55472,55474],{"class":10276,"line":17520},[40,55467,17523],{"class":10446},[40,55469,10772],{"class":10732},[40,55471,15163],{"class":10430},[40,55473,13655],{"class":10363},[40,55475,17532],{"class":10280},[40,55477,55478,55480,55482,55484],{"class":10276,"line":17535},[40,55479,13684],{"class":11903},[40,55481,162],{"class":10732},[40,55483,13689],{"class":10287},[40,55485,10507],{"class":10363},[40,55487,55488,55490,55492,55494,55496,55498,55500,55502,55504,55506,55508,55510,55512,55514,55516,55518],{"class":10276,"line":17546},[40,55489,13696],{"class":11903},[40,55491,162],{"class":10732},[40,55493,13701],{"class":10446},[40,55495,10484],{"class":10363},[40,55497,11816],{"class":11903},[40,55499,162],{"class":10732},[40,55501,17561],{"class":10446},[40,55503,10484],{"class":10363},[40,55505,11825],{"class":11903},[40,55507,162],{"class":10732},[40,55509,17570],{"class":10446},[40,55511,10484],{"class":10363},[40,55513,13588],{"class":11903},[40,55515,162],{"class":10732},[40,55517,13742],{"class":10446},[40,55519,10507],{"class":10363},[40,55521,55522,55524,55526,55528,55530,55532,55534,55536],{"class":10276,"line":17583},[40,55523,17586],{"class":11903},[40,55525,162],{"class":10732},[40,55527,17591],{"class":10446},[40,55529,10484],{"class":10363},[40,55531,16439],{"class":11903},[40,55533,17598],{"class":10732},[40,55535,17601],{"class":10446},[40,55537,10507],{"class":10363},[40,55539,55540,55542,55544,55546],{"class":10276,"line":17606},[40,55541,13749],{"class":11903},[40,55543,162],{"class":10732},[40,55545,13754],{"class":10287},[40,55547,10507],{"class":10363},[40,55549,55550,55552,55554,55556,55558,55560,55562],{"class":10276,"line":17617},[40,55551,13761],{"class":11903},[40,55553,162],{"class":10732},[40,55555,167],{"class":10363},[40,55557,10481],{"class":10446},[40,55559,10484],{"class":10363},[40,55561,15378],{"class":10446},[40,55563,11061],{"class":10363},[40,55565,55566,55568,55570,55572],{"class":10276,"line":17634},[40,55567,17637],{"class":11903},[40,55569,162],{"class":10732},[40,55571,13793],{"class":10446},[40,55573,10507],{"class":10363},[40,55575,55576],{"class":10276,"line":17646},[40,55577,10761],{"class":10363},[40,55579,55580],{"class":10276,"line":17651},[40,55581,10301],{"emptyLinePlaceholder":10300},[40,55583,55584,55586,55588,55590,55592,55594,55596,55598,55600,55602,55604,55606,55608,55610,55612],{"class":10276,"line":17656},[40,55585,17659],{"class":10732},[40,55587,167],{"class":10363},[40,55589,17664],{"class":10430},[40,55591,167],{"class":10363},[40,55593,17523],{"class":10446},[40,55595,10484],{"class":10363},[40,55597,17673],{"class":11903},[40,55599,162],{"class":10732},[40,55601,167],{"class":10363},[40,55603,10748],{"class":10287},[40,55605,10484],{"class":10363},[40,55607,10753],{"class":10287},[40,55609,10484],{"class":10363},[40,55611,10758],{"class":10287},[40,55613,17690],{"class":10363},[10082,55615,55617],{"className":55616,"code":17694,"language":2932,"meta":10090},[10085],[10088,55618,17694],{"__ignoreMap":10090},[10082,55620,55622],{"className":55621,"code":17700,"language":2932,"meta":10090},[10085],[10088,55623,17700],{"__ignoreMap":10090},[10082,55625,55627],{"className":55626,"code":17706,"language":2932,"meta":10090},[10085],[10088,55628,17706],{"__ignoreMap":10090},[13997,55630],{"alt":17711,"src":17712},[13997,55632],{"alt":17715,"src":17716},[13997,55634],{"alt":17719,"src":17720},[10082,55636,55637],{"className":10269,"code":17723,"language":10271,"meta":10090,"style":10090},[10088,55638,55639,55643,55647,55651,55661,55665,55675,55685,55695,55699,55709,55719,55729,55739,55749,55759,55763,55767,55771,55779,55783,55817,55827,55837,55869,55879,55889,55899,55909,55919,55931,55941,55965,55975,55979,55983,55987,56019,56027,56063,56099,56135,56139,56163,56179,56189,56193,56197,56229,56245,56257,56269,56279,56291,56355,56371,56387,56423,56427,56431,56455,56471,56493,56513,56533,56553,56565,56585,56591,56595,56599,56615,56675,56679,56683,56695,56699,56709,56719,56723,56743,56763,56775,56779,56805,56831,56853,56865,56887,56895,56899,56925,56949,56969,56981,57001,57005,57009,57017,57031,57041,57069,57073,57085,57089,57099,57117,57135,57145,57149,57153,57187,57199,57203,57215,57227,57239,57251,57265,57297,57337,57347,57351,57363,57399,57407,57447,57457,57461,57465,57481,57507,57521,57531,57559,57587,57595,57647,57677,57681,57687,57691,57723,57733,57757,57783,57799,57807,57813,57817,57821,57825,57835,57863,57881,57885,57899,57919],{"__ignoreMap":10090},[40,55640,55641],{"class":10276,"line":10277},[40,55642,17730],{"class":10280},[40,55644,55645],{"class":10276,"line":10284},[40,55646,17735],{"class":10280},[40,55648,55649],{"class":10276,"line":10291},[40,55650,10301],{"emptyLinePlaceholder":10300},[40,55652,55653,55655,55657,55659],{"class":10276,"line":10297},[40,55654,10353],{"class":10352},[40,55656,10357],{"class":10356},[40,55658,10360],{"class":10352},[40,55660,10364],{"class":10363},[40,55662,55663],{"class":10276,"line":10304},[40,55664,10301],{"emptyLinePlaceholder":10300},[40,55666,55667,55669,55671,55673],{"class":10276,"line":10310},[40,55668,10353],{"class":10352},[40,55670,10530],{"class":10363},[40,55672,10375],{"class":10352},[40,55674,10535],{"class":10363},[40,55676,55677,55679,55681,55683],{"class":10276,"line":10316},[40,55678,10353],{"class":10352},[40,55680,10542],{"class":10363},[40,55682,10375],{"class":10352},[40,55684,10547],{"class":10363},[40,55686,55687,55689,55691,55693],{"class":10276,"line":10322},[40,55688,10353],{"class":10352},[40,55690,10554],{"class":10363},[40,55692,10375],{"class":10352},[40,55694,17784],{"class":10363},[40,55696,55697],{"class":10276,"line":10327},[40,55698,10301],{"emptyLinePlaceholder":10300},[40,55700,55701,55703,55705,55707],{"class":10276,"line":10333},[40,55702,10375],{"class":10352},[40,55704,10378],{"class":10363},[40,55706,10381],{"class":10352},[40,55708,10384],{"class":10363},[40,55710,55711,55713,55715,55717],{"class":10276,"line":10339},[40,55712,10375],{"class":10352},[40,55714,10405],{"class":10363},[40,55716,10381],{"class":10352},[40,55718,10410],{"class":10363},[40,55720,55721,55723,55725,55727],{"class":10276,"line":10344},[40,55722,10353],{"class":10352},[40,55724,10597],{"class":10363},[40,55726,10375],{"class":10352},[40,55728,10602],{"class":10363},[40,55730,55731,55733,55735,55737],{"class":10276,"line":10349},[40,55732,10353],{"class":10352},[40,55734,15117],{"class":10363},[40,55736,10375],{"class":10352},[40,55738,15122],{"class":10363},[40,55740,55741,55743,55745,55747],{"class":10276,"line":10367},[40,55742,10353],{"class":10352},[40,55744,15129],{"class":10363},[40,55746,10375],{"class":10352},[40,55748,15134],{"class":10363},[40,55750,55751,55753,55755,55757],{"class":10276,"line":10372},[40,55752,10353],{"class":10352},[40,55754,10609],{"class":10363},[40,55756,10375],{"class":10352},[40,55758,10614],{"class":10363},[40,55760,55761],{"class":10276,"line":10387},[40,55762,10301],{"emptyLinePlaceholder":10300},[40,55764,55765],{"class":10276,"line":10400},[40,55766,10301],{"emptyLinePlaceholder":10300},[40,55768,55769],{"class":10276,"line":10413},[40,55770,10627],{"class":10430},[40,55772,55773,55775,55777],{"class":10276,"line":10418},[40,55774,10632],{"class":10352},[40,55776,17867],{"class":10635},[40,55778,10639],{"class":10363},[40,55780,55781],{"class":10276,"line":10424},[40,55782,17874],{"class":10287},[40,55784,55785,55787,55789,55791,55793,55795,55797,55799,55801,55803,55805,55807,55809,55811,55813,55815],{"class":10276,"line":10437},[40,55786,17879],{"class":10363},[40,55788,10784],{"class":10732},[40,55790,10484],{"class":10363},[40,55792,10784],{"class":10732},[40,55794,10484],{"class":10363},[40,55796,10784],{"class":10732},[40,55798,10741],{"class":10363},[40,55800,162],{"class":10732},[40,55802,920],{"class":10363},[40,55804,10801],{"class":10446},[40,55806,10484],{"class":10363},[40,55808,10810],{"class":10446},[40,55810,10484],{"class":10363},[40,55812,10801],{"class":10446},[40,55814,17908],{"class":10363},[40,55816,17911],{"class":10280},[40,55818,55819,55821,55823,55825],{"class":10276,"line":10456},[40,55820,17916],{"class":10363},[40,55822,10784],{"class":10732},[40,55824,10772],{"class":10732},[40,55826,17923],{"class":10446},[40,55828,55829,55831,55833,55835],{"class":10276,"line":10472},[40,55830,10767],{"class":10363},[40,55832,10733],{"class":10732},[40,55834,10772],{"class":10732},[40,55836,10775],{"class":10287},[40,55838,55839,55841,55843,55845,55847,55849,55851,55853,55855,55857,55859,55861,55863,55865,55867],{"class":10276,"line":10496},[40,55840,10781],{"class":10363},[40,55842,10784],{"class":10732},[40,55844,10484],{"class":10363},[40,55846,10784],{"class":10732},[40,55848,10484],{"class":10363},[40,55850,10784],{"class":10732},[40,55852,10741],{"class":10363},[40,55854,162],{"class":10732},[40,55856,920],{"class":10363},[40,55858,10801],{"class":10446},[40,55860,10484],{"class":10363},[40,55862,10801],{"class":10446},[40,55864,10484],{"class":10363},[40,55866,10810],{"class":10446},[40,55868,10761],{"class":10363},[40,55870,55871,55873,55875,55877],{"class":10276,"line":10510},[40,55872,10818],{"class":10363},[40,55874,10821],{"class":10732},[40,55876,10772],{"class":10732},[40,55878,10826],{"class":10446},[40,55880,55881,55883,55885,55887],{"class":10276,"line":10696},[40,55882,10832],{"class":10363},[40,55884,10821],{"class":10732},[40,55886,10772],{"class":10732},[40,55888,15298],{"class":10446},[40,55890,55891,55893,55895,55897],{"class":10276,"line":10702},[40,55892,10845],{"class":10363},[40,55894,10821],{"class":10732},[40,55896,10772],{"class":10732},[40,55898,15309],{"class":10446},[40,55900,55901,55903,55905,55907],{"class":10276,"line":10708},[40,55902,10858],{"class":10363},[40,55904,10821],{"class":10732},[40,55906,10772],{"class":10732},[40,55908,10865],{"class":10446},[40,55910,55911,55913,55915,55917],{"class":10276,"line":10714},[40,55912,15324],{"class":10363},[40,55914,10784],{"class":10732},[40,55916,10772],{"class":10732},[40,55918,15331],{"class":10446},[40,55920,55921,55923,55925,55927,55929],{"class":10276,"line":10720},[40,55922,15336],{"class":10363},[40,55924,10784],{"class":10732},[40,55926,10772],{"class":10732},[40,55928,11764],{"class":10732},[40,55930,15345],{"class":10446},[40,55932,55933,55935,55937,55939],{"class":10276,"line":10726},[40,55934,18032],{"class":10363},[40,55936,10733],{"class":10732},[40,55938,10772],{"class":10732},[40,55940,18039],{"class":10287},[40,55942,55943,55945,55947,55949,55951,55953,55955,55957,55959,55961,55963],{"class":10276,"line":10764},[40,55944,10884],{"class":10363},[40,55946,10784],{"class":10732},[40,55948,10484],{"class":10363},[40,55950,10784],{"class":10732},[40,55952,10741],{"class":10363},[40,55954,162],{"class":10732},[40,55956,920],{"class":10363},[40,55958,10481],{"class":10446},[40,55960,10484],{"class":10363},[40,55962,15378],{"class":10446},[40,55964,10761],{"class":10363},[40,55966,55967,55969,55971,55973],{"class":10276,"line":10778},[40,55968,10923],{"class":10363},[40,55970,10821],{"class":10732},[40,55972,10772],{"class":10732},[40,55974,10930],{"class":10446},[40,55976,55977],{"class":10276,"line":10815},[40,55978,10301],{"emptyLinePlaceholder":10300},[40,55980,55981],{"class":10276,"line":10829},[40,55982,10301],{"emptyLinePlaceholder":10300},[40,55984,55985],{"class":10276,"line":10842},[40,55986,18086],{"class":10280},[40,55988,55989,55991,55993,55995,55997,55999,56001,56003,56005,56007,56009,56011,56013,56015,56017],{"class":10276,"line":10855},[40,55990,10965],{"class":10352},[40,55992,15410],{"class":10430},[40,55994,167],{"class":10363},[40,55996,15415],{"class":10973},[40,55998,10443],{"class":10363},[40,56000,10733],{"class":10732},[40,56002,10484],{"class":10363},[40,56004,15424],{"class":10973},[40,56006,15427],{"class":10363},[40,56008,10801],{"class":10446},[40,56010,10484],{"class":10363},[40,56012,10801],{"class":10446},[40,56014,10484],{"class":10363},[40,56016,10810],{"class":10446},[40,56018,15440],{"class":10363},[40,56020,56021,56023,56025],{"class":10276,"line":10868},[40,56022,11014],{"class":10363},[40,56024,162],{"class":10732},[40,56026,11019],{"class":10363},[40,56028,56029,56031,56033,56035,56037,56039,56041,56043,56045,56047,56049,56051,56053,56055,56057,56059,56061],{"class":10276,"line":10881},[40,56030,11025],{"class":10287},[40,56032,10478],{"class":10363},[40,56034,10801],{"class":10446},[40,56036,10484],{"class":10363},[40,56038,10801],{"class":10446},[40,56040,10484],{"class":10363},[40,56042,10810],{"class":10446},[40,56044,11040],{"class":10363},[40,56046,11043],{"class":10287},[40,56048,10478],{"class":10363},[40,56050,10801],{"class":10446},[40,56052,10484],{"class":10363},[40,56054,10801],{"class":10446},[40,56056,10484],{"class":10363},[40,56058,11056],{"class":10732},[40,56060,10810],{"class":10446},[40,56062,11061],{"class":10363},[40,56064,56065,56067,56069,56071,56073,56075,56077,56079,56081,56083,56085,56087,56089,56091,56093,56095,56097],{"class":10276,"line":10907},[40,56066,11067],{"class":10287},[40,56068,10478],{"class":10363},[40,56070,10810],{"class":10446},[40,56072,10484],{"class":10363},[40,56074,10801],{"class":10446},[40,56076,10484],{"class":10363},[40,56078,10801],{"class":10446},[40,56080,11040],{"class":10363},[40,56082,11084],{"class":10287},[40,56084,10478],{"class":10363},[40,56086,11056],{"class":10732},[40,56088,10810],{"class":10446},[40,56090,10484],{"class":10363},[40,56092,10801],{"class":10446},[40,56094,10484],{"class":10363},[40,56096,10801],{"class":10446},[40,56098,11061],{"class":10363},[40,56100,56101,56103,56105,56107,56109,56111,56113,56115,56117,56119,56121,56123,56125,56127,56129,56131,56133],{"class":10276,"line":10920},[40,56102,11106],{"class":10287},[40,56104,10478],{"class":10363},[40,56106,10801],{"class":10446},[40,56108,10484],{"class":10363},[40,56110,10810],{"class":10446},[40,56112,10484],{"class":10363},[40,56114,10801],{"class":10446},[40,56116,11040],{"class":10363},[40,56118,11123],{"class":10287},[40,56120,10478],{"class":10363},[40,56122,10801],{"class":10446},[40,56124,10484],{"class":10363},[40,56126,11056],{"class":10732},[40,56128,10810],{"class":10446},[40,56130,10484],{"class":10363},[40,56132,10801],{"class":10446},[40,56134,11061],{"class":10363},[40,56136,56137],{"class":10276,"line":10933},[40,56138,11145],{"class":10363},[40,56140,56141,56143,56145,56147,56149,56151,56153,56155,56157,56159,56161],{"class":10276,"line":10946},[40,56142,11151],{"class":10363},[40,56144,162],{"class":10732},[40,56146,11156],{"class":10363},[40,56148,11159],{"class":10430},[40,56150,11162],{"class":10363},[40,56152,11165],{"class":10430},[40,56154,15577],{"class":10363},[40,56156,11171],{"class":10430},[40,56158,15582],{"class":10363},[40,56160,10784],{"class":10732},[40,56162,10761],{"class":10363},[40,56164,56165,56167,56169,56171,56173,56175,56177],{"class":10276,"line":10951},[40,56166,11184],{"class":10363},[40,56168,162],{"class":10732},[40,56170,11189],{"class":10363},[40,56172,11192],{"class":10430},[40,56174,15599],{"class":10363},[40,56176,15602],{"class":10352},[40,56178,15605],{"class":10446},[40,56180,56181,56183,56185,56187],{"class":10276,"line":10956},[40,56182,11201],{"class":10352},[40,56184,11240],{"class":10363},[40,56186,11243],{"class":10732},[40,56188,11246],{"class":10363},[40,56190,56191],{"class":10276,"line":10962},[40,56192,10301],{"emptyLinePlaceholder":10300},[40,56194,56195],{"class":10276,"line":10983},[40,56196,10301],{"emptyLinePlaceholder":10300},[40,56198,56199,56201,56203,56205,56207,56209,56211,56213,56215,56217,56219,56221,56223,56225,56227],{"class":10276,"line":11005},[40,56200,10965],{"class":10352},[40,56202,18303],{"class":10430},[40,56204,167],{"class":10363},[40,56206,11527],{"class":10973},[40,56208,10989],{"class":10363},[40,56210,10784],{"class":10732},[40,56212,10484],{"class":10363},[40,56214,10784],{"class":10732},[40,56216,10484],{"class":10363},[40,56218,10784],{"class":10732},[40,56220,11672],{"class":10363},[40,56222,11284],{"class":10973},[40,56224,10443],{"class":10363},[40,56226,10784],{"class":10732},[40,56228,11291],{"class":10363},[40,56230,56231,56233,56235,56237,56239,56241,56243],{"class":10276,"line":11011},[40,56232,11566],{"class":10363},[40,56234,162],{"class":10732},[40,56236,11156],{"class":10363},[40,56238,11159],{"class":10430},[40,56240,11575],{"class":10363},[40,56242,10784],{"class":10732},[40,56244,10761],{"class":10363},[40,56246,56247,56249,56251,56253,56255],{"class":10276,"line":11022},[40,56248,11184],{"class":10363},[40,56250,162],{"class":10732},[40,56252,11189],{"class":10363},[40,56254,11192],{"class":10430},[40,56256,11593],{"class":10363},[40,56258,56259,56261,56263,56265,56267],{"class":10276,"line":11064},[40,56260,11323],{"class":10352},[40,56262,11228],{"class":10363},[40,56264,11231],{"class":10732},[40,56266,11234],{"class":10446},[40,56268,10639],{"class":10363},[40,56270,56271,56273,56275,56277],{"class":10276,"line":11103},[40,56272,11612],{"class":10352},[40,56274,11499],{"class":10363},[40,56276,11617],{"class":10287},[40,56278,10761],{"class":10363},[40,56280,56281,56283,56285,56287,56289],{"class":10276,"line":11142},[40,56282,11625],{"class":10363},[40,56284,162],{"class":10732},[40,56286,11630],{"class":10363},[40,56288,11243],{"class":10732},[40,56290,11246],{"class":10363},[40,56292,56293,56295,56297,56299,56301,56303,56305,56307,56309,56311,56313,56315,56317,56319,56321,56323,56325,56327,56329,56331,56333,56335,56337,56339,56341,56343,56345,56347,56349,56351,56353],{"class":10276,"line":11148},[40,56294,11640],{"class":10363},[40,56296,162],{"class":10732},[40,56298,11156],{"class":10363},[40,56300,11206],{"class":10430},[40,56302,11346],{"class":10363},[40,56304,10801],{"class":10446},[40,56306,10484],{"class":10363},[40,56308,11056],{"class":10732},[40,56310,11657],{"class":10363},[40,56312,1044],{"class":10446},[40,56314,11662],{"class":10363},[40,56316,78],{"class":10446},[40,56318,11667],{"class":10363},[40,56320,1044],{"class":10446},[40,56322,11672],{"class":10363},[40,56324,10801],{"class":10446},[40,56326,10484],{"class":10363},[40,56328,11056],{"class":10732},[40,56330,11657],{"class":10363},[40,56332,66],{"class":10446},[40,56334,11685],{"class":10363},[40,56336,11056],{"class":10732},[40,56338,11657],{"class":10363},[40,56340,78],{"class":10446},[40,56342,11662],{"class":10363},[40,56344,66],{"class":10446},[40,56346,11672],{"class":10363},[40,56348,10801],{"class":10446},[40,56350,11486],{"class":10363},[40,56352,10784],{"class":10732},[40,56354,10761],{"class":10363},[40,56356,56357,56359,56361,56363,56365,56367,56369],{"class":10276,"line":11181},[40,56358,11711],{"class":10363},[40,56360,162],{"class":10732},[40,56362,11156],{"class":10363},[40,56364,11718],{"class":10430},[40,56366,167],{"class":10363},[40,56368,949],{"class":10446},[40,56370,10761],{"class":10363},[40,56372,56373,56375,56377,56379,56381,56383,56385],{"class":10276,"line":11198},[40,56374,11303],{"class":10363},[40,56376,162],{"class":10732},[40,56378,11156],{"class":10363},[40,56380,1820],{"class":10430},[40,56382,11312],{"class":10363},[40,56384,1813],{"class":10430},[40,56386,11317],{"class":10363},[40,56388,56389,56391,56393,56395,56397,56399,56401,56403,56405,56407,56409,56411,56413,56415,56417,56419,56421],{"class":10276,"line":11249},[40,56390,11201],{"class":10352},[40,56392,11749],{"class":10363},[40,56394,11752],{"class":10732},[40,56396,11755],{"class":10363},[40,56398,1053],{"class":10732},[40,56400,920],{"class":10363},[40,56402,78],{"class":10446},[40,56404,11764],{"class":10732},[40,56406,11767],{"class":10363},[40,56408,11752],{"class":10732},[40,56410,11156],{"class":10363},[40,56412,11774],{"class":10430},[40,56414,11777],{"class":10363},[40,56416,1053],{"class":10732},[40,56418,11782],{"class":10363},[40,56420,11752],{"class":10732},[40,56422,11787],{"class":10363},[40,56424,56425],{"class":10276,"line":11254},[40,56426,10301],{"emptyLinePlaceholder":10300},[40,56428,56429],{"class":10276,"line":11259},[40,56430,10301],{"emptyLinePlaceholder":10300},[40,56432,56433,56435,56437,56439,56441,56443,56445,56447,56449,56451,56453],{"class":10276,"line":11265},[40,56434,10965],{"class":10352},[40,56436,15840],{"class":10430},[40,56438,167],{"class":10363},[40,56440,11816],{"class":10973},[40,56442,10443],{"class":10363},[40,56444,10821],{"class":10732},[40,56446,10484],{"class":10363},[40,56448,11825],{"class":10973},[40,56450,10443],{"class":10363},[40,56452,10821],{"class":10732},[40,56454,13353],{"class":10363},[40,56456,56457,56459,56461,56463,56465,56467,56469],{"class":10276,"line":11294},[40,56458,11843],{"class":10363},[40,56460,162],{"class":10732},[40,56462,11156],{"class":10363},[40,56464,11850],{"class":10430},[40,56466,167],{"class":10363},[40,56468,10801],{"class":10446},[40,56470,11857],{"class":10363},[40,56472,56473,56475,56477,56479,56481,56483,56485,56487,56489,56491],{"class":10276,"line":11300},[40,56474,11863],{"class":10363},[40,56476,162],{"class":10732},[40,56478,11156],{"class":10363},[40,56480,11850],{"class":10430},[40,56482,167],{"class":10363},[40,56484,10801],{"class":10446},[40,56486,10484],{"class":10363},[40,56488,11878],{"class":10446},[40,56490,11881],{"class":10732},[40,56492,11884],{"class":10363},[40,56494,56495,56497,56499,56501,56503,56505,56507,56509,56511],{"class":10276,"line":11320},[40,56496,11890],{"class":10363},[40,56498,162],{"class":10732},[40,56500,11156],{"class":10363},[40,56502,11897],{"class":10430},[40,56504,11900],{"class":10363},[40,56506,11904],{"class":11903},[40,56508,162],{"class":10732},[40,56510,11909],{"class":10287},[40,56512,10761],{"class":10363},[40,56514,56515,56517,56519,56521,56523,56525,56527,56529,56531],{"class":10276,"line":11336},[40,56516,11917],{"class":10363},[40,56518,162],{"class":10732},[40,56520,11156],{"class":10363},[40,56522,1813],{"class":10430},[40,56524,11926],{"class":10363},[40,56526,11752],{"class":10732},[40,56528,11156],{"class":10363},[40,56530,1820],{"class":10430},[40,56532,11935],{"class":10363},[40,56534,56535,56537,56539,56541,56543,56545,56547,56549,56551],{"class":10276,"line":11381},[40,56536,11941],{"class":10363},[40,56538,162],{"class":10732},[40,56540,11156],{"class":10363},[40,56542,1813],{"class":10430},[40,56544,11926],{"class":10363},[40,56546,11752],{"class":10732},[40,56548,11156],{"class":10363},[40,56550,1813],{"class":10430},[40,56552,11935],{"class":10363},[40,56554,56555,56557,56559,56561,56563],{"class":10276,"line":11395},[40,56556,11963],{"class":10363},[40,56558,162],{"class":10732},[40,56560,11156],{"class":10363},[40,56562,1820],{"class":10430},[40,56564,11972],{"class":10363},[40,56566,56567,56569,56571,56573,56575,56577,56579,56581,56583],{"class":10276,"line":11438},[40,56568,15978],{"class":10363},[40,56570,162],{"class":10732},[40,56572,11156],{"class":10363},[40,56574,11982],{"class":10430},[40,56576,11985],{"class":10363},[40,56578,11275],{"class":11903},[40,56580,162],{"class":10732},[40,56582,66],{"class":10446},[40,56584,10761],{"class":10363},[40,56586,56587,56589],{"class":10276,"line":11452},[40,56588,11201],{"class":10352},[40,56590,16001],{"class":10363},[40,56592,56593],{"class":10276,"line":11493},[40,56594,10301],{"emptyLinePlaceholder":10300},[40,56596,56597],{"class":10276,"line":11507},[40,56598,10301],{"emptyLinePlaceholder":10300},[40,56600,56601,56603,56605,56607,56609,56611,56613],{"class":10276,"line":11512},[40,56602,10965],{"class":10352},[40,56604,16016],{"class":10430},[40,56606,167],{"class":10363},[40,56608,12016],{"class":10973},[40,56610,12019],{"class":10363},[40,56612,1796],{"class":10973},[40,56614,12025],{"class":10363},[40,56616,56617,56619,56621,56623,56625,56627,56629,56631,56633,56635,56637,56639,56641,56643,56645,56647,56649,56651,56653,56655,56657,56659,56661,56663,56665,56667,56669,56671,56673],{"class":10276,"line":11517},[40,56618,11201],{"class":10352},[40,56620,12039],{"class":10446},[40,56622,12042],{"class":10732},[40,56624,12045],{"class":10363},[40,56626,2218],{"class":10430},[40,56628,167],{"class":10363},[40,56630,12052],{"class":10446},[40,56632,12055],{"class":10363},[40,56634,11243],{"class":10732},[40,56636,12060],{"class":10446},[40,56638,12055],{"class":10363},[40,56640,11752],{"class":10732},[40,56642,12067],{"class":10363},[40,56644,12070],{"class":10430},[40,56646,167],{"class":10363},[40,56648,949],{"class":10446},[40,56650,10484],{"class":10363},[40,56652,78],{"class":10446},[40,56654,10484],{"class":10363},[40,56656,78],{"class":10446},[40,56658,12055],{"class":10363},[40,56660,11752],{"class":10732},[40,56662,16080],{"class":10363},[40,56664,12092],{"class":10430},[40,56666,167],{"class":10363},[40,56668,11275],{"class":11903},[40,56670,162],{"class":10732},[40,56672,66],{"class":10446},[40,56674,10761],{"class":10363},[40,56676,56677],{"class":10276,"line":11545},[40,56678,10301],{"emptyLinePlaceholder":10300},[40,56680,56681],{"class":10276,"line":11557},[40,56682,10301],{"emptyLinePlaceholder":10300},[40,56684,56685,56687,56689,56691,56693],{"class":10276,"line":11563},[40,56686,10965],{"class":10352},[40,56688,18790],{"class":10430},[40,56690,167],{"class":10363},[40,56692,13346],{"class":10973},[40,56694,18797],{"class":10363},[40,56696,56697],{"class":10276,"line":11582},[40,56698,18802],{"class":10287},[40,56700,56701,56703,56705,56707],{"class":10276,"line":11596},[40,56702,16132],{"class":10363},[40,56704,162],{"class":10732},[40,56706,15410],{"class":10430},[40,56708,16139],{"class":10363},[40,56710,56711,56713,56715,56717],{"class":10276,"line":11609},[40,56712,16144],{"class":10363},[40,56714,162],{"class":10732},[40,56716,15840],{"class":10430},[40,56718,16151],{"class":10363},[40,56720,56721],{"class":10276,"line":11622},[40,56722,10301],{"emptyLinePlaceholder":10300},[40,56724,56725,56727,56729,56731,56733,56735,56737,56739,56741],{"class":10276,"line":11637},[40,56726,16165],{"class":10363},[40,56728,162],{"class":10732},[40,56730,11156],{"class":10363},[40,56732,1813],{"class":10430},[40,56734,11926],{"class":10363},[40,56736,11752],{"class":10732},[40,56738,11156],{"class":10363},[40,56740,1820],{"class":10430},[40,56742,11935],{"class":10363},[40,56744,56745,56747,56749,56751,56753,56755,56757,56759,56761],{"class":10276,"line":11708},[40,56746,16186],{"class":10363},[40,56748,162],{"class":10732},[40,56750,11156],{"class":10363},[40,56752,1813],{"class":10430},[40,56754,11926],{"class":10363},[40,56756,11752],{"class":10732},[40,56758,11156],{"class":10363},[40,56760,1813],{"class":10430},[40,56762,11935],{"class":10363},[40,56764,56765,56767,56769,56771,56773],{"class":10276,"line":11727},[40,56766,16207],{"class":10363},[40,56768,162],{"class":10732},[40,56770,11156],{"class":10363},[40,56772,1820],{"class":10430},[40,56774,11972],{"class":10363},[40,56776,56777],{"class":10276,"line":11744},[40,56778,10301],{"emptyLinePlaceholder":10300},[40,56780,56781,56783,56785,56787,56789,56791,56793,56795,56797,56799,56801,56803],{"class":10276,"line":11790},[40,56782,12982],{"class":10363},[40,56784,162],{"class":10732},[40,56786,12039],{"class":10446},[40,56788,11764],{"class":10732},[40,56790,12045],{"class":10363},[40,56792,2218],{"class":10430},[40,56794,167],{"class":10363},[40,56796,12052],{"class":10446},[40,56798,12055],{"class":10363},[40,56800,11243],{"class":10732},[40,56802,12060],{"class":10446},[40,56804,10761],{"class":10363},[40,56806,56807,56809,56811,56813,56815,56817,56819,56821,56823,56825,56827,56829],{"class":10276,"line":11795},[40,56808,13009],{"class":10363},[40,56810,162],{"class":10732},[40,56812,12039],{"class":10446},[40,56814,12042],{"class":10732},[40,56816,12045],{"class":10363},[40,56818,2218],{"class":10430},[40,56820,167],{"class":10363},[40,56822,12052],{"class":10446},[40,56824,12055],{"class":10363},[40,56826,11243],{"class":10732},[40,56828,12060],{"class":10446},[40,56830,10761],{"class":10363},[40,56832,56833,56835,56837,56839,56841,56843,56845,56847,56849,56851],{"class":10276,"line":11800},[40,56834,16281],{"class":10363},[40,56836,162],{"class":10732},[40,56838,16286],{"class":10430},[40,56840,167],{"class":10363},[40,56842,16291],{"class":11903},[40,56844,162],{"class":10732},[40,56846,13142],{"class":10363},[40,56848,13145],{"class":11903},[40,56850,162],{"class":10732},[40,56852,16302],{"class":10363},[40,56854,56855,56857,56859,56861,56863],{"class":10276,"line":11806},[40,56856,16307],{"class":10363},[40,56858,162],{"class":10732},[40,56860,13070],{"class":10363},[40,56862,16314],{"class":10430},[40,56864,12174],{"class":10363},[40,56866,56867,56869,56871,56873,56875,56877,56879,56881,56883,56885],{"class":10276,"line":11834},[40,56868,16321],{"class":10363},[40,56870,162],{"class":10732},[40,56872,16326],{"class":10430},[40,56874,167],{"class":10363},[40,56876,11192],{"class":11903},[40,56878,162],{"class":10732},[40,56880,16335],{"class":10363},[40,56882,16338],{"class":11903},[40,56884,162],{"class":10732},[40,56886,16343],{"class":10363},[40,56888,56889,56891,56893],{"class":10276,"line":11840},[40,56890,16348],{"class":10363},[40,56892,16351],{"class":10430},[40,56894,18999],{"class":10363},[40,56896,56897],{"class":10276,"line":11860},[40,56898,10301],{"emptyLinePlaceholder":10300},[40,56900,56901,56903,56905,56907,56909,56911,56913,56915,56917,56919,56921,56923],{"class":10276,"line":11887},[40,56902,16366],{"class":10363},[40,56904,162],{"class":10732},[40,56906,13070],{"class":10363},[40,56908,16373],{"class":10430},[40,56910,167],{"class":10363},[40,56912,13078],{"class":11903},[40,56914,162],{"class":10732},[40,56916,16382],{"class":10363},[40,56918,13086],{"class":11903},[40,56920,162],{"class":10732},[40,56922,13091],{"class":10446},[40,56924,10761],{"class":10363},[40,56926,56927,56929,56931,56933,56935,56937,56939,56941,56943,56945,56947],{"class":10276,"line":11914},[40,56928,16395],{"class":10363},[40,56930,162],{"class":10732},[40,56932,13170],{"class":10363},[40,56934,16402],{"class":10430},[40,56936,167],{"class":10363},[40,56938,16407],{"class":10446},[40,56940,10484],{"class":10363},[40,56942,16412],{"class":11903},[40,56944,162],{"class":10732},[40,56946,16417],{"class":10287},[40,56948,10761],{"class":10363},[40,56950,56951,56953,56955,56957,56959,56961,56963,56965,56967],{"class":10276,"line":11938},[40,56952,13227],{"class":10363},[40,56954,16426],{"class":10430},[40,56956,167],{"class":10363},[40,56958,16431],{"class":11903},[40,56960,162],{"class":10732},[40,56962,16436],{"class":10363},[40,56964,16439],{"class":11903},[40,56966,162],{"class":10732},[40,56968,16444],{"class":10363},[40,56970,56971,56973,56975,56977,56979],{"class":10276,"line":11960},[40,56972,11323],{"class":10352},[40,56974,16451],{"class":10732},[40,56976,13259],{"class":10363},[40,56978,16456],{"class":10287},[40,56980,13353],{"class":10363},[40,56982,56983,56985,56987,56989,56991,56993,56995,56997,56999],{"class":10276,"line":11975},[40,56984,16463],{"class":10363},[40,56986,16466],{"class":10430},[40,56988,16469],{"class":10363},[40,56990,78],{"class":10446},[40,56992,10484],{"class":10363},[40,56994,78],{"class":10446},[40,56996,10484],{"class":10363},[40,56998,78],{"class":10446},[40,57000,13612],{"class":10363},[40,57002,57003],{"class":10276,"line":11996},[40,57004,10301],{"emptyLinePlaceholder":10300},[40,57006,57007],{"class":10276,"line":12001},[40,57008,19114],{"class":10280},[40,57010,57011,57013,57015],{"class":10276,"line":12006},[40,57012,19119],{"class":10363},[40,57014,162],{"class":10732},[40,57016,19124],{"class":10446},[40,57018,57019,57021,57023,57025,57027,57029],{"class":10276,"line":12028},[40,57020,19129],{"class":10363},[40,57022,162],{"class":10732},[40,57024,18303],{"class":10430},[40,57026,19136],{"class":10363},[40,57028,13404],{"class":10732},[40,57030,13407],{"class":10363},[40,57032,57033,57035,57037,57039],{"class":10276,"line":12034},[40,57034,16495],{"class":10363},[40,57036,162],{"class":10732},[40,57038,16016],{"class":10430},[40,57040,19151],{"class":10363},[40,57042,57043,57045,57047,57049,57051,57053,57055,57057,57059,57061,57063,57065,57067],{"class":10276,"line":12105},[40,57044,16521],{"class":10446},[40,57046,10772],{"class":10732},[40,57048,16526],{"class":10430},[40,57050,167],{"class":10363},[40,57052,11192],{"class":10430},[40,57054,16533],{"class":10363},[40,57056,11056],{"class":10732},[40,57058,78],{"class":10446},[40,57060,16540],{"class":10363},[40,57062,11056],{"class":10732},[40,57064,78],{"class":10446},[40,57066,16547],{"class":10363},[40,57068,19180],{"class":10280},[40,57070,57071],{"class":10276,"line":12110},[40,57072,10301],{"emptyLinePlaceholder":10300},[40,57074,57075,57077,57079,57081,57083],{"class":10276,"line":12115},[40,57076,16559],{"class":10363},[40,57078,162],{"class":10732},[40,57080,12522],{"class":10363},[40,57082,16566],{"class":10430},[40,57084,12527],{"class":10363},[40,57086,57087],{"class":10276,"line":12121},[40,57088,16573],{"class":10363},[40,57090,57091,57093,57095,57097],{"class":10276,"line":12155},[40,57092,16578],{"class":11903},[40,57094,162],{"class":10732},[40,57096,16583],{"class":10446},[40,57098,10507],{"class":10363},[40,57100,57101,57103,57105,57107,57109,57111,57113,57115],{"class":10276,"line":12161},[40,57102,16590],{"class":11903},[40,57104,162],{"class":10732},[40,57106,78],{"class":10446},[40,57108,10484],{"class":10363},[40,57110,16599],{"class":11903},[40,57112,162],{"class":10732},[40,57114,78],{"class":10446},[40,57116,10507],{"class":10363},[40,57118,57119,57121,57123,57125,57127,57129,57131,57133],{"class":10276,"line":12177},[40,57120,16610],{"class":11903},[40,57122,162],{"class":10732},[40,57124,10504],{"class":10446},[40,57126,10484],{"class":10363},[40,57128,16619],{"class":11903},[40,57130,162],{"class":10732},[40,57132,66],{"class":10446},[40,57134,10507],{"class":10363},[40,57136,57137,57139,57141,57143],{"class":10276,"line":12192},[40,57138,16630],{"class":11903},[40,57140,162],{"class":10732},[40,57142,10504],{"class":10446},[40,57144,10507],{"class":10363},[40,57146,57147],{"class":10276,"line":12231},[40,57148,12631],{"class":10363},[40,57150,57151],{"class":10276,"line":12245},[40,57152,10301],{"emptyLinePlaceholder":10300},[40,57154,57155,57157,57159,57161,57163,57165,57167,57169,57171,57173,57175,57177,57179,57181,57183,57185],{"class":10276,"line":12280},[40,57156,13165],{"class":10363},[40,57158,162],{"class":10732},[40,57160,13170],{"class":10363},[40,57162,13173],{"class":10430},[40,57164,16666],{"class":10363},[40,57166,12429],{"class":11903},[40,57168,162],{"class":10732},[40,57170,13183],{"class":10363},[40,57172,13186],{"class":11903},[40,57174,162],{"class":10732},[40,57176,13191],{"class":10446},[40,57178,10484],{"class":10363},[40,57180,13196],{"class":11903},[40,57182,162],{"class":10732},[40,57184,13201],{"class":10446},[40,57186,10761],{"class":10363},[40,57188,57189,57191,57193,57195,57197],{"class":10276,"line":12294},[40,57190,13208],{"class":10363},[40,57192,13211],{"class":10430},[40,57194,167],{"class":10363},[40,57196,13216],{"class":10287},[40,57198,10761],{"class":10363},[40,57200,57201],{"class":10276,"line":12329},[40,57202,10301],{"emptyLinePlaceholder":10300},[40,57204,57205,57207,57209,57211,57213],{"class":10276,"line":12343},[40,57206,13227],{"class":10363},[40,57208,13230],{"class":10430},[40,57210,167],{"class":10363},[40,57212,10748],{"class":10287},[40,57214,10761],{"class":10363},[40,57216,57217,57219,57221,57223,57225],{"class":10276,"line":12376},[40,57218,13227],{"class":10363},[40,57220,13244],{"class":10430},[40,57222,167],{"class":10363},[40,57224,10753],{"class":10287},[40,57226,10761],{"class":10363},[40,57228,57229,57231,57233,57235,57237],{"class":10276,"line":12409},[40,57230,13227],{"class":10363},[40,57232,16740],{"class":10430},[40,57234,167],{"class":10363},[40,57236,10758],{"class":10287},[40,57238,10761],{"class":10363},[40,57240,57241,57243,57245,57247,57249],{"class":10276,"line":12414},[40,57242,16751],{"class":10363},[40,57244,162],{"class":10732},[40,57246,12522],{"class":10363},[40,57248,16758],{"class":10430},[40,57250,12527],{"class":10363},[40,57252,57253,57255,57257,57259,57261,57263],{"class":10276,"line":12419},[40,57254,12687],{"class":10446},[40,57256,10484],{"class":10363},[40,57258,16769],{"class":10446},[40,57260,10484],{"class":10363},[40,57262,19375],{"class":10287},[40,57264,10507],{"class":10363},[40,57266,57267,57269,57271,57273,57275,57277,57279,57281,57283,57285,57287,57289,57291,57293,57295],{"class":10276,"line":12435},[40,57268,12545],{"class":11903},[40,57270,162],{"class":10732},[40,57272,12550],{"class":10363},[40,57274,12211],{"class":11903},[40,57276,162],{"class":10732},[40,57278,12710],{"class":10287},[40,57280,10484],{"class":10363},[40,57282,12221],{"class":11903},[40,57284,162],{"class":10732},[40,57286,12226],{"class":10287},[40,57288,10484],{"class":10363},[40,57290,12553],{"class":11903},[40,57292,162],{"class":10732},[40,57294,12731],{"class":10446},[40,57296,10507],{"class":10363},[40,57298,57299,57301,57303,57305,57307,57309,57311,57313,57315,57317,57319,57321,57323,57325,57327,57329,57331,57333,57335],{"class":10276,"line":12447},[40,57300,12574],{"class":11903},[40,57302,12577],{"class":10732},[40,57304,167],{"class":10363},[40,57306,12582],{"class":11903},[40,57308,162],{"class":10732},[40,57310,12587],{"class":10287},[40,57312,10484],{"class":10363},[40,57314,12592],{"class":11903},[40,57316,162],{"class":10732},[40,57318,12597],{"class":10287},[40,57320,10484],{"class":10363},[40,57322,12602],{"class":11903},[40,57324,162],{"class":10732},[40,57326,12607],{"class":10446},[40,57328,10484],{"class":10363},[40,57330,12612],{"class":11903},[40,57332,162],{"class":10732},[40,57334,12617],{"class":10287},[40,57336,11061],{"class":10363},[40,57338,57339,57341,57343,57345],{"class":10276,"line":12463},[40,57340,12780],{"class":11903},[40,57342,162],{"class":10732},[40,57344,12785],{"class":10446},[40,57346,10507],{"class":10363},[40,57348,57349],{"class":10276,"line":12480},[40,57350,12631],{"class":10363},[40,57352,57353,57355,57357,57359,57361],{"class":10276,"line":12498},[40,57354,16881],{"class":10363},[40,57356,162],{"class":10732},[40,57358,12522],{"class":10363},[40,57360,16758],{"class":10430},[40,57362,12527],{"class":10363},[40,57364,57365,57367,57369,57371,57373,57375,57377,57379,57381,57383,57385,57387,57389,57391,57393,57395,57397],{"class":10276,"line":12504},[40,57366,16894],{"class":10446},[40,57368,10484],{"class":10363},[40,57370,16899],{"class":10446},[40,57372,10484],{"class":10363},[40,57374,13303],{"class":10287},[40,57376,10484],{"class":10363},[40,57378,16908],{"class":11903},[40,57380,162],{"class":10732},[40,57382,12550],{"class":10363},[40,57384,12211],{"class":11903},[40,57386,162],{"class":10732},[40,57388,12216],{"class":10287},[40,57390,10484],{"class":10363},[40,57392,12221],{"class":11903},[40,57394,162],{"class":10732},[40,57396,12226],{"class":10287},[40,57398,10507],{"class":10363},[40,57400,57401,57403,57405],{"class":10276,"line":12517},[40,57402,12726],{"class":11903},[40,57404,162],{"class":10732},[40,57406,16937],{"class":10363},[40,57408,57409,57411,57413,57415,57417,57419,57421,57423,57425,57427,57429,57431,57433,57435,57437,57439,57441,57443,57445],{"class":10276,"line":12530},[40,57410,12574],{"class":11903},[40,57412,12577],{"class":10732},[40,57414,167],{"class":10363},[40,57416,12582],{"class":11903},[40,57418,162],{"class":10732},[40,57420,12587],{"class":10287},[40,57422,10484],{"class":10363},[40,57424,12592],{"class":11903},[40,57426,162],{"class":10732},[40,57428,12597],{"class":10287},[40,57430,10484],{"class":10363},[40,57432,12602],{"class":11903},[40,57434,162],{"class":10732},[40,57436,12607],{"class":10446},[40,57438,10484],{"class":10363},[40,57440,12612],{"class":11903},[40,57442,162],{"class":10732},[40,57444,12617],{"class":10287},[40,57446,11061],{"class":10363},[40,57448,57449,57451,57453,57455],{"class":10276,"line":12542},[40,57450,12780],{"class":11903},[40,57452,162],{"class":10732},[40,57454,12785],{"class":10446},[40,57456,10507],{"class":10363},[40,57458,57459],{"class":10276,"line":12571},[40,57460,12631],{"class":10363},[40,57462,57463],{"class":10276,"line":12622},[40,57464,10301],{"emptyLinePlaceholder":10300},[40,57466,57467,57469,57471,57473,57475,57477,57479],{"class":10276,"line":12628},[40,57468,13338],{"class":10352},[40,57470,13341],{"class":10430},[40,57472,167],{"class":10363},[40,57474,17010],{"class":10973},[40,57476,10443],{"class":10363},[40,57478,10821],{"class":10732},[40,57480,13353],{"class":10363},[40,57482,57483,57485,57487,57489,57491,57493,57495,57497,57499,57501,57503,57505],{"class":10276,"line":12634},[40,57484,13358],{"class":10363},[40,57486,162],{"class":10732},[40,57488,17027],{"class":10363},[40,57490,11243],{"class":10732},[40,57492,17032],{"class":10363},[40,57494,11056],{"class":10732},[40,57496,13373],{"class":10446},[40,57498,13376],{"class":10363},[40,57500,11752],{"class":10732},[40,57502,11156],{"class":10363},[40,57504,14527],{"class":10430},[40,57506,19620],{"class":10363},[40,57508,57509,57511,57513,57515,57517,57519],{"class":10276,"line":12639},[40,57510,17050],{"class":10363},[40,57512,162],{"class":10732},[40,57514,18303],{"class":10430},[40,57516,19631],{"class":10363},[40,57518,13404],{"class":10732},[40,57520,13407],{"class":10363},[40,57522,57523,57525,57527,57529],{"class":10276,"line":12644},[40,57524,17067],{"class":10363},[40,57526,162],{"class":10732},[40,57528,16016],{"class":10430},[40,57530,17074],{"class":10363},[40,57532,57533,57535,57537,57539,57541,57543,57545,57547,57549,57551,57553,57555,57557],{"class":10276,"line":12667},[40,57534,17080],{"class":10446},[40,57536,10772],{"class":10732},[40,57538,16526],{"class":10430},[40,57540,167],{"class":10363},[40,57542,11192],{"class":10430},[40,57544,17091],{"class":10363},[40,57546,11056],{"class":10732},[40,57548,78],{"class":10446},[40,57550,16540],{"class":10363},[40,57552,11056],{"class":10732},[40,57554,78],{"class":10446},[40,57556,19672],{"class":10363},[40,57558,19180],{"class":10280},[40,57560,57561,57563,57565,57567,57569,57571,57573,57575,57577,57579,57581,57583,57585],{"class":10276,"line":12673},[40,57562,17113],{"class":10363},[40,57564,17116],{"class":10430},[40,57566,167],{"class":10363},[40,57568,17121],{"class":10446},[40,57570,1123],{"class":10363},[40,57572,12070],{"class":10430},[40,57574,167],{"class":10363},[40,57576,11056],{"class":10732},[40,57578,78],{"class":10446},[40,57580,10484],{"class":10363},[40,57582,17136],{"class":10446},[40,57584,19701],{"class":10363},[40,57586,19704],{"class":10280},[40,57588,57589,57591,57593],{"class":10276,"line":12684},[40,57590,17148],{"class":10363},[40,57592,13431],{"class":10430},[40,57594,12527],{"class":10363},[40,57596,57597,57599,57601,57603,57605,57607,57609,57611,57613,57615,57617,57619,57621,57623,57625,57627,57629,57631,57633,57635,57637,57639,57641,57643,57645],{"class":10276,"line":12697},[40,57598,14580],{"class":10352},[40,57600,14583],{"class":10287},[40,57602,13268],{"class":10446},[40,57604,19723],{"class":10363},[40,57606,66],{"class":10446},[40,57608,1664],{"class":10363},[40,57610,14595],{"class":10352},[40,57612,13280],{"class":10446},[40,57614,10484],{"class":10287},[40,57616,13268],{"class":10446},[40,57618,19723],{"class":10363},[40,57620,78],{"class":10446},[40,57622,1664],{"class":10363},[40,57624,14595],{"class":10352},[40,57626,13280],{"class":10446},[40,57628,10484],{"class":10287},[40,57630,13268],{"class":10446},[40,57632,19723],{"class":10363},[40,57634,1044],{"class":10446},[40,57636,1664],{"class":10363},[40,57638,14595],{"class":10352},[40,57640,13280],{"class":10446},[40,57642,207],{"class":10287},[40,57644,13454],{"class":10732},[40,57646,14632],{"class":10287},[40,57648,57649,57651,57653,57655,57657,57659,57661,57663,57665,57667,57669,57671,57673,57675],{"class":10276,"line":12723},[40,57650,14580],{"class":10352},[40,57652,13438],{"class":10287},[40,57654,13268],{"class":10446},[40,57656,14643],{"class":10363},[40,57658,14646],{"class":10430},[40,57660,14649],{"class":10363},[40,57662,14652],{"class":10352},[40,57664,13280],{"class":10446},[40,57666,14657],{"class":10287},[40,57668,13268],{"class":10446},[40,57670,19790],{"class":10363},[40,57672,14652],{"class":10352},[40,57674,13280],{"class":10446},[40,57676,14669],{"class":10287},[40,57678,57679],{"class":10276,"line":12736},[40,57680,14674],{"class":10363},[40,57682,57683,57685],{"class":10276,"line":12777},[40,57684,11339],{"class":10352},[40,57686,17199],{"class":10363},[40,57688,57689],{"class":10276,"line":12790},[40,57690,10301],{"emptyLinePlaceholder":10300},[40,57692,57693,57695,57697,57699,57701,57703,57705,57707,57709,57711,57713,57715,57717,57719,57721],{"class":10276,"line":16980},[40,57694,13496],{"class":10363},[40,57696,162],{"class":10732},[40,57698,13501],{"class":10430},[40,57700,13504],{"class":10363},[40,57702,13473],{"class":11903},[40,57704,162],{"class":10732},[40,57706,17222],{"class":10363},[40,57708,13514],{"class":11903},[40,57710,162],{"class":10732},[40,57712,13519],{"class":10446},[40,57714,10484],{"class":10363},[40,57716,13524],{"class":11903},[40,57718,162],{"class":10732},[40,57720,10504],{"class":10446},[40,57722,10761],{"class":10363},[40,57724,57725,57727,57729,57731],{"class":10276,"line":16991},[40,57726,17244],{"class":10363},[40,57728,162],{"class":10732},[40,57730,13859],{"class":10430},[40,57732,19853],{"class":10363},[40,57734,57735,57737,57739,57741,57743,57745,57747,57749,57751,57753,57755],{"class":10276,"line":16996},[40,57736,17272],{"class":10363},[40,57738,13538],{"class":10430},[40,57740,167],{"class":10363},[40,57742,13543],{"class":11903},[40,57744,162],{"class":10732},[40,57746,13091],{"class":10446},[40,57748,10484],{"class":10363},[40,57750,13552],{"class":11903},[40,57752,162],{"class":10732},[40,57754,13091],{"class":10446},[40,57756,10761],{"class":10363},[40,57758,57759,57761,57763,57765,57767,57769,57771,57773,57775,57777,57779,57781],{"class":10276,"line":17001},[40,57760,13563],{"class":10363},[40,57762,13566],{"class":10430},[40,57764,17302],{"class":10363},[40,57766,13572],{"class":10430},[40,57768,13575],{"class":10363},[40,57770,13578],{"class":11903},[40,57772,162],{"class":10732},[40,57774,13583],{"class":10430},[40,57776,167],{"class":10363},[40,57778,13588],{"class":11903},[40,57780,162],{"class":10732},[40,57782,17321],{"class":10363},[40,57784,57785,57787,57789,57791,57793,57795,57797],{"class":10276,"line":17019},[40,57786,13598],{"class":11903},[40,57788,12577],{"class":10732},[40,57790,167],{"class":10363},[40,57792,13605],{"class":11903},[40,57794,162],{"class":10732},[40,57796,12597],{"class":10287},[40,57798,13612],{"class":10363},[40,57800,57801,57803,57805],{"class":10276,"line":17047},[40,57802,13617],{"class":10363},[40,57804,13620],{"class":10430},[40,57806,13623],{"class":10363},[40,57808,57809,57811],{"class":10276,"line":17064},[40,57810,11201],{"class":10352},[40,57812,17355],{"class":10363},[40,57814,57815],{"class":10276,"line":17077},[40,57816,10301],{"emptyLinePlaceholder":10300},[40,57818,57819],{"class":10276,"line":17110},[40,57820,10301],{"emptyLinePlaceholder":10300},[40,57822,57823],{"class":10276,"line":17145},[40,57824,19946],{"class":10280},[40,57826,57827,57829,57831,57833],{"class":10276,"line":17194},[40,57828,19951],{"class":10446},[40,57830,10772],{"class":10732},[40,57832,17867],{"class":10430},[40,57834,12527],{"class":10363},[40,57836,57837,57839,57841,57843,57845,57847,57849,57851,57853,57855,57857,57859,57861],{"class":10276,"line":17202},[40,57838,14837],{"class":11903},[40,57840,162],{"class":10732},[40,57842,167],{"class":10363},[40,57844,10801],{"class":10446},[40,57846,10484],{"class":10363},[40,57848,10810],{"class":10446},[40,57850,10484],{"class":10363},[40,57852,10801],{"class":10446},[40,57854,11040],{"class":10363},[40,57856,14662],{"class":11903},[40,57858,162],{"class":10732},[40,57860,14863],{"class":10446},[40,57862,10507],{"class":10363},[40,57864,57865,57867,57869,57871,57873,57875,57877,57879],{"class":10276,"line":17207},[40,57866,13684],{"class":11903},[40,57868,162],{"class":10732},[40,57870,13689],{"class":10287},[40,57872,10484],{"class":10363},[40,57874,19998],{"class":11903},[40,57876,162],{"class":10732},[40,57878,20003],{"class":10287},[40,57880,10507],{"class":10363},[40,57882,57883],{"class":10276,"line":17241},[40,57884,10761],{"class":10363},[40,57886,57887,57889,57891,57893,57895,57897],{"class":10276,"line":17269},[40,57888,20014],{"class":10363},[40,57890,162],{"class":10732},[40,57892,18790],{"class":10430},[40,57894,167],{"class":10363},[40,57896,19951],{"class":10446},[40,57898,10761],{"class":10363},[40,57900,57901,57903,57905,57907,57909,57911,57913,57915,57917],{"class":10276,"line":17295},[40,57902,14975],{"class":10430},[40,57904,167],{"class":10363},[40,57906,13953],{"class":10430},[40,57908,167],{"class":10363},[40,57910,13958],{"class":11903},[40,57912,162],{"class":10732},[40,57914,20041],{"class":10363},[40,57916,13572],{"class":10430},[40,57918,20046],{"class":10363},[40,57920,57921,57923,57925,57927],{"class":10276,"line":17324},[40,57922,14993],{"class":10732},[40,57924,167],{"class":10363},[40,57926,13974],{"class":10287},[40,57928,20057],{"class":10363},[10082,57930,57932],{"className":57931,"code":20061,"language":2932,"meta":10090},[10085],[10088,57933,20061],{"__ignoreMap":10090},[13997,57935],{"alt":20066,"src":20067},[20069,57937,20071],{},{"title":10090,"searchDepth":10284,"depth":10284,"links":57939},[57940,57941,57942,57943,57944,57945,57946,57947,57948,57949],{"id":30,"depth":10284,"text":31},{"id":1994,"depth":10284,"text":1995},{"id":4114,"depth":10284,"text":20077},{"id":5126,"depth":10284,"text":5127},{"id":5935,"depth":10284,"text":20080},{"id":8078,"depth":10284,"text":20082},{"id":8992,"depth":10284,"text":8993},{"id":9740,"depth":10284,"text":9741},{"id":10076,"depth":10284,"text":10077},{"id":10095,"depth":10284,"text":10096},[20088,20089,20090,20091],[57952],{"username":7,"name":20094,"role":20095,"bio":20096,"links":57953},[57954,57955],{"label":20099,"href":20100},{"label":20102,"href":21},{"username":7,"name":20094,"role":20104},{},[],{"title":5,"description":20106},[20120,20121],{"id":57962,"title":57963,"authors":57964,"body":57965,"breadcrumb":83800,"builders":83802,"byline":83807,"challenge":20105,"courseAuthor":20105,"courseLead":20105,"dek":20105,"description":83808,"draft":20107,"extension":20108,"eyebrow":27279,"featured":10300,"finish":20105,"fork":20105,"hero":20105,"heroAlt":20105,"heroCta":20105,"heroImage":20105,"homepageFeatured":20107,"kind":20110,"lessonCount":20105,"meta":83809,"navigation":10300,"newsItems":20105,"next":20105,"ogImage":83810,"order":20105,"outcomes":20105,"path":83811,"publishDate":83812,"readingTime":83813,"related":83814,"relatedProjects":20105,"seo":83815,"stem":83816,"tags":83817,"track":20105,"trackName":20105,"__hash__":83818},"blog\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization.md","Rabi Oscillations: Visualizing Excited-State Probability",[7],{"type":9,"value":57966,"toc":83783},[57967,57974,57977,58086,58149,58153,58156,59621,59625,59706,59918,59921,60072,60075,60198,60307,60407,60410,60414,60417,60538,60598,60756,60759,60981,60984,61713,61750,61754,61757,62091,62221,62531,62534,62761,62879,62883,62953,63423,63425,63657,63660,63839,63894,64095,64098,64101,64352,64384,64664,64667,64817,64849,64969,64972,65239,65242,65360,65364,65367,65733,65736,65870,65877,65881,65884,66026,66029,66571,66574,66800,66976,66980,67054,67325,67327,67392,67424,67591,67660,67885,67888,67891,68252,68254,68578,68581,68585,68588,68695,68727,68911,68954,68958,69089,69335,69338,69731,69734,69945,69948,69952,70425,70885,71004,71123,71126,71130,71231,71234,71333,71416,71601,71604,71804,71807,71908,72305,72309,72312,72797,72800,72804,72807,73066,73069,73282,73285,73387,73390,73420,73731,73737,74217,74308,74314,74490,74496,74649,74653,79477,79481,79487,80189,80193,80839,80843,80847,82002,82008,82208,82489,82493,83298,83302,83758,83762,83766,83770,83780],[12,57968,57969],{},[15,57970,17,57971,23],{},[19,57972,22],{"href":57973},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb",[12,57975,57976],{},"This notebook provides a primer on the quantum dynamics of a driven two-level system by visualizing the excited-state probability under the rotating-wave approximation (RWA). The model treats a qubit as an ideal two-state system driven by a classical microwave tone. This formulation maps the probability of measuring the excited state as a function of drive frequency and pulse duration. The central computational object is the probability surface",[40,57978,57980],{"className":57979},[136],[40,57981,57983,58015],{"className":57982},[43],[40,57984,57986],{"className":57985},[47],[49,57987,57988],{"xmlns":51,"display":145},[53,57989,57990,58012],{},[56,57991,57992,57999,58001,58003,58005,58008,58010],{},[151,57993,57994,57997],{},[154,57995,57996],{},"P",[154,57998,4203],{},[59,58000,167],{"stretchy":61},[154,58002,13262],{},[59,58004,73],{"separator":72},[154,58006,58007],{},"τ",[59,58009,207],{"stretchy":61},[59,58011,73],{"separator":72},[82,58013,58014],{"encoding":84},"P_e(f,\\tau),",[40,58016,58018],{"className":58017,"ariaHidden":72},[89],[40,58019,58021,58024,58064,58067,58070,58073,58076,58080,58083],{"className":58020},[93],[40,58022],{"className":58023,"style":98},[97],[40,58025,58027,58030],{"className":58026},[102],[40,58028,57996],{"className":58029,"style":2541},[102,335],[40,58031,58033],{"className":58032},[340],[40,58034,58036,58056],{"className":58035},[344,345],[40,58037,58039,58053],{"className":58038},[349],[40,58040,58042],{"className":58041,"style":354},[353],[40,58043,58044,58047],{"style":2556},[40,58045],{"className":58046,"style":362},[361],[40,58048,58050],{"className":58049},[366,367,368,369],[40,58051,4203],{"className":58052},[102,335,369],[40,58054,377],{"className":58055},[376],[40,58057,58059],{"className":58058},[349],[40,58060,58062],{"className":58061,"style":384},[353],[40,58063],{},[40,58065,167],{"className":58066},[106],[40,58068,13262],{"className":58069,"style":33300},[102,335],[40,58071,73],{"className":58072},[117],[40,58074],{"className":58075,"style":122},[121],[40,58077,58007],{"className":58078,"style":58079},[102,335],"margin-right:0.1132em;",[40,58081,207],{"className":58082},[113],[40,58084,73],{"className":58085},[117],[12,58087,58088,58089,58118,58119,58148],{},"where ",[40,58090,58092,58105],{"className":58091},[43],[40,58093,58095],{"className":58094},[47],[49,58096,58097],{"xmlns":51},[53,58098,58099,58103],{},[56,58100,58101],{},[154,58102,13262],{},[82,58104,13262],{"encoding":84},[40,58106,58108],{"className":58107,"ariaHidden":72},[89],[40,58109,58111,58115],{"className":58110},[93],[40,58112],{"className":58113,"style":58114},[97],"height:0.8889em;vertical-align:-0.1944em;",[40,58116,13262],{"className":58117,"style":33300},[102,335]," denotes the applied drive frequency and ",[40,58120,58122,58136],{"className":58121},[43],[40,58123,58125],{"className":58124},[47],[49,58126,58127],{"xmlns":51},[53,58128,58129,58133],{},[56,58130,58131],{},[154,58132,58007],{},[82,58134,58135],{"encoding":84},"\\tau",[40,58137,58139],{"className":58138,"ariaHidden":72},[89],[40,58140,58142,58145],{"className":58141},[93],[40,58143],{"className":58144,"style":4142},[97],[40,58146,58007],{"className":58147,"style":58079},[102,335]," denotes the microwave pulse duration. The notebook renders this probability surface with two-dimensional heatmaps, three-dimensional surfaces, one-dimensional cross sections, and Fast Fourier transform (FFT) spectrograms. Fourier analysis exposes the generalized Rabi frequency as a theoretical ridge in the frequency domain, which connects the time-domain quantum oscillation to its spectral structure.",[28,58150,58152],{"id":58151},"nomenclature-and-definitions","Nomenclature and Definitions",[12,58154,58155],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[20142,58157,58158,58168],{},[20145,58159,58160],{},[20148,58161,58162,58165],{},[20151,58163,58164],{},"Symbol or Term",[20151,58166,58167],{},"Meaning",[20162,58169,58170,58217,58263,58309,58436,58513,58590,58625,58845,58942,58977,59023,59173,59250,59475,59510,59605,59613],{},[20148,58171,58172,58214],{},[20167,58173,58174],{},[40,58175,58177,58196],{"className":58176},[43],[40,58178,58180],{"className":58179},[47],[49,58181,58182],{"xmlns":51},[53,58183,58184,58193],{},[56,58185,58186,58188,58191],{},[59,58187,62],{"stretchy":61},[154,58189,58190],{},"g",[59,58192,69],{"stretchy":61},[82,58194,58195],{"encoding":84},"\\lvert g\\rangle",[40,58197,58199],{"className":58198,"ariaHidden":72},[89],[40,58200,58202,58205,58208,58211],{"className":58201},[93],[40,58203],{"className":58204,"style":98},[97],[40,58206,62],{"className":58207},[106],[40,58209,58190],{"className":58210,"style":336},[102,335],[40,58212,69],{"className":58213},[113],[20167,58215,58216],{},"Ground state of the qubit",[20148,58218,58219,58260],{},[20167,58220,58221],{},[40,58222,58224,58242],{"className":58223},[43],[40,58225,58227],{"className":58226},[47],[49,58228,58229],{"xmlns":51},[53,58230,58231,58239],{},[56,58232,58233,58235,58237],{},[59,58234,62],{"stretchy":61},[154,58236,4203],{},[59,58238,69],{"stretchy":61},[82,58240,58241],{"encoding":84},"\\lvert e\\rangle",[40,58243,58245],{"className":58244,"ariaHidden":72},[89],[40,58246,58248,58251,58254,58257],{"className":58247},[93],[40,58249],{"className":58250,"style":98},[97],[40,58252,62],{"className":58253},[106],[40,58255,4203],{"className":58256},[102,335],[40,58258,69],{"className":58259},[113],[20167,58261,58262],{},"Excited state of the qubit",[20148,58264,58265,58306],{},[20167,58266,58267],{},[40,58268,58270,58288],{"className":58269},[43],[40,58271,58273],{"className":58272},[47],[49,58274,58275],{"xmlns":51},[53,58276,58277,58285],{},[56,58278,58279,58281,58283],{},[59,58280,62],{"stretchy":61},[154,58282,1506],{},[59,58284,69],{"stretchy":61},[82,58286,58287],{"encoding":84},"\\lvert \\psi\\rangle",[40,58289,58291],{"className":58290,"ariaHidden":72},[89],[40,58292,58294,58297,58300,58303],{"className":58293},[93],[40,58295],{"className":58296,"style":98},[97],[40,58298,62],{"className":58299},[106],[40,58301,1506],{"className":58302,"style":336},[102,335],[40,58304,69],{"className":58305},[113],[20167,58307,58308],{},"Quantum state vector in a two-dimensional Hilbert space",[20148,58310,58311,58357],{},[20167,58312,58313],{},[40,58314,58316,58335],{"className":58315},[43],[40,58317,58319],{"className":58318},[47],[49,58320,58321],{"xmlns":51},[53,58322,58323,58332],{},[56,58324,58325,58327,58329],{},[154,58326,5215],{},[59,58328,73],{"separator":72},[154,58330,58331],{},"β",[82,58333,58334],{"encoding":84},"\\alpha,\\beta",[40,58336,58338],{"className":58337,"ariaHidden":72},[89],[40,58339,58341,58344,58347,58350,58353],{"className":58340},[93],[40,58342],{"className":58343,"style":58114},[97],[40,58345,5215],{"className":58346,"style":5231},[102,335],[40,58348,73],{"className":58349},[117],[40,58351],{"className":58352,"style":122},[121],[40,58354,58331],{"className":58355,"style":58356},[102,335],"margin-right:0.0528em;",[20167,58358,58359,58360,916,58398],{},"Complex probability amplitudes for ",[40,58361,58363,58380],{"className":58362},[43],[40,58364,58366],{"className":58365},[47],[49,58367,58368],{"xmlns":51},[53,58369,58370,58378],{},[56,58371,58372,58374,58376],{},[59,58373,62],{"stretchy":61},[154,58375,58190],{},[59,58377,69],{"stretchy":61},[82,58379,58195],{"encoding":84},[40,58381,58383],{"className":58382,"ariaHidden":72},[89],[40,58384,58386,58389,58392,58395],{"className":58385},[93],[40,58387],{"className":58388,"style":98},[97],[40,58390,62],{"className":58391},[106],[40,58393,58190],{"className":58394,"style":336},[102,335],[40,58396,69],{"className":58397},[113],[40,58399,58401,58418],{"className":58400},[43],[40,58402,58404],{"className":58403},[47],[49,58405,58406],{"xmlns":51},[53,58407,58408,58416],{},[56,58409,58410,58412,58414],{},[59,58411,62],{"stretchy":61},[154,58413,4203],{},[59,58415,69],{"stretchy":61},[82,58417,58241],{"encoding":84},[40,58419,58421],{"className":58420,"ariaHidden":72},[89],[40,58422,58424,58427,58430,58433],{"className":58423},[93],[40,58425],{"className":58426,"style":98},[97],[40,58428,62],{"className":58429},[106],[40,58431,4203],{"className":58432},[102,335],[40,58434,69],{"className":58435},[113],[20148,58437,58438,58510],{},[20167,58439,58440],{},[40,58441,58443,58461],{"className":58442},[43],[40,58444,58446],{"className":58445},[47],[49,58447,58448],{"xmlns":51},[53,58449,58450,58458],{},[56,58451,58452],{},[151,58453,58454,58456],{},[154,58455,57996],{},[154,58457,4203],{},[82,58459,58460],{"encoding":84},"P_e",[40,58462,58464],{"className":58463,"ariaHidden":72},[89],[40,58465,58467,58470],{"className":58466},[93],[40,58468],{"className":58469,"style":3106},[97],[40,58471,58473,58476],{"className":58472},[102],[40,58474,57996],{"className":58475,"style":2541},[102,335],[40,58477,58479],{"className":58478},[340],[40,58480,58482,58502],{"className":58481},[344,345],[40,58483,58485,58499],{"className":58484},[349],[40,58486,58488],{"className":58487,"style":354},[353],[40,58489,58490,58493],{"style":2556},[40,58491],{"className":58492,"style":362},[361],[40,58494,58496],{"className":58495},[366,367,368,369],[40,58497,4203],{"className":58498},[102,335,369],[40,58500,377],{"className":58501},[376],[40,58503,58505],{"className":58504},[349],[40,58506,58508],{"className":58507,"style":384},[353],[40,58509],{},[20167,58511,58512],{},"Probability of measuring the qubit in the excited state",[20148,58514,58515,58587],{},[20167,58516,58517],{},[40,58518,58520,58538],{"className":58519},[43],[40,58521,58523],{"className":58522},[47],[49,58524,58525],{"xmlns":51},[53,58526,58527,58535],{},[56,58528,58529],{},[151,58530,58531,58533],{},[154,58532,13262],{},[64,58534,66],{},[82,58536,58537],{"encoding":84},"f_0",[40,58539,58541],{"className":58540,"ariaHidden":72},[89],[40,58542,58544,58547],{"className":58543},[93],[40,58545],{"className":58546,"style":58114},[97],[40,58548,58550,58553],{"className":58549},[102],[40,58551,13262],{"className":58552,"style":33300},[102,335],[40,58554,58556],{"className":58555},[340],[40,58557,58559,58579],{"className":58558},[344,345],[40,58560,58562,58576],{"className":58561},[349],[40,58563,58565],{"className":58564,"style":7201},[353],[40,58566,58567,58570],{"style":33315},[40,58568],{"className":58569,"style":362},[361],[40,58571,58573],{"className":58572},[366,367,368,369],[40,58574,66],{"className":58575},[102,369],[40,58577,377],{"className":58578},[376],[40,58580,58582],{"className":58581},[349],[40,58583,58585],{"className":58584,"style":384},[353],[40,58586],{},[20167,58588,58589],{},"Natural transition frequency of the qubit",[20148,58591,58592,58622],{},[20167,58593,58594],{},[40,58595,58597,58610],{"className":58596},[43],[40,58598,58600],{"className":58599},[47],[49,58601,58602],{"xmlns":51},[53,58603,58604,58608],{},[56,58605,58606],{},[154,58607,13262],{},[82,58609,13262],{"encoding":84},[40,58611,58613],{"className":58612,"ariaHidden":72},[89],[40,58614,58616,58619],{"className":58615},[93],[40,58617],{"className":58618,"style":58114},[97],[40,58620,13262],{"className":58621,"style":33300},[102,335],[20167,58623,58624],{},"Applied drive frequency",[20148,58626,58627,58699],{},[20167,58628,58629],{},[40,58630,58632,58650],{"className":58631},[43],[40,58633,58635],{"className":58634},[47],[49,58636,58637],{"xmlns":51},[53,58638,58639,58647],{},[56,58640,58641],{},[151,58642,58643,58645],{},[154,58644,20187],{},[64,58646,66],{},[82,58648,58649],{"encoding":84},"\\omega_0",[40,58651,58653],{"className":58652,"ariaHidden":72},[89],[40,58654,58656,58659],{"className":58655},[93],[40,58657],{"className":58658,"style":328},[97],[40,58660,58662,58665],{"className":58661},[102],[40,58663,20187],{"className":58664,"style":336},[102,335],[40,58666,58668],{"className":58667},[340],[40,58669,58671,58691],{"className":58670},[344,345],[40,58672,58674,58688],{"className":58673},[349],[40,58675,58677],{"className":58676,"style":7201},[353],[40,58678,58679,58682],{"style":357},[40,58680],{"className":58681,"style":362},[361],[40,58683,58685],{"className":58684},[366,367,368,369],[40,58686,66],{"className":58687},[102,369],[40,58689,377],{"className":58690},[376],[40,58692,58694],{"className":58693},[349],[40,58695,58697],{"className":58696,"style":384},[353],[40,58698],{},[20167,58700,58701,58702],{},"Angular transition frequency, ",[40,58703,58705,58735],{"className":58704},[43],[40,58706,58708],{"className":58707},[47],[49,58709,58710],{"xmlns":51},[53,58711,58712,58732],{},[56,58713,58714,58720,58722,58724,58726],{},[151,58715,58716,58718],{},[154,58717,20187],{},[64,58719,66],{},[59,58721,162],{},[64,58723,1044],{},[154,58725,1661],{},[151,58727,58728,58730],{},[154,58729,13262],{},[64,58731,66],{},[82,58733,58734],{"encoding":84},"\\omega_0=2\\pi f_0",[40,58736,58738,58793],{"className":58737,"ariaHidden":72},[89],[40,58739,58741,58744,58784,58787,58790],{"className":58740},[93],[40,58742],{"className":58743,"style":328},[97],[40,58745,58747,58750],{"className":58746},[102],[40,58748,20187],{"className":58749,"style":336},[102,335],[40,58751,58753],{"className":58752},[340],[40,58754,58756,58776],{"className":58755},[344,345],[40,58757,58759,58773],{"className":58758},[349],[40,58760,58762],{"className":58761,"style":7201},[353],[40,58763,58764,58767],{"style":357},[40,58765],{"className":58766,"style":362},[361],[40,58768,58770],{"className":58769},[366,367,368,369],[40,58771,66],{"className":58772},[102,369],[40,58774,377],{"className":58775},[376],[40,58777,58779],{"className":58778},[349],[40,58780,58782],{"className":58781,"style":384},[353],[40,58783],{},[40,58785],{"className":58786,"style":390},[121],[40,58788,162],{"className":58789},[394],[40,58791],{"className":58792,"style":390},[121],[40,58794,58796,58799,58802,58805],{"className":58795},[93],[40,58797],{"className":58798,"style":58114},[97],[40,58800,1044],{"className":58801},[102],[40,58803,1661],{"className":58804,"style":336},[102,335],[40,58806,58808,58811],{"className":58807},[102],[40,58809,13262],{"className":58810,"style":33300},[102,335],[40,58812,58814],{"className":58813},[340],[40,58815,58817,58837],{"className":58816},[344,345],[40,58818,58820,58834],{"className":58819},[349],[40,58821,58823],{"className":58822,"style":7201},[353],[40,58824,58825,58828],{"style":33315},[40,58826],{"className":58827,"style":362},[361],[40,58829,58831],{"className":58830},[366,367,368,369],[40,58832,66],{"className":58833},[102,369],[40,58835,377],{"className":58836},[376],[40,58838,58840],{"className":58839},[349],[40,58841,58843],{"className":58842,"style":384},[353],[40,58844],{},[20148,58846,58847,58878],{},[20167,58848,58849],{},[40,58850,58852,58866],{"className":58851},[43],[40,58853,58855],{"className":58854},[47],[49,58856,58857],{"xmlns":51},[53,58858,58859,58863],{},[56,58860,58861],{},[154,58862,20187],{},[82,58864,58865],{"encoding":84},"\\omega",[40,58867,58869],{"className":58868,"ariaHidden":72},[89],[40,58870,58872,58875],{"className":58871},[93],[40,58873],{"className":58874,"style":4142},[97],[40,58876,20187],{"className":58877,"style":336},[102,335],[20167,58879,58880,58881],{},"Angular drive frequency, ",[40,58882,58884,58906],{"className":58883},[43],[40,58885,58887],{"className":58886},[47],[49,58888,58889],{"xmlns":51},[53,58890,58891,58903],{},[56,58892,58893,58895,58897,58899,58901],{},[154,58894,20187],{},[59,58896,162],{},[64,58898,1044],{},[154,58900,1661],{},[154,58902,13262],{},[82,58904,58905],{"encoding":84},"\\omega=2\\pi f",[40,58907,58909,58927],{"className":58908,"ariaHidden":72},[89],[40,58910,58912,58915,58918,58921,58924],{"className":58911},[93],[40,58913],{"className":58914,"style":4142},[97],[40,58916,20187],{"className":58917,"style":336},[102,335],[40,58919],{"className":58920,"style":390},[121],[40,58922,162],{"className":58923},[394],[40,58925],{"className":58926,"style":390},[121],[40,58928,58930,58933,58936,58939],{"className":58929},[93],[40,58931],{"className":58932,"style":58114},[97],[40,58934,1044],{"className":58935},[102],[40,58937,1661],{"className":58938,"style":336},[102,335],[40,58940,13262],{"className":58941,"style":33300},[102,335],[20148,58943,58944,58974],{},[20167,58945,58946],{},[40,58947,58949,58962],{"className":58948},[43],[40,58950,58952],{"className":58951},[47],[49,58953,58954],{"xmlns":51},[53,58955,58956,58960],{},[56,58957,58958],{},[154,58959,20352],{"mathvariant":1080},[82,58961,20355],{"encoding":84},[40,58963,58965],{"className":58964,"ariaHidden":72},[89],[40,58966,58968,58971],{"className":58967},[93],[40,58969],{"className":58970,"style":4304},[97],[40,58972,20352],{"className":58973},[102],[20167,58975,58976],{},"On-resonance Rabi angular frequency",[20148,58978,58979,59020],{},[20167,58980,58981],{},[40,58982,58984,59004],{"className":58983},[43],[40,58985,58987],{"className":58986},[47],[49,58988,58989],{"xmlns":51},[53,58990,58991,59001],{},[56,58992,58993,58995,58997,58999],{},[154,58994,20352],{"mathvariant":1080},[154,58996,11243],{"mathvariant":1080},[64,58998,1044],{},[154,59000,1661],{},[82,59002,59003],{"encoding":84},"\\Omega\u002F2\\pi",[40,59005,59007],{"className":59006,"ariaHidden":72},[89],[40,59008,59010,59013,59017],{"className":59009},[93],[40,59011],{"className":59012,"style":98},[97],[40,59014,59016],{"className":59015},[102],"Ω\u002F2",[40,59018,1661],{"className":59019,"style":336},[102,335],[20167,59021,59022],{},"On-resonance Rabi frequency in ordinary frequency units",[20148,59024,59025,59055],{},[20167,59026,59027],{},[40,59028,59030,59043],{"className":59029},[43],[40,59031,59033],{"className":59032},[47],[49,59034,59035],{"xmlns":51},[53,59036,59037,59041],{},[56,59038,59039],{},[154,59040,2016],{"mathvariant":1080},[82,59042,10118],{"encoding":84},[40,59044,59046],{"className":59045,"ariaHidden":72},[89],[40,59047,59049,59052],{"className":59048},[93],[40,59050],{"className":59051,"style":4304},[97],[40,59053,2016],{"className":59054},[102],[20167,59056,59057,59058],{},"Drive detuning, ",[40,59059,59061,59087],{"className":59060},[43],[40,59062,59064],{"className":59063},[47],[49,59065,59066],{"xmlns":51},[53,59067,59068,59084],{},[56,59069,59070,59072,59074,59076,59078],{},[154,59071,2016],{"mathvariant":1080},[59,59073,162],{},[154,59075,20187],{},[59,59077,243],{},[151,59079,59080,59082],{},[154,59081,20187],{},[64,59083,66],{},[82,59085,59086],{"encoding":84},"\\Delta=\\omega-\\omega_0",[40,59088,59090,59108,59127],{"className":59089,"ariaHidden":72},[89],[40,59091,59093,59096,59099,59102,59105],{"className":59092},[93],[40,59094],{"className":59095,"style":4304},[97],[40,59097,2016],{"className":59098},[102],[40,59100],{"className":59101,"style":390},[121],[40,59103,162],{"className":59104},[394],[40,59106],{"className":59107,"style":390},[121],[40,59109,59111,59115,59118,59121,59124],{"className":59110},[93],[40,59112],{"className":59113,"style":59114},[97],"height:0.6667em;vertical-align:-0.0833em;",[40,59116,20187],{"className":59117,"style":336},[102,335],[40,59119],{"className":59120,"style":1246},[121],[40,59122,243],{"className":59123},[1250],[40,59125],{"className":59126,"style":1246},[121],[40,59128,59130,59133],{"className":59129},[93],[40,59131],{"className":59132,"style":328},[97],[40,59134,59136,59139],{"className":59135},[102],[40,59137,20187],{"className":59138,"style":336},[102,335],[40,59140,59142],{"className":59141},[340],[40,59143,59145,59165],{"className":59144},[344,345],[40,59146,59148,59162],{"className":59147},[349],[40,59149,59151],{"className":59150,"style":7201},[353],[40,59152,59153,59156],{"style":357},[40,59154],{"className":59155,"style":362},[361],[40,59157,59159],{"className":59158},[366,367,368,369],[40,59160,66],{"className":59161},[102,369],[40,59163,377],{"className":59164},[376],[40,59166,59168],{"className":59167},[349],[40,59169,59171],{"className":59170,"style":384},[353],[40,59172],{},[20148,59174,59175,59247],{},[20167,59176,59177],{},[40,59178,59180,59198],{"className":59179},[43],[40,59181,59183],{"className":59182},[47],[49,59184,59185],{"xmlns":51},[53,59186,59187,59195],{},[56,59188,59189],{},[151,59190,59191,59193],{},[154,59192,20352],{"mathvariant":1080},[154,59194,946],{},[82,59196,59197],{"encoding":84},"\\Omega_R",[40,59199,59201],{"className":59200,"ariaHidden":72},[89],[40,59202,59204,59207],{"className":59203},[93],[40,59205],{"className":59206,"style":3106},[97],[40,59208,59210,59213],{"className":59209},[102],[40,59211,20352],{"className":59212},[102],[40,59214,59216],{"className":59215},[340],[40,59217,59219,59239],{"className":59218},[344,345],[40,59220,59222,59236],{"className":59221},[349],[40,59223,59225],{"className":59224,"style":20224},[353],[40,59226,59227,59230],{"style":6098},[40,59228],{"className":59229,"style":362},[361],[40,59231,59233],{"className":59232},[366,367,368,369],[40,59234,946],{"className":59235,"style":3946},[102,335,369],[40,59237,377],{"className":59238},[376],[40,59240,59242],{"className":59241},[349],[40,59243,59245],{"className":59244,"style":384},[353],[40,59246],{},[20167,59248,59249],{},"Generalized Rabi angular frequency",[20148,59251,59252,59327],{},[20167,59253,59254],{},[40,59255,59257,59276],{"className":59256},[43],[40,59258,59260],{"className":59259},[47],[49,59261,59262],{"xmlns":51},[53,59263,59264,59273],{},[56,59265,59266],{},[151,59267,59268,59271],{},[154,59269,59270],{},"ν",[154,59272,946],{},[82,59274,59275],{"encoding":84},"\\nu_R",[40,59277,59279],{"className":59278,"ariaHidden":72},[89],[40,59280,59282,59285],{"className":59281},[93],[40,59283],{"className":59284,"style":328},[97],[40,59286,59288,59292],{"className":59287},[102],[40,59289,59270],{"className":59290,"style":59291},[102,335],"margin-right:0.0637em;",[40,59293,59295],{"className":59294},[340],[40,59296,59298,59319],{"className":59297},[344,345],[40,59299,59301,59316],{"className":59300},[349],[40,59302,59304],{"className":59303,"style":20224},[353],[40,59305,59307,59310],{"style":59306},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[40,59308],{"className":59309,"style":362},[361],[40,59311,59313],{"className":59312},[366,367,368,369],[40,59314,946],{"className":59315,"style":3946},[102,335,369],[40,59317,377],{"className":59318},[376],[40,59320,59322],{"className":59321},[349],[40,59323,59325],{"className":59324,"style":384},[353],[40,59326],{},[20167,59328,59329,59330],{},"Generalized Rabi frequency, ",[40,59331,59333,59365],{"className":59332},[43],[40,59334,59336],{"className":59335},[47],[49,59337,59338],{"xmlns":51},[53,59339,59340,59362],{},[56,59341,59342,59348,59350,59356,59358,59360],{},[151,59343,59344,59346],{},[154,59345,59270],{},[154,59347,946],{},[59,59349,162],{},[151,59351,59352,59354],{},[154,59353,20352],{"mathvariant":1080},[154,59355,946],{},[154,59357,11243],{"mathvariant":1080},[64,59359,1044],{},[154,59361,1661],{},[82,59363,59364],{"encoding":84},"\\nu_R=\\Omega_R\u002F2\\pi",[40,59366,59368,59423],{"className":59367,"ariaHidden":72},[89],[40,59369,59371,59374,59414,59417,59420],{"className":59370},[93],[40,59372],{"className":59373,"style":328},[97],[40,59375,59377,59380],{"className":59376},[102],[40,59378,59270],{"className":59379,"style":59291},[102,335],[40,59381,59383],{"className":59382},[340],[40,59384,59386,59406],{"className":59385},[344,345],[40,59387,59389,59403],{"className":59388},[349],[40,59390,59392],{"className":59391,"style":20224},[353],[40,59393,59394,59397],{"style":59306},[40,59395],{"className":59396,"style":362},[361],[40,59398,59400],{"className":59399},[366,367,368,369],[40,59401,946],{"className":59402,"style":3946},[102,335,369],[40,59404,377],{"className":59405},[376],[40,59407,59409],{"className":59408},[349],[40,59410,59412],{"className":59411,"style":384},[353],[40,59413],{},[40,59415],{"className":59416,"style":390},[121],[40,59418,162],{"className":59419},[394],[40,59421],{"className":59422,"style":390},[121],[40,59424,59426,59429,59469,59472],{"className":59425},[93],[40,59427],{"className":59428,"style":98},[97],[40,59430,59432,59435],{"className":59431},[102],[40,59433,20352],{"className":59434},[102],[40,59436,59438],{"className":59437},[340],[40,59439,59441,59461],{"className":59440},[344,345],[40,59442,59444,59458],{"className":59443},[349],[40,59445,59447],{"className":59446,"style":20224},[353],[40,59448,59449,59452],{"style":6098},[40,59450],{"className":59451,"style":362},[361],[40,59453,59455],{"className":59454},[366,367,368,369],[40,59456,946],{"className":59457,"style":3946},[102,335,369],[40,59459,377],{"className":59460},[376],[40,59462,59464],{"className":59463},[349],[40,59465,59467],{"className":59466,"style":384},[353],[40,59468],{},[40,59470,33653],{"className":59471},[102],[40,59473,1661],{"className":59474,"style":336},[102,335],[20148,59476,59477,59507],{},[20167,59478,59479],{},[40,59480,59482,59495],{"className":59481},[43],[40,59483,59485],{"className":59484},[47],[49,59486,59487],{"xmlns":51},[53,59488,59489,59493],{},[56,59490,59491],{},[154,59492,58007],{},[82,59494,58135],{"encoding":84},[40,59496,59498],{"className":59497,"ariaHidden":72},[89],[40,59499,59501,59504],{"className":59500},[93],[40,59502],{"className":59503,"style":4142},[97],[40,59505,58007],{"className":59506,"style":58079},[102,335],[20167,59508,59509],{},"Pulse duration",[20148,59511,59512,59602],{},[20167,59513,59514],{},[40,59515,59517,59538],{"className":59516},[43],[40,59518,59520],{"className":59519},[47],[49,59521,59522],{"xmlns":51},[53,59523,59524,59535],{},[56,59525,59526],{},[8360,59527,59528,59530,59532],{},[154,59529,7059],{},[64,59531,1044],{},[59,59533,59534],{},"∗",[82,59536,59537],{"encoding":84},"T_2^\\ast",[40,59539,59541],{"className":59540,"ariaHidden":72},[89],[40,59542,59544,59548],{"className":59543},[93],[40,59545],{"className":59546,"style":59547},[97],"height:0.9368em;vertical-align:-0.2481em;",[40,59549,59551,59554],{"className":59550},[102],[40,59552,7059],{"className":59553,"style":2541},[102,335],[40,59555,59557],{"className":59556},[340],[40,59558,59560,59593],{"className":59559},[344,345],[40,59561,59563,59590],{"className":59562},[349],[40,59564,59567,59579],{"className":59565,"style":59566},[353],"height:0.6887em;",[40,59568,59570,59573],{"style":59569},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[40,59571],{"className":59572,"style":362},[361],[40,59574,59576],{"className":59575},[366,367,368,369],[40,59577,1044],{"className":59578},[102,369],[40,59580,59581,59584],{"style":1004},[40,59582],{"className":59583,"style":362},[361],[40,59585,59587],{"className":59586},[366,367,368,369],[40,59588,59534],{"className":59589},[1250,369],[40,59591,377],{"className":59592},[376],[40,59594,59596],{"className":59595},[349],[40,59597,59600],{"className":59598,"style":59599},[353],"height:0.2481em;",[40,59601],{},[20167,59603,59604],{},"Phenomenological dephasing time used as an optional damping parameter",[20148,59606,59607,59610],{},[20167,59608,59609],{},"RWA",[20167,59611,59612],{},"Rotating-wave approximation",[20148,59614,59615,59618],{},[20167,59616,59617],{},"FFT",[20167,59619,59620],{},"Fast Fourier transform",[28,59622,59624],{"id":59623},"physical-model","Physical Model",[12,59626,59627,59628,59666,59667,59705],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[40,59629,59631,59648],{"className":59630},[43],[40,59632,59634],{"className":59633},[47],[49,59635,59636],{"xmlns":51},[53,59637,59638,59646],{},[56,59639,59640,59642,59644],{},[59,59641,62],{"stretchy":61},[154,59643,58190],{},[59,59645,69],{"stretchy":61},[82,59647,58195],{"encoding":84},[40,59649,59651],{"className":59650,"ariaHidden":72},[89],[40,59652,59654,59657,59660,59663],{"className":59653},[93],[40,59655],{"className":59656,"style":98},[97],[40,59658,62],{"className":59659},[106],[40,59661,58190],{"className":59662,"style":336},[102,335],[40,59664,69],{"className":59665},[113]," and an excited state ",[40,59668,59670,59687],{"className":59669},[43],[40,59671,59673],{"className":59672},[47],[49,59674,59675],{"xmlns":51},[53,59676,59677,59685],{},[56,59678,59679,59681,59683],{},[59,59680,62],{"stretchy":61},[154,59682,4203],{},[59,59684,69],{"stretchy":61},[82,59686,58241],{"encoding":84},[40,59688,59690],{"className":59689,"ariaHidden":72},[89],[40,59691,59693,59696,59699,59702],{"className":59692},[93],[40,59694],{"className":59695,"style":98},[97],[40,59697,62],{"className":59698},[106],[40,59700,4203],{"className":59701},[102,335],[40,59703,69],{"className":59704},[113],". In the absence of a microwave drive, the energy splitting is",[40,59707,59709],{"className":59708},[136],[40,59710,59712,59751],{"className":59711},[43],[40,59713,59715],{"className":59714},[47],[49,59716,59717],{"xmlns":51,"display":145},[53,59718,59719,59748],{},[56,59720,59721,59727,59729,59735,59737,59740,59746],{},[151,59722,59723,59725],{},[154,59724,32874],{},[154,59726,4203],{},[59,59728,243],{},[151,59730,59731,59733],{},[154,59732,32874],{},[154,59734,58190],{},[59,59736,162],{},[154,59738,59739],{"mathvariant":1080},"ℏ",[151,59741,59742,59744],{},[154,59743,20187],{},[64,59745,66],{},[59,59747,73],{"separator":72},[82,59749,59750],{"encoding":84},"E_e-E_g=\\hbar\\omega_0,",[40,59752,59754,59809,59865],{"className":59753,"ariaHidden":72},[89],[40,59755,59757,59760,59800,59803,59806],{"className":59756},[93],[40,59758],{"className":59759,"style":3106},[97],[40,59761,59763,59766],{"className":59762},[102],[40,59764,32874],{"className":59765,"style":2383},[102,335],[40,59767,59769],{"className":59768},[340],[40,59770,59772,59792],{"className":59771},[344,345],[40,59773,59775,59789],{"className":59774},[349],[40,59776,59778],{"className":59777,"style":354},[353],[40,59779,59780,59783],{"style":32995},[40,59781],{"className":59782,"style":362},[361],[40,59784,59786],{"className":59785},[366,367,368,369],[40,59787,4203],{"className":59788},[102,335,369],[40,59790,377],{"className":59791},[376],[40,59793,59795],{"className":59794},[349],[40,59796,59798],{"className":59797,"style":384},[353],[40,59799],{},[40,59801],{"className":59802,"style":1246},[121],[40,59804,243],{"className":59805},[1250],[40,59807],{"className":59808,"style":1246},[121],[40,59810,59812,59816,59856,59859,59862],{"className":59811},[93],[40,59813],{"className":59814,"style":59815},[97],"height:0.9694em;vertical-align:-0.2861em;",[40,59817,59819,59822],{"className":59818},[102],[40,59820,32874],{"className":59821,"style":2383},[102,335],[40,59823,59825],{"className":59824},[340],[40,59826,59828,59848],{"className":59827},[344,345],[40,59829,59831,59845],{"className":59830},[349],[40,59832,59834],{"className":59833,"style":354},[353],[40,59835,59836,59839],{"style":32995},[40,59837],{"className":59838,"style":362},[361],[40,59840,59842],{"className":59841},[366,367,368,369],[40,59843,58190],{"className":59844,"style":336},[102,335,369],[40,59846,377],{"className":59847},[376],[40,59849,59851],{"className":59850},[349],[40,59852,59854],{"className":59853,"style":585},[353],[40,59855],{},[40,59857],{"className":59858,"style":390},[121],[40,59860,162],{"className":59861},[394],[40,59863],{"className":59864,"style":390},[121],[40,59866,59868,59872,59875,59915],{"className":59867},[93],[40,59869],{"className":59870,"style":59871},[97],"height:0.8833em;vertical-align:-0.1944em;",[40,59873,59739],{"className":59874},[102],[40,59876,59878,59881],{"className":59877},[102],[40,59879,20187],{"className":59880,"style":336},[102,335],[40,59882,59884],{"className":59883},[340],[40,59885,59887,59907],{"className":59886},[344,345],[40,59888,59890,59904],{"className":59889},[349],[40,59891,59893],{"className":59892,"style":7201},[353],[40,59894,59895,59898],{"style":357},[40,59896],{"className":59897,"style":362},[361],[40,59899,59901],{"className":59900},[366,367,368,369],[40,59902,66],{"className":59903},[102,369],[40,59905,377],{"className":59906},[376],[40,59908,59910],{"className":59909},[349],[40,59911,59913],{"className":59912,"style":384},[353],[40,59914],{},[40,59916,73],{"className":59917},[117],[12,59919,59920],{},"where the angular transition frequency is",[40,59922,59924],{"className":59923},[136],[40,59925,59927,59959],{"className":59926},[43],[40,59928,59930],{"className":59929},[47],[49,59931,59932],{"xmlns":51,"display":145},[53,59933,59934,59956],{},[56,59935,59936,59942,59944,59946,59948,59954],{},[151,59937,59938,59940],{},[154,59939,20187],{},[64,59941,66],{},[59,59943,162],{},[64,59945,1044],{},[154,59947,1661],{},[151,59949,59950,59952],{},[154,59951,13262],{},[64,59953,66],{},[154,59955,1123],{"mathvariant":1080},[82,59957,59958],{"encoding":84},"\\omega_0=2\\pi f_0.",[40,59960,59962,60017],{"className":59961,"ariaHidden":72},[89],[40,59963,59965,59968,60008,60011,60014],{"className":59964},[93],[40,59966],{"className":59967,"style":328},[97],[40,59969,59971,59974],{"className":59970},[102],[40,59972,20187],{"className":59973,"style":336},[102,335],[40,59975,59977],{"className":59976},[340],[40,59978,59980,60000],{"className":59979},[344,345],[40,59981,59983,59997],{"className":59982},[349],[40,59984,59986],{"className":59985,"style":7201},[353],[40,59987,59988,59991],{"style":357},[40,59989],{"className":59990,"style":362},[361],[40,59992,59994],{"className":59993},[366,367,368,369],[40,59995,66],{"className":59996},[102,369],[40,59998,377],{"className":59999},[376],[40,60001,60003],{"className":60002},[349],[40,60004,60006],{"className":60005,"style":384},[353],[40,60007],{},[40,60009],{"className":60010,"style":390},[121],[40,60012,162],{"className":60013},[394],[40,60015],{"className":60016,"style":390},[121],[40,60018,60020,60023,60026,60029,60069],{"className":60019},[93],[40,60021],{"className":60022,"style":58114},[97],[40,60024,1044],{"className":60025},[102],[40,60027,1661],{"className":60028,"style":336},[102,335],[40,60030,60032,60035],{"className":60031},[102],[40,60033,13262],{"className":60034,"style":33300},[102,335],[40,60036,60038],{"className":60037},[340],[40,60039,60041,60061],{"className":60040},[344,345],[40,60042,60044,60058],{"className":60043},[349],[40,60045,60047],{"className":60046,"style":7201},[353],[40,60048,60049,60052],{"style":33315},[40,60050],{"className":60051,"style":362},[361],[40,60053,60055],{"className":60054},[366,367,368,369],[40,60056,66],{"className":60057},[102,369],[40,60059,377],{"className":60060},[376],[40,60062,60064],{"className":60063},[349],[40,60065,60067],{"className":60066,"style":384},[353],[40,60068],{},[40,60070,1123],{"className":60071},[102],[12,60073,60074],{},"The default configuration uses a transition frequency of",[40,60076,60078],{"className":60077},[136],[40,60079,60081,60117],{"className":60080},[43],[40,60082,60084],{"className":60083},[47],[49,60085,60086],{"xmlns":51,"display":145},[53,60087,60088,60114],{},[56,60089,60090,60096,60098,60101,60103,60112],{},[151,60091,60092,60094],{},[154,60093,13262],{},[64,60095,66],{},[59,60097,162],{},[64,60099,60100],{},"5.000",[1102,60102,1104],{},[56,60104,60105,60108,60110],{},[154,60106,60107],{"mathvariant":1080},"G",[154,60109,26630],{"mathvariant":1080},[154,60111,273],{"mathvariant":1080},[154,60113,1123],{"mathvariant":1080},[82,60115,60116],{"encoding":84},"f_0=5.000~\\mathrm{GHz}.",[40,60118,60120,60175],{"className":60119,"ariaHidden":72},[89],[40,60121,60123,60126,60166,60169,60172],{"className":60122},[93],[40,60124],{"className":60125,"style":58114},[97],[40,60127,60129,60132],{"className":60128},[102],[40,60130,13262],{"className":60131,"style":33300},[102,335],[40,60133,60135],{"className":60134},[340],[40,60136,60138,60158],{"className":60137},[344,345],[40,60139,60141,60155],{"className":60140},[349],[40,60142,60144],{"className":60143,"style":7201},[353],[40,60145,60146,60149],{"style":33315},[40,60147],{"className":60148,"style":362},[361],[40,60150,60152],{"className":60151},[366,367,368,369],[40,60153,66],{"className":60154},[102,369],[40,60156,377],{"className":60157},[376],[40,60159,60161],{"className":60160},[349],[40,60162,60164],{"className":60163,"style":384},[353],[40,60165],{},[40,60167],{"className":60168,"style":390},[121],[40,60170,162],{"className":60171},[394],[40,60173],{"className":60174,"style":390},[121],[40,60176,60178,60181,60184,60188,60195],{"className":60177},[93],[40,60179],{"className":60180,"style":4304},[97],[40,60182,60100],{"className":60183},[102],[40,60185,1104],{"className":60186},[121,60187],"nobreak",[40,60189,60191],{"className":60190},[102],[40,60192,60194],{"className":60193},[102,1373],"GHz",[40,60196,1123],{"className":60197},[102],[12,60199,60200,60201,60229,60230,916,60268,60306],{},"Applying a microwave drive with frequency ",[40,60202,60204,60217],{"className":60203},[43],[40,60205,60207],{"className":60206},[47],[49,60208,60209],{"xmlns":51},[53,60210,60211,60215],{},[56,60212,60213],{},[154,60214,13262],{},[82,60216,13262],{"encoding":84},[40,60218,60220],{"className":60219,"ariaHidden":72},[89],[40,60221,60223,60226],{"className":60222},[93],[40,60224],{"className":60225,"style":58114},[97],[40,60227,13262],{"className":60228,"style":33300},[102,335]," induces coherent rotations between ",[40,60231,60233,60250],{"className":60232},[43],[40,60234,60236],{"className":60235},[47],[49,60237,60238],{"xmlns":51},[53,60239,60240,60248],{},[56,60241,60242,60244,60246],{},[59,60243,62],{"stretchy":61},[154,60245,58190],{},[59,60247,69],{"stretchy":61},[82,60249,58195],{"encoding":84},[40,60251,60253],{"className":60252,"ariaHidden":72},[89],[40,60254,60256,60259,60262,60265],{"className":60255},[93],[40,60257],{"className":60258,"style":98},[97],[40,60260,62],{"className":60261},[106],[40,60263,58190],{"className":60264,"style":336},[102,335],[40,60266,69],{"className":60267},[113],[40,60269,60271,60288],{"className":60270},[43],[40,60272,60274],{"className":60273},[47],[49,60275,60276],{"xmlns":51},[53,60277,60278,60286],{},[56,60279,60280,60282,60284],{},[59,60281,62],{"stretchy":61},[154,60283,4203],{},[59,60285,69],{"stretchy":61},[82,60287,58241],{"encoding":84},[40,60289,60291],{"className":60290,"ariaHidden":72},[89],[40,60292,60294,60297,60300,60303],{"className":60293},[93],[40,60295],{"className":60296,"style":98},[97],[40,60298,62],{"className":60299},[106],[40,60301,4203],{"className":60302},[102,335],[40,60304,69],{"className":60305},[113],". Exact resonance occurs when",[40,60308,60310],{"className":60309},[136],[40,60311,60313,60337],{"className":60312},[43],[40,60314,60316],{"className":60315},[47],[49,60317,60318],{"xmlns":51,"display":145},[53,60319,60320,60334],{},[56,60321,60322,60324,60326,60332],{},[154,60323,13262],{},[59,60325,162],{},[151,60327,60328,60330],{},[154,60329,13262],{},[64,60331,66],{},[154,60333,1123],{"mathvariant":1080},[82,60335,60336],{"encoding":84},"f=f_0.",[40,60338,60340,60358],{"className":60339,"ariaHidden":72},[89],[40,60341,60343,60346,60349,60352,60355],{"className":60342},[93],[40,60344],{"className":60345,"style":58114},[97],[40,60347,13262],{"className":60348,"style":33300},[102,335],[40,60350],{"className":60351,"style":390},[121],[40,60353,162],{"className":60354},[394],[40,60356],{"className":60357,"style":390},[121],[40,60359,60361,60364,60404],{"className":60360},[93],[40,60362],{"className":60363,"style":58114},[97],[40,60365,60367,60370],{"className":60366},[102],[40,60368,13262],{"className":60369,"style":33300},[102,335],[40,60371,60373],{"className":60372},[340],[40,60374,60376,60396],{"className":60375},[344,345],[40,60377,60379,60393],{"className":60378},[349],[40,60380,60382],{"className":60381,"style":7201},[353],[40,60383,60384,60387],{"style":33315},[40,60385],{"className":60386,"style":362},[361],[40,60388,60390],{"className":60389},[366,367,368,369],[40,60391,66],{"className":60392},[102,369],[40,60394,377],{"className":60395},[376],[40,60397,60399],{"className":60398},[349],[40,60400,60402],{"className":60401,"style":384},[353],[40,60403],{},[40,60405,1123],{"className":60406},[102],[12,60408,60409],{},"At resonance, the drive produces ideal Rabi oscillations. Away from resonance, detuning changes the oscillation frequency and reduces the maximum achievable excited-state probability. This same two-level control structure appears across superconducting qubits, spin qubits, trapped ions, neutral atoms, and nitrogen-vacancy centers whenever a selected transition can be treated as an effective qubit.",[28,60411,60413],{"id":60412},"mathematical-foundation","Mathematical Foundation",[12,60415,60416],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[40,60418,60420],{"className":60419},[136],[40,60421,60423,60463],{"className":60422},[43],[40,60424,60426],{"className":60425},[47],[49,60427,60428],{"xmlns":51,"display":145},[53,60429,60430,60460],{},[56,60431,60432,60434,60436,60438,60440,60442,60444,60446,60448,60450,60452,60454,60456,60458],{},[59,60433,62],{"stretchy":61},[154,60435,1506],{},[59,60437,69],{"stretchy":61},[59,60439,162],{},[154,60441,5215],{},[59,60443,62],{"stretchy":61},[154,60445,58190],{},[59,60447,69],{"stretchy":61},[59,60449,1053],{},[154,60451,58331],{},[59,60453,62],{"stretchy":61},[154,60455,4203],{},[59,60457,69],{"stretchy":61},[59,60459,73],{"separator":72},[82,60461,60462],{"encoding":84},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[40,60464,60466,60490,60517],{"className":60465,"ariaHidden":72},[89],[40,60467,60469,60472,60475,60478,60481,60484,60487],{"className":60468},[93],[40,60470],{"className":60471,"style":98},[97],[40,60473,62],{"className":60474},[106],[40,60476,1506],{"className":60477,"style":336},[102,335],[40,60479,69],{"className":60480},[113],[40,60482],{"className":60483,"style":390},[121],[40,60485,162],{"className":60486},[394],[40,60488],{"className":60489,"style":390},[121],[40,60491,60493,60496,60499,60502,60505,60508,60511,60514],{"className":60492},[93],[40,60494],{"className":60495,"style":98},[97],[40,60497,5215],{"className":60498,"style":5231},[102,335],[40,60500,62],{"className":60501},[106],[40,60503,58190],{"className":60504,"style":336},[102,335],[40,60506,69],{"className":60507},[113],[40,60509],{"className":60510,"style":1246},[121],[40,60512,1053],{"className":60513},[1250],[40,60515],{"className":60516,"style":1246},[121],[40,60518,60520,60523,60526,60529,60532,60535],{"className":60519},[93],[40,60521],{"className":60522,"style":98},[97],[40,60524,58331],{"className":60525,"style":58356},[102,335],[40,60527,62],{"className":60528},[106],[40,60530,4203],{"className":60531},[102,335],[40,60533,69],{"className":60534},[113],[40,60536,73],{"className":60537},[117],[12,60539,58088,60540,916,60568,60597],{},[40,60541,60543,60556],{"className":60542},[43],[40,60544,60546],{"className":60545},[47],[49,60547,60548],{"xmlns":51},[53,60549,60550,60554],{},[56,60551,60552],{},[154,60553,5215],{},[82,60555,5218],{"encoding":84},[40,60557,60559],{"className":60558,"ariaHidden":72},[89],[40,60560,60562,60565],{"className":60561},[93],[40,60563],{"className":60564,"style":4142},[97],[40,60566,5215],{"className":60567,"style":5231},[102,335],[40,60569,60571,60585],{"className":60570},[43],[40,60572,60574],{"className":60573},[47],[49,60575,60576],{"xmlns":51},[53,60577,60578,60582],{},[56,60579,60580],{},[154,60581,58331],{},[82,60583,60584],{"encoding":84},"\\beta",[40,60586,60588],{"className":60587,"ariaHidden":72},[89],[40,60589,60591,60594],{"className":60590},[93],[40,60592],{"className":60593,"style":58114},[97],[40,60595,58331],{"className":60596,"style":58356},[102,335]," are complex probability amplitudes. Normalization requires",[40,60599,60601],{"className":60600},[136],[40,60602,60604,60643],{"className":60603},[43],[40,60605,60607],{"className":60606},[47],[49,60608,60609],{"xmlns":51,"display":145},[53,60610,60611,60640],{},[56,60612,60613,60615,60617,60623,60625,60627,60629,60635,60637],{},[154,60614,62],{"mathvariant":1080},[154,60616,5215],{},[941,60618,60619,60621],{},[154,60620,62],{"mathvariant":1080},[64,60622,1044],{},[59,60624,1053],{},[154,60626,62],{"mathvariant":1080},[154,60628,58331],{},[941,60630,60631,60633],{},[154,60632,62],{"mathvariant":1080},[64,60634,1044],{},[59,60636,162],{},[64,60638,60639],{},"1.",[82,60641,60642],{"encoding":84},"|\\alpha|^2+|\\beta|^2=1.",[40,60644,60646,60697,60747],{"className":60645,"ariaHidden":72},[89],[40,60647,60649,60653,60656,60659,60688,60691,60694],{"className":60648},[93],[40,60650],{"className":60651,"style":60652},[97],"height:1.1141em;vertical-align:-0.25em;",[40,60654,62],{"className":60655},[102],[40,60657,5215],{"className":60658,"style":5231},[102,335],[40,60660,60662,60665],{"className":60661},[102],[40,60663,62],{"className":60664},[102],[40,60666,60668],{"className":60667},[340],[40,60669,60671],{"className":60670},[344],[40,60672,60674],{"className":60673},[349],[40,60675,60677],{"className":60676,"style":3959},[353],[40,60678,60679,60682],{"style":3962},[40,60680],{"className":60681,"style":362},[361],[40,60683,60685],{"className":60684},[366,367,368,369],[40,60686,1044],{"className":60687},[102,369],[40,60689],{"className":60690,"style":1246},[121],[40,60692,1053],{"className":60693},[1250],[40,60695],{"className":60696,"style":1246},[121],[40,60698,60700,60703,60706,60709,60738,60741,60744],{"className":60699},[93],[40,60701],{"className":60702,"style":60652},[97],[40,60704,62],{"className":60705},[102],[40,60707,58331],{"className":60708,"style":58356},[102,335],[40,60710,60712,60715],{"className":60711},[102],[40,60713,62],{"className":60714},[102],[40,60716,60718],{"className":60717},[340],[40,60719,60721],{"className":60720},[344],[40,60722,60724],{"className":60723},[349],[40,60725,60727],{"className":60726,"style":3959},[353],[40,60728,60729,60732],{"style":3962},[40,60730],{"className":60731,"style":362},[361],[40,60733,60735],{"className":60734},[366,367,368,369],[40,60736,1044],{"className":60737},[102,369],[40,60739],{"className":60740,"style":390},[121],[40,60742,162],{"className":60743},[394],[40,60745],{"className":60746,"style":390},[121],[40,60748,60750,60753],{"className":60749},[93],[40,60751],{"className":60752,"style":1624},[97],[40,60754,60639],{"className":60755},[102],[12,60757,60758],{},"The Born rule gives the probability of measuring the excited state,",[40,60760,60762],{"className":60761},[136],[40,60763,60765,60817],{"className":60764},[43],[40,60766,60768],{"className":60767},[47],[49,60769,60770],{"xmlns":51,"display":145},[53,60771,60772,60814],{},[56,60773,60774,60780,60782,60784,60786,60788,60790,60792,60794,60800,60802,60804,60806,60812],{},[151,60775,60776,60778],{},[154,60777,57996],{},[154,60779,4203],{},[59,60781,162],{},[154,60783,62],{"mathvariant":1080},[59,60785,1511],{"stretchy":61},[154,60787,4203],{},[59,60789,62],{"stretchy":61},[154,60791,1506],{},[59,60793,69],{"stretchy":61},[941,60795,60796,60798],{},[154,60797,62],{"mathvariant":1080},[64,60799,1044],{},[59,60801,162],{},[154,60803,62],{"mathvariant":1080},[154,60805,58331],{},[941,60807,60808,60810],{},[154,60809,62],{"mathvariant":1080},[64,60811,1044],{},[154,60813,1123],{"mathvariant":1080},[82,60815,60816],{"encoding":84},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[40,60818,60820,60875,60937],{"className":60819,"ariaHidden":72},[89],[40,60821,60823,60826,60866,60869,60872],{"className":60822},[93],[40,60824],{"className":60825,"style":3106},[97],[40,60827,60829,60832],{"className":60828},[102],[40,60830,57996],{"className":60831,"style":2541},[102,335],[40,60833,60835],{"className":60834},[340],[40,60836,60838,60858],{"className":60837},[344,345],[40,60839,60841,60855],{"className":60840},[349],[40,60842,60844],{"className":60843,"style":354},[353],[40,60845,60846,60849],{"style":2556},[40,60847],{"className":60848,"style":362},[361],[40,60850,60852],{"className":60851},[366,367,368,369],[40,60853,4203],{"className":60854},[102,335,369],[40,60856,377],{"className":60857},[376],[40,60859,60861],{"className":60860},[349],[40,60862,60864],{"className":60863,"style":384},[353],[40,60865],{},[40,60867],{"className":60868,"style":390},[121],[40,60870,162],{"className":60871},[394],[40,60873],{"className":60874,"style":390},[121],[40,60876,60878,60881,60884,60887,60890,60893,60896,60899,60928,60931,60934],{"className":60877},[93],[40,60879],{"className":60880,"style":60652},[97],[40,60882,62],{"className":60883},[102],[40,60885,1511],{"className":60886},[106],[40,60888,4203],{"className":60889},[102,335],[40,60891,62],{"className":60892},[106],[40,60894,1506],{"className":60895,"style":336},[102,335],[40,60897,69],{"className":60898},[113],[40,60900,60902,60905],{"className":60901},[102],[40,60903,62],{"className":60904},[102],[40,60906,60908],{"className":60907},[340],[40,60909,60911],{"className":60910},[344],[40,60912,60914],{"className":60913},[349],[40,60915,60917],{"className":60916,"style":3959},[353],[40,60918,60919,60922],{"style":3962},[40,60920],{"className":60921,"style":362},[361],[40,60923,60925],{"className":60924},[366,367,368,369],[40,60926,1044],{"className":60927},[102,369],[40,60929],{"className":60930,"style":390},[121],[40,60932,162],{"className":60933},[394],[40,60935],{"className":60936,"style":390},[121],[40,60938,60940,60943,60946,60949,60978],{"className":60939},[93],[40,60941],{"className":60942,"style":60652},[97],[40,60944,62],{"className":60945},[102],[40,60947,58331],{"className":60948,"style":58356},[102,335],[40,60950,60952,60955],{"className":60951},[102],[40,60953,62],{"className":60954},[102],[40,60956,60958],{"className":60957},[340],[40,60959,60961],{"className":60960},[344],[40,60962,60964],{"className":60963},[349],[40,60965,60967],{"className":60966,"style":3959},[353],[40,60968,60969,60972],{"style":3962},[40,60970],{"className":60971,"style":362},[361],[40,60973,60975],{"className":60974},[366,367,368,369],[40,60976,1044],{"className":60977},[102,369],[40,60979,1123],{"className":60980},[102],[12,60982,60983],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[40,60985,60987],{"className":60986},[136],[40,60988,60990,61152],{"className":60989},[43],[40,60991,60993],{"className":60992},[47],[49,60994,60995],{"xmlns":51,"display":145},[53,60996,60997,61149],{},[56,60998,60999,61005,61007,61043,61045,61047,61053,61055,61095,61097,61099,61105,61107,61147],{},[151,61000,61001,61003],{},[154,61002,156],{},[154,61004,159],{},[59,61006,162],{},[56,61008,61009,61011,61041],{},[59,61010,1654],{"fence":72},[169,61012,61013,61027],{"rowspacing":171,"columnalign":172,"columnspacing":173},[175,61014,61015,61021],{},[178,61016,61017],{},[181,61018,61019],{"scriptlevel":66,"displaystyle":61},[64,61020,66],{},[178,61022,61023],{},[181,61024,61025],{"scriptlevel":66,"displaystyle":61},[64,61026,78],{},[175,61028,61029,61035],{},[178,61030,61031],{},[181,61032,61033],{"scriptlevel":66,"displaystyle":61},[64,61034,78],{},[178,61036,61037],{},[181,61038,61039],{"scriptlevel":66,"displaystyle":61},[64,61040,66],{},[59,61042,1664],{"fence":72},[59,61044,73],{"separator":72},[121,61046],{"width":1068},[151,61048,61049,61051],{},[154,61050,156],{},[154,61052,218],{},[59,61054,162],{},[56,61056,61057,61059,61093],{},[59,61058,1654],{"fence":72},[169,61060,61061,61079],{"rowspacing":171,"columnalign":172,"columnspacing":173},[175,61062,61063,61069],{},[178,61064,61065],{},[181,61066,61067],{"scriptlevel":66,"displaystyle":61},[64,61068,66],{},[178,61070,61071],{},[181,61072,61073],{"scriptlevel":66,"displaystyle":61},[56,61074,61075,61077],{},[59,61076,243],{},[154,61078,246],{},[175,61080,61081,61087],{},[178,61082,61083],{},[181,61084,61085],{"scriptlevel":66,"displaystyle":61},[154,61086,246],{},[178,61088,61089],{},[181,61090,61091],{"scriptlevel":66,"displaystyle":61},[64,61092,66],{},[59,61094,1664],{"fence":72},[59,61096,73],{"separator":72},[121,61098],{"width":1068},[151,61100,61101,61103],{},[154,61102,156],{},[154,61104,273],{},[59,61106,162],{},[56,61108,61109,61111,61145],{},[59,61110,1654],{"fence":72},[169,61112,61113,61127],{"rowspacing":171,"columnalign":172,"columnspacing":173},[175,61114,61115,61121],{},[178,61116,61117],{},[181,61118,61119],{"scriptlevel":66,"displaystyle":61},[64,61120,78],{},[178,61122,61123],{},[181,61124,61125],{"scriptlevel":66,"displaystyle":61},[64,61126,66],{},[175,61128,61129,61135],{},[178,61130,61131],{},[181,61132,61133],{"scriptlevel":66,"displaystyle":61},[64,61134,66],{},[178,61136,61137],{},[181,61138,61139],{"scriptlevel":66,"displaystyle":61},[56,61140,61141,61143],{},[59,61142,243],{},[64,61144,78],{},[59,61146,1664],{"fence":72},[154,61148,1123],{"mathvariant":1080},[82,61150,61151],{"encoding":84},"\\sigma_x=\\begin{bmatrix}0 & 1 \\\\ 1 & 0\\end{bmatrix}, \\qquad\n\\sigma_y=\\begin{bmatrix}0 & -i \\\\ i & 0\\end{bmatrix}, \\qquad\n\\sigma_z=\\begin{bmatrix}1 & 0 \\\\ 0 & -1\\end{bmatrix}.",[40,61153,61155,61210,61394,61581],{"className":61154,"ariaHidden":72},[89],[40,61156,61158,61161,61201,61204,61207],{"className":61157},[93],[40,61159],{"className":61160,"style":328},[97],[40,61162,61164,61167],{"className":61163},[102],[40,61165,156],{"className":61166,"style":336},[102,335],[40,61168,61170],{"className":61169},[340],[40,61171,61173,61193],{"className":61172},[344,345],[40,61174,61176,61190],{"className":61175},[349],[40,61177,61179],{"className":61178,"style":354},[353],[40,61180,61181,61184],{"style":357},[40,61182],{"className":61183,"style":362},[361],[40,61185,61187],{"className":61186},[366,367,368,369],[40,61188,159],{"className":61189},[102,335,369],[40,61191,377],{"className":61192},[376],[40,61194,61196],{"className":61195},[349],[40,61197,61199],{"className":61198,"style":384},[353],[40,61200],{},[40,61202],{"className":61203,"style":390},[121],[40,61205,162],{"className":61206},[394],[40,61208],{"className":61209,"style":390},[121],[40,61211,61213,61216,61333,61336,61339,61342,61345,61385,61388,61391],{"className":61212},[93],[40,61214],{"className":61215,"style":404},[97],[40,61217,61219,61225,61327],{"className":61218},[408],[40,61220,61222],{"className":61221,"style":413},[106,412],[40,61223,1654],{"className":61224},[417,368],[40,61226,61228],{"className":61227},[102],[40,61229,61231,61276,61279,61282],{"className":61230},[169],[40,61232,61234],{"className":61233},[427],[40,61235,61237,61268],{"className":61236},[344,345],[40,61238,61240,61265],{"className":61239},[349],[40,61241,61243,61254],{"className":61242,"style":437},[353],[40,61244,61245,61248],{"style":440},[40,61246],{"className":61247,"style":444},[361],[40,61249,61251],{"className":61250},[102],[40,61252,66],{"className":61253},[102],[40,61255,61256,61259],{"style":453},[40,61257],{"className":61258,"style":444},[361],[40,61260,61262],{"className":61261},[102],[40,61263,78],{"className":61264},[102],[40,61266,377],{"className":61267},[376],[40,61269,61271],{"className":61270},[349],[40,61272,61274],{"className":61273,"style":472},[353],[40,61275],{},[40,61277],{"className":61278,"style":479},[478],[40,61280],{"className":61281,"style":479},[478],[40,61283,61285],{"className":61284},[427],[40,61286,61288,61319],{"className":61287},[344,345],[40,61289,61291,61316],{"className":61290},[349],[40,61292,61294,61305],{"className":61293,"style":437},[353],[40,61295,61296,61299],{"style":440},[40,61297],{"className":61298,"style":444},[361],[40,61300,61302],{"className":61301},[102],[40,61303,78],{"className":61304},[102],[40,61306,61307,61310],{"style":453},[40,61308],{"className":61309,"style":444},[361],[40,61311,61313],{"className":61312},[102],[40,61314,66],{"className":61315},[102],[40,61317,377],{"className":61318},[376],[40,61320,61322],{"className":61321},[349],[40,61323,61325],{"className":61324,"style":472},[353],[40,61326],{},[40,61328,61330],{"className":61329,"style":413},[113,412],[40,61331,1664],{"className":61332},[417,368],[40,61334],{"className":61335,"style":122},[121],[40,61337,73],{"className":61338},[117],[40,61340],{"className":61341,"style":1302},[121],[40,61343],{"className":61344,"style":122},[121],[40,61346,61348,61351],{"className":61347},[102],[40,61349,156],{"className":61350,"style":336},[102,335],[40,61352,61354],{"className":61353},[340],[40,61355,61357,61377],{"className":61356},[344,345],[40,61358,61360,61374],{"className":61359},[349],[40,61361,61363],{"className":61362,"style":354},[353],[40,61364,61365,61368],{"style":357},[40,61366],{"className":61367,"style":362},[361],[40,61369,61371],{"className":61370},[366,367,368,369],[40,61372,218],{"className":61373,"style":336},[102,335,369],[40,61375,377],{"className":61376},[376],[40,61378,61380],{"className":61379},[349],[40,61381,61383],{"className":61382,"style":585},[353],[40,61384],{},[40,61386],{"className":61387,"style":390},[121],[40,61389,162],{"className":61390},[394],[40,61392],{"className":61393,"style":390},[121],[40,61395,61397,61400,61520,61523,61526,61529,61532,61572,61575,61578],{"className":61396},[93],[40,61398],{"className":61399,"style":404},[97],[40,61401,61403,61409,61514],{"className":61402},[408],[40,61404,61406],{"className":61405,"style":413},[106,412],[40,61407,1654],{"className":61408},[417,368],[40,61410,61412],{"className":61411},[102],[40,61413,61415,61460,61463,61466],{"className":61414},[169],[40,61416,61418],{"className":61417},[427],[40,61419,61421,61452],{"className":61420},[344,345],[40,61422,61424,61449],{"className":61423},[349],[40,61425,61427,61438],{"className":61426,"style":437},[353],[40,61428,61429,61432],{"style":440},[40,61430],{"className":61431,"style":444},[361],[40,61433,61435],{"className":61434},[102],[40,61436,66],{"className":61437},[102],[40,61439,61440,61443],{"style":453},[40,61441],{"className":61442,"style":444},[361],[40,61444,61446],{"className":61445},[102],[40,61447,246],{"className":61448},[102,335],[40,61450,377],{"className":61451},[376],[40,61453,61455],{"className":61454},[349],[40,61456,61458],{"className":61457,"style":472},[353],[40,61459],{},[40,61461],{"className":61462,"style":479},[478],[40,61464],{"className":61465,"style":479},[478],[40,61467,61469],{"className":61468},[427],[40,61470,61472,61506],{"className":61471},[344,345],[40,61473,61475,61503],{"className":61474},[349],[40,61476,61478,61492],{"className":61477,"style":437},[353],[40,61479,61480,61483],{"style":440},[40,61481],{"className":61482,"style":444},[361],[40,61484,61486,61489],{"className":61485},[102],[40,61487,243],{"className":61488},[102],[40,61490,246],{"className":61491},[102,335],[40,61493,61494,61497],{"style":453},[40,61495],{"className":61496,"style":444},[361],[40,61498,61500],{"className":61499},[102],[40,61501,66],{"className":61502},[102],[40,61504,377],{"className":61505},[376],[40,61507,61509],{"className":61508},[349],[40,61510,61512],{"className":61511,"style":472},[353],[40,61513],{},[40,61515,61517],{"className":61516,"style":413},[113,412],[40,61518,1664],{"className":61519},[417,368],[40,61521],{"className":61522,"style":122},[121],[40,61524,73],{"className":61525},[117],[40,61527],{"className":61528,"style":1302},[121],[40,61530],{"className":61531,"style":122},[121],[40,61533,61535,61538],{"className":61534},[102],[40,61536,156],{"className":61537,"style":336},[102,335],[40,61539,61541],{"className":61540},[340],[40,61542,61544,61564],{"className":61543},[344,345],[40,61545,61547,61561],{"className":61546},[349],[40,61548,61550],{"className":61549,"style":354},[353],[40,61551,61552,61555],{"style":357},[40,61553],{"className":61554,"style":362},[361],[40,61556,61558],{"className":61557},[366,367,368,369],[40,61559,273],{"className":61560,"style":764},[102,335,369],[40,61562,377],{"className":61563},[376],[40,61565,61567],{"className":61566},[349],[40,61568,61570],{"className":61569,"style":384},[353],[40,61571],{},[40,61573],{"className":61574,"style":390},[121],[40,61576,162],{"className":61577},[394],[40,61579],{"className":61580,"style":390},[121],[40,61582,61584,61587,61707,61710],{"className":61583},[93],[40,61585],{"className":61586,"style":404},[97],[40,61588,61590,61596,61701],{"className":61589},[408],[40,61591,61593],{"className":61592,"style":413},[106,412],[40,61594,1654],{"className":61595},[417,368],[40,61597,61599],{"className":61598},[102],[40,61600,61602,61647,61650,61653],{"className":61601},[169],[40,61603,61605],{"className":61604},[427],[40,61606,61608,61639],{"className":61607},[344,345],[40,61609,61611,61636],{"className":61610},[349],[40,61612,61614,61625],{"className":61613,"style":437},[353],[40,61615,61616,61619],{"style":440},[40,61617],{"className":61618,"style":444},[361],[40,61620,61622],{"className":61621},[102],[40,61623,78],{"className":61624},[102],[40,61626,61627,61630],{"style":453},[40,61628],{"className":61629,"style":444},[361],[40,61631,61633],{"className":61632},[102],[40,61634,66],{"className":61635},[102],[40,61637,377],{"className":61638},[376],[40,61640,61642],{"className":61641},[349],[40,61643,61645],{"className":61644,"style":472},[353],[40,61646],{},[40,61648],{"className":61649,"style":479},[478],[40,61651],{"className":61652,"style":479},[478],[40,61654,61656],{"className":61655},[427],[40,61657,61659,61693],{"className":61658},[344,345],[40,61660,61662,61690],{"className":61661},[349],[40,61663,61665,61676],{"className":61664,"style":437},[353],[40,61666,61667,61670],{"style":440},[40,61668],{"className":61669,"style":444},[361],[40,61671,61673],{"className":61672},[102],[40,61674,66],{"className":61675},[102],[40,61677,61678,61681],{"style":453},[40,61679],{"className":61680,"style":444},[361],[40,61682,61684,61687],{"className":61683},[102],[40,61685,243],{"className":61686},[102],[40,61688,78],{"className":61689},[102],[40,61691,377],{"className":61692},[376],[40,61694,61696],{"className":61695},[349],[40,61697,61699],{"className":61698,"style":472},[353],[40,61700],{},[40,61702,61704],{"className":61703,"style":413},[113,412],[40,61705,1664],{"className":61706},[417,368],[40,61708],{"className":61709,"style":122},[121],[40,61711,1123],{"className":61712},[102],[12,61714,61715,61716,61749],{},"Within this representation, a driven two-level system behaves like a spin-",[40,61717,61719,61737],{"className":61718},[43],[40,61720,61722],{"className":61721},[47],[49,61723,61724],{"xmlns":51},[53,61725,61726,61734],{},[56,61727,61728,61730,61732],{},[64,61729,78],{},[154,61731,11243],{"mathvariant":1080},[64,61733,1044],{},[82,61735,61736],{"encoding":84},"1\u002F2",[40,61738,61740],{"className":61739,"ariaHidden":72},[89],[40,61741,61743,61746],{"className":61742},[93],[40,61744],{"className":61745,"style":98},[97],[40,61747,61736],{"className":61748},[102]," particle precessing around an effective magnetic field. The drive strength supplies the transverse control component. Applied detuning supplies the longitudinal control component.",[28,61751,61753],{"id":61752},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[12,61755,61756],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit is",[40,61758,61760],{"className":61759},[136],[40,61761,61763,61831],{"className":61762},[43],[40,61764,61766],{"className":61765},[47],[49,61767,61768],{"xmlns":51,"display":145},[53,61769,61770,61828],{},[56,61771,61772,61774,61776,61778,61780,61782,61796,61802,61804,61806,61808,61810,61812,61814,61816,61818,61820,61826],{},[154,61773,26630],{},[59,61775,167],{"stretchy":61},[154,61777,20277],{},[59,61779,207],{"stretchy":61},[59,61781,162],{},[1038,61783,61784,61794],{},[56,61785,61786,61788],{},[154,61787,59739],{"mathvariant":1080},[151,61789,61790,61792],{},[154,61791,20187],{},[64,61793,66],{},[64,61795,1044],{},[151,61797,61798,61800],{},[154,61799,156],{},[154,61801,273],{},[59,61803,1053],{},[154,61805,59739],{"mathvariant":1080},[154,61807,20352],{"mathvariant":1080},[154,61809,1820],{},[59,61811,1084],{},[59,61813,167],{"stretchy":61},[154,61815,20187],{},[154,61817,20277],{},[59,61819,207],{"stretchy":61},[151,61821,61822,61824],{},[154,61823,156],{},[154,61825,159],{},[59,61827,73],{"separator":72},[82,61829,61830],{"encoding":84},"H(t)=\\frac{\\hbar\\omega_0}{2}\\sigma_z+\\hbar\\Omega\\cos(\\omega t)\\sigma_x,",[40,61832,61834,61861,62020],{"className":61833,"ariaHidden":72},[89],[40,61835,61837,61840,61843,61846,61849,61852,61855,61858],{"className":61836},[93],[40,61838],{"className":61839,"style":98},[97],[40,61841,26630],{"className":61842,"style":36265},[102,335],[40,61844,167],{"className":61845},[106],[40,61847,20277],{"className":61848},[102,335],[40,61850,207],{"className":61851},[113],[40,61853],{"className":61854,"style":390},[121],[40,61856,162],{"className":61857},[394],[40,61859],{"className":61860,"style":390},[121],[40,61862,61864,61868,61971,62011,62014,62017],{"className":61863},[93],[40,61865],{"className":61866,"style":61867},[97],"height:2.0519em;vertical-align:-0.686em;",[40,61869,61871,61874,61968],{"className":61870},[102],[40,61872],{"className":61873},[106,1162],[40,61875,61877],{"className":61876},[1038],[40,61878,61880,61960],{"className":61879},[344,345],[40,61881,61883,61957],{"className":61882},[349],[40,61884,61887,61898,61906],{"className":61885,"style":61886},[353],"height:1.3659em;",[40,61888,61889,61892],{"style":2782},[40,61890],{"className":61891,"style":444},[361],[40,61893,61895],{"className":61894},[102],[40,61896,1044],{"className":61897},[102],[40,61899,61900,61903],{"style":1193},[40,61901],{"className":61902,"style":444},[361],[40,61904],{"className":61905,"style":1201},[1200],[40,61907,61908,61911],{"style":2802},[40,61909],{"className":61910,"style":444},[361],[40,61912,61914,61917],{"className":61913},[102],[40,61915,59739],{"className":61916},[102],[40,61918,61920,61923],{"className":61919},[102],[40,61921,20187],{"className":61922,"style":336},[102,335],[40,61924,61926],{"className":61925},[340],[40,61927,61929,61949],{"className":61928},[344,345],[40,61930,61932,61946],{"className":61931},[349],[40,61933,61935],{"className":61934,"style":7201},[353],[40,61936,61937,61940],{"style":357},[40,61938],{"className":61939,"style":362},[361],[40,61941,61943],{"className":61942},[366,367,368,369],[40,61944,66],{"className":61945},[102,369],[40,61947,377],{"className":61948},[376],[40,61950,61952],{"className":61951},[349],[40,61953,61955],{"className":61954,"style":384},[353],[40,61956],{},[40,61958,377],{"className":61959},[376],[40,61961,61963],{"className":61962},[349],[40,61964,61966],{"className":61965,"style":2868},[353],[40,61967],{},[40,61969],{"className":61970},[113,1162],[40,61972,61974,61977],{"className":61973},[102],[40,61975,156],{"className":61976,"style":336},[102,335],[40,61978,61980],{"className":61979},[340],[40,61981,61983,62003],{"className":61982},[344,345],[40,61984,61986,62000],{"className":61985},[349],[40,61987,61989],{"className":61988,"style":354},[353],[40,61990,61991,61994],{"style":357},[40,61992],{"className":61993,"style":362},[361],[40,61995,61997],{"className":61996},[366,367,368,369],[40,61998,273],{"className":61999,"style":764},[102,335,369],[40,62001,377],{"className":62002},[376],[40,62004,62006],{"className":62005},[349],[40,62007,62009],{"className":62008,"style":384},[353],[40,62010],{},[40,62012],{"className":62013,"style":1246},[121],[40,62015,1053],{"className":62016},[1250],[40,62018],{"className":62019,"style":1246},[121],[40,62021,62023,62026,62030,62033,62036,62039,62042,62045,62048,62088],{"className":62022},[93],[40,62024],{"className":62025,"style":98},[97],[40,62027,62029],{"className":62028},[102],"ℏΩ",[40,62031],{"className":62032,"style":122},[121],[40,62034,1820],{"className":62035},[1369],[40,62037,167],{"className":62038},[106],[40,62040,20187],{"className":62041,"style":336},[102,335],[40,62043,20277],{"className":62044},[102,335],[40,62046,207],{"className":62047},[113],[40,62049,62051,62054],{"className":62050},[102],[40,62052,156],{"className":62053,"style":336},[102,335],[40,62055,62057],{"className":62056},[340],[40,62058,62060,62080],{"className":62059},[344,345],[40,62061,62063,62077],{"className":62062},[349],[40,62064,62066],{"className":62065,"style":354},[353],[40,62067,62068,62071],{"style":357},[40,62069],{"className":62070,"style":362},[361],[40,62072,62074],{"className":62073},[366,367,368,369],[40,62075,159],{"className":62076},[102,335,369],[40,62078,377],{"className":62079},[376],[40,62081,62083],{"className":62082},[349],[40,62084,62086],{"className":62085,"style":384},[353],[40,62087],{},[40,62089,73],{"className":62090},[117],[12,62092,58088,62093,62153,62154,62182,62183,62220],{},[40,62094,62096,62117],{"className":62095},[43],[40,62097,62099],{"className":62098},[47],[49,62100,62101],{"xmlns":51},[53,62102,62103,62115],{},[56,62104,62105,62107,62109,62111,62113],{},[154,62106,20187],{},[59,62108,162],{},[64,62110,1044],{},[154,62112,1661],{},[154,62114,13262],{},[82,62116,58905],{"encoding":84},[40,62118,62120,62138],{"className":62119,"ariaHidden":72},[89],[40,62121,62123,62126,62129,62132,62135],{"className":62122},[93],[40,62124],{"className":62125,"style":4142},[97],[40,62127,20187],{"className":62128,"style":336},[102,335],[40,62130],{"className":62131,"style":390},[121],[40,62133,162],{"className":62134},[394],[40,62136],{"className":62137,"style":390},[121],[40,62139,62141,62144,62147,62150],{"className":62140},[93],[40,62142],{"className":62143,"style":58114},[97],[40,62145,1044],{"className":62146},[102],[40,62148,1661],{"className":62149,"style":336},[102,335],[40,62151,13262],{"className":62152,"style":33300},[102,335]," is the drive angular frequency and ",[40,62155,62157,62170],{"className":62156},[43],[40,62158,62160],{"className":62159},[47],[49,62161,62162],{"xmlns":51},[53,62163,62164,62168],{},[56,62165,62166],{},[154,62167,20352],{"mathvariant":1080},[82,62169,20355],{"encoding":84},[40,62171,62173],{"className":62172,"ariaHidden":72},[89],[40,62174,62176,62179],{"className":62175},[93],[40,62177],{"className":62178,"style":4304},[97],[40,62180,20352],{"className":62181},[102]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[40,62184,62186,62205],{"className":62185},[43],[40,62187,62189],{"className":62188},[47],[49,62190,62191],{"xmlns":51},[53,62192,62193,62203],{},[56,62194,62195,62197,62199,62201],{},[154,62196,20352],{"mathvariant":1080},[154,62198,11243],{"mathvariant":1080},[64,62200,1044],{},[154,62202,1661],{},[82,62204,59003],{"encoding":84},[40,62206,62208],{"className":62207,"ariaHidden":72},[89],[40,62209,62211,62214,62217],{"className":62210},[93],[40,62212],{"className":62213,"style":98},[97],[40,62215,59016],{"className":62216},[102],[40,62218,1661],{"className":62219,"style":336},[102,335]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[40,62222,62224],{"className":62223},[136],[40,62225,62227,62285],{"className":62226},[43],[40,62228,62230],{"className":62229},[47],[49,62231,62232],{"xmlns":51,"display":145},[53,62233,62234,62282],{},[56,62235,62236,62248,62250,62256,62280],{},[151,62237,62238,62240],{},[154,62239,26630],{},[56,62241,62242,62244,62246],{},[154,62243,946],{"mathvariant":1080},[154,62245,2437],{"mathvariant":1080},[154,62247,8234],{"mathvariant":1080},[59,62249,162],{},[1038,62251,62252,62254],{},[154,62253,59739],{"mathvariant":1080},[64,62255,1044],{},[56,62257,62258,62260,62262,62268,62270,62272,62278],{},[59,62259,167],{"fence":72},[154,62261,2016],{"mathvariant":1080},[151,62263,62264,62266],{},[154,62265,156],{},[154,62267,273],{},[59,62269,1053],{},[154,62271,20352],{"mathvariant":1080},[151,62273,62274,62276],{},[154,62275,156],{},[154,62277,159],{},[59,62279,207],{"fence":72},[59,62281,73],{"separator":72},[82,62283,62284],{"encoding":84},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[40,62286,62288,62350],{"className":62287,"ariaHidden":72},[89],[40,62289,62291,62294,62341,62344,62347],{"className":62290},[93],[40,62292],{"className":62293,"style":3106},[97],[40,62295,62297,62300],{"className":62296},[102],[40,62298,26630],{"className":62299,"style":36265},[102,335],[40,62301,62303],{"className":62302},[340],[40,62304,62306,62333],{"className":62305},[344,345],[40,62307,62309,62330],{"className":62308},[349],[40,62310,62312],{"className":62311,"style":20224},[353],[40,62313,62315,62318],{"style":62314},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[40,62316],{"className":62317,"style":362},[361],[40,62319,62321],{"className":62320},[366,367,368,369],[40,62322,62324],{"className":62323},[102,369],[40,62325,62327],{"className":62326},[102,369],[40,62328,59609],{"className":62329},[102,1373,369],[40,62331,377],{"className":62332},[376],[40,62334,62336],{"className":62335},[349],[40,62337,62339],{"className":62338,"style":384},[353],[40,62340],{},[40,62342],{"className":62343,"style":390},[121],[40,62345,162],{"className":62346},[394],[40,62348],{"className":62349,"style":390},[121],[40,62351,62353,62356,62418,62421,62525,62528],{"className":62352},[93],[40,62354],{"className":62355,"style":61867},[97],[40,62357,62359,62362,62415],{"className":62358},[102],[40,62360],{"className":62361},[106,1162],[40,62363,62365],{"className":62364},[1038],[40,62366,62368,62407],{"className":62367},[344,345],[40,62369,62371,62404],{"className":62370},[349],[40,62372,62374,62385,62393],{"className":62373,"style":61886},[353],[40,62375,62376,62379],{"style":2782},[40,62377],{"className":62378,"style":444},[361],[40,62380,62382],{"className":62381},[102],[40,62383,1044],{"className":62384},[102],[40,62386,62387,62390],{"style":1193},[40,62388],{"className":62389,"style":444},[361],[40,62391],{"className":62392,"style":1201},[1200],[40,62394,62395,62398],{"style":2802},[40,62396],{"className":62397,"style":444},[361],[40,62399,62401],{"className":62400},[102],[40,62402,59739],{"className":62403},[102],[40,62405,377],{"className":62406},[376],[40,62408,62410],{"className":62409},[349],[40,62411,62413],{"className":62412,"style":2868},[353],[40,62414],{},[40,62416],{"className":62417},[113,1162],[40,62419],{"className":62420,"style":122},[121],[40,62422,62424,62427,62430,62470,62473,62476,62479,62482,62522],{"className":62423},[408],[40,62425,167],{"className":62426,"style":413},[106,412],[40,62428,2016],{"className":62429},[102],[40,62431,62433,62436],{"className":62432},[102],[40,62434,156],{"className":62435,"style":336},[102,335],[40,62437,62439],{"className":62438},[340],[40,62440,62442,62462],{"className":62441},[344,345],[40,62443,62445,62459],{"className":62444},[349],[40,62446,62448],{"className":62447,"style":354},[353],[40,62449,62450,62453],{"style":357},[40,62451],{"className":62452,"style":362},[361],[40,62454,62456],{"className":62455},[366,367,368,369],[40,62457,273],{"className":62458,"style":764},[102,335,369],[40,62460,377],{"className":62461},[376],[40,62463,62465],{"className":62464},[349],[40,62466,62468],{"className":62467,"style":384},[353],[40,62469],{},[40,62471],{"className":62472,"style":1246},[121],[40,62474,1053],{"className":62475},[1250],[40,62477],{"className":62478,"style":1246},[121],[40,62480,20352],{"className":62481},[102],[40,62483,62485,62488],{"className":62484},[102],[40,62486,156],{"className":62487,"style":336},[102,335],[40,62489,62491],{"className":62490},[340],[40,62492,62494,62514],{"className":62493},[344,345],[40,62495,62497,62511],{"className":62496},[349],[40,62498,62500],{"className":62499,"style":354},[353],[40,62501,62502,62505],{"style":357},[40,62503],{"className":62504,"style":362},[361],[40,62506,62508],{"className":62507},[366,367,368,369],[40,62509,159],{"className":62510},[102,335,369],[40,62512,377],{"className":62513},[376],[40,62515,62517],{"className":62516},[349],[40,62518,62520],{"className":62519,"style":384},[353],[40,62521],{},[40,62523,207],{"className":62524,"style":413},[113,412],[40,62526],{"className":62527,"style":122},[121],[40,62529,73],{"className":62530},[117],[12,62532,62533],{},"with detuning",[40,62535,62537],{"className":62536},[136],[40,62538,62540,62588],{"className":62539},[43],[40,62541,62543],{"className":62542},[47],[49,62544,62545],{"xmlns":51,"display":145},[53,62546,62547,62585],{},[56,62548,62549,62551,62553,62555,62557,62563,62565,62567,62569,62571,62573,62575,62581,62583],{},[154,62550,2016],{"mathvariant":1080},[59,62552,162],{},[154,62554,20187],{},[59,62556,243],{},[151,62558,62559,62561],{},[154,62560,20187],{},[64,62562,66],{},[59,62564,162],{},[64,62566,1044],{},[154,62568,1661],{},[59,62570,167],{"stretchy":61},[154,62572,13262],{},[59,62574,243],{},[151,62576,62577,62579],{},[154,62578,13262],{},[64,62580,66],{},[59,62582,207],{"stretchy":61},[154,62584,1123],{"mathvariant":1080},[82,62586,62587],{"encoding":84},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[40,62589,62591,62609,62627,62682,62709],{"className":62590,"ariaHidden":72},[89],[40,62592,62594,62597,62600,62603,62606],{"className":62593},[93],[40,62595],{"className":62596,"style":4304},[97],[40,62598,2016],{"className":62599},[102],[40,62601],{"className":62602,"style":390},[121],[40,62604,162],{"className":62605},[394],[40,62607],{"className":62608,"style":390},[121],[40,62610,62612,62615,62618,62621,62624],{"className":62611},[93],[40,62613],{"className":62614,"style":59114},[97],[40,62616,20187],{"className":62617,"style":336},[102,335],[40,62619],{"className":62620,"style":1246},[121],[40,62622,243],{"className":62623},[1250],[40,62625],{"className":62626,"style":1246},[121],[40,62628,62630,62633,62673,62676,62679],{"className":62629},[93],[40,62631],{"className":62632,"style":328},[97],[40,62634,62636,62639],{"className":62635},[102],[40,62637,20187],{"className":62638,"style":336},[102,335],[40,62640,62642],{"className":62641},[340],[40,62643,62645,62665],{"className":62644},[344,345],[40,62646,62648,62662],{"className":62647},[349],[40,62649,62651],{"className":62650,"style":7201},[353],[40,62652,62653,62656],{"style":357},[40,62654],{"className":62655,"style":362},[361],[40,62657,62659],{"className":62658},[366,367,368,369],[40,62660,66],{"className":62661},[102,369],[40,62663,377],{"className":62664},[376],[40,62666,62668],{"className":62667},[349],[40,62669,62671],{"className":62670,"style":384},[353],[40,62672],{},[40,62674],{"className":62675,"style":390},[121],[40,62677,162],{"className":62678},[394],[40,62680],{"className":62681,"style":390},[121],[40,62683,62685,62688,62691,62694,62697,62700,62703,62706],{"className":62684},[93],[40,62686],{"className":62687,"style":98},[97],[40,62689,1044],{"className":62690},[102],[40,62692,1661],{"className":62693,"style":336},[102,335],[40,62695,167],{"className":62696},[106],[40,62698,13262],{"className":62699,"style":33300},[102,335],[40,62701],{"className":62702,"style":1246},[121],[40,62704,243],{"className":62705},[1250],[40,62707],{"className":62708,"style":1246},[121],[40,62710,62712,62715,62755,62758],{"className":62711},[93],[40,62713],{"className":62714,"style":98},[97],[40,62716,62718,62721],{"className":62717},[102],[40,62719,13262],{"className":62720,"style":33300},[102,335],[40,62722,62724],{"className":62723},[340],[40,62725,62727,62747],{"className":62726},[344,345],[40,62728,62730,62744],{"className":62729},[349],[40,62731,62733],{"className":62732,"style":7201},[353],[40,62734,62735,62738],{"style":33315},[40,62736],{"className":62737,"style":362},[361],[40,62739,62741],{"className":62740},[366,367,368,369],[40,62742,66],{"className":62743},[102,369],[40,62745,377],{"className":62746},[376],[40,62748,62750],{"className":62749},[349],[40,62751,62753],{"className":62752,"style":384},[353],[40,62754],{},[40,62756,207],{"className":62757},[113],[40,62759,1123],{"className":62760},[102],[12,62762,62763,62764,916,62792,62820,62821,62849,62850,62878],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[40,62765,62767,62780],{"className":62766},[43],[40,62768,62770],{"className":62769},[47],[49,62771,62772],{"xmlns":51},[53,62773,62774,62778],{},[56,62775,62776],{},[154,62777,20352],{"mathvariant":1080},[82,62779,20355],{"encoding":84},[40,62781,62783],{"className":62782,"ariaHidden":72},[89],[40,62784,62786,62789],{"className":62785},[93],[40,62787],{"className":62788,"style":4304},[97],[40,62790,20352],{"className":62791},[102],[40,62793,62795,62808],{"className":62794},[43],[40,62796,62798],{"className":62797},[47],[49,62799,62800],{"xmlns":51},[53,62801,62802,62806],{},[56,62803,62804],{},[154,62805,2016],{"mathvariant":1080},[82,62807,10118],{"encoding":84},[40,62809,62811],{"className":62810,"ariaHidden":72},[89],[40,62812,62814,62817],{"className":62813},[93],[40,62815],{"className":62816,"style":4304},[97],[40,62818,2016],{"className":62819},[102],". The transverse component ",[40,62822,62824,62837],{"className":62823},[43],[40,62825,62827],{"className":62826},[47],[49,62828,62829],{"xmlns":51},[53,62830,62831,62835],{},[56,62832,62833],{},[154,62834,20352],{"mathvariant":1080},[82,62836,20355],{"encoding":84},[40,62838,62840],{"className":62839,"ariaHidden":72},[89],[40,62841,62843,62846],{"className":62842},[93],[40,62844],{"className":62845,"style":4304},[97],[40,62847,20352],{"className":62848},[102]," drives population transfer. A longitudinal component ",[40,62851,62853,62866],{"className":62852},[43],[40,62854,62856],{"className":62855},[47],[49,62857,62858],{"xmlns":51},[53,62859,62860,62864],{},[56,62861,62862],{},[154,62863,2016],{"mathvariant":1080},[82,62865,10118],{"encoding":84},[40,62867,62869],{"className":62868,"ariaHidden":72},[89],[40,62870,62872,62875],{"className":62871},[93],[40,62873],{"className":62874,"style":4304},[97],[40,62876,2016],{"className":62877},[102]," tilts the rotation axis away from the resonant direction.",[28,62880,62882],{"id":62881},"excited-state-probability","Excited-State Probability",[12,62884,62885,62886,62924,62925,8711],{},"The simulation assumes that the qubit begins in the ground state ",[40,62887,62889,62906],{"className":62888},[43],[40,62890,62892],{"className":62891},[47],[49,62893,62894],{"xmlns":51},[53,62895,62896,62904],{},[56,62897,62898,62900,62902],{},[59,62899,62],{"stretchy":61},[154,62901,58190],{},[59,62903,69],{"stretchy":61},[82,62905,58195],{"encoding":84},[40,62907,62909],{"className":62908,"ariaHidden":72},[89],[40,62910,62912,62915,62918,62921],{"className":62911},[93],[40,62913],{"className":62914,"style":98},[97],[40,62916,62],{"className":62917},[106],[40,62919,58190],{"className":62920,"style":336},[102,335],[40,62922,69],{"className":62923},[113],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration ",[40,62926,62928,62941],{"className":62927},[43],[40,62929,62931],{"className":62930},[47],[49,62932,62933],{"xmlns":51},[53,62934,62935,62939],{},[56,62936,62937],{},[154,62938,58007],{},[82,62940,58135],{"encoding":84},[40,62942,62944],{"className":62943,"ariaHidden":72},[89],[40,62945,62947,62950],{"className":62946},[93],[40,62948],{"className":62949,"style":4142},[97],[40,62951,58007],{"className":62952,"style":58079},[102,335],[40,62954,62956],{"className":62955},[136],[40,62957,62959,63037],{"className":62958},[43],[40,62960,62962],{"className":62961},[47],[49,62963,62964],{"xmlns":51,"display":145},[53,62965,62966,63034],{},[56,62967,62968,62974,62976,62978,62980,62982,62984,62986,63002,63012,63032],{},[151,62969,62970,62972],{},[154,62971,57996],{},[154,62973,4203],{},[59,62975,167],{"stretchy":61},[154,62977,13262],{},[59,62979,73],{"separator":72},[154,62981,58007],{},[59,62983,207],{"stretchy":61},[59,62985,162],{},[1038,62987,62988,62994],{},[941,62989,62990,62992],{},[154,62991,20352],{"mathvariant":1080},[64,62993,1044],{},[8360,62995,62996,62998,63000],{},[154,62997,20352],{"mathvariant":1080},[154,62999,946],{},[64,63001,1044],{},[941,63003,63004,63010],{},[56,63005,63006,63008],{},[154,63007,1813],{},[59,63009,1084],{},[64,63011,1044],{},[56,63013,63014,63016,63030],{},[59,63015,167],{"fence":72},[1038,63017,63018,63028],{},[56,63019,63020,63026],{},[151,63021,63022,63024],{},[154,63023,20352],{"mathvariant":1080},[154,63025,946],{},[154,63027,58007],{},[64,63029,1044],{},[59,63031,207],{"fence":72},[59,63033,73],{"separator":72},[82,63035,63036],{"encoding":84},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right),",[40,63038,63040,63113],{"className":63039,"ariaHidden":72},[89],[40,63041,63043,63046,63086,63089,63092,63095,63098,63101,63104,63107,63110],{"className":63042},[93],[40,63044],{"className":63045,"style":98},[97],[40,63047,63049,63052],{"className":63048},[102],[40,63050,57996],{"className":63051,"style":2541},[102,335],[40,63053,63055],{"className":63054},[340],[40,63056,63058,63078],{"className":63057},[344,345],[40,63059,63061,63075],{"className":63060},[349],[40,63062,63064],{"className":63063,"style":354},[353],[40,63065,63066,63069],{"style":2556},[40,63067],{"className":63068,"style":362},[361],[40,63070,63072],{"className":63071},[366,367,368,369],[40,63073,4203],{"className":63074},[102,335,369],[40,63076,377],{"className":63077},[376],[40,63079,63081],{"className":63080},[349],[40,63082,63084],{"className":63083,"style":384},[353],[40,63085],{},[40,63087,167],{"className":63088},[106],[40,63090,13262],{"className":63091,"style":33300},[102,335],[40,63093,73],{"className":63094},[117],[40,63096],{"className":63097,"style":122},[121],[40,63099,58007],{"className":63100,"style":58079},[102,335],[40,63102,207],{"className":63103},[113],[40,63105],{"className":63106,"style":390},[121],[40,63108,162],{"className":63109},[394],[40,63111],{"className":63112,"style":390},[121],[40,63114,63116,63120,63262,63265,63296,63299,63417,63420],{"className":63115},[93],[40,63117],{"className":63118,"style":63119},[97],"height:2.4706em;vertical-align:-0.9795em;",[40,63121,63123,63126,63259],{"className":63122},[102],[40,63124],{"className":63125},[106,1162],[40,63127,63129],{"className":63128},[1038],[40,63130,63132,63250],{"className":63131},[344,345],[40,63133,63135,63247],{"className":63134},[349],[40,63136,63139,63202,63210],{"className":63137,"style":63138},[353],"height:1.4911em;",[40,63140,63141,63144],{"style":2782},[40,63142],{"className":63143,"style":444},[361],[40,63145,63147],{"className":63146},[102],[40,63148,63150,63153],{"className":63149},[102],[40,63151,20352],{"className":63152},[102],[40,63154,63156],{"className":63155},[340],[40,63157,63159,63193],{"className":63158},[344,345],[40,63160,63162,63190],{"className":63161},[349],[40,63163,63166,63178],{"className":63164,"style":63165},[353],"height:0.7959em;",[40,63167,63169,63172],{"style":63168},"top:-2.4065em;margin-left:0em;margin-right:0.05em;",[40,63170],{"className":63171,"style":362},[361],[40,63173,63175],{"className":63174},[366,367,368,369],[40,63176,946],{"className":63177,"style":3946},[102,335,369],[40,63179,63181,63184],{"style":63180},"top:-3.0448em;margin-right:0.05em;",[40,63182],{"className":63183,"style":362},[361],[40,63185,63187],{"className":63186},[366,367,368,369],[40,63188,1044],{"className":63189},[102,369],[40,63191,377],{"className":63192},[376],[40,63194,63196],{"className":63195},[349],[40,63197,63200],{"className":63198,"style":63199},[353],"height:0.2935em;",[40,63201],{},[40,63203,63204,63207],{"style":1193},[40,63205],{"className":63206,"style":444},[361],[40,63208],{"className":63209,"style":1201},[1200],[40,63211,63212,63215],{"style":2802},[40,63213],{"className":63214,"style":444},[361],[40,63216,63218],{"className":63217},[102],[40,63219,63221,63224],{"className":63220},[102],[40,63222,20352],{"className":63223},[102],[40,63225,63227],{"className":63226},[340],[40,63228,63230],{"className":63229},[344],[40,63231,63233],{"className":63232},[349],[40,63234,63236],{"className":63235,"style":982},[353],[40,63237,63238,63241],{"style":1004},[40,63239],{"className":63240,"style":362},[361],[40,63242,63244],{"className":63243},[366,367,368,369],[40,63245,1044],{"className":63246},[102,369],[40,63248,377],{"className":63249},[376],[40,63251,63253],{"className":63252},[349],[40,63254,63257],{"className":63255,"style":63256},[353],"height:0.9795em;",[40,63258],{},[40,63260],{"className":63261},[113,1162],[40,63263],{"className":63264,"style":122},[121],[40,63266,63268,63271],{"className":63267},[1369],[40,63269,1813],{"className":63270},[1369],[40,63272,63274],{"className":63273},[340],[40,63275,63277],{"className":63276},[344],[40,63278,63280],{"className":63279},[349],[40,63281,63284],{"className":63282,"style":63283},[353],"height:0.8719em;",[40,63285,63287,63290],{"style":63286},"top:-3.1208em;margin-right:0.05em;",[40,63288],{"className":63289,"style":362},[361],[40,63291,63293],{"className":63292},[366,367,368,369],[40,63294,1044],{"className":63295},[102,369],[40,63297],{"className":63298,"style":122},[121],[40,63300,63302,63308,63411],{"className":63301},[408],[40,63303,63305],{"className":63304,"style":413},[106,412],[40,63306,167],{"className":63307},[417,368],[40,63309,63311,63314,63408],{"className":63310},[102],[40,63312],{"className":63313},[106,1162],[40,63315,63317],{"className":63316},[1038],[40,63318,63320,63400],{"className":63319},[344,345],[40,63321,63323,63397],{"className":63322},[349],[40,63324,63327,63338,63346],{"className":63325,"style":63326},[353],"height:1.3603em;",[40,63328,63329,63332],{"style":2782},[40,63330],{"className":63331,"style":444},[361],[40,63333,63335],{"className":63334},[102],[40,63336,1044],{"className":63337},[102],[40,63339,63340,63343],{"style":1193},[40,63341],{"className":63342,"style":444},[361],[40,63344],{"className":63345,"style":1201},[1200],[40,63347,63348,63351],{"style":2802},[40,63349],{"className":63350,"style":444},[361],[40,63352,63354,63394],{"className":63353},[102],[40,63355,63357,63360],{"className":63356},[102],[40,63358,20352],{"className":63359},[102],[40,63361,63363],{"className":63362},[340],[40,63364,63366,63386],{"className":63365},[344,345],[40,63367,63369,63383],{"className":63368},[349],[40,63370,63372],{"className":63371,"style":20224},[353],[40,63373,63374,63377],{"style":6098},[40,63375],{"className":63376,"style":362},[361],[40,63378,63380],{"className":63379},[366,367,368,369],[40,63381,946],{"className":63382,"style":3946},[102,335,369],[40,63384,377],{"className":63385},[376],[40,63387,63389],{"className":63388},[349],[40,63390,63392],{"className":63391,"style":384},[353],[40,63393],{},[40,63395,58007],{"className":63396,"style":58079},[102,335],[40,63398,377],{"className":63399},[376],[40,63401,63403],{"className":63402},[349],[40,63404,63406],{"className":63405,"style":2868},[353],[40,63407],{},[40,63409],{"className":63410},[113,1162],[40,63412,63414],{"className":63413,"style":413},[113,412],[40,63415,207],{"className":63416},[417,368],[40,63418],{"className":63419,"style":122},[121],[40,63421,73],{"className":63422},[117],[12,63424,23436],{},[40,63426,63428],{"className":63427},[136],[40,63429,63431,63469],{"className":63430},[43],[40,63432,63434],{"className":63433},[47],[49,63435,63436],{"xmlns":51,"display":145},[53,63437,63438,63466],{},[56,63439,63440,63446,63448],{},[151,63441,63442,63444],{},[154,63443,20352],{"mathvariant":1080},[154,63445,946],{},[59,63447,162],{},[2045,63449,63450],{},[56,63451,63452,63458,63460],{},[941,63453,63454,63456],{},[154,63455,20352],{"mathvariant":1080},[64,63457,1044],{},[59,63459,1053],{},[941,63461,63462,63464],{},[154,63463,2016],{"mathvariant":1080},[64,63465,1044],{},[82,63467,63468],{"encoding":84},"\\Omega_R=\\sqrt{\\Omega^2+\\Delta^2}",[40,63470,63472,63527],{"className":63471,"ariaHidden":72},[89],[40,63473,63475,63478,63518,63521,63524],{"className":63474},[93],[40,63476],{"className":63477,"style":3106},[97],[40,63479,63481,63484],{"className":63480},[102],[40,63482,20352],{"className":63483},[102],[40,63485,63487],{"className":63486},[340],[40,63488,63490,63510],{"className":63489},[344,345],[40,63491,63493,63507],{"className":63492},[349],[40,63494,63496],{"className":63495,"style":20224},[353],[40,63497,63498,63501],{"style":6098},[40,63499],{"className":63500,"style":362},[361],[40,63502,63504],{"className":63503},[366,367,368,369],[40,63505,946],{"className":63506,"style":3946},[102,335,369],[40,63508,377],{"className":63509},[376],[40,63511,63513],{"className":63512},[349],[40,63514,63516],{"className":63515,"style":384},[353],[40,63517],{},[40,63519],{"className":63520,"style":390},[121],[40,63522,162],{"className":63523},[394],[40,63525],{"className":63526,"style":390},[121],[40,63528,63530,63534],{"className":63529},[93],[40,63531],{"className":63532,"style":63533},[97],"height:1.24em;vertical-align:-0.1777em;",[40,63535,63537],{"className":63536},[102,2218],[40,63538,63540,63648],{"className":63539},[344,345],[40,63541,63543,63645],{"className":63542},[349],[40,63544,63547,63628],{"className":63545,"style":63546},[353],"height:1.0623em;",[40,63548,63551,63555],{"className":63549,"style":63550},[2232],"top:-3.2em;",[40,63552],{"className":63553,"style":63554},[361],"height:3.2em;",[40,63556,63559,63590,63593,63596,63599],{"className":63557,"style":63558},[102],"padding-left:1em;",[40,63560,63562,63565],{"className":63561},[102],[40,63563,20352],{"className":63564},[102],[40,63566,63568],{"className":63567},[340],[40,63569,63571],{"className":63570},[344],[40,63572,63574],{"className":63573},[349],[40,63575,63578],{"className":63576,"style":63577},[353],"height:0.7401em;",[40,63579,63581,63584],{"style":63580},"top:-2.989em;margin-right:0.05em;",[40,63582],{"className":63583,"style":362},[361],[40,63585,63587],{"className":63586},[366,367,368,369],[40,63588,1044],{"className":63589},[102,369],[40,63591],{"className":63592,"style":1246},[121],[40,63594,1053],{"className":63595},[1250],[40,63597],{"className":63598,"style":1246},[121],[40,63600,63602,63605],{"className":63601},[102],[40,63603,2016],{"className":63604},[102],[40,63606,63608],{"className":63607},[340],[40,63609,63611],{"className":63610},[344],[40,63612,63614],{"className":63613},[349],[40,63615,63617],{"className":63616,"style":63577},[353],[40,63618,63619,63622],{"style":63580},[40,63620],{"className":63621,"style":362},[361],[40,63623,63625],{"className":63624},[366,367,368,369],[40,63626,1044],{"className":63627},[102,369],[40,63629,63631,63634],{"style":63630},"top:-3.0223em;",[40,63632],{"className":63633,"style":63554},[361],[40,63635,63638],{"className":63636,"style":63637},[2253],"min-width:1.02em;height:1.28em;",[2256,63639,63642],{"xmlns":2258,"width":2259,"height":63640,"viewBox":63641,"preserveAspectRatio":2262},"1.28em","0 0 400000 1296",[2264,63643],{"d":63644},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[40,63646,377],{"className":63647},[376],[40,63649,63651],{"className":63650},[349],[40,63652,63655],{"className":63653,"style":63654},[353],"height:0.1777em;",[40,63656],{},[12,63658,63659],{},"is the generalized Rabi angular frequency. The amplitude coefficient",[40,63661,63663],{"className":63662},[136],[40,63664,63666,63694],{"className":63665},[43],[40,63667,63669],{"className":63668},[47],[49,63670,63671],{"xmlns":51,"display":145},[53,63672,63673,63691],{},[56,63674,63675],{},[1038,63676,63677,63683],{},[941,63678,63679,63681],{},[154,63680,20352],{"mathvariant":1080},[64,63682,1044],{},[8360,63684,63685,63687,63689],{},[154,63686,20352],{"mathvariant":1080},[154,63688,946],{},[64,63690,1044],{},[82,63692,63693],{"encoding":84},"\\frac{\\Omega^2}{\\Omega_R^2}",[40,63695,63697],{"className":63696,"ariaHidden":72},[89],[40,63698,63700,63703],{"className":63699},[93],[40,63701],{"className":63702,"style":63119},[97],[40,63704,63706,63709,63836],{"className":63705},[102],[40,63707],{"className":63708},[106,1162],[40,63710,63712],{"className":63711},[1038],[40,63713,63715,63828],{"className":63714},[344,345],[40,63716,63718,63825],{"className":63717},[349],[40,63719,63721,63780,63788],{"className":63720,"style":63138},[353],[40,63722,63723,63726],{"style":2782},[40,63724],{"className":63725,"style":444},[361],[40,63727,63729],{"className":63728},[102],[40,63730,63732,63735],{"className":63731},[102],[40,63733,20352],{"className":63734},[102],[40,63736,63738],{"className":63737},[340],[40,63739,63741,63772],{"className":63740},[344,345],[40,63742,63744,63769],{"className":63743},[349],[40,63745,63747,63758],{"className":63746,"style":63165},[353],[40,63748,63749,63752],{"style":63168},[40,63750],{"className":63751,"style":362},[361],[40,63753,63755],{"className":63754},[366,367,368,369],[40,63756,946],{"className":63757,"style":3946},[102,335,369],[40,63759,63760,63763],{"style":63180},[40,63761],{"className":63762,"style":362},[361],[40,63764,63766],{"className":63765},[366,367,368,369],[40,63767,1044],{"className":63768},[102,369],[40,63770,377],{"className":63771},[376],[40,63773,63775],{"className":63774},[349],[40,63776,63778],{"className":63777,"style":63199},[353],[40,63779],{},[40,63781,63782,63785],{"style":1193},[40,63783],{"className":63784,"style":444},[361],[40,63786],{"className":63787,"style":1201},[1200],[40,63789,63790,63793],{"style":2802},[40,63791],{"className":63792,"style":444},[361],[40,63794,63796],{"className":63795},[102],[40,63797,63799,63802],{"className":63798},[102],[40,63800,20352],{"className":63801},[102],[40,63803,63805],{"className":63804},[340],[40,63806,63808],{"className":63807},[344],[40,63809,63811],{"className":63810},[349],[40,63812,63814],{"className":63813,"style":982},[353],[40,63815,63816,63819],{"style":1004},[40,63817],{"className":63818,"style":362},[361],[40,63820,63822],{"className":63821},[366,367,368,369],[40,63823,1044],{"className":63824},[102,369],[40,63826,377],{"className":63827},[376],[40,63829,63831],{"className":63830},[349],[40,63832,63834],{"className":63833,"style":63256},[353],[40,63835],{},[40,63837],{"className":63838},[113,1162],[12,63840,63841,63842,63893],{},"sets the maximum possible population transfer. At exact resonance, ",[40,63843,63845,63863],{"className":63844},[43],[40,63846,63848],{"className":63847},[47],[49,63849,63850],{"xmlns":51},[53,63851,63852,63860],{},[56,63853,63854,63856,63858],{},[154,63855,2016],{"mathvariant":1080},[59,63857,162],{},[64,63859,66],{},[82,63861,63862],{"encoding":84},"\\Delta=0",[40,63864,63866,63884],{"className":63865,"ariaHidden":72},[89],[40,63867,63869,63872,63875,63878,63881],{"className":63868},[93],[40,63870],{"className":63871,"style":4304},[97],[40,63873,2016],{"className":63874},[102],[40,63876],{"className":63877,"style":390},[121],[40,63879,162],{"className":63880},[394],[40,63882],{"className":63883,"style":390},[121],[40,63885,63887,63890],{"className":63886},[93],[40,63888],{"className":63889,"style":1624},[97],[40,63891,66],{"className":63892},[102],", so",[40,63895,63897],{"className":63896},[136],[40,63898,63900,63932],{"className":63899},[43],[40,63901,63903],{"className":63902},[47],[49,63904,63905],{"xmlns":51,"display":145},[53,63906,63907,63929],{},[56,63908,63909,63925,63927],{},[1038,63910,63911,63917],{},[941,63912,63913,63915],{},[154,63914,20352],{"mathvariant":1080},[64,63916,1044],{},[8360,63918,63919,63921,63923],{},[154,63920,20352],{"mathvariant":1080},[154,63922,946],{},[64,63924,1044],{},[59,63926,162],{},[64,63928,60639],{},[82,63930,63931],{"encoding":84},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[40,63933,63935,64086],{"className":63934,"ariaHidden":72},[89],[40,63936,63938,63941,64077,64080,64083],{"className":63937},[93],[40,63939],{"className":63940,"style":63119},[97],[40,63942,63944,63947,64074],{"className":63943},[102],[40,63945],{"className":63946},[106,1162],[40,63948,63950],{"className":63949},[1038],[40,63951,63953,64066],{"className":63952},[344,345],[40,63954,63956,64063],{"className":63955},[349],[40,63957,63959,64018,64026],{"className":63958,"style":63138},[353],[40,63960,63961,63964],{"style":2782},[40,63962],{"className":63963,"style":444},[361],[40,63965,63967],{"className":63966},[102],[40,63968,63970,63973],{"className":63969},[102],[40,63971,20352],{"className":63972},[102],[40,63974,63976],{"className":63975},[340],[40,63977,63979,64010],{"className":63978},[344,345],[40,63980,63982,64007],{"className":63981},[349],[40,63983,63985,63996],{"className":63984,"style":63165},[353],[40,63986,63987,63990],{"style":63168},[40,63988],{"className":63989,"style":362},[361],[40,63991,63993],{"className":63992},[366,367,368,369],[40,63994,946],{"className":63995,"style":3946},[102,335,369],[40,63997,63998,64001],{"style":63180},[40,63999],{"className":64000,"style":362},[361],[40,64002,64004],{"className":64003},[366,367,368,369],[40,64005,1044],{"className":64006},[102,369],[40,64008,377],{"className":64009},[376],[40,64011,64013],{"className":64012},[349],[40,64014,64016],{"className":64015,"style":63199},[353],[40,64017],{},[40,64019,64020,64023],{"style":1193},[40,64021],{"className":64022,"style":444},[361],[40,64024],{"className":64025,"style":1201},[1200],[40,64027,64028,64031],{"style":2802},[40,64029],{"className":64030,"style":444},[361],[40,64032,64034],{"className":64033},[102],[40,64035,64037,64040],{"className":64036},[102],[40,64038,20352],{"className":64039},[102],[40,64041,64043],{"className":64042},[340],[40,64044,64046],{"className":64045},[344],[40,64047,64049],{"className":64048},[349],[40,64050,64052],{"className":64051,"style":982},[353],[40,64053,64054,64057],{"style":1004},[40,64055],{"className":64056,"style":362},[361],[40,64058,64060],{"className":64059},[366,367,368,369],[40,64061,1044],{"className":64062},[102,369],[40,64064,377],{"className":64065},[376],[40,64067,64069],{"className":64068},[349],[40,64070,64072],{"className":64071,"style":63256},[353],[40,64073],{},[40,64075],{"className":64076},[113,1162],[40,64078],{"className":64079,"style":390},[121],[40,64081,162],{"className":64082},[394],[40,64084],{"className":64085,"style":390},[121],[40,64087,64089,64092],{"className":64088},[93],[40,64090],{"className":64091,"style":1624},[97],[40,64093,60639],{"className":64094},[102],[12,64096,64097],{},"Full population inversion is achievable in the ideal resonant model. For any nonzero detuning, the coefficient falls below one, so the state generally cannot reach unit excited-state probability.",[12,64099,64100],{},"At resonance, the excited-state probability simplifies to",[40,64102,64104],{"className":64103},[136],[40,64105,64107,64161],{"className":64106},[43],[40,64108,64110],{"className":64109},[47],[49,64111,64112],{"xmlns":51,"display":145},[53,64113,64114,64158],{},[56,64115,64116,64122,64124,64126,64128,64130,64140,64156],{},[151,64117,64118,64120],{},[154,64119,57996],{},[154,64121,4203],{},[59,64123,167],{"stretchy":61},[154,64125,58007],{},[59,64127,207],{"stretchy":61},[59,64129,162],{},[941,64131,64132,64138],{},[56,64133,64134,64136],{},[154,64135,1813],{},[59,64137,1084],{},[64,64139,1044],{},[56,64141,64142,64144,64154],{},[59,64143,167],{"fence":72},[1038,64145,64146,64152],{},[56,64147,64148,64150],{},[154,64149,20352],{"mathvariant":1080},[154,64151,58007],{},[64,64153,1044],{},[59,64155,207],{"fence":72},[154,64157,1123],{"mathvariant":1080},[82,64159,64160],{"encoding":84},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[40,64162,64164,64228],{"className":64163,"ariaHidden":72},[89],[40,64165,64167,64170,64210,64213,64216,64219,64222,64225],{"className":64166},[93],[40,64168],{"className":64169,"style":98},[97],[40,64171,64173,64176],{"className":64172},[102],[40,64174,57996],{"className":64175,"style":2541},[102,335],[40,64177,64179],{"className":64178},[340],[40,64180,64182,64202],{"className":64181},[344,345],[40,64183,64185,64199],{"className":64184},[349],[40,64186,64188],{"className":64187,"style":354},[353],[40,64189,64190,64193],{"style":2556},[40,64191],{"className":64192,"style":362},[361],[40,64194,64196],{"className":64195},[366,367,368,369],[40,64197,4203],{"className":64198},[102,335,369],[40,64200,377],{"className":64201},[376],[40,64203,64205],{"className":64204},[349],[40,64206,64208],{"className":64207,"style":384},[353],[40,64209],{},[40,64211,167],{"className":64212},[106],[40,64214,58007],{"className":64215,"style":58079},[102,335],[40,64217,207],{"className":64218},[113],[40,64220],{"className":64221,"style":390},[121],[40,64223,162],{"className":64224},[394],[40,64226],{"className":64227,"style":390},[121],[40,64229,64231,64234,64263,64266,64346,64349],{"className":64230},[93],[40,64232],{"className":64233,"style":404},[97],[40,64235,64237,64240],{"className":64236},[1369],[40,64238,1813],{"className":64239},[1369],[40,64241,64243],{"className":64242},[340],[40,64244,64246],{"className":64245},[344],[40,64247,64249],{"className":64248},[349],[40,64250,64252],{"className":64251,"style":63283},[353],[40,64253,64254,64257],{"style":63286},[40,64255],{"className":64256,"style":362},[361],[40,64258,64260],{"className":64259},[366,367,368,369],[40,64261,1044],{"className":64262},[102,369],[40,64264],{"className":64265,"style":122},[121],[40,64267,64269,64275,64340],{"className":64268},[408],[40,64270,64272],{"className":64271,"style":413},[106,412],[40,64273,167],{"className":64274},[417,368],[40,64276,64278,64281,64337],{"className":64277},[102],[40,64279],{"className":64280},[106,1162],[40,64282,64284],{"className":64283},[1038],[40,64285,64287,64329],{"className":64286},[344,345],[40,64288,64290,64326],{"className":64289},[349],[40,64291,64293,64304,64312],{"className":64292,"style":63326},[353],[40,64294,64295,64298],{"style":2782},[40,64296],{"className":64297,"style":444},[361],[40,64299,64301],{"className":64300},[102],[40,64302,1044],{"className":64303},[102],[40,64305,64306,64309],{"style":1193},[40,64307],{"className":64308,"style":444},[361],[40,64310],{"className":64311,"style":1201},[1200],[40,64313,64314,64317],{"style":2802},[40,64315],{"className":64316,"style":444},[361],[40,64318,64320,64323],{"className":64319},[102],[40,64321,20352],{"className":64322},[102],[40,64324,58007],{"className":64325,"style":58079},[102,335],[40,64327,377],{"className":64328},[376],[40,64330,64332],{"className":64331},[349],[40,64333,64335],{"className":64334,"style":2868},[353],[40,64336],{},[40,64338],{"className":64339},[113,1162],[40,64341,64343],{"className":64342,"style":413},[113,412],[40,64344,207],{"className":64345},[417,368],[40,64347],{"className":64348,"style":122},[121],[40,64350,1123],{"className":64351},[102],[12,64353,64354,64355,64383],{},"The pulse duration required for ideal population inversion defines the standard ",[40,64356,64358,64371],{"className":64357},[43],[40,64359,64361],{"className":64360},[47],[49,64362,64363],{"xmlns":51},[53,64364,64365,64369],{},[56,64366,64367],{},[154,64368,1661],{},[82,64370,4132],{"encoding":84},[40,64372,64374],{"className":64373,"ariaHidden":72},[89],[40,64375,64377,64380],{"className":64376},[93],[40,64378],{"className":64379,"style":4142},[97],[40,64381,1661],{"className":64382,"style":336},[102,335],"-pulse time,",[40,64385,64387],{"className":64386},[136],[40,64388,64390,64440],{"className":64389},[43],[40,64391,64393],{"className":64392},[47],[49,64394,64395],{"xmlns":51,"display":145},[53,64396,64397,64437],{},[56,64398,64399,64405,64407,64413,64415,64435],{},[151,64400,64401,64403],{},[154,64402,58007],{},[154,64404,1661],{},[59,64406,162],{},[1038,64408,64409,64411],{},[154,64410,1661],{},[154,64412,20352],{"mathvariant":1080},[59,64414,162],{},[1038,64416,64417,64419],{},[64,64418,78],{},[56,64420,64421,64423,64425,64427,64429,64431,64433],{},[64,64422,1044],{},[59,64424,167],{"stretchy":61},[154,64426,20352],{"mathvariant":1080},[154,64428,11243],{"mathvariant":1080},[64,64430,1044],{},[154,64432,1661],{},[59,64434,207],{"stretchy":61},[154,64436,1123],{"mathvariant":1080},[82,64438,64439],{"encoding":84},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[40,64441,64443,64499,64578],{"className":64442,"ariaHidden":72},[89],[40,64444,64446,64449,64490,64493,64496],{"className":64445},[93],[40,64447],{"className":64448,"style":328},[97],[40,64450,64452,64455],{"className":64451},[102],[40,64453,58007],{"className":64454,"style":58079},[102,335],[40,64456,64458],{"className":64457},[340],[40,64459,64461,64482],{"className":64460},[344,345],[40,64462,64464,64479],{"className":64463},[349],[40,64465,64467],{"className":64466,"style":354},[353],[40,64468,64470,64473],{"style":64469},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[40,64471],{"className":64472,"style":362},[361],[40,64474,64476],{"className":64475},[366,367,368,369],[40,64477,1661],{"className":64478,"style":336},[102,335,369],[40,64480,377],{"className":64481},[376],[40,64483,64485],{"className":64484},[349],[40,64486,64488],{"className":64487,"style":384},[353],[40,64489],{},[40,64491],{"className":64492,"style":390},[121],[40,64494,162],{"className":64495},[394],[40,64497],{"className":64498,"style":390},[121],[40,64500,64502,64506,64569,64572,64575],{"className":64501},[93],[40,64503],{"className":64504,"style":64505},[97],"height:1.7936em;vertical-align:-0.686em;",[40,64507,64509,64512,64566],{"className":64508},[102],[40,64510],{"className":64511},[106,1162],[40,64513,64515],{"className":64514},[1038],[40,64516,64518,64558],{"className":64517},[344,345],[40,64519,64521,64555],{"className":64520},[349],[40,64522,64525,64536,64544],{"className":64523,"style":64524},[353],"height:1.1076em;",[40,64526,64527,64530],{"style":2782},[40,64528],{"className":64529,"style":444},[361],[40,64531,64533],{"className":64532},[102],[40,64534,20352],{"className":64535},[102],[40,64537,64538,64541],{"style":1193},[40,64539],{"className":64540,"style":444},[361],[40,64542],{"className":64543,"style":1201},[1200],[40,64545,64546,64549],{"style":2802},[40,64547],{"className":64548,"style":444},[361],[40,64550,64552],{"className":64551},[102],[40,64553,1661],{"className":64554,"style":336},[102,335],[40,64556,377],{"className":64557},[376],[40,64559,64561],{"className":64560},[349],[40,64562,64564],{"className":64563,"style":2868},[353],[40,64565],{},[40,64567],{"className":64568},[113,1162],[40,64570],{"className":64571,"style":390},[121],[40,64573,162],{"className":64574},[394],[40,64576],{"className":64577,"style":390},[121],[40,64579,64581,64585,64661],{"className":64580},[93],[40,64582],{"className":64583,"style":64584},[97],"height:2.2574em;vertical-align:-0.936em;",[40,64586,64588,64591,64658],{"className":64587},[102],[40,64589],{"className":64590},[106,1162],[40,64592,64594],{"className":64593},[1038],[40,64595,64597,64649],{"className":64596},[344,345],[40,64598,64600,64646],{"className":64599},[349],[40,64601,64604,64627,64635],{"className":64602,"style":64603},[353],"height:1.3214em;",[40,64605,64606,64609],{"style":2782},[40,64607],{"className":64608,"style":444},[361],[40,64610,64612,64615,64618,64621,64624],{"className":64611},[102],[40,64613,1044],{"className":64614},[102],[40,64616,167],{"className":64617},[106],[40,64619,59016],{"className":64620},[102],[40,64622,1661],{"className":64623,"style":336},[102,335],[40,64625,207],{"className":64626},[113],[40,64628,64629,64632],{"style":1193},[40,64630],{"className":64631,"style":444},[361],[40,64633],{"className":64634,"style":1201},[1200],[40,64636,64637,64640],{"style":2802},[40,64638],{"className":64639,"style":444},[361],[40,64641,64643],{"className":64642},[102],[40,64644,78],{"className":64645},[102],[40,64647,377],{"className":64648},[376],[40,64650,64652],{"className":64651},[349],[40,64653,64656],{"className":64654,"style":64655},[353],"height:0.936em;",[40,64657],{},[40,64659],{"className":64660},[113,1162],[40,64662,1123],{"className":64663},[102],[12,64665,64666],{},"Given the default drive strength",[40,64668,64670],{"className":64669},[136],[40,64671,64673,64712],{"className":64672},[43],[40,64674,64676],{"className":64675},[47],[49,64677,64678],{"xmlns":51,"display":145},[53,64679,64680,64709],{},[56,64681,64682,64692,64694,64696,64698,64707],{},[1038,64683,64684,64686],{},[154,64685,20352],{"mathvariant":1080},[56,64687,64688,64690],{},[64,64689,1044],{},[154,64691,1661],{},[59,64693,162],{},[64,64695,13742],{},[1102,64697,1104],{},[56,64699,64700,64703,64705],{},[154,64701,64702],{"mathvariant":1080},"M",[154,64704,26630],{"mathvariant":1080},[154,64706,273],{"mathvariant":1080},[59,64708,73],{"separator":72},[82,64710,64711],{"encoding":84},"\\frac{\\Omega}{2\\pi}=20~\\mathrm{MHz},",[40,64713,64715,64796],{"className":64714,"ariaHidden":72},[89],[40,64716,64718,64722,64787,64790,64793],{"className":64717},[93],[40,64719],{"className":64720,"style":64721},[97],"height:2.0463em;vertical-align:-0.686em;",[40,64723,64725,64728,64784],{"className":64724},[102],[40,64726],{"className":64727},[106,1162],[40,64729,64731],{"className":64730},[1038],[40,64732,64734,64776],{"className":64733},[344,345],[40,64735,64737,64773],{"className":64736},[349],[40,64738,64740,64754,64762],{"className":64739,"style":63326},[353],[40,64741,64742,64745],{"style":2782},[40,64743],{"className":64744,"style":444},[361],[40,64746,64748,64751],{"className":64747},[102],[40,64749,1044],{"className":64750},[102],[40,64752,1661],{"className":64753,"style":336},[102,335],[40,64755,64756,64759],{"style":1193},[40,64757],{"className":64758,"style":444},[361],[40,64760],{"className":64761,"style":1201},[1200],[40,64763,64764,64767],{"style":2802},[40,64765],{"className":64766,"style":444},[361],[40,64768,64770],{"className":64769},[102],[40,64771,20352],{"className":64772},[102],[40,64774,377],{"className":64775},[376],[40,64777,64779],{"className":64778},[349],[40,64780,64782],{"className":64781,"style":2868},[353],[40,64783],{},[40,64785],{"className":64786},[113,1162],[40,64788],{"className":64789,"style":390},[121],[40,64791,162],{"className":64792},[394],[40,64794],{"className":64795,"style":390},[121],[40,64797,64799,64802,64805,64808,64814],{"className":64798},[93],[40,64800],{"className":64801,"style":7167},[97],[40,64803,13742],{"className":64804},[102],[40,64806,1104],{"className":64807},[121,60187],[40,64809,64811],{"className":64810},[102],[40,64812,23612],{"className":64813},[102,1373],[40,64815,73],{"className":64816},[117],[12,64818,64819,64820,64848],{},"the ideal ",[40,64821,64823,64836],{"className":64822},[43],[40,64824,64826],{"className":64825},[47],[49,64827,64828],{"xmlns":51},[53,64829,64830,64834],{},[56,64831,64832],{},[154,64833,1661],{},[82,64835,4132],{"encoding":84},[40,64837,64839],{"className":64838,"ariaHidden":72},[89],[40,64840,64842,64845],{"className":64841},[93],[40,64843],{"className":64844,"style":4142},[97],[40,64846,1661],{"className":64847,"style":336},[102,335],"-pulse time is",[40,64850,64852],{"className":64851},[136],[40,64853,64855,64889],{"className":64854},[43],[40,64856,64858],{"className":64857},[47],[49,64859,64860],{"xmlns":51,"display":145},[53,64861,64862,64886],{},[56,64863,64864,64870,64872,64875,64877,64884],{},[151,64865,64866,64868],{},[154,64867,58007],{},[154,64869,1661],{},[59,64871,162],{},[64,64873,64874],{},"25",[1102,64876,1104],{},[56,64878,64879,64881],{},[154,64880,1796],{"mathvariant":1080},[154,64882,64883],{"mathvariant":1080},"s",[154,64885,1123],{"mathvariant":1080},[82,64887,64888],{"encoding":84},"\\tau_\\pi=25~\\mathrm{ns}.",[40,64890,64892,64947],{"className":64891,"ariaHidden":72},[89],[40,64893,64895,64898,64938,64941,64944],{"className":64894},[93],[40,64896],{"className":64897,"style":328},[97],[40,64899,64901,64904],{"className":64900},[102],[40,64902,58007],{"className":64903,"style":58079},[102,335],[40,64905,64907],{"className":64906},[340],[40,64908,64910,64930],{"className":64909},[344,345],[40,64911,64913,64927],{"className":64912},[349],[40,64914,64916],{"className":64915,"style":354},[353],[40,64917,64918,64921],{"style":64469},[40,64919],{"className":64920,"style":362},[361],[40,64922,64924],{"className":64923},[366,367,368,369],[40,64925,1661],{"className":64926,"style":336},[102,335,369],[40,64928,377],{"className":64929},[376],[40,64931,64933],{"className":64932},[349],[40,64934,64936],{"className":64935,"style":384},[353],[40,64937],{},[40,64939],{"className":64940,"style":390},[121],[40,64942,162],{"className":64943},[394],[40,64945],{"className":64946,"style":390},[121],[40,64948,64950,64953,64956,64959,64966],{"className":64949},[93],[40,64951],{"className":64952,"style":1624},[97],[40,64954,64874],{"className":64955},[102],[40,64957,1104],{"className":64958},[121,60187],[40,64960,64962],{"className":64961},[102],[40,64963,64965],{"className":64964},[102,1373],"ns",[40,64967,1123],{"className":64968},[102],[12,64970,64971],{},"A full period of the resonant excited-state probability oscillation is",[40,64973,64975],{"className":64974},[136],[40,64976,64978,65026],{"className":64977},[43],[40,64979,64981],{"className":64980},[47],[49,64982,64983],{"xmlns":51,"display":145},[53,64984,64985,65023],{},[56,64986,64987,64993,64995,65005,65007,65021],{},[151,64988,64989,64991],{},[154,64990,7059],{},[154,64992,946],{},[59,64994,162],{},[1038,64996,64997,65003],{},[56,64998,64999,65001],{},[64,65000,1044],{},[154,65002,1661],{},[154,65004,20352],{"mathvariant":1080},[59,65006,162],{},[1038,65008,65009,65011],{},[64,65010,78],{},[56,65012,65013,65015,65017,65019],{},[154,65014,20352],{"mathvariant":1080},[154,65016,11243],{"mathvariant":1080},[64,65018,1044],{},[154,65020,1661],{},[154,65022,1123],{"mathvariant":1080},[82,65024,65025],{"encoding":84},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[40,65027,65029,65084,65165],{"className":65028,"ariaHidden":72},[89],[40,65030,65032,65035,65075,65078,65081],{"className":65031},[93],[40,65033],{"className":65034,"style":3106},[97],[40,65036,65038,65041],{"className":65037},[102],[40,65039,7059],{"className":65040,"style":2541},[102,335],[40,65042,65044],{"className":65043},[340],[40,65045,65047,65067],{"className":65046},[344,345],[40,65048,65050,65064],{"className":65049},[349],[40,65051,65053],{"className":65052,"style":20224},[353],[40,65054,65055,65058],{"style":2556},[40,65056],{"className":65057,"style":362},[361],[40,65059,65061],{"className":65060},[366,367,368,369],[40,65062,946],{"className":65063,"style":3946},[102,335,369],[40,65065,377],{"className":65066},[376],[40,65068,65070],{"className":65069},[349],[40,65071,65073],{"className":65072,"style":384},[353],[40,65074],{},[40,65076],{"className":65077,"style":390},[121],[40,65079,162],{"className":65080},[394],[40,65082],{"className":65083,"style":390},[121],[40,65085,65087,65091,65156,65159,65162],{"className":65086},[93],[40,65088],{"className":65089,"style":65090},[97],"height:2.0074em;vertical-align:-0.686em;",[40,65092,65094,65097,65153],{"className":65093},[102],[40,65095],{"className":65096},[106,1162],[40,65098,65100],{"className":65099},[1038],[40,65101,65103,65145],{"className":65102},[344,345],[40,65104,65106,65142],{"className":65105},[349],[40,65107,65109,65120,65128],{"className":65108,"style":64603},[353],[40,65110,65111,65114],{"style":2782},[40,65112],{"className":65113,"style":444},[361],[40,65115,65117],{"className":65116},[102],[40,65118,20352],{"className":65119},[102],[40,65121,65122,65125],{"style":1193},[40,65123],{"className":65124,"style":444},[361],[40,65126],{"className":65127,"style":1201},[1200],[40,65129,65130,65133],{"style":2802},[40,65131],{"className":65132,"style":444},[361],[40,65134,65136,65139],{"className":65135},[102],[40,65137,1044],{"className":65138},[102],[40,65140,1661],{"className":65141,"style":336},[102,335],[40,65143,377],{"className":65144},[376],[40,65146,65148],{"className":65147},[349],[40,65149,65151],{"className":65150,"style":2868},[353],[40,65152],{},[40,65154],{"className":65155},[113,1162],[40,65157],{"className":65158,"style":390},[121],[40,65160,162],{"className":65161},[394],[40,65163],{"className":65164,"style":390},[121],[40,65166,65168,65171,65236],{"className":65167},[93],[40,65169],{"className":65170,"style":64584},[97],[40,65172,65174,65177,65233],{"className":65173},[102],[40,65175],{"className":65176},[106,1162],[40,65178,65180],{"className":65179},[1038],[40,65181,65183,65225],{"className":65182},[344,345],[40,65184,65186,65222],{"className":65185},[349],[40,65187,65189,65203,65211],{"className":65188,"style":64603},[353],[40,65190,65191,65194],{"style":2782},[40,65192],{"className":65193,"style":444},[361],[40,65195,65197,65200],{"className":65196},[102],[40,65198,59016],{"className":65199},[102],[40,65201,1661],{"className":65202,"style":336},[102,335],[40,65204,65205,65208],{"style":1193},[40,65206],{"className":65207,"style":444},[361],[40,65209],{"className":65210,"style":1201},[1200],[40,65212,65213,65216],{"style":2802},[40,65214],{"className":65215,"style":444},[361],[40,65217,65219],{"className":65218},[102],[40,65220,78],{"className":65221},[102],[40,65223,377],{"className":65224},[376],[40,65226,65228],{"className":65227},[349],[40,65229,65231],{"className":65230,"style":64655},[353],[40,65232],{},[40,65234],{"className":65235},[113,1162],[40,65237,1123],{"className":65238},[102],[12,65240,65241],{},"For the same default drive strength, the full temporal period is",[40,65243,65245],{"className":65244},[136],[40,65246,65248,65281],{"className":65247},[43],[40,65249,65251],{"className":65250},[47],[49,65252,65253],{"xmlns":51,"display":145},[53,65254,65255,65278],{},[56,65256,65257,65263,65265,65268,65270,65276],{},[151,65258,65259,65261],{},[154,65260,7059],{},[154,65262,946],{},[59,65264,162],{},[64,65266,65267],{},"50",[1102,65269,1104],{},[56,65271,65272,65274],{},[154,65273,1796],{"mathvariant":1080},[154,65275,64883],{"mathvariant":1080},[154,65277,1123],{"mathvariant":1080},[82,65279,65280],{"encoding":84},"T_R=50~\\mathrm{ns}.",[40,65282,65284,65339],{"className":65283,"ariaHidden":72},[89],[40,65285,65287,65290,65330,65333,65336],{"className":65286},[93],[40,65288],{"className":65289,"style":3106},[97],[40,65291,65293,65296],{"className":65292},[102],[40,65294,7059],{"className":65295,"style":2541},[102,335],[40,65297,65299],{"className":65298},[340],[40,65300,65302,65322],{"className":65301},[344,345],[40,65303,65305,65319],{"className":65304},[349],[40,65306,65308],{"className":65307,"style":20224},[353],[40,65309,65310,65313],{"style":2556},[40,65311],{"className":65312,"style":362},[361],[40,65314,65316],{"className":65315},[366,367,368,369],[40,65317,946],{"className":65318,"style":3946},[102,335,369],[40,65320,377],{"className":65321},[376],[40,65323,65325],{"className":65324},[349],[40,65326,65328],{"className":65327,"style":384},[353],[40,65329],{},[40,65331],{"className":65332,"style":390},[121],[40,65334,162],{"className":65335},[394],[40,65337],{"className":65338,"style":390},[121],[40,65340,65342,65345,65348,65351,65357],{"className":65341},[93],[40,65343],{"className":65344,"style":1624},[97],[40,65346,65267],{"className":65347},[102],[40,65349,1104],{"className":65350},[121,60187],[40,65352,65354],{"className":65353},[102],[40,65355,64965],{"className":65356},[102,1373],[40,65358,1123],{"className":65359},[102],[28,65361,65363],{"id":65362},"phenomenological-dephasing","Phenomenological Dephasing",[12,65365,65366],{},"The computational model includes an optional phenomenological damping factor,",[40,65368,65370],{"className":65369},[136],[40,65371,65373,65445],{"className":65372},[43],[40,65374,65376],{"className":65375},[47],[49,65377,65378],{"xmlns":51,"display":145},[53,65379,65380,65442],{},[56,65381,65382,65388,65390,65392,65394,65396,65398,65400,65406,65408,65410,65412,65414,65416,65418,65420,65440],{},[151,65383,65384,65386],{},[154,65385,57996],{},[154,65387,4203],{},[59,65389,167],{"stretchy":61},[154,65391,13262],{},[59,65393,73],{"separator":72},[154,65395,58007],{},[59,65397,207],{"stretchy":61},[59,65399,5620],{},[151,65401,65402,65404],{},[154,65403,57996],{},[154,65405,4203],{},[59,65407,167],{"stretchy":61},[154,65409,13262],{},[59,65411,73],{"separator":72},[154,65413,58007],{},[59,65415,207],{"stretchy":61},[154,65417,5268],{},[59,65419,1084],{},[56,65421,65422,65424,65426,65438],{},[59,65423,167],{"fence":72},[59,65425,243],{},[1038,65427,65428,65430],{},[154,65429,58007],{},[8360,65431,65432,65434,65436],{},[154,65433,7059],{},[64,65435,1044],{},[59,65437,59534],{},[59,65439,207],{"fence":72},[154,65441,1123],{"mathvariant":1080},[82,65443,65444],{"encoding":84},"P_e(f,\\tau)\\rightarrow P_e(f,\\tau)\\exp\\left(-\\frac{\\tau}{T_2^\\ast}\\right).",[40,65446,65448,65521],{"className":65447,"ariaHidden":72},[89],[40,65449,65451,65454,65494,65497,65500,65503,65506,65509,65512,65515,65518],{"className":65450},[93],[40,65452],{"className":65453,"style":98},[97],[40,65455,65457,65460],{"className":65456},[102],[40,65458,57996],{"className":65459,"style":2541},[102,335],[40,65461,65463],{"className":65462},[340],[40,65464,65466,65486],{"className":65465},[344,345],[40,65467,65469,65483],{"className":65468},[349],[40,65470,65472],{"className":65471,"style":354},[353],[40,65473,65474,65477],{"style":2556},[40,65475],{"className":65476,"style":362},[361],[40,65478,65480],{"className":65479},[366,367,368,369],[40,65481,4203],{"className":65482},[102,335,369],[40,65484,377],{"className":65485},[376],[40,65487,65489],{"className":65488},[349],[40,65490,65492],{"className":65491,"style":384},[353],[40,65493],{},[40,65495,167],{"className":65496},[106],[40,65498,13262],{"className":65499,"style":33300},[102,335],[40,65501,73],{"className":65502},[117],[40,65504],{"className":65505,"style":122},[121],[40,65507,58007],{"className":65508,"style":58079},[102,335],[40,65510,207],{"className":65511},[113],[40,65513],{"className":65514,"style":390},[121],[40,65516,5620],{"className":65517},[394],[40,65519],{"className":65520,"style":390},[121],[40,65522,65524,65528,65568,65571,65574,65577,65580,65583,65586,65589,65592,65595,65727,65730],{"className":65523},[93],[40,65525],{"className":65526,"style":65527},[97],"height:2.4023em;vertical-align:-0.9523em;",[40,65529,65531,65534],{"className":65530},[102],[40,65532,57996],{"className":65533,"style":2541},[102,335],[40,65535,65537],{"className":65536},[340],[40,65538,65540,65560],{"className":65539},[344,345],[40,65541,65543,65557],{"className":65542},[349],[40,65544,65546],{"className":65545,"style":354},[353],[40,65547,65548,65551],{"style":2556},[40,65549],{"className":65550,"style":362},[361],[40,65552,65554],{"className":65553},[366,367,368,369],[40,65555,4203],{"className":65556},[102,335,369],[40,65558,377],{"className":65559},[376],[40,65561,65563],{"className":65562},[349],[40,65564,65566],{"className":65565,"style":384},[353],[40,65567],{},[40,65569,167],{"className":65570},[106],[40,65572,13262],{"className":65573,"style":33300},[102,335],[40,65575,73],{"className":65576},[117],[40,65578],{"className":65579,"style":122},[121],[40,65581,58007],{"className":65582,"style":58079},[102,335],[40,65584,207],{"className":65585},[113],[40,65587],{"className":65588,"style":122},[121],[40,65590,5268],{"className":65591},[1369],[40,65593],{"className":65594,"style":122},[121],[40,65596,65598,65604,65607,65721],{"className":65597},[408],[40,65599,65601],{"className":65600,"style":413},[106,412],[40,65602,167],{"className":65603},[417,368],[40,65605,243],{"className":65606},[102],[40,65608,65610,65613,65718],{"className":65609},[102],[40,65611],{"className":65612},[106,1162],[40,65614,65616],{"className":65615},[1038],[40,65617,65619,65709],{"className":65618},[344,345],[40,65620,65622,65706],{"className":65621},[349],[40,65623,65625,65687,65695],{"className":65624,"style":64524},[353],[40,65626,65627,65630],{"style":2782},[40,65628],{"className":65629,"style":444},[361],[40,65631,65633],{"className":65632},[102],[40,65634,65636,65639],{"className":65635},[102],[40,65637,7059],{"className":65638,"style":2541},[102,335],[40,65640,65642],{"className":65641},[340],[40,65643,65645,65678],{"className":65644},[344,345],[40,65646,65648,65675],{"className":65647},[349],[40,65649,65652,65664],{"className":65650,"style":65651},[353],"height:0.6705em;",[40,65653,65655,65658],{"style":65654},"top:-2.4337em;margin-left:-0.1389em;margin-right:0.05em;",[40,65656],{"className":65657,"style":362},[361],[40,65659,65661],{"className":65660},[366,367,368,369],[40,65662,1044],{"className":65663},[102,369],[40,65665,65666,65669],{"style":63180},[40,65667],{"className":65668,"style":362},[361],[40,65670,65672],{"className":65671},[366,367,368,369],[40,65673,59534],{"className":65674},[1250,369],[40,65676,377],{"className":65677},[376],[40,65679,65681],{"className":65680},[349],[40,65682,65685],{"className":65683,"style":65684},[353],"height:0.2663em;",[40,65686],{},[40,65688,65689,65692],{"style":1193},[40,65690],{"className":65691,"style":444},[361],[40,65693],{"className":65694,"style":1201},[1200],[40,65696,65697,65700],{"style":2802},[40,65698],{"className":65699,"style":444},[361],[40,65701,65703],{"className":65702},[102],[40,65704,58007],{"className":65705,"style":58079},[102,335],[40,65707,377],{"className":65708},[376],[40,65710,65712],{"className":65711},[349],[40,65713,65716],{"className":65714,"style":65715},[353],"height:0.9523em;",[40,65717],{},[40,65719],{"className":65720},[113,1162],[40,65722,65724],{"className":65723,"style":413},[113,412],[40,65725,207],{"className":65726},[417,368],[40,65728],{"className":65729,"style":122},[121],[40,65731,1123],{"className":65732},[102],[12,65734,65735],{},"This exponential envelope attenuates the plotted Rabi fringes as pulse duration increases. The factor is a compact visualization device for contrast decay. A rigorous open-quantum-system treatment would require a density matrix model with relaxation channels, dephasing channels, finite temperature effects, shaped pulse envelopes, and stochastic dynamics. The default dephasing parameter is",[40,65737,65739],{"className":65738},[136],[40,65740,65742,65777],{"className":65741},[43],[40,65743,65745],{"className":65744},[47],[49,65746,65747],{"xmlns":51,"display":145},[53,65748,65749,65774],{},[56,65750,65751,65759,65761,65764,65766,65772],{},[8360,65752,65753,65755,65757],{},[154,65754,7059],{},[64,65756,1044],{},[59,65758,59534],{},[59,65760,162],{},[64,65762,65763],{},"500",[1102,65765,1104],{},[56,65767,65768,65770],{},[154,65769,1796],{"mathvariant":1080},[154,65771,64883],{"mathvariant":1080},[154,65773,1123],{"mathvariant":1080},[82,65775,65776],{"encoding":84},"T_2^\\ast=500~\\mathrm{ns}.",[40,65778,65780,65849],{"className":65779,"ariaHidden":72},[89],[40,65781,65783,65787,65840,65843,65846],{"className":65782},[93],[40,65784],{"className":65785,"style":65786},[97],"height:0.9857em;vertical-align:-0.247em;",[40,65788,65790,65793],{"className":65789},[102],[40,65791,7059],{"className":65792,"style":2541},[102,335],[40,65794,65796],{"className":65795},[340],[40,65797,65799,65832],{"className":65798},[344,345],[40,65800,65802,65829],{"className":65801},[349],[40,65803,65806,65818],{"className":65804,"style":65805},[353],"height:0.7387em;",[40,65807,65809,65812],{"style":65808},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[40,65810],{"className":65811,"style":362},[361],[40,65813,65815],{"className":65814},[366,367,368,369],[40,65816,1044],{"className":65817},[102,369],[40,65819,65820,65823],{"style":3962},[40,65821],{"className":65822,"style":362},[361],[40,65824,65826],{"className":65825},[366,367,368,369],[40,65827,59534],{"className":65828},[1250,369],[40,65830,377],{"className":65831},[376],[40,65833,65835],{"className":65834},[349],[40,65836,65838],{"className":65837,"style":8699},[353],[40,65839],{},[40,65841],{"className":65842,"style":390},[121],[40,65844,162],{"className":65845},[394],[40,65847],{"className":65848,"style":390},[121],[40,65850,65852,65855,65858,65861,65867],{"className":65851},[93],[40,65853],{"className":65854,"style":1624},[97],[40,65856,65763],{"className":65857},[102],[40,65859,1104],{"className":65860},[121,60187],[40,65862,65864],{"className":65863},[102],[40,65865,64965],{"className":65866},[102,1373],[40,65868,1123],{"className":65869},[102],[12,65871,65872,65873,65876],{},"Setting ",[10088,65874,65875],{},"T2STAR_NS = None"," disables the phenomenological damping factor.",[28,65878,65880],{"id":65879},"detuning-and-generalized-rabi-frequency","Detuning and Generalized Rabi Frequency",[12,65882,65883],{},"Detuning quantifies the spectral mismatch between the applied drive and the natural qubit transition,",[40,65885,65887],{"className":65886},[136],[40,65888,65890,65926],{"className":65889},[43],[40,65891,65893],{"className":65892},[47],[49,65894,65895],{"xmlns":51,"display":145},[53,65896,65897,65923],{},[56,65898,65899,65901,65903,65905,65907,65909,65911,65913,65919,65921],{},[154,65900,2016],{"mathvariant":1080},[59,65902,162],{},[64,65904,1044],{},[154,65906,1661],{},[59,65908,167],{"stretchy":61},[154,65910,13262],{},[59,65912,243],{},[151,65914,65915,65917],{},[154,65916,13262],{},[64,65918,66],{},[59,65920,207],{"stretchy":61},[154,65922,1123],{"mathvariant":1080},[82,65924,65925],{"encoding":84},"\\Delta=2\\pi(f-f_0).",[40,65927,65929,65947,65974],{"className":65928,"ariaHidden":72},[89],[40,65930,65932,65935,65938,65941,65944],{"className":65931},[93],[40,65933],{"className":65934,"style":4304},[97],[40,65936,2016],{"className":65937},[102],[40,65939],{"className":65940,"style":390},[121],[40,65942,162],{"className":65943},[394],[40,65945],{"className":65946,"style":390},[121],[40,65948,65950,65953,65956,65959,65962,65965,65968,65971],{"className":65949},[93],[40,65951],{"className":65952,"style":98},[97],[40,65954,1044],{"className":65955},[102],[40,65957,1661],{"className":65958,"style":336},[102,335],[40,65960,167],{"className":65961},[106],[40,65963,13262],{"className":65964,"style":33300},[102,335],[40,65966],{"className":65967,"style":1246},[121],[40,65969,243],{"className":65970},[1250],[40,65972],{"className":65973,"style":1246},[121],[40,65975,65977,65980,66020,66023],{"className":65976},[93],[40,65978],{"className":65979,"style":98},[97],[40,65981,65983,65986],{"className":65982},[102],[40,65984,13262],{"className":65985,"style":33300},[102,335],[40,65987,65989],{"className":65988},[340],[40,65990,65992,66012],{"className":65991},[344,345],[40,65993,65995,66009],{"className":65994},[349],[40,65996,65998],{"className":65997,"style":7201},[353],[40,65999,66000,66003],{"style":33315},[40,66001],{"className":66002,"style":362},[361],[40,66004,66006],{"className":66005},[366,367,368,369],[40,66007,66],{"className":66008},[102,369],[40,66010,377],{"className":66011},[376],[40,66013,66015],{"className":66014},[349],[40,66016,66018],{"className":66017,"style":384},[353],[40,66019],{},[40,66021,207],{"className":66022},[113],[40,66024,1123],{"className":66025},[102],[12,66027,66028],{},"Expressed in ordinary frequency units, the generalized Rabi frequency is",[40,66030,66032],{"className":66031},[136],[40,66033,66035,66123],{"className":66034},[43],[40,66036,66038],{"className":66037},[47],[49,66039,66040],{"xmlns":51,"display":145},[53,66041,66042,66120],{},[56,66043,66044,66050,66052,66054,66056,66058,66072,66074,66118],{},[151,66045,66046,66048],{},[154,66047,59270],{},[154,66049,946],{},[59,66051,167],{"stretchy":61},[154,66053,13262],{},[59,66055,207],{"stretchy":61},[59,66057,162],{},[1038,66059,66060,66066],{},[151,66061,66062,66064],{},[154,66063,20352],{"mathvariant":1080},[154,66065,946],{},[56,66067,66068,66070],{},[64,66069,1044],{},[154,66071,1661],{},[59,66073,162],{},[2045,66075,66076],{},[56,66077,66078,66098,66100,66102,66104,66106,66112],{},[941,66079,66080,66096],{},[56,66081,66082,66084,66094],{},[59,66083,167],{"fence":72},[1038,66085,66086,66088],{},[154,66087,20352],{"mathvariant":1080},[56,66089,66090,66092],{},[64,66091,1044],{},[154,66093,1661],{},[59,66095,207],{"fence":72},[64,66097,1044],{},[59,66099,1053],{},[59,66101,167],{"stretchy":61},[154,66103,13262],{},[59,66105,243],{},[151,66107,66108,66110],{},[154,66109,13262],{},[64,66111,66],{},[941,66113,66114,66116],{},[59,66115,207],{"stretchy":61},[64,66117,1044],{},[154,66119,1123],{"mathvariant":1080},[82,66121,66122],{"encoding":84},"\\nu_R(f)=\\frac{\\Omega_R}{2\\pi}=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[40,66124,66126,66190,66307],{"className":66125,"ariaHidden":72},[89],[40,66127,66129,66132,66172,66175,66178,66181,66184,66187],{"className":66128},[93],[40,66130],{"className":66131,"style":98},[97],[40,66133,66135,66138],{"className":66134},[102],[40,66136,59270],{"className":66137,"style":59291},[102,335],[40,66139,66141],{"className":66140},[340],[40,66142,66144,66164],{"className":66143},[344,345],[40,66145,66147,66161],{"className":66146},[349],[40,66148,66150],{"className":66149,"style":20224},[353],[40,66151,66152,66155],{"style":59306},[40,66153],{"className":66154,"style":362},[361],[40,66156,66158],{"className":66157},[366,367,368,369],[40,66159,946],{"className":66160,"style":3946},[102,335,369],[40,66162,377],{"className":66163},[376],[40,66165,66167],{"className":66166},[349],[40,66168,66170],{"className":66169,"style":384},[353],[40,66171],{},[40,66173,167],{"className":66174},[106],[40,66176,13262],{"className":66177,"style":33300},[102,335],[40,66179,207],{"className":66180},[113],[40,66182],{"className":66183,"style":390},[121],[40,66185,162],{"className":66186},[394],[40,66188],{"className":66189,"style":390},[121],[40,66191,66193,66196,66298,66301,66304],{"className":66192},[93],[40,66194],{"className":66195,"style":64721},[97],[40,66197,66199,66202,66295],{"className":66198},[102],[40,66200],{"className":66201},[106,1162],[40,66203,66205],{"className":66204},[1038],[40,66206,66208,66287],{"className":66207},[344,345],[40,66209,66211,66284],{"className":66210},[349],[40,66212,66214,66228,66236],{"className":66213,"style":63326},[353],[40,66215,66216,66219],{"style":2782},[40,66217],{"className":66218,"style":444},[361],[40,66220,66222,66225],{"className":66221},[102],[40,66223,1044],{"className":66224},[102],[40,66226,1661],{"className":66227,"style":336},[102,335],[40,66229,66230,66233],{"style":1193},[40,66231],{"className":66232,"style":444},[361],[40,66234],{"className":66235,"style":1201},[1200],[40,66237,66238,66241],{"style":2802},[40,66239],{"className":66240,"style":444},[361],[40,66242,66244],{"className":66243},[102],[40,66245,66247,66250],{"className":66246},[102],[40,66248,20352],{"className":66249},[102],[40,66251,66253],{"className":66252},[340],[40,66254,66256,66276],{"className":66255},[344,345],[40,66257,66259,66273],{"className":66258},[349],[40,66260,66262],{"className":66261,"style":20224},[353],[40,66263,66264,66267],{"style":6098},[40,66265],{"className":66266,"style":362},[361],[40,66268,66270],{"className":66269},[366,367,368,369],[40,66271,946],{"className":66272,"style":3946},[102,335,369],[40,66274,377],{"className":66275},[376],[40,66277,66279],{"className":66278},[349],[40,66280,66282],{"className":66281,"style":384},[353],[40,66283],{},[40,66285,377],{"className":66286},[376],[40,66288,66290],{"className":66289},[349],[40,66291,66293],{"className":66292,"style":2868},[353],[40,66294],{},[40,66296],{"className":66297},[113,1162],[40,66299],{"className":66300,"style":390},[121],[40,66302,162],{"className":66303},[394],[40,66305],{"className":66306,"style":390},[121],[40,66308,66310,66314,66568],{"className":66309},[93],[40,66311],{"className":66312,"style":66313},[97],"height:3.04em;vertical-align:-1.0541em;",[40,66315,66317],{"className":66316},[102,2218],[40,66318,66320,66559],{"className":66319},[344,345],[40,66321,66323,66556],{"className":66322},[349],[40,66324,66327,66539],{"className":66325,"style":66326},[353],"height:1.9859em;",[40,66328,66331,66335],{"className":66329,"style":66330},[2232],"top:-5em;",[40,66332],{"className":66333,"style":66334},[361],"height:5em;",[40,66336,66338,66446,66449,66452,66455,66458,66461,66464,66467,66470,66510],{"className":66337,"style":63558},[102],[40,66339,66341,66421],{"className":66340},[408],[40,66342,66344,66350,66415],{"className":66343},[408],[40,66345,66347],{"className":66346,"style":413},[106,412],[40,66348,167],{"className":66349},[417,368],[40,66351,66353,66356,66412],{"className":66352},[102],[40,66354],{"className":66355},[106,1162],[40,66357,66359],{"className":66358},[1038],[40,66360,66362,66404],{"className":66361},[344,345],[40,66363,66365,66401],{"className":66364},[349],[40,66366,66368,66382,66390],{"className":66367,"style":63326},[353],[40,66369,66370,66373],{"style":2782},[40,66371],{"className":66372,"style":444},[361],[40,66374,66376,66379],{"className":66375},[102],[40,66377,1044],{"className":66378},[102],[40,66380,1661],{"className":66381,"style":336},[102,335],[40,66383,66384,66387],{"style":1193},[40,66385],{"className":66386,"style":444},[361],[40,66388],{"className":66389,"style":1201},[1200],[40,66391,66392,66395],{"style":2802},[40,66393],{"className":66394,"style":444},[361],[40,66396,66398],{"className":66397},[102],[40,66399,20352],{"className":66400},[102],[40,66402,377],{"className":66403},[376],[40,66405,66407],{"className":66406},[349],[40,66408,66410],{"className":66409,"style":2868},[353],[40,66411],{},[40,66413],{"className":66414},[113,1162],[40,66416,66418],{"className":66417,"style":413},[113,412],[40,66419,207],{"className":66420},[417,368],[40,66422,66424],{"className":66423},[340],[40,66425,66427],{"className":66426},[344],[40,66428,66430],{"className":66429},[349],[40,66431,66434],{"className":66432,"style":66433},[353],"height:1.654em;",[40,66435,66437,66440],{"style":66436},"top:-3.9029em;margin-right:0.05em;",[40,66438],{"className":66439,"style":362},[361],[40,66441,66443],{"className":66442},[366,367,368,369],[40,66444,1044],{"className":66445},[102,369],[40,66447],{"className":66448,"style":1246},[121],[40,66450,1053],{"className":66451},[1250],[40,66453],{"className":66454,"style":1246},[121],[40,66456,167],{"className":66457},[106],[40,66459,13262],{"className":66460,"style":33300},[102,335],[40,66462],{"className":66463,"style":1246},[121],[40,66465,243],{"className":66466},[1250],[40,66468],{"className":66469,"style":1246},[121],[40,66471,66473,66476],{"className":66472},[102],[40,66474,13262],{"className":66475,"style":33300},[102,335],[40,66477,66479],{"className":66478},[340],[40,66480,66482,66502],{"className":66481},[344,345],[40,66483,66485,66499],{"className":66484},[349],[40,66486,66488],{"className":66487,"style":7201},[353],[40,66489,66490,66493],{"style":33315},[40,66491],{"className":66492,"style":362},[361],[40,66494,66496],{"className":66495},[366,367,368,369],[40,66497,66],{"className":66498},[102,369],[40,66500,377],{"className":66501},[376],[40,66503,66505],{"className":66504},[349],[40,66506,66508],{"className":66507,"style":384},[353],[40,66509],{},[40,66511,66513,66516],{"className":66512},[113],[40,66514,207],{"className":66515},[113],[40,66517,66519],{"className":66518},[340],[40,66520,66522],{"className":66521},[344],[40,66523,66525],{"className":66524},[349],[40,66526,66528],{"className":66527,"style":63577},[353],[40,66529,66530,66533],{"style":63580},[40,66531],{"className":66532,"style":362},[361],[40,66534,66536],{"className":66535},[366,367,368,369],[40,66537,1044],{"className":66538},[102,369],[40,66540,66542,66545],{"style":66541},"top:-3.9459em;",[40,66543],{"className":66544,"style":66334},[361],[40,66546,66549],{"className":66547,"style":66548},[2253],"min-width:1.02em;height:3.08em;",[2256,66550,66553],{"xmlns":2258,"width":2259,"height":66551,"viewBox":66552,"preserveAspectRatio":2262},"3.08em","0 0 400000 3240",[2264,66554],{"d":66555},"M473,2793\nc339.3,-1799.3,509.3,-2700,510,-2702 l0 -0\nc3.3,-7.3,9.3,-11,18,-11 H400000v40H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM1001 80h400000v40H1017.7z",[40,66557,377],{"className":66558},[376],[40,66560,66562],{"className":66561},[349],[40,66563,66566],{"className":66564,"style":66565},[353],"height:1.0541em;",[40,66567],{},[40,66569,1123],{"className":66570},[102],[12,66572,66573],{},"This expression explains the ridge visible in the Fourier-domain heatmap. At resonance, the generalized frequency equals the applied drive strength,",[40,66575,66577],{"className":66576},[136],[40,66578,66580,66622],{"className":66579},[43],[40,66581,66583],{"className":66582},[47],[49,66584,66585],{"xmlns":51,"display":145},[53,66586,66587,66619],{},[56,66588,66589,66595,66597,66603,66605,66607,66617],{},[151,66590,66591,66593],{},[154,66592,59270],{},[154,66594,946],{},[59,66596,167],{"stretchy":61},[151,66598,66599,66601],{},[154,66600,13262],{},[64,66602,66],{},[59,66604,207],{"stretchy":61},[59,66606,162],{},[1038,66608,66609,66611],{},[154,66610,20352],{"mathvariant":1080},[56,66612,66613,66615],{},[64,66614,1044],{},[154,66616,1661],{},[154,66618,1123],{"mathvariant":1080},[82,66620,66621],{"encoding":84},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[40,66623,66625,66726],{"className":66624,"ariaHidden":72},[89],[40,66626,66628,66631,66671,66674,66714,66717,66720,66723],{"className":66627},[93],[40,66629],{"className":66630,"style":98},[97],[40,66632,66634,66637],{"className":66633},[102],[40,66635,59270],{"className":66636,"style":59291},[102,335],[40,66638,66640],{"className":66639},[340],[40,66641,66643,66663],{"className":66642},[344,345],[40,66644,66646,66660],{"className":66645},[349],[40,66647,66649],{"className":66648,"style":20224},[353],[40,66650,66651,66654],{"style":59306},[40,66652],{"className":66653,"style":362},[361],[40,66655,66657],{"className":66656},[366,367,368,369],[40,66658,946],{"className":66659,"style":3946},[102,335,369],[40,66661,377],{"className":66662},[376],[40,66664,66666],{"className":66665},[349],[40,66667,66669],{"className":66668,"style":384},[353],[40,66670],{},[40,66672,167],{"className":66673},[106],[40,66675,66677,66680],{"className":66676},[102],[40,66678,13262],{"className":66679,"style":33300},[102,335],[40,66681,66683],{"className":66682},[340],[40,66684,66686,66706],{"className":66685},[344,345],[40,66687,66689,66703],{"className":66688},[349],[40,66690,66692],{"className":66691,"style":7201},[353],[40,66693,66694,66697],{"style":33315},[40,66695],{"className":66696,"style":362},[361],[40,66698,66700],{"className":66699},[366,367,368,369],[40,66701,66],{"className":66702},[102,369],[40,66704,377],{"className":66705},[376],[40,66707,66709],{"className":66708},[349],[40,66710,66712],{"className":66711,"style":384},[353],[40,66713],{},[40,66715,207],{"className":66716},[113],[40,66718],{"className":66719,"style":390},[121],[40,66721,162],{"className":66722},[394],[40,66724],{"className":66725,"style":390},[121],[40,66727,66729,66732,66797],{"className":66728},[93],[40,66730],{"className":66731,"style":64721},[97],[40,66733,66735,66738,66794],{"className":66734},[102],[40,66736],{"className":66737},[106,1162],[40,66739,66741],{"className":66740},[1038],[40,66742,66744,66786],{"className":66743},[344,345],[40,66745,66747,66783],{"className":66746},[349],[40,66748,66750,66764,66772],{"className":66749,"style":63326},[353],[40,66751,66752,66755],{"style":2782},[40,66753],{"className":66754,"style":444},[361],[40,66756,66758,66761],{"className":66757},[102],[40,66759,1044],{"className":66760},[102],[40,66762,1661],{"className":66763,"style":336},[102,335],[40,66765,66766,66769],{"style":1193},[40,66767],{"className":66768,"style":444},[361],[40,66770],{"className":66771,"style":1201},[1200],[40,66773,66774,66777],{"style":2802},[40,66775],{"className":66776,"style":444},[361],[40,66778,66780],{"className":66779},[102],[40,66781,20352],{"className":66782},[102],[40,66784,377],{"className":66785},[376],[40,66787,66789],{"className":66788},[349],[40,66790,66792],{"className":66791,"style":2868},[353],[40,66793],{},[40,66795],{"className":66796},[113,1162],[40,66798,1123],{"className":66799},[102],[12,66801,66802,66803,66872,66873,66975],{},"As the system moves away from resonance, ",[40,66804,66806,66823],{"className":66805},[43],[40,66807,66809],{"className":66808},[47],[49,66810,66811],{"xmlns":51},[53,66812,66813,66821],{},[56,66814,66815],{},[151,66816,66817,66819],{},[154,66818,59270],{},[154,66820,946],{},[82,66822,59275],{"encoding":84},[40,66824,66826],{"className":66825,"ariaHidden":72},[89],[40,66827,66829,66832],{"className":66828},[93],[40,66830],{"className":66831,"style":328},[97],[40,66833,66835,66838],{"className":66834},[102],[40,66836,59270],{"className":66837,"style":59291},[102,335],[40,66839,66841],{"className":66840},[340],[40,66842,66844,66864],{"className":66843},[344,345],[40,66845,66847,66861],{"className":66846},[349],[40,66848,66850],{"className":66849,"style":20224},[353],[40,66851,66852,66855],{"style":59306},[40,66853],{"className":66854,"style":362},[361],[40,66856,66858],{"className":66857},[366,367,368,369],[40,66859,946],{"className":66860,"style":3946},[102,335,369],[40,66862,377],{"className":66863},[376],[40,66865,66867],{"className":66866},[349],[40,66868,66870],{"className":66869,"style":384},[353],[40,66871],{}," increases symmetrically with ",[40,66874,66876,66902],{"className":66875},[43],[40,66877,66879],{"className":66878},[47],[49,66880,66881],{"xmlns":51},[53,66882,66883,66899],{},[56,66884,66885,66887,66889,66891,66897],{},[59,66886,62],{"stretchy":61},[154,66888,13262],{},[59,66890,243],{},[151,66892,66893,66895],{},[154,66894,13262],{},[64,66896,66],{},[59,66898,62],{"stretchy":61},[82,66900,66901],{"encoding":84},"\\lvert f-f_0\\rvert",[40,66903,66905,66926],{"className":66904,"ariaHidden":72},[89],[40,66906,66908,66911,66914,66917,66920,66923],{"className":66907},[93],[40,66909],{"className":66910,"style":98},[97],[40,66912,62],{"className":66913},[106],[40,66915,13262],{"className":66916,"style":33300},[102,335],[40,66918],{"className":66919,"style":1246},[121],[40,66921,243],{"className":66922},[1250],[40,66924],{"className":66925,"style":1246},[121],[40,66927,66929,66932,66972],{"className":66928},[93],[40,66930],{"className":66931,"style":98},[97],[40,66933,66935,66938],{"className":66934},[102],[40,66936,13262],{"className":66937,"style":33300},[102,335],[40,66939,66941],{"className":66940},[340],[40,66942,66944,66964],{"className":66943},[344,345],[40,66945,66947,66961],{"className":66946},[349],[40,66948,66950],{"className":66949,"style":7201},[353],[40,66951,66952,66955],{"style":33315},[40,66953],{"className":66954,"style":362},[361],[40,66956,66958],{"className":66957},[366,367,368,369],[40,66959,66],{"className":66960},[102,369],[40,66962,377],{"className":66963},[376],[40,66965,66967],{"className":66966},[349],[40,66968,66970],{"className":66969,"style":384},[353],[40,66971],{},[40,66973,62],{"className":66974},[113],". The oscillations accelerate along the pulse-duration axis, and the maximum achievable excited-state probability decreases. This tradeoff is the defining visual signature of detuned Rabi dynamics.",[28,66977,66979],{"id":66978},"quantum-information-interpretation","Quantum Information Interpretation",[12,66981,66982,66983,67053],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to ",[40,66984,66986,67004],{"className":66985},[43],[40,66987,66989],{"className":66988},[47],[49,66990,66991],{"xmlns":51},[53,66992,66993,67001],{},[56,66994,66995],{},[151,66996,66997,66999],{},[154,66998,156],{},[154,67000,159],{},[82,67002,67003],{"encoding":84},"\\sigma_x",[40,67005,67007],{"className":67006,"ariaHidden":72},[89],[40,67008,67010,67013],{"className":67009},[93],[40,67011],{"className":67012,"style":328},[97],[40,67014,67016,67019],{"className":67015},[102],[40,67017,156],{"className":67018,"style":336},[102,335],[40,67020,67022],{"className":67021},[340],[40,67023,67025,67045],{"className":67024},[344,345],[40,67026,67028,67042],{"className":67027},[349],[40,67029,67031],{"className":67030,"style":354},[353],[40,67032,67033,67036],{"style":357},[40,67034],{"className":67035,"style":362},[361],[40,67037,67039],{"className":67038},[366,367,368,369],[40,67040,159],{"className":67041},[102,335,369],[40,67043,377],{"className":67044},[376],[40,67046,67048],{"className":67047},[349],[40,67049,67051],{"className":67050,"style":384},[353],[40,67052],{},", so the microwave pulse implements",[40,67055,67057],{"className":67056},[136],[40,67058,67060,67116],{"className":67059},[43],[40,67061,67063],{"className":67062},[47],[49,67064,67065],{"xmlns":51,"display":145},[53,67066,67067,67113],{},[56,67068,67069,67075,67077,67079,67081,67083,67085,67087,67111],{},[151,67070,67071,67073],{},[154,67072,946],{},[154,67074,159],{},[59,67076,167],{"stretchy":61},[154,67078,1649],{},[59,67080,207],{"stretchy":61},[59,67082,162],{},[154,67084,5268],{},[59,67086,1084],{},[56,67088,67089,67091,67093,67109],{},[59,67090,167],{"fence":72},[59,67092,243],{},[1038,67094,67095,67107],{},[56,67096,67097,67099,67101],{},[154,67098,246],{},[154,67100,1649],{},[151,67102,67103,67105],{},[154,67104,156],{},[154,67106,159],{},[64,67108,1044],{},[59,67110,207],{"fence":72},[59,67112,73],{"separator":72},[82,67114,67115],{"encoding":84},"R_x(\\theta)=\\exp\\left(-\\frac{i\\theta\\sigma_x}{2}\\right),",[40,67117,67119,67183],{"className":67118,"ariaHidden":72},[89],[40,67120,67122,67125,67165,67168,67171,67174,67177,67180],{"className":67121},[93],[40,67123],{"className":67124,"style":98},[97],[40,67126,67128,67131],{"className":67127},[102],[40,67129,946],{"className":67130,"style":3946},[102,335],[40,67132,67134],{"className":67133},[340],[40,67135,67137,67157],{"className":67136},[344,345],[40,67138,67140,67154],{"className":67139},[349],[40,67141,67143],{"className":67142,"style":354},[353],[40,67144,67145,67148],{"style":4920},[40,67146],{"className":67147,"style":362},[361],[40,67149,67151],{"className":67150},[366,367,368,369],[40,67152,159],{"className":67153},[102,335,369],[40,67155,377],{"className":67156},[376],[40,67158,67160],{"className":67159},[349],[40,67161,67163],{"className":67162,"style":384},[353],[40,67164],{},[40,67166,167],{"className":67167},[106],[40,67169,1649],{"className":67170,"style":1312},[102,335],[40,67172,207],{"className":67173},[113],[40,67175],{"className":67176,"style":390},[121],[40,67178,162],{"className":67179},[394],[40,67181],{"className":67182,"style":390},[121],[40,67184,67186,67189,67192,67195,67319,67322],{"className":67185},[93],[40,67187],{"className":67188,"style":404},[97],[40,67190,5268],{"className":67191},[1369],[40,67193],{"className":67194,"style":122},[121],[40,67196,67198,67204,67207,67313],{"className":67197},[408],[40,67199,67201],{"className":67200,"style":413},[106,412],[40,67202,167],{"className":67203},[417,368],[40,67205,243],{"className":67206},[102],[40,67208,67210,67213,67310],{"className":67209},[102],[40,67211],{"className":67212},[106,1162],[40,67214,67216],{"className":67215},[1038],[40,67217,67219,67302],{"className":67218},[344,345],[40,67220,67222,67299],{"className":67221},[349],[40,67223,67226,67237,67245],{"className":67224,"style":67225},[353],"height:1.3714em;",[40,67227,67228,67231],{"style":2782},[40,67229],{"className":67230,"style":444},[361],[40,67232,67234],{"className":67233},[102],[40,67235,1044],{"className":67236},[102],[40,67238,67239,67242],{"style":1193},[40,67240],{"className":67241,"style":444},[361],[40,67243],{"className":67244,"style":1201},[1200],[40,67246,67247,67250],{"style":2802},[40,67248],{"className":67249,"style":444},[361],[40,67251,67253,67256,67259],{"className":67252},[102],[40,67254,246],{"className":67255},[102,335],[40,67257,1649],{"className":67258,"style":1312},[102,335],[40,67260,67262,67265],{"className":67261},[102],[40,67263,156],{"className":67264,"style":336},[102,335],[40,67266,67268],{"className":67267},[340],[40,67269,67271,67291],{"className":67270},[344,345],[40,67272,67274,67288],{"className":67273},[349],[40,67275,67277],{"className":67276,"style":354},[353],[40,67278,67279,67282],{"style":357},[40,67280],{"className":67281,"style":362},[361],[40,67283,67285],{"className":67284},[366,367,368,369],[40,67286,159],{"className":67287},[102,335,369],[40,67289,377],{"className":67290},[376],[40,67292,67294],{"className":67293},[349],[40,67295,67297],{"className":67296,"style":384},[353],[40,67298],{},[40,67300,377],{"className":67301},[376],[40,67303,67305],{"className":67304},[349],[40,67306,67308],{"className":67307,"style":2868},[353],[40,67309],{},[40,67311],{"className":67312},[113,1162],[40,67314,67316],{"className":67315,"style":413},[113,412],[40,67317,207],{"className":67318},[417,368],[40,67320],{"className":67321,"style":122},[121],[40,67323,73],{"className":67324},[117],[12,67326,23436],{},[40,67328,67330],{"className":67329},[136],[40,67331,67333,67355],{"className":67332},[43],[40,67334,67336],{"className":67335},[47],[49,67337,67338],{"xmlns":51,"display":145},[53,67339,67340,67352],{},[56,67341,67342,67344,67346,67348,67350],{},[154,67343,1649],{},[59,67345,162],{},[154,67347,20352],{"mathvariant":1080},[154,67349,58007],{},[154,67351,1123],{"mathvariant":1080},[82,67353,67354],{"encoding":84},"\\theta=\\Omega\\tau.",[40,67356,67358,67377],{"className":67357,"ariaHidden":72},[89],[40,67359,67361,67365,67368,67371,67374],{"className":67360},[93],[40,67362],{"className":67363,"style":67364},[97],"height:0.6944em;",[40,67366,1649],{"className":67367,"style":1312},[102,335],[40,67369],{"className":67370,"style":390},[121],[40,67372,162],{"className":67373},[394],[40,67375],{"className":67376,"style":390},[121],[40,67378,67380,67383,67386,67389],{"className":67379},[93],[40,67381],{"className":67382,"style":4304},[97],[40,67384,20352],{"className":67385},[102],[40,67387,58007],{"className":67388,"style":58079},[102,335],[40,67390,1123],{"className":67391},[102],[12,67393,67394,67395,67423],{},"A calibrated resonant ",[40,67396,67398,67411],{"className":67397},[43],[40,67399,67401],{"className":67400},[47],[49,67402,67403],{"xmlns":51},[53,67404,67405,67409],{},[56,67406,67407],{},[154,67408,1661],{},[82,67410,4132],{"encoding":84},[40,67412,67414],{"className":67413,"ariaHidden":72},[89],[40,67415,67417,67420],{"className":67416},[93],[40,67418],{"className":67419,"style":4142},[97],[40,67421,1661],{"className":67422,"style":336},[102,335],"-pulse enacts",[40,67425,67427],{"className":67426},[136],[40,67428,67430,67468],{"className":67429},[43],[40,67431,67433],{"className":67432},[47],[49,67434,67435],{"xmlns":51,"display":145},[53,67436,67437,67465],{},[56,67438,67439,67445,67447,67449,67451,67453,67455,67457,67463],{},[151,67440,67441,67443],{},[154,67442,946],{},[154,67444,159],{},[59,67446,167],{"stretchy":61},[154,67448,1661],{},[59,67450,207],{"stretchy":61},[59,67452,162],{},[59,67454,243],{},[154,67456,246],{},[151,67458,67459,67461],{},[154,67460,156],{},[154,67462,159],{},[59,67464,73],{"separator":72},[82,67466,67467],{"encoding":84},"R_x(\\pi)=-i\\sigma_x,",[40,67469,67471,67535],{"className":67470,"ariaHidden":72},[89],[40,67472,67474,67477,67517,67520,67523,67526,67529,67532],{"className":67473},[93],[40,67475],{"className":67476,"style":98},[97],[40,67478,67480,67483],{"className":67479},[102],[40,67481,946],{"className":67482,"style":3946},[102,335],[40,67484,67486],{"className":67485},[340],[40,67487,67489,67509],{"className":67488},[344,345],[40,67490,67492,67506],{"className":67491},[349],[40,67493,67495],{"className":67494,"style":354},[353],[40,67496,67497,67500],{"style":4920},[40,67498],{"className":67499,"style":362},[361],[40,67501,67503],{"className":67502},[366,367,368,369],[40,67504,159],{"className":67505},[102,335,369],[40,67507,377],{"className":67508},[376],[40,67510,67512],{"className":67511},[349],[40,67513,67515],{"className":67514,"style":384},[353],[40,67516],{},[40,67518,167],{"className":67519},[106],[40,67521,1661],{"className":67522,"style":336},[102,335],[40,67524,207],{"className":67525},[113],[40,67527],{"className":67528,"style":390},[121],[40,67530,162],{"className":67531},[394],[40,67533],{"className":67534,"style":390},[121],[40,67536,67538,67542,67545,67548,67588],{"className":67537},[93],[40,67539],{"className":67540,"style":67541},[97],"height:0.854em;vertical-align:-0.1944em;",[40,67543,243],{"className":67544},[102],[40,67546,246],{"className":67547},[102,335],[40,67549,67551,67554],{"className":67550},[102],[40,67552,156],{"className":67553,"style":336},[102,335],[40,67555,67557],{"className":67556},[340],[40,67558,67560,67580],{"className":67559},[344,345],[40,67561,67563,67577],{"className":67562},[349],[40,67564,67566],{"className":67565,"style":354},[353],[40,67567,67568,67571],{"style":357},[40,67569],{"className":67570,"style":362},[361],[40,67572,67574],{"className":67573},[366,367,368,369],[40,67575,159],{"className":67576},[102,335,369],[40,67578,377],{"className":67579},[376],[40,67581,67583],{"className":67582},[349],[40,67584,67586],{"className":67585,"style":384},[353],[40,67587],{},[40,67589,73],{"className":67590},[117],[12,67592,67593,67594,67622,67623,67659],{},"which corresponds to a quantum ",[40,67595,67597,67610],{"className":67596},[43],[40,67598,67600],{"className":67599},[47],[49,67601,67602],{"xmlns":51},[53,67603,67604,67608],{},[56,67605,67606],{},[154,67607,4196],{},[82,67609,4196],{"encoding":84},[40,67611,67613],{"className":67612,"ariaHidden":72},[89],[40,67614,67616,67619],{"className":67615},[93],[40,67617],{"className":67618,"style":4304},[97],[40,67620,4196],{"className":67621,"style":1242},[102,335]," gate up to a global phase. A resonant ",[40,67624,67626,67644],{"className":67625},[43],[40,67627,67629],{"className":67628},[47],[49,67630,67631],{"xmlns":51},[53,67632,67633,67641],{},[56,67634,67635,67637,67639],{},[154,67636,1661],{},[154,67638,11243],{"mathvariant":1080},[64,67640,1044],{},[82,67642,67643],{"encoding":84},"\\pi\u002F2",[40,67645,67647],{"className":67646,"ariaHidden":72},[89],[40,67648,67650,67653,67656],{"className":67649},[93],[40,67651],{"className":67652,"style":98},[97],[40,67654,1661],{"className":67655,"style":336},[102,335],[40,67657,33653],{"className":67658},[102],"-pulse prepares a balanced coherent superposition,",[40,67661,67663],{"className":67662},[136],[40,67664,67666,67712],{"className":67665},[43],[40,67667,67669],{"className":67668},[47],[49,67670,67671],{"xmlns":51,"display":145},[53,67672,67673,67709],{},[56,67674,67675,67677,67679,67681,67683,67707],{},[59,67676,62],{"stretchy":61},[154,67678,58190],{},[59,67680,69],{"stretchy":61},[59,67682,5620],{},[1038,67684,67685,67703],{},[56,67686,67687,67689,67691,67693,67695,67697,67699,67701],{},[59,67688,62],{"stretchy":61},[154,67690,58190],{},[59,67692,69],{"stretchy":61},[59,67694,243],{},[154,67696,246],{},[59,67698,62],{"stretchy":61},[154,67700,4203],{},[59,67702,69],{"stretchy":61},[2045,67704,67705],{},[64,67706,1044],{},[59,67708,73],{"separator":72},[82,67710,67711],{"encoding":84},"\\lvert g\\rangle\\rightarrow\\frac{\\lvert g\\rangle-i\\lvert e\\rangle}{\\sqrt{2}},",[40,67713,67715,67739],{"className":67714,"ariaHidden":72},[89],[40,67716,67718,67721,67724,67727,67730,67733,67736],{"className":67717},[93],[40,67719],{"className":67720,"style":98},[97],[40,67722,62],{"className":67723},[106],[40,67725,58190],{"className":67726,"style":336},[102,335],[40,67728,69],{"className":67729},[113],[40,67731],{"className":67732,"style":390},[121],[40,67734,5620],{"className":67735},[394],[40,67737],{"className":67738,"style":390},[121],[40,67740,67742,67746,67882],{"className":67741},[93],[40,67743],{"className":67744,"style":67745},[97],"height:2.357em;vertical-align:-0.93em;",[40,67747,67749,67752,67879],{"className":67748},[102],[40,67750],{"className":67751},[106,1162],[40,67753,67755],{"className":67754},[1038],[40,67756,67758,67870],{"className":67757},[344,345],[40,67759,67761,67867],{"className":67760},[349],[40,67762,67765,67821,67829],{"className":67763,"style":67764},[353],"height:1.427em;",[40,67766,67768,67771],{"style":67767},"top:-2.2028em;",[40,67769],{"className":67770,"style":444},[361],[40,67772,67774],{"className":67773},[102],[40,67775,67777],{"className":67776},[102,2218],[40,67778,67780,67813],{"className":67779},[344,345],[40,67781,67783,67810],{"className":67782},[349],[40,67784,67786,67798],{"className":67785,"style":2821},[353],[40,67787,67789,67792],{"className":67788,"style":2233},[2232],[40,67790],{"className":67791,"style":444},[361],[40,67793,67795],{"className":67794,"style":2240},[102],[40,67796,1044],{"className":67797},[102],[40,67799,67800,67803],{"style":2836},[40,67801],{"className":67802,"style":444},[361],[40,67804,67806],{"className":67805,"style":2254},[2253],[2256,67807,67808],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,67809],{"d":2266},[40,67811,377],{"className":67812},[376],[40,67814,67816],{"className":67815},[349],[40,67817,67819],{"className":67818,"style":2856},[353],[40,67820],{},[40,67822,67823,67826],{"style":1193},[40,67824],{"className":67825,"style":444},[361],[40,67827],{"className":67828,"style":1201},[1200],[40,67830,67831,67834],{"style":2802},[40,67832],{"className":67833,"style":444},[361],[40,67835,67837,67840,67843,67846,67849,67852,67855,67858,67861,67864],{"className":67836},[102],[40,67838,62],{"className":67839},[106],[40,67841,58190],{"className":67842,"style":336},[102,335],[40,67844,69],{"className":67845},[113],[40,67847],{"className":67848,"style":1246},[121],[40,67850,243],{"className":67851},[1250],[40,67853],{"className":67854,"style":1246},[121],[40,67856,246],{"className":67857},[102,335],[40,67859,62],{"className":67860},[106],[40,67862,4203],{"className":67863},[102,335],[40,67865,69],{"className":67866},[113],[40,67868,377],{"className":67869},[376],[40,67871,67873],{"className":67872},[349],[40,67874,67877],{"className":67875,"style":67876},[353],"height:0.93em;",[40,67878],{},[40,67880],{"className":67881},[113,1162],[40,67883,73],{"className":67884},[117],[12,67886,67887],{},"with a phase convention set by the microwave drive phase.",[12,67889,67890],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[40,67892,67894],{"className":67893},[136],[40,67895,67897,67966],{"className":67896},[43],[40,67898,67900],{"className":67899},[47],[49,67901,67902],{"xmlns":51,"display":145},[53,67903,67904,67963],{},[56,67905,67906,67916,67918,67920,67922,67924,67926,67928,67961],{},[151,67907,67908,67910],{},[154,67909,946],{},[5144,67911,67912,67914],{"accent":72},[154,67913,1796],{},[59,67915,5151],{},[59,67917,167],{"stretchy":61},[154,67919,1649],{},[59,67921,207],{"stretchy":61},[59,67923,162],{},[154,67925,5268],{},[59,67927,1084],{},[56,67929,67930,67932,67934,67944,67950,67952,67959],{},[59,67931,167],{"fence":72},[59,67933,243],{},[1038,67935,67936,67942],{},[56,67937,67938,67940],{},[154,67939,246],{},[154,67941,1649],{},[64,67943,1044],{},[5144,67945,67946,67948],{"accent":72},[154,67947,1796],{},[59,67949,5151],{},[59,67951,1058],{},[5144,67953,67954,67956],{"accent":72},[154,67955,156],{},[59,67957,67958],{},"⃗",[59,67960,207],{"fence":72},[59,67962,73],{"separator":72},[82,67964,67965],{"encoding":84},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[40,67967,67969,68066],{"className":67968,"ariaHidden":72},[89],[40,67970,67972,67975,68048,68051,68054,68057,68060,68063],{"className":67971},[93],[40,67973],{"className":67974,"style":98},[97],[40,67976,67978,67981],{"className":67977},[102],[40,67979,946],{"className":67980,"style":3946},[102,335],[40,67982,67984],{"className":67983},[340],[40,67985,67987,68040],{"className":67986},[344,345],[40,67988,67990,68037],{"className":67989},[349],[40,67991,67994],{"className":67992,"style":67993},[353],"height:0.3361em;",[40,67995,67996,67999],{"style":4920},[40,67997],{"className":67998,"style":362},[361],[40,68000,68002],{"className":68001},[366,367,368,369],[40,68003,68005],{"className":68004},[102,369],[40,68006,68008],{"className":68007},[102,5168,369],[40,68009,68011],{"className":68010},[344],[40,68012,68014],{"className":68013},[349],[40,68015,68017,68026],{"className":68016,"style":67364},[353],[40,68018,68020,68023],{"style":68019},"top:-2.7em;",[40,68021],{"className":68022,"style":362},[361],[40,68024,1796],{"className":68025},[102,335,369],[40,68027,68028,68031],{"style":68019},[40,68029],{"className":68030,"style":362},[361],[40,68032,68034],{"className":68033,"style":5196},[5195],[40,68035,5151],{"className":68036},[102,369],[40,68038,377],{"className":68039},[376],[40,68041,68043],{"className":68042},[349],[40,68044,68046],{"className":68045,"style":384},[353],[40,68047],{},[40,68049,167],{"className":68050},[106],[40,68052,1649],{"className":68053,"style":1312},[102,335],[40,68055,207],{"className":68056},[113],[40,68058],{"className":68059,"style":390},[121],[40,68061,162],{"className":68062},[394],[40,68064],{"className":68065,"style":390},[121],[40,68067,68069,68072,68075,68078,68246,68249],{"className":68068},[93],[40,68070],{"className":68071,"style":404},[97],[40,68073,5268],{"className":68074},[1369],[40,68076],{"className":68077,"style":122},[121],[40,68079,68081,68087,68090,68155,68186,68189,68192,68195,68240],{"className":68080},[408],[40,68082,68084],{"className":68083,"style":413},[106,412],[40,68085,167],{"className":68086},[417,368],[40,68088,243],{"className":68089},[102],[40,68091,68093,68096,68152],{"className":68092},[102],[40,68094],{"className":68095},[106,1162],[40,68097,68099],{"className":68098},[1038],[40,68100,68102,68144],{"className":68101},[344,345],[40,68103,68105,68141],{"className":68104},[349],[40,68106,68108,68119,68127],{"className":68107,"style":67225},[353],[40,68109,68110,68113],{"style":2782},[40,68111],{"className":68112,"style":444},[361],[40,68114,68116],{"className":68115},[102],[40,68117,1044],{"className":68118},[102],[40,68120,68121,68124],{"style":1193},[40,68122],{"className":68123,"style":444},[361],[40,68125],{"className":68126,"style":1201},[1200],[40,68128,68129,68132],{"style":2802},[40,68130],{"className":68131,"style":444},[361],[40,68133,68135,68138],{"className":68134},[102],[40,68136,246],{"className":68137},[102,335],[40,68139,1649],{"className":68140,"style":1312},[102,335],[40,68142,377],{"className":68143},[376],[40,68145,68147],{"className":68146},[349],[40,68148,68150],{"className":68149,"style":2868},[353],[40,68151],{},[40,68153],{"className":68154},[113,1162],[40,68156,68158],{"className":68157},[102,5168],[40,68159,68161],{"className":68160},[344],[40,68162,68164],{"className":68163},[349],[40,68165,68167,68175],{"className":68166,"style":67364},[353],[40,68168,68169,68172],{"style":2233},[40,68170],{"className":68171,"style":444},[361],[40,68173,1796],{"className":68174},[102,335],[40,68176,68177,68180],{"style":2233},[40,68178],{"className":68179,"style":444},[361],[40,68181,68183],{"className":68182,"style":5196},[5195],[40,68184,5151],{"className":68185},[102],[40,68187],{"className":68188,"style":1246},[121],[40,68190,1058],{"className":68191},[1250],[40,68193],{"className":68194,"style":1246},[121],[40,68196,68198],{"className":68197},[102,5168],[40,68199,68201],{"className":68200},[344],[40,68202,68204],{"className":68203},[349],[40,68205,68208,68216],{"className":68206,"style":68207},[353],"height:0.714em;",[40,68209,68210,68213],{"style":2233},[40,68211],{"className":68212,"style":444},[361],[40,68214,156],{"className":68215,"style":336},[102,335],[40,68217,68218,68221],{"style":2233},[40,68219],{"className":68220,"style":444},[361],[40,68222,68225],{"className":68223,"style":68224},[5195],"left:-0.2355em;",[40,68226,68230],{"className":68227,"style":68229},[68228],"overlay","height:0.714em;width:0.471em;",[2256,68231,68237],{"xmlns":2258,"width":68232,"height":68233,"style":68234,"viewBox":68235,"preserveAspectRatio":68236},"0.471em","0.714em","width:0.471em","0 0 471 714","xMinYMin",[2264,68238],{"d":68239},"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z",[40,68241,68243],{"className":68242,"style":413},[113,412],[40,68244,207],{"className":68245},[417,368],[40,68247],{"className":68248,"style":122},[121],[40,68250,73],{"className":68251},[117],[12,68253,23436],{},[40,68255,68257],{"className":68256},[136],[40,68258,68260,68322],{"className":68259},[43],[40,68261,68263],{"className":68262},[47],[49,68264,68265],{"xmlns":51,"display":145},[53,68266,68267,68319],{},[56,68268,68269,68275,68277,68287,68289,68291,68293,68295,68297,68299,68301,68303,68305,68307,68309,68315,68317],{},[5144,68270,68271,68273],{"accent":72},[154,68272,1796],{},[59,68274,5151],{},[59,68276,162],{},[1038,68278,68279,68281],{},[64,68280,78],{},[151,68282,68283,68285],{},[154,68284,20352],{"mathvariant":1080},[154,68286,946],{},[59,68288,167],{"stretchy":61},[154,68290,20352],{"mathvariant":1080},[59,68292,73],{"separator":72},[64,68294,66],{},[59,68296,73],{"separator":72},[154,68298,2016],{"mathvariant":1080},[59,68300,207],{"stretchy":61},[59,68302,73],{"separator":72},[121,68304],{"width":1068},[154,68306,1649],{},[59,68308,162],{},[151,68310,68311,68313],{},[154,68312,20352],{"mathvariant":1080},[154,68314,946],{},[154,68316,58007],{},[154,68318,1123],{"mathvariant":1080},[82,68320,68321],{"encoding":84},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[40,68323,68325,68371,68526],{"className":68324,"ariaHidden":72},[89],[40,68326,68328,68331,68362,68365,68368],{"className":68327},[93],[40,68329],{"className":68330,"style":67364},[97],[40,68332,68334],{"className":68333},[102,5168],[40,68335,68337],{"className":68336},[344],[40,68338,68340],{"className":68339},[349],[40,68341,68343,68351],{"className":68342,"style":67364},[353],[40,68344,68345,68348],{"style":2233},[40,68346],{"className":68347,"style":444},[361],[40,68349,1796],{"className":68350},[102,335],[40,68352,68353,68356],{"style":2233},[40,68354],{"className":68355,"style":444},[361],[40,68357,68359],{"className":68358,"style":5196},[5195],[40,68360,5151],{"className":68361},[102],[40,68363],{"className":68364,"style":390},[121],[40,68366,162],{"className":68367},[394],[40,68369],{"className":68370,"style":390},[121],[40,68372,68374,68378,68478,68481,68484,68487,68490,68493,68496,68499,68502,68505,68508,68511,68514,68517,68520,68523],{"className":68373},[93],[40,68375],{"className":68376,"style":68377},[97],"height:2.1574em;vertical-align:-0.836em;",[40,68379,68381,68384,68475],{"className":68380},[102],[40,68382],{"className":68383},[106,1162],[40,68385,68387],{"className":68386},[1038],[40,68388,68390,68466],{"className":68389},[344,345],[40,68391,68393,68463],{"className":68392},[349],[40,68394,68396,68444,68452],{"className":68395,"style":64603},[353],[40,68397,68398,68401],{"style":2782},[40,68399],{"className":68400,"style":444},[361],[40,68402,68404],{"className":68403},[102],[40,68405,68407,68410],{"className":68406},[102],[40,68408,20352],{"className":68409},[102],[40,68411,68413],{"className":68412},[340],[40,68414,68416,68436],{"className":68415},[344,345],[40,68417,68419,68433],{"className":68418},[349],[40,68420,68422],{"className":68421,"style":20224},[353],[40,68423,68424,68427],{"style":6098},[40,68425],{"className":68426,"style":362},[361],[40,68428,68430],{"className":68429},[366,367,368,369],[40,68431,946],{"className":68432,"style":3946},[102,335,369],[40,68434,377],{"className":68435},[376],[40,68437,68439],{"className":68438},[349],[40,68440,68442],{"className":68441,"style":384},[353],[40,68443],{},[40,68445,68446,68449],{"style":1193},[40,68447],{"className":68448,"style":444},[361],[40,68450],{"className":68451,"style":1201},[1200],[40,68453,68454,68457],{"style":2802},[40,68455],{"className":68456,"style":444},[361],[40,68458,68460],{"className":68459},[102],[40,68461,78],{"className":68462},[102],[40,68464,377],{"className":68465},[376],[40,68467,68469],{"className":68468},[349],[40,68470,68473],{"className":68471,"style":68472},[353],"height:0.836em;",[40,68474],{},[40,68476],{"className":68477},[113,1162],[40,68479,167],{"className":68480},[106],[40,68482,20352],{"className":68483},[102],[40,68485,73],{"className":68486},[117],[40,68488],{"className":68489,"style":122},[121],[40,68491,66],{"className":68492},[102],[40,68494,73],{"className":68495},[117],[40,68497],{"className":68498,"style":122},[121],[40,68500,2016],{"className":68501},[102],[40,68503,207],{"className":68504},[113],[40,68506,73],{"className":68507},[117],[40,68509],{"className":68510,"style":1302},[121],[40,68512],{"className":68513,"style":122},[121],[40,68515,1649],{"className":68516,"style":1312},[102,335],[40,68518],{"className":68519,"style":390},[121],[40,68521,162],{"className":68522},[394],[40,68524],{"className":68525,"style":390},[121],[40,68527,68529,68532,68572,68575],{"className":68528},[93],[40,68530],{"className":68531,"style":3106},[97],[40,68533,68535,68538],{"className":68534},[102],[40,68536,20352],{"className":68537},[102],[40,68539,68541],{"className":68540},[340],[40,68542,68544,68564],{"className":68543},[344,345],[40,68545,68547,68561],{"className":68546},[349],[40,68548,68550],{"className":68549,"style":20224},[353],[40,68551,68552,68555],{"style":6098},[40,68553],{"className":68554,"style":362},[361],[40,68556,68558],{"className":68557},[366,367,368,369],[40,68559,946],{"className":68560,"style":3946},[102,335,369],[40,68562,377],{"className":68563},[376],[40,68565,68567],{"className":68566},[349],[40,68568,68570],{"className":68569,"style":384},[353],[40,68571],{},[40,68573,58007],{"className":68574,"style":58079},[102,335],[40,68576,1123],{"className":68577},[102],[12,68579,68580],{},"This representation shows that detuning increases the rotation rate and tilts the control axis. Under this idealized model, the qubit still evolves coherently, although the trajectory generally misses the pure excited-state pole.",[28,68582,68584],{"id":68583},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[12,68586,68587],{},"A Bloch sphere maps any pure qubit state to a unit vector in three-dimensional space. Ground and excited states occupy opposite poles. Equal-amplitude coherent superpositions lie on the equator.",[12,68589,68590,68591,68629,68630,68658,68659,68694],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[40,68592,68594,68611],{"className":68593},[43],[40,68595,68597],{"className":68596},[47],[49,68598,68599],{"xmlns":51},[53,68600,68601,68609],{},[56,68602,68603,68605,68607],{},[59,68604,62],{"stretchy":61},[154,68606,58190],{},[59,68608,69],{"stretchy":61},[82,68610,58195],{"encoding":84},[40,68612,68614],{"className":68613,"ariaHidden":72},[89],[40,68615,68617,68620,68623,68626],{"className":68616},[93],[40,68618],{"className":68619,"style":98},[97],[40,68621,62],{"className":68622},[106],[40,68624,58190],{"className":68625,"style":336},[102,335],[40,68627,69],{"className":68628},[113],", a properly timed ",[40,68631,68633,68646],{"className":68632},[43],[40,68634,68636],{"className":68635},[47],[49,68637,68638],{"xmlns":51},[53,68639,68640,68644],{},[56,68641,68642],{},[154,68643,1661],{},[82,68645,4132],{"encoding":84},[40,68647,68649],{"className":68648,"ariaHidden":72},[89],[40,68650,68652,68655],{"className":68651},[93],[40,68653],{"className":68654,"style":4142},[97],[40,68656,1661],{"className":68657,"style":336},[102,335],"-pulse moves the state to the excited-state pole. A ",[40,68660,68662,68679],{"className":68661},[43],[40,68663,68665],{"className":68664},[47],[49,68666,68667],{"xmlns":51},[53,68668,68669,68677],{},[56,68670,68671,68673,68675],{},[154,68672,1661],{},[154,68674,11243],{"mathvariant":1080},[64,68676,1044],{},[82,68678,67643],{"encoding":84},[40,68680,68682],{"className":68681,"ariaHidden":72},[89],[40,68683,68685,68688,68691],{"className":68684},[93],[40,68686],{"className":68687,"style":98},[97],[40,68689,1661],{"className":68690,"style":336},[102,335],[40,68692,33653],{"className":68693},[102],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[12,68696,68697,68698,68726],{},"Detuning tilts the rotation axis toward the ",[40,68699,68701,68714],{"className":68700},[43],[40,68702,68704],{"className":68703},[47],[49,68705,68706],{"xmlns":51},[53,68707,68708,68712],{},[56,68709,68710],{},[154,68711,273],{},[82,68713,273],{"encoding":84},[40,68715,68717],{"className":68716,"ariaHidden":72},[89],[40,68718,68720,68723],{"className":68719},[93],[40,68721],{"className":68722,"style":4142},[97],[40,68724,273],{"className":68725,"style":764},[102,335],"-axis. The state vector then precesses around the tilted axis and generally fails to reach the excited-state pole. This geometric picture explains the amplitude coefficient",[40,68728,68730],{"className":68729},[136],[40,68731,68733,68763],{"className":68732},[43],[40,68734,68736],{"className":68735},[47],[49,68737,68738],{"xmlns":51,"display":145},[53,68739,68740,68760],{},[56,68741,68742,68758],{},[1038,68743,68744,68750],{},[941,68745,68746,68748],{},[154,68747,20352],{"mathvariant":1080},[64,68749,1044],{},[8360,68751,68752,68754,68756],{},[154,68753,20352],{"mathvariant":1080},[154,68755,946],{},[64,68757,1044],{},[154,68759,1123],{"mathvariant":1080},[82,68761,68762],{"encoding":84},"\\frac{\\Omega^2}{\\Omega_R^2}.",[40,68764,68766],{"className":68765,"ariaHidden":72},[89],[40,68767,68769,68772,68908],{"className":68768},[93],[40,68770],{"className":68771,"style":63119},[97],[40,68773,68775,68778,68905],{"className":68774},[102],[40,68776],{"className":68777},[106,1162],[40,68779,68781],{"className":68780},[1038],[40,68782,68784,68897],{"className":68783},[344,345],[40,68785,68787,68894],{"className":68786},[349],[40,68788,68790,68849,68857],{"className":68789,"style":63138},[353],[40,68791,68792,68795],{"style":2782},[40,68793],{"className":68794,"style":444},[361],[40,68796,68798],{"className":68797},[102],[40,68799,68801,68804],{"className":68800},[102],[40,68802,20352],{"className":68803},[102],[40,68805,68807],{"className":68806},[340],[40,68808,68810,68841],{"className":68809},[344,345],[40,68811,68813,68838],{"className":68812},[349],[40,68814,68816,68827],{"className":68815,"style":63165},[353],[40,68817,68818,68821],{"style":63168},[40,68819],{"className":68820,"style":362},[361],[40,68822,68824],{"className":68823},[366,367,368,369],[40,68825,946],{"className":68826,"style":3946},[102,335,369],[40,68828,68829,68832],{"style":63180},[40,68830],{"className":68831,"style":362},[361],[40,68833,68835],{"className":68834},[366,367,368,369],[40,68836,1044],{"className":68837},[102,369],[40,68839,377],{"className":68840},[376],[40,68842,68844],{"className":68843},[349],[40,68845,68847],{"className":68846,"style":63199},[353],[40,68848],{},[40,68850,68851,68854],{"style":1193},[40,68852],{"className":68853,"style":444},[361],[40,68855],{"className":68856,"style":1201},[1200],[40,68858,68859,68862],{"style":2802},[40,68860],{"className":68861,"style":444},[361],[40,68863,68865],{"className":68864},[102],[40,68866,68868,68871],{"className":68867},[102],[40,68869,20352],{"className":68870},[102],[40,68872,68874],{"className":68873},[340],[40,68875,68877],{"className":68876},[344],[40,68878,68880],{"className":68879},[349],[40,68881,68883],{"className":68882,"style":982},[353],[40,68884,68885,68888],{"style":1004},[40,68886],{"className":68887,"style":362},[361],[40,68889,68891],{"className":68890},[366,367,368,369],[40,68892,1044],{"className":68893},[102,369],[40,68895,377],{"className":68896},[376],[40,68898,68900],{"className":68899},[349],[40,68901,68903],{"className":68902,"style":63256},[353],[40,68904],{},[40,68906],{"className":68907},[113,1162],[40,68909,1123],{"className":68910},[102],[12,68912,68913,68914,68953],{},"As ",[40,68915,68917,68935],{"className":68916},[43],[40,68918,68920],{"className":68919},[47],[49,68921,68922],{"xmlns":51},[53,68923,68924,68932],{},[56,68925,68926,68928,68930],{},[59,68927,62],{"stretchy":61},[154,68929,2016],{"mathvariant":1080},[59,68931,62],{"stretchy":61},[82,68933,68934],{"encoding":84},"\\lvert\\Delta\\rvert",[40,68936,68938],{"className":68937,"ariaHidden":72},[89],[40,68939,68941,68944,68947,68950],{"className":68940},[93],[40,68942],{"className":68943,"style":98},[97],[40,68945,62],{"className":68946},[106],[40,68948,2016],{"className":68949},[102],[40,68951,62],{"className":68952},[113]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[28,68955,68957],{"id":68956},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[12,68959,68960,68961,68989,68990,69088],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[40,68962,68964,68977],{"className":68963},[43],[40,68965,68967],{"className":68966},[47],[49,68968,68969],{"xmlns":51},[53,68970,68971,68975],{},[56,68972,68973],{},[154,68974,13262],{},[82,68976,13262],{"encoding":84},[40,68978,68980],{"className":68979,"ariaHidden":72},[89],[40,68981,68983,68986],{"className":68982},[93],[40,68984],{"className":68985,"style":58114},[97],[40,68987,13262],{"className":68988,"style":33300},[102,335],", the trace ",[40,68991,68993,69021],{"className":68992},[43],[40,68994,68996],{"className":68995},[47],[49,68997,68998],{"xmlns":51},[53,68999,69000,69018],{},[56,69001,69002,69008,69010,69012,69014,69016],{},[151,69003,69004,69006],{},[154,69005,57996],{},[154,69007,4203],{},[59,69009,167],{"stretchy":61},[154,69011,58007],{},[59,69013,73],{"separator":72},[154,69015,13262],{},[59,69017,207],{"stretchy":61},[82,69019,69020],{"encoding":84},"P_e(\\tau,f)",[40,69022,69024],{"className":69023,"ariaHidden":72},[89],[40,69025,69027,69030,69070,69073,69076,69079,69082,69085],{"className":69026},[93],[40,69028],{"className":69029,"style":98},[97],[40,69031,69033,69036],{"className":69032},[102],[40,69034,57996],{"className":69035,"style":2541},[102,335],[40,69037,69039],{"className":69038},[340],[40,69040,69042,69062],{"className":69041},[344,345],[40,69043,69045,69059],{"className":69044},[349],[40,69046,69048],{"className":69047,"style":354},[353],[40,69049,69050,69053],{"style":2556},[40,69051],{"className":69052,"style":362},[361],[40,69054,69056],{"className":69055},[366,367,368,369],[40,69057,4203],{"className":69058},[102,335,369],[40,69060,377],{"className":69061},[376],[40,69063,69065],{"className":69064},[349],[40,69066,69068],{"className":69067,"style":384},[353],[40,69069],{},[40,69071,167],{"className":69072},[106],[40,69074,58007],{"className":69075,"style":58079},[102,335],[40,69077,73],{"className":69078},[117],[40,69080],{"className":69081,"style":122},[121],[40,69083,13262],{"className":69084,"style":33300},[102,335],[40,69086,207],{"className":69087},[113]," is treated as a time-domain signal. The spectral amplitude is",[40,69090,69092],{"className":69091},[136],[40,69093,69095,69159],{"className":69094},[43],[40,69096,69098],{"className":69097},[47],[49,69099,69100],{"xmlns":51,"display":145},[53,69101,69102,69156],{},[56,69103,69104,69106,69108,69110,69112,69114,69116,69118,69154],{},[154,69105,2362],{},[59,69107,167],{"stretchy":61},[154,69109,59270],{},[59,69111,73],{"separator":72},[154,69113,13262],{},[59,69115,207],{"stretchy":61},[59,69117,162],{},[56,69119,69120,69122,69130,69152],{},[59,69121,62],{"fence":72},[151,69123,69124,69128],{},[154,69125,69127],{"mathvariant":69126},"script","F",[154,69129,58007],{},[56,69131,69132,69134,69140,69142,69144,69146,69148,69150],{},[59,69133,13268],{"fence":72},[151,69135,69136,69138],{},[154,69137,57996],{},[154,69139,4203],{},[59,69141,167],{"stretchy":61},[154,69143,58007],{},[59,69145,73],{"separator":72},[154,69147,13262],{},[59,69149,207],{"stretchy":61},[59,69151,13280],{"fence":72},[59,69153,62],{"fence":72},[154,69155,1123],{"mathvariant":1080},[82,69157,69158],{"encoding":84},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[40,69160,69162,69198],{"className":69161,"ariaHidden":72},[89],[40,69163,69165,69168,69171,69174,69177,69180,69183,69186,69189,69192,69195],{"className":69164},[93],[40,69166],{"className":69167,"style":98},[97],[40,69169,2362],{"className":69170,"style":2383},[102,335],[40,69172,167],{"className":69173},[106],[40,69175,59270],{"className":69176,"style":59291},[102,335],[40,69178,73],{"className":69179},[117],[40,69181],{"className":69182,"style":122},[121],[40,69184,13262],{"className":69185,"style":33300},[102,335],[40,69187,207],{"className":69188},[113],[40,69190],{"className":69191,"style":390},[121],[40,69193,162],{"className":69194},[394],[40,69196],{"className":69197,"style":390},[121],[40,69199,69201,69204,69329,69332],{"className":69200},[93],[40,69202],{"className":69203,"style":98},[97],[40,69205,69207,69210,69256,69259,69326],{"className":69206},[408],[40,69208,62],{"className":69209,"style":413},[106,412],[40,69211,69213,69218],{"className":69212},[102],[40,69214,69127],{"className":69215,"style":69217},[102,69216],"mathcal","margin-right:0.0993em;",[40,69219,69221],{"className":69220},[340],[40,69222,69224,69248],{"className":69223},[344,345],[40,69225,69227,69245],{"className":69226},[349],[40,69228,69230],{"className":69229,"style":354},[353],[40,69231,69233,69236],{"style":69232},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[40,69234],{"className":69235,"style":362},[361],[40,69237,69239],{"className":69238},[366,367,368,369],[40,69240,69242],{"className":69241},[102,369],[40,69243,58007],{"className":69244,"style":58079},[102,335,369],[40,69246,377],{"className":69247},[376],[40,69249,69251],{"className":69250},[349],[40,69252,69254],{"className":69253,"style":384},[353],[40,69255],{},[40,69257],{"className":69258,"style":122},[121],[40,69260,69262,69265,69305,69308,69311,69314,69317,69320,69323],{"className":69261},[408],[40,69263,13268],{"className":69264,"style":413},[106,412],[40,69266,69268,69271],{"className":69267},[102],[40,69269,57996],{"className":69270,"style":2541},[102,335],[40,69272,69274],{"className":69273},[340],[40,69275,69277,69297],{"className":69276},[344,345],[40,69278,69280,69294],{"className":69279},[349],[40,69281,69283],{"className":69282,"style":354},[353],[40,69284,69285,69288],{"style":2556},[40,69286],{"className":69287,"style":362},[361],[40,69289,69291],{"className":69290},[366,367,368,369],[40,69292,4203],{"className":69293},[102,335,369],[40,69295,377],{"className":69296},[376],[40,69298,69300],{"className":69299},[349],[40,69301,69303],{"className":69302,"style":384},[353],[40,69304],{},[40,69306,167],{"className":69307},[106],[40,69309,58007],{"className":69310,"style":58079},[102,335],[40,69312,73],{"className":69313},[117],[40,69315],{"className":69316,"style":122},[121],[40,69318,13262],{"className":69319,"style":33300},[102,335],[40,69321,207],{"className":69322},[113],[40,69324,13280],{"className":69325,"style":413},[113,412],[40,69327,62],{"className":69328,"style":413},[113,412],[40,69330],{"className":69331,"style":122},[121],[40,69333,1123],{"className":69334},[102],[12,69336,69337],{},"This probability expression contains a squared sinusoid. Using the identity",[40,69339,69341],{"className":69340},[136],[40,69342,69344,69420],{"className":69343},[43],[40,69345,69347],{"className":69346},[47],[49,69348,69349],{"xmlns":51,"display":145},[53,69350,69351,69417],{},[56,69352,69353,69363,69383,69385,69391],{},[941,69354,69355,69361],{},[56,69356,69357,69359],{},[154,69358,1813],{},[59,69360,1084],{},[64,69362,1044],{},[56,69364,69365,69367,69381],{},[59,69366,167],{"fence":72},[1038,69368,69369,69379],{},[56,69370,69371,69377],{},[151,69372,69373,69375],{},[154,69374,20352],{"mathvariant":1080},[154,69376,946],{},[154,69378,58007],{},[64,69380,1044],{},[59,69382,207],{"fence":72},[59,69384,162],{},[1038,69386,69387,69389],{},[64,69388,78],{},[64,69390,1044],{},[56,69392,69393,69395,69397,69399,69401,69403,69405,69411,69413,69415],{},[59,69394,1654],{"fence":72},[64,69396,78],{},[59,69398,243],{},[154,69400,1820],{},[59,69402,1084],{},[59,69404,167],{"stretchy":61},[151,69406,69407,69409],{},[154,69408,20352],{"mathvariant":1080},[154,69410,946],{},[154,69412,58007],{},[59,69414,207],{"stretchy":61},[59,69416,1664],{"fence":72},[82,69418,69419],{"encoding":84},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[40,69421,69423,69587],{"className":69422,"ariaHidden":72},[89],[40,69424,69426,69429,69458,69461,69578,69581,69584],{"className":69425},[93],[40,69427],{"className":69428,"style":404},[97],[40,69430,69432,69435],{"className":69431},[1369],[40,69433,1813],{"className":69434},[1369],[40,69436,69438],{"className":69437},[340],[40,69439,69441],{"className":69440},[344],[40,69442,69444],{"className":69443},[349],[40,69445,69447],{"className":69446,"style":63283},[353],[40,69448,69449,69452],{"style":63286},[40,69450],{"className":69451,"style":362},[361],[40,69453,69455],{"className":69454},[366,367,368,369],[40,69456,1044],{"className":69457},[102,369],[40,69459],{"className":69460,"style":122},[121],[40,69462,69464,69470,69572],{"className":69463},[408],[40,69465,69467],{"className":69466,"style":413},[106,412],[40,69468,167],{"className":69469},[417,368],[40,69471,69473,69476,69569],{"className":69472},[102],[40,69474],{"className":69475},[106,1162],[40,69477,69479],{"className":69478},[1038],[40,69480,69482,69561],{"className":69481},[344,345],[40,69483,69485,69558],{"className":69484},[349],[40,69486,69488,69499,69507],{"className":69487,"style":63326},[353],[40,69489,69490,69493],{"style":2782},[40,69491],{"className":69492,"style":444},[361],[40,69494,69496],{"className":69495},[102],[40,69497,1044],{"className":69498},[102],[40,69500,69501,69504],{"style":1193},[40,69502],{"className":69503,"style":444},[361],[40,69505],{"className":69506,"style":1201},[1200],[40,69508,69509,69512],{"style":2802},[40,69510],{"className":69511,"style":444},[361],[40,69513,69515,69555],{"className":69514},[102],[40,69516,69518,69521],{"className":69517},[102],[40,69519,20352],{"className":69520},[102],[40,69522,69524],{"className":69523},[340],[40,69525,69527,69547],{"className":69526},[344,345],[40,69528,69530,69544],{"className":69529},[349],[40,69531,69533],{"className":69532,"style":20224},[353],[40,69534,69535,69538],{"style":6098},[40,69536],{"className":69537,"style":362},[361],[40,69539,69541],{"className":69540},[366,367,368,369],[40,69542,946],{"className":69543,"style":3946},[102,335,369],[40,69545,377],{"className":69546},[376],[40,69548,69550],{"className":69549},[349],[40,69551,69553],{"className":69552,"style":384},[353],[40,69554],{},[40,69556,58007],{"className":69557,"style":58079},[102,335],[40,69559,377],{"className":69560},[376],[40,69562,69564],{"className":69563},[349],[40,69565,69567],{"className":69566,"style":2868},[353],[40,69568],{},[40,69570],{"className":69571},[113,1162],[40,69573,69575],{"className":69574,"style":413},[113,412],[40,69576,207],{"className":69577},[417,368],[40,69579],{"className":69580,"style":390},[121],[40,69582,162],{"className":69583},[394],[40,69585],{"className":69586,"style":390},[121],[40,69588,69590,69593,69655,69658],{"className":69589},[93],[40,69591],{"className":69592,"style":65090},[97],[40,69594,69596,69599,69652],{"className":69595},[102],[40,69597],{"className":69598},[106,1162],[40,69600,69602],{"className":69601},[1038],[40,69603,69605,69644],{"className":69604},[344,345],[40,69606,69608,69641],{"className":69607},[349],[40,69609,69611,69622,69630],{"className":69610,"style":64603},[353],[40,69612,69613,69616],{"style":2782},[40,69614],{"className":69615,"style":444},[361],[40,69617,69619],{"className":69618},[102],[40,69620,1044],{"className":69621},[102],[40,69623,69624,69627],{"style":1193},[40,69625],{"className":69626,"style":444},[361],[40,69628],{"className":69629,"style":1201},[1200],[40,69631,69632,69635],{"style":2802},[40,69633],{"className":69634,"style":444},[361],[40,69636,69638],{"className":69637},[102],[40,69639,78],{"className":69640},[102],[40,69642,377],{"className":69643},[376],[40,69645,69647],{"className":69646},[349],[40,69648,69650],{"className":69649,"style":2868},[353],[40,69651],{},[40,69653],{"className":69654},[113,1162],[40,69656],{"className":69657,"style":122},[121],[40,69659,69661,69664,69667,69670,69673,69676,69679,69682,69722,69725,69728],{"className":69660},[408],[40,69662,1654],{"className":69663,"style":413},[106,412],[40,69665,78],{"className":69666},[102],[40,69668],{"className":69669,"style":1246},[121],[40,69671,243],{"className":69672},[1250],[40,69674],{"className":69675,"style":1246},[121],[40,69677,1820],{"className":69678},[1369],[40,69680,167],{"className":69681},[106],[40,69683,69685,69688],{"className":69684},[102],[40,69686,20352],{"className":69687},[102],[40,69689,69691],{"className":69690},[340],[40,69692,69694,69714],{"className":69693},[344,345],[40,69695,69697,69711],{"className":69696},[349],[40,69698,69700],{"className":69699,"style":20224},[353],[40,69701,69702,69705],{"style":6098},[40,69703],{"className":69704,"style":362},[361],[40,69706,69708],{"className":69707},[366,367,368,369],[40,69709,946],{"className":69710,"style":3946},[102,335,369],[40,69712,377],{"className":69713},[376],[40,69715,69717],{"className":69716},[349],[40,69718,69720],{"className":69719,"style":384},[353],[40,69721],{},[40,69723,58007],{"className":69724,"style":58079},[102,335],[40,69726,207],{"className":69727},[113],[40,69729,1664],{"className":69730,"style":413},[113,412],[12,69732,69733],{},"shows that the dominant oscillatory spectral component appears at the generalized frequency",[40,69735,69737],{"className":69736},[136],[40,69738,69740,69776],{"className":69739},[43],[40,69741,69743],{"className":69742},[47],[49,69744,69745],{"xmlns":51,"display":145},[53,69746,69747,69773],{},[56,69748,69749,69755,69757,69771],{},[151,69750,69751,69753],{},[154,69752,59270],{},[154,69754,946],{},[59,69756,162],{},[1038,69758,69759,69765],{},[151,69760,69761,69763],{},[154,69762,20352],{"mathvariant":1080},[154,69764,946],{},[56,69766,69767,69769],{},[64,69768,1044],{},[154,69770,1661],{},[154,69772,1123],{"mathvariant":1080},[82,69774,69775],{"encoding":84},"\\nu_R=\\frac{\\Omega_R}{2\\pi}.",[40,69777,69779,69834],{"className":69778,"ariaHidden":72},[89],[40,69780,69782,69785,69825,69828,69831],{"className":69781},[93],[40,69783],{"className":69784,"style":328},[97],[40,69786,69788,69791],{"className":69787},[102],[40,69789,59270],{"className":69790,"style":59291},[102,335],[40,69792,69794],{"className":69793},[340],[40,69795,69797,69817],{"className":69796},[344,345],[40,69798,69800,69814],{"className":69799},[349],[40,69801,69803],{"className":69802,"style":20224},[353],[40,69804,69805,69808],{"style":59306},[40,69806],{"className":69807,"style":362},[361],[40,69809,69811],{"className":69810},[366,367,368,369],[40,69812,946],{"className":69813,"style":3946},[102,335,369],[40,69815,377],{"className":69816},[376],[40,69818,69820],{"className":69819},[349],[40,69821,69823],{"className":69822,"style":384},[353],[40,69824],{},[40,69826],{"className":69827,"style":390},[121],[40,69829,162],{"className":69830},[394],[40,69832],{"className":69833,"style":390},[121],[40,69835,69837,69840,69942],{"className":69836},[93],[40,69838],{"className":69839,"style":64721},[97],[40,69841,69843,69846,69939],{"className":69842},[102],[40,69844],{"className":69845},[106,1162],[40,69847,69849],{"className":69848},[1038],[40,69850,69852,69931],{"className":69851},[344,345],[40,69853,69855,69928],{"className":69854},[349],[40,69856,69858,69872,69880],{"className":69857,"style":63326},[353],[40,69859,69860,69863],{"style":2782},[40,69861],{"className":69862,"style":444},[361],[40,69864,69866,69869],{"className":69865},[102],[40,69867,1044],{"className":69868},[102],[40,69870,1661],{"className":69871,"style":336},[102,335],[40,69873,69874,69877],{"style":1193},[40,69875],{"className":69876,"style":444},[361],[40,69878],{"className":69879,"style":1201},[1200],[40,69881,69882,69885],{"style":2802},[40,69883],{"className":69884,"style":444},[361],[40,69886,69888],{"className":69887},[102],[40,69889,69891,69894],{"className":69890},[102],[40,69892,20352],{"className":69893},[102],[40,69895,69897],{"className":69896},[340],[40,69898,69900,69920],{"className":69899},[344,345],[40,69901,69903,69917],{"className":69902},[349],[40,69904,69906],{"className":69905,"style":20224},[353],[40,69907,69908,69911],{"style":6098},[40,69909],{"className":69910,"style":362},[361],[40,69912,69914],{"className":69913},[366,367,368,369],[40,69915,946],{"className":69916,"style":3946},[102,335,369],[40,69918,377],{"className":69919},[376],[40,69921,69923],{"className":69922},[349],[40,69924,69926],{"className":69925,"style":384},[353],[40,69927],{},[40,69929,377],{"className":69930},[376],[40,69932,69934],{"className":69933},[349],[40,69935,69937],{"className":69936,"style":2868},[353],[40,69938],{},[40,69940],{"className":69941},[113,1162],[40,69943,1123],{"className":69944},[102],[12,69946,69947],{},"The computational workflow subtracts the mean signal along the pulse-duration axis, applies a Hann window to reduce spectral leakage, uses zero-padding to smooth the displayed Fourier grid, and overlays the exact generalized Rabi ridge on the Fourier heatmap. Zero-padding improves visual interpolation of the plotted spectrum, although the physical frequency resolution remains set by the sampled temporal window.",[28,69949,69951],{"id":69950},"computational-methodology","Computational Methodology",[12,69953,69954,69955,69983,69984,70012,70013,916,70041,70069,70070,70204,70205,70424],{},"The notebook uses a deterministic closed-form workflow, so it does not require numerical time stepping of the Schrödinger equation. It defines a drive-frequency grid ",[40,69956,69958,69971],{"className":69957},[43],[40,69959,69961],{"className":69960},[47],[49,69962,69963],{"xmlns":51},[53,69964,69965,69969],{},[56,69966,69967],{},[154,69968,13262],{},[82,69970,13262],{"encoding":84},[40,69972,69974],{"className":69973,"ariaHidden":72},[89],[40,69975,69977,69980],{"className":69976},[93],[40,69978],{"className":69979,"style":58114},[97],[40,69981,13262],{"className":69982,"style":33300},[102,335]," in GHz and a pulse-duration grid ",[40,69985,69987,70000],{"className":69986},[43],[40,69988,69990],{"className":69989},[47],[49,69991,69992],{"xmlns":51},[53,69993,69994,69998],{},[56,69995,69996],{},[154,69997,58007],{},[82,69999,58135],{"encoding":84},[40,70001,70003],{"className":70002,"ariaHidden":72},[89],[40,70004,70006,70009],{"className":70005},[93],[40,70007],{"className":70008,"style":4142},[97],[40,70010,58007],{"className":70011,"style":58079},[102,335]," in ns, then converts both to SI units. From a dense two-dimensional mesh over ",[40,70014,70016,70029],{"className":70015},[43],[40,70017,70019],{"className":70018},[47],[49,70020,70021],{"xmlns":51},[53,70022,70023,70027],{},[56,70024,70025],{},[154,70026,13262],{},[82,70028,13262],{"encoding":84},[40,70030,70032],{"className":70031,"ariaHidden":72},[89],[40,70033,70035,70038],{"className":70034},[93],[40,70036],{"className":70037,"style":58114},[97],[40,70039,13262],{"className":70040,"style":33300},[102,335],[40,70042,70044,70057],{"className":70043},[43],[40,70045,70047],{"className":70046},[47],[49,70048,70049],{"xmlns":51},[53,70050,70051,70055],{},[56,70052,70053],{},[154,70054,58007],{},[82,70056,58135],{"encoding":84},[40,70058,70060],{"className":70059,"ariaHidden":72},[89],[40,70061,70063,70066],{"className":70062},[93],[40,70064],{"className":70065,"style":4142},[97],[40,70067,58007],{"className":70068,"style":58079},[102,335],", the workflow computes the detuning ",[40,70071,70073,70107],{"className":70072},[43],[40,70074,70076],{"className":70075},[47],[49,70077,70078],{"xmlns":51},[53,70079,70080,70104],{},[56,70081,70082,70084,70086,70088,70090,70092,70094,70096,70102],{},[154,70083,2016],{"mathvariant":1080},[59,70085,162],{},[64,70087,1044],{},[154,70089,1661],{},[59,70091,167],{"stretchy":61},[154,70093,13262],{},[59,70095,243],{},[151,70097,70098,70100],{},[154,70099,13262],{},[64,70101,66],{},[59,70103,207],{"stretchy":61},[82,70105,70106],{"encoding":84},"\\Delta=2\\pi(f-f_0)",[40,70108,70110,70128,70155],{"className":70109,"ariaHidden":72},[89],[40,70111,70113,70116,70119,70122,70125],{"className":70112},[93],[40,70114],{"className":70115,"style":4304},[97],[40,70117,2016],{"className":70118},[102],[40,70120],{"className":70121,"style":390},[121],[40,70123,162],{"className":70124},[394],[40,70126],{"className":70127,"style":390},[121],[40,70129,70131,70134,70137,70140,70143,70146,70149,70152],{"className":70130},[93],[40,70132],{"className":70133,"style":98},[97],[40,70135,1044],{"className":70136},[102],[40,70138,1661],{"className":70139,"style":336},[102,335],[40,70141,167],{"className":70142},[106],[40,70144,13262],{"className":70145,"style":33300},[102,335],[40,70147],{"className":70148,"style":1246},[121],[40,70150,243],{"className":70151},[1250],[40,70153],{"className":70154,"style":1246},[121],[40,70156,70158,70161,70201],{"className":70157},[93],[40,70159],{"className":70160,"style":98},[97],[40,70162,70164,70167],{"className":70163},[102],[40,70165,13262],{"className":70166,"style":33300},[102,335],[40,70168,70170],{"className":70169},[340],[40,70171,70173,70193],{"className":70172},[344,345],[40,70174,70176,70190],{"className":70175},[349],[40,70177,70179],{"className":70178,"style":7201},[353],[40,70180,70181,70184],{"style":33315},[40,70182],{"className":70183,"style":362},[361],[40,70185,70187],{"className":70186},[366,367,368,369],[40,70188,66],{"className":70189},[102,369],[40,70191,377],{"className":70192},[376],[40,70194,70196],{"className":70195},[349],[40,70197,70199],{"className":70198,"style":384},[353],[40,70200],{},[40,70202,207],{"className":70203},[113],", evaluates the generalized Rabi angular frequency ",[40,70206,70208,70245],{"className":70207},[43],[40,70209,70211],{"className":70210},[47],[49,70212,70213],{"xmlns":51},[53,70214,70215,70243],{},[56,70216,70217,70223,70225],{},[151,70218,70219,70221],{},[154,70220,20352],{"mathvariant":1080},[154,70222,946],{},[59,70224,162],{},[2045,70226,70227],{},[56,70228,70229,70235,70237],{},[941,70230,70231,70233],{},[154,70232,20352],{"mathvariant":1080},[64,70234,1044],{},[59,70236,1053],{},[941,70238,70239,70241],{},[154,70240,2016],{"mathvariant":1080},[64,70242,1044],{},[82,70244,63468],{"encoding":84},[40,70246,70248,70303],{"className":70247,"ariaHidden":72},[89],[40,70249,70251,70254,70294,70297,70300],{"className":70250},[93],[40,70252],{"className":70253,"style":3106},[97],[40,70255,70257,70260],{"className":70256},[102],[40,70258,20352],{"className":70259},[102],[40,70261,70263],{"className":70262},[340],[40,70264,70266,70286],{"className":70265},[344,345],[40,70267,70269,70283],{"className":70268},[349],[40,70270,70272],{"className":70271,"style":20224},[353],[40,70273,70274,70277],{"style":6098},[40,70275],{"className":70276,"style":362},[361],[40,70278,70280],{"className":70279},[366,367,368,369],[40,70281,946],{"className":70282,"style":3946},[102,335,369],[40,70284,377],{"className":70285},[376],[40,70287,70289],{"className":70288},[349],[40,70290,70292],{"className":70291,"style":384},[353],[40,70293],{},[40,70295],{"className":70296,"style":390},[121],[40,70298,162],{"className":70299},[394],[40,70301],{"className":70302,"style":390},[121],[40,70304,70306,70310],{"className":70305},[93],[40,70307],{"className":70308,"style":70309},[97],"height:1.04em;vertical-align:-0.1266em;",[40,70311,70313],{"className":70312},[102,2218],[40,70314,70316,70415],{"className":70315},[344,345],[40,70317,70319,70412],{"className":70318},[349],[40,70320,70323,70399],{"className":70321,"style":70322},[353],"height:0.9134em;",[40,70324,70326,70329],{"className":70325,"style":2233},[2232],[40,70327],{"className":70328,"style":444},[361],[40,70330,70332,70361,70364,70367,70370],{"className":70331,"style":2240},[102],[40,70333,70335,70338],{"className":70334},[102],[40,70336,20352],{"className":70337},[102],[40,70339,70341],{"className":70340},[340],[40,70342,70344],{"className":70343},[344],[40,70345,70347],{"className":70346},[349],[40,70348,70350],{"className":70349,"style":63577},[353],[40,70351,70352,70355],{"style":63580},[40,70353],{"className":70354,"style":362},[361],[40,70356,70358],{"className":70357},[366,367,368,369],[40,70359,1044],{"className":70360},[102,369],[40,70362],{"className":70363,"style":1246},[121],[40,70365,1053],{"className":70366},[1250],[40,70368],{"className":70369,"style":1246},[121],[40,70371,70373,70376],{"className":70372},[102],[40,70374,2016],{"className":70375},[102],[40,70377,70379],{"className":70378},[340],[40,70380,70382],{"className":70381},[344],[40,70383,70385],{"className":70384},[349],[40,70386,70388],{"className":70387,"style":63577},[353],[40,70389,70390,70393],{"style":63580},[40,70391],{"className":70392,"style":362},[361],[40,70394,70396],{"className":70395},[366,367,368,369],[40,70397,1044],{"className":70398},[102,369],[40,70400,70402,70405],{"style":70401},"top:-2.8734em;",[40,70403],{"className":70404,"style":444},[361],[40,70406,70408],{"className":70407,"style":2254},[2253],[2256,70409,70410],{"xmlns":2258,"width":2259,"height":2260,"viewBox":2261,"preserveAspectRatio":2262},[2264,70411],{"d":2266},[40,70413,377],{"className":70414},[376],[40,70416,70418],{"className":70417},[349],[40,70419,70422],{"className":70420,"style":70421},[353],"height:0.1266em;",[40,70423],{},", and calculates the analytic excited-state probability",[40,70426,70428],{"className":70427},[136],[40,70429,70431,70509],{"className":70430},[43],[40,70432,70434],{"className":70433},[47],[49,70435,70436],{"xmlns":51,"display":145},[53,70437,70438,70506],{},[56,70439,70440,70446,70448,70450,70452,70454,70456,70458,70474,70484,70504],{},[151,70441,70442,70444],{},[154,70443,57996],{},[154,70445,4203],{},[59,70447,167],{"stretchy":61},[154,70449,13262],{},[59,70451,73],{"separator":72},[154,70453,58007],{},[59,70455,207],{"stretchy":61},[59,70457,162],{},[1038,70459,70460,70466],{},[941,70461,70462,70464],{},[154,70463,20352],{"mathvariant":1080},[64,70465,1044],{},[8360,70467,70468,70470,70472],{},[154,70469,20352],{"mathvariant":1080},[154,70471,946],{},[64,70473,1044],{},[941,70475,70476,70482],{},[56,70477,70478,70480],{},[154,70479,1813],{},[59,70481,1084],{},[64,70483,1044],{},[56,70485,70486,70488,70502],{},[59,70487,167],{"fence":72},[1038,70489,70490,70500],{},[56,70491,70492,70498],{},[151,70493,70494,70496],{},[154,70495,20352],{"mathvariant":1080},[154,70497,946],{},[154,70499,58007],{},[64,70501,1044],{},[59,70503,207],{"fence":72},[154,70505,1123],{"mathvariant":1080},[82,70507,70508],{"encoding":84},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[40,70510,70512,70585],{"className":70511,"ariaHidden":72},[89],[40,70513,70515,70518,70558,70561,70564,70567,70570,70573,70576,70579,70582],{"className":70514},[93],[40,70516],{"className":70517,"style":98},[97],[40,70519,70521,70524],{"className":70520},[102],[40,70522,57996],{"className":70523,"style":2541},[102,335],[40,70525,70527],{"className":70526},[340],[40,70528,70530,70550],{"className":70529},[344,345],[40,70531,70533,70547],{"className":70532},[349],[40,70534,70536],{"className":70535,"style":354},[353],[40,70537,70538,70541],{"style":2556},[40,70539],{"className":70540,"style":362},[361],[40,70542,70544],{"className":70543},[366,367,368,369],[40,70545,4203],{"className":70546},[102,335,369],[40,70548,377],{"className":70549},[376],[40,70551,70553],{"className":70552},[349],[40,70554,70556],{"className":70555,"style":384},[353],[40,70557],{},[40,70559,167],{"className":70560},[106],[40,70562,13262],{"className":70563,"style":33300},[102,335],[40,70565,73],{"className":70566},[117],[40,70568],{"className":70569,"style":122},[121],[40,70571,58007],{"className":70572,"style":58079},[102,335],[40,70574,207],{"className":70575},[113],[40,70577],{"className":70578,"style":390},[121],[40,70580,162],{"className":70581},[394],[40,70583],{"className":70584,"style":390},[121],[40,70586,70588,70591,70727,70730,70759,70762,70879,70882],{"className":70587},[93],[40,70589],{"className":70590,"style":63119},[97],[40,70592,70594,70597,70724],{"className":70593},[102],[40,70595],{"className":70596},[106,1162],[40,70598,70600],{"className":70599},[1038],[40,70601,70603,70716],{"className":70602},[344,345],[40,70604,70606,70713],{"className":70605},[349],[40,70607,70609,70668,70676],{"className":70608,"style":63138},[353],[40,70610,70611,70614],{"style":2782},[40,70612],{"className":70613,"style":444},[361],[40,70615,70617],{"className":70616},[102],[40,70618,70620,70623],{"className":70619},[102],[40,70621,20352],{"className":70622},[102],[40,70624,70626],{"className":70625},[340],[40,70627,70629,70660],{"className":70628},[344,345],[40,70630,70632,70657],{"className":70631},[349],[40,70633,70635,70646],{"className":70634,"style":63165},[353],[40,70636,70637,70640],{"style":63168},[40,70638],{"className":70639,"style":362},[361],[40,70641,70643],{"className":70642},[366,367,368,369],[40,70644,946],{"className":70645,"style":3946},[102,335,369],[40,70647,70648,70651],{"style":63180},[40,70649],{"className":70650,"style":362},[361],[40,70652,70654],{"className":70653},[366,367,368,369],[40,70655,1044],{"className":70656},[102,369],[40,70658,377],{"className":70659},[376],[40,70661,70663],{"className":70662},[349],[40,70664,70666],{"className":70665,"style":63199},[353],[40,70667],{},[40,70669,70670,70673],{"style":1193},[40,70671],{"className":70672,"style":444},[361],[40,70674],{"className":70675,"style":1201},[1200],[40,70677,70678,70681],{"style":2802},[40,70679],{"className":70680,"style":444},[361],[40,70682,70684],{"className":70683},[102],[40,70685,70687,70690],{"className":70686},[102],[40,70688,20352],{"className":70689},[102],[40,70691,70693],{"className":70692},[340],[40,70694,70696],{"className":70695},[344],[40,70697,70699],{"className":70698},[349],[40,70700,70702],{"className":70701,"style":982},[353],[40,70703,70704,70707],{"style":1004},[40,70705],{"className":70706,"style":362},[361],[40,70708,70710],{"className":70709},[366,367,368,369],[40,70711,1044],{"className":70712},[102,369],[40,70714,377],{"className":70715},[376],[40,70717,70719],{"className":70718},[349],[40,70720,70722],{"className":70721,"style":63256},[353],[40,70723],{},[40,70725],{"className":70726},[113,1162],[40,70728],{"className":70729,"style":122},[121],[40,70731,70733,70736],{"className":70732},[1369],[40,70734,1813],{"className":70735},[1369],[40,70737,70739],{"className":70738},[340],[40,70740,70742],{"className":70741},[344],[40,70743,70745],{"className":70744},[349],[40,70746,70748],{"className":70747,"style":63283},[353],[40,70749,70750,70753],{"style":63286},[40,70751],{"className":70752,"style":362},[361],[40,70754,70756],{"className":70755},[366,367,368,369],[40,70757,1044],{"className":70758},[102,369],[40,70760],{"className":70761,"style":122},[121],[40,70763,70765,70771,70873],{"className":70764},[408],[40,70766,70768],{"className":70767,"style":413},[106,412],[40,70769,167],{"className":70770},[417,368],[40,70772,70774,70777,70870],{"className":70773},[102],[40,70775],{"className":70776},[106,1162],[40,70778,70780],{"className":70779},[1038],[40,70781,70783,70862],{"className":70782},[344,345],[40,70784,70786,70859],{"className":70785},[349],[40,70787,70789,70800,70808],{"className":70788,"style":63326},[353],[40,70790,70791,70794],{"style":2782},[40,70792],{"className":70793,"style":444},[361],[40,70795,70797],{"className":70796},[102],[40,70798,1044],{"className":70799},[102],[40,70801,70802,70805],{"style":1193},[40,70803],{"className":70804,"style":444},[361],[40,70806],{"className":70807,"style":1201},[1200],[40,70809,70810,70813],{"style":2802},[40,70811],{"className":70812,"style":444},[361],[40,70814,70816,70856],{"className":70815},[102],[40,70817,70819,70822],{"className":70818},[102],[40,70820,20352],{"className":70821},[102],[40,70823,70825],{"className":70824},[340],[40,70826,70828,70848],{"className":70827},[344,345],[40,70829,70831,70845],{"className":70830},[349],[40,70832,70834],{"className":70833,"style":20224},[353],[40,70835,70836,70839],{"style":6098},[40,70837],{"className":70838,"style":362},[361],[40,70840,70842],{"className":70841},[366,367,368,369],[40,70843,946],{"className":70844,"style":3946},[102,335,369],[40,70846,377],{"className":70847},[376],[40,70849,70851],{"className":70850},[349],[40,70852,70854],{"className":70853,"style":384},[353],[40,70855],{},[40,70857,58007],{"className":70858,"style":58079},[102,335],[40,70860,377],{"className":70861},[376],[40,70863,70865],{"className":70864},[349],[40,70866,70868],{"className":70867,"style":2868},[353],[40,70869],{},[40,70871],{"className":70872},[113,1162],[40,70874,70876],{"className":70875,"style":413},[113,412],[40,70877,207],{"className":70878},[417,368],[40,70880],{"className":70881,"style":122},[121],[40,70883,1123],{"className":70884},[102],[12,70886,70887,70888,71003],{},"The routine optionally applies the damping envelope ",[40,70889,70891,70925],{"className":70890},[43],[40,70892,70894],{"className":70893},[47],[49,70895,70896],{"xmlns":51},[53,70897,70898,70922],{},[56,70899,70900,70902,70904,70906,70908,70910,70912,70920],{},[154,70901,5268],{},[59,70903,1084],{},[59,70905,167],{"stretchy":61},[59,70907,243],{},[154,70909,58007],{},[154,70911,11243],{"mathvariant":1080},[8360,70913,70914,70916,70918],{},[154,70915,7059],{},[64,70917,1044],{},[59,70919,59534],{},[59,70921,207],{"stretchy":61},[82,70923,70924],{"encoding":84},"\\exp(-\\tau\u002FT_2^\\ast)",[40,70926,70928],{"className":70927,"ariaHidden":72},[89],[40,70929,70931,70934,70937,70940,70943,70946,70949,71000],{"className":70930},[93],[40,70932],{"className":70933,"style":98},[97],[40,70935,5268],{"className":70936},[1369],[40,70938,167],{"className":70939},[106],[40,70941,243],{"className":70942},[102],[40,70944,58007],{"className":70945,"style":58079},[102,335],[40,70947,11243],{"className":70948},[102],[40,70950,70952,70955],{"className":70951},[102],[40,70953,7059],{"className":70954,"style":2541},[102,335],[40,70956,70958],{"className":70957},[340],[40,70959,70961,70992],{"className":70960},[344,345],[40,70962,70964,70989],{"className":70963},[349],[40,70965,70967,70978],{"className":70966,"style":59566},[353],[40,70968,70969,70972],{"style":59569},[40,70970],{"className":70971,"style":362},[361],[40,70973,70975],{"className":70974},[366,367,368,369],[40,70976,1044],{"className":70977},[102,369],[40,70979,70980,70983],{"style":1004},[40,70981],{"className":70982,"style":362},[361],[40,70984,70986],{"className":70985},[366,367,368,369],[40,70987,59534],{"className":70988},[1250,369],[40,70990,377],{"className":70991},[376],[40,70993,70995],{"className":70994},[349],[40,70996,70998],{"className":70997,"style":59599},[353],[40,70999],{},[40,71001,207],{"className":71002},[113]," and clips the result to the physical interval",[40,71005,71007],{"className":71006},[136],[40,71008,71010,71037],{"className":71009},[43],[40,71011,71013],{"className":71012},[47],[49,71014,71015],{"xmlns":51,"display":145},[53,71016,71017,71034],{},[56,71018,71019,71021,71024,71030,71032],{},[64,71020,66],{},[59,71022,71023],{},"≤",[151,71025,71026,71028],{},[154,71027,57996],{},[154,71029,4203],{},[59,71031,71023],{},[64,71033,60639],{},[82,71035,71036],{"encoding":84},"0\\le P_e\\le 1.",[40,71038,71040,71059,71114],{"className":71039,"ariaHidden":72},[89],[40,71041,71043,71047,71050,71053,71056],{"className":71042},[93],[40,71044],{"className":71045,"style":71046},[97],"height:0.7804em;vertical-align:-0.136em;",[40,71048,66],{"className":71049},[102],[40,71051],{"className":71052,"style":390},[121],[40,71054,71023],{"className":71055},[394],[40,71057],{"className":71058,"style":390},[121],[40,71060,71062,71065,71105,71108,71111],{"className":71061},[93],[40,71063],{"className":71064,"style":3106},[97],[40,71066,71068,71071],{"className":71067},[102],[40,71069,57996],{"className":71070,"style":2541},[102,335],[40,71072,71074],{"className":71073},[340],[40,71075,71077,71097],{"className":71076},[344,345],[40,71078,71080,71094],{"className":71079},[349],[40,71081,71083],{"className":71082,"style":354},[353],[40,71084,71085,71088],{"style":2556},[40,71086],{"className":71087,"style":362},[361],[40,71089,71091],{"className":71090},[366,367,368,369],[40,71092,4203],{"className":71093},[102,335,369],[40,71095,377],{"className":71096},[376],[40,71098,71100],{"className":71099},[349],[40,71101,71103],{"className":71102,"style":384},[353],[40,71104],{},[40,71106],{"className":71107,"style":390},[121],[40,71109,71023],{"className":71110},[394],[40,71112],{"className":71113,"style":390},[121],[40,71115,71117,71120],{"className":71116},[93],[40,71118],{"className":71119,"style":1624},[97],[40,71121,60639],{"className":71122},[102],[12,71124,71125],{},"Finally, the output is rendered as heatmaps, three-dimensional surfaces, one-dimensional slices, and Fourier-domain spectra. Standard scientific Python libraries provide the numerical arrays, visualization routines, and optional interactive rendering tools.",[28,71127,71129],{"id":71128},"graphical-interpretation","Graphical Interpretation",[12,71131,8707,71132,71230],{},[40,71133,71135,71163],{"className":71134},[43],[40,71136,71138],{"className":71137},[47],[49,71139,71140],{"xmlns":51},[53,71141,71142,71160],{},[56,71143,71144,71150,71152,71154,71156,71158],{},[151,71145,71146,71148],{},[154,71147,57996],{},[154,71149,4203],{},[59,71151,167],{"stretchy":61},[154,71153,13262],{},[59,71155,73],{"separator":72},[154,71157,58007],{},[59,71159,207],{"stretchy":61},[82,71161,71162],{"encoding":84},"P_e(f,\\tau)",[40,71164,71166],{"className":71165,"ariaHidden":72},[89],[40,71167,71169,71172,71212,71215,71218,71221,71224,71227],{"className":71168},[93],[40,71170],{"className":71171,"style":98},[97],[40,71173,71175,71178],{"className":71174},[102],[40,71176,57996],{"className":71177,"style":2541},[102,335],[40,71179,71181],{"className":71180},[340],[40,71182,71184,71204],{"className":71183},[344,345],[40,71185,71187,71201],{"className":71186},[349],[40,71188,71190],{"className":71189,"style":354},[353],[40,71191,71192,71195],{"style":2556},[40,71193],{"className":71194,"style":362},[361],[40,71196,71198],{"className":71197},[366,367,368,369],[40,71199,4203],{"className":71200},[102,335,369],[40,71202,377],{"className":71203},[376],[40,71205,71207],{"className":71206},[349],[40,71208,71210],{"className":71209,"style":384},[353],[40,71211],{},[40,71213,167],{"className":71214},[106],[40,71216,13262],{"className":71217,"style":33300},[102,335],[40,71219,73],{"className":71220},[117],[40,71222],{"className":71223,"style":122},[121],[40,71225,58007],{"className":71226,"style":58079},[102,335],[40,71228,207],{"className":71229},[113]," heatmap shows the predicted probability of detecting the qubit in the excited state after applying a control pulse. Its horizontal axis gives the applied drive frequency. The vertical axis gives the pulse duration. Brighter regions indicate higher excited-state probability.",[12,71232,71233],{},"The resonant interaction region appears near",[40,71235,71237],{"className":71236},[136],[40,71238,71240,71263],{"className":71239},[43],[40,71241,71243],{"className":71242},[47],[49,71244,71245],{"xmlns":51,"display":145},[53,71246,71247,71261],{},[56,71248,71249,71251,71253,71259],{},[154,71250,13262],{},[59,71252,162],{},[151,71254,71255,71257],{},[154,71256,13262],{},[64,71258,66],{},[154,71260,1123],{"mathvariant":1080},[82,71262,60336],{"encoding":84},[40,71264,71266,71284],{"className":71265,"ariaHidden":72},[89],[40,71267,71269,71272,71275,71278,71281],{"className":71268},[93],[40,71270],{"className":71271,"style":58114},[97],[40,71273,13262],{"className":71274,"style":33300},[102,335],[40,71276],{"className":71277,"style":390},[121],[40,71279,162],{"className":71280},[394],[40,71282],{"className":71283,"style":390},[121],[40,71285,71287,71290,71330],{"className":71286},[93],[40,71288],{"className":71289,"style":58114},[97],[40,71291,71293,71296],{"className":71292},[102],[40,71294,13262],{"className":71295,"style":33300},[102,335],[40,71297,71299],{"className":71298},[340],[40,71300,71302,71322],{"className":71301},[344,345],[40,71303,71305,71319],{"className":71304},[349],[40,71306,71308],{"className":71307,"style":7201},[353],[40,71309,71310,71313],{"style":33315},[40,71311],{"className":71312,"style":362},[361],[40,71314,71316],{"className":71315},[366,367,368,369],[40,71317,66],{"className":71318},[102,369],[40,71320,377],{"className":71321},[376],[40,71323,71325],{"className":71324},[349],[40,71326,71328],{"className":71327,"style":384},[353],[40,71329],{},[40,71331,1123],{"className":71332},[102],[12,71334,71335,71336,71415],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[40,71337,71339,71373],{"className":71338},[43],[40,71340,71342],{"className":71341},[47],[49,71343,71344],{"xmlns":51},[53,71345,71346,71370],{},[56,71347,71348,71350,71352,71354,71356,71358,71360,71362],{},[154,71349,20352],{"mathvariant":1080},[154,71351,11243],{"mathvariant":1080},[64,71353,1044],{},[154,71355,1661],{},[59,71357,162],{},[64,71359,13742],{},[1102,71361,1104],{},[56,71363,71364,71366,71368],{},[154,71365,64702],{"mathvariant":1080},[154,71367,26630],{"mathvariant":1080},[154,71369,273],{"mathvariant":1080},[82,71371,71372],{"encoding":84},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[40,71374,71376,71397],{"className":71375,"ariaHidden":72},[89],[40,71377,71379,71382,71385,71388,71391,71394],{"className":71378},[93],[40,71380],{"className":71381,"style":98},[97],[40,71383,59016],{"className":71384},[102],[40,71386,1661],{"className":71387,"style":336},[102,335],[40,71389],{"className":71390,"style":390},[121],[40,71392,162],{"className":71393},[394],[40,71395],{"className":71396,"style":390},[121],[40,71398,71400,71403,71406,71409],{"className":71399},[93],[40,71401],{"className":71402,"style":4304},[97],[40,71404,13742],{"className":71405},[102],[40,71407,1104],{"className":71408},[121,60187],[40,71410,71412],{"className":71411},[102],[40,71413,23612],{"className":71414},[102,1373],", the ideal resonant maxima appear near",[40,71417,71419],{"className":71418},[136],[40,71420,71422,71495],{"className":71421},[43],[40,71423,71425],{"className":71424},[47],[49,71426,71427],{"xmlns":51,"display":145},[53,71428,71429,71492],{},[56,71430,71431,71433,71435,71437,71439,71445,71447,71449,71452,71454,71460,71462,71464,71467,71469,71475,71477,71479,71482,71484,71490],{},[154,71432,58007],{},[59,71434,162],{},[64,71436,64874],{},[1102,71438,1104],{},[56,71440,71441,71443],{},[154,71442,1796],{"mathvariant":1080},[154,71444,64883],{"mathvariant":1080},[59,71446,73],{"separator":72},[1102,71448,1104],{},[64,71450,71451],{},"75",[1102,71453,1104],{},[56,71455,71456,71458],{},[154,71457,1796],{"mathvariant":1080},[154,71459,64883],{"mathvariant":1080},[59,71461,73],{"separator":72},[1102,71463,1104],{},[64,71465,71466],{},"125",[1102,71468,1104],{},[56,71470,71471,71473],{},[154,71472,1796],{"mathvariant":1080},[154,71474,64883],{"mathvariant":1080},[59,71476,73],{"separator":72},[1102,71478,1104],{},[64,71480,71481],{},"175",[1102,71483,1104],{},[56,71485,71486,71488],{},[154,71487,1796],{"mathvariant":1080},[154,71489,64883],{"mathvariant":1080},[59,71491,73],{"separator":72},[82,71493,71494],{"encoding":84},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[40,71496,71498,71516],{"className":71497,"ariaHidden":72},[89],[40,71499,71501,71504,71507,71510,71513],{"className":71500},[93],[40,71502],{"className":71503,"style":4142},[97],[40,71505,58007],{"className":71506,"style":58079},[102,335],[40,71508],{"className":71509,"style":390},[121],[40,71511,162],{"className":71512},[394],[40,71514],{"className":71515,"style":390},[121],[40,71517,71519,71523,71526,71529,71535,71538,71541,71544,71547,71550,71556,71559,71562,71565,71568,71571,71577,71580,71583,71586,71589,71592,71598],{"className":71518},[93],[40,71520],{"className":71521,"style":71522},[97],"height:0.8389em;vertical-align:-0.1944em;",[40,71524,64874],{"className":71525},[102],[40,71527,1104],{"className":71528},[121,60187],[40,71530,71532],{"className":71531},[102],[40,71533,64965],{"className":71534},[102,1373],[40,71536,73],{"className":71537},[117],[40,71539,1104],{"className":71540},[121],[40,71542],{"className":71543,"style":122},[121],[40,71545,71451],{"className":71546},[102],[40,71548,1104],{"className":71549},[121,60187],[40,71551,71553],{"className":71552},[102],[40,71554,64965],{"className":71555},[102,1373],[40,71557,73],{"className":71558},[117],[40,71560,1104],{"className":71561},[121],[40,71563],{"className":71564,"style":122},[121],[40,71566,71466],{"className":71567},[102],[40,71569,1104],{"className":71570},[121,60187],[40,71572,71574],{"className":71573},[102],[40,71575,64965],{"className":71576},[102,1373],[40,71578,73],{"className":71579},[117],[40,71581,1104],{"className":71582},[121],[40,71584],{"className":71585,"style":122},[121],[40,71587,71481],{"className":71588},[102],[40,71590,1104],{"className":71591},[121,60187],[40,71593,71595],{"className":71594},[102],[40,71596,64965],{"className":71597},[102,1373],[40,71599,73],{"className":71600},[117],[12,71602,71603],{},"before the optional phenomenological damping factor is applied.",[12,71605,71606,71607,71676,71677,71803],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[40,71608,71610,71627],{"className":71609},[43],[40,71611,71613],{"className":71612},[47],[49,71614,71615],{"xmlns":51},[53,71616,71617,71625],{},[56,71618,71619],{},[151,71620,71621,71623],{},[154,71622,20352],{"mathvariant":1080},[154,71624,946],{},[82,71626,59197],{"encoding":84},[40,71628,71630],{"className":71629,"ariaHidden":72},[89],[40,71631,71633,71636],{"className":71632},[93],[40,71634],{"className":71635,"style":3106},[97],[40,71637,71639,71642],{"className":71638},[102],[40,71640,20352],{"className":71641},[102],[40,71643,71645],{"className":71644},[340],[40,71646,71648,71668],{"className":71647},[344,345],[40,71649,71651,71665],{"className":71650},[349],[40,71652,71654],{"className":71653,"style":20224},[353],[40,71655,71656,71659],{"style":6098},[40,71657],{"className":71658,"style":362},[361],[40,71660,71662],{"className":71661},[366,367,368,369],[40,71663,946],{"className":71664,"style":3946},[102,335,369],[40,71666,377],{"className":71667},[376],[40,71669,71671],{"className":71670},[349],[40,71672,71674],{"className":71673,"style":384},[353],[40,71675],{},", which increases with detuning, and from the amplitude coefficient ",[40,71678,71680,71708],{"className":71679},[43],[40,71681,71683],{"className":71682},[47],[49,71684,71685],{"xmlns":51},[53,71686,71687,71705],{},[56,71688,71689,71695,71697],{},[941,71690,71691,71693],{},[154,71692,20352],{"mathvariant":1080},[64,71694,1044],{},[154,71696,11243],{"mathvariant":1080},[8360,71698,71699,71701,71703],{},[154,71700,20352],{"mathvariant":1080},[154,71702,946],{},[64,71704,1044],{},[82,71706,71707],{"encoding":84},"\\Omega^2\u002F\\Omega_R^2",[40,71709,71711],{"className":71710,"ariaHidden":72},[89],[40,71712,71714,71718,71747,71750],{"className":71713},[93],[40,71715],{"className":71716,"style":71717},[97],"height:1.0894em;vertical-align:-0.2753em;",[40,71719,71721,71724],{"className":71720},[102],[40,71722,20352],{"className":71723},[102],[40,71725,71727],{"className":71726},[340],[40,71728,71730],{"className":71729},[344],[40,71731,71733],{"className":71732},[349],[40,71734,71736],{"className":71735,"style":982},[353],[40,71737,71738,71741],{"style":1004},[40,71739],{"className":71740,"style":362},[361],[40,71742,71744],{"className":71743},[366,367,368,369],[40,71745,1044],{"className":71746},[102,369],[40,71748,11243],{"className":71749},[102],[40,71751,71753,71756],{"className":71752},[102],[40,71754,20352],{"className":71755},[102],[40,71757,71759],{"className":71758},[340],[40,71760,71762,71794],{"className":71761},[344,345],[40,71763,71765,71791],{"className":71764},[349],[40,71766,71768,71780],{"className":71767,"style":982},[353],[40,71769,71771,71774],{"style":71770},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[40,71772],{"className":71773,"style":362},[361],[40,71775,71777],{"className":71776},[366,367,368,369],[40,71778,946],{"className":71779,"style":3946},[102,335,369],[40,71781,71782,71785],{"style":1004},[40,71783],{"className":71784,"style":362},[361],[40,71786,71788],{"className":71787},[366,367,368,369],[40,71789,1044],{"className":71790},[102,369],[40,71792,377],{"className":71793},[376],[40,71795,71797],{"className":71796},[349],[40,71798,71801],{"className":71799,"style":71800},[353],"height:0.2753em;",[40,71802],{},", which decreases with detuning.",[12,71805,71806],{},"The three-dimensional surface plot represents the same probability data as a height field. This view emphasizes the central resonant ridge and the suppressed off-resonant oscillations. Cross-section plots isolate one-dimensional behavior. A fixed-frequency slice shows how probability evolves with pulse duration. At a fixed duration, a complementary slice shows how probability changes across the drive-frequency sweep.",[12,71808,71809,71810,71907],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[40,71811,71813,71840],{"className":71812},[43],[40,71814,71816],{"className":71815},[47],[49,71817,71818],{"xmlns":51},[53,71819,71820,71838],{},[56,71821,71822,71828,71830,71832,71834,71836],{},[151,71823,71824,71826],{},[154,71825,57996],{},[154,71827,4203],{},[59,71829,167],{"stretchy":61},[154,71831,58007],{},[59,71833,73],{"separator":72},[154,71835,13262],{},[59,71837,207],{"stretchy":61},[82,71839,69020],{"encoding":84},[40,71841,71843],{"className":71842,"ariaHidden":72},[89],[40,71844,71846,71849,71889,71892,71895,71898,71901,71904],{"className":71845},[93],[40,71847],{"className":71848,"style":98},[97],[40,71850,71852,71855],{"className":71851},[102],[40,71853,57996],{"className":71854,"style":2541},[102,335],[40,71856,71858],{"className":71857},[340],[40,71859,71861,71881],{"className":71860},[344,345],[40,71862,71864,71878],{"className":71863},[349],[40,71865,71867],{"className":71866,"style":354},[353],[40,71868,71869,71872],{"style":2556},[40,71870],{"className":71871,"style":362},[361],[40,71873,71875],{"className":71874},[366,367,368,369],[40,71876,4203],{"className":71877},[102,335,369],[40,71879,377],{"className":71880},[376],[40,71882,71884],{"className":71883},[349],[40,71885,71887],{"className":71886,"style":384},[353],[40,71888],{},[40,71890,167],{"className":71891},[106],[40,71893,58007],{"className":71894,"style":58079},[102,335],[40,71896,73],{"className":71897},[117],[40,71899],{"className":71900,"style":122},[121],[40,71902,13262],{"className":71903,"style":33300},[102,335],[40,71905,207],{"className":71906},[113],". An overlaid theoretical curve follows",[40,71909,71911],{"className":71910},[136],[40,71912,71914,71986],{"className":71913},[43],[40,71915,71917],{"className":71916},[47],[49,71918,71919],{"xmlns":51,"display":145},[53,71920,71921,71983],{},[56,71922,71923,71929,71931,71933,71935,71937,71981],{},[151,71924,71925,71927],{},[154,71926,59270],{},[154,71928,946],{},[59,71930,167],{"stretchy":61},[154,71932,13262],{},[59,71934,207],{"stretchy":61},[59,71936,162],{},[2045,71938,71939],{},[56,71940,71941,71961,71963,71965,71967,71969,71975],{},[941,71942,71943,71959],{},[56,71944,71945,71947,71957],{},[59,71946,167],{"fence":72},[1038,71948,71949,71951],{},[154,71950,20352],{"mathvariant":1080},[56,71952,71953,71955],{},[64,71954,1044],{},[154,71956,1661],{},[59,71958,207],{"fence":72},[64,71960,1044],{},[59,71962,1053],{},[59,71964,167],{"stretchy":61},[154,71966,13262],{},[59,71968,243],{},[151,71970,71971,71973],{},[154,71972,13262],{},[64,71974,66],{},[941,71976,71977,71979],{},[59,71978,207],{"stretchy":61},[64,71980,1044],{},[154,71982,1123],{"mathvariant":1080},[82,71984,71985],{"encoding":84},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[40,71987,71989,72053],{"className":71988,"ariaHidden":72},[89],[40,71990,71992,71995,72035,72038,72041,72044,72047,72050],{"className":71991},[93],[40,71993],{"className":71994,"style":98},[97],[40,71996,71998,72001],{"className":71997},[102],[40,71999,59270],{"className":72000,"style":59291},[102,335],[40,72002,72004],{"className":72003},[340],[40,72005,72007,72027],{"className":72006},[344,345],[40,72008,72010,72024],{"className":72009},[349],[40,72011,72013],{"className":72012,"style":20224},[353],[40,72014,72015,72018],{"style":59306},[40,72016],{"className":72017,"style":362},[361],[40,72019,72021],{"className":72020},[366,367,368,369],[40,72022,946],{"className":72023,"style":3946},[102,335,369],[40,72025,377],{"className":72026},[376],[40,72028,72030],{"className":72029},[349],[40,72031,72033],{"className":72032,"style":384},[353],[40,72034],{},[40,72036,167],{"className":72037},[106],[40,72039,13262],{"className":72040,"style":33300},[102,335],[40,72042,207],{"className":72043},[113],[40,72045],{"className":72046,"style":390},[121],[40,72048,162],{"className":72049},[394],[40,72051],{"className":72052,"style":390},[121],[40,72054,72056,72059,72302],{"className":72055},[93],[40,72057],{"className":72058,"style":66313},[97],[40,72060,72062],{"className":72061},[102,2218],[40,72063,72065,72294],{"className":72064},[344,345],[40,72066,72068,72291],{"className":72067},[349],[40,72069,72071,72279],{"className":72070,"style":66326},[353],[40,72072,72074,72077],{"className":72073,"style":66330},[2232],[40,72075],{"className":72076,"style":66334},[361],[40,72078,72080,72186,72189,72192,72195,72198,72201,72204,72207,72210,72250],{"className":72079,"style":63558},[102],[40,72081,72083,72163],{"className":72082},[408],[40,72084,72086,72092,72157],{"className":72085},[408],[40,72087,72089],{"className":72088,"style":413},[106,412],[40,72090,167],{"className":72091},[417,368],[40,72093,72095,72098,72154],{"className":72094},[102],[40,72096],{"className":72097},[106,1162],[40,72099,72101],{"className":72100},[1038],[40,72102,72104,72146],{"className":72103},[344,345],[40,72105,72107,72143],{"className":72106},[349],[40,72108,72110,72124,72132],{"className":72109,"style":63326},[353],[40,72111,72112,72115],{"style":2782},[40,72113],{"className":72114,"style":444},[361],[40,72116,72118,72121],{"className":72117},[102],[40,72119,1044],{"className":72120},[102],[40,72122,1661],{"className":72123,"style":336},[102,335],[40,72125,72126,72129],{"style":1193},[40,72127],{"className":72128,"style":444},[361],[40,72130],{"className":72131,"style":1201},[1200],[40,72133,72134,72137],{"style":2802},[40,72135],{"className":72136,"style":444},[361],[40,72138,72140],{"className":72139},[102],[40,72141,20352],{"className":72142},[102],[40,72144,377],{"className":72145},[376],[40,72147,72149],{"className":72148},[349],[40,72150,72152],{"className":72151,"style":2868},[353],[40,72153],{},[40,72155],{"className":72156},[113,1162],[40,72158,72160],{"className":72159,"style":413},[113,412],[40,72161,207],{"className":72162},[417,368],[40,72164,72166],{"className":72165},[340],[40,72167,72169],{"className":72168},[344],[40,72170,72172],{"className":72171},[349],[40,72173,72175],{"className":72174,"style":66433},[353],[40,72176,72177,72180],{"style":66436},[40,72178],{"className":72179,"style":362},[361],[40,72181,72183],{"className":72182},[366,367,368,369],[40,72184,1044],{"className":72185},[102,369],[40,72187],{"className":72188,"style":1246},[121],[40,72190,1053],{"className":72191},[1250],[40,72193],{"className":72194,"style":1246},[121],[40,72196,167],{"className":72197},[106],[40,72199,13262],{"className":72200,"style":33300},[102,335],[40,72202],{"className":72203,"style":1246},[121],[40,72205,243],{"className":72206},[1250],[40,72208],{"className":72209,"style":1246},[121],[40,72211,72213,72216],{"className":72212},[102],[40,72214,13262],{"className":72215,"style":33300},[102,335],[40,72217,72219],{"className":72218},[340],[40,72220,72222,72242],{"className":72221},[344,345],[40,72223,72225,72239],{"className":72224},[349],[40,72226,72228],{"className":72227,"style":7201},[353],[40,72229,72230,72233],{"style":33315},[40,72231],{"className":72232,"style":362},[361],[40,72234,72236],{"className":72235},[366,367,368,369],[40,72237,66],{"className":72238},[102,369],[40,72240,377],{"className":72241},[376],[40,72243,72245],{"className":72244},[349],[40,72246,72248],{"className":72247,"style":384},[353],[40,72249],{},[40,72251,72253,72256],{"className":72252},[113],[40,72254,207],{"className":72255},[113],[40,72257,72259],{"className":72258},[340],[40,72260,72262],{"className":72261},[344],[40,72263,72265],{"className":72264},[349],[40,72266,72268],{"className":72267,"style":63577},[353],[40,72269,72270,72273],{"style":63580},[40,72271],{"className":72272,"style":362},[361],[40,72274,72276],{"className":72275},[366,367,368,369],[40,72277,1044],{"className":72278},[102,369],[40,72280,72281,72284],{"style":66541},[40,72282],{"className":72283,"style":66334},[361],[40,72285,72287],{"className":72286,"style":66548},[2253],[2256,72288,72289],{"xmlns":2258,"width":2259,"height":66551,"viewBox":66552,"preserveAspectRatio":2262},[2264,72290],{"d":66555},[40,72292,377],{"className":72293},[376],[40,72295,72297],{"className":72296},[349],[40,72298,72300],{"className":72299,"style":66565},[353],[40,72301],{},[40,72303,1123],{"className":72304},[102],[28,72306,72308],{"id":72307},"primary-modeling-assumptions","Primary Modeling Assumptions",[12,72310,72311],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[20142,72313,72314,72324],{},[20145,72315,72316],{},[20148,72317,72318,72321],{},[20151,72319,72320],{},"Assumption",[20151,72322,72323],{},"Implication",[20162,72325,72326,72410,72418,72425,72433,72470,72560,72717,72765,72773,72781,72789],{},[20148,72327,72328,72331],{},[20167,72329,72330],{},"Two-level truncation",[20167,72332,72333,72334,916,72372,1123],{},"The simulated model includes only ",[40,72335,72337,72354],{"className":72336},[43],[40,72338,72340],{"className":72339},[47],[49,72341,72342],{"xmlns":51},[53,72343,72344,72352],{},[56,72345,72346,72348,72350],{},[59,72347,62],{"stretchy":61},[154,72349,58190],{},[59,72351,69],{"stretchy":61},[82,72353,58195],{"encoding":84},[40,72355,72357],{"className":72356,"ariaHidden":72},[89],[40,72358,72360,72363,72366,72369],{"className":72359},[93],[40,72361],{"className":72362,"style":98},[97],[40,72364,62],{"className":72365},[106],[40,72367,58190],{"className":72368,"style":336},[102,335],[40,72370,69],{"className":72371},[113],[40,72373,72375,72392],{"className":72374},[43],[40,72376,72378],{"className":72377},[47],[49,72379,72380],{"xmlns":51},[53,72381,72382,72390],{},[56,72383,72384,72386,72388],{},[59,72385,62],{"stretchy":61},[154,72387,4203],{},[59,72389,69],{"stretchy":61},[82,72391,58241],{"encoding":84},[40,72393,72395],{"className":72394,"ariaHidden":72},[89],[40,72396,72398,72401,72404,72407],{"className":72397},[93],[40,72399],{"className":72400,"style":98},[97],[40,72402,62],{"className":72403},[106],[40,72405,4203],{"className":72406},[102,335],[40,72408,69],{"className":72409},[113],[20148,72411,72412,72415],{},[20167,72413,72414],{},"Classical drive",[20167,72416,72417],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[20148,72419,72420,72422],{},[20167,72421,59612],{},[20167,72423,72424],{},"Fast counter-rotating terms are omitted.",[20148,72426,72427,72430],{},[20167,72428,72429],{},"Square pulse",[20167,72431,72432],{},"The drive is assumed to turn on and off instantaneously.",[20148,72434,72435,72438],{},[20167,72436,72437],{},"Uniform drive amplitude",[20167,72439,72440,72441,72469],{},"The parameter ",[40,72442,72444,72457],{"className":72443},[43],[40,72445,72447],{"className":72446},[47],[49,72448,72449],{"xmlns":51},[53,72450,72451,72455],{},[56,72452,72453],{},[154,72454,20352],{"mathvariant":1080},[82,72456,20355],{"encoding":84},[40,72458,72460],{"className":72459,"ariaHidden":72},[89],[40,72461,72463,72466],{"className":72462},[93],[40,72464],{"className":72465,"style":4304},[97],[40,72467,20352],{"className":72468},[102]," remains constant across the frequency sweep.",[20148,72471,72472,72475],{},[20167,72473,72474],{},"Optional damping envelope",[20167,72476,72440,72477,72559],{},[40,72478,72480,72499],{"className":72479},[43],[40,72481,72483],{"className":72482},[47],[49,72484,72485],{"xmlns":51},[53,72486,72487,72497],{},[56,72488,72489],{},[8360,72490,72491,72493,72495],{},[154,72492,7059],{},[64,72494,1044],{},[59,72496,59534],{},[82,72498,59537],{"encoding":84},[40,72500,72502],{"className":72501,"ariaHidden":72},[89],[40,72503,72505,72508],{"className":72504},[93],[40,72506],{"className":72507,"style":59547},[97],[40,72509,72511,72514],{"className":72510},[102],[40,72512,7059],{"className":72513,"style":2541},[102,335],[40,72515,72517],{"className":72516},[340],[40,72518,72520,72551],{"className":72519},[344,345],[40,72521,72523,72548],{"className":72522},[349],[40,72524,72526,72537],{"className":72525,"style":59566},[353],[40,72527,72528,72531],{"style":59569},[40,72529],{"className":72530,"style":362},[361],[40,72532,72534],{"className":72533},[366,367,368,369],[40,72535,1044],{"className":72536},[102,369],[40,72538,72539,72542],{"style":1004},[40,72540],{"className":72541,"style":362},[361],[40,72543,72545],{"className":72544},[366,367,368,369],[40,72546,59534],{"className":72547},[1250,369],[40,72549,377],{"className":72550},[376],[40,72552,72554],{"className":72553},[349],[40,72555,72557],{"className":72556,"style":59599},[353],[40,72558],{}," enters as a simple exponential contrast envelope.",[20148,72561,72562,72636],{},[20167,72563,72564,72565,72635],{},"No explicit ",[40,72566,72568,72586],{"className":72567},[43],[40,72569,72571],{"className":72570},[47],[49,72572,72573],{"xmlns":51},[53,72574,72575,72583],{},[56,72576,72577],{},[151,72578,72579,72581],{},[154,72580,7059],{},[64,72582,78],{},[82,72584,72585],{"encoding":84},"T_1",[40,72587,72589],{"className":72588,"ariaHidden":72},[89],[40,72590,72592,72595],{"className":72591},[93],[40,72593],{"className":72594,"style":3106},[97],[40,72596,72598,72601],{"className":72597},[102],[40,72599,7059],{"className":72600,"style":2541},[102,335],[40,72602,72604],{"className":72603},[340],[40,72605,72607,72627],{"className":72606},[344,345],[40,72608,72610,72624],{"className":72609},[349],[40,72611,72613],{"className":72612,"style":7201},[353],[40,72614,72615,72618],{"style":2556},[40,72616],{"className":72617,"style":362},[361],[40,72619,72621],{"className":72620},[366,367,368,369],[40,72622,78],{"className":72623},[102,369],[40,72625,377],{"className":72626},[376],[40,72628,72630],{"className":72629},[349],[40,72631,72633],{"className":72632,"style":384},[353],[40,72634],{}," relaxation",[20167,72637,72638,72639,72677,72678,72716],{},"Energy decay from ",[40,72640,72642,72659],{"className":72641},[43],[40,72643,72645],{"className":72644},[47],[49,72646,72647],{"xmlns":51},[53,72648,72649,72657],{},[56,72650,72651,72653,72655],{},[59,72652,62],{"stretchy":61},[154,72654,4203],{},[59,72656,69],{"stretchy":61},[82,72658,58241],{"encoding":84},[40,72660,72662],{"className":72661,"ariaHidden":72},[89],[40,72663,72665,72668,72671,72674],{"className":72664},[93],[40,72666],{"className":72667,"style":98},[97],[40,72669,62],{"className":72670},[106],[40,72672,4203],{"className":72673},[102,335],[40,72675,69],{"className":72676},[113]," to ",[40,72679,72681,72698],{"className":72680},[43],[40,72682,72684],{"className":72683},[47],[49,72685,72686],{"xmlns":51},[53,72687,72688,72696],{},[56,72689,72690,72692,72694],{},[59,72691,62],{"stretchy":61},[154,72693,58190],{},[59,72695,69],{"stretchy":61},[82,72697,58195],{"encoding":84},[40,72699,72701],{"className":72700,"ariaHidden":72},[89],[40,72702,72704,72707,72710,72713],{"className":72703},[93],[40,72705],{"className":72706,"style":98},[97],[40,72708,62],{"className":72709},[106],[40,72711,58190],{"className":72712,"style":336},[102,335],[40,72714,69],{"className":72715},[113]," is omitted.",[20148,72718,72719,72722],{},[20167,72720,72721],{},"No higher-state leakage",[20167,72723,72724,72725,72764],{},"Higher levels, including a transmon ",[40,72726,72728,72746],{"className":72727},[43],[40,72729,72731],{"className":72730},[47],[49,72732,72733],{"xmlns":51},[53,72734,72735,72743],{},[56,72736,72737,72739,72741],{},[59,72738,62],{"stretchy":61},[154,72740,13262],{},[59,72742,69],{"stretchy":61},[82,72744,72745],{"encoding":84},"\\lvert f\\rangle",[40,72747,72749],{"className":72748,"ariaHidden":72},[89],[40,72750,72752,72755,72758,72761],{"className":72751},[93],[40,72753],{"className":72754,"style":98},[97],[40,72756,62],{"className":72757},[106],[40,72759,13262],{"className":72760,"style":33300},[102,335],[40,72762,69],{"className":72763},[113]," state, are excluded.",[20148,72766,72767,72770],{},[20167,72768,72769],{},"No AC Stark shift",[20167,72771,72772],{},"Drive-induced transition-frequency shifts are excluded.",[20148,72774,72775,72778],{},[20167,72776,72777],{},"No Bloch-Siegert shift",[20167,72779,72780],{},"Counter-rotating corrections are excluded.",[20148,72782,72783,72786],{},[20167,72784,72785],{},"No experimental readout model",[20167,72787,72788],{},"Assignment errors and finite signal-to-noise effects are omitted.",[20148,72790,72791,72794],{},[20167,72792,72793],{},"No pulse-envelope shaping",[20167,72795,72796],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[12,72798,72799],{},"A hardware-accurate superconducting-qubit simulation requires additional structure, including anharmonic multilevel dynamics, calibrated microwave pulse envelopes, drive phase control, amplitude-dependent frequency shifts, finite thermal populations, leakage channels, and measurement infidelity modeled with density matrix dynamics.",[28,72801,72803],{"id":72802},"notebook-control-knobs","Notebook Control Knobs",[12,72805,72806],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[20142,72808,72809,72821],{},[20145,72810,72811],{},[20148,72812,72813,72816,72819],{},[20151,72814,72815],{},"Parameter",[20151,72817,72818],{"align":20153},"Default",[20151,72820,58167],{},[20162,72822,72823,72907,72960,72975,72990,73005,73019,73034,73048],{},[20148,72824,72825,72830,72834],{},[20167,72826,72827],{},[10088,72828,72829],{},"F0_GHZ",[20167,72831,72832],{"align":20153},[10088,72833,60100],{},[20167,72835,72836,72837,72906],{},"Qubit transition frequency ",[40,72838,72840,72857],{"className":72839},[43],[40,72841,72843],{"className":72842},[47],[49,72844,72845],{"xmlns":51},[53,72846,72847,72855],{},[56,72848,72849],{},[151,72850,72851,72853],{},[154,72852,13262],{},[64,72854,66],{},[82,72856,58537],{"encoding":84},[40,72858,72860],{"className":72859,"ariaHidden":72},[89],[40,72861,72863,72866],{"className":72862},[93],[40,72864],{"className":72865,"style":58114},[97],[40,72867,72869,72872],{"className":72868},[102],[40,72870,13262],{"className":72871,"style":33300},[102,335],[40,72873,72875],{"className":72874},[340],[40,72876,72878,72898],{"className":72877},[344,345],[40,72879,72881,72895],{"className":72880},[349],[40,72882,72884],{"className":72883,"style":7201},[353],[40,72885,72886,72889],{"style":33315},[40,72887],{"className":72888,"style":362},[361],[40,72890,72892],{"className":72891},[366,367,368,369],[40,72893,66],{"className":72894},[102,369],[40,72896,377],{"className":72897},[376],[40,72899,72901],{"className":72900},[349],[40,72902,72904],{"className":72903,"style":384},[353],[40,72905],{}," in GHz",[20148,72908,72909,72914,72919],{},[20167,72910,72911],{},[10088,72912,72913],{},"OMEGA_ONRESONANCE_MHZ",[20167,72915,72916],{"align":20153},[10088,72917,72918],{},"20.0",[20167,72920,72921,72922,72959],{},"On-resonance Rabi rate ",[40,72923,72925,72944],{"className":72924},[43],[40,72926,72928],{"className":72927},[47],[49,72929,72930],{"xmlns":51},[53,72931,72932,72942],{},[56,72933,72934,72936,72938,72940],{},[154,72935,20352],{"mathvariant":1080},[154,72937,11243],{"mathvariant":1080},[64,72939,1044],{},[154,72941,1661],{},[82,72943,59003],{"encoding":84},[40,72945,72947],{"className":72946,"ariaHidden":72},[89],[40,72948,72950,72953,72956],{"className":72949},[93],[40,72951],{"className":72952,"style":98},[97],[40,72954,59016],{"className":72955},[102],[40,72957,1661],{"className":72958,"style":336},[102,335]," in MHz",[20148,72961,72962,72967,72972],{},[20167,72963,72964],{},[10088,72965,72966],{},"FREQ_MIN_GHZ",[20167,72968,72969],{"align":20153},[10088,72970,72971],{},"4.90",[20167,72973,72974],{},"Minimum swept drive frequency in GHz",[20148,72976,72977,72982,72987],{},[20167,72978,72979],{},[10088,72980,72981],{},"FREQ_MAX_GHZ",[20167,72983,72984],{"align":20153},[10088,72985,72986],{},"5.10",[20167,72988,72989],{},"Maximum swept drive frequency in GHz",[20148,72991,72992,72997,73002],{},[20167,72993,72994],{},[10088,72995,72996],{},"N_FREQ",[20167,72998,72999],{"align":20153},[10088,73000,73001],{},"401",[20167,73003,73004],{},"Number of drive-frequency samples",[20148,73006,73007,73012,73016],{},[20167,73008,73009],{},[10088,73010,73011],{},"DUR_MIN_NS",[20167,73013,73014],{"align":20153},[10088,73015,10801],{},[20167,73017,73018],{},"Minimum pulse duration in ns",[20148,73020,73021,73026,73031],{},[20167,73022,73023],{},[10088,73024,73025],{},"DUR_MAX_NS",[20167,73027,73028],{"align":20153},[10088,73029,73030],{},"200.0",[20167,73032,73033],{},"Maximum pulse duration in ns",[20148,73035,73036,73041,73045],{},[20167,73037,73038],{},[10088,73039,73040],{},"N_DUR",[20167,73042,73043],{"align":20153},[10088,73044,73001],{},[20167,73046,73047],{},"Number of pulse-duration samples",[20148,73049,73050,73055,73059],{},[20167,73051,73052],{},[10088,73053,73054],{},"T2STAR_NS",[20167,73056,73057],{"align":20153},[10088,73058,21989],{},[20167,73060,73061,73062,73065],{},"Optional damping time in ns, with ",[10088,73063,73064],{},"None"," disabling damping",[12,73067,73068],{},"Additional cross-section plotting cells expose localized slice controls.",[20142,73070,73071,73081],{},[20145,73072,73073],{},[20148,73074,73075,73077,73079],{},[20151,73076,72815],{},[20151,73078,72818],{"align":20153},[20151,73080,58167],{},[20162,73082,73083,73182],{},[20148,73084,73085,73090,73094],{},[20167,73086,73087],{},[10088,73088,73089],{},"F_CROSS_GHZ",[20167,73091,73092],{"align":20153},[10088,73093,60100],{},[20167,73095,73096,73097],{},"Drive-frequency slice used to plot ",[40,73098,73100,73124],{"className":73099},[43],[40,73101,73103],{"className":73102},[47],[49,73104,73105],{"xmlns":51},[53,73106,73107,73121],{},[56,73108,73109,73115,73117,73119],{},[151,73110,73111,73113],{},[154,73112,57996],{},[154,73114,4203],{},[59,73116,167],{"stretchy":61},[154,73118,58007],{},[59,73120,207],{"stretchy":61},[82,73122,73123],{"encoding":84},"P_e(\\tau)",[40,73125,73127],{"className":73126,"ariaHidden":72},[89],[40,73128,73130,73133,73173,73176,73179],{"className":73129},[93],[40,73131],{"className":73132,"style":98},[97],[40,73134,73136,73139],{"className":73135},[102],[40,73137,57996],{"className":73138,"style":2541},[102,335],[40,73140,73142],{"className":73141},[340],[40,73143,73145,73165],{"className":73144},[344,345],[40,73146,73148,73162],{"className":73147},[349],[40,73149,73151],{"className":73150,"style":354},[353],[40,73152,73153,73156],{"style":2556},[40,73154],{"className":73155,"style":362},[361],[40,73157,73159],{"className":73158},[366,367,368,369],[40,73160,4203],{"className":73161},[102,335,369],[40,73163,377],{"className":73164},[376],[40,73166,73168],{"className":73167},[349],[40,73169,73171],{"className":73170,"style":384},[353],[40,73172],{},[40,73174,167],{"className":73175},[106],[40,73177,58007],{"className":73178,"style":58079},[102,335],[40,73180,207],{"className":73181},[113],[20148,73183,73184,73189,73194],{},[20167,73185,73186],{},[10088,73187,73188],{},"T_CROSS_NS",[20167,73190,73191],{"align":20153},[10088,73192,73193],{},"100.0",[20167,73195,73196,73197],{},"Pulse-duration slice used to plot ",[40,73198,73200,73224],{"className":73199},[43],[40,73201,73203],{"className":73202},[47],[49,73204,73205],{"xmlns":51},[53,73206,73207,73221],{},[56,73208,73209,73215,73217,73219],{},[151,73210,73211,73213],{},[154,73212,57996],{},[154,73214,4203],{},[59,73216,167],{"stretchy":61},[154,73218,13262],{},[59,73220,207],{"stretchy":61},[82,73222,73223],{"encoding":84},"P_e(f)",[40,73225,73227],{"className":73226,"ariaHidden":72},[89],[40,73228,73230,73233,73273,73276,73279],{"className":73229},[93],[40,73231],{"className":73232,"style":98},[97],[40,73234,73236,73239],{"className":73235},[102],[40,73237,57996],{"className":73238,"style":2541},[102,335],[40,73240,73242],{"className":73241},[340],[40,73243,73245,73265],{"className":73244},[344,345],[40,73246,73248,73262],{"className":73247},[349],[40,73249,73251],{"className":73250,"style":354},[353],[40,73252,73253,73256],{"style":2556},[40,73254],{"className":73255,"style":362},[361],[40,73257,73259],{"className":73258},[366,367,368,369],[40,73260,4203],{"className":73261},[102,335,369],[40,73263,377],{"className":73264},[376],[40,73266,73268],{"className":73267},[349],[40,73269,73271],{"className":73270,"style":384},[353],[40,73272],{},[40,73274,167],{"className":73275},[106],[40,73277,13262],{"className":73278,"style":33300},[102,335],[40,73280,207],{"className":73281},[113],[12,73283,73284],{},"Separate FFT processing controls support spectral analysis.",[20142,73286,73287,73297],{},[20145,73288,73289],{},[20148,73290,73291,73293,73295],{},[20151,73292,72815],{},[20151,73294,72818],{"align":20153},[20151,73296,58167],{},[20162,73298,73299,73313,73328,73342,73357,73372],{},[20148,73300,73301,73306,73310],{},[20167,73302,73303],{},[10088,73304,73305],{},"FFT_PAD",[20167,73307,73308],{"align":20153},[10088,73309,23482],{},[20167,73311,73312],{},"Zero-padding factor used in the Fourier transform",[20148,73314,73315,73320,73325],{},[20167,73316,73317],{},[10088,73318,73319],{},"WINDOW",[20167,73321,73322],{"align":20153},[10088,73323,73324],{},"\"hann\"",[20167,73326,73327],{},"Window function applied along the pulse-duration axis",[20148,73329,73330,73335,73339],{},[20167,73331,73332],{},[10088,73333,73334],{},"DETREND_MEAN",[20167,73336,73337],{"align":20153},[10088,73338,13091],{},[20167,73340,73341],{},"Flag that subtracts the mean before the FFT",[20148,73343,73344,73349,73354],{},[20167,73345,73346],{},[10088,73347,73348],{},"AMP_MODE",[20167,73350,73351],{"align":20153},[10088,73352,73353],{},"\"magnitude\"",[20167,73355,73356],{},"Display mode for magnitude, power, or dB-scaled amplitude",[20148,73358,73359,73364,73369],{},[20167,73360,73361],{},[10088,73362,73363],{},"NORM_MODE",[20167,73365,73366],{"align":20153},[10088,73367,73368],{},"\"global\"",[20167,73370,73371],{},"Normalization mode applied globally or per drive-frequency column",[20148,73373,73374,73379,73384],{},[20167,73375,73376],{},[10088,73377,73378],{},"FREQ_UNITS",[20167,73380,73381],{"align":20153},[10088,73382,73383],{},"\"MHz\"",[20167,73385,73386],{},"Frequency units used on the Fourier-frequency axis",[12,73388,73389],{},"Three-dimensional visualization cells add rendering controls for upsampling, colormaps, face-count limits, and viewing angles. These settings change the appearance of the rendered figures while preserving the analytic probability model.",[10082,73391,73393],{"className":10269,"code":73392,"language":10271,"meta":10090,"style":10090},"# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n",[10088,73394,73395,73400,73407],{"__ignoreMap":10090},[40,73396,73397],{"class":10276,"line":10277},[40,73398,73399],{"class":10280},"# @title Install dependencies\n",[40,73401,73402,73404],{"class":10276,"line":10284},[40,73403,10375],{"class":10352},[40,73405,73406],{"class":10363}," sys\n",[40,73408,73409,73412,73415,73417],{"class":10276,"line":10291},[40,73410,73411],{"class":10732},"%",[40,73413,73414],{"class":10363},"pip ",[40,73416,11056],{"class":10732},[40,73418,73419],{"class":10363},"q install pyvista pythreejs trame\n",[10082,73421,73423],{"className":10269,"code":73422,"language":10271,"meta":10090,"style":10090},"# @title Controls & imports\nimport math\nfrom typing import Optional, Tuple\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\n\ntry:\n    import pyvista as pv  # optional; only for 3D interactive\n    PV_AVAILABLE = True\nexcept Exception:\n    PV_AVAILABLE = False\n\nplt.rcParams['figure.dpi'] = 250\nplt.rcParams['axes.grid'] = False\n\n# ---------- Control knobs ----------\nF0_GHZ: float = 5.000        # Qubit transition freq f0 [GHz]\nOMEGA_ONRESONANCE_MHZ: float = 20.0  # Ω\u002F2π on-resonance [MHz]\nFREQ_MIN_GHZ: float = 4.90   # Sweep min f [GHz]\nFREQ_MAX_GHZ: float = 5.10   # Sweep max f [GHz]\nN_FREQ: int = 401            # # of frequency points\nDUR_MIN_NS: float = 0.0      # Sweep min duration τ [ns]\nDUR_MAX_NS: float = 200.0    # Sweep max duration τ [ns]\nN_DUR: int = 401             # # of duration points\nT2STAR_NS: Optional[float] = 500.0  # None to disable exp(-τ\u002FT2*)\n\nprint('Controls loaded.')\n",[10088,73424,73425,73430,73436,73447,73457,73467,73479,73483,73490,73506,73516,73524,73533,73537,73550,73563,73567,73572,73588,73603,73619,73635,73651,73666,73682,73697,73716,73720],{"__ignoreMap":10090},[40,73426,73427],{"class":10276,"line":10277},[40,73428,73429],{"class":10280},"# @title Controls & imports\n",[40,73431,73432,73434],{"class":10276,"line":10284},[40,73433,10375],{"class":10352},[40,73435,21798],{"class":10363},[40,73437,73438,73440,73442,73444],{"class":10276,"line":10291},[40,73439,10353],{"class":10352},[40,73441,10554],{"class":10363},[40,73443,10375],{"class":10352},[40,73445,73446],{"class":10363}," Optional, Tuple\n",[40,73448,73449,73451,73453,73455],{"class":10276,"line":10297},[40,73450,10375],{"class":10352},[40,73452,10378],{"class":10363},[40,73454,10381],{"class":10352},[40,73456,10384],{"class":10363},[40,73458,73459,73461,73463,73465],{"class":10276,"line":10304},[40,73460,10375],{"class":10352},[40,73462,10405],{"class":10363},[40,73464,10381],{"class":10352},[40,73466,10410],{"class":10363},[40,73468,73469,73471,73474,73476],{"class":10276,"line":10310},[40,73470,10375],{"class":10352},[40,73472,73473],{"class":10363}," pandas ",[40,73475,10381],{"class":10352},[40,73477,73478],{"class":10363}," pd\n",[40,73480,73481],{"class":10276,"line":10316},[40,73482,10301],{"emptyLinePlaceholder":10300},[40,73484,73485,73488],{"class":10276,"line":10322},[40,73486,73487],{"class":10352},"try",[40,73489,10639],{"class":10363},[40,73491,73492,73495,73498,73500,73503],{"class":10276,"line":10327},[40,73493,73494],{"class":10352},"    import",[40,73496,73497],{"class":10363}," pyvista ",[40,73499,10381],{"class":10352},[40,73501,73502],{"class":10363}," pv  ",[40,73504,73505],{"class":10280},"# optional; only for 3D interactive\n",[40,73507,73508,73511,73513],{"class":10276,"line":10333},[40,73509,73510],{"class":10446},"    PV_AVAILABLE",[40,73512,10772],{"class":10732},[40,73514,73515],{"class":10446}," True\n",[40,73517,73518,73521],{"class":10276,"line":10339},[40,73519,73520],{"class":10352},"except",[40,73522,73523],{"class":10363}," Exception:\n",[40,73525,73526,73528,73530],{"class":10276,"line":10344},[40,73527,73510],{"class":10446},[40,73529,10772],{"class":10732},[40,73531,73532],{"class":10446}," False\n",[40,73534,73535],{"class":10276,"line":10349},[40,73536,10301],{"emptyLinePlaceholder":10300},[40,73538,73539,73541,73543,73545,73547],{"class":10276,"line":10367},[40,73540,27384],{"class":10363},[40,73542,27387],{"class":10287},[40,73544,10741],{"class":10363},[40,73546,162],{"class":10732},[40,73548,73549],{"class":10446}," 250\n",[40,73551,73552,73554,73557,73559,73561],{"class":10276,"line":10372},[40,73553,27384],{"class":10363},[40,73555,73556],{"class":10287},"'axes.grid'",[40,73558,10741],{"class":10363},[40,73560,162],{"class":10732},[40,73562,73532],{"class":10446},[40,73564,73565],{"class":10276,"line":10387},[40,73566,10301],{"emptyLinePlaceholder":10300},[40,73568,73569],{"class":10276,"line":10400},[40,73570,73571],{"class":10280},"# ---------- Control knobs ----------\n",[40,73573,73574,73576,73578,73580,73582,73585],{"class":10276,"line":10413},[40,73575,72829],{"class":10446},[40,73577,10443],{"class":10363},[40,73579,10784],{"class":10732},[40,73581,10772],{"class":10732},[40,73583,73584],{"class":10446}," 5.000",[40,73586,73587],{"class":10280},"        # Qubit transition freq f0 [GHz]\n",[40,73589,73590,73592,73594,73596,73598,73600],{"class":10276,"line":10418},[40,73591,72913],{"class":10446},[40,73593,10443],{"class":10363},[40,73595,10784],{"class":10732},[40,73597,10772],{"class":10732},[40,73599,25770],{"class":10446},[40,73601,73602],{"class":10280},"  # Ω\u002F2π on-resonance [MHz]\n",[40,73604,73605,73607,73609,73611,73613,73616],{"class":10276,"line":10424},[40,73606,72966],{"class":10446},[40,73608,10443],{"class":10363},[40,73610,10784],{"class":10732},[40,73612,10772],{"class":10732},[40,73614,73615],{"class":10446}," 4.90",[40,73617,73618],{"class":10280},"   # Sweep min f [GHz]\n",[40,73620,73621,73623,73625,73627,73629,73632],{"class":10276,"line":10437},[40,73622,72981],{"class":10446},[40,73624,10443],{"class":10363},[40,73626,10784],{"class":10732},[40,73628,10772],{"class":10732},[40,73630,73631],{"class":10446}," 5.10",[40,73633,73634],{"class":10280},"   # Sweep max f [GHz]\n",[40,73636,73637,73639,73641,73643,73645,73648],{"class":10276,"line":10456},[40,73638,72996],{"class":10446},[40,73640,10443],{"class":10363},[40,73642,10821],{"class":10732},[40,73644,10772],{"class":10732},[40,73646,73647],{"class":10446}," 401",[40,73649,73650],{"class":10280},"            # # of frequency points\n",[40,73652,73653,73655,73657,73659,73661,73663],{"class":10276,"line":10472},[40,73654,73011],{"class":10446},[40,73656,10443],{"class":10363},[40,73658,10784],{"class":10732},[40,73660,10772],{"class":10732},[40,73662,22435],{"class":10446},[40,73664,73665],{"class":10280},"      # Sweep min duration τ [ns]\n",[40,73667,73668,73670,73672,73674,73676,73679],{"class":10276,"line":10496},[40,73669,73025],{"class":10446},[40,73671,10443],{"class":10363},[40,73673,10784],{"class":10732},[40,73675,10772],{"class":10732},[40,73677,73678],{"class":10446}," 200.0",[40,73680,73681],{"class":10280},"    # Sweep max duration τ [ns]\n",[40,73683,73684,73686,73688,73690,73692,73694],{"class":10276,"line":10510},[40,73685,73040],{"class":10446},[40,73687,10443],{"class":10363},[40,73689,10821],{"class":10732},[40,73691,10772],{"class":10732},[40,73693,73647],{"class":10446},[40,73695,73696],{"class":10280},"             # # of duration points\n",[40,73698,73699,73701,73704,73706,73708,73710,73713],{"class":10276,"line":10696},[40,73700,73054],{"class":10446},[40,73702,73703],{"class":10363},": Optional[",[40,73705,10784],{"class":10732},[40,73707,10741],{"class":10363},[40,73709,162],{"class":10732},[40,73711,73712],{"class":10446}," 500.0",[40,73714,73715],{"class":10280},"  # None to disable exp(-τ\u002FT2*)\n",[40,73717,73718],{"class":10276,"line":10702},[40,73719,10301],{"emptyLinePlaceholder":10300},[40,73721,73722,73724,73726,73729],{"class":10276,"line":10708},[40,73723,14993],{"class":10732},[40,73725,167],{"class":10363},[40,73727,73728],{"class":10287},"'Controls loaded.'",[40,73730,10761],{"class":10363},[10082,73732,73735],{"className":73733,"code":73734,"language":2932,"meta":10090},[10085],"Controls loaded.\n",[10088,73736,73734],{"__ignoreMap":10090},[10082,73738,73740],{"className":10269,"code":73739,"language":10271,"meta":10090,"style":10090},"def rabi_probability(\n    f_ghz: np.ndarray,\n    tau_ns: np.ndarray,\n    f0_ghz: float,\n    omega_onres_mhz: float,\n    t2star_ns: Optional[float] = None,\n) -> np.ndarray:\n    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n\n    Args:\n        f_ghz: 1D array of drive frequencies [GHz].\n        tau_ns: 1D array of pulse durations [ns].\n        f0_ghz: Qubit frequency on resonance [GHz].\n        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n\n    Returns:\n        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n    \"\"\"\n    two_pi = 2.0 * math.pi\n    f = f_ghz * 1e9\n    f0 = f0_ghz * 1e9\n    tau = tau_ns * 1e-9\n    omega = two_pi * omega_onres_mhz * 1e6  # rad\u002Fs\n\n    F, TAU = np.meshgrid(f, tau, indexing='xy')\n    Delta = two_pi * (F - f0)               # rad\u002Fs\n    Omega_R = np.sqrt(omega**2 + Delta**2)\n\n    with np.errstate(divide='ignore', invalid='ignore'):\n        amp = (omega \u002F Omega_R) ** 2\n        phase = 0.5 * Omega_R * TAU\n        Pe = amp * np.sin(phase) ** 2\n\n    if t2star_ns is not None and t2star_ns > 0.0:\n        decay = np.exp(-TAU \u002F (t2star_ns * 1e-9))\n        Pe = Pe * decay\n\n    return np.clip(Pe, 0.0, 1.0)\n",[10088,73741,73742,73751,73758,73765,73776,73787,73805,73809,73814,73818,73823,73828,73833,73838,73843,73848,73852,73857,73862,73866,73880,73895,73909,73923,73945,73949,73975,73997,74026,74030,74061,74081,74100,74123,74127,74152,74180,74194,74198],{"__ignoreMap":10090},[40,73743,73744,73746,73749],{"class":10276,"line":10277},[40,73745,10965],{"class":10352},[40,73747,73748],{"class":10430}," rabi_probability",[40,73750,12527],{"class":10363},[40,73752,73753,73756],{"class":10276,"line":10284},[40,73754,73755],{"class":10973},"    f_ghz",[40,73757,12841],{"class":10363},[40,73759,73760,73763],{"class":10276,"line":10291},[40,73761,73762],{"class":10973},"    tau_ns",[40,73764,12841],{"class":10363},[40,73766,73767,73770,73772,73774],{"class":10276,"line":10297},[40,73768,73769],{"class":10973},"    f0_ghz",[40,73771,10443],{"class":10363},[40,73773,10784],{"class":10732},[40,73775,10507],{"class":10363},[40,73777,73778,73781,73783,73785],{"class":10276,"line":10304},[40,73779,73780],{"class":10973},"    omega_onres_mhz",[40,73782,10443],{"class":10363},[40,73784,10784],{"class":10732},[40,73786,10507],{"class":10363},[40,73788,73789,73792,73794,73796,73798,73800,73803],{"class":10276,"line":10310},[40,73790,73791],{"class":10973},"    t2star_ns",[40,73793,73703],{"class":10363},[40,73795,10784],{"class":10732},[40,73797,10741],{"class":10363},[40,73799,162],{"class":10732},[40,73801,73802],{"class":10446}," None",[40,73804,10507],{"class":10363},[40,73806,73807],{"class":10276,"line":10316},[40,73808,11291],{"class":10363},[40,73810,73811],{"class":10276,"line":10322},[40,73812,73813],{"class":10287},"    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[40,73815,73816],{"class":10276,"line":10327},[40,73817,10301],{"emptyLinePlaceholder":10300},[40,73819,73820],{"class":10276,"line":10333},[40,73821,73822],{"class":10287},"    Args:\n",[40,73824,73825],{"class":10276,"line":10339},[40,73826,73827],{"class":10287},"        f_ghz: 1D array of drive frequencies [GHz].\n",[40,73829,73830],{"class":10276,"line":10344},[40,73831,73832],{"class":10287},"        tau_ns: 1D array of pulse durations [ns].\n",[40,73834,73835],{"class":10276,"line":10349},[40,73836,73837],{"class":10287},"        f0_ghz: Qubit frequency on resonance [GHz].\n",[40,73839,73840],{"class":10276,"line":10367},[40,73841,73842],{"class":10287},"        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[40,73844,73845],{"class":10276,"line":10372},[40,73846,73847],{"class":10287},"        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[40,73849,73850],{"class":10276,"line":10387},[40,73851,10301],{"emptyLinePlaceholder":10300},[40,73853,73854],{"class":10276,"line":10400},[40,73855,73856],{"class":10287},"    Returns:\n",[40,73858,73859],{"class":10276,"line":10413},[40,73860,73861],{"class":10287},"        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[40,73863,73864],{"class":10276,"line":10418},[40,73865,10723],{"class":10287},[40,73867,73868,73871,73873,73875,73877],{"class":10276,"line":10424},[40,73869,73870],{"class":10363},"    two_pi ",[40,73872,162],{"class":10732},[40,73874,12060],{"class":10446},[40,73876,11881],{"class":10732},[40,73878,73879],{"class":10363}," math.pi\n",[40,73881,73882,73885,73887,73890,73892],{"class":10276,"line":10437},[40,73883,73884],{"class":10363},"    f ",[40,73886,162],{"class":10732},[40,73888,73889],{"class":10363}," f_ghz ",[40,73891,11752],{"class":10732},[40,73893,73894],{"class":10446}," 1e9\n",[40,73896,73897,73900,73902,73905,73907],{"class":10276,"line":10456},[40,73898,73899],{"class":10363},"    f0 ",[40,73901,162],{"class":10732},[40,73903,73904],{"class":10363}," f0_ghz ",[40,73906,11752],{"class":10732},[40,73908,73894],{"class":10446},[40,73910,73911,73914,73916,73919,73921],{"class":10276,"line":10472},[40,73912,73913],{"class":10363},"    tau ",[40,73915,162],{"class":10732},[40,73917,73918],{"class":10363}," tau_ns ",[40,73920,11752],{"class":10732},[40,73922,25987],{"class":10446},[40,73924,73925,73928,73930,73933,73935,73938,73940,73942],{"class":10276,"line":10496},[40,73926,73927],{"class":10363},"    omega ",[40,73929,162],{"class":10732},[40,73931,73932],{"class":10363}," two_pi ",[40,73934,11752],{"class":10732},[40,73936,73937],{"class":10363}," omega_onres_mhz ",[40,73939,11752],{"class":10732},[40,73941,26027],{"class":10446},[40,73943,73944],{"class":10280},"  # rad\u002Fs\n",[40,73946,73947],{"class":10276,"line":10510},[40,73948,10301],{"emptyLinePlaceholder":10300},[40,73950,73951,73954,73957,73959,73961,73963,73966,73968,73970,73973],{"class":10276,"line":10696},[40,73952,73953],{"class":10363},"    F, ",[40,73955,73956],{"class":10446},"TAU",[40,73958,10772],{"class":10732},[40,73960,11156],{"class":10363},[40,73962,11897],{"class":10430},[40,73964,73965],{"class":10363},"(f, tau, ",[40,73967,11904],{"class":11903},[40,73969,162],{"class":10732},[40,73971,73972],{"class":10287},"'xy'",[40,73974,10761],{"class":10363},[40,73976,73977,73980,73982,73984,73986,73989,73991,73994],{"class":10276,"line":10702},[40,73978,73979],{"class":10363},"    Delta ",[40,73981,162],{"class":10732},[40,73983,73932],{"class":10363},[40,73985,11752],{"class":10732},[40,73987,73988],{"class":10363}," (F ",[40,73990,11056],{"class":10732},[40,73992,73993],{"class":10363}," f0)               ",[40,73995,73996],{"class":10280},"# rad\u002Fs\n",[40,73998,73999,74002,74004,74006,74008,74011,74013,74015,74017,74020,74022,74024],{"class":10276,"line":10708},[40,74000,74001],{"class":10363},"    Omega_R ",[40,74003,162],{"class":10732},[40,74005,11156],{"class":10363},[40,74007,2218],{"class":10430},[40,74009,74010],{"class":10363},"(omega",[40,74012,22571],{"class":10732},[40,74014,1044],{"class":10446},[40,74016,12042],{"class":10732},[40,74018,74019],{"class":10363}," Delta",[40,74021,22571],{"class":10732},[40,74023,1044],{"class":10446},[40,74025,10761],{"class":10363},[40,74027,74028],{"class":10276,"line":10714},[40,74029,10301],{"emptyLinePlaceholder":10300},[40,74031,74032,74035,74037,74040,74042,74045,74047,74050,74052,74055,74057,74059],{"class":10276,"line":10720},[40,74033,74034],{"class":10352},"    with",[40,74036,11156],{"class":10363},[40,74038,74039],{"class":10430},"errstate",[40,74041,167],{"class":10363},[40,74043,74044],{"class":11903},"divide",[40,74046,162],{"class":10732},[40,74048,74049],{"class":10287},"'ignore'",[40,74051,10484],{"class":10363},[40,74053,74054],{"class":11903},"invalid",[40,74056,162],{"class":10732},[40,74058,74049],{"class":10287},[40,74060,13353],{"class":10363},[40,74062,74063,74066,74068,74071,74073,74076,74078],{"class":10276,"line":10726},[40,74064,74065],{"class":10363},"        amp ",[40,74067,162],{"class":10732},[40,74069,74070],{"class":10363}," (omega ",[40,74072,11243],{"class":10732},[40,74074,74075],{"class":10363}," Omega_R) ",[40,74077,22571],{"class":10732},[40,74079,74080],{"class":10446}," 2\n",[40,74082,74083,74086,74088,74090,74092,74095,74097],{"class":10276,"line":10764},[40,74084,74085],{"class":10363},"        phase ",[40,74087,162],{"class":10732},[40,74089,12039],{"class":10446},[40,74091,11881],{"class":10732},[40,74093,74094],{"class":10363}," Omega_R ",[40,74096,11752],{"class":10732},[40,74098,74099],{"class":10446}," TAU\n",[40,74101,74102,74105,74107,74110,74112,74114,74116,74119,74121],{"class":10276,"line":10778},[40,74103,74104],{"class":10363},"        Pe ",[40,74106,162],{"class":10732},[40,74108,74109],{"class":10363}," amp ",[40,74111,11752],{"class":10732},[40,74113,11156],{"class":10363},[40,74115,1813],{"class":10430},[40,74117,74118],{"class":10363},"(phase) ",[40,74120,22571],{"class":10732},[40,74122,74080],{"class":10446},[40,74124,74125],{"class":10276,"line":10815},[40,74126,10301],{"emptyLinePlaceholder":10300},[40,74128,74129,74131,74134,74137,74140,74142,74144,74146,74148,74150],{"class":10276,"line":10829},[40,74130,11323],{"class":10352},[40,74132,74133],{"class":10363}," t2star_ns ",[40,74135,74136],{"class":10352},"is",[40,74138,74139],{"class":10352}," not",[40,74141,73802],{"class":10446},[40,74143,22438],{"class":10352},[40,74145,74133],{"class":10363},[40,74147,22432],{"class":10732},[40,74149,22435],{"class":10446},[40,74151,10639],{"class":10363},[40,74153,74154,74157,74159,74161,74163,74165,74167,74169,74171,74174,74176,74178],{"class":10276,"line":10842},[40,74155,74156],{"class":10363},"        decay ",[40,74158,162],{"class":10732},[40,74160,11156],{"class":10363},[40,74162,5268],{"class":10430},[40,74164,167],{"class":10363},[40,74166,11056],{"class":10732},[40,74168,73956],{"class":10446},[40,74170,22539],{"class":10732},[40,74172,74173],{"class":10363}," (t2star_ns ",[40,74175,11752],{"class":10732},[40,74177,38902],{"class":10446},[40,74179,13612],{"class":10363},[40,74181,74182,74184,74186,74189,74191],{"class":10276,"line":10855},[40,74183,74104],{"class":10363},[40,74185,162],{"class":10732},[40,74187,74188],{"class":10363}," Pe ",[40,74190,11752],{"class":10732},[40,74192,74193],{"class":10363}," decay\n",[40,74195,74196],{"class":10276,"line":10868},[40,74197,10301],{"emptyLinePlaceholder":10300},[40,74199,74200,74202,74204,74206,74209,74211,74213,74215],{"class":10276,"line":10881},[40,74201,11201],{"class":10352},[40,74203,11156],{"class":10363},[40,74205,24368],{"class":10430},[40,74207,74208],{"class":10363},"(Pe, ",[40,74210,10801],{"class":10446},[40,74212,10484],{"class":10363},[40,74214,10810],{"class":10446},[40,74216,10761],{"class":10363},[10082,74218,74220],{"className":10269,"code":74219,"language":10271,"meta":10090,"style":10090},"freq_ghz = np.linspace(FREQ_MIN_GHZ, FREQ_MAX_GHZ, N_FREQ)\ndur_ns = np.linspace(DUR_MIN_NS, DUR_MAX_NS, N_DUR)\nPe = rabi_probability(freq_ghz, dur_ns, F0_GHZ, OMEGA_ONRESONANCE_MHZ, T2STAR_NS)\nprint('Grid shape (τ × f):', Pe.shape)\n",[10088,74221,74222,74247,74272,74296],{"__ignoreMap":10090},[40,74223,74224,74227,74229,74231,74233,74235,74237,74239,74241,74243,74245],{"class":10276,"line":10277},[40,74225,74226],{"class":10363},"freq_ghz ",[40,74228,162],{"class":10732},[40,74230,11156],{"class":10363},[40,74232,11850],{"class":10430},[40,74234,167],{"class":10363},[40,74236,72966],{"class":10446},[40,74238,10484],{"class":10363},[40,74240,72981],{"class":10446},[40,74242,10484],{"class":10363},[40,74244,72996],{"class":10446},[40,74246,10761],{"class":10363},[40,74248,74249,74252,74254,74256,74258,74260,74262,74264,74266,74268,74270],{"class":10276,"line":10284},[40,74250,74251],{"class":10363},"dur_ns ",[40,74253,162],{"class":10732},[40,74255,11156],{"class":10363},[40,74257,11850],{"class":10430},[40,74259,167],{"class":10363},[40,74261,73011],{"class":10446},[40,74263,10484],{"class":10363},[40,74265,73025],{"class":10446},[40,74267,10484],{"class":10363},[40,74269,73040],{"class":10446},[40,74271,10761],{"class":10363},[40,74273,74274,74277,74279,74281,74284,74286,74288,74290,74292,74294],{"class":10276,"line":10291},[40,74275,74276],{"class":10363},"Pe ",[40,74278,162],{"class":10732},[40,74280,73748],{"class":10430},[40,74282,74283],{"class":10363},"(freq_ghz, dur_ns, ",[40,74285,72829],{"class":10446},[40,74287,10484],{"class":10363},[40,74289,72913],{"class":10446},[40,74291,10484],{"class":10363},[40,74293,73054],{"class":10446},[40,74295,10761],{"class":10363},[40,74297,74298,74300,74302,74305],{"class":10276,"line":10297},[40,74299,14993],{"class":10732},[40,74301,167],{"class":10363},[40,74303,74304],{"class":10287},"'Grid shape (τ × f):'",[40,74306,74307],{"class":10363},", Pe.shape)\n",[10082,74309,74312],{"className":74310,"code":74311,"language":2932,"meta":10090},[10085],"Grid shape (τ × f): (401, 401)\n",[10088,74313,74311],{"__ignoreMap":10090},[10082,74315,74317],{"className":10269,"code":74316,"language":10271,"meta":10090,"style":10090},"controls_table = pd.DataFrame(\n    [\n        ['f0 (GHz)', F0_GHZ],\n        ['Ω\u002F2π on-res (MHz)', OMEGA_ONRESONANCE_MHZ],\n        ['f sweep min (GHz)', FREQ_MIN_GHZ],\n        ['f sweep max (GHz)', FREQ_MAX_GHZ],\n        ['N_f', N_FREQ],\n        ['τ sweep min (ns)', DUR_MIN_NS],\n        ['τ sweep max (ns)', DUR_MAX_NS],\n        ['N_τ', N_DUR],\n        ['T2* (ns; None=off)', T2STAR_NS],\n    ],\n    columns=['Parameter', 'Value']\n)\ncontrols_table\n",[10088,74318,74319,74334,74339,74353,74366,74379,74392,74405,74418,74431,74444,74457,74462,74481,74485],{"__ignoreMap":10090},[40,74320,74321,74324,74326,74329,74332],{"class":10276,"line":10277},[40,74322,74323],{"class":10363},"controls_table ",[40,74325,162],{"class":10732},[40,74327,74328],{"class":10363}," pd.",[40,74330,74331],{"class":10430},"DataFrame",[40,74333,12527],{"class":10363},[40,74335,74336],{"class":10276,"line":10284},[40,74337,74338],{"class":10363},"    [\n",[40,74340,74341,74344,74347,74349,74351],{"class":10276,"line":10291},[40,74342,74343],{"class":10363},"        [",[40,74345,74346],{"class":10287},"'f0 (GHz)'",[40,74348,10484],{"class":10363},[40,74350,72829],{"class":10446},[40,74352,11542],{"class":10363},[40,74354,74355,74357,74360,74362,74364],{"class":10276,"line":10297},[40,74356,74343],{"class":10363},[40,74358,74359],{"class":10287},"'Ω\u002F2π on-res (MHz)'",[40,74361,10484],{"class":10363},[40,74363,72913],{"class":10446},[40,74365,11542],{"class":10363},[40,74367,74368,74370,74373,74375,74377],{"class":10276,"line":10304},[40,74369,74343],{"class":10363},[40,74371,74372],{"class":10287},"'f sweep min (GHz)'",[40,74374,10484],{"class":10363},[40,74376,72966],{"class":10446},[40,74378,11542],{"class":10363},[40,74380,74381,74383,74386,74388,74390],{"class":10276,"line":10310},[40,74382,74343],{"class":10363},[40,74384,74385],{"class":10287},"'f sweep max (GHz)'",[40,74387,10484],{"class":10363},[40,74389,72981],{"class":10446},[40,74391,11542],{"class":10363},[40,74393,74394,74396,74399,74401,74403],{"class":10276,"line":10316},[40,74395,74343],{"class":10363},[40,74397,74398],{"class":10287},"'N_f'",[40,74400,10484],{"class":10363},[40,74402,72996],{"class":10446},[40,74404,11542],{"class":10363},[40,74406,74407,74409,74412,74414,74416],{"class":10276,"line":10322},[40,74408,74343],{"class":10363},[40,74410,74411],{"class":10287},"'τ sweep min (ns)'",[40,74413,10484],{"class":10363},[40,74415,73011],{"class":10446},[40,74417,11542],{"class":10363},[40,74419,74420,74422,74425,74427,74429],{"class":10276,"line":10327},[40,74421,74343],{"class":10363},[40,74423,74424],{"class":10287},"'τ sweep max (ns)'",[40,74426,10484],{"class":10363},[40,74428,73025],{"class":10446},[40,74430,11542],{"class":10363},[40,74432,74433,74435,74438,74440,74442],{"class":10276,"line":10333},[40,74434,74343],{"class":10363},[40,74436,74437],{"class":10287},"'N_τ'",[40,74439,10484],{"class":10363},[40,74441,73040],{"class":10446},[40,74443,11542],{"class":10363},[40,74445,74446,74448,74451,74453,74455],{"class":10276,"line":10339},[40,74447,74343],{"class":10363},[40,74449,74450],{"class":10287},"'T2* (ns; None=off)'",[40,74452,10484],{"class":10363},[40,74454,73054],{"class":10446},[40,74456,11542],{"class":10363},[40,74458,74459],{"class":10276,"line":10344},[40,74460,74461],{"class":10363},"    ],\n",[40,74463,74464,74467,74469,74471,74474,74476,74479],{"class":10276,"line":10349},[40,74465,74466],{"class":11903},"    columns",[40,74468,162],{"class":10732},[40,74470,1654],{"class":10363},[40,74472,74473],{"class":10287},"'Parameter'",[40,74475,10484],{"class":10363},[40,74477,74478],{"class":10287},"'Value'",[40,74480,25466],{"class":10363},[40,74482,74483],{"class":10276,"line":10367},[40,74484,10761],{"class":10363},[40,74486,74487],{"class":10276,"line":10372},[40,74488,74489],{"class":10363},"controls_table\n",[10082,74491,74494],{"className":74492,"code":74493,"language":2932,"meta":10090},[10085],"Parameter  Value\n0            f0 (GHz)    5.0\n1   Ω\u002F2π on-res (MHz)   20.0\n2   f sweep min (GHz)    4.9\n3   f sweep max (GHz)    5.1\n4                 N_f  401.0\n5    τ sweep min (ns)    0.0\n6    τ sweep max (ns)  200.0\n7                 N_τ  401.0\n8  T2* (ns; None=off)  500.0\n",[10088,74495,74493],{"__ignoreMap":10090},[10082,74497,74499],{"className":10269,"code":74498,"language":10271,"meta":10090,"style":10090},"fig, ax = plt.subplots(figsize=(7, 5))\nc = ax.pcolormesh(freq_ghz, dur_ns, Pe, shading='auto')\ncb = fig.colorbar(c, ax=ax, label=r'$P_e$')\nax.set_xlabel('Drive frequency f (GHz)')\nax.set_ylabel('Pulse duration τ (ns)')\nax.set_title('Rabi excited-state probability $P_e(f,\\\\,\\\\tau)$')\nfig.tight_layout()\nplt.show()\n",[10088,74500,74501,74527,74552,74583,74596,74609,74632,74641],{"__ignoreMap":10090},[40,74502,74503,74505,74507,74509,74511,74513,74515,74517,74519,74521,74523,74525],{"class":10276,"line":10277},[40,74504,38079],{"class":10363},[40,74506,162],{"class":10732},[40,74508,13070],{"class":10363},[40,74510,13073],{"class":10430},[40,74512,167],{"class":10363},[40,74514,13078],{"class":11903},[40,74516,162],{"class":10732},[40,74518,167],{"class":10363},[40,74520,21520],{"class":10446},[40,74522,10484],{"class":10363},[40,74524,12566],{"class":10446},[40,74526,13612],{"class":10363},[40,74528,74529,74532,74534,74536,74539,74542,74545,74547,74550],{"class":10276,"line":10284},[40,74530,74531],{"class":10363},"c ",[40,74533,162],{"class":10732},[40,74535,12522],{"class":10363},[40,74537,74538],{"class":10430},"pcolormesh",[40,74540,74541],{"class":10363},"(freq_ghz, dur_ns, Pe, ",[40,74543,74544],{"class":11903},"shading",[40,74546,162],{"class":10732},[40,74548,74549],{"class":10287},"'auto'",[40,74551,10761],{"class":10363},[40,74553,74554,74557,74559,74561,74563,74566,74568,74570,74572,74574,74576,74578,74581],{"class":10276,"line":10291},[40,74555,74556],{"class":10363},"cb ",[40,74558,162],{"class":10732},[40,74560,13170],{"class":10363},[40,74562,13173],{"class":10430},[40,74564,74565],{"class":10363},"(c, ",[40,74567,12429],{"class":11903},[40,74569,162],{"class":10732},[40,74571,13183],{"class":10363},[40,74573,15415],{"class":11903},[40,74575,162],{"class":10732},[40,74577,936],{"class":10352},[40,74579,74580],{"class":10356},"'$P_e$'",[40,74582,10761],{"class":10363},[40,74584,74585,74587,74589,74591,74594],{"class":10276,"line":10297},[40,74586,38114],{"class":10363},[40,74588,13230],{"class":10430},[40,74590,167],{"class":10363},[40,74592,74593],{"class":10287},"'Drive frequency f (GHz)'",[40,74595,10761],{"class":10363},[40,74597,74598,74600,74602,74604,74607],{"class":10276,"line":10304},[40,74599,38114],{"class":10363},[40,74601,13244],{"class":10430},[40,74603,167],{"class":10363},[40,74605,74606],{"class":10287},"'Pulse duration τ (ns)'",[40,74608,10761],{"class":10363},[40,74610,74611,74613,74615,74617,74620,74623,74625,74627,74630],{"class":10276,"line":10310},[40,74612,38114],{"class":10363},[40,74614,30334],{"class":10430},[40,74616,167],{"class":10363},[40,74618,74619],{"class":10287},"'Rabi excited-state probability $P_e(f,",[40,74621,74622],{"class":10732},"\\\\",[40,74624,73],{"class":10287},[40,74626,74622],{"class":10732},[40,74628,74629],{"class":10287},"tau)$'",[40,74631,10761],{"class":10363},[40,74633,74634,74637,74639],{"class":10276,"line":10316},[40,74635,74636],{"class":10363},"fig.",[40,74638,30425],{"class":10430},[40,74640,12174],{"class":10363},[40,74642,74643,74645,74647],{"class":10276,"line":10322},[40,74644,23469],{"class":10363},[40,74646,23686],{"class":10430},[40,74648,12174],{"class":10363},[13997,74650],{"alt":74651,"src":74652},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[10082,74654,74656],{"className":10269,"code":74655,"language":10271,"meta":10090,"style":10090},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n\nimport importlib.util\nimport subprocess\nimport sys\n\n# -------------------------------------------------------------------------\n# Control knobs\n# -------------------------------------------------------------------------\nINSTALL_BLOCH_DEPENDENCIES = True\n\nBLOCH_REQUIRED_PACKAGES = {\n    \"numpy\": \"numpy\",\n    \"matplotlib\": \"matplotlib\",\n    \"scipy\": \"scipy\",\n    \"qutip\": \"qutip\",\n}\n\nBLOCH_F0_GHZ = globals().get(\"F0_GHZ\", 5.000)\nBLOCH_OMEGA_ONRESONANCE_MHZ = globals().get(\n    \"OMEGA_ONRESONANCE_MHZ\",\n    20.0,\n)\n\nBLOCH_RESONANT_DETUNING_MHZ = 0.0\nBLOCH_DETUNED_DETUNING_MHZ = 40.0\n\n# Use None to automatically plot one resonant pi pulse.\nBLOCH_DURATION_NS = None\n\nBLOCH_NUM_TIME_POINTS = 300\nBLOCH_FIGSIZE = (8.8, 7.8)\nBLOCH_DPI = 250\nBLOCH_VIEW = [-60, 25]\n\nBLOCH_SHOW_CONTROL_AXIS_ARROWS = True\nBLOCH_SHOW_ARROW_TIP_LABELS = True\nBLOCH_SHOW_ENDPOINT_MARKERS = True\nBLOCH_SHOW_LEGEND = True\nBLOCH_PRINT_SUMMARY = True\n\nCOLOR_RESONANT_TRAJECTORY = \"tab:blue\"\nCOLOR_DETUNED_TRAJECTORY = \"tab:orange\"\nCOLOR_RESONANT_AXIS = \"tab:green\"\nCOLOR_DETUNED_AXIS = \"tab:red\"\nCOLOR_INITIAL_STATE = \"black\"\n\nTRAJECTORY_LINEWIDTH = 2.4\nCONTROL_AXIS_LINEWIDTH = 2.2\nENDPOINT_MARKER_SIZE = 45\nARROW_LENGTH_RATIO = 0.12\nARROW_LABEL_SCALE = 1.16\n\nLEGEND_LOCATION = \"upper left\"\nLEGEND_BBOX_TO_ANCHOR = (1.02, 1.02)\n\n\ndef package_is_available(package_name: str) -> bool:\n    \"\"\"Return True when a package can be imported.\"\"\"\n    return importlib.util.find_spec(package_name) is not None\n\n\ndef install_missing_packages(\n    required_packages: dict[str, str],\n) -> None:\n    \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n    if not INSTALL_BLOCH_DEPENDENCIES:\n        return\n\n    missing_package_names = [\n        pip_name\n        for import_name, pip_name in required_packages.items()\n        if not package_is_available(import_name)\n    ]\n\n    if not missing_package_names:\n        return\n\n    try:\n        subprocess.run(\n            [sys.executable, \"-m\", \"pip\", \"install\", \"-q\", \"uv\"],\n            check=True,\n        )\n        subprocess.run(\n            [\n                sys.executable,\n                \"-m\",\n                \"uv\",\n                \"pip\",\n                \"install\",\n                \"--system\",\n                \"-q\",\n                *missing_package_names,\n            ],\n            check=True,\n        )\n    except (subprocess.CalledProcessError, FileNotFoundError):\n        subprocess.run(\n            [\n                sys.executable,\n                \"-m\",\n                \"pip\",\n                \"install\",\n                \"-q\",\n                *missing_package_names,\n            ],\n            check=True,\n        )\n\n\ninstall_missing_packages(BLOCH_REQUIRED_PACKAGES)\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport qutip as qt\nfrom matplotlib.lines import Line2D\n\ntry:\n    from IPython import get_ipython\n\n    ipython = get_ipython()\n    if ipython is not None:\n        ipython.run_line_magic(\"matplotlib\", \"inline\")\nexcept ImportError:\n    pass\n\nplt.rcParams.update(\n    {\n        \"figure.dpi\": BLOCH_DPI,\n    }\n)\n\n\ndef get_pi_pulse_duration_ns(omega_onresonance_mhz: float) -> float:\n    \"\"\"Return the resonant pi-pulse duration in nanoseconds.\"\"\"\n    return 1.0e3 \u002F (2.0 * omega_onresonance_mhz)\n\n\ndef solve_rwa_bloch_trajectory(\n    detuning_mhz: float,\n    duration_ns: float,\n    num_time_points: int,\n) -> dict[str, np.ndarray | float]:\n    \"\"\"Solve the RWA two-level trajectory and return Bloch observables.\"\"\"\n    omega_rad_s = 2.0 * np.pi * BLOCH_OMEGA_ONRESONANCE_MHZ * 1.0e6\n    delta_rad_s = 2.0 * np.pi * detuning_mhz * 1.0e6\n    omega_generalized_rad_s = np.hypot(omega_rad_s, delta_rad_s)\n\n    time_s = np.linspace(0.0, duration_ns * 1.0e-9, num_time_points)\n\n    ground_state = qt.basis(2, 0)\n    excited_state = qt.basis(2, 1)\n    excited_projector = excited_state * excited_state.dag()\n\n    hamiltonian = 0.5 * (\n        delta_rad_s * qt.sigmaz()\n        + omega_rad_s * qt.sigmax()\n    )\n\n    result = qt.sesolve(\n        hamiltonian,\n        ground_state,\n        time_s,\n    )\n\n    x_expectation = np.real(np.asarray(qt.expect(qt.sigmax(), result.states)))\n    y_expectation = np.real(np.asarray(qt.expect(qt.sigmay(), result.states)))\n    z_expectation = np.real(np.asarray(qt.expect(qt.sigmaz(), result.states)))\n\n    excited_probability = np.real(\n        np.asarray(qt.expect(excited_projector, result.states))\n    )\n\n    bloch_vectors = np.vstack(\n        [\n            x_expectation,\n            y_expectation,\n            z_expectation,\n        ]\n    )\n\n    control_axis = np.array(\n        [\n            omega_rad_s \u002F omega_generalized_rad_s,\n            0.0,\n            delta_rad_s \u002F omega_generalized_rad_s,\n        ]\n    )\n\n    return {\n        \"time_ns\": time_s * 1.0e9,\n        \"vectors\": bloch_vectors,\n        \"excited_probability\": excited_probability,\n        \"control_axis\": control_axis,\n        \"generalized_rabi_mhz\": (\n            omega_generalized_rad_s \u002F (2.0 * np.pi * 1.0e6)\n        ),\n    }\n\n\ndef render_empty_bloch_sphere(\n    bloch_sphere: qt.Bloch,\n    fallback_figure: plt.Figure,\n):\n    \"\"\"Render an empty QuTiP Bloch sphere and return its Matplotlib axis.\"\"\"\n    if hasattr(bloch_sphere, \"render\"):\n        bloch_sphere.render()\n    elif hasattr(bloch_sphere, \"make_sphere\"):\n        bloch_sphere.make_sphere()\n    else:\n        bloch_sphere.show()\n\n    axis = getattr(bloch_sphere, \"axes\", None)\n\n    if axis is None and fallback_figure.axes:\n        axis = fallback_figure.axes[-1]\n\n    if axis is None:\n        raise RuntimeError(\"Could not access the Matplotlib Bloch axis.\")\n\n    return axis\n\n\ndef draw_bloch_trajectory(\n    axis,\n    vectors: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a Bloch-vector trajectory on an existing 3D axis.\"\"\"\n    axis.plot(\n        vectors[0],\n        vectors[1],\n        vectors[2],\n        color=color,\n        linewidth=TRAJECTORY_LINEWIDTH,\n        label=label,\n    )\n\n\ndef draw_endpoint_marker(\n    axis,\n    vector: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a marker at the final point of a Bloch trajectory.\"\"\"\n    axis.scatter(\n        vector[0],\n        vector[1],\n        vector[2],\n        color=color,\n        s=ENDPOINT_MARKER_SIZE,\n        depthshade=True,\n        label=label,\n    )\n\n\ndef draw_control_axis_arrow(\n    axis,\n    vector: np.ndarray,\n    color: str,\n    label: str,\n) -> None:\n    \"\"\"Draw a normalized effective-control-axis arrow.\"\"\"\n    axis.quiver(\n        0.0,\n        0.0,\n        0.0,\n        vector[0],\n        vector[1],\n        vector[2],\n        color=color,\n        linewidth=CONTROL_AXIS_LINEWIDTH,\n        arrow_length_ratio=ARROW_LENGTH_RATIO,\n        normalize=False,\n        label=label,\n    )\n\n\ndef add_arrow_tip_label(\n    axis,\n    vector: np.ndarray,\n    text: str,\n    color: str,\n) -> None:\n    \"\"\"Place a text label slightly beyond the arrow tip.\"\"\"\n    label_position = ARROW_LABEL_SCALE * vector\n\n    axis.text(\n        label_position[0],\n        label_position[1],\n        label_position[2],\n        text,\n        color=color,\n        fontsize=10,\n        ha=\"center\",\n        va=\"center\",\n    )\n\n\ndef add_initial_state_label(axis) -> None:\n    \"\"\"Label the initialized ground-state pole.\"\"\"\n    axis.scatter(\n        0.0,\n        0.0,\n        1.0,\n        color=COLOR_INITIAL_STATE,\n        s=ENDPOINT_MARKER_SIZE,\n        depthshade=True,\n    )\n    axis.text(\n        0.0,\n        0.0,\n        1.16,\n        r\"initial $|g\\rangle$\",\n        color=COLOR_INITIAL_STATE,\n        fontsize=9,\n        ha=\"center\",\n        va=\"center\",\n    )\n\n\nif BLOCH_DURATION_NS is None:\n    bloch_duration_ns = get_pi_pulse_duration_ns(\n        BLOCH_OMEGA_ONRESONANCE_MHZ\n    )\nelse:\n    bloch_duration_ns = BLOCH_DURATION_NS\n\nresonant_trajectory = solve_rwa_bloch_trajectory(\n    detuning_mhz=BLOCH_RESONANT_DETUNING_MHZ,\n    duration_ns=bloch_duration_ns,\n    num_time_points=BLOCH_NUM_TIME_POINTS,\n)\n\ndetuned_trajectory = solve_rwa_bloch_trajectory(\n    detuning_mhz=BLOCH_DETUNED_DETUNING_MHZ,\n    duration_ns=bloch_duration_ns,\n    num_time_points=BLOCH_NUM_TIME_POINTS,\n)\n\nresonant_drive_ghz = (\n    BLOCH_F0_GHZ\n    + BLOCH_RESONANT_DETUNING_MHZ * 1.0e-3\n)\ndetuned_drive_ghz = (\n    BLOCH_F0_GHZ\n    + BLOCH_DETUNED_DETUNING_MHZ * 1.0e-3\n)\n\nfigure = plt.figure(figsize=BLOCH_FIGSIZE)\n\nbloch = qt.Bloch(fig=figure)\nbloch.view = BLOCH_VIEW\nbloch.zlabel = [r\"$|g\\rangle$\", r\"$|e\\rangle$\"]\nbloch.title = (\n    \"Bloch-sphere Rabi trajectories\\n\"\n    rf\"resonant $\\Delta\u002F2\\pi={BLOCH_RESONANT_DETUNING_MHZ:.1f}$ MHz, \"\n    rf\"detuned $\\Delta\u002F2\\pi={BLOCH_DETUNED_DETUNING_MHZ:.1f}$ MHz\"\n)\n\nbloch_axis = render_empty_bloch_sphere(\n    bloch_sphere=bloch,\n    fallback_figure=figure,\n)\n\ndraw_bloch_trajectory(\n    axis=bloch_axis,\n    vectors=resonant_trajectory[\"vectors\"],\n    color=COLOR_RESONANT_TRAJECTORY,\n    label=\"Resonant state trajectory\",\n)\ndraw_bloch_trajectory(\n    axis=bloch_axis,\n    vectors=detuned_trajectory[\"vectors\"],\n    color=COLOR_DETUNED_TRAJECTORY,\n    label=\"Detuned state trajectory\",\n)\n\nif BLOCH_SHOW_ENDPOINT_MARKERS:\n    draw_endpoint_marker(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"vectors\"][:, -1],\n        color=COLOR_RESONANT_TRAJECTORY,\n        label=\"Resonant final state\",\n    )\n    draw_endpoint_marker(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"vectors\"][:, -1],\n        color=COLOR_DETUNED_TRAJECTORY,\n        label=\"Detuned final state\",\n    )\n    add_initial_state_label(bloch_axis)\n\nif BLOCH_SHOW_CONTROL_AXIS_ARROWS:\n    draw_control_axis_arrow(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"control_axis\"],\n        color=COLOR_RESONANT_AXIS,\n        label=r\"Resonant control axis $\\hat{n}_{\\mathrm{res}}$\",\n    )\n    draw_control_axis_arrow(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"control_axis\"],\n        color=COLOR_DETUNED_AXIS,\n        label=r\"Detuned control axis $\\hat{n}_{\\mathrm{det}}$\",\n    )\n\nif BLOCH_SHOW_ARROW_TIP_LABELS and BLOCH_SHOW_CONTROL_AXIS_ARROWS:\n    add_arrow_tip_label(\n        axis=bloch_axis,\n        vector=resonant_trajectory[\"control_axis\"],\n        text=r\"$\\hat{n}_{\\mathrm{res}}$\",\n        color=COLOR_RESONANT_AXIS,\n    )\n    add_arrow_tip_label(\n        axis=bloch_axis,\n        vector=detuned_trajectory[\"control_axis\"],\n        text=r\"$\\hat{n}_{\\mathrm{det}}$\",\n        color=COLOR_DETUNED_AXIS,\n    )\n\nif BLOCH_SHOW_LEGEND:\n    legend_handles = [\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_RESONANT_TRAJECTORY,\n            linewidth=TRAJECTORY_LINEWIDTH,\n            label=\"Resonant state trajectory\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_DETUNED_TRAJECTORY,\n            linewidth=TRAJECTORY_LINEWIDTH,\n            label=\"Detuned state trajectory\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_RESONANT_AXIS,\n            linewidth=CONTROL_AXIS_LINEWIDTH,\n            label=r\"Resonant control axis $\\hat{n}_{\\mathrm{res}}$\",\n        ),\n        Line2D(\n            [0],\n            [0],\n            color=COLOR_DETUNED_AXIS,\n            linewidth=CONTROL_AXIS_LINEWIDTH,\n            label=r\"Detuned control axis $\\hat{n}_{\\mathrm{det}}$\",\n        ),\n    ]\n\n    bloch_axis.legend(\n        handles=legend_handles,\n        loc=LEGEND_LOCATION,\n        bbox_to_anchor=LEGEND_BBOX_TO_ANCHOR,\n        frameon=True,\n        borderaxespad=0.0,\n    )\n\nplt.show()\n\nif BLOCH_PRINT_SUMMARY:\n    resonant_axis = resonant_trajectory[\"control_axis\"]\n    detuned_axis = detuned_trajectory[\"control_axis\"]\n\n    print(\"Bloch-sphere trajectory summary\")\n    print(f\"Natural transition frequency f0: {BLOCH_F0_GHZ:.6f} GHz\")\n    print(f\"Resonant drive frequency: {resonant_drive_ghz:.6f} GHz\")\n    print(f\"Detuned drive frequency: {detuned_drive_ghz:.6f} GHz\")\n    print(f\"On-resonance Rabi rate: {BLOCH_OMEGA_ONRESONANCE_MHZ:.3f} MHz\")\n    print(f\"Plotted duration: {bloch_duration_ns:.3f} ns\")\n    print(\n        \"Resonant control axis n_res: \"\n        f\"({resonant_axis[0]:.6f}, \"\n        f\"{resonant_axis[1]:.6f}, \"\n        f\"{resonant_axis[2]:.6f})\"\n    )\n    print(\n        \"Detuned control axis n_det: \"\n        f\"({detuned_axis[0]:.6f}, \"\n        f\"{detuned_axis[1]:.6f}, \"\n        f\"{detuned_axis[2]:.6f})\"\n    )\n    print(\n        \"Resonant final excited-state probability: \"\n        f\"{resonant_trajectory['excited_probability'][-1]:.6f}\"\n    )\n    print(\n        \"Detuned final excited-state probability: \"\n        f\"{detuned_trajectory['excited_probability'][-1]:.6f}\"\n    )\n    print(\n        \"Detuned generalized Rabi frequency: \"\n        f\"{detuned_trajectory['generalized_rabi_mhz']:.6f} MHz\"\n    )\n",[10088,74657,74658,74663,74668,74672,74679,74686,74692,74696,74701,74706,74710,74719,74723,74732,74744,74756,74768,74780,74785,74789,74815,74830,74837,74844,74848,74852,74861,74871,74875,74880,74890,74894,74904,74923,74932,74951,74955,74964,74973,74982,74991,75000,75004,75014,75024,75034,75044,75054,75058,75068,75078,75088,75098,75108,75112,75122,75140,75144,75148,75170,75175,75194,75198,75202,75211,75227,75235,75240,75251,75256,75260,75269,75274,75292,75304,75309,75313,75322,75326,75330,75337,75347,75377,75388,75392,75400,75405,75410,75417,75424,75431,75438,75445,75452,75460,75465,75475,75479,75487,75495,75499,75503,75509,75515,75521,75527,75533,75537,75547,75551,75555,75559,75570,75574,75584,75594,75606,75618,75622,75628,75641,75645,75657,75672,75691,75698,75703,75707,75716,75721,75732,75736,75740,75744,75748,75770,75775,75793,75797,75801,75810,75821,75832,75843,75860,75865,75888,75910,75925,75929,75955,75959,75982,76003,76023,76027,76040,76054,76071,76075,76079,76093,76098,76103,76108,76112,76116,76145,76171,76196,76200,76213,76227,76231,76235,76249,76254,76259,76264,76269,76274,76278,76282,76295,76300,76311,76319,76329,76334,76339,76344,76351,76367,76376,76385,76394,76403,76426,76432,76437,76442,76447,76457,76466,76475,76480,76486,76501,76512,76527,76537,76545,76554,76559,76581,76586,76602,76619,76624,76637,76650,76655,76663,76668,76673,76683,76691,76699,76711,76723,76732,76738,76748,76758,76767,76776,76787,76799,76810,76815,76820,76825,76835,76842,76850,76861,76872,76881,76887,76897,76907,76916,76925,76934,76946,76958,76967,76972,76977,76982,76992,76999,77006,77017,77028,77037,77043,77053,77061,77068,77075,77084,77093,77102,77111,77122,77134,77146,77155,77160,77165,77170,77180,77187,77194,77206,77217,77226,77232,77248,77253,77262,77272,77281,77290,77296,77305,77317,77329,77341,77346,77351,77356,77374,77380,77389,77396,77403,77411,77422,77433,77444,77449,77458,77465,77472,77480,77501,77512,77523,77534,77545,77550,77555,77560,77574,77586,77592,77597,77604,77614,77619,77631,77642,77652,77663,77668,77673,77685,77696,77705,77716,77721,77726,77736,77742,77756,77761,77771,77776,77788,77793,77798,77820,77825,77848,77859,77898,77908,77918,77937,77955,77960,77965,77977,77987,77997,78002,78007,78015,78025,78040,78051,78063,78068,78075,78084,78098,78109,78121,78126,78131,78141,78149,78159,78180,78191,78203,78208,78215,78224,78243,78254,78266,78271,78280,78285,78295,78303,78312,78326,78337,78363,78368,78375,78384,78397,78408,78431,78436,78441,78455,78463,78472,78485,78507,78518,78523,78530,78539,78552,78573,78584,78589,78594,78604,78614,78622,78632,78641,78653,78665,78677,78682,78689,78698,78707,78718,78729,78740,78745,78752,78761,78770,78781,78792,78813,78818,78825,78834,78843,78854,78865,78886,78891,78896,78901,78911,78922,78934,78946,78958,78970,78975,78980,78989,78994,79004,79019,79034,79039,79051,79076,79101,79126,79150,79176,79183,79189,79214,79235,79257,79262,79269,79275,79297,79318,79339,79344,79351,79357,79386,79391,79398,79404,79431,79436,79443,79449,79472],{"__ignoreMap":10090},[40,74659,74660],{"class":10276,"line":10277},[40,74661,74662],{"class":10280},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[40,74664,74665],{"class":10276,"line":10284},[40,74666,74667],{"class":10287},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[40,74669,74670],{"class":10276,"line":10291},[40,74671,10301],{"emptyLinePlaceholder":10300},[40,74673,74674,74676],{"class":10276,"line":10297},[40,74675,10375],{"class":10352},[40,74677,74678],{"class":10363}," importlib.util\n",[40,74680,74681,74683],{"class":10276,"line":10304},[40,74682,10375],{"class":10352},[40,74684,74685],{"class":10363}," subprocess\n",[40,74687,74688,74690],{"class":10276,"line":10310},[40,74689,10375],{"class":10352},[40,74691,73406],{"class":10363},[40,74693,74694],{"class":10276,"line":10316},[40,74695,10301],{"emptyLinePlaceholder":10300},[40,74697,74698],{"class":10276,"line":10322},[40,74699,74700],{"class":10280},"# -------------------------------------------------------------------------\n",[40,74702,74703],{"class":10276,"line":10327},[40,74704,74705],{"class":10280},"# Control knobs\n",[40,74707,74708],{"class":10276,"line":10333},[40,74709,74700],{"class":10280},[40,74711,74712,74715,74717],{"class":10276,"line":10339},[40,74713,74714],{"class":10446},"INSTALL_BLOCH_DEPENDENCIES",[40,74716,10772],{"class":10732},[40,74718,73515],{"class":10446},[40,74720,74721],{"class":10276,"line":10344},[40,74722,10301],{"emptyLinePlaceholder":10300},[40,74724,74725,74728,74730],{"class":10276,"line":10349},[40,74726,74727],{"class":10446},"BLOCH_REQUIRED_PACKAGES",[40,74729,10772],{"class":10732},[40,74731,11019],{"class":10363},[40,74733,74734,74737,74739,74742],{"class":10276,"line":10367},[40,74735,74736],{"class":10287},"    \"numpy\"",[40,74738,10443],{"class":10363},[40,74740,74741],{"class":10287},"\"numpy\"",[40,74743,10507],{"class":10363},[40,74745,74746,74749,74751,74754],{"class":10276,"line":10372},[40,74747,74748],{"class":10287},"    \"matplotlib\"",[40,74750,10443],{"class":10363},[40,74752,74753],{"class":10287},"\"matplotlib\"",[40,74755,10507],{"class":10363},[40,74757,74758,74761,74763,74766],{"class":10276,"line":10387},[40,74759,74760],{"class":10287},"    \"scipy\"",[40,74762,10443],{"class":10363},[40,74764,74765],{"class":10287},"\"scipy\"",[40,74767,10507],{"class":10363},[40,74769,74770,74773,74775,74778],{"class":10276,"line":10400},[40,74771,74772],{"class":10287},"    \"qutip\"",[40,74774,10443],{"class":10363},[40,74776,74777],{"class":10287},"\"qutip\"",[40,74779,10507],{"class":10363},[40,74781,74782],{"class":10276,"line":10413},[40,74783,74784],{"class":10363},"}\n",[40,74786,74787],{"class":10276,"line":10418},[40,74788,10301],{"emptyLinePlaceholder":10300},[40,74790,74791,74794,74796,74799,74802,74804,74806,74809,74811,74813],{"class":10276,"line":10424},[40,74792,74793],{"class":10446},"BLOCH_F0_GHZ",[40,74795,10772],{"class":10732},[40,74797,74798],{"class":10732}," globals",[40,74800,74801],{"class":10363},"().",[40,74803,11165],{"class":10430},[40,74805,167],{"class":10363},[40,74807,74808],{"class":10287},"\"F0_GHZ\"",[40,74810,10484],{"class":10363},[40,74812,60100],{"class":10446},[40,74814,10761],{"class":10363},[40,74816,74817,74820,74822,74824,74826,74828],{"class":10276,"line":10437},[40,74818,74819],{"class":10446},"BLOCH_OMEGA_ONRESONANCE_MHZ",[40,74821,10772],{"class":10732},[40,74823,74798],{"class":10732},[40,74825,74801],{"class":10363},[40,74827,11165],{"class":10430},[40,74829,12527],{"class":10363},[40,74831,74832,74835],{"class":10276,"line":10456},[40,74833,74834],{"class":10287},"    \"OMEGA_ONRESONANCE_MHZ\"",[40,74836,10507],{"class":10363},[40,74838,74839,74842],{"class":10276,"line":10472},[40,74840,74841],{"class":10446},"    20.0",[40,74843,10507],{"class":10363},[40,74845,74846],{"class":10276,"line":10496},[40,74847,10761],{"class":10363},[40,74849,74850],{"class":10276,"line":10510},[40,74851,10301],{"emptyLinePlaceholder":10300},[40,74853,74854,74857,74859],{"class":10276,"line":10696},[40,74855,74856],{"class":10446},"BLOCH_RESONANT_DETUNING_MHZ",[40,74858,10772],{"class":10732},[40,74860,19124],{"class":10446},[40,74862,74863,74866,74868],{"class":10276,"line":10702},[40,74864,74865],{"class":10446},"BLOCH_DETUNED_DETUNING_MHZ",[40,74867,10772],{"class":10732},[40,74869,74870],{"class":10446}," 40.0\n",[40,74872,74873],{"class":10276,"line":10708},[40,74874,10301],{"emptyLinePlaceholder":10300},[40,74876,74877],{"class":10276,"line":10714},[40,74878,74879],{"class":10280},"# Use None to automatically plot one resonant pi pulse.\n",[40,74881,74882,74885,74887],{"class":10276,"line":10720},[40,74883,74884],{"class":10446},"BLOCH_DURATION_NS",[40,74886,10772],{"class":10732},[40,74888,74889],{"class":10446}," None\n",[40,74891,74892],{"class":10276,"line":10726},[40,74893,10301],{"emptyLinePlaceholder":10300},[40,74895,74896,74899,74901],{"class":10276,"line":10764},[40,74897,74898],{"class":10446},"BLOCH_NUM_TIME_POINTS",[40,74900,10772],{"class":10732},[40,74902,74903],{"class":10446}," 300\n",[40,74905,74906,74909,74911,74913,74916,74918,74921],{"class":10276,"line":10778},[40,74907,74908],{"class":10446},"BLOCH_FIGSIZE",[40,74910,10772],{"class":10732},[40,74912,920],{"class":10363},[40,74914,74915],{"class":10446},"8.8",[40,74917,10484],{"class":10363},[40,74919,74920],{"class":10446},"7.8",[40,74922,10761],{"class":10363},[40,74924,74925,74928,74930],{"class":10276,"line":10815},[40,74926,74927],{"class":10446},"BLOCH_DPI",[40,74929,10772],{"class":10732},[40,74931,73549],{"class":10446},[40,74933,74934,74937,74939,74941,74943,74945,74947,74949],{"class":10276,"line":10829},[40,74935,74936],{"class":10446},"BLOCH_VIEW",[40,74938,10772],{"class":10732},[40,74940,24010],{"class":10363},[40,74942,11056],{"class":10732},[40,74944,13519],{"class":10446},[40,74946,10484],{"class":10363},[40,74948,64874],{"class":10446},[40,74950,25466],{"class":10363},[40,74952,74953],{"class":10276,"line":10842},[40,74954,10301],{"emptyLinePlaceholder":10300},[40,74956,74957,74960,74962],{"class":10276,"line":10855},[40,74958,74959],{"class":10446},"BLOCH_SHOW_CONTROL_AXIS_ARROWS",[40,74961,10772],{"class":10732},[40,74963,73515],{"class":10446},[40,74965,74966,74969,74971],{"class":10276,"line":10868},[40,74967,74968],{"class":10446},"BLOCH_SHOW_ARROW_TIP_LABELS",[40,74970,10772],{"class":10732},[40,74972,73515],{"class":10446},[40,74974,74975,74978,74980],{"class":10276,"line":10881},[40,74976,74977],{"class":10446},"BLOCH_SHOW_ENDPOINT_MARKERS",[40,74979,10772],{"class":10732},[40,74981,73515],{"class":10446},[40,74983,74984,74987,74989],{"class":10276,"line":10907},[40,74985,74986],{"class":10446},"BLOCH_SHOW_LEGEND",[40,74988,10772],{"class":10732},[40,74990,73515],{"class":10446},[40,74992,74993,74996,74998],{"class":10276,"line":10920},[40,74994,74995],{"class":10446},"BLOCH_PRINT_SUMMARY",[40,74997,10772],{"class":10732},[40,74999,73515],{"class":10446},[40,75001,75002],{"class":10276,"line":10933},[40,75003,10301],{"emptyLinePlaceholder":10300},[40,75005,75006,75009,75011],{"class":10276,"line":10946},[40,75007,75008],{"class":10446},"COLOR_RESONANT_TRAJECTORY",[40,75010,10772],{"class":10732},[40,75012,75013],{"class":10287}," \"tab:blue\"\n",[40,75015,75016,75019,75021],{"class":10276,"line":10951},[40,75017,75018],{"class":10446},"COLOR_DETUNED_TRAJECTORY",[40,75020,10772],{"class":10732},[40,75022,75023],{"class":10287}," \"tab:orange\"\n",[40,75025,75026,75029,75031],{"class":10276,"line":10956},[40,75027,75028],{"class":10446},"COLOR_RESONANT_AXIS",[40,75030,10772],{"class":10732},[40,75032,75033],{"class":10287}," \"tab:green\"\n",[40,75035,75036,75039,75041],{"class":10276,"line":10962},[40,75037,75038],{"class":10446},"COLOR_DETUNED_AXIS",[40,75040,10772],{"class":10732},[40,75042,75043],{"class":10287}," \"tab:red\"\n",[40,75045,75046,75049,75051],{"class":10276,"line":10983},[40,75047,75048],{"class":10446},"COLOR_INITIAL_STATE",[40,75050,10772],{"class":10732},[40,75052,75053],{"class":10287}," \"black\"\n",[40,75055,75056],{"class":10276,"line":11005},[40,75057,10301],{"emptyLinePlaceholder":10300},[40,75059,75060,75063,75065],{"class":10276,"line":11011},[40,75061,75062],{"class":10446},"TRAJECTORY_LINEWIDTH",[40,75064,10772],{"class":10732},[40,75066,75067],{"class":10446}," 2.4\n",[40,75069,75070,75073,75075],{"class":10276,"line":11022},[40,75071,75072],{"class":10446},"CONTROL_AXIS_LINEWIDTH",[40,75074,10772],{"class":10732},[40,75076,75077],{"class":10446}," 2.2\n",[40,75079,75080,75083,75085],{"class":10276,"line":11064},[40,75081,75082],{"class":10446},"ENDPOINT_MARKER_SIZE",[40,75084,10772],{"class":10732},[40,75086,75087],{"class":10446}," 45\n",[40,75089,75090,75093,75095],{"class":10276,"line":11103},[40,75091,75092],{"class":10446},"ARROW_LENGTH_RATIO",[40,75094,10772],{"class":10732},[40,75096,75097],{"class":10446}," 0.12\n",[40,75099,75100,75103,75105],{"class":10276,"line":11142},[40,75101,75102],{"class":10446},"ARROW_LABEL_SCALE",[40,75104,10772],{"class":10732},[40,75106,75107],{"class":10446}," 1.16\n",[40,75109,75110],{"class":10276,"line":11148},[40,75111,10301],{"emptyLinePlaceholder":10300},[40,75113,75114,75117,75119],{"class":10276,"line":11181},[40,75115,75116],{"class":10446},"LEGEND_LOCATION",[40,75118,10772],{"class":10732},[40,75120,75121],{"class":10287}," \"upper left\"\n",[40,75123,75124,75127,75129,75131,75134,75136,75138],{"class":10276,"line":11198},[40,75125,75126],{"class":10446},"LEGEND_BBOX_TO_ANCHOR",[40,75128,10772],{"class":10732},[40,75130,920],{"class":10363},[40,75132,75133],{"class":10446},"1.02",[40,75135,10484],{"class":10363},[40,75137,75133],{"class":10446},[40,75139,10761],{"class":10363},[40,75141,75142],{"class":10276,"line":11249},[40,75143,10301],{"emptyLinePlaceholder":10300},[40,75145,75146],{"class":10276,"line":11254},[40,75147,10301],{"emptyLinePlaceholder":10300},[40,75149,75150,75152,75155,75157,75160,75162,75164,75166,75168],{"class":10276,"line":11259},[40,75151,10965],{"class":10352},[40,75153,75154],{"class":10430}," package_is_available",[40,75156,167],{"class":10363},[40,75158,75159],{"class":10973},"package_name",[40,75161,10443],{"class":10363},[40,75163,10733],{"class":10732},[40,75165,22117],{"class":10363},[40,75167,21950],{"class":10732},[40,75169,10639],{"class":10363},[40,75171,75172],{"class":10276,"line":11265},[40,75173,75174],{"class":10287},"    \"\"\"Return True when a package can be imported.\"\"\"\n",[40,75176,75177,75179,75182,75185,75188,75190,75192],{"class":10276,"line":11294},[40,75178,11201],{"class":10352},[40,75180,75181],{"class":10363}," importlib.util.",[40,75183,75184],{"class":10430},"find_spec",[40,75186,75187],{"class":10363},"(package_name) ",[40,75189,74136],{"class":10352},[40,75191,74139],{"class":10352},[40,75193,74889],{"class":10446},[40,75195,75196],{"class":10276,"line":11300},[40,75197,10301],{"emptyLinePlaceholder":10300},[40,75199,75200],{"class":10276,"line":11320},[40,75201,10301],{"emptyLinePlaceholder":10300},[40,75203,75204,75206,75209],{"class":10276,"line":11336},[40,75205,10965],{"class":10352},[40,75207,75208],{"class":10430}," install_missing_packages",[40,75210,12527],{"class":10363},[40,75212,75213,75216,75219,75221,75223,75225],{"class":10276,"line":11381},[40,75214,75215],{"class":10973},"    required_packages",[40,75217,75218],{"class":10363},": dict[",[40,75220,10733],{"class":10732},[40,75222,10484],{"class":10363},[40,75224,10733],{"class":10732},[40,75226,11542],{"class":10363},[40,75228,75229,75231,75233],{"class":10276,"line":11395},[40,75230,22117],{"class":10363},[40,75232,73064],{"class":10446},[40,75234,10639],{"class":10363},[40,75236,75237],{"class":10276,"line":11438},[40,75238,75239],{"class":10287},"    \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n",[40,75241,75242,75244,75246,75249],{"class":10276,"line":11452},[40,75243,11323],{"class":10352},[40,75245,74139],{"class":10352},[40,75247,75248],{"class":10446}," INSTALL_BLOCH_DEPENDENCIES",[40,75250,10639],{"class":10363},[40,75252,75253],{"class":10276,"line":11493},[40,75254,75255],{"class":10352},"        return\n",[40,75257,75258],{"class":10276,"line":11507},[40,75259,10301],{"emptyLinePlaceholder":10300},[40,75261,75262,75265,75267],{"class":10276,"line":11512},[40,75263,75264],{"class":10363},"    missing_package_names ",[40,75266,162],{"class":10732},[40,75268,37422],{"class":10363},[40,75270,75271],{"class":10276,"line":11517},[40,75272,75273],{"class":10363},"        pip_name\n",[40,75275,75276,75279,75282,75284,75287,75290],{"class":10276,"line":11545},[40,75277,75278],{"class":10352},"        for",[40,75280,75281],{"class":10363}," import_name, pip_name ",[40,75283,13846],{"class":10352},[40,75285,75286],{"class":10363}," required_packages.",[40,75288,75289],{"class":10430},"items",[40,75291,12174],{"class":10363},[40,75293,75294,75297,75299,75301],{"class":10276,"line":11557},[40,75295,75296],{"class":10352},"        if",[40,75298,74139],{"class":10352},[40,75300,75154],{"class":10430},[40,75302,75303],{"class":10363},"(import_name)\n",[40,75305,75306],{"class":10276,"line":11563},[40,75307,75308],{"class":10363},"    ]\n",[40,75310,75311],{"class":10276,"line":11582},[40,75312,10301],{"emptyLinePlaceholder":10300},[40,75314,75315,75317,75319],{"class":10276,"line":11596},[40,75316,11323],{"class":10352},[40,75318,74139],{"class":10352},[40,75320,75321],{"class":10363}," missing_package_names:\n",[40,75323,75324],{"class":10276,"line":11609},[40,75325,75255],{"class":10352},[40,75327,75328],{"class":10276,"line":11622},[40,75329,10301],{"emptyLinePlaceholder":10300},[40,75331,75332,75335],{"class":10276,"line":11637},[40,75333,75334],{"class":10352},"    try",[40,75336,10639],{"class":10363},[40,75338,75339,75342,75345],{"class":10276,"line":11708},[40,75340,75341],{"class":10363},"        subprocess.",[40,75343,75344],{"class":10430},"run",[40,75346,12527],{"class":10363},[40,75348,75349,75352,75355,75357,75360,75362,75365,75367,75370,75372,75375],{"class":10276,"line":11727},[40,75350,75351],{"class":10363},"            [sys.executable, ",[40,75353,75354],{"class":10287},"\"-m\"",[40,75356,10484],{"class":10363},[40,75358,75359],{"class":10287},"\"pip\"",[40,75361,10484],{"class":10363},[40,75363,75364],{"class":10287},"\"install\"",[40,75366,10484],{"class":10363},[40,75368,75369],{"class":10287},"\"-q\"",[40,75371,10484],{"class":10363},[40,75373,75374],{"class":10287},"\"uv\"",[40,75376,11542],{"class":10363},[40,75378,75379,75382,75384,75386],{"class":10276,"line":11744},[40,75380,75381],{"class":11903},"            check",[40,75383,162],{"class":10732},[40,75385,13091],{"class":10446},[40,75387,10507],{"class":10363},[40,75389,75390],{"class":10276,"line":11790},[40,75391,14674],{"class":10363},[40,75393,75394,75396,75398],{"class":10276,"line":11795},[40,75395,75341],{"class":10363},[40,75397,75344],{"class":10430},[40,75399,12527],{"class":10363},[40,75401,75402],{"class":10276,"line":11800},[40,75403,75404],{"class":10363},"            [\n",[40,75406,75407],{"class":10276,"line":11806},[40,75408,75409],{"class":10363},"                sys.executable,\n",[40,75411,75412,75415],{"class":10276,"line":11834},[40,75413,75414],{"class":10287},"                \"-m\"",[40,75416,10507],{"class":10363},[40,75418,75419,75422],{"class":10276,"line":11840},[40,75420,75421],{"class":10287},"                \"uv\"",[40,75423,10507],{"class":10363},[40,75425,75426,75429],{"class":10276,"line":11860},[40,75427,75428],{"class":10287},"                \"pip\"",[40,75430,10507],{"class":10363},[40,75432,75433,75436],{"class":10276,"line":11887},[40,75434,75435],{"class":10287},"                \"install\"",[40,75437,10507],{"class":10363},[40,75439,75440,75443],{"class":10276,"line":11914},[40,75441,75442],{"class":10287},"                \"--system\"",[40,75444,10507],{"class":10363},[40,75446,75447,75450],{"class":10276,"line":11938},[40,75448,75449],{"class":10287},"                \"-q\"",[40,75451,10507],{"class":10363},[40,75453,75454,75457],{"class":10276,"line":11960},[40,75455,75456],{"class":10732},"                *",[40,75458,75459],{"class":10363},"missing_package_names,\n",[40,75461,75462],{"class":10276,"line":11975},[40,75463,75464],{"class":10363},"            ],\n",[40,75466,75467,75469,75471,75473],{"class":10276,"line":11996},[40,75468,75381],{"class":11903},[40,75470,162],{"class":10732},[40,75472,13091],{"class":10446},[40,75474,10507],{"class":10363},[40,75476,75477],{"class":10276,"line":12001},[40,75478,14674],{"class":10363},[40,75480,75481,75484],{"class":10276,"line":12006},[40,75482,75483],{"class":10352},"    except",[40,75485,75486],{"class":10363}," (subprocess.CalledProcessError, FileNotFoundError):\n",[40,75488,75489,75491,75493],{"class":10276,"line":12028},[40,75490,75341],{"class":10363},[40,75492,75344],{"class":10430},[40,75494,12527],{"class":10363},[40,75496,75497],{"class":10276,"line":12034},[40,75498,75404],{"class":10363},[40,75500,75501],{"class":10276,"line":12105},[40,75502,75409],{"class":10363},[40,75504,75505,75507],{"class":10276,"line":12110},[40,75506,75414],{"class":10287},[40,75508,10507],{"class":10363},[40,75510,75511,75513],{"class":10276,"line":12115},[40,75512,75428],{"class":10287},[40,75514,10507],{"class":10363},[40,75516,75517,75519],{"class":10276,"line":12121},[40,75518,75435],{"class":10287},[40,75520,10507],{"class":10363},[40,75522,75523,75525],{"class":10276,"line":12155},[40,75524,75449],{"class":10287},[40,75526,10507],{"class":10363},[40,75528,75529,75531],{"class":10276,"line":12161},[40,75530,75456],{"class":10732},[40,75532,75459],{"class":10363},[40,75534,75535],{"class":10276,"line":12177},[40,75536,75464],{"class":10363},[40,75538,75539,75541,75543,75545],{"class":10276,"line":12192},[40,75540,75381],{"class":11903},[40,75542,162],{"class":10732},[40,75544,13091],{"class":10446},[40,75546,10507],{"class":10363},[40,75548,75549],{"class":10276,"line":12231},[40,75550,14674],{"class":10363},[40,75552,75553],{"class":10276,"line":12245},[40,75554,10301],{"emptyLinePlaceholder":10300},[40,75556,75557],{"class":10276,"line":12280},[40,75558,10301],{"emptyLinePlaceholder":10300},[40,75560,75561,75564,75566,75568],{"class":10276,"line":12294},[40,75562,75563],{"class":10430},"install_missing_packages",[40,75565,167],{"class":10363},[40,75567,74727],{"class":10446},[40,75569,10761],{"class":10363},[40,75571,75572],{"class":10276,"line":12329},[40,75573,10301],{"emptyLinePlaceholder":10300},[40,75575,75576,75578,75580,75582],{"class":10276,"line":12343},[40,75577,10375],{"class":10352},[40,75579,10405],{"class":10363},[40,75581,10381],{"class":10352},[40,75583,10410],{"class":10363},[40,75585,75586,75588,75590,75592],{"class":10276,"line":12376},[40,75587,10375],{"class":10352},[40,75589,10378],{"class":10363},[40,75591,10381],{"class":10352},[40,75593,10384],{"class":10363},[40,75595,75596,75598,75601,75603],{"class":10276,"line":12409},[40,75597,10375],{"class":10352},[40,75599,75600],{"class":10363}," qutip ",[40,75602,10381],{"class":10352},[40,75604,75605],{"class":10363}," qt\n",[40,75607,75608,75610,75613,75615],{"class":10276,"line":12414},[40,75609,10353],{"class":10352},[40,75611,75612],{"class":10363}," matplotlib.lines ",[40,75614,10375],{"class":10352},[40,75616,75617],{"class":10363}," Line2D\n",[40,75619,75620],{"class":10276,"line":12419},[40,75621,10301],{"emptyLinePlaceholder":10300},[40,75623,75624,75626],{"class":10276,"line":12435},[40,75625,73487],{"class":10352},[40,75627,10639],{"class":10363},[40,75629,75630,75633,75636,75638],{"class":10276,"line":12447},[40,75631,75632],{"class":10352},"    from",[40,75634,75635],{"class":10363}," IPython ",[40,75637,10375],{"class":10352},[40,75639,75640],{"class":10363}," get_ipython\n",[40,75642,75643],{"class":10276,"line":12463},[40,75644,10301],{"emptyLinePlaceholder":10300},[40,75646,75647,75650,75652,75655],{"class":10276,"line":12480},[40,75648,75649],{"class":10363},"    ipython ",[40,75651,162],{"class":10732},[40,75653,75654],{"class":10430}," get_ipython",[40,75656,12174],{"class":10363},[40,75658,75659,75661,75664,75666,75668,75670],{"class":10276,"line":12498},[40,75660,11323],{"class":10352},[40,75662,75663],{"class":10363}," ipython ",[40,75665,74136],{"class":10352},[40,75667,74139],{"class":10352},[40,75669,73802],{"class":10446},[40,75671,10639],{"class":10363},[40,75673,75674,75677,75680,75682,75684,75686,75689],{"class":10276,"line":12504},[40,75675,75676],{"class":10363},"        ipython.",[40,75678,75679],{"class":10430},"run_line_magic",[40,75681,167],{"class":10363},[40,75683,74753],{"class":10287},[40,75685,10484],{"class":10363},[40,75687,75688],{"class":10287},"\"inline\"",[40,75690,10761],{"class":10363},[40,75692,75693,75695],{"class":10276,"line":12517},[40,75694,73520],{"class":10352},[40,75696,75697],{"class":10363}," ImportError:\n",[40,75699,75700],{"class":10276,"line":12530},[40,75701,75702],{"class":10352},"    pass\n",[40,75704,75705],{"class":10276,"line":12542},[40,75706,10301],{"emptyLinePlaceholder":10300},[40,75708,75709,75712,75714],{"class":10276,"line":12571},[40,75710,75711],{"class":10363},"plt.rcParams.",[40,75713,10431],{"class":10430},[40,75715,12527],{"class":10363},[40,75717,75718],{"class":10276,"line":12622},[40,75719,75720],{"class":10363},"    {\n",[40,75722,75723,75726,75728,75730],{"class":10276,"line":12628},[40,75724,75725],{"class":10287},"        \"figure.dpi\"",[40,75727,10443],{"class":10363},[40,75729,74927],{"class":10446},[40,75731,10507],{"class":10363},[40,75733,75734],{"class":10276,"line":12634},[40,75735,11145],{"class":10363},[40,75737,75738],{"class":10276,"line":12639},[40,75739,10761],{"class":10363},[40,75741,75742],{"class":10276,"line":12644},[40,75743,10301],{"emptyLinePlaceholder":10300},[40,75745,75746],{"class":10276,"line":12667},[40,75747,10301],{"emptyLinePlaceholder":10300},[40,75749,75750,75752,75755,75757,75760,75762,75764,75766,75768],{"class":10276,"line":12673},[40,75751,10965],{"class":10352},[40,75753,75754],{"class":10430}," get_pi_pulse_duration_ns",[40,75756,167],{"class":10363},[40,75758,75759],{"class":10973},"omega_onresonance_mhz",[40,75761,10443],{"class":10363},[40,75763,10784],{"class":10732},[40,75765,22117],{"class":10363},[40,75767,10784],{"class":10732},[40,75769,10639],{"class":10363},[40,75771,75772],{"class":10276,"line":12684},[40,75773,75774],{"class":10287},"    \"\"\"Return the resonant pi-pulse duration in nanoseconds.\"\"\"\n",[40,75776,75777,75779,75782,75784,75786,75788,75790],{"class":10276,"line":12697},[40,75778,11201],{"class":10352},[40,75780,75781],{"class":10446}," 1.0e3",[40,75783,22539],{"class":10732},[40,75785,920],{"class":10363},[40,75787,11878],{"class":10446},[40,75789,11881],{"class":10732},[40,75791,75792],{"class":10363}," omega_onresonance_mhz)\n",[40,75794,75795],{"class":10276,"line":12723},[40,75796,10301],{"emptyLinePlaceholder":10300},[40,75798,75799],{"class":10276,"line":12736},[40,75800,10301],{"emptyLinePlaceholder":10300},[40,75802,75803,75805,75808],{"class":10276,"line":12777},[40,75804,10965],{"class":10352},[40,75806,75807],{"class":10430}," solve_rwa_bloch_trajectory",[40,75809,12527],{"class":10363},[40,75811,75812,75815,75817,75819],{"class":10276,"line":12790},[40,75813,75814],{"class":10973},"    detuning_mhz",[40,75816,10443],{"class":10363},[40,75818,10784],{"class":10732},[40,75820,10507],{"class":10363},[40,75822,75823,75826,75828,75830],{"class":10276,"line":16980},[40,75824,75825],{"class":10973},"    duration_ns",[40,75827,10443],{"class":10363},[40,75829,10784],{"class":10732},[40,75831,10507],{"class":10363},[40,75833,75834,75837,75839,75841],{"class":10276,"line":16991},[40,75835,75836],{"class":10973},"    num_time_points",[40,75838,10443],{"class":10363},[40,75840,10821],{"class":10732},[40,75842,10507],{"class":10363},[40,75844,75845,75848,75850,75853,75855,75858],{"class":10276,"line":16996},[40,75846,75847],{"class":10363},") -> dict[",[40,75849,10733],{"class":10732},[40,75851,75852],{"class":10363},", np.ndarray ",[40,75854,24461],{"class":10732},[40,75856,75857],{"class":10732}," float",[40,75859,12152],{"class":10363},[40,75861,75862],{"class":10276,"line":17001},[40,75863,75864],{"class":10287},"    \"\"\"Solve the RWA two-level trajectory and return Bloch observables.\"\"\"\n",[40,75866,75867,75870,75872,75874,75876,75878,75880,75883,75885],{"class":10276,"line":17019},[40,75868,75869],{"class":10363},"    omega_rad_s ",[40,75871,162],{"class":10732},[40,75873,12060],{"class":10446},[40,75875,11881],{"class":10732},[40,75877,28376],{"class":10363},[40,75879,11752],{"class":10732},[40,75881,75882],{"class":10446}," BLOCH_OMEGA_ONRESONANCE_MHZ",[40,75884,11881],{"class":10732},[40,75886,75887],{"class":10446}," 1.0e6\n",[40,75889,75890,75893,75895,75897,75899,75901,75903,75906,75908],{"class":10276,"line":17047},[40,75891,75892],{"class":10363},"    delta_rad_s ",[40,75894,162],{"class":10732},[40,75896,12060],{"class":10446},[40,75898,11881],{"class":10732},[40,75900,28376],{"class":10363},[40,75902,11752],{"class":10732},[40,75904,75905],{"class":10363}," detuning_mhz ",[40,75907,11752],{"class":10732},[40,75909,75887],{"class":10446},[40,75911,75912,75915,75917,75919,75922],{"class":10276,"line":17064},[40,75913,75914],{"class":10363},"    omega_generalized_rad_s ",[40,75916,162],{"class":10732},[40,75918,11156],{"class":10363},[40,75920,75921],{"class":10430},"hypot",[40,75923,75924],{"class":10363},"(omega_rad_s, delta_rad_s)\n",[40,75926,75927],{"class":10276,"line":17077},[40,75928,10301],{"emptyLinePlaceholder":10300},[40,75930,75931,75934,75936,75938,75940,75942,75944,75947,75949,75952],{"class":10276,"line":17110},[40,75932,75933],{"class":10363},"    time_s ",[40,75935,162],{"class":10732},[40,75937,11156],{"class":10363},[40,75939,11850],{"class":10430},[40,75941,167],{"class":10363},[40,75943,10801],{"class":10446},[40,75945,75946],{"class":10363},", duration_ns ",[40,75948,11752],{"class":10732},[40,75950,75951],{"class":10446}," 1.0e-9",[40,75953,75954],{"class":10363},", num_time_points)\n",[40,75956,75957],{"class":10276,"line":17145},[40,75958,10301],{"emptyLinePlaceholder":10300},[40,75960,75961,75964,75966,75969,75972,75974,75976,75978,75980],{"class":10276,"line":17194},[40,75962,75963],{"class":10363},"    ground_state ",[40,75965,162],{"class":10732},[40,75967,75968],{"class":10363}," qt.",[40,75970,75971],{"class":10430},"basis",[40,75973,167],{"class":10363},[40,75975,1044],{"class":10446},[40,75977,10484],{"class":10363},[40,75979,66],{"class":10446},[40,75981,10761],{"class":10363},[40,75983,75984,75987,75989,75991,75993,75995,75997,75999,76001],{"class":10276,"line":17202},[40,75985,75986],{"class":10363},"    excited_state ",[40,75988,162],{"class":10732},[40,75990,75968],{"class":10363},[40,75992,75971],{"class":10430},[40,75994,167],{"class":10363},[40,75996,1044],{"class":10446},[40,75998,10484],{"class":10363},[40,76000,78],{"class":10446},[40,76002,10761],{"class":10363},[40,76004,76005,76008,76010,76013,76015,76018,76021],{"class":10276,"line":17207},[40,76006,76007],{"class":10363},"    excited_projector ",[40,76009,162],{"class":10732},[40,76011,76012],{"class":10363}," excited_state ",[40,76014,11752],{"class":10732},[40,76016,76017],{"class":10363}," excited_state.",[40,76019,76020],{"class":10430},"dag",[40,76022,12174],{"class":10363},[40,76024,76025],{"class":10276,"line":17241},[40,76026,10301],{"emptyLinePlaceholder":10300},[40,76028,76029,76032,76034,76036,76038],{"class":10276,"line":17269},[40,76030,76031],{"class":10363},"    hamiltonian ",[40,76033,162],{"class":10732},[40,76035,12039],{"class":10446},[40,76037,11881],{"class":10732},[40,76039,22894],{"class":10363},[40,76041,76042,76045,76047,76049,76052],{"class":10276,"line":17295},[40,76043,76044],{"class":10363},"        delta_rad_s ",[40,76046,11752],{"class":10732},[40,76048,75968],{"class":10363},[40,76050,76051],{"class":10430},"sigmaz",[40,76053,12174],{"class":10363},[40,76055,76056,76059,76062,76064,76066,76069],{"class":10276,"line":17324},[40,76057,76058],{"class":10732},"        +",[40,76060,76061],{"class":10363}," omega_rad_s ",[40,76063,11752],{"class":10732},[40,76065,75968],{"class":10363},[40,76067,76068],{"class":10430},"sigmax",[40,76070,12174],{"class":10363},[40,76072,76073],{"class":10276,"line":17341},[40,76074,12631],{"class":10363},[40,76076,76077],{"class":10276,"line":17350},[40,76078,10301],{"emptyLinePlaceholder":10300},[40,76080,76081,76084,76086,76088,76091],{"class":10276,"line":17358},[40,76082,76083],{"class":10363},"    result ",[40,76085,162],{"class":10732},[40,76087,75968],{"class":10363},[40,76089,76090],{"class":10430},"sesolve",[40,76092,12527],{"class":10363},[40,76094,76095],{"class":10276,"line":17363},[40,76096,76097],{"class":10363},"        hamiltonian,\n",[40,76099,76100],{"class":10276,"line":17368},[40,76101,76102],{"class":10363},"        ground_state,\n",[40,76104,76105],{"class":10276,"line":17383},[40,76106,76107],{"class":10363},"        time_s,\n",[40,76109,76110],{"class":10276,"line":17416},[40,76111,12631],{"class":10363},[40,76113,76114],{"class":10276,"line":17422},[40,76115,10301],{"emptyLinePlaceholder":10300},[40,76117,76118,76121,76123,76125,76128,76130,76132,76135,76138,76140,76142],{"class":10276,"line":17435},[40,76119,76120],{"class":10363},"    x_expectation ",[40,76122,162],{"class":10732},[40,76124,11156],{"class":10363},[40,76126,76127],{"class":10430},"real",[40,76129,13108],{"class":10363},[40,76131,11159],{"class":10430},[40,76133,76134],{"class":10363},"(qt.",[40,76136,76137],{"class":10430},"expect",[40,76139,76134],{"class":10363},[40,76141,76068],{"class":10430},[40,76143,76144],{"class":10363},"(), result.states)))\n",[40,76146,76147,76150,76152,76154,76156,76158,76160,76162,76164,76166,76169],{"class":10276,"line":17453},[40,76148,76149],{"class":10363},"    y_expectation ",[40,76151,162],{"class":10732},[40,76153,11156],{"class":10363},[40,76155,76127],{"class":10430},[40,76157,13108],{"class":10363},[40,76159,11159],{"class":10430},[40,76161,76134],{"class":10363},[40,76163,76137],{"class":10430},[40,76165,76134],{"class":10363},[40,76167,76168],{"class":10430},"sigmay",[40,76170,76144],{"class":10363},[40,76172,76173,76176,76178,76180,76182,76184,76186,76188,76190,76192,76194],{"class":10276,"line":17472},[40,76174,76175],{"class":10363},"    z_expectation ",[40,76177,162],{"class":10732},[40,76179,11156],{"class":10363},[40,76181,76127],{"class":10430},[40,76183,13108],{"class":10363},[40,76185,11159],{"class":10430},[40,76187,76134],{"class":10363},[40,76189,76137],{"class":10430},[40,76191,76134],{"class":10363},[40,76193,76051],{"class":10430},[40,76195,76144],{"class":10363},[40,76197,76198],{"class":10276,"line":17495},[40,76199,10301],{"emptyLinePlaceholder":10300},[40,76201,76202,76205,76207,76209,76211],{"class":10276,"line":17504},[40,76203,76204],{"class":10363},"    excited_probability ",[40,76206,162],{"class":10732},[40,76208,11156],{"class":10363},[40,76210,76127],{"class":10430},[40,76212,12527],{"class":10363},[40,76214,76215,76218,76220,76222,76224],{"class":10276,"line":17509},[40,76216,76217],{"class":10363},"        np.",[40,76219,11159],{"class":10430},[40,76221,76134],{"class":10363},[40,76223,76137],{"class":10430},[40,76225,76226],{"class":10363},"(excited_projector, result.states))\n",[40,76228,76229],{"class":10276,"line":17514},[40,76230,12631],{"class":10363},[40,76232,76233],{"class":10276,"line":17520},[40,76234,10301],{"emptyLinePlaceholder":10300},[40,76236,76237,76240,76242,76244,76247],{"class":10276,"line":17535},[40,76238,76239],{"class":10363},"    bloch_vectors ",[40,76241,162],{"class":10732},[40,76243,11156],{"class":10363},[40,76245,76246],{"class":10430},"vstack",[40,76248,12527],{"class":10363},[40,76250,76251],{"class":10276,"line":17546},[40,76252,76253],{"class":10363},"        [\n",[40,76255,76256],{"class":10276,"line":17583},[40,76257,76258],{"class":10363},"            x_expectation,\n",[40,76260,76261],{"class":10276,"line":17606},[40,76262,76263],{"class":10363},"            y_expectation,\n",[40,76265,76266],{"class":10276,"line":17617},[40,76267,76268],{"class":10363},"            z_expectation,\n",[40,76270,76271],{"class":10276,"line":17634},[40,76272,76273],{"class":10363},"        ]\n",[40,76275,76276],{"class":10276,"line":17646},[40,76277,12631],{"class":10363},[40,76279,76280],{"class":10276,"line":17651},[40,76281,10301],{"emptyLinePlaceholder":10300},[40,76283,76284,76287,76289,76291,76293],{"class":10276,"line":17656},[40,76285,76286],{"class":10363},"    control_axis ",[40,76288,162],{"class":10732},[40,76290,11156],{"class":10363},[40,76292,11206],{"class":10430},[40,76294,12527],{"class":10363},[40,76296,76298],{"class":10276,"line":76297},184,[40,76299,76253],{"class":10363},[40,76301,76303,76306,76308],{"class":10276,"line":76302},185,[40,76304,76305],{"class":10363},"            omega_rad_s ",[40,76307,11243],{"class":10732},[40,76309,76310],{"class":10363}," omega_generalized_rad_s,\n",[40,76312,76314,76317],{"class":10276,"line":76313},186,[40,76315,76316],{"class":10446},"            0.0",[40,76318,10507],{"class":10363},[40,76320,76322,76325,76327],{"class":10276,"line":76321},187,[40,76323,76324],{"class":10363},"            delta_rad_s ",[40,76326,11243],{"class":10732},[40,76328,76310],{"class":10363},[40,76330,76332],{"class":10276,"line":76331},188,[40,76333,76273],{"class":10363},[40,76335,76337],{"class":10276,"line":76336},189,[40,76338,12631],{"class":10363},[40,76340,76342],{"class":10276,"line":76341},190,[40,76343,10301],{"emptyLinePlaceholder":10300},[40,76345,76347,76349],{"class":10276,"line":76346},191,[40,76348,11201],{"class":10352},[40,76350,11019],{"class":10363},[40,76352,76354,76357,76360,76362,76365],{"class":10276,"line":76353},192,[40,76355,76356],{"class":10287},"        \"time_ns\"",[40,76358,76359],{"class":10363},": time_s ",[40,76361,11752],{"class":10732},[40,76363,76364],{"class":10446}," 1.0e9",[40,76366,10507],{"class":10363},[40,76368,76370,76373],{"class":10276,"line":76369},193,[40,76371,76372],{"class":10287},"        \"vectors\"",[40,76374,76375],{"class":10363},": bloch_vectors,\n",[40,76377,76379,76382],{"class":10276,"line":76378},194,[40,76380,76381],{"class":10287},"        \"excited_probability\"",[40,76383,76384],{"class":10363},": excited_probability,\n",[40,76386,76388,76391],{"class":10276,"line":76387},195,[40,76389,76390],{"class":10287},"        \"control_axis\"",[40,76392,76393],{"class":10363},": control_axis,\n",[40,76395,76397,76400],{"class":10276,"line":76396},196,[40,76398,76399],{"class":10287},"        \"generalized_rabi_mhz\"",[40,76401,76402],{"class":10363},": (\n",[40,76404,76406,76409,76411,76413,76415,76417,76419,76421,76424],{"class":10276,"line":76405},197,[40,76407,76408],{"class":10363},"            omega_generalized_rad_s ",[40,76410,11243],{"class":10732},[40,76412,920],{"class":10363},[40,76414,11878],{"class":10446},[40,76416,11881],{"class":10732},[40,76418,28376],{"class":10363},[40,76420,11752],{"class":10732},[40,76422,76423],{"class":10446}," 1.0e6",[40,76425,10761],{"class":10363},[40,76427,76429],{"class":10276,"line":76428},198,[40,76430,76431],{"class":10363},"        ),\n",[40,76433,76435],{"class":10276,"line":76434},199,[40,76436,11145],{"class":10363},[40,76438,76440],{"class":10276,"line":76439},200,[40,76441,10301],{"emptyLinePlaceholder":10300},[40,76443,76445],{"class":10276,"line":76444},201,[40,76446,10301],{"emptyLinePlaceholder":10300},[40,76448,76450,76452,76455],{"class":10276,"line":76449},202,[40,76451,10965],{"class":10352},[40,76453,76454],{"class":10430}," render_empty_bloch_sphere",[40,76456,12527],{"class":10363},[40,76458,76460,76463],{"class":10276,"line":76459},203,[40,76461,76462],{"class":10973},"    bloch_sphere",[40,76464,76465],{"class":10363},": qt.Bloch,\n",[40,76467,76469,76472],{"class":10276,"line":76468},204,[40,76470,76471],{"class":10973},"    fallback_figure",[40,76473,76474],{"class":10363},": plt.Figure,\n",[40,76476,76478],{"class":10276,"line":76477},205,[40,76479,13353],{"class":10363},[40,76481,76483],{"class":10276,"line":76482},206,[40,76484,76485],{"class":10287},"    \"\"\"Render an empty QuTiP Bloch sphere and return its Matplotlib axis.\"\"\"\n",[40,76487,76489,76491,76493,76496,76499],{"class":10276,"line":76488},207,[40,76490,11323],{"class":10352},[40,76492,16451],{"class":10732},[40,76494,76495],{"class":10363},"(bloch_sphere, ",[40,76497,76498],{"class":10287},"\"render\"",[40,76500,13353],{"class":10363},[40,76502,76504,76507,76510],{"class":10276,"line":76503},208,[40,76505,76506],{"class":10363},"        bloch_sphere.",[40,76508,76509],{"class":10430},"render",[40,76511,12174],{"class":10363},[40,76513,76515,76518,76520,76522,76525],{"class":10276,"line":76514},209,[40,76516,76517],{"class":10352},"    elif",[40,76519,16451],{"class":10732},[40,76521,76495],{"class":10363},[40,76523,76524],{"class":10287},"\"make_sphere\"",[40,76526,13353],{"class":10363},[40,76528,76530,76532,76535],{"class":10276,"line":76529},210,[40,76531,76506],{"class":10363},[40,76533,76534],{"class":10430},"make_sphere",[40,76536,12174],{"class":10363},[40,76538,76540,76543],{"class":10276,"line":76539},211,[40,76541,76542],{"class":10352},"    else",[40,76544,10639],{"class":10363},[40,76546,76548,76550,76552],{"class":10276,"line":76547},212,[40,76549,76506],{"class":10363},[40,76551,23686],{"class":10430},[40,76553,12174],{"class":10363},[40,76555,76557],{"class":10276,"line":76556},213,[40,76558,10301],{"emptyLinePlaceholder":10300},[40,76560,76562,76565,76567,76570,76572,76575,76577,76579],{"class":10276,"line":76561},214,[40,76563,76564],{"class":10363},"    axis ",[40,76566,162],{"class":10732},[40,76568,76569],{"class":10732}," getattr",[40,76571,76495],{"class":10363},[40,76573,76574],{"class":10287},"\"axes\"",[40,76576,10484],{"class":10363},[40,76578,73064],{"class":10446},[40,76580,10761],{"class":10363},[40,76582,76584],{"class":10276,"line":76583},215,[40,76585,10301],{"emptyLinePlaceholder":10300},[40,76587,76589,76591,76593,76595,76597,76599],{"class":10276,"line":76588},216,[40,76590,11323],{"class":10352},[40,76592,11326],{"class":10363},[40,76594,74136],{"class":10352},[40,76596,73802],{"class":10446},[40,76598,22438],{"class":10352},[40,76600,76601],{"class":10363}," fallback_figure.axes:\n",[40,76603,76605,76608,76610,76613,76615,76617],{"class":10276,"line":76604},217,[40,76606,76607],{"class":10363},"        axis ",[40,76609,162],{"class":10732},[40,76611,76612],{"class":10363}," fallback_figure.axes[",[40,76614,11056],{"class":10732},[40,76616,78],{"class":10446},[40,76618,25466],{"class":10363},[40,76620,76622],{"class":10276,"line":76621},218,[40,76623,10301],{"emptyLinePlaceholder":10300},[40,76625,76627,76629,76631,76633,76635],{"class":10276,"line":76626},219,[40,76628,11323],{"class":10352},[40,76630,11326],{"class":10363},[40,76632,74136],{"class":10352},[40,76634,73802],{"class":10446},[40,76636,10639],{"class":10363},[40,76638,76640,76642,76645,76648],{"class":10276,"line":76639},220,[40,76641,11612],{"class":10352},[40,76643,76644],{"class":10363}," RuntimeError(",[40,76646,76647],{"class":10287},"\"Could not access the Matplotlib Bloch axis.\"",[40,76649,10761],{"class":10363},[40,76651,76653],{"class":10276,"line":76652},221,[40,76654,10301],{"emptyLinePlaceholder":10300},[40,76656,76658,76660],{"class":10276,"line":76657},222,[40,76659,11201],{"class":10352},[40,76661,76662],{"class":10363}," axis\n",[40,76664,76666],{"class":10276,"line":76665},223,[40,76667,10301],{"emptyLinePlaceholder":10300},[40,76669,76671],{"class":10276,"line":76670},224,[40,76672,10301],{"emptyLinePlaceholder":10300},[40,76674,76676,76678,76681],{"class":10276,"line":76675},225,[40,76677,10965],{"class":10352},[40,76679,76680],{"class":10430}," draw_bloch_trajectory",[40,76682,12527],{"class":10363},[40,76684,76686,76689],{"class":10276,"line":76685},226,[40,76687,76688],{"class":10973},"    axis",[40,76690,10507],{"class":10363},[40,76692,76694,76697],{"class":10276,"line":76693},227,[40,76695,76696],{"class":10973},"    vectors",[40,76698,12841],{"class":10363},[40,76700,76702,76705,76707,76709],{"class":10276,"line":76701},228,[40,76703,76704],{"class":10973},"    color",[40,76706,10443],{"class":10363},[40,76708,10733],{"class":10732},[40,76710,10507],{"class":10363},[40,76712,76714,76717,76719,76721],{"class":10276,"line":76713},229,[40,76715,76716],{"class":10973},"    label",[40,76718,10443],{"class":10363},[40,76720,10733],{"class":10732},[40,76722,10507],{"class":10363},[40,76724,76726,76728,76730],{"class":10276,"line":76725},230,[40,76727,22117],{"class":10363},[40,76729,73064],{"class":10446},[40,76731,10639],{"class":10363},[40,76733,76735],{"class":10276,"line":76734},231,[40,76736,76737],{"class":10287},"    \"\"\"Draw a Bloch-vector trajectory on an existing 3D axis.\"\"\"\n",[40,76739,76741,76744,76746],{"class":10276,"line":76740},232,[40,76742,76743],{"class":10363},"    axis.",[40,76745,23505],{"class":10430},[40,76747,12527],{"class":10363},[40,76749,76751,76754,76756],{"class":10276,"line":76750},233,[40,76752,76753],{"class":10363},"        vectors[",[40,76755,66],{"class":10446},[40,76757,11542],{"class":10363},[40,76759,76761,76763,76765],{"class":10276,"line":76760},234,[40,76762,76753],{"class":10363},[40,76764,78],{"class":10446},[40,76766,11542],{"class":10363},[40,76768,76770,76772,76774],{"class":10276,"line":76769},235,[40,76771,76753],{"class":10363},[40,76773,1044],{"class":10446},[40,76775,11542],{"class":10363},[40,76777,76779,76782,76784],{"class":10276,"line":76778},236,[40,76780,76781],{"class":11903},"        color",[40,76783,162],{"class":10732},[40,76785,76786],{"class":10363},"color,\n",[40,76788,76790,76793,76795,76797],{"class":10276,"line":76789},237,[40,76791,76792],{"class":11903},"        linewidth",[40,76794,162],{"class":10732},[40,76796,75062],{"class":10446},[40,76798,10507],{"class":10363},[40,76800,76802,76805,76807],{"class":10276,"line":76801},238,[40,76803,76804],{"class":11903},"        label",[40,76806,162],{"class":10732},[40,76808,76809],{"class":10363},"label,\n",[40,76811,76813],{"class":10276,"line":76812},239,[40,76814,12631],{"class":10363},[40,76816,76818],{"class":10276,"line":76817},240,[40,76819,10301],{"emptyLinePlaceholder":10300},[40,76821,76823],{"class":10276,"line":76822},241,[40,76824,10301],{"emptyLinePlaceholder":10300},[40,76826,76828,76830,76833],{"class":10276,"line":76827},242,[40,76829,10965],{"class":10352},[40,76831,76832],{"class":10430}," draw_endpoint_marker",[40,76834,12527],{"class":10363},[40,76836,76838,76840],{"class":10276,"line":76837},243,[40,76839,76688],{"class":10973},[40,76841,10507],{"class":10363},[40,76843,76845,76848],{"class":10276,"line":76844},244,[40,76846,76847],{"class":10973},"    vector",[40,76849,12841],{"class":10363},[40,76851,76853,76855,76857,76859],{"class":10276,"line":76852},245,[40,76854,76704],{"class":10973},[40,76856,10443],{"class":10363},[40,76858,10733],{"class":10732},[40,76860,10507],{"class":10363},[40,76862,76864,76866,76868,76870],{"class":10276,"line":76863},246,[40,76865,76716],{"class":10973},[40,76867,10443],{"class":10363},[40,76869,10733],{"class":10732},[40,76871,10507],{"class":10363},[40,76873,76875,76877,76879],{"class":10276,"line":76874},247,[40,76876,22117],{"class":10363},[40,76878,73064],{"class":10446},[40,76880,10639],{"class":10363},[40,76882,76884],{"class":10276,"line":76883},248,[40,76885,76886],{"class":10287},"    \"\"\"Draw a marker at the final point of a Bloch trajectory.\"\"\"\n",[40,76888,76890,76892,76895],{"class":10276,"line":76889},249,[40,76891,76743],{"class":10363},[40,76893,76894],{"class":10430},"scatter",[40,76896,12527],{"class":10363},[40,76898,76900,76903,76905],{"class":10276,"line":76899},250,[40,76901,76902],{"class":10363},"        vector[",[40,76904,66],{"class":10446},[40,76906,11542],{"class":10363},[40,76908,76910,76912,76914],{"class":10276,"line":76909},251,[40,76911,76902],{"class":10363},[40,76913,78],{"class":10446},[40,76915,11542],{"class":10363},[40,76917,76919,76921,76923],{"class":10276,"line":76918},252,[40,76920,76902],{"class":10363},[40,76922,1044],{"class":10446},[40,76924,11542],{"class":10363},[40,76926,76928,76930,76932],{"class":10276,"line":76927},253,[40,76929,76781],{"class":11903},[40,76931,162],{"class":10732},[40,76933,76786],{"class":10363},[40,76935,76937,76940,76942,76944],{"class":10276,"line":76936},254,[40,76938,76939],{"class":11903},"        s",[40,76941,162],{"class":10732},[40,76943,75082],{"class":10446},[40,76945,10507],{"class":10363},[40,76947,76949,76952,76954,76956],{"class":10276,"line":76948},255,[40,76950,76951],{"class":11903},"        depthshade",[40,76953,162],{"class":10732},[40,76955,13091],{"class":10446},[40,76957,10507],{"class":10363},[40,76959,76961,76963,76965],{"class":10276,"line":76960},256,[40,76962,76804],{"class":11903},[40,76964,162],{"class":10732},[40,76966,76809],{"class":10363},[40,76968,76970],{"class":10276,"line":76969},257,[40,76971,12631],{"class":10363},[40,76973,76975],{"class":10276,"line":76974},258,[40,76976,10301],{"emptyLinePlaceholder":10300},[40,76978,76980],{"class":10276,"line":76979},259,[40,76981,10301],{"emptyLinePlaceholder":10300},[40,76983,76985,76987,76990],{"class":10276,"line":76984},260,[40,76986,10965],{"class":10352},[40,76988,76989],{"class":10430}," draw_control_axis_arrow",[40,76991,12527],{"class":10363},[40,76993,76995,76997],{"class":10276,"line":76994},261,[40,76996,76688],{"class":10973},[40,76998,10507],{"class":10363},[40,77000,77002,77004],{"class":10276,"line":77001},262,[40,77003,76847],{"class":10973},[40,77005,12841],{"class":10363},[40,77007,77009,77011,77013,77015],{"class":10276,"line":77008},263,[40,77010,76704],{"class":10973},[40,77012,10443],{"class":10363},[40,77014,10733],{"class":10732},[40,77016,10507],{"class":10363},[40,77018,77020,77022,77024,77026],{"class":10276,"line":77019},264,[40,77021,76716],{"class":10973},[40,77023,10443],{"class":10363},[40,77025,10733],{"class":10732},[40,77027,10507],{"class":10363},[40,77029,77031,77033,77035],{"class":10276,"line":77030},265,[40,77032,22117],{"class":10363},[40,77034,73064],{"class":10446},[40,77036,10639],{"class":10363},[40,77038,77040],{"class":10276,"line":77039},266,[40,77041,77042],{"class":10287},"    \"\"\"Draw a normalized effective-control-axis arrow.\"\"\"\n",[40,77044,77046,77048,77051],{"class":10276,"line":77045},267,[40,77047,76743],{"class":10363},[40,77049,77050],{"class":10430},"quiver",[40,77052,12527],{"class":10363},[40,77054,77056,77059],{"class":10276,"line":77055},268,[40,77057,77058],{"class":10446},"        0.0",[40,77060,10507],{"class":10363},[40,77062,77064,77066],{"class":10276,"line":77063},269,[40,77065,77058],{"class":10446},[40,77067,10507],{"class":10363},[40,77069,77071,77073],{"class":10276,"line":77070},270,[40,77072,77058],{"class":10446},[40,77074,10507],{"class":10363},[40,77076,77078,77080,77082],{"class":10276,"line":77077},271,[40,77079,76902],{"class":10363},[40,77081,66],{"class":10446},[40,77083,11542],{"class":10363},[40,77085,77087,77089,77091],{"class":10276,"line":77086},272,[40,77088,76902],{"class":10363},[40,77090,78],{"class":10446},[40,77092,11542],{"class":10363},[40,77094,77096,77098,77100],{"class":10276,"line":77095},273,[40,77097,76902],{"class":10363},[40,77099,1044],{"class":10446},[40,77101,11542],{"class":10363},[40,77103,77105,77107,77109],{"class":10276,"line":77104},274,[40,77106,76781],{"class":11903},[40,77108,162],{"class":10732},[40,77110,76786],{"class":10363},[40,77112,77114,77116,77118,77120],{"class":10276,"line":77113},275,[40,77115,76792],{"class":11903},[40,77117,162],{"class":10732},[40,77119,75072],{"class":10446},[40,77121,10507],{"class":10363},[40,77123,77125,77128,77130,77132],{"class":10276,"line":77124},276,[40,77126,77127],{"class":11903},"        arrow_length_ratio",[40,77129,162],{"class":10732},[40,77131,75092],{"class":10446},[40,77133,10507],{"class":10363},[40,77135,77137,77140,77142,77144],{"class":10276,"line":77136},277,[40,77138,77139],{"class":11903},"        normalize",[40,77141,162],{"class":10732},[40,77143,10504],{"class":10446},[40,77145,10507],{"class":10363},[40,77147,77149,77151,77153],{"class":10276,"line":77148},278,[40,77150,76804],{"class":11903},[40,77152,162],{"class":10732},[40,77154,76809],{"class":10363},[40,77156,77158],{"class":10276,"line":77157},279,[40,77159,12631],{"class":10363},[40,77161,77163],{"class":10276,"line":77162},280,[40,77164,10301],{"emptyLinePlaceholder":10300},[40,77166,77168],{"class":10276,"line":77167},281,[40,77169,10301],{"emptyLinePlaceholder":10300},[40,77171,77173,77175,77178],{"class":10276,"line":77172},282,[40,77174,10965],{"class":10352},[40,77176,77177],{"class":10430}," add_arrow_tip_label",[40,77179,12527],{"class":10363},[40,77181,77183,77185],{"class":10276,"line":77182},283,[40,77184,76688],{"class":10973},[40,77186,10507],{"class":10363},[40,77188,77190,77192],{"class":10276,"line":77189},284,[40,77191,76847],{"class":10973},[40,77193,12841],{"class":10363},[40,77195,77197,77200,77202,77204],{"class":10276,"line":77196},285,[40,77198,77199],{"class":10973},"    text",[40,77201,10443],{"class":10363},[40,77203,10733],{"class":10732},[40,77205,10507],{"class":10363},[40,77207,77209,77211,77213,77215],{"class":10276,"line":77208},286,[40,77210,76704],{"class":10973},[40,77212,10443],{"class":10363},[40,77214,10733],{"class":10732},[40,77216,10507],{"class":10363},[40,77218,77220,77222,77224],{"class":10276,"line":77219},287,[40,77221,22117],{"class":10363},[40,77223,73064],{"class":10446},[40,77225,10639],{"class":10363},[40,77227,77229],{"class":10276,"line":77228},288,[40,77230,77231],{"class":10287},"    \"\"\"Place a text label slightly beyond the arrow tip.\"\"\"\n",[40,77233,77235,77238,77240,77243,77245],{"class":10276,"line":77234},289,[40,77236,77237],{"class":10363},"    label_position ",[40,77239,162],{"class":10732},[40,77241,77242],{"class":10446}," ARROW_LABEL_SCALE",[40,77244,11881],{"class":10732},[40,77246,77247],{"class":10363}," vector\n",[40,77249,77251],{"class":10276,"line":77250},290,[40,77252,10301],{"emptyLinePlaceholder":10300},[40,77254,77256,77258,77260],{"class":10276,"line":77255},291,[40,77257,76743],{"class":10363},[40,77259,2932],{"class":10430},[40,77261,12527],{"class":10363},[40,77263,77265,77268,77270],{"class":10276,"line":77264},292,[40,77266,77267],{"class":10363},"        label_position[",[40,77269,66],{"class":10446},[40,77271,11542],{"class":10363},[40,77273,77275,77277,77279],{"class":10276,"line":77274},293,[40,77276,77267],{"class":10363},[40,77278,78],{"class":10446},[40,77280,11542],{"class":10363},[40,77282,77284,77286,77288],{"class":10276,"line":77283},294,[40,77285,77267],{"class":10363},[40,77287,1044],{"class":10446},[40,77289,11542],{"class":10363},[40,77291,77293],{"class":10276,"line":77292},295,[40,77294,77295],{"class":10363},"        text,\n",[40,77297,77299,77301,77303],{"class":10276,"line":77298},296,[40,77300,76781],{"class":11903},[40,77302,162],{"class":10732},[40,77304,76786],{"class":10363},[40,77306,77308,77310,77312,77315],{"class":10276,"line":77307},297,[40,77309,12726],{"class":11903},[40,77311,162],{"class":10732},[40,77313,77314],{"class":10446},"10",[40,77316,10507],{"class":10363},[40,77318,77320,77323,77325,77327],{"class":10276,"line":77319},298,[40,77321,77322],{"class":11903},"        ha",[40,77324,162],{"class":10732},[40,77326,12710],{"class":10287},[40,77328,10507],{"class":10363},[40,77330,77332,77335,77337,77339],{"class":10276,"line":77331},299,[40,77333,77334],{"class":11903},"        va",[40,77336,162],{"class":10732},[40,77338,12710],{"class":10287},[40,77340,10507],{"class":10363},[40,77342,77344],{"class":10276,"line":77343},300,[40,77345,12631],{"class":10363},[40,77347,77349],{"class":10276,"line":77348},301,[40,77350,10301],{"emptyLinePlaceholder":10300},[40,77352,77354],{"class":10276,"line":77353},302,[40,77355,10301],{"emptyLinePlaceholder":10300},[40,77357,77359,77361,77364,77366,77368,77370,77372],{"class":10276,"line":77358},303,[40,77360,10965],{"class":10352},[40,77362,77363],{"class":10430}," add_initial_state_label",[40,77365,167],{"class":10363},[40,77367,11275],{"class":10973},[40,77369,22117],{"class":10363},[40,77371,73064],{"class":10446},[40,77373,10639],{"class":10363},[40,77375,77377],{"class":10276,"line":77376},304,[40,77378,77379],{"class":10287},"    \"\"\"Label the initialized ground-state pole.\"\"\"\n",[40,77381,77383,77385,77387],{"class":10276,"line":77382},305,[40,77384,76743],{"class":10363},[40,77386,76894],{"class":10430},[40,77388,12527],{"class":10363},[40,77390,77392,77394],{"class":10276,"line":77391},306,[40,77393,77058],{"class":10446},[40,77395,10507],{"class":10363},[40,77397,77399,77401],{"class":10276,"line":77398},307,[40,77400,77058],{"class":10446},[40,77402,10507],{"class":10363},[40,77404,77406,77409],{"class":10276,"line":77405},308,[40,77407,77408],{"class":10446},"        1.0",[40,77410,10507],{"class":10363},[40,77412,77414,77416,77418,77420],{"class":10276,"line":77413},309,[40,77415,76781],{"class":11903},[40,77417,162],{"class":10732},[40,77419,75048],{"class":10446},[40,77421,10507],{"class":10363},[40,77423,77425,77427,77429,77431],{"class":10276,"line":77424},310,[40,77426,76939],{"class":11903},[40,77428,162],{"class":10732},[40,77430,75082],{"class":10446},[40,77432,10507],{"class":10363},[40,77434,77436,77438,77440,77442],{"class":10276,"line":77435},311,[40,77437,76951],{"class":11903},[40,77439,162],{"class":10732},[40,77441,13091],{"class":10446},[40,77443,10507],{"class":10363},[40,77445,77447],{"class":10276,"line":77446},312,[40,77448,12631],{"class":10363},[40,77450,77452,77454,77456],{"class":10276,"line":77451},313,[40,77453,76743],{"class":10363},[40,77455,2932],{"class":10430},[40,77457,12527],{"class":10363},[40,77459,77461,77463],{"class":10276,"line":77460},314,[40,77462,77058],{"class":10446},[40,77464,10507],{"class":10363},[40,77466,77468,77470],{"class":10276,"line":77467},315,[40,77469,77058],{"class":10446},[40,77471,10507],{"class":10363},[40,77473,77475,77478],{"class":10276,"line":77474},316,[40,77476,77477],{"class":10446},"        1.16",[40,77479,10507],{"class":10363},[40,77481,77483,77486,77489,77491,77493,77496,77499],{"class":10276,"line":77482},317,[40,77484,77485],{"class":10352},"        r",[40,77487,77488],{"class":10356},"\"initial $",[40,77490,24461],{"class":10363},[40,77492,58190],{"class":10356},[40,77494,77495],{"class":10732},"\\r",[40,77497,77498],{"class":10356},"angle$\"",[40,77500,10507],{"class":10363},[40,77502,77504,77506,77508,77510],{"class":10276,"line":77503},318,[40,77505,76781],{"class":11903},[40,77507,162],{"class":10732},[40,77509,75048],{"class":10446},[40,77511,10507],{"class":10363},[40,77513,77515,77517,77519,77521],{"class":10276,"line":77514},319,[40,77516,12726],{"class":11903},[40,77518,162],{"class":10732},[40,77520,13793],{"class":10446},[40,77522,10507],{"class":10363},[40,77524,77526,77528,77530,77532],{"class":10276,"line":77525},320,[40,77527,77322],{"class":11903},[40,77529,162],{"class":10732},[40,77531,12710],{"class":10287},[40,77533,10507],{"class":10363},[40,77535,77537,77539,77541,77543],{"class":10276,"line":77536},321,[40,77538,77334],{"class":11903},[40,77540,162],{"class":10732},[40,77542,12710],{"class":10287},[40,77544,10507],{"class":10363},[40,77546,77548],{"class":10276,"line":77547},322,[40,77549,12631],{"class":10363},[40,77551,77553],{"class":10276,"line":77552},323,[40,77554,10301],{"emptyLinePlaceholder":10300},[40,77556,77558],{"class":10276,"line":77557},324,[40,77559,10301],{"emptyLinePlaceholder":10300},[40,77561,77563,77565,77568,77570,77572],{"class":10276,"line":77562},325,[40,77564,11225],{"class":10352},[40,77566,77567],{"class":10446}," BLOCH_DURATION_NS",[40,77569,8711],{"class":10352},[40,77571,73802],{"class":10446},[40,77573,10639],{"class":10363},[40,77575,77577,77580,77582,77584],{"class":10276,"line":77576},326,[40,77578,77579],{"class":10363},"    bloch_duration_ns ",[40,77581,162],{"class":10732},[40,77583,75754],{"class":10430},[40,77585,12527],{"class":10363},[40,77587,77589],{"class":10276,"line":77588},327,[40,77590,77591],{"class":10446},"        BLOCH_OMEGA_ONRESONANCE_MHZ\n",[40,77593,77595],{"class":10276,"line":77594},328,[40,77596,12631],{"class":10363},[40,77598,77600,77602],{"class":10276,"line":77599},329,[40,77601,22464],{"class":10352},[40,77603,10639],{"class":10363},[40,77605,77607,77609,77611],{"class":10276,"line":77606},330,[40,77608,77579],{"class":10363},[40,77610,162],{"class":10732},[40,77612,77613],{"class":10446}," BLOCH_DURATION_NS\n",[40,77615,77617],{"class":10276,"line":77616},331,[40,77618,10301],{"emptyLinePlaceholder":10300},[40,77620,77622,77625,77627,77629],{"class":10276,"line":77621},332,[40,77623,77624],{"class":10363},"resonant_trajectory ",[40,77626,162],{"class":10732},[40,77628,75807],{"class":10430},[40,77630,12527],{"class":10363},[40,77632,77634,77636,77638,77640],{"class":10276,"line":77633},333,[40,77635,75814],{"class":11903},[40,77637,162],{"class":10732},[40,77639,74856],{"class":10446},[40,77641,10507],{"class":10363},[40,77643,77645,77647,77649],{"class":10276,"line":77644},334,[40,77646,75825],{"class":11903},[40,77648,162],{"class":10732},[40,77650,77651],{"class":10363},"bloch_duration_ns,\n",[40,77653,77655,77657,77659,77661],{"class":10276,"line":77654},335,[40,77656,75836],{"class":11903},[40,77658,162],{"class":10732},[40,77660,74898],{"class":10446},[40,77662,10507],{"class":10363},[40,77664,77666],{"class":10276,"line":77665},336,[40,77667,10761],{"class":10363},[40,77669,77671],{"class":10276,"line":77670},337,[40,77672,10301],{"emptyLinePlaceholder":10300},[40,77674,77676,77679,77681,77683],{"class":10276,"line":77675},338,[40,77677,77678],{"class":10363},"detuned_trajectory ",[40,77680,162],{"class":10732},[40,77682,75807],{"class":10430},[40,77684,12527],{"class":10363},[40,77686,77688,77690,77692,77694],{"class":10276,"line":77687},339,[40,77689,75814],{"class":11903},[40,77691,162],{"class":10732},[40,77693,74865],{"class":10446},[40,77695,10507],{"class":10363},[40,77697,77699,77701,77703],{"class":10276,"line":77698},340,[40,77700,75825],{"class":11903},[40,77702,162],{"class":10732},[40,77704,77651],{"class":10363},[40,77706,77708,77710,77712,77714],{"class":10276,"line":77707},341,[40,77709,75836],{"class":11903},[40,77711,162],{"class":10732},[40,77713,74898],{"class":10446},[40,77715,10507],{"class":10363},[40,77717,77719],{"class":10276,"line":77718},342,[40,77720,10761],{"class":10363},[40,77722,77724],{"class":10276,"line":77723},343,[40,77725,10301],{"emptyLinePlaceholder":10300},[40,77727,77729,77732,77734],{"class":10276,"line":77728},344,[40,77730,77731],{"class":10363},"resonant_drive_ghz ",[40,77733,162],{"class":10732},[40,77735,22894],{"class":10363},[40,77737,77739],{"class":10276,"line":77738},345,[40,77740,77741],{"class":10446},"    BLOCH_F0_GHZ\n",[40,77743,77745,77748,77751,77753],{"class":10276,"line":77744},346,[40,77746,77747],{"class":10732},"    +",[40,77749,77750],{"class":10446}," BLOCH_RESONANT_DETUNING_MHZ",[40,77752,11881],{"class":10732},[40,77754,77755],{"class":10446}," 1.0e-3\n",[40,77757,77759],{"class":10276,"line":77758},347,[40,77760,10761],{"class":10363},[40,77762,77764,77767,77769],{"class":10276,"line":77763},348,[40,77765,77766],{"class":10363},"detuned_drive_ghz ",[40,77768,162],{"class":10732},[40,77770,22894],{"class":10363},[40,77772,77774],{"class":10276,"line":77773},349,[40,77775,77741],{"class":10446},[40,77777,77779,77781,77784,77786],{"class":10276,"line":77778},350,[40,77780,77747],{"class":10732},[40,77782,77783],{"class":10446}," BLOCH_DETUNED_DETUNING_MHZ",[40,77785,11881],{"class":10732},[40,77787,77755],{"class":10446},[40,77789,77791],{"class":10276,"line":77790},351,[40,77792,10761],{"class":10363},[40,77794,77796],{"class":10276,"line":77795},352,[40,77797,10301],{"emptyLinePlaceholder":10300},[40,77799,77801,77804,77806,77808,77810,77812,77814,77816,77818],{"class":10276,"line":77800},353,[40,77802,77803],{"class":10363},"figure ",[40,77805,162],{"class":10732},[40,77807,13070],{"class":10363},[40,77809,16373],{"class":10430},[40,77811,167],{"class":10363},[40,77813,13078],{"class":11903},[40,77815,162],{"class":10732},[40,77817,74908],{"class":10446},[40,77819,10761],{"class":10363},[40,77821,77823],{"class":10276,"line":77822},354,[40,77824,10301],{"emptyLinePlaceholder":10300},[40,77826,77828,77831,77833,77835,77838,77840,77843,77845],{"class":10276,"line":77827},355,[40,77829,77830],{"class":10363},"bloch ",[40,77832,162],{"class":10732},[40,77834,75968],{"class":10363},[40,77836,77837],{"class":10430},"Bloch",[40,77839,167],{"class":10363},[40,77841,77842],{"class":11903},"fig",[40,77844,162],{"class":10732},[40,77846,77847],{"class":10363},"figure)\n",[40,77849,77851,77854,77856],{"class":10276,"line":77850},356,[40,77852,77853],{"class":10363},"bloch.view ",[40,77855,162],{"class":10732},[40,77857,77858],{"class":10446}," BLOCH_VIEW\n",[40,77860,77862,77865,77867,77869,77871,77874,77876,77878,77880,77882,77884,77886,77888,77890,77892,77894,77896],{"class":10276,"line":77861},357,[40,77863,77864],{"class":10363},"bloch.zlabel ",[40,77866,162],{"class":10732},[40,77868,24010],{"class":10363},[40,77870,936],{"class":10352},[40,77872,77873],{"class":10356},"\"$",[40,77875,24461],{"class":10363},[40,77877,58190],{"class":10356},[40,77879,77495],{"class":10732},[40,77881,77498],{"class":10356},[40,77883,10484],{"class":10363},[40,77885,936],{"class":10352},[40,77887,77873],{"class":10356},[40,77889,24461],{"class":10363},[40,77891,4203],{"class":10356},[40,77893,77495],{"class":10732},[40,77895,77498],{"class":10356},[40,77897,25466],{"class":10363},[40,77899,77901,77904,77906],{"class":10276,"line":77900},358,[40,77902,77903],{"class":10363},"bloch.title ",[40,77905,162],{"class":10732},[40,77907,22894],{"class":10363},[40,77909,77911,77914,77916],{"class":10276,"line":77910},359,[40,77912,77913],{"class":10287},"    \"Bloch-sphere Rabi trajectories",[40,77915,13454],{"class":10732},[40,77917,14632],{"class":10287},[40,77919,77921,77924,77927,77930,77932,77934],{"class":10276,"line":77920},360,[40,77922,77923],{"class":10352},"    rf",[40,77925,77926],{"class":10287},"\"resonant $\\Delta\u002F2\\pi=",[40,77928,77929],{"class":10446},"{BLOCH_RESONANT_DETUNING_MHZ",[40,77931,14652],{"class":10352},[40,77933,13280],{"class":10446},[40,77935,77936],{"class":10287},"$ MHz, \"\n",[40,77938,77940,77942,77945,77948,77950,77952],{"class":10276,"line":77939},361,[40,77941,77923],{"class":10352},[40,77943,77944],{"class":10287},"\"detuned $\\Delta\u002F2\\pi=",[40,77946,77947],{"class":10446},"{BLOCH_DETUNED_DETUNING_MHZ",[40,77949,14652],{"class":10352},[40,77951,13280],{"class":10446},[40,77953,77954],{"class":10287},"$ MHz\"\n",[40,77956,77958],{"class":10276,"line":77957},362,[40,77959,10761],{"class":10363},[40,77961,77963],{"class":10276,"line":77962},363,[40,77964,10301],{"emptyLinePlaceholder":10300},[40,77966,77968,77971,77973,77975],{"class":10276,"line":77967},364,[40,77969,77970],{"class":10363},"bloch_axis ",[40,77972,162],{"class":10732},[40,77974,76454],{"class":10430},[40,77976,12527],{"class":10363},[40,77978,77980,77982,77984],{"class":10276,"line":77979},365,[40,77981,76462],{"class":11903},[40,77983,162],{"class":10732},[40,77985,77986],{"class":10363},"bloch,\n",[40,77988,77990,77992,77994],{"class":10276,"line":77989},366,[40,77991,76471],{"class":11903},[40,77993,162],{"class":10732},[40,77995,77996],{"class":10363},"figure,\n",[40,77998,78000],{"class":10276,"line":77999},367,[40,78001,10761],{"class":10363},[40,78003,78005],{"class":10276,"line":78004},368,[40,78006,10301],{"emptyLinePlaceholder":10300},[40,78008,78010,78013],{"class":10276,"line":78009},369,[40,78011,78012],{"class":10430},"draw_bloch_trajectory",[40,78014,12527],{"class":10363},[40,78016,78018,78020,78022],{"class":10276,"line":78017},370,[40,78019,76688],{"class":11903},[40,78021,162],{"class":10732},[40,78023,78024],{"class":10363},"bloch_axis,\n",[40,78026,78028,78030,78032,78035,78038],{"class":10276,"line":78027},371,[40,78029,76696],{"class":11903},[40,78031,162],{"class":10732},[40,78033,78034],{"class":10363},"resonant_trajectory[",[40,78036,78037],{"class":10287},"\"vectors\"",[40,78039,11542],{"class":10363},[40,78041,78043,78045,78047,78049],{"class":10276,"line":78042},372,[40,78044,76704],{"class":11903},[40,78046,162],{"class":10732},[40,78048,75008],{"class":10446},[40,78050,10507],{"class":10363},[40,78052,78054,78056,78058,78061],{"class":10276,"line":78053},373,[40,78055,76716],{"class":11903},[40,78057,162],{"class":10732},[40,78059,78060],{"class":10287},"\"Resonant state trajectory\"",[40,78062,10507],{"class":10363},[40,78064,78066],{"class":10276,"line":78065},374,[40,78067,10761],{"class":10363},[40,78069,78071,78073],{"class":10276,"line":78070},375,[40,78072,78012],{"class":10430},[40,78074,12527],{"class":10363},[40,78076,78078,78080,78082],{"class":10276,"line":78077},376,[40,78079,76688],{"class":11903},[40,78081,162],{"class":10732},[40,78083,78024],{"class":10363},[40,78085,78087,78089,78091,78094,78096],{"class":10276,"line":78086},377,[40,78088,76696],{"class":11903},[40,78090,162],{"class":10732},[40,78092,78093],{"class":10363},"detuned_trajectory[",[40,78095,78037],{"class":10287},[40,78097,11542],{"class":10363},[40,78099,78101,78103,78105,78107],{"class":10276,"line":78100},378,[40,78102,76704],{"class":11903},[40,78104,162],{"class":10732},[40,78106,75018],{"class":10446},[40,78108,10507],{"class":10363},[40,78110,78112,78114,78116,78119],{"class":10276,"line":78111},379,[40,78113,76716],{"class":11903},[40,78115,162],{"class":10732},[40,78117,78118],{"class":10287},"\"Detuned state trajectory\"",[40,78120,10507],{"class":10363},[40,78122,78124],{"class":10276,"line":78123},380,[40,78125,10761],{"class":10363},[40,78127,78129],{"class":10276,"line":78128},381,[40,78130,10301],{"emptyLinePlaceholder":10300},[40,78132,78134,78136,78139],{"class":10276,"line":78133},382,[40,78135,11225],{"class":10352},[40,78137,78138],{"class":10446}," BLOCH_SHOW_ENDPOINT_MARKERS",[40,78140,10639],{"class":10363},[40,78142,78144,78147],{"class":10276,"line":78143},383,[40,78145,78146],{"class":10430},"    draw_endpoint_marker",[40,78148,12527],{"class":10363},[40,78150,78152,78155,78157],{"class":10276,"line":78151},384,[40,78153,78154],{"class":11903},"        axis",[40,78156,162],{"class":10732},[40,78158,78024],{"class":10363},[40,78160,78162,78165,78167,78169,78171,78174,78176,78178],{"class":10276,"line":78161},385,[40,78163,78164],{"class":11903},"        vector",[40,78166,162],{"class":10732},[40,78168,78034],{"class":10363},[40,78170,78037],{"class":10287},[40,78172,78173],{"class":10363},"][:, ",[40,78175,11056],{"class":10732},[40,78177,78],{"class":10446},[40,78179,11542],{"class":10363},[40,78181,78183,78185,78187,78189],{"class":10276,"line":78182},386,[40,78184,76781],{"class":11903},[40,78186,162],{"class":10732},[40,78188,75008],{"class":10446},[40,78190,10507],{"class":10363},[40,78192,78194,78196,78198,78201],{"class":10276,"line":78193},387,[40,78195,76804],{"class":11903},[40,78197,162],{"class":10732},[40,78199,78200],{"class":10287},"\"Resonant final state\"",[40,78202,10507],{"class":10363},[40,78204,78206],{"class":10276,"line":78205},388,[40,78207,12631],{"class":10363},[40,78209,78211,78213],{"class":10276,"line":78210},389,[40,78212,78146],{"class":10430},[40,78214,12527],{"class":10363},[40,78216,78218,78220,78222],{"class":10276,"line":78217},390,[40,78219,78154],{"class":11903},[40,78221,162],{"class":10732},[40,78223,78024],{"class":10363},[40,78225,78227,78229,78231,78233,78235,78237,78239,78241],{"class":10276,"line":78226},391,[40,78228,78164],{"class":11903},[40,78230,162],{"class":10732},[40,78232,78093],{"class":10363},[40,78234,78037],{"class":10287},[40,78236,78173],{"class":10363},[40,78238,11056],{"class":10732},[40,78240,78],{"class":10446},[40,78242,11542],{"class":10363},[40,78244,78246,78248,78250,78252],{"class":10276,"line":78245},392,[40,78247,76781],{"class":11903},[40,78249,162],{"class":10732},[40,78251,75018],{"class":10446},[40,78253,10507],{"class":10363},[40,78255,78257,78259,78261,78264],{"class":10276,"line":78256},393,[40,78258,76804],{"class":11903},[40,78260,162],{"class":10732},[40,78262,78263],{"class":10287},"\"Detuned final state\"",[40,78265,10507],{"class":10363},[40,78267,78269],{"class":10276,"line":78268},394,[40,78270,12631],{"class":10363},[40,78272,78274,78277],{"class":10276,"line":78273},395,[40,78275,78276],{"class":10430},"    add_initial_state_label",[40,78278,78279],{"class":10363},"(bloch_axis)\n",[40,78281,78283],{"class":10276,"line":78282},396,[40,78284,10301],{"emptyLinePlaceholder":10300},[40,78286,78288,78290,78293],{"class":10276,"line":78287},397,[40,78289,11225],{"class":10352},[40,78291,78292],{"class":10446}," BLOCH_SHOW_CONTROL_AXIS_ARROWS",[40,78294,10639],{"class":10363},[40,78296,78298,78301],{"class":10276,"line":78297},398,[40,78299,78300],{"class":10430},"    draw_control_axis_arrow",[40,78302,12527],{"class":10363},[40,78304,78306,78308,78310],{"class":10276,"line":78305},399,[40,78307,78154],{"class":11903},[40,78309,162],{"class":10732},[40,78311,78024],{"class":10363},[40,78313,78315,78317,78319,78321,78324],{"class":10276,"line":78314},400,[40,78316,78164],{"class":11903},[40,78318,162],{"class":10732},[40,78320,78034],{"class":10363},[40,78322,78323],{"class":10287},"\"control_axis\"",[40,78325,11542],{"class":10363},[40,78327,78329,78331,78333,78335],{"class":10276,"line":78328},401,[40,78330,76781],{"class":11903},[40,78332,162],{"class":10732},[40,78334,75028],{"class":10446},[40,78336,10507],{"class":10363},[40,78338,78340,78342,78344,78346,78349,78352,78355,78358,78361],{"class":10276,"line":78339},402,[40,78341,76804],{"class":11903},[40,78343,162],{"class":10732},[40,78345,936],{"class":10352},[40,78347,78348],{"class":10356},"\"Resonant control axis $",[40,78350,78351],{"class":10732},"\\h",[40,78353,78354],{"class":10356},"at{n}_{",[40,78356,78357],{"class":10732},"\\m",[40,78359,78360],{"class":10356},"athrm{res}}$\"",[40,78362,10507],{"class":10363},[40,78364,78366],{"class":10276,"line":78365},403,[40,78367,12631],{"class":10363},[40,78369,78371,78373],{"class":10276,"line":78370},404,[40,78372,78300],{"class":10430},[40,78374,12527],{"class":10363},[40,78376,78378,78380,78382],{"class":10276,"line":78377},405,[40,78379,78154],{"class":11903},[40,78381,162],{"class":10732},[40,78383,78024],{"class":10363},[40,78385,78387,78389,78391,78393,78395],{"class":10276,"line":78386},406,[40,78388,78164],{"class":11903},[40,78390,162],{"class":10732},[40,78392,78093],{"class":10363},[40,78394,78323],{"class":10287},[40,78396,11542],{"class":10363},[40,78398,78400,78402,78404,78406],{"class":10276,"line":78399},407,[40,78401,76781],{"class":11903},[40,78403,162],{"class":10732},[40,78405,75038],{"class":10446},[40,78407,10507],{"class":10363},[40,78409,78411,78413,78415,78417,78420,78422,78424,78426,78429],{"class":10276,"line":78410},408,[40,78412,76804],{"class":11903},[40,78414,162],{"class":10732},[40,78416,936],{"class":10352},[40,78418,78419],{"class":10356},"\"Detuned control axis $",[40,78421,78351],{"class":10732},[40,78423,78354],{"class":10356},[40,78425,78357],{"class":10732},[40,78427,78428],{"class":10356},"athrm{det}}$\"",[40,78430,10507],{"class":10363},[40,78432,78434],{"class":10276,"line":78433},409,[40,78435,12631],{"class":10363},[40,78437,78439],{"class":10276,"line":78438},410,[40,78440,10301],{"emptyLinePlaceholder":10300},[40,78442,78444,78446,78449,78451,78453],{"class":10276,"line":78443},411,[40,78445,11225],{"class":10352},[40,78447,78448],{"class":10446}," BLOCH_SHOW_ARROW_TIP_LABELS",[40,78450,22438],{"class":10352},[40,78452,78292],{"class":10446},[40,78454,10639],{"class":10363},[40,78456,78458,78461],{"class":10276,"line":78457},412,[40,78459,78460],{"class":10430},"    add_arrow_tip_label",[40,78462,12527],{"class":10363},[40,78464,78466,78468,78470],{"class":10276,"line":78465},413,[40,78467,78154],{"class":11903},[40,78469,162],{"class":10732},[40,78471,78024],{"class":10363},[40,78473,78475,78477,78479,78481,78483],{"class":10276,"line":78474},414,[40,78476,78164],{"class":11903},[40,78478,162],{"class":10732},[40,78480,78034],{"class":10363},[40,78482,78323],{"class":10287},[40,78484,11542],{"class":10363},[40,78486,78488,78491,78493,78495,78497,78499,78501,78503,78505],{"class":10276,"line":78487},415,[40,78489,78490],{"class":11903},"        text",[40,78492,162],{"class":10732},[40,78494,936],{"class":10352},[40,78496,77873],{"class":10356},[40,78498,78351],{"class":10732},[40,78500,78354],{"class":10356},[40,78502,78357],{"class":10732},[40,78504,78360],{"class":10356},[40,78506,10507],{"class":10363},[40,78508,78510,78512,78514,78516],{"class":10276,"line":78509},416,[40,78511,76781],{"class":11903},[40,78513,162],{"class":10732},[40,78515,75028],{"class":10446},[40,78517,10507],{"class":10363},[40,78519,78521],{"class":10276,"line":78520},417,[40,78522,12631],{"class":10363},[40,78524,78526,78528],{"class":10276,"line":78525},418,[40,78527,78460],{"class":10430},[40,78529,12527],{"class":10363},[40,78531,78533,78535,78537],{"class":10276,"line":78532},419,[40,78534,78154],{"class":11903},[40,78536,162],{"class":10732},[40,78538,78024],{"class":10363},[40,78540,78542,78544,78546,78548,78550],{"class":10276,"line":78541},420,[40,78543,78164],{"class":11903},[40,78545,162],{"class":10732},[40,78547,78093],{"class":10363},[40,78549,78323],{"class":10287},[40,78551,11542],{"class":10363},[40,78553,78555,78557,78559,78561,78563,78565,78567,78569,78571],{"class":10276,"line":78554},421,[40,78556,78490],{"class":11903},[40,78558,162],{"class":10732},[40,78560,936],{"class":10352},[40,78562,77873],{"class":10356},[40,78564,78351],{"class":10732},[40,78566,78354],{"class":10356},[40,78568,78357],{"class":10732},[40,78570,78428],{"class":10356},[40,78572,10507],{"class":10363},[40,78574,78576,78578,78580,78582],{"class":10276,"line":78575},422,[40,78577,76781],{"class":11903},[40,78579,162],{"class":10732},[40,78581,75038],{"class":10446},[40,78583,10507],{"class":10363},[40,78585,78587],{"class":10276,"line":78586},423,[40,78588,12631],{"class":10363},[40,78590,78592],{"class":10276,"line":78591},424,[40,78593,10301],{"emptyLinePlaceholder":10300},[40,78595,78597,78599,78602],{"class":10276,"line":78596},425,[40,78598,11225],{"class":10352},[40,78600,78601],{"class":10446}," BLOCH_SHOW_LEGEND",[40,78603,10639],{"class":10363},[40,78605,78607,78610,78612],{"class":10276,"line":78606},426,[40,78608,78609],{"class":10363},"    legend_handles ",[40,78611,162],{"class":10732},[40,78613,37422],{"class":10363},[40,78615,78617,78620],{"class":10276,"line":78616},427,[40,78618,78619],{"class":10430},"        Line2D",[40,78621,12527],{"class":10363},[40,78623,78625,78628,78630],{"class":10276,"line":78624},428,[40,78626,78627],{"class":10363},"            [",[40,78629,66],{"class":10446},[40,78631,11542],{"class":10363},[40,78633,78635,78637,78639],{"class":10276,"line":78634},429,[40,78636,78627],{"class":10363},[40,78638,66],{"class":10446},[40,78640,11542],{"class":10363},[40,78642,78644,78647,78649,78651],{"class":10276,"line":78643},430,[40,78645,78646],{"class":11903},"            color",[40,78648,162],{"class":10732},[40,78650,75008],{"class":10446},[40,78652,10507],{"class":10363},[40,78654,78656,78659,78661,78663],{"class":10276,"line":78655},431,[40,78657,78658],{"class":11903},"            linewidth",[40,78660,162],{"class":10732},[40,78662,75062],{"class":10446},[40,78664,10507],{"class":10363},[40,78666,78668,78671,78673,78675],{"class":10276,"line":78667},432,[40,78669,78670],{"class":11903},"            label",[40,78672,162],{"class":10732},[40,78674,78060],{"class":10287},[40,78676,10507],{"class":10363},[40,78678,78680],{"class":10276,"line":78679},433,[40,78681,76431],{"class":10363},[40,78683,78685,78687],{"class":10276,"line":78684},434,[40,78686,78619],{"class":10430},[40,78688,12527],{"class":10363},[40,78690,78692,78694,78696],{"class":10276,"line":78691},435,[40,78693,78627],{"class":10363},[40,78695,66],{"class":10446},[40,78697,11542],{"class":10363},[40,78699,78701,78703,78705],{"class":10276,"line":78700},436,[40,78702,78627],{"class":10363},[40,78704,66],{"class":10446},[40,78706,11542],{"class":10363},[40,78708,78710,78712,78714,78716],{"class":10276,"line":78709},437,[40,78711,78646],{"class":11903},[40,78713,162],{"class":10732},[40,78715,75018],{"class":10446},[40,78717,10507],{"class":10363},[40,78719,78721,78723,78725,78727],{"class":10276,"line":78720},438,[40,78722,78658],{"class":11903},[40,78724,162],{"class":10732},[40,78726,75062],{"class":10446},[40,78728,10507],{"class":10363},[40,78730,78732,78734,78736,78738],{"class":10276,"line":78731},439,[40,78733,78670],{"class":11903},[40,78735,162],{"class":10732},[40,78737,78118],{"class":10287},[40,78739,10507],{"class":10363},[40,78741,78743],{"class":10276,"line":78742},440,[40,78744,76431],{"class":10363},[40,78746,78748,78750],{"class":10276,"line":78747},441,[40,78749,78619],{"class":10430},[40,78751,12527],{"class":10363},[40,78753,78755,78757,78759],{"class":10276,"line":78754},442,[40,78756,78627],{"class":10363},[40,78758,66],{"class":10446},[40,78760,11542],{"class":10363},[40,78762,78764,78766,78768],{"class":10276,"line":78763},443,[40,78765,78627],{"class":10363},[40,78767,66],{"class":10446},[40,78769,11542],{"class":10363},[40,78771,78773,78775,78777,78779],{"class":10276,"line":78772},444,[40,78774,78646],{"class":11903},[40,78776,162],{"class":10732},[40,78778,75028],{"class":10446},[40,78780,10507],{"class":10363},[40,78782,78784,78786,78788,78790],{"class":10276,"line":78783},445,[40,78785,78658],{"class":11903},[40,78787,162],{"class":10732},[40,78789,75072],{"class":10446},[40,78791,10507],{"class":10363},[40,78793,78795,78797,78799,78801,78803,78805,78807,78809,78811],{"class":10276,"line":78794},446,[40,78796,78670],{"class":11903},[40,78798,162],{"class":10732},[40,78800,936],{"class":10352},[40,78802,78348],{"class":10356},[40,78804,78351],{"class":10732},[40,78806,78354],{"class":10356},[40,78808,78357],{"class":10732},[40,78810,78360],{"class":10356},[40,78812,10507],{"class":10363},[40,78814,78816],{"class":10276,"line":78815},447,[40,78817,76431],{"class":10363},[40,78819,78821,78823],{"class":10276,"line":78820},448,[40,78822,78619],{"class":10430},[40,78824,12527],{"class":10363},[40,78826,78828,78830,78832],{"class":10276,"line":78827},449,[40,78829,78627],{"class":10363},[40,78831,66],{"class":10446},[40,78833,11542],{"class":10363},[40,78835,78837,78839,78841],{"class":10276,"line":78836},450,[40,78838,78627],{"class":10363},[40,78840,66],{"class":10446},[40,78842,11542],{"class":10363},[40,78844,78846,78848,78850,78852],{"class":10276,"line":78845},451,[40,78847,78646],{"class":11903},[40,78849,162],{"class":10732},[40,78851,75038],{"class":10446},[40,78853,10507],{"class":10363},[40,78855,78857,78859,78861,78863],{"class":10276,"line":78856},452,[40,78858,78658],{"class":11903},[40,78860,162],{"class":10732},[40,78862,75072],{"class":10446},[40,78864,10507],{"class":10363},[40,78866,78868,78870,78872,78874,78876,78878,78880,78882,78884],{"class":10276,"line":78867},453,[40,78869,78670],{"class":11903},[40,78871,162],{"class":10732},[40,78873,936],{"class":10352},[40,78875,78419],{"class":10356},[40,78877,78351],{"class":10732},[40,78879,78354],{"class":10356},[40,78881,78357],{"class":10732},[40,78883,78428],{"class":10356},[40,78885,10507],{"class":10363},[40,78887,78889],{"class":10276,"line":78888},454,[40,78890,76431],{"class":10363},[40,78892,78894],{"class":10276,"line":78893},455,[40,78895,75308],{"class":10363},[40,78897,78899],{"class":10276,"line":78898},456,[40,78900,10301],{"emptyLinePlaceholder":10300},[40,78902,78904,78907,78909],{"class":10276,"line":78903},457,[40,78905,78906],{"class":10363},"    bloch_axis.",[40,78908,23677],{"class":10430},[40,78910,12527],{"class":10363},[40,78912,78914,78917,78919],{"class":10276,"line":78913},458,[40,78915,78916],{"class":11903},"        handles",[40,78918,162],{"class":10732},[40,78920,78921],{"class":10363},"legend_handles,\n",[40,78923,78925,78928,78930,78932],{"class":10276,"line":78924},459,[40,78926,78927],{"class":11903},"        loc",[40,78929,162],{"class":10732},[40,78931,75116],{"class":10446},[40,78933,10507],{"class":10363},[40,78935,78937,78940,78942,78944],{"class":10276,"line":78936},460,[40,78938,78939],{"class":11903},"        bbox_to_anchor",[40,78941,162],{"class":10732},[40,78943,75126],{"class":10446},[40,78945,10507],{"class":10363},[40,78947,78949,78952,78954,78956],{"class":10276,"line":78948},461,[40,78950,78951],{"class":11903},"        frameon",[40,78953,162],{"class":10732},[40,78955,13091],{"class":10446},[40,78957,10507],{"class":10363},[40,78959,78961,78964,78966,78968],{"class":10276,"line":78960},462,[40,78962,78963],{"class":11903},"        borderaxespad",[40,78965,162],{"class":10732},[40,78967,10801],{"class":10446},[40,78969,10507],{"class":10363},[40,78971,78973],{"class":10276,"line":78972},463,[40,78974,12631],{"class":10363},[40,78976,78978],{"class":10276,"line":78977},464,[40,78979,10301],{"emptyLinePlaceholder":10300},[40,78981,78983,78985,78987],{"class":10276,"line":78982},465,[40,78984,23469],{"class":10363},[40,78986,23686],{"class":10430},[40,78988,12174],{"class":10363},[40,78990,78992],{"class":10276,"line":78991},466,[40,78993,10301],{"emptyLinePlaceholder":10300},[40,78995,78997,78999,79002],{"class":10276,"line":78996},467,[40,78998,11225],{"class":10352},[40,79000,79001],{"class":10446}," BLOCH_PRINT_SUMMARY",[40,79003,10639],{"class":10363},[40,79005,79007,79010,79012,79015,79017],{"class":10276,"line":79006},468,[40,79008,79009],{"class":10363},"    resonant_axis ",[40,79011,162],{"class":10732},[40,79013,79014],{"class":10363}," resonant_trajectory[",[40,79016,78323],{"class":10287},[40,79018,25466],{"class":10363},[40,79020,79022,79025,79027,79030,79032],{"class":10276,"line":79021},469,[40,79023,79024],{"class":10363},"    detuned_axis ",[40,79026,162],{"class":10732},[40,79028,79029],{"class":10363}," detuned_trajectory[",[40,79031,78323],{"class":10287},[40,79033,25466],{"class":10363},[40,79035,79037],{"class":10276,"line":79036},470,[40,79038,10301],{"emptyLinePlaceholder":10300},[40,79040,79042,79044,79046,79049],{"class":10276,"line":79041},471,[40,79043,13969],{"class":10732},[40,79045,167],{"class":10363},[40,79047,79048],{"class":10287},"\"Bloch-sphere trajectory summary\"",[40,79050,10761],{"class":10363},[40,79052,79054,79056,79058,79060,79063,79066,79069,79071,79074],{"class":10276,"line":79053},472,[40,79055,13969],{"class":10732},[40,79057,167],{"class":10363},[40,79059,13262],{"class":10352},[40,79061,79062],{"class":10287},"\"Natural transition frequency f0: ",[40,79064,79065],{"class":10446},"{BLOCH_F0_GHZ",[40,79067,79068],{"class":10352},":.6f",[40,79070,13280],{"class":10446},[40,79072,79073],{"class":10287}," GHz\"",[40,79075,10761],{"class":10363},[40,79077,79079,79081,79083,79085,79088,79090,79093,79095,79097,79099],{"class":10276,"line":79078},473,[40,79080,13969],{"class":10732},[40,79082,167],{"class":10363},[40,79084,13262],{"class":10352},[40,79086,79087],{"class":10287},"\"Resonant drive frequency: ",[40,79089,13268],{"class":10446},[40,79091,79092],{"class":10363},"resonant_drive_ghz",[40,79094,79068],{"class":10352},[40,79096,13280],{"class":10446},[40,79098,79073],{"class":10287},[40,79100,10761],{"class":10363},[40,79102,79104,79106,79108,79110,79113,79115,79118,79120,79122,79124],{"class":10276,"line":79103},474,[40,79105,13969],{"class":10732},[40,79107,167],{"class":10363},[40,79109,13262],{"class":10352},[40,79111,79112],{"class":10287},"\"Detuned drive frequency: ",[40,79114,13268],{"class":10446},[40,79116,79117],{"class":10363},"detuned_drive_ghz",[40,79119,79068],{"class":10352},[40,79121,13280],{"class":10446},[40,79123,79073],{"class":10287},[40,79125,10761],{"class":10363},[40,79127,79129,79131,79133,79135,79138,79141,79143,79145,79148],{"class":10276,"line":79128},475,[40,79130,13969],{"class":10732},[40,79132,167],{"class":10363},[40,79134,13262],{"class":10352},[40,79136,79137],{"class":10287},"\"On-resonance Rabi rate: ",[40,79139,79140],{"class":10446},"{BLOCH_OMEGA_ONRESONANCE_MHZ",[40,79142,13446],{"class":10352},[40,79144,13280],{"class":10446},[40,79146,79147],{"class":10287}," MHz\"",[40,79149,10761],{"class":10363},[40,79151,79153,79155,79157,79159,79162,79164,79167,79169,79171,79174],{"class":10276,"line":79152},476,[40,79154,13969],{"class":10732},[40,79156,167],{"class":10363},[40,79158,13262],{"class":10352},[40,79160,79161],{"class":10287},"\"Plotted duration: ",[40,79163,13268],{"class":10446},[40,79165,79166],{"class":10363},"bloch_duration_ns",[40,79168,13446],{"class":10352},[40,79170,13280],{"class":10446},[40,79172,79173],{"class":10287}," ns\"",[40,79175,10761],{"class":10363},[40,79177,79179,79181],{"class":10276,"line":79178},477,[40,79180,13969],{"class":10732},[40,79182,12527],{"class":10363},[40,79184,79186],{"class":10276,"line":79185},478,[40,79187,79188],{"class":10287},"        \"Resonant control axis n_res: \"\n",[40,79190,79192,79195,79198,79200,79203,79205,79207,79209,79211],{"class":10276,"line":79191},479,[40,79193,79194],{"class":10352},"        f",[40,79196,79197],{"class":10287},"\"(",[40,79199,13268],{"class":10446},[40,79201,79202],{"class":10363},"resonant_axis[",[40,79204,66],{"class":10446},[40,79206,1664],{"class":10363},[40,79208,79068],{"class":10352},[40,79210,13280],{"class":10446},[40,79212,79213],{"class":10287},", \"\n",[40,79215,79217,79219,79221,79223,79225,79227,79229,79231,79233],{"class":10276,"line":79216},480,[40,79218,79194],{"class":10352},[40,79220,13478],{"class":10287},[40,79222,13268],{"class":10446},[40,79224,79202],{"class":10363},[40,79226,78],{"class":10446},[40,79228,1664],{"class":10363},[40,79230,79068],{"class":10352},[40,79232,13280],{"class":10446},[40,79234,79213],{"class":10287},[40,79236,79238,79240,79242,79244,79246,79248,79250,79252,79254],{"class":10276,"line":79237},481,[40,79239,79194],{"class":10352},[40,79241,13478],{"class":10287},[40,79243,13268],{"class":10446},[40,79245,79202],{"class":10363},[40,79247,1044],{"class":10446},[40,79249,1664],{"class":10363},[40,79251,79068],{"class":10352},[40,79253,13280],{"class":10446},[40,79255,79256],{"class":10287},")\"\n",[40,79258,79260],{"class":10276,"line":79259},482,[40,79261,12631],{"class":10363},[40,79263,79265,79267],{"class":10276,"line":79264},483,[40,79266,13969],{"class":10732},[40,79268,12527],{"class":10363},[40,79270,79272],{"class":10276,"line":79271},484,[40,79273,79274],{"class":10287},"        \"Detuned control axis n_det: \"\n",[40,79276,79278,79280,79282,79284,79287,79289,79291,79293,79295],{"class":10276,"line":79277},485,[40,79279,79194],{"class":10352},[40,79281,79197],{"class":10287},[40,79283,13268],{"class":10446},[40,79285,79286],{"class":10363},"detuned_axis[",[40,79288,66],{"class":10446},[40,79290,1664],{"class":10363},[40,79292,79068],{"class":10352},[40,79294,13280],{"class":10446},[40,79296,79213],{"class":10287},[40,79298,79300,79302,79304,79306,79308,79310,79312,79314,79316],{"class":10276,"line":79299},486,[40,79301,79194],{"class":10352},[40,79303,13478],{"class":10287},[40,79305,13268],{"class":10446},[40,79307,79286],{"class":10363},[40,79309,78],{"class":10446},[40,79311,1664],{"class":10363},[40,79313,79068],{"class":10352},[40,79315,13280],{"class":10446},[40,79317,79213],{"class":10287},[40,79319,79321,79323,79325,79327,79329,79331,79333,79335,79337],{"class":10276,"line":79320},487,[40,79322,79194],{"class":10352},[40,79324,13478],{"class":10287},[40,79326,13268],{"class":10446},[40,79328,79286],{"class":10363},[40,79330,1044],{"class":10446},[40,79332,1664],{"class":10363},[40,79334,79068],{"class":10352},[40,79336,13280],{"class":10446},[40,79338,79256],{"class":10287},[40,79340,79342],{"class":10276,"line":79341},488,[40,79343,12631],{"class":10363},[40,79345,79347,79349],{"class":10276,"line":79346},489,[40,79348,13969],{"class":10732},[40,79350,12527],{"class":10363},[40,79352,79354],{"class":10276,"line":79353},490,[40,79355,79356],{"class":10287},"        \"Resonant final excited-state probability: \"\n",[40,79358,79360,79362,79364,79366,79368,79371,79374,79376,79378,79380,79382,79384],{"class":10276,"line":79359},491,[40,79361,79194],{"class":10352},[40,79363,13478],{"class":10287},[40,79365,13268],{"class":10446},[40,79367,78034],{"class":10363},[40,79369,79370],{"class":10287},"'excited_probability'",[40,79372,79373],{"class":10363},"][",[40,79375,11056],{"class":10732},[40,79377,78],{"class":10446},[40,79379,1664],{"class":10363},[40,79381,79068],{"class":10352},[40,79383,13280],{"class":10446},[40,79385,14632],{"class":10287},[40,79387,79389],{"class":10276,"line":79388},492,[40,79390,12631],{"class":10363},[40,79392,79394,79396],{"class":10276,"line":79393},493,[40,79395,13969],{"class":10732},[40,79397,12527],{"class":10363},[40,79399,79401],{"class":10276,"line":79400},494,[40,79402,79403],{"class":10287},"        \"Detuned final excited-state probability: \"\n",[40,79405,79407,79409,79411,79413,79415,79417,79419,79421,79423,79425,79427,79429],{"class":10276,"line":79406},495,[40,79408,79194],{"class":10352},[40,79410,13478],{"class":10287},[40,79412,13268],{"class":10446},[40,79414,78093],{"class":10363},[40,79416,79370],{"class":10287},[40,79418,79373],{"class":10363},[40,79420,11056],{"class":10732},[40,79422,78],{"class":10446},[40,79424,1664],{"class":10363},[40,79426,79068],{"class":10352},[40,79428,13280],{"class":10446},[40,79430,14632],{"class":10287},[40,79432,79434],{"class":10276,"line":79433},496,[40,79435,12631],{"class":10363},[40,79437,79439,79441],{"class":10276,"line":79438},497,[40,79440,13969],{"class":10732},[40,79442,12527],{"class":10363},[40,79444,79446],{"class":10276,"line":79445},498,[40,79447,79448],{"class":10287},"        \"Detuned generalized Rabi frequency: \"\n",[40,79450,79452,79454,79456,79458,79460,79463,79465,79467,79469],{"class":10276,"line":79451},499,[40,79453,79194],{"class":10352},[40,79455,13478],{"class":10287},[40,79457,13268],{"class":10446},[40,79459,78093],{"class":10363},[40,79461,79462],{"class":10287},"'generalized_rabi_mhz'",[40,79464,1664],{"class":10363},[40,79466,79068],{"class":10352},[40,79468,13280],{"class":10446},[40,79470,79471],{"class":10287}," MHz\"\n",[40,79473,79475],{"class":10276,"line":79474},500,[40,79476,12631],{"class":10363},[13997,79478],{"alt":79479,"src":79480},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[10082,79482,79485],{"className":79483,"code":79484,"language":2932,"meta":10090},[10085],"Bloch-sphere trajectory summary\nNatural transition frequency f0: 5.000000 GHz\nResonant drive frequency: 5.000000 GHz\nDetuned drive frequency: 5.040000 GHz\nOn-resonance Rabi rate: 20.000 MHz\nPlotted duration: 25.000 ns\nResonant control axis n_res: (1.000000, 0.000000, 0.000000)\nDetuned control axis n_det: (0.447214, 0.000000, 0.894427)\nResonant final excited-state probability: 1.000000\nDetuned final excited-state probability: 0.026264\nDetuned generalized Rabi frequency: 44.721360 MHz\n",[10088,79486,79484],{"__ignoreMap":10090},[10082,79488,79490],{"className":10269,"code":79489,"language":10271,"meta":10090,"style":10090},"# --- High-resolution 3D surface with colormap & legend  ---\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# Controls for surface quality and appearance\nUPSAMPLE_FACTOR: int = 2           # 2=~4× more faces; try 3 for even smoother\nSURFACE_CMAP: str = 'inferno'      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\nZ_MIN, Z_MAX = 0.0, 1.0            # clamp\u002Fnormalize expected Pe range\n\n# --- Bilinear upsampling (SciPy-free) ---\n# 1) Upsample frequency axis for each fixed τ\nfreq_hi = np.linspace(freq_ghz.min(), freq_ghz.max(),\n                      UPSAMPLE_FACTOR * (len(freq_ghz) - 1) + 1)\nPe_tau_f = np.empty((len(dur_ns), len(freq_hi)), dtype=float)\nfor i in range(len(dur_ns)):\n    Pe_tau_f[i, :] = np.interp(freq_hi, freq_ghz, Pe[i, :])\n\n# 2) Upsample τ axis for each fixed frequency\ndur_hi = np.linspace(dur_ns.min(), dur_ns.max(),\n                     UPSAMPLE_FACTOR * (len(dur_ns) - 1) + 1)\nPe_hi = np.empty((len(dur_hi), len(freq_hi)), dtype=float)\nfor j in range(len(freq_hi)):\n    Pe_hi[:, j] = np.interp(dur_hi, dur_ns, Pe_tau_f[:, j])\n\n# Mesh for plotting\nF_hi, T_hi = np.meshgrid(freq_hi, dur_hi, indexing='xy')\n\n# --- Plot ---\nfig = plt.figure(figsize=(8, 6), dpi=200)\nax = fig.add_subplot(111, projection='3d')\n\nsurf = ax.plot_surface(\n    F_hi, T_hi, Pe_hi,\n    rstride=1, cstride=1,             # use every row\u002Fcolumn (after upsampling)\n    linewidth=0, antialiased=True,\n    cmap=SURFACE_CMAP,\n    vmin=Z_MIN, vmax=Z_MAX,\n)\n\nax.set_xlabel('f (GHz)')\nax.set_ylabel(r'$\\tau$ (ns)')\nax.set_zlabel(r'$P_e$')\nax.set_title('Rabi $P_e$ surface')\n\n# Colorbar (legend for the colormap)\ncb = fig.colorbar(surf, ax=ax, shrink=0.7, pad=0.08)\ncb.set_label(r'$P_e$')\n\nfig.tight_layout()\nplt.show()\n",[10088,79491,79492,79497,79509,79519,79529,79533,79538,79554,79571,79592,79596,79601,79606,79630,79656,79686,79704,79719,79723,79728,79751,79777,79805,79822,79836,79840,79845,79867,79871,79876,79911,79937,79941,79954,79959,79982,80002,80013,80032,80036,80040,80053,80081,80095,80108,80112,80117,80154,80169,80173,80181],{"__ignoreMap":10090},[40,79493,79494],{"class":10276,"line":10277},[40,79495,79496],{"class":10280},"# --- High-resolution 3D surface with colormap & legend  ---\n",[40,79498,79499,79501,79504,79506],{"class":10276,"line":10284},[40,79500,10353],{"class":10352},[40,79502,79503],{"class":10363}," mpl_toolkits.mplot3d ",[40,79505,10375],{"class":10352},[40,79507,79508],{"class":10363}," Axes3D\n",[40,79510,79511,79513,79515,79517],{"class":10276,"line":10291},[40,79512,10375],{"class":10352},[40,79514,10378],{"class":10363},[40,79516,10381],{"class":10352},[40,79518,10384],{"class":10363},[40,79520,79521,79523,79525,79527],{"class":10276,"line":10297},[40,79522,10375],{"class":10352},[40,79524,10405],{"class":10363},[40,79526,10381],{"class":10352},[40,79528,10410],{"class":10363},[40,79530,79531],{"class":10276,"line":10304},[40,79532,10301],{"emptyLinePlaceholder":10300},[40,79534,79535],{"class":10276,"line":10310},[40,79536,79537],{"class":10280},"# Controls for surface quality and appearance\n",[40,79539,79540,79543,79545,79547,79549,79551],{"class":10276,"line":10316},[40,79541,79542],{"class":10446},"UPSAMPLE_FACTOR",[40,79544,10443],{"class":10363},[40,79546,10821],{"class":10732},[40,79548,10772],{"class":10732},[40,79550,12960],{"class":10446},[40,79552,79553],{"class":10280},"           # 2=~4× more faces; try 3 for even smoother\n",[40,79555,79556,79559,79561,79563,79565,79568],{"class":10276,"line":10322},[40,79557,79558],{"class":10446},"SURFACE_CMAP",[40,79560,10443],{"class":10363},[40,79562,10733],{"class":10732},[40,79564,10772],{"class":10732},[40,79566,79567],{"class":10287}," 'inferno'",[40,79569,79570],{"class":10280},"      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[40,79572,79573,79576,79578,79581,79583,79585,79587,79589],{"class":10276,"line":10327},[40,79574,79575],{"class":10446},"Z_MIN",[40,79577,10484],{"class":10363},[40,79579,79580],{"class":10446},"Z_MAX",[40,79582,10772],{"class":10732},[40,79584,22435],{"class":10446},[40,79586,10484],{"class":10363},[40,79588,10810],{"class":10446},[40,79590,79591],{"class":10280},"            # clamp\u002Fnormalize expected Pe range\n",[40,79593,79594],{"class":10276,"line":10333},[40,79595,10301],{"emptyLinePlaceholder":10300},[40,79597,79598],{"class":10276,"line":10339},[40,79599,79600],{"class":10280},"# --- Bilinear upsampling (SciPy-free) ---\n",[40,79602,79603],{"class":10276,"line":10344},[40,79604,79605],{"class":10280},"# 1) Upsample frequency axis for each fixed τ\n",[40,79607,79608,79611,79613,79615,79617,79620,79622,79625,79627],{"class":10276,"line":10349},[40,79609,79610],{"class":10363},"freq_hi ",[40,79612,162],{"class":10732},[40,79614,11156],{"class":10363},[40,79616,11850],{"class":10430},[40,79618,79619],{"class":10363},"(freq_ghz.",[40,79621,3067],{"class":10430},[40,79623,79624],{"class":10363},"(), freq_ghz.",[40,79626,3032],{"class":10430},[40,79628,79629],{"class":10363},"(),\n",[40,79631,79632,79635,79637,79639,79641,79644,79646,79648,79650,79652,79654],{"class":10276,"line":10367},[40,79633,79634],{"class":10446},"                      UPSAMPLE_FACTOR",[40,79636,11881],{"class":10732},[40,79638,920],{"class":10363},[40,79640,25507],{"class":10732},[40,79642,79643],{"class":10363},"(freq_ghz) ",[40,79645,11056],{"class":10732},[40,79647,13373],{"class":10446},[40,79649,12055],{"class":10363},[40,79651,1053],{"class":10732},[40,79653,13373],{"class":10446},[40,79655,10761],{"class":10363},[40,79657,79658,79661,79663,79665,79667,79669,79671,79674,79676,79679,79681,79684],{"class":10276,"line":10372},[40,79659,79660],{"class":10363},"Pe_tau_f ",[40,79662,162],{"class":10732},[40,79664,11156],{"class":10363},[40,79666,23907],{"class":10430},[40,79668,16469],{"class":10363},[40,79670,25507],{"class":10732},[40,79672,79673],{"class":10363},"(dur_ns), ",[40,79675,25507],{"class":10732},[40,79677,79678],{"class":10363},"(freq_hi)), ",[40,79680,26704],{"class":11903},[40,79682,79683],{"class":10732},"=float",[40,79685,10761],{"class":10363},[40,79687,79688,79690,79693,79695,79697,79699,79701],{"class":10276,"line":10387},[40,79689,13840],{"class":10352},[40,79691,79692],{"class":10363}," i ",[40,79694,13846],{"class":10352},[40,79696,30389],{"class":10732},[40,79698,167],{"class":10363},[40,79700,25507],{"class":10732},[40,79702,79703],{"class":10363},"(dur_ns)):\n",[40,79705,79706,79709,79711,79713,79716],{"class":10276,"line":10400},[40,79707,79708],{"class":10363},"    Pe_tau_f[i, :] ",[40,79710,162],{"class":10732},[40,79712,11156],{"class":10363},[40,79714,79715],{"class":10430},"interp",[40,79717,79718],{"class":10363},"(freq_hi, freq_ghz, Pe[i, :])\n",[40,79720,79721],{"class":10276,"line":10413},[40,79722,10301],{"emptyLinePlaceholder":10300},[40,79724,79725],{"class":10276,"line":10418},[40,79726,79727],{"class":10280},"# 2) Upsample τ axis for each fixed frequency\n",[40,79729,79730,79733,79735,79737,79739,79742,79744,79747,79749],{"class":10276,"line":10424},[40,79731,79732],{"class":10363},"dur_hi ",[40,79734,162],{"class":10732},[40,79736,11156],{"class":10363},[40,79738,11850],{"class":10430},[40,79740,79741],{"class":10363},"(dur_ns.",[40,79743,3067],{"class":10430},[40,79745,79746],{"class":10363},"(), dur_ns.",[40,79748,3032],{"class":10430},[40,79750,79629],{"class":10363},[40,79752,79753,79756,79758,79760,79762,79765,79767,79769,79771,79773,79775],{"class":10276,"line":10437},[40,79754,79755],{"class":10446},"                     UPSAMPLE_FACTOR",[40,79757,11881],{"class":10732},[40,79759,920],{"class":10363},[40,79761,25507],{"class":10732},[40,79763,79764],{"class":10363},"(dur_ns) ",[40,79766,11056],{"class":10732},[40,79768,13373],{"class":10446},[40,79770,12055],{"class":10363},[40,79772,1053],{"class":10732},[40,79774,13373],{"class":10446},[40,79776,10761],{"class":10363},[40,79778,79779,79782,79784,79786,79788,79790,79792,79795,79797,79799,79801,79803],{"class":10276,"line":10456},[40,79780,79781],{"class":10363},"Pe_hi ",[40,79783,162],{"class":10732},[40,79785,11156],{"class":10363},[40,79787,23907],{"class":10430},[40,79789,16469],{"class":10363},[40,79791,25507],{"class":10732},[40,79793,79794],{"class":10363},"(dur_hi), ",[40,79796,25507],{"class":10732},[40,79798,79678],{"class":10363},[40,79800,26704],{"class":11903},[40,79802,79683],{"class":10732},[40,79804,10761],{"class":10363},[40,79806,79807,79809,79811,79813,79815,79817,79819],{"class":10276,"line":10472},[40,79808,13840],{"class":10352},[40,79810,30384],{"class":10363},[40,79812,13846],{"class":10352},[40,79814,30389],{"class":10732},[40,79816,167],{"class":10363},[40,79818,25507],{"class":10732},[40,79820,79821],{"class":10363},"(freq_hi)):\n",[40,79823,79824,79827,79829,79831,79833],{"class":10276,"line":10496},[40,79825,79826],{"class":10363},"    Pe_hi[:, j] ",[40,79828,162],{"class":10732},[40,79830,11156],{"class":10363},[40,79832,79715],{"class":10430},[40,79834,79835],{"class":10363},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[40,79837,79838],{"class":10276,"line":10510},[40,79839,10301],{"emptyLinePlaceholder":10300},[40,79841,79842],{"class":10276,"line":10696},[40,79843,79844],{"class":10280},"# Mesh for plotting\n",[40,79846,79847,79850,79852,79854,79856,79859,79861,79863,79865],{"class":10276,"line":10702},[40,79848,79849],{"class":10363},"F_hi, T_hi ",[40,79851,162],{"class":10732},[40,79853,11156],{"class":10363},[40,79855,11897],{"class":10430},[40,79857,79858],{"class":10363},"(freq_hi, dur_hi, ",[40,79860,11904],{"class":11903},[40,79862,162],{"class":10732},[40,79864,73972],{"class":10287},[40,79866,10761],{"class":10363},[40,79868,79869],{"class":10276,"line":10708},[40,79870,10301],{"emptyLinePlaceholder":10300},[40,79872,79873],{"class":10276,"line":10714},[40,79874,79875],{"class":10280},"# --- Plot ---\n",[40,79877,79878,79881,79883,79885,79887,79889,79891,79893,79895,79897,79899,79901,79903,79905,79907,79909],{"class":10276,"line":10720},[40,79879,79880],{"class":10363},"fig ",[40,79882,162],{"class":10732},[40,79884,13070],{"class":10363},[40,79886,16373],{"class":10430},[40,79888,167],{"class":10363},[40,79890,13078],{"class":11903},[40,79892,162],{"class":10732},[40,79894,167],{"class":10363},[40,79896,23482],{"class":10446},[40,79898,10484],{"class":10363},[40,79900,12785],{"class":10446},[40,79902,11040],{"class":10363},[40,79904,23491],{"class":11903},[40,79906,162],{"class":10732},[40,79908,10447],{"class":10446},[40,79910,10761],{"class":10363},[40,79912,79913,79916,79918,79920,79922,79924,79926,79928,79930,79932,79935],{"class":10276,"line":10726},[40,79914,79915],{"class":10363},"ax ",[40,79917,162],{"class":10732},[40,79919,13170],{"class":10363},[40,79921,16402],{"class":10430},[40,79923,167],{"class":10363},[40,79925,16407],{"class":10446},[40,79927,10484],{"class":10363},[40,79929,16412],{"class":11903},[40,79931,162],{"class":10732},[40,79933,79934],{"class":10287},"'3d'",[40,79936,10761],{"class":10363},[40,79938,79939],{"class":10276,"line":10764},[40,79940,10301],{"emptyLinePlaceholder":10300},[40,79942,79943,79946,79948,79950,79952],{"class":10276,"line":10778},[40,79944,79945],{"class":10363},"surf ",[40,79947,162],{"class":10732},[40,79949,12522],{"class":10363},[40,79951,16566],{"class":10430},[40,79953,12527],{"class":10363},[40,79955,79956],{"class":10276,"line":10815},[40,79957,79958],{"class":10363},"    F_hi, T_hi, Pe_hi,\n",[40,79960,79961,79964,79966,79968,79970,79972,79974,79976,79979],{"class":10276,"line":10829},[40,79962,79963],{"class":11903},"    rstride",[40,79965,162],{"class":10732},[40,79967,78],{"class":10446},[40,79969,10484],{"class":10363},[40,79971,16599],{"class":11903},[40,79973,162],{"class":10732},[40,79975,78],{"class":10446},[40,79977,79978],{"class":10363},",             ",[40,79980,79981],{"class":10280},"# use every row\u002Fcolumn (after upsampling)\n",[40,79983,79984,79987,79989,79991,79993,79996,79998,80000],{"class":10276,"line":10842},[40,79985,79986],{"class":11903},"    linewidth",[40,79988,162],{"class":10732},[40,79990,66],{"class":10446},[40,79992,10484],{"class":10363},[40,79994,79995],{"class":11903},"antialiased",[40,79997,162],{"class":10732},[40,79999,13091],{"class":10446},[40,80001,10507],{"class":10363},[40,80003,80004,80007,80009,80011],{"class":10276,"line":10855},[40,80005,80006],{"class":11903},"    cmap",[40,80008,162],{"class":10732},[40,80010,79558],{"class":10446},[40,80012,10507],{"class":10363},[40,80014,80015,80018,80020,80022,80024,80026,80028,80030],{"class":10276,"line":10868},[40,80016,80017],{"class":11903},"    vmin",[40,80019,162],{"class":10732},[40,80021,79575],{"class":10446},[40,80023,10484],{"class":10363},[40,80025,13145],{"class":11903},[40,80027,162],{"class":10732},[40,80029,79580],{"class":10446},[40,80031,10507],{"class":10363},[40,80033,80034],{"class":10276,"line":10881},[40,80035,10761],{"class":10363},[40,80037,80038],{"class":10276,"line":10907},[40,80039,10301],{"emptyLinePlaceholder":10300},[40,80041,80042,80044,80046,80048,80051],{"class":10276,"line":10920},[40,80043,38114],{"class":10363},[40,80045,13230],{"class":10430},[40,80047,167],{"class":10363},[40,80049,80050],{"class":10287},"'f (GHz)'",[40,80052,10761],{"class":10363},[40,80054,80055,80057,80059,80061,80063,80065,80068,80071,80073,80075,80077,80079],{"class":10276,"line":10933},[40,80056,38114],{"class":10363},[40,80058,13244],{"class":10430},[40,80060,167],{"class":10363},[40,80062,936],{"class":10352},[40,80064,23633],{"class":10356},[40,80066,80067],{"class":10732},"\\t",[40,80069,80070],{"class":10356},"au$ ",[40,80072,167],{"class":10446},[40,80074,64965],{"class":10356},[40,80076,207],{"class":10446},[40,80078,23617],{"class":10356},[40,80080,10761],{"class":10363},[40,80082,80083,80085,80087,80089,80091,80093],{"class":10276,"line":10946},[40,80084,38114],{"class":10363},[40,80086,16740],{"class":10430},[40,80088,167],{"class":10363},[40,80090,936],{"class":10352},[40,80092,74580],{"class":10356},[40,80094,10761],{"class":10363},[40,80096,80097,80099,80101,80103,80106],{"class":10276,"line":10951},[40,80098,38114],{"class":10363},[40,80100,30334],{"class":10430},[40,80102,167],{"class":10363},[40,80104,80105],{"class":10287},"'Rabi $P_e$ surface'",[40,80107,10761],{"class":10363},[40,80109,80110],{"class":10276,"line":10956},[40,80111,10301],{"emptyLinePlaceholder":10300},[40,80113,80114],{"class":10276,"line":10962},[40,80115,80116],{"class":10280},"# Colorbar (legend for the colormap)\n",[40,80118,80119,80121,80123,80125,80127,80130,80132,80134,80136,80139,80141,80143,80145,80147,80149,80152],{"class":10276,"line":10983},[40,80120,74556],{"class":10363},[40,80122,162],{"class":10732},[40,80124,13170],{"class":10363},[40,80126,13173],{"class":10430},[40,80128,80129],{"class":10363},"(surf, ",[40,80131,12429],{"class":11903},[40,80133,162],{"class":10732},[40,80135,13183],{"class":10363},[40,80137,80138],{"class":11903},"shrink",[40,80140,162],{"class":10732},[40,80142,38131],{"class":10446},[40,80144,10484],{"class":10363},[40,80146,13196],{"class":11903},[40,80148,162],{"class":10732},[40,80150,80151],{"class":10446},"0.08",[40,80153,10761],{"class":10363},[40,80155,80156,80159,80161,80163,80165,80167],{"class":10276,"line":11005},[40,80157,80158],{"class":10363},"cb.",[40,80160,13211],{"class":10430},[40,80162,167],{"class":10363},[40,80164,936],{"class":10352},[40,80166,74580],{"class":10356},[40,80168,10761],{"class":10363},[40,80170,80171],{"class":10276,"line":11011},[40,80172,10301],{"emptyLinePlaceholder":10300},[40,80174,80175,80177,80179],{"class":10276,"line":11022},[40,80176,74636],{"class":10363},[40,80178,30425],{"class":10430},[40,80180,12174],{"class":10363},[40,80182,80183,80185,80187],{"class":10276,"line":11064},[40,80184,23469],{"class":10363},[40,80186,23686],{"class":10430},[40,80188,12174],{"class":10363},[13997,80190],{"alt":80191,"src":80192},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[10082,80194,80196],{"className":10269,"code":80195,"language":10271,"meta":10090,"style":10090},"# @title Cross-sections\n# Slicing controls\nF_CROSS_GHZ: float = 5.000  # drive frequency slice [GHz]\nT_CROSS_NS: float = 100.0   # duration slice [ns]\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\ndef _nearest_index(vec: np.ndarray, value: float, name: str) -> int:\n    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n    vmin, vmax = float(vec.min()), float(vec.max())\n    if value \u003C vmin or value > vmax:\n        print(f\"[Note] {name}={value} is outside sweep; clamping to [{vmin}, {vmax}].\")\n        value = np.clip(value, vmin, vmax)\n    return int(np.argmin(np.abs(vec - value)))\n\n# --- Slice at f = F_CROSS_GHZ ---\nj = _nearest_index(freq_ghz, F_CROSS_GHZ, \"F_CROSS_GHZ\")\nPe_vs_tau = Pe[:, j]\n\nfig, ax = plt.subplots(figsize=(6, 3.5))\nax.plot(dur_ns, Pe_vs_tau, color='blue')\nax.set_xlabel(r'$\\tau$ (ns)')\nax.set_ylabel(r'$P_e$')\n# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\nax.set_title(r'$P_e(\\tau)$ at f = ' + f'{freq_ghz[j]:.6f} GHz')\nax.set_ylim(0.0, 1.0)\nfig.tight_layout()\nplt.show()\n\n# --- Slice at τ = T_CROSS_NS ---\ni = _nearest_index(dur_ns, T_CROSS_NS, \"T_CROSS_NS\")\nPe_vs_f = Pe[i, :]\n\nfig, ax = plt.subplots(figsize=(6, 3.5))\nax.plot(freq_ghz, Pe_vs_f, color='blue')\nax.set_xlabel('f (GHz)')\nax.set_ylabel(r'$P_e$')\nax.set_title(r'$P_e(f)$ at $\\tau$ = ' + f'{dur_ns[i]:.3f} ns')\nax.set_ylim(0.0, 1.0)\nfig.tight_layout()\nplt.show()\n",[10088,80197,80198,80203,80208,80223,80238,80242,80252,80262,80266,80302,80307,80332,80353,80400,80414,80437,80441,80446,80467,80477,80481,80508,80525,80551,80565,80570,80615,80632,80640,80648,80652,80657,80678,80688,80692,80718,80735,80747,80761,80807,80823,80831],{"__ignoreMap":10090},[40,80199,80200],{"class":10276,"line":10277},[40,80201,80202],{"class":10280},"# @title Cross-sections\n",[40,80204,80205],{"class":10276,"line":10284},[40,80206,80207],{"class":10280},"# Slicing controls\n",[40,80209,80210,80212,80214,80216,80218,80220],{"class":10276,"line":10291},[40,80211,73089],{"class":10446},[40,80213,10443],{"class":10363},[40,80215,10784],{"class":10732},[40,80217,10772],{"class":10732},[40,80219,73584],{"class":10446},[40,80221,80222],{"class":10280},"  # drive frequency slice [GHz]\n",[40,80224,80225,80227,80229,80231,80233,80235],{"class":10276,"line":10297},[40,80226,73188],{"class":10446},[40,80228,10443],{"class":10363},[40,80230,10784],{"class":10732},[40,80232,10772],{"class":10732},[40,80234,25800],{"class":10446},[40,80236,80237],{"class":10280},"   # duration slice [ns]\n",[40,80239,80240],{"class":10276,"line":10304},[40,80241,10301],{"emptyLinePlaceholder":10300},[40,80243,80244,80246,80248,80250],{"class":10276,"line":10310},[40,80245,10375],{"class":10352},[40,80247,10378],{"class":10363},[40,80249,10381],{"class":10352},[40,80251,10384],{"class":10363},[40,80253,80254,80256,80258,80260],{"class":10276,"line":10316},[40,80255,10375],{"class":10352},[40,80257,10405],{"class":10363},[40,80259,10381],{"class":10352},[40,80261,10410],{"class":10363},[40,80263,80264],{"class":10276,"line":10322},[40,80265,10301],{"emptyLinePlaceholder":10300},[40,80267,80268,80270,80273,80275,80278,80280,80283,80285,80287,80289,80292,80294,80296,80298,80300],{"class":10276,"line":10327},[40,80269,10965],{"class":10352},[40,80271,80272],{"class":10430}," _nearest_index",[40,80274,167],{"class":10363},[40,80276,80277],{"class":10973},"vec",[40,80279,12019],{"class":10363},[40,80281,80282],{"class":10973},"value",[40,80284,10443],{"class":10363},[40,80286,10784],{"class":10732},[40,80288,10484],{"class":10363},[40,80290,80291],{"class":10973},"name",[40,80293,10443],{"class":10363},[40,80295,10733],{"class":10732},[40,80297,22117],{"class":10363},[40,80299,10821],{"class":10732},[40,80301,10639],{"class":10363},[40,80303,80304],{"class":10276,"line":10333},[40,80305,80306],{"class":10287},"    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[40,80308,80309,80312,80314,80316,80319,80321,80324,80326,80328,80330],{"class":10276,"line":10339},[40,80310,80311],{"class":10363},"    vmin, vmax ",[40,80313,162],{"class":10732},[40,80315,75857],{"class":10732},[40,80317,80318],{"class":10363},"(vec.",[40,80320,3067],{"class":10430},[40,80322,80323],{"class":10363},"()), ",[40,80325,10784],{"class":10732},[40,80327,80318],{"class":10363},[40,80329,3032],{"class":10430},[40,80331,23593],{"class":10363},[40,80333,80334,80336,80339,80341,80344,80346,80348,80350],{"class":10276,"line":10344},[40,80335,11323],{"class":10352},[40,80337,80338],{"class":10363}," value ",[40,80340,10184],{"class":10732},[40,80342,80343],{"class":10363}," vmin ",[40,80345,15602],{"class":10352},[40,80347,80338],{"class":10363},[40,80349,22432],{"class":10732},[40,80351,80352],{"class":10363}," vmax:\n",[40,80354,80355,80357,80359,80361,80364,80366,80368,80370,80372,80374,80376,80378,80381,80383,80385,80387,80389,80391,80393,80395,80398],{"class":10276,"line":10349},[40,80356,17475],{"class":10732},[40,80358,167],{"class":10363},[40,80360,13262],{"class":10352},[40,80362,80363],{"class":10287},"\"[Note] ",[40,80365,13268],{"class":10446},[40,80367,80291],{"class":10363},[40,80369,13280],{"class":10446},[40,80371,162],{"class":10287},[40,80373,13268],{"class":10446},[40,80375,80282],{"class":10363},[40,80377,13280],{"class":10446},[40,80379,80380],{"class":10287}," is outside sweep; clamping to [",[40,80382,13268],{"class":10446},[40,80384,16291],{"class":10363},[40,80386,13280],{"class":10446},[40,80388,10484],{"class":10287},[40,80390,13268],{"class":10446},[40,80392,13145],{"class":10363},[40,80394,13280],{"class":10446},[40,80396,80397],{"class":10287},"].\"",[40,80399,10761],{"class":10363},[40,80401,80402,80405,80407,80409,80411],{"class":10276,"line":10367},[40,80403,80404],{"class":10363},"        value ",[40,80406,162],{"class":10732},[40,80408,11156],{"class":10363},[40,80410,24368],{"class":10430},[40,80412,80413],{"class":10363},"(value, vmin, vmax)\n",[40,80415,80416,80418,80420,80422,80425,80427,80429,80432,80434],{"class":10276,"line":10372},[40,80417,11201],{"class":10352},[40,80419,37310],{"class":10732},[40,80421,13108],{"class":10363},[40,80423,80424],{"class":10430},"argmin",[40,80426,13108],{"class":10363},[40,80428,23431],{"class":10430},[40,80430,80431],{"class":10363},"(vec ",[40,80433,11056],{"class":10732},[40,80435,80436],{"class":10363}," value)))\n",[40,80438,80439],{"class":10276,"line":10387},[40,80440,10301],{"emptyLinePlaceholder":10300},[40,80442,80443],{"class":10276,"line":10400},[40,80444,80445],{"class":10280},"# --- Slice at f = F_CROSS_GHZ ---\n",[40,80447,80448,80451,80453,80455,80458,80460,80462,80465],{"class":10276,"line":10413},[40,80449,80450],{"class":10363},"j ",[40,80452,162],{"class":10732},[40,80454,80272],{"class":10430},[40,80456,80457],{"class":10363},"(freq_ghz, ",[40,80459,73089],{"class":10446},[40,80461,10484],{"class":10363},[40,80463,80464],{"class":10287},"\"F_CROSS_GHZ\"",[40,80466,10761],{"class":10363},[40,80468,80469,80472,80474],{"class":10276,"line":10418},[40,80470,80471],{"class":10363},"Pe_vs_tau ",[40,80473,162],{"class":10732},[40,80475,80476],{"class":10363}," Pe[:, j]\n",[40,80478,80479],{"class":10276,"line":10424},[40,80480,10301],{"emptyLinePlaceholder":10300},[40,80482,80483,80485,80487,80489,80491,80493,80495,80497,80499,80501,80503,80506],{"class":10276,"line":10437},[40,80484,38079],{"class":10363},[40,80486,162],{"class":10732},[40,80488,13070],{"class":10363},[40,80490,13073],{"class":10430},[40,80492,167],{"class":10363},[40,80494,13078],{"class":11903},[40,80496,162],{"class":10732},[40,80498,167],{"class":10363},[40,80500,12785],{"class":10446},[40,80502,10484],{"class":10363},[40,80504,80505],{"class":10446},"3.5",[40,80507,13612],{"class":10363},[40,80509,80510,80512,80514,80517,80519,80521,80523],{"class":10276,"line":10456},[40,80511,38114],{"class":10363},[40,80513,23505],{"class":10430},[40,80515,80516],{"class":10363},"(dur_ns, Pe_vs_tau, ",[40,80518,23550],{"class":11903},[40,80520,162],{"class":10732},[40,80522,23555],{"class":10287},[40,80524,10761],{"class":10363},[40,80526,80527,80529,80531,80533,80535,80537,80539,80541,80543,80545,80547,80549],{"class":10276,"line":10472},[40,80528,38114],{"class":10363},[40,80530,13230],{"class":10430},[40,80532,167],{"class":10363},[40,80534,936],{"class":10352},[40,80536,23633],{"class":10356},[40,80538,80067],{"class":10732},[40,80540,80070],{"class":10356},[40,80542,167],{"class":10446},[40,80544,64965],{"class":10356},[40,80546,207],{"class":10446},[40,80548,23617],{"class":10356},[40,80550,10761],{"class":10363},[40,80552,80553,80555,80557,80559,80561,80563],{"class":10276,"line":10496},[40,80554,38114],{"class":10363},[40,80556,13244],{"class":10430},[40,80558,167],{"class":10363},[40,80560,936],{"class":10352},[40,80562,74580],{"class":10356},[40,80564,10761],{"class":10363},[40,80566,80567],{"class":10276,"line":10510},[40,80568,80569],{"class":10280},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[40,80571,80572,80574,80576,80578,80580,80583,80585,80587,80590,80592,80595,80597,80599,80601,80603,80606,80608,80610,80613],{"class":10276,"line":10696},[40,80573,38114],{"class":10363},[40,80575,30334],{"class":10430},[40,80577,167],{"class":10363},[40,80579,936],{"class":10352},[40,80581,80582],{"class":10356},"'$P_e",[40,80584,167],{"class":10446},[40,80586,80067],{"class":10732},[40,80588,80589],{"class":10356},"au",[40,80591,207],{"class":10446},[40,80593,80594],{"class":10356},"$ at f = '",[40,80596,12042],{"class":10732},[40,80598,13867],{"class":10352},[40,80600,23617],{"class":10287},[40,80602,13268],{"class":10446},[40,80604,80605],{"class":10363},"freq_ghz[j]",[40,80607,79068],{"class":10352},[40,80609,13280],{"class":10446},[40,80611,80612],{"class":10287}," GHz'",[40,80614,10761],{"class":10363},[40,80616,80617,80619,80622,80624,80626,80628,80630],{"class":10276,"line":10702},[40,80618,38114],{"class":10363},[40,80620,80621],{"class":10430},"set_ylim",[40,80623,167],{"class":10363},[40,80625,10801],{"class":10446},[40,80627,10484],{"class":10363},[40,80629,10810],{"class":10446},[40,80631,10761],{"class":10363},[40,80633,80634,80636,80638],{"class":10276,"line":10708},[40,80635,74636],{"class":10363},[40,80637,30425],{"class":10430},[40,80639,12174],{"class":10363},[40,80641,80642,80644,80646],{"class":10276,"line":10714},[40,80643,23469],{"class":10363},[40,80645,23686],{"class":10430},[40,80647,12174],{"class":10363},[40,80649,80650],{"class":10276,"line":10720},[40,80651,10301],{"emptyLinePlaceholder":10300},[40,80653,80654],{"class":10276,"line":10726},[40,80655,80656],{"class":10280},"# --- Slice at τ = T_CROSS_NS ---\n",[40,80658,80659,80662,80664,80666,80669,80671,80673,80676],{"class":10276,"line":10764},[40,80660,80661],{"class":10363},"i ",[40,80663,162],{"class":10732},[40,80665,80272],{"class":10430},[40,80667,80668],{"class":10363},"(dur_ns, ",[40,80670,73188],{"class":10446},[40,80672,10484],{"class":10363},[40,80674,80675],{"class":10287},"\"T_CROSS_NS\"",[40,80677,10761],{"class":10363},[40,80679,80680,80683,80685],{"class":10276,"line":10778},[40,80681,80682],{"class":10363},"Pe_vs_f ",[40,80684,162],{"class":10732},[40,80686,80687],{"class":10363}," Pe[i, :]\n",[40,80689,80690],{"class":10276,"line":10815},[40,80691,10301],{"emptyLinePlaceholder":10300},[40,80693,80694,80696,80698,80700,80702,80704,80706,80708,80710,80712,80714,80716],{"class":10276,"line":10829},[40,80695,38079],{"class":10363},[40,80697,162],{"class":10732},[40,80699,13070],{"class":10363},[40,80701,13073],{"class":10430},[40,80703,167],{"class":10363},[40,80705,13078],{"class":11903},[40,80707,162],{"class":10732},[40,80709,167],{"class":10363},[40,80711,12785],{"class":10446},[40,80713,10484],{"class":10363},[40,80715,80505],{"class":10446},[40,80717,13612],{"class":10363},[40,80719,80720,80722,80724,80727,80729,80731,80733],{"class":10276,"line":10842},[40,80721,38114],{"class":10363},[40,80723,23505],{"class":10430},[40,80725,80726],{"class":10363},"(freq_ghz, Pe_vs_f, ",[40,80728,23550],{"class":11903},[40,80730,162],{"class":10732},[40,80732,23555],{"class":10287},[40,80734,10761],{"class":10363},[40,80736,80737,80739,80741,80743,80745],{"class":10276,"line":10855},[40,80738,38114],{"class":10363},[40,80740,13230],{"class":10430},[40,80742,167],{"class":10363},[40,80744,80050],{"class":10287},[40,80746,10761],{"class":10363},[40,80748,80749,80751,80753,80755,80757,80759],{"class":10276,"line":10868},[40,80750,38114],{"class":10363},[40,80752,13244],{"class":10430},[40,80754,167],{"class":10363},[40,80756,936],{"class":10352},[40,80758,74580],{"class":10356},[40,80760,10761],{"class":10363},[40,80762,80763,80765,80767,80769,80771,80773,80775,80777,80779,80782,80784,80787,80789,80791,80793,80795,80798,80800,80802,80805],{"class":10276,"line":10881},[40,80764,38114],{"class":10363},[40,80766,30334],{"class":10430},[40,80768,167],{"class":10363},[40,80770,936],{"class":10352},[40,80772,80582],{"class":10356},[40,80774,167],{"class":10446},[40,80776,13262],{"class":10356},[40,80778,207],{"class":10446},[40,80780,80781],{"class":10356},"$ at $",[40,80783,80067],{"class":10732},[40,80785,80786],{"class":10356},"au$ = '",[40,80788,12042],{"class":10732},[40,80790,13867],{"class":10352},[40,80792,23617],{"class":10287},[40,80794,13268],{"class":10446},[40,80796,80797],{"class":10363},"dur_ns[i]",[40,80799,13446],{"class":10352},[40,80801,13280],{"class":10446},[40,80803,80804],{"class":10287}," ns'",[40,80806,10761],{"class":10363},[40,80808,80809,80811,80813,80815,80817,80819,80821],{"class":10276,"line":10907},[40,80810,38114],{"class":10363},[40,80812,80621],{"class":10430},[40,80814,167],{"class":10363},[40,80816,10801],{"class":10446},[40,80818,10484],{"class":10363},[40,80820,10810],{"class":10446},[40,80822,10761],{"class":10363},[40,80824,80825,80827,80829],{"class":10276,"line":10920},[40,80826,74636],{"class":10363},[40,80828,30425],{"class":10430},[40,80830,12174],{"class":10363},[40,80832,80833,80835,80837],{"class":10276,"line":10933},[40,80834,23469],{"class":10363},[40,80836,23686],{"class":10430},[40,80838,12174],{"class":10363},[13997,80840],{"alt":80841,"src":80842},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[13997,80844],{"alt":80845,"src":80846},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[10082,80848,80850],{"className":10269,"code":80849,"language":10271,"meta":10090,"style":10090},"# @title FFT of Rabi oscillations along tau (controls + computation)\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# -------- Control knobs (FFT & display) --------\nFFT_PAD: int = 8                 # zero-padding factor (2, 4, 8, ...)\nWINDOW: str = \"hann\"             # \"hann\", \"boxcar\"\nDETREND_MEAN: bool = True        # subtract mean along τ before FFT\nAMP_MODE: str = \"magnitude\"      # \"magnitude\" | \"power\" | \"db\"\nNORM_MODE: str = \"global\"        # \"global\" | \"per_f\" (column-wise)\nFREQ_UNITS: str = \"MHz\"          # \"MHz\" or \"GHz\" for Fourier axis\n\n# -------- Helpers --------\ndef _make_window(n: int, kind: str) -> np.ndarray:\n    if kind.lower() == \"hann\":\n        return np.hanning(n)\n    return np.ones(n, dtype=float)  # boxcar\n\ndef rfft_spectrogram(\n    Pe: np.ndarray,\n    dur_ns: np.ndarray,\n    pad: int = 8,\n    window: str = \"hann\",\n    detrend_mean: bool = True,\n    amp_mode: str = \"magnitude\",\n    norm_mode: str = \"global\",\n    freq_units: str = \"MHz\",\n) -> tuple[np.ndarray, np.ndarray]:\n    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n\n    Args:\n        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n        pad: zero-padding factor (≥1).\n        window: \"hann\" or \"boxcar\".\n        detrend_mean: subtract column-wise mean before FFT.\n        amp_mode: \"magnitude\", \"power\", or \"db\".\n        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n\n    Returns:\n        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n        S:     2D nonnegative spectrogram (N_w, N_f).\n    \"\"\"\n    Pe = np.asarray(Pe, float)\n    dur_ns = np.asarray(dur_ns, float)\n    assert Pe.shape[0] == dur_ns.size, \"Pe must be shaped (N_tau, N_f).\"\n\n    # Uniform sampling assumption (true for np.linspace)\n    dt = float(dur_ns[1] - dur_ns[0]) * 1e-9               # [s]\n    n_tau = dur_ns.size\n    nfft = int(2 ** int(np.ceil(np.log2(n_tau))) * max(1, pad))\n    w = _make_window(n_tau, window)\n\n    # Prepare output\n    S_list = []\n    X = Pe.copy()\n    if detrend_mean:\n        X = X - X.mean(axis=0, keepdims=True)\n    X *= w[:, None]\n\n    # Real FFT along τ (axis=0)\n    Xf = np.fft.rfft(X, n=nfft, axis=0)                     # shape (N_w, N_f)\n    f_fft_hz = np.fft.rfftfreq(nfft, d=dt)                  # [Hz]\n\n    # Amplitude selection\n    if amp_mode.lower() == \"power\":\n        S = np.abs(Xf) ** 2\n    else:\n        S = np.abs(Xf)\n\n    # Normalization\n    if norm_mode.lower() == \"per_f\":\n        col_max = np.maximum(S.max(axis=0, keepdims=True), 1e-15)\n        S = S \u002F col_max\n    else:\n        S = S \u002F max(S.max(), 1e-15)\n\n    if amp_mode.lower() == \"db\":\n        S = 20.0 * np.log10(np.maximum(S, 1e-12))  # dB, clipped floor\n\n    # Units\n    if freq_units.upper() == \"GHz\":\n        f_fft = f_fft_hz * 1e-9\n    else:\n        f_fft = f_fft_hz * 1e-6\n\n    return f_fft, S\n\n# Compute spectrogram with current grid\nf_fft_axis, S_fft = rfft_spectrogram(\n    Pe, dur_ns,\n    pad=FFT_PAD,\n    window=WINDOW,\n    detrend_mean=DETREND_MEAN,\n    amp_mode=AMP_MODE,\n    norm_mode=NORM_MODE,\n    freq_units=FREQ_UNITS,\n)\nprint(\"FFT grid:\", S_fft.shape, \"| Fourier freq range:\", f_fft_axis[0], \"→\", f_fft_axis[-1], FREQ_UNITS)\n",[10088,80851,80852,80857,80867,80877,80881,80886,80902,80918,80934,80950,80966,80982,80986,80991,81017,81034,81046,81067,81071,81080,81087,81094,81109,81124,81139,81154,81169,81184,81189,81194,81198,81202,81207,81212,81217,81222,81227,81232,81237,81242,81246,81250,81255,81260,81264,81281,81298,81317,81321,81326,81358,81368,81410,81422,81426,81431,81440,81454,81461,81498,81512,81516,81521,81556,81582,81586,81591,81609,81627,81633,81646,81650,81655,81673,81713,81727,81733,81755,81759,81776,81805,81809,81814,81832,81846,81852,81865,81869,81876,81880,81885,81896,81901,81911,81921,81931,81941,81951,81961,81965],{"__ignoreMap":10090},[40,80853,80854],{"class":10276,"line":10277},[40,80855,80856],{"class":10280},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[40,80858,80859,80861,80863,80865],{"class":10276,"line":10284},[40,80860,10375],{"class":10352},[40,80862,10378],{"class":10363},[40,80864,10381],{"class":10352},[40,80866,10384],{"class":10363},[40,80868,80869,80871,80873,80875],{"class":10276,"line":10291},[40,80870,10375],{"class":10352},[40,80872,10405],{"class":10363},[40,80874,10381],{"class":10352},[40,80876,10410],{"class":10363},[40,80878,80879],{"class":10276,"line":10297},[40,80880,10301],{"emptyLinePlaceholder":10300},[40,80882,80883],{"class":10276,"line":10304},[40,80884,80885],{"class":10280},"# -------- Control knobs (FFT & display) --------\n",[40,80887,80888,80890,80892,80894,80896,80899],{"class":10276,"line":10310},[40,80889,73305],{"class":10446},[40,80891,10443],{"class":10363},[40,80893,10821],{"class":10732},[40,80895,10772],{"class":10732},[40,80897,80898],{"class":10446}," 8",[40,80900,80901],{"class":10280},"                 # zero-padding factor (2, 4, 8, ...)\n",[40,80903,80904,80906,80908,80910,80912,80915],{"class":10276,"line":10316},[40,80905,73319],{"class":10446},[40,80907,10443],{"class":10363},[40,80909,10733],{"class":10732},[40,80911,10772],{"class":10732},[40,80913,80914],{"class":10287}," \"hann\"",[40,80916,80917],{"class":10280},"             # \"hann\", \"boxcar\"\n",[40,80919,80920,80922,80924,80926,80928,80931],{"class":10276,"line":10322},[40,80921,73334],{"class":10446},[40,80923,10443],{"class":10363},[40,80925,21950],{"class":10732},[40,80927,10772],{"class":10732},[40,80929,80930],{"class":10446}," True",[40,80932,80933],{"class":10280},"        # subtract mean along τ before FFT\n",[40,80935,80936,80938,80940,80942,80944,80947],{"class":10276,"line":10327},[40,80937,73348],{"class":10446},[40,80939,10443],{"class":10363},[40,80941,10733],{"class":10732},[40,80943,10772],{"class":10732},[40,80945,80946],{"class":10287}," \"magnitude\"",[40,80948,80949],{"class":10280},"      # \"magnitude\" | \"power\" | \"db\"\n",[40,80951,80952,80954,80956,80958,80960,80963],{"class":10276,"line":10333},[40,80953,73363],{"class":10446},[40,80955,10443],{"class":10363},[40,80957,10733],{"class":10732},[40,80959,10772],{"class":10732},[40,80961,80962],{"class":10287}," \"global\"",[40,80964,80965],{"class":10280},"        # \"global\" | \"per_f\" (column-wise)\n",[40,80967,80968,80970,80972,80974,80976,80979],{"class":10276,"line":10339},[40,80969,73378],{"class":10446},[40,80971,10443],{"class":10363},[40,80973,10733],{"class":10732},[40,80975,10772],{"class":10732},[40,80977,80978],{"class":10287}," \"MHz\"",[40,80980,80981],{"class":10280},"          # \"MHz\" or \"GHz\" for Fourier axis\n",[40,80983,80984],{"class":10276,"line":10344},[40,80985,10301],{"emptyLinePlaceholder":10300},[40,80987,80988],{"class":10276,"line":10349},[40,80989,80990],{"class":10280},"# -------- Helpers --------\n",[40,80992,80993,80995,80998,81000,81002,81004,81006,81008,81011,81013,81015],{"class":10276,"line":10367},[40,80994,10965],{"class":10352},[40,80996,80997],{"class":10430}," _make_window",[40,80999,167],{"class":10363},[40,81001,1796],{"class":10973},[40,81003,10443],{"class":10363},[40,81005,10821],{"class":10732},[40,81007,10484],{"class":10363},[40,81009,81010],{"class":10973},"kind",[40,81012,10443],{"class":10363},[40,81014,10733],{"class":10732},[40,81016,11291],{"class":10363},[40,81018,81019,81021,81024,81026,81028,81030,81032],{"class":10276,"line":10372},[40,81020,11323],{"class":10352},[40,81022,81023],{"class":10363}," kind.",[40,81025,11171],{"class":10430},[40,81027,27025],{"class":10363},[40,81029,11231],{"class":10732},[40,81031,80914],{"class":10287},[40,81033,10639],{"class":10363},[40,81035,81036,81038,81040,81043],{"class":10276,"line":10387},[40,81037,11339],{"class":10352},[40,81039,11156],{"class":10363},[40,81041,81042],{"class":10430},"hanning",[40,81044,81045],{"class":10363},"(n)\n",[40,81047,81048,81050,81052,81055,81058,81060,81062,81064],{"class":10276,"line":10400},[40,81049,11201],{"class":10352},[40,81051,11156],{"class":10363},[40,81053,81054],{"class":10430},"ones",[40,81056,81057],{"class":10363},"(n, ",[40,81059,26704],{"class":11903},[40,81061,79683],{"class":10732},[40,81063,17908],{"class":10363},[40,81065,81066],{"class":10280},"# boxcar\n",[40,81068,81069],{"class":10276,"line":10413},[40,81070,10301],{"emptyLinePlaceholder":10300},[40,81072,81073,81075,81078],{"class":10276,"line":10418},[40,81074,10965],{"class":10352},[40,81076,81077],{"class":10430}," rfft_spectrogram",[40,81079,12527],{"class":10363},[40,81081,81082,81085],{"class":10276,"line":10424},[40,81083,81084],{"class":10973},"    Pe",[40,81086,12841],{"class":10363},[40,81088,81089,81092],{"class":10276,"line":10437},[40,81090,81091],{"class":10973},"    dur_ns",[40,81093,12841],{"class":10363},[40,81095,81096,81099,81101,81103,81105,81107],{"class":10276,"line":10456},[40,81097,81098],{"class":10973},"    pad",[40,81100,10443],{"class":10363},[40,81102,10821],{"class":10732},[40,81104,10772],{"class":10732},[40,81106,80898],{"class":10446},[40,81108,10507],{"class":10363},[40,81110,81111,81114,81116,81118,81120,81122],{"class":10276,"line":10472},[40,81112,81113],{"class":10973},"    window",[40,81115,10443],{"class":10363},[40,81117,10733],{"class":10732},[40,81119,10772],{"class":10732},[40,81121,80914],{"class":10287},[40,81123,10507],{"class":10363},[40,81125,81126,81129,81131,81133,81135,81137],{"class":10276,"line":10496},[40,81127,81128],{"class":10973},"    detrend_mean",[40,81130,10443],{"class":10363},[40,81132,21950],{"class":10732},[40,81134,10772],{"class":10732},[40,81136,80930],{"class":10446},[40,81138,10507],{"class":10363},[40,81140,81141,81144,81146,81148,81150,81152],{"class":10276,"line":10510},[40,81142,81143],{"class":10973},"    amp_mode",[40,81145,10443],{"class":10363},[40,81147,10733],{"class":10732},[40,81149,10772],{"class":10732},[40,81151,80946],{"class":10287},[40,81153,10507],{"class":10363},[40,81155,81156,81159,81161,81163,81165,81167],{"class":10276,"line":10696},[40,81157,81158],{"class":10973},"    norm_mode",[40,81160,10443],{"class":10363},[40,81162,10733],{"class":10732},[40,81164,10772],{"class":10732},[40,81166,80962],{"class":10287},[40,81168,10507],{"class":10363},[40,81170,81171,81174,81176,81178,81180,81182],{"class":10276,"line":10702},[40,81172,81173],{"class":10973},"    freq_units",[40,81175,10443],{"class":10363},[40,81177,10733],{"class":10732},[40,81179,10772],{"class":10732},[40,81181,80978],{"class":10287},[40,81183,10507],{"class":10363},[40,81185,81186],{"class":10276,"line":10708},[40,81187,81188],{"class":10363},") -> tuple[np.ndarray, np.ndarray]:\n",[40,81190,81191],{"class":10276,"line":10714},[40,81192,81193],{"class":10287},"    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[40,81195,81196],{"class":10276,"line":10720},[40,81197,10301],{"emptyLinePlaceholder":10300},[40,81199,81200],{"class":10276,"line":10726},[40,81201,73822],{"class":10287},[40,81203,81204],{"class":10276,"line":10764},[40,81205,81206],{"class":10287},"        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[40,81208,81209],{"class":10276,"line":10778},[40,81210,81211],{"class":10287},"        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[40,81213,81214],{"class":10276,"line":10815},[40,81215,81216],{"class":10287},"        pad: zero-padding factor (≥1).\n",[40,81218,81219],{"class":10276,"line":10829},[40,81220,81221],{"class":10287},"        window: \"hann\" or \"boxcar\".\n",[40,81223,81224],{"class":10276,"line":10842},[40,81225,81226],{"class":10287},"        detrend_mean: subtract column-wise mean before FFT.\n",[40,81228,81229],{"class":10276,"line":10855},[40,81230,81231],{"class":10287},"        amp_mode: \"magnitude\", \"power\", or \"db\".\n",[40,81233,81234],{"class":10276,"line":10868},[40,81235,81236],{"class":10287},"        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[40,81238,81239],{"class":10276,"line":10881},[40,81240,81241],{"class":10287},"        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[40,81243,81244],{"class":10276,"line":10907},[40,81245,10301],{"emptyLinePlaceholder":10300},[40,81247,81248],{"class":10276,"line":10920},[40,81249,73856],{"class":10287},[40,81251,81252],{"class":10276,"line":10933},[40,81253,81254],{"class":10287},"        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[40,81256,81257],{"class":10276,"line":10946},[40,81258,81259],{"class":10287},"        S:     2D nonnegative spectrogram (N_w, N_f).\n",[40,81261,81262],{"class":10276,"line":10951},[40,81263,10723],{"class":10287},[40,81265,81266,81269,81271,81273,81275,81277,81279],{"class":10276,"line":10956},[40,81267,81268],{"class":10363},"    Pe ",[40,81270,162],{"class":10732},[40,81272,11156],{"class":10363},[40,81274,11159],{"class":10430},[40,81276,74208],{"class":10363},[40,81278,10784],{"class":10732},[40,81280,10761],{"class":10363},[40,81282,81283,81286,81288,81290,81292,81294,81296],{"class":10276,"line":10962},[40,81284,81285],{"class":10363},"    dur_ns ",[40,81287,162],{"class":10732},[40,81289,11156],{"class":10363},[40,81291,11159],{"class":10430},[40,81293,80668],{"class":10363},[40,81295,10784],{"class":10732},[40,81297,10761],{"class":10363},[40,81299,81300,81302,81305,81307,81309,81311,81314],{"class":10276,"line":10983},[40,81301,12951],{"class":10352},[40,81303,81304],{"class":10363}," Pe.shape[",[40,81306,66],{"class":10446},[40,81308,10741],{"class":10363},[40,81310,11231],{"class":10732},[40,81312,81313],{"class":10363}," dur_ns.size, ",[40,81315,81316],{"class":10287},"\"Pe must be shaped (N_tau, N_f).\"\n",[40,81318,81319],{"class":10276,"line":11005},[40,81320,10301],{"emptyLinePlaceholder":10300},[40,81322,81323],{"class":10276,"line":11011},[40,81324,81325],{"class":10280},"    # Uniform sampling assumption (true for np.linspace)\n",[40,81327,81328,81331,81333,81335,81338,81340,81342,81344,81347,81349,81351,81353,81355],{"class":10276,"line":11022},[40,81329,81330],{"class":10363},"    dt ",[40,81332,162],{"class":10732},[40,81334,75857],{"class":10732},[40,81336,81337],{"class":10363},"(dur_ns[",[40,81339,78],{"class":10446},[40,81341,10741],{"class":10363},[40,81343,11056],{"class":10732},[40,81345,81346],{"class":10363}," dur_ns[",[40,81348,66],{"class":10446},[40,81350,11222],{"class":10363},[40,81352,11752],{"class":10732},[40,81354,38902],{"class":10446},[40,81356,81357],{"class":10280},"               # [s]\n",[40,81359,81360,81363,81365],{"class":10276,"line":11064},[40,81361,81362],{"class":10363},"    n_tau ",[40,81364,162],{"class":10732},[40,81366,81367],{"class":10363}," dur_ns.size\n",[40,81369,81370,81373,81375,81377,81379,81381,81384,81386,81388,81391,81393,81396,81399,81401,81403,81405,81407],{"class":10276,"line":11103},[40,81371,81372],{"class":10363},"    nfft ",[40,81374,162],{"class":10732},[40,81376,37310],{"class":10732},[40,81378,167],{"class":10363},[40,81380,1044],{"class":10446},[40,81382,81383],{"class":10732}," **",[40,81385,37310],{"class":10732},[40,81387,13108],{"class":10363},[40,81389,81390],{"class":10430},"ceil",[40,81392,13108],{"class":10363},[40,81394,81395],{"class":10430},"log2",[40,81397,81398],{"class":10363},"(n_tau))) ",[40,81400,11752],{"class":10732},[40,81402,22290],{"class":10732},[40,81404,167],{"class":10363},[40,81406,78],{"class":10446},[40,81408,81409],{"class":10363},", pad))\n",[40,81411,81412,81415,81417,81419],{"class":10276,"line":11142},[40,81413,81414],{"class":10363},"    w ",[40,81416,162],{"class":10732},[40,81418,80997],{"class":10430},[40,81420,81421],{"class":10363},"(n_tau, window)\n",[40,81423,81424],{"class":10276,"line":11148},[40,81425,10301],{"emptyLinePlaceholder":10300},[40,81427,81428],{"class":10276,"line":11181},[40,81429,81430],{"class":10280},"    # Prepare output\n",[40,81432,81433,81436,81438],{"class":10276,"line":11198},[40,81434,81435],{"class":10363},"    S_list ",[40,81437,162],{"class":10732},[40,81439,13835],{"class":10363},[40,81441,81442,81444,81446,81449,81452],{"class":10276,"line":11249},[40,81443,16165],{"class":10363},[40,81445,162],{"class":10732},[40,81447,81448],{"class":10363}," Pe.",[40,81450,81451],{"class":10430},"copy",[40,81453,12174],{"class":10363},[40,81455,81456,81458],{"class":10276,"line":11254},[40,81457,11323],{"class":10352},[40,81459,81460],{"class":10363}," detrend_mean:\n",[40,81462,81463,81466,81468,81471,81473,81476,81479,81481,81483,81485,81487,81489,81492,81494,81496],{"class":10276,"line":11259},[40,81464,81465],{"class":10363},"        X ",[40,81467,162],{"class":10732},[40,81469,81470],{"class":10363}," X ",[40,81472,11056],{"class":10732},[40,81474,81475],{"class":10363}," X.",[40,81477,81478],{"class":10430},"mean",[40,81480,167],{"class":10363},[40,81482,11275],{"class":11903},[40,81484,162],{"class":10732},[40,81486,66],{"class":10446},[40,81488,10484],{"class":10363},[40,81490,81491],{"class":11903},"keepdims",[40,81493,162],{"class":10732},[40,81495,13091],{"class":10446},[40,81497,10761],{"class":10363},[40,81499,81500,81502,81505,81508,81510],{"class":10276,"line":11265},[40,81501,16165],{"class":10363},[40,81503,81504],{"class":10732},"*=",[40,81506,81507],{"class":10363}," w[:, ",[40,81509,73064],{"class":10446},[40,81511,25466],{"class":10363},[40,81513,81514],{"class":10276,"line":11294},[40,81515,10301],{"emptyLinePlaceholder":10300},[40,81517,81518],{"class":10276,"line":11300},[40,81519,81520],{"class":10280},"    # Real FFT along τ (axis=0)\n",[40,81522,81523,81526,81528,81531,81534,81537,81539,81541,81544,81546,81548,81550,81553],{"class":10276,"line":11320},[40,81524,81525],{"class":10363},"    Xf ",[40,81527,162],{"class":10732},[40,81529,81530],{"class":10363}," np.fft.",[40,81532,81533],{"class":10430},"rfft",[40,81535,81536],{"class":10363},"(X, ",[40,81538,1796],{"class":11903},[40,81540,162],{"class":10732},[40,81542,81543],{"class":10363},"nfft, ",[40,81545,11275],{"class":11903},[40,81547,162],{"class":10732},[40,81549,66],{"class":10446},[40,81551,81552],{"class":10363},")                     ",[40,81554,81555],{"class":10280},"# shape (N_w, N_f)\n",[40,81557,81558,81561,81563,81565,81568,81571,81574,81576,81579],{"class":10276,"line":11336},[40,81559,81560],{"class":10363},"    f_fft_hz ",[40,81562,162],{"class":10732},[40,81564,81530],{"class":10363},[40,81566,81567],{"class":10430},"rfftfreq",[40,81569,81570],{"class":10363},"(nfft, ",[40,81572,81573],{"class":11903},"d",[40,81575,162],{"class":10732},[40,81577,81578],{"class":10363},"dt)                  ",[40,81580,81581],{"class":10280},"# [Hz]\n",[40,81583,81584],{"class":10276,"line":11381},[40,81585,10301],{"emptyLinePlaceholder":10300},[40,81587,81588],{"class":10276,"line":11395},[40,81589,81590],{"class":10280},"    # Amplitude selection\n",[40,81592,81593,81595,81598,81600,81602,81604,81607],{"class":10276,"line":11438},[40,81594,11323],{"class":10352},[40,81596,81597],{"class":10363}," amp_mode.",[40,81599,11171],{"class":10430},[40,81601,27025],{"class":10363},[40,81603,11231],{"class":10732},[40,81605,81606],{"class":10287}," \"power\"",[40,81608,10639],{"class":10363},[40,81610,81611,81614,81616,81618,81620,81623,81625],{"class":10276,"line":11452},[40,81612,81613],{"class":10363},"        S ",[40,81615,162],{"class":10732},[40,81617,11156],{"class":10363},[40,81619,23431],{"class":10430},[40,81621,81622],{"class":10363},"(Xf) ",[40,81624,22571],{"class":10732},[40,81626,74080],{"class":10446},[40,81628,81629,81631],{"class":10276,"line":11493},[40,81630,76542],{"class":10352},[40,81632,10639],{"class":10363},[40,81634,81635,81637,81639,81641,81643],{"class":10276,"line":11507},[40,81636,81613],{"class":10363},[40,81638,162],{"class":10732},[40,81640,11156],{"class":10363},[40,81642,23431],{"class":10430},[40,81644,81645],{"class":10363},"(Xf)\n",[40,81647,81648],{"class":10276,"line":11512},[40,81649,10301],{"emptyLinePlaceholder":10300},[40,81651,81652],{"class":10276,"line":11517},[40,81653,81654],{"class":10280},"    # Normalization\n",[40,81656,81657,81659,81662,81664,81666,81668,81671],{"class":10276,"line":11545},[40,81658,11323],{"class":10352},[40,81660,81661],{"class":10363}," norm_mode.",[40,81663,11171],{"class":10430},[40,81665,27025],{"class":10363},[40,81667,11231],{"class":10732},[40,81669,81670],{"class":10287}," \"per_f\"",[40,81672,10639],{"class":10363},[40,81674,81675,81678,81680,81682,81685,81688,81690,81692,81694,81696,81698,81700,81702,81704,81706,81708,81711],{"class":10276,"line":11557},[40,81676,81677],{"class":10363},"        col_max ",[40,81679,162],{"class":10732},[40,81681,11156],{"class":10363},[40,81683,81684],{"class":10430},"maximum",[40,81686,81687],{"class":10363},"(S.",[40,81689,3032],{"class":10430},[40,81691,167],{"class":10363},[40,81693,11275],{"class":11903},[40,81695,162],{"class":10732},[40,81697,66],{"class":10446},[40,81699,10484],{"class":10363},[40,81701,81491],{"class":11903},[40,81703,162],{"class":10732},[40,81705,13091],{"class":10446},[40,81707,11040],{"class":10363},[40,81709,81710],{"class":10446},"1e-15",[40,81712,10761],{"class":10363},[40,81714,81715,81717,81719,81722,81724],{"class":10276,"line":11563},[40,81716,81613],{"class":10363},[40,81718,162],{"class":10732},[40,81720,81721],{"class":10363}," S ",[40,81723,11243],{"class":10732},[40,81725,81726],{"class":10363}," col_max\n",[40,81728,81729,81731],{"class":10276,"line":11582},[40,81730,76542],{"class":10352},[40,81732,10639],{"class":10363},[40,81734,81735,81737,81739,81741,81743,81745,81747,81749,81751,81753],{"class":10276,"line":11596},[40,81736,81613],{"class":10363},[40,81738,162],{"class":10732},[40,81740,81721],{"class":10363},[40,81742,11243],{"class":10732},[40,81744,22290],{"class":10732},[40,81746,81687],{"class":10363},[40,81748,3032],{"class":10430},[40,81750,13575],{"class":10363},[40,81752,81710],{"class":10446},[40,81754,10761],{"class":10363},[40,81756,81757],{"class":10276,"line":11609},[40,81758,10301],{"emptyLinePlaceholder":10300},[40,81760,81761,81763,81765,81767,81769,81771,81774],{"class":10276,"line":11622},[40,81762,11323],{"class":10352},[40,81764,81597],{"class":10363},[40,81766,11171],{"class":10430},[40,81768,27025],{"class":10363},[40,81770,11231],{"class":10732},[40,81772,81773],{"class":10287}," \"db\"",[40,81775,10639],{"class":10363},[40,81777,81778,81780,81782,81784,81786,81788,81791,81793,81795,81798,81800,81802],{"class":10276,"line":11637},[40,81779,81613],{"class":10363},[40,81781,162],{"class":10732},[40,81783,25770],{"class":10446},[40,81785,11881],{"class":10732},[40,81787,11156],{"class":10363},[40,81789,81790],{"class":10430},"log10",[40,81792,13108],{"class":10363},[40,81794,81684],{"class":10430},[40,81796,81797],{"class":10363},"(S, ",[40,81799,25310],{"class":10446},[40,81801,17139],{"class":10363},[40,81803,81804],{"class":10280},"# dB, clipped floor\n",[40,81806,81807],{"class":10276,"line":11708},[40,81808,10301],{"emptyLinePlaceholder":10300},[40,81810,81811],{"class":10276,"line":11727},[40,81812,81813],{"class":10280},"    # Units\n",[40,81815,81816,81818,81821,81823,81825,81827,81830],{"class":10276,"line":11744},[40,81817,11323],{"class":10352},[40,81819,81820],{"class":10363}," freq_units.",[40,81822,13274],{"class":10430},[40,81824,27025],{"class":10363},[40,81826,11231],{"class":10732},[40,81828,81829],{"class":10287}," \"GHz\"",[40,81831,10639],{"class":10363},[40,81833,81834,81837,81839,81842,81844],{"class":10276,"line":11790},[40,81835,81836],{"class":10363},"        f_fft ",[40,81838,162],{"class":10732},[40,81840,81841],{"class":10363}," f_fft_hz ",[40,81843,11752],{"class":10732},[40,81845,25987],{"class":10446},[40,81847,81848,81850],{"class":10276,"line":11795},[40,81849,76542],{"class":10352},[40,81851,10639],{"class":10363},[40,81853,81854,81856,81858,81860,81862],{"class":10276,"line":11800},[40,81855,81836],{"class":10363},[40,81857,162],{"class":10732},[40,81859,81841],{"class":10363},[40,81861,11752],{"class":10732},[40,81863,81864],{"class":10446}," 1e-6\n",[40,81866,81867],{"class":10276,"line":11806},[40,81868,10301],{"emptyLinePlaceholder":10300},[40,81870,81871,81873],{"class":10276,"line":11834},[40,81872,11201],{"class":10352},[40,81874,81875],{"class":10363}," f_fft, S\n",[40,81877,81878],{"class":10276,"line":11840},[40,81879,10301],{"emptyLinePlaceholder":10300},[40,81881,81882],{"class":10276,"line":11860},[40,81883,81884],{"class":10280},"# Compute spectrogram with current grid\n",[40,81886,81887,81890,81892,81894],{"class":10276,"line":11887},[40,81888,81889],{"class":10363},"f_fft_axis, S_fft ",[40,81891,162],{"class":10732},[40,81893,81077],{"class":10430},[40,81895,12527],{"class":10363},[40,81897,81898],{"class":10276,"line":11914},[40,81899,81900],{"class":10363},"    Pe, dur_ns,\n",[40,81902,81903,81905,81907,81909],{"class":10276,"line":11938},[40,81904,81098],{"class":11903},[40,81906,162],{"class":10732},[40,81908,73305],{"class":10446},[40,81910,10507],{"class":10363},[40,81912,81913,81915,81917,81919],{"class":10276,"line":11960},[40,81914,81113],{"class":11903},[40,81916,162],{"class":10732},[40,81918,73319],{"class":10446},[40,81920,10507],{"class":10363},[40,81922,81923,81925,81927,81929],{"class":10276,"line":11975},[40,81924,81128],{"class":11903},[40,81926,162],{"class":10732},[40,81928,73334],{"class":10446},[40,81930,10507],{"class":10363},[40,81932,81933,81935,81937,81939],{"class":10276,"line":11996},[40,81934,81143],{"class":11903},[40,81936,162],{"class":10732},[40,81938,73348],{"class":10446},[40,81940,10507],{"class":10363},[40,81942,81943,81945,81947,81949],{"class":10276,"line":12001},[40,81944,81158],{"class":11903},[40,81946,162],{"class":10732},[40,81948,73363],{"class":10446},[40,81950,10507],{"class":10363},[40,81952,81953,81955,81957,81959],{"class":10276,"line":12006},[40,81954,81173],{"class":11903},[40,81956,162],{"class":10732},[40,81958,73378],{"class":10446},[40,81960,10507],{"class":10363},[40,81962,81963],{"class":10276,"line":12028},[40,81964,10761],{"class":10363},[40,81966,81967,81969,81971,81974,81977,81980,81983,81985,81987,81990,81992,81994,81996,81998,82000],{"class":10276,"line":12034},[40,81968,14993],{"class":10732},[40,81970,167],{"class":10363},[40,81972,81973],{"class":10287},"\"FFT grid:\"",[40,81975,81976],{"class":10363},", S_fft.shape, ",[40,81978,81979],{"class":10287},"\"| Fourier freq range:\"",[40,81981,81982],{"class":10363},", f_fft_axis[",[40,81984,66],{"class":10446},[40,81986,11672],{"class":10363},[40,81988,81989],{"class":10287},"\"→\"",[40,81991,81982],{"class":10363},[40,81993,11056],{"class":10732},[40,81995,78],{"class":10446},[40,81997,11672],{"class":10363},[40,81999,73378],{"class":10446},[40,82001,10761],{"class":10363},[10082,82003,82006],{"className":82004,"code":82005,"language":2932,"meta":10090},[10085],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[10088,82007,82005],{"__ignoreMap":10090},[10082,82009,82011],{"className":10269,"code":82010,"language":10271,"meta":10090,"style":10090},"# @title Theoretical Rabi ridge\nimport math\ntwo_pi = 2.0 * math.pi\nf0 = F0_GHZ * 1e9\ndelta_hz = two_pi * (freq_ghz*1e9 - f0)   # rad\u002Fs detuning\nomega_onres = two_pi * OMEGA_ONRESONANCE_MHZ * 1e6  # rad\u002Fs\nOmega_R = np.sqrt(omega_onres**2 + delta_hz**2)     # rad\u002Fs\nridge_hz = Omega_R \u002F (2.0 * math.pi)                # Hz\nif FREQ_UNITS.upper() == \"GHz\":\n    ridge_axis = ridge_hz * 1e-9\nelse:\n    ridge_axis = ridge_hz * 1e-6  # MHz\n",[10088,82012,82013,82018,82024,82037,82051,82078,82098,82130,82153,82172,82186,82192],{"__ignoreMap":10090},[40,82014,82015],{"class":10276,"line":10277},[40,82016,82017],{"class":10280},"# @title Theoretical Rabi ridge\n",[40,82019,82020,82022],{"class":10276,"line":10284},[40,82021,10375],{"class":10352},[40,82023,21798],{"class":10363},[40,82025,82026,82029,82031,82033,82035],{"class":10276,"line":10291},[40,82027,82028],{"class":10363},"two_pi ",[40,82030,162],{"class":10732},[40,82032,12060],{"class":10446},[40,82034,11881],{"class":10732},[40,82036,73879],{"class":10363},[40,82038,82039,82042,82044,82047,82049],{"class":10276,"line":10297},[40,82040,82041],{"class":10363},"f0 ",[40,82043,162],{"class":10732},[40,82045,82046],{"class":10446}," F0_GHZ",[40,82048,11881],{"class":10732},[40,82050,73894],{"class":10446},[40,82052,82053,82056,82058,82060,82062,82065,82067,82070,82072,82075],{"class":10276,"line":10304},[40,82054,82055],{"class":10363},"delta_hz ",[40,82057,162],{"class":10732},[40,82059,73932],{"class":10363},[40,82061,11752],{"class":10732},[40,82063,82064],{"class":10363}," (freq_ghz",[40,82066,11752],{"class":10732},[40,82068,82069],{"class":10446},"1e9",[40,82071,11764],{"class":10732},[40,82073,82074],{"class":10363}," f0)   ",[40,82076,82077],{"class":10280},"# rad\u002Fs detuning\n",[40,82079,82080,82083,82085,82087,82089,82092,82094,82096],{"class":10276,"line":10310},[40,82081,82082],{"class":10363},"omega_onres ",[40,82084,162],{"class":10732},[40,82086,73932],{"class":10363},[40,82088,11752],{"class":10732},[40,82090,82091],{"class":10446}," OMEGA_ONRESONANCE_MHZ",[40,82093,11881],{"class":10732},[40,82095,26027],{"class":10446},[40,82097,73944],{"class":10280},[40,82099,82100,82103,82105,82107,82109,82112,82114,82116,82118,82121,82123,82125,82128],{"class":10276,"line":10316},[40,82101,82102],{"class":10363},"Omega_R ",[40,82104,162],{"class":10732},[40,82106,11156],{"class":10363},[40,82108,2218],{"class":10430},[40,82110,82111],{"class":10363},"(omega_onres",[40,82113,22571],{"class":10732},[40,82115,1044],{"class":10446},[40,82117,12042],{"class":10732},[40,82119,82120],{"class":10363}," delta_hz",[40,82122,22571],{"class":10732},[40,82124,1044],{"class":10446},[40,82126,82127],{"class":10363},")     ",[40,82129,73996],{"class":10280},[40,82131,82132,82135,82137,82139,82141,82143,82145,82147,82150],{"class":10276,"line":10322},[40,82133,82134],{"class":10363},"ridge_hz ",[40,82136,162],{"class":10732},[40,82138,74094],{"class":10363},[40,82140,11243],{"class":10732},[40,82142,920],{"class":10363},[40,82144,11878],{"class":10446},[40,82146,11881],{"class":10732},[40,82148,82149],{"class":10363}," math.pi)                ",[40,82151,82152],{"class":10280},"# Hz\n",[40,82154,82155,82157,82160,82162,82164,82166,82168,82170],{"class":10276,"line":10327},[40,82156,11225],{"class":10352},[40,82158,82159],{"class":10446}," FREQ_UNITS",[40,82161,1123],{"class":10363},[40,82163,13274],{"class":10430},[40,82165,27025],{"class":10363},[40,82167,11231],{"class":10732},[40,82169,81829],{"class":10287},[40,82171,10639],{"class":10363},[40,82173,82174,82177,82179,82182,82184],{"class":10276,"line":10333},[40,82175,82176],{"class":10363},"    ridge_axis ",[40,82178,162],{"class":10732},[40,82180,82181],{"class":10363}," ridge_hz ",[40,82183,11752],{"class":10732},[40,82185,25987],{"class":10446},[40,82187,82188,82190],{"class":10276,"line":10339},[40,82189,22464],{"class":10352},[40,82191,10639],{"class":10363},[40,82193,82194,82196,82198,82200,82202,82205],{"class":10276,"line":10344},[40,82195,82176],{"class":10363},[40,82197,162],{"class":10732},[40,82199,82181],{"class":10363},[40,82201,11752],{"class":10732},[40,82203,82204],{"class":10446}," 1e-6",[40,82206,82207],{"class":10280},"  # MHz\n",[10082,82209,82211],{"className":10269,"code":82210,"language":10271,"meta":10090,"style":10090},"# @title 2D heatmap\nfig, ax = plt.subplots(figsize=(7.5, 5.5))\nim = ax.pcolormesh(freq_ghz, f_fft_axis, S_fft, shading=\"auto\", cmap=\"inferno\")\ncb = fig.colorbar(im, ax=ax)\ncb.set_label(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\n\nax.plot(freq_ghz, ridge_axis, \"--\", lw=1.5, label=r\"$\\Omega_R\u002F2\\pi$\")\nax.set_xlabel(\"Drive frequency f (GHz)\")\nax.set_ylabel(f\"Fourier frequency ({FREQ_UNITS})\")\nax.set_title(\"Fourier transform along pulse length: $|\\\\mathcal{F}_\\\\tau\\\\{P_e\\\\}|$\")\nax.legend(loc=\"upper right\")\nfig.tight_layout()\nplt.show()\n",[10088,82212,82213,82218,82245,82276,82295,82337,82341,82384,82397,82418,82456,82473,82481],{"__ignoreMap":10090},[40,82214,82215],{"class":10276,"line":10277},[40,82216,82217],{"class":10280},"# @title 2D heatmap\n",[40,82219,82220,82222,82224,82226,82228,82230,82232,82234,82236,82238,82240,82243],{"class":10276,"line":10284},[40,82221,38079],{"class":10363},[40,82223,162],{"class":10732},[40,82225,13070],{"class":10363},[40,82227,13073],{"class":10430},[40,82229,167],{"class":10363},[40,82231,13078],{"class":11903},[40,82233,162],{"class":10732},[40,82235,167],{"class":10363},[40,82237,25397],{"class":10446},[40,82239,10484],{"class":10363},[40,82241,82242],{"class":10446},"5.5",[40,82244,13612],{"class":10363},[40,82246,82247,82250,82252,82254,82256,82259,82261,82263,82265,82267,82269,82271,82274],{"class":10276,"line":10291},[40,82248,82249],{"class":10363},"im ",[40,82251,162],{"class":10732},[40,82253,12522],{"class":10363},[40,82255,74538],{"class":10430},[40,82257,82258],{"class":10363},"(freq_ghz, f_fft_axis, S_fft, ",[40,82260,74544],{"class":11903},[40,82262,162],{"class":10732},[40,82264,13158],{"class":10287},[40,82266,10484],{"class":10363},[40,82268,16338],{"class":11903},[40,82270,162],{"class":10732},[40,82272,82273],{"class":10287},"\"inferno\"",[40,82275,10761],{"class":10363},[40,82277,82278,82280,82282,82284,82286,82288,82290,82292],{"class":10276,"line":10297},[40,82279,74556],{"class":10363},[40,82281,162],{"class":10732},[40,82283,13170],{"class":10363},[40,82285,13173],{"class":10430},[40,82287,13176],{"class":10363},[40,82289,12429],{"class":11903},[40,82291,162],{"class":10732},[40,82293,82294],{"class":10363},"ax)\n",[40,82296,82297,82299,82301,82303,82306,82308,82310,82313,82316,82319,82321,82323,82325,82327,82330,82332,82335],{"class":10276,"line":10304},[40,82298,80158],{"class":10363},[40,82300,13211],{"class":10430},[40,82302,167],{"class":10363},[40,82304,82305],{"class":10287},"\"FFT amplitude\"",[40,82307,12042],{"class":10732},[40,82309,920],{"class":10363},[40,82311,82312],{"class":10287},"\" (dB)\"",[40,82314,82315],{"class":10352}," if",[40,82317,82318],{"class":10446}," AMP_MODE",[40,82320,1123],{"class":10363},[40,82322,11171],{"class":10430},[40,82324,13277],{"class":10363},[40,82326,11231],{"class":10732},[40,82328,82329],{"class":10287},"\"db\"",[40,82331,11237],{"class":10352},[40,82333,82334],{"class":10287}," \" (arb.)\"",[40,82336,13612],{"class":10363},[40,82338,82339],{"class":10276,"line":10310},[40,82340,10301],{"emptyLinePlaceholder":10300},[40,82342,82343,82345,82347,82350,82352,82354,82357,82359,82361,82363,82365,82367,82369,82371,82373,82376,82379,82382],{"class":10276,"line":10316},[40,82344,38114],{"class":10363},[40,82346,23505],{"class":10430},[40,82348,82349],{"class":10363},"(freq_ghz, ridge_axis, ",[40,82351,23279],{"class":10287},[40,82353,10484],{"class":10363},[40,82355,82356],{"class":11903},"lw",[40,82358,162],{"class":10732},[40,82360,23229],{"class":10446},[40,82362,10484],{"class":10363},[40,82364,15415],{"class":11903},[40,82366,162],{"class":10732},[40,82368,936],{"class":10352},[40,82370,77873],{"class":10356},[40,82372,24129],{"class":10732},[40,82374,82375],{"class":10356},"mega_R\u002F2",[40,82377,82378],{"class":10732},"\\p",[40,82380,82381],{"class":10356},"i$\"",[40,82383,10761],{"class":10363},[40,82385,82386,82388,82390,82392,82395],{"class":10276,"line":10322},[40,82387,38114],{"class":10363},[40,82389,13230],{"class":10430},[40,82391,167],{"class":10363},[40,82393,82394],{"class":10287},"\"Drive frequency f (GHz)\"",[40,82396,10761],{"class":10363},[40,82398,82399,82401,82403,82405,82407,82410,82413,82416],{"class":10276,"line":10327},[40,82400,38114],{"class":10363},[40,82402,13244],{"class":10430},[40,82404,167],{"class":10363},[40,82406,13262],{"class":10352},[40,82408,82409],{"class":10287},"\"Fourier frequency (",[40,82411,82412],{"class":10446},"{FREQ_UNITS}",[40,82414,82415],{"class":10287},")\"",[40,82417,10761],{"class":10363},[40,82419,82420,82422,82424,82426,82429,82431,82433,82436,82439,82441,82444,82446,82449,82451,82454],{"class":10276,"line":10333},[40,82421,38114],{"class":10363},[40,82423,30334],{"class":10430},[40,82425,167],{"class":10363},[40,82427,82428],{"class":10287},"\"Fourier transform along pulse length: $|",[40,82430,74622],{"class":10732},[40,82432,69216],{"class":10287},[40,82434,82435],{"class":10446},"{F}",[40,82437,82438],{"class":10287},"_",[40,82440,74622],{"class":10732},[40,82442,82443],{"class":10287},"tau",[40,82445,74622],{"class":10732},[40,82447,82448],{"class":10287},"{P_e",[40,82450,74622],{"class":10732},[40,82452,82453],{"class":10287},"}|$\"",[40,82455,10761],{"class":10363},[40,82457,82458,82460,82462,82464,82466,82468,82471],{"class":10276,"line":10339},[40,82459,38114],{"class":10363},[40,82461,23677],{"class":10430},[40,82463,167],{"class":10363},[40,82465,12131],{"class":11903},[40,82467,162],{"class":10732},[40,82469,82470],{"class":10287},"\"upper right\"",[40,82472,10761],{"class":10363},[40,82474,82475,82477,82479],{"class":10276,"line":10344},[40,82476,74636],{"class":10363},[40,82478,30425],{"class":10430},[40,82480,12174],{"class":10363},[40,82482,82483,82485,82487],{"class":10276,"line":10349},[40,82484,23469],{"class":10363},[40,82486,23686],{"class":10430},[40,82488,12174],{"class":10363},[13997,82490],{"alt":82491,"src":82492},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[10082,82494,82496],{"className":10269,"code":82495,"language":10271,"meta":10090,"style":10090},"# --- Adaptive high-res 3D surface ---\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# ------ Control knobs ------\nUPSAMPLE_F_FREQ: int = 3       # try 2–3; upsample the drive-frequency axis only\nMAX_FACES: int = 300_000       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\nMIN_ROWS: int = 128            # keep at least this many Fourier-frequency rows\nMAX_ROWS: int = 1024           # hard cap for rows to keep plotting fast\n\n# Preserve global color limits so the legend stays consistent\nvmin = float(np.nanmin(S_fft))\nvmax = float(np.nanmax(S_fft))\n\n# Shapes\nn_rows, n_cols = S_fft.shape        # rows=Fourier freq (ν), cols=drive freq (f)\n\n# 1) Upsample along the typically smaller axis (drive frequency) only\nn_cols_up = min(UPSAMPLE_F_FREQ * (n_cols - 1) + 1, 1200)  # sensible upper bound\nfreq_hi = np.linspace(freq_ghz.min(), freq_ghz.max(), n_cols_up)\n\nS_up_f = np.empty((n_rows, n_cols_up), dtype=float)\nfor i in range(n_rows):\n    S_up_f[i, :] = np.interp(freq_hi, freq_ghz, S_fft[i, :])\n\n# 2) Determine how many Fourier rows we can afford for MAX_FACES\n# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\nn_rows_cap = int(MAX_FACES \u002F max(n_cols_up - 1, 1)) + 1\nn_rows_plot = int(np.clip(n_rows_cap, MIN_ROWS, min(MAX_ROWS, n_rows)))\n\n# Down-sample ν-axis evenly to n_rows_plot\nrow_idx = np.linspace(0, n_rows - 1, n_rows_plot).astype(int)\nf_fft_plot = f_fft_axis[row_idx]\nS_final = S_up_f[row_idx, :]\n\n# Mesh for plotting\nF2_hi, FFOUR_hi = np.meshgrid(freq_hi, f_fft_plot, indexing=\"xy\")\n\n# 3) Plot (keeps 'inferno')\nfig = plt.figure(figsize=(8, 6), dpi=200)\nax = fig.add_subplot(111, projection=\"3d\")\nsurf = ax.plot_surface(\n    F2_hi, FFOUR_hi, S_final,\n    rstride=1, cstride=1,\n    linewidth=0, antialiased=True,\n    cmap=\"inferno\",\n    vmin=vmin, vmax=vmax\n)\n\nax.set_xlabel(\"f (GHz)\")\nax.set_ylabel(f\"Fourier freq ({FREQ_UNITS})\")\nax.set_zlabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower() == \"db\" else \" (arb.)\"))\nax.set_title(\"Rabi FFT surface (adaptive resolution)\")\nfig.colorbar(surf, ax=ax, shrink=0.7, pad=0.08, label=\"Amplitude\")\n\n# A clearer default view\nax.view_init(elev=25, azim=-60)\n\nfig.tight_layout()\nplt.show()\n",[10088,82497,82498,82503,82513,82523,82533,82537,82542,82559,82576,82593,82610,82614,82619,82636,82652,82656,82661,82674,82678,82683,82721,82742,82746,82766,82779,82793,82797,82802,82807,82841,82870,82874,82879,82913,82923,82933,82937,82941,82967,82971,82976,83010,83034,83046,83056,83074,83092,83102,83118,83122,83126,83139,83158,83194,83207,83245,83249,83254,83278,83282,83290],{"__ignoreMap":10090},[40,82499,82500],{"class":10276,"line":10277},[40,82501,82502],{"class":10280},"# --- Adaptive high-res 3D surface ---\n",[40,82504,82505,82507,82509,82511],{"class":10276,"line":10284},[40,82506,10353],{"class":10352},[40,82508,79503],{"class":10363},[40,82510,10375],{"class":10352},[40,82512,79508],{"class":10363},[40,82514,82515,82517,82519,82521],{"class":10276,"line":10291},[40,82516,10375],{"class":10352},[40,82518,10378],{"class":10363},[40,82520,10381],{"class":10352},[40,82522,10384],{"class":10363},[40,82524,82525,82527,82529,82531],{"class":10276,"line":10297},[40,82526,10375],{"class":10352},[40,82528,10405],{"class":10363},[40,82530,10381],{"class":10352},[40,82532,10410],{"class":10363},[40,82534,82535],{"class":10276,"line":10304},[40,82536,10301],{"emptyLinePlaceholder":10300},[40,82538,82539],{"class":10276,"line":10310},[40,82540,82541],{"class":10280},"# ------ Control knobs ------\n",[40,82543,82544,82547,82549,82551,82553,82556],{"class":10276,"line":10316},[40,82545,82546],{"class":10446},"UPSAMPLE_F_FREQ",[40,82548,10443],{"class":10363},[40,82550,10821],{"class":10732},[40,82552,10772],{"class":10732},[40,82554,82555],{"class":10446}," 3",[40,82557,82558],{"class":10280},"       # try 2–3; upsample the drive-frequency axis only\n",[40,82560,82561,82564,82566,82568,82570,82573],{"class":10276,"line":10322},[40,82562,82563],{"class":10446},"MAX_FACES",[40,82565,10443],{"class":10363},[40,82567,10821],{"class":10732},[40,82569,10772],{"class":10732},[40,82571,82572],{"class":10446}," 300_000",[40,82574,82575],{"class":10280},"       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[40,82577,82578,82581,82583,82585,82587,82590],{"class":10276,"line":10327},[40,82579,82580],{"class":10446},"MIN_ROWS",[40,82582,10443],{"class":10363},[40,82584,10821],{"class":10732},[40,82586,10772],{"class":10732},[40,82588,82589],{"class":10446}," 128",[40,82591,82592],{"class":10280},"            # keep at least this many Fourier-frequency rows\n",[40,82594,82595,82598,82600,82602,82604,82607],{"class":10276,"line":10333},[40,82596,82597],{"class":10446},"MAX_ROWS",[40,82599,10443],{"class":10363},[40,82601,10821],{"class":10732},[40,82603,10772],{"class":10732},[40,82605,82606],{"class":10446}," 1024",[40,82608,82609],{"class":10280},"           # hard cap for rows to keep plotting fast\n",[40,82611,82612],{"class":10276,"line":10339},[40,82613,10301],{"emptyLinePlaceholder":10300},[40,82615,82616],{"class":10276,"line":10344},[40,82617,82618],{"class":10280},"# Preserve global color limits so the legend stays consistent\n",[40,82620,82621,82624,82626,82628,82630,82633],{"class":10276,"line":10349},[40,82622,82623],{"class":10363},"vmin ",[40,82625,162],{"class":10732},[40,82627,75857],{"class":10732},[40,82629,13108],{"class":10363},[40,82631,82632],{"class":10430},"nanmin",[40,82634,82635],{"class":10363},"(S_fft))\n",[40,82637,82638,82641,82643,82645,82647,82650],{"class":10276,"line":10367},[40,82639,82640],{"class":10363},"vmax ",[40,82642,162],{"class":10732},[40,82644,75857],{"class":10732},[40,82646,13108],{"class":10363},[40,82648,82649],{"class":10430},"nanmax",[40,82651,82635],{"class":10363},[40,82653,82654],{"class":10276,"line":10372},[40,82655,10301],{"emptyLinePlaceholder":10300},[40,82657,82658],{"class":10276,"line":10387},[40,82659,82660],{"class":10280},"# Shapes\n",[40,82662,82663,82666,82668,82671],{"class":10276,"line":10400},[40,82664,82665],{"class":10363},"n_rows, n_cols ",[40,82667,162],{"class":10732},[40,82669,82670],{"class":10363}," S_fft.shape        ",[40,82672,82673],{"class":10280},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[40,82675,82676],{"class":10276,"line":10413},[40,82677,10301],{"emptyLinePlaceholder":10300},[40,82679,82680],{"class":10276,"line":10418},[40,82681,82682],{"class":10280},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[40,82684,82685,82688,82690,82692,82694,82696,82698,82701,82703,82705,82707,82709,82711,82713,82716,82718],{"class":10276,"line":10424},[40,82686,82687],{"class":10363},"n_cols_up ",[40,82689,162],{"class":10732},[40,82691,22453],{"class":10732},[40,82693,167],{"class":10363},[40,82695,82546],{"class":10446},[40,82697,11881],{"class":10732},[40,82699,82700],{"class":10363}," (n_cols ",[40,82702,11056],{"class":10732},[40,82704,13373],{"class":10446},[40,82706,12055],{"class":10363},[40,82708,1053],{"class":10732},[40,82710,13373],{"class":10446},[40,82712,10484],{"class":10363},[40,82714,82715],{"class":10446},"1200",[40,82717,17908],{"class":10363},[40,82719,82720],{"class":10280},"# sensible upper bound\n",[40,82722,82723,82725,82727,82729,82731,82733,82735,82737,82739],{"class":10276,"line":10437},[40,82724,79610],{"class":10363},[40,82726,162],{"class":10732},[40,82728,11156],{"class":10363},[40,82730,11850],{"class":10430},[40,82732,79619],{"class":10363},[40,82734,3067],{"class":10430},[40,82736,79624],{"class":10363},[40,82738,3032],{"class":10430},[40,82740,82741],{"class":10363},"(), n_cols_up)\n",[40,82743,82744],{"class":10276,"line":10456},[40,82745,10301],{"emptyLinePlaceholder":10300},[40,82747,82748,82751,82753,82755,82757,82760,82762,82764],{"class":10276,"line":10472},[40,82749,82750],{"class":10363},"S_up_f ",[40,82752,162],{"class":10732},[40,82754,11156],{"class":10363},[40,82756,23907],{"class":10430},[40,82758,82759],{"class":10363},"((n_rows, n_cols_up), ",[40,82761,26704],{"class":11903},[40,82763,79683],{"class":10732},[40,82765,10761],{"class":10363},[40,82767,82768,82770,82772,82774,82776],{"class":10276,"line":10496},[40,82769,13840],{"class":10352},[40,82771,79692],{"class":10363},[40,82773,13846],{"class":10352},[40,82775,30389],{"class":10732},[40,82777,82778],{"class":10363},"(n_rows):\n",[40,82780,82781,82784,82786,82788,82790],{"class":10276,"line":10510},[40,82782,82783],{"class":10363},"    S_up_f[i, :] ",[40,82785,162],{"class":10732},[40,82787,11156],{"class":10363},[40,82789,79715],{"class":10430},[40,82791,82792],{"class":10363},"(freq_hi, freq_ghz, S_fft[i, :])\n",[40,82794,82795],{"class":10276,"line":10696},[40,82796,10301],{"emptyLinePlaceholder":10300},[40,82798,82799],{"class":10276,"line":10702},[40,82800,82801],{"class":10280},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[40,82803,82804],{"class":10276,"line":10708},[40,82805,82806],{"class":10280},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[40,82808,82809,82812,82814,82816,82818,82820,82822,82824,82827,82829,82831,82833,82835,82837,82839],{"class":10276,"line":10714},[40,82810,82811],{"class":10363},"n_rows_cap ",[40,82813,162],{"class":10732},[40,82815,37310],{"class":10732},[40,82817,167],{"class":10363},[40,82819,82563],{"class":10446},[40,82821,22539],{"class":10732},[40,82823,22290],{"class":10732},[40,82825,82826],{"class":10363},"(n_cols_up ",[40,82828,11056],{"class":10732},[40,82830,13373],{"class":10446},[40,82832,10484],{"class":10363},[40,82834,78],{"class":10446},[40,82836,13376],{"class":10363},[40,82838,1053],{"class":10732},[40,82840,27423],{"class":10446},[40,82842,82843,82846,82848,82850,82852,82854,82857,82859,82861,82863,82865,82867],{"class":10276,"line":10720},[40,82844,82845],{"class":10363},"n_rows_plot ",[40,82847,162],{"class":10732},[40,82849,37310],{"class":10732},[40,82851,13108],{"class":10363},[40,82853,24368],{"class":10430},[40,82855,82856],{"class":10363},"(n_rows_cap, ",[40,82858,82580],{"class":10446},[40,82860,10484],{"class":10363},[40,82862,3067],{"class":10732},[40,82864,167],{"class":10363},[40,82866,82597],{"class":10446},[40,82868,82869],{"class":10363},", n_rows)))\n",[40,82871,82872],{"class":10276,"line":10726},[40,82873,10301],{"emptyLinePlaceholder":10300},[40,82875,82876],{"class":10276,"line":10764},[40,82877,82878],{"class":10280},"# Down-sample ν-axis evenly to n_rows_plot\n",[40,82880,82881,82884,82886,82888,82890,82892,82894,82897,82899,82901,82904,82907,82909,82911],{"class":10276,"line":10778},[40,82882,82883],{"class":10363},"row_idx ",[40,82885,162],{"class":10732},[40,82887,11156],{"class":10363},[40,82889,11850],{"class":10430},[40,82891,167],{"class":10363},[40,82893,66],{"class":10446},[40,82895,82896],{"class":10363},", n_rows ",[40,82898,11056],{"class":10732},[40,82900,13373],{"class":10446},[40,82902,82903],{"class":10363},", n_rows_plot).",[40,82905,82906],{"class":10430},"astype",[40,82908,167],{"class":10363},[40,82910,10821],{"class":10732},[40,82912,10761],{"class":10363},[40,82914,82915,82918,82920],{"class":10276,"line":10815},[40,82916,82917],{"class":10363},"f_fft_plot ",[40,82919,162],{"class":10732},[40,82921,82922],{"class":10363}," f_fft_axis[row_idx]\n",[40,82924,82925,82928,82930],{"class":10276,"line":10829},[40,82926,82927],{"class":10363},"S_final ",[40,82929,162],{"class":10732},[40,82931,82932],{"class":10363}," S_up_f[row_idx, :]\n",[40,82934,82935],{"class":10276,"line":10842},[40,82936,10301],{"emptyLinePlaceholder":10300},[40,82938,82939],{"class":10276,"line":10855},[40,82940,79844],{"class":10280},[40,82942,82943,82946,82949,82951,82953,82955,82958,82960,82962,82965],{"class":10276,"line":10868},[40,82944,82945],{"class":10363},"F2_hi, ",[40,82947,82948],{"class":10446},"FFOUR_hi",[40,82950,10772],{"class":10732},[40,82952,11156],{"class":10363},[40,82954,11897],{"class":10430},[40,82956,82957],{"class":10363},"(freq_hi, f_fft_plot, ",[40,82959,11904],{"class":11903},[40,82961,162],{"class":10732},[40,82963,82964],{"class":10287},"\"xy\"",[40,82966,10761],{"class":10363},[40,82968,82969],{"class":10276,"line":10881},[40,82970,10301],{"emptyLinePlaceholder":10300},[40,82972,82973],{"class":10276,"line":10907},[40,82974,82975],{"class":10280},"# 3) Plot (keeps 'inferno')\n",[40,82977,82978,82980,82982,82984,82986,82988,82990,82992,82994,82996,82998,83000,83002,83004,83006,83008],{"class":10276,"line":10920},[40,82979,79880],{"class":10363},[40,82981,162],{"class":10732},[40,82983,13070],{"class":10363},[40,82985,16373],{"class":10430},[40,82987,167],{"class":10363},[40,82989,13078],{"class":11903},[40,82991,162],{"class":10732},[40,82993,167],{"class":10363},[40,82995,23482],{"class":10446},[40,82997,10484],{"class":10363},[40,82999,12785],{"class":10446},[40,83001,11040],{"class":10363},[40,83003,23491],{"class":11903},[40,83005,162],{"class":10732},[40,83007,10447],{"class":10446},[40,83009,10761],{"class":10363},[40,83011,83012,83014,83016,83018,83020,83022,83024,83026,83028,83030,83032],{"class":10276,"line":10933},[40,83013,79915],{"class":10363},[40,83015,162],{"class":10732},[40,83017,13170],{"class":10363},[40,83019,16402],{"class":10430},[40,83021,167],{"class":10363},[40,83023,16407],{"class":10446},[40,83025,10484],{"class":10363},[40,83027,16412],{"class":11903},[40,83029,162],{"class":10732},[40,83031,16417],{"class":10287},[40,83033,10761],{"class":10363},[40,83035,83036,83038,83040,83042,83044],{"class":10276,"line":10946},[40,83037,79945],{"class":10363},[40,83039,162],{"class":10732},[40,83041,12522],{"class":10363},[40,83043,16566],{"class":10430},[40,83045,12527],{"class":10363},[40,83047,83048,83051,83053],{"class":10276,"line":10951},[40,83049,83050],{"class":10363},"    F2_hi, ",[40,83052,82948],{"class":10446},[40,83054,83055],{"class":10363},", S_final,\n",[40,83057,83058,83060,83062,83064,83066,83068,83070,83072],{"class":10276,"line":10956},[40,83059,79963],{"class":11903},[40,83061,162],{"class":10732},[40,83063,78],{"class":10446},[40,83065,10484],{"class":10363},[40,83067,16599],{"class":11903},[40,83069,162],{"class":10732},[40,83071,78],{"class":10446},[40,83073,10507],{"class":10363},[40,83075,83076,83078,83080,83082,83084,83086,83088,83090],{"class":10276,"line":10962},[40,83077,79986],{"class":11903},[40,83079,162],{"class":10732},[40,83081,66],{"class":10446},[40,83083,10484],{"class":10363},[40,83085,79995],{"class":11903},[40,83087,162],{"class":10732},[40,83089,13091],{"class":10446},[40,83091,10507],{"class":10363},[40,83093,83094,83096,83098,83100],{"class":10276,"line":10983},[40,83095,80006],{"class":11903},[40,83097,162],{"class":10732},[40,83099,82273],{"class":10287},[40,83101,10507],{"class":10363},[40,83103,83104,83106,83108,83111,83113,83115],{"class":10276,"line":11005},[40,83105,80017],{"class":11903},[40,83107,162],{"class":10732},[40,83109,83110],{"class":10363},"vmin, ",[40,83112,13145],{"class":11903},[40,83114,162],{"class":10732},[40,83116,83117],{"class":10363},"vmax\n",[40,83119,83120],{"class":10276,"line":11011},[40,83121,10761],{"class":10363},[40,83123,83124],{"class":10276,"line":11022},[40,83125,10301],{"emptyLinePlaceholder":10300},[40,83127,83128,83130,83132,83134,83137],{"class":10276,"line":11064},[40,83129,38114],{"class":10363},[40,83131,13230],{"class":10430},[40,83133,167],{"class":10363},[40,83135,83136],{"class":10287},"\"f (GHz)\"",[40,83138,10761],{"class":10363},[40,83140,83141,83143,83145,83147,83149,83152,83154,83156],{"class":10276,"line":11103},[40,83142,38114],{"class":10363},[40,83144,13244],{"class":10430},[40,83146,167],{"class":10363},[40,83148,13262],{"class":10352},[40,83150,83151],{"class":10287},"\"Fourier freq (",[40,83153,82412],{"class":10446},[40,83155,82415],{"class":10287},[40,83157,10761],{"class":10363},[40,83159,83160,83162,83164,83166,83168,83170,83172,83174,83176,83178,83180,83182,83184,83186,83188,83190,83192],{"class":10276,"line":11142},[40,83161,38114],{"class":10363},[40,83163,16740],{"class":10430},[40,83165,167],{"class":10363},[40,83167,82305],{"class":10287},[40,83169,12042],{"class":10732},[40,83171,920],{"class":10363},[40,83173,82312],{"class":10287},[40,83175,82315],{"class":10352},[40,83177,82318],{"class":10446},[40,83179,1123],{"class":10363},[40,83181,11171],{"class":10430},[40,83183,27025],{"class":10363},[40,83185,11231],{"class":10732},[40,83187,81773],{"class":10287},[40,83189,11237],{"class":10352},[40,83191,82334],{"class":10287},[40,83193,13612],{"class":10363},[40,83195,83196,83198,83200,83202,83205],{"class":10276,"line":11148},[40,83197,38114],{"class":10363},[40,83199,30334],{"class":10430},[40,83201,167],{"class":10363},[40,83203,83204],{"class":10287},"\"Rabi FFT surface (adaptive resolution)\"",[40,83206,10761],{"class":10363},[40,83208,83209,83211,83213,83215,83217,83219,83221,83223,83225,83227,83229,83231,83233,83235,83237,83239,83241,83243],{"class":10276,"line":11181},[40,83210,74636],{"class":10363},[40,83212,13173],{"class":10430},[40,83214,80129],{"class":10363},[40,83216,12429],{"class":11903},[40,83218,162],{"class":10732},[40,83220,13183],{"class":10363},[40,83222,80138],{"class":11903},[40,83224,162],{"class":10732},[40,83226,38131],{"class":10446},[40,83228,10484],{"class":10363},[40,83230,13196],{"class":11903},[40,83232,162],{"class":10732},[40,83234,80151],{"class":10446},[40,83236,10484],{"class":10363},[40,83238,15415],{"class":11903},[40,83240,162],{"class":10732},[40,83242,30361],{"class":10287},[40,83244,10761],{"class":10363},[40,83246,83247],{"class":10276,"line":11198},[40,83248,10301],{"emptyLinePlaceholder":10300},[40,83250,83251],{"class":10276,"line":11249},[40,83252,83253],{"class":10280},"# A clearer default view\n",[40,83255,83256,83258,83260,83262,83264,83266,83268,83270,83272,83274,83276],{"class":10276,"line":11254},[40,83257,38114],{"class":10363},[40,83259,16426],{"class":10430},[40,83261,167],{"class":10363},[40,83263,16431],{"class":11903},[40,83265,162],{"class":10732},[40,83267,64874],{"class":10446},[40,83269,10484],{"class":10363},[40,83271,16439],{"class":11903},[40,83273,17598],{"class":10732},[40,83275,13519],{"class":10446},[40,83277,10761],{"class":10363},[40,83279,83280],{"class":10276,"line":11259},[40,83281,10301],{"emptyLinePlaceholder":10300},[40,83283,83284,83286,83288],{"class":10276,"line":11265},[40,83285,74636],{"class":10363},[40,83287,30425],{"class":10430},[40,83289,12174],{"class":10363},[40,83291,83292,83294,83296],{"class":10276,"line":11294},[40,83293,23469],{"class":10363},[40,83295,23686],{"class":10430},[40,83297,12174],{"class":10363},[13997,83299],{"alt":83300,"src":83301},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[10082,83303,83305],{"className":10269,"code":83304,"language":10271,"meta":10090,"style":10090},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\nF_STAR_GHZ: float = float(F0_GHZ)   # slice near resonance\nNU_STAR = float(ridge_axis[np.argmin(np.abs(freq_ghz - F_STAR_GHZ))])  # near Ω_R\u002F2π\n\n# Slice at f = F_STAR_GHZ\nj = int(np.argmin(np.abs(freq_ghz - F_STAR_GHZ)))\nfig, ax = plt.subplots(figsize=(6.0, 3.8))\nax.plot(f_fft_axis, S_fft[:, j], color='blue')\nax.axvline(ridge_axis[j], ls=\"--\", lw=1.0, label=r\"$\\Omega_R\u002F2\\pi$\")\nax.set_xlabel(f\"Fourier frequency ({FREQ_UNITS})\")\nax.set_ylabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\nax.set_title(fr\"Slice at $f={freq_ghz[j]:.6f}$ GHz\")\nax.legend()\nfig.tight_layout(); plt.show()\n\n# Slice at ν = NU_STAR\ni = int(np.argmin(np.abs(f_fft_axis - NU_STAR)))\nfig, ax = plt.subplots(figsize=(6.0, 3.8))\nax.plot(freq_ghz, S_fft[i, :], color='blue')\nax.set_xlabel(\"Drive frequency f (GHz)\")\nax.set_ylabel(\"FFT amplitude\" + (\" (dB)\" if AMP_MODE.lower()==\"db\" else \" (arb.)\"))\nax.set_title(fr\"Slice at Fourier freq ≈ {f_fft_axis[i]:.3f} {FREQ_UNITS}\")\nfig.tight_layout(); plt.show()\n",[10088,83306,83307,83312,83335,83367,83371,83376,83400,83428,83445,83489,83507,83543,83570,83578,83591,83595,83600,83626,83652,83669,83681,83717,83746],{"__ignoreMap":10090},[40,83308,83309],{"class":10276,"line":10277},[40,83310,83311],{"class":10280},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[40,83313,83314,83317,83319,83321,83323,83325,83327,83329,83332],{"class":10276,"line":10284},[40,83315,83316],{"class":10446},"F_STAR_GHZ",[40,83318,10443],{"class":10363},[40,83320,10784],{"class":10732},[40,83322,10772],{"class":10732},[40,83324,75857],{"class":10732},[40,83326,167],{"class":10363},[40,83328,72829],{"class":10446},[40,83330,83331],{"class":10363},")   ",[40,83333,83334],{"class":10280},"# slice near resonance\n",[40,83336,83337,83340,83342,83344,83347,83349,83351,83353,83356,83358,83361,83364],{"class":10276,"line":10291},[40,83338,83339],{"class":10446},"NU_STAR",[40,83341,10772],{"class":10732},[40,83343,75857],{"class":10732},[40,83345,83346],{"class":10363},"(ridge_axis[np.",[40,83348,80424],{"class":10430},[40,83350,13108],{"class":10363},[40,83352,23431],{"class":10430},[40,83354,83355],{"class":10363},"(freq_ghz ",[40,83357,11056],{"class":10732},[40,83359,83360],{"class":10446}," F_STAR_GHZ",[40,83362,83363],{"class":10363},"))])  ",[40,83365,83366],{"class":10280},"# near Ω_R\u002F2π\n",[40,83368,83369],{"class":10276,"line":10297},[40,83370,10301],{"emptyLinePlaceholder":10300},[40,83372,83373],{"class":10276,"line":10304},[40,83374,83375],{"class":10280},"# Slice at f = F_STAR_GHZ\n",[40,83377,83378,83380,83382,83384,83386,83388,83390,83392,83394,83396,83398],{"class":10276,"line":10310},[40,83379,80450],{"class":10363},[40,83381,162],{"class":10732},[40,83383,37310],{"class":10732},[40,83385,13108],{"class":10363},[40,83387,80424],{"class":10430},[40,83389,13108],{"class":10363},[40,83391,23431],{"class":10430},[40,83393,83355],{"class":10363},[40,83395,11056],{"class":10732},[40,83397,83360],{"class":10446},[40,83399,17690],{"class":10363},[40,83401,83402,83404,83406,83408,83410,83412,83414,83416,83418,83421,83423,83426],{"class":10276,"line":10316},[40,83403,38079],{"class":10363},[40,83405,162],{"class":10732},[40,83407,13070],{"class":10363},[40,83409,13073],{"class":10430},[40,83411,167],{"class":10363},[40,83413,13078],{"class":11903},[40,83415,162],{"class":10732},[40,83417,167],{"class":10363},[40,83419,83420],{"class":10446},"6.0",[40,83422,10484],{"class":10363},[40,83424,83425],{"class":10446},"3.8",[40,83427,13612],{"class":10363},[40,83429,83430,83432,83434,83437,83439,83441,83443],{"class":10276,"line":10322},[40,83431,38114],{"class":10363},[40,83433,23505],{"class":10430},[40,83435,83436],{"class":10363},"(f_fft_axis, S_fft[:, j], ",[40,83438,23550],{"class":11903},[40,83440,162],{"class":10732},[40,83442,23555],{"class":10287},[40,83444,10761],{"class":10363},[40,83446,83447,83449,83451,83454,83457,83459,83461,83463,83465,83467,83469,83471,83473,83475,83477,83479,83481,83483,83485,83487],{"class":10276,"line":10327},[40,83448,38114],{"class":10363},[40,83450,23268],{"class":10430},[40,83452,83453],{"class":10363},"(ridge_axis[j], ",[40,83455,83456],{"class":11903},"ls",[40,83458,162],{"class":10732},[40,83460,23279],{"class":10287},[40,83462,10484],{"class":10363},[40,83464,82356],{"class":11903},[40,83466,162],{"class":10732},[40,83468,10810],{"class":10446},[40,83470,10484],{"class":10363},[40,83472,15415],{"class":11903},[40,83474,162],{"class":10732},[40,83476,936],{"class":10352},[40,83478,77873],{"class":10356},[40,83480,24129],{"class":10732},[40,83482,82375],{"class":10356},[40,83484,82378],{"class":10732},[40,83486,82381],{"class":10356},[40,83488,10761],{"class":10363},[40,83490,83491,83493,83495,83497,83499,83501,83503,83505],{"class":10276,"line":10333},[40,83492,38114],{"class":10363},[40,83494,13230],{"class":10430},[40,83496,167],{"class":10363},[40,83498,13262],{"class":10352},[40,83500,82409],{"class":10287},[40,83502,82412],{"class":10446},[40,83504,82415],{"class":10287},[40,83506,10761],{"class":10363},[40,83508,83509,83511,83513,83515,83517,83519,83521,83523,83525,83527,83529,83531,83533,83535,83537,83539,83541],{"class":10276,"line":10339},[40,83510,38114],{"class":10363},[40,83512,13244],{"class":10430},[40,83514,167],{"class":10363},[40,83516,82305],{"class":10287},[40,83518,12042],{"class":10732},[40,83520,920],{"class":10363},[40,83522,82312],{"class":10287},[40,83524,82315],{"class":10352},[40,83526,82318],{"class":10446},[40,83528,1123],{"class":10363},[40,83530,11171],{"class":10430},[40,83532,13277],{"class":10363},[40,83534,11231],{"class":10732},[40,83536,82329],{"class":10287},[40,83538,11237],{"class":10352},[40,83540,82334],{"class":10287},[40,83542,13612],{"class":10363},[40,83544,83545,83547,83549,83551,83554,83557,83559,83561,83563,83565,83568],{"class":10276,"line":10344},[40,83546,38114],{"class":10363},[40,83548,30334],{"class":10430},[40,83550,167],{"class":10363},[40,83552,83553],{"class":10352},"fr",[40,83555,83556],{"class":10287},"\"Slice at $f=",[40,83558,13268],{"class":10446},[40,83560,80605],{"class":10363},[40,83562,79068],{"class":10352},[40,83564,13280],{"class":10446},[40,83566,83567],{"class":10287},"$ GHz\"",[40,83569,10761],{"class":10363},[40,83571,83572,83574,83576],{"class":10276,"line":10349},[40,83573,38114],{"class":10363},[40,83575,23677],{"class":10430},[40,83577,12174],{"class":10363},[40,83579,83580,83582,83584,83587,83589],{"class":10276,"line":10367},[40,83581,74636],{"class":10363},[40,83583,30425],{"class":10430},[40,83585,83586],{"class":10363},"(); plt.",[40,83588,23686],{"class":10430},[40,83590,12174],{"class":10363},[40,83592,83593],{"class":10276,"line":10372},[40,83594,10301],{"emptyLinePlaceholder":10300},[40,83596,83597],{"class":10276,"line":10387},[40,83598,83599],{"class":10280},"# Slice at ν = NU_STAR\n",[40,83601,83602,83604,83606,83608,83610,83612,83614,83616,83619,83621,83624],{"class":10276,"line":10400},[40,83603,80661],{"class":10363},[40,83605,162],{"class":10732},[40,83607,37310],{"class":10732},[40,83609,13108],{"class":10363},[40,83611,80424],{"class":10430},[40,83613,13108],{"class":10363},[40,83615,23431],{"class":10430},[40,83617,83618],{"class":10363},"(f_fft_axis ",[40,83620,11056],{"class":10732},[40,83622,83623],{"class":10446}," NU_STAR",[40,83625,17690],{"class":10363},[40,83627,83628,83630,83632,83634,83636,83638,83640,83642,83644,83646,83648,83650],{"class":10276,"line":10413},[40,83629,38079],{"class":10363},[40,83631,162],{"class":10732},[40,83633,13070],{"class":10363},[40,83635,13073],{"class":10430},[40,83637,167],{"class":10363},[40,83639,13078],{"class":11903},[40,83641,162],{"class":10732},[40,83643,167],{"class":10363},[40,83645,83420],{"class":10446},[40,83647,10484],{"class":10363},[40,83649,83425],{"class":10446},[40,83651,13612],{"class":10363},[40,83653,83654,83656,83658,83661,83663,83665,83667],{"class":10276,"line":10418},[40,83655,38114],{"class":10363},[40,83657,23505],{"class":10430},[40,83659,83660],{"class":10363},"(freq_ghz, S_fft[i, :], ",[40,83662,23550],{"class":11903},[40,83664,162],{"class":10732},[40,83666,23555],{"class":10287},[40,83668,10761],{"class":10363},[40,83670,83671,83673,83675,83677,83679],{"class":10276,"line":10424},[40,83672,38114],{"class":10363},[40,83674,13230],{"class":10430},[40,83676,167],{"class":10363},[40,83678,82394],{"class":10287},[40,83680,10761],{"class":10363},[40,83682,83683,83685,83687,83689,83691,83693,83695,83697,83699,83701,83703,83705,83707,83709,83711,83713,83715],{"class":10276,"line":10437},[40,83684,38114],{"class":10363},[40,83686,13244],{"class":10430},[40,83688,167],{"class":10363},[40,83690,82305],{"class":10287},[40,83692,12042],{"class":10732},[40,83694,920],{"class":10363},[40,83696,82312],{"class":10287},[40,83698,82315],{"class":10352},[40,83700,82318],{"class":10446},[40,83702,1123],{"class":10363},[40,83704,11171],{"class":10430},[40,83706,13277],{"class":10363},[40,83708,11231],{"class":10732},[40,83710,82329],{"class":10287},[40,83712,11237],{"class":10352},[40,83714,82334],{"class":10287},[40,83716,13612],{"class":10363},[40,83718,83719,83721,83723,83725,83727,83730,83732,83735,83737,83739,83742,83744],{"class":10276,"line":10456},[40,83720,38114],{"class":10363},[40,83722,30334],{"class":10430},[40,83724,167],{"class":10363},[40,83726,83553],{"class":10352},[40,83728,83729],{"class":10287},"\"Slice at Fourier freq ≈ ",[40,83731,13268],{"class":10446},[40,83733,83734],{"class":10363},"f_fft_axis[i]",[40,83736,13446],{"class":10352},[40,83738,13280],{"class":10446},[40,83740,83741],{"class":10446}," {FREQ_UNITS}",[40,83743,13478],{"class":10287},[40,83745,10761],{"class":10363},[40,83747,83748,83750,83752,83754,83756],{"class":10276,"line":10472},[40,83749,74636],{"class":10363},[40,83751,30425],{"class":10430},[40,83753,83586],{"class":10363},[40,83755,23686],{"class":10430},[40,83757,12174],{"class":10363},[13997,83759],{"alt":83760,"src":83761},"Output 8 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-08.webp",[13997,83763],{"alt":83764,"src":83765},"Output 9 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-09.webp",[28,83767,83769],{"id":83768},"run-it-yourself","Run it yourself",[12,83771,83772,83773,83775,83776,1123],{},"Get ",[19,83774,22],{"href":57973}," or ",[19,83777,83779],{"href":83778},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb","open it in Colab",[20069,83781,83782],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}",{"title":10090,"searchDepth":10284,"depth":10284,"links":83784},[83785,83786,83787,83788,83789,83790,83791,83792,83793,83794,83795,83796,83797,83798,83799],{"id":58151,"depth":10284,"text":58152},{"id":59623,"depth":10284,"text":59624},{"id":60412,"depth":10284,"text":60413},{"id":61752,"depth":10284,"text":61753},{"id":62881,"depth":10284,"text":62882},{"id":65362,"depth":10284,"text":65363},{"id":65879,"depth":10284,"text":65880},{"id":66978,"depth":10284,"text":66979},{"id":68583,"depth":10284,"text":68584},{"id":68956,"depth":10284,"text":68957},{"id":69950,"depth":10284,"text":69951},{"id":71128,"depth":10284,"text":71129},{"id":72307,"depth":10284,"text":72308},{"id":72802,"depth":10284,"text":72803},{"id":83768,"depth":10284,"text":83769},[20088,20089,20090,83801],"Rabi oscillations",[83803],{"username":7,"name":20094,"role":20095,"bio":20096,"links":83804},[83805,83806],{"label":20099,"href":20100},{"label":20102,"href":57973},{"username":7,"name":20094,"role":20104},"A primer on the quantum dynamics of a driven two-level system, visualizing excited-state probability under the rotating-wave approximation (RWA).",{},"\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization","2026-07-15","6 min read",[],{"title":57963,"description":83808},"blog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[27288,20120],"IAzB0ZaRHqWwL_egvokIhbtpcSM2VZ_XYnWDf-q-QyM",{"id":83820,"title":83821,"authors":83822,"body":83823,"breadcrumb":88391,"builders":88393,"byline":88398,"challenge":20105,"courseAuthor":20105,"courseLead":20105,"dek":20105,"description":88399,"draft":20107,"extension":20108,"eyebrow":27279,"featured":20107,"finish":20105,"fork":20105,"hero":20105,"heroAlt":20105,"heroCta":20105,"heroImage":20105,"homepageFeatured":20107,"kind":20110,"lessonCount":20105,"meta":88400,"navigation":10300,"newsItems":20105,"next":20105,"ogImage":88401,"order":20105,"outcomes":20105,"path":88402,"publishDate":83812,"readingTime":88403,"related":88404,"relatedProjects":20105,"seo":88405,"stem":88406,"tags":88407,"track":20105,"trackName":20105,"__hash__":88408},"blog\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications.md","A Sampling-Budget & Bit-String Analogy for Qubit Classifications",[7],{"type":9,"value":83824,"toc":88387},[83825,83832,88228,88234,88237,88240,88246,88252,88254,88256,88375,88377,88384],[12,83826,83827],{},[15,83828,17,83829,23],{},[19,83830,22],{"href":83831},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[10082,83833,83835],{"className":10269,"code":83834,"language":10271,"meta":10090,"style":10090},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n# generates tables and figures, so you can iterate further.\n#\n# Contents created in the current directory:\n#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n#  - summary.csv                                 (summary table)\n#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n#  - fig_required_shots.png                      (required shots for ±ε)\n#  - simulation_config.json                      (run parameters)\n\nimport json\nimport math\nfrom dataclasses import dataclass\nfrom typing import List, Dict, Tuple\n\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom IPython.display import display, Markdown\nimport nbformat as nbf\n\nplt.rcParams['figure.dpi'] = 200\n\n# ---------- Helper math ----------\n\ndef wilson_ci(k: int, n: int, z: float = 1.96) -> Tuple[float, float]:\n    \"\"\"\n    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n    Returns (low, high). Works for edge cases k=0 or k=n.\n    \"\"\"\n    if n == 0:\n        return (0.0, 1.0)\n    p_hat = k \u002F n\n    denom = 1.0 + (z ** 2) \u002F n\n    center = (p_hat + (z ** 2) \u002F (2 * n)) \u002F denom\n    span = (z * math.sqrt((p_hat * (1 - p_hat) \u002F n) + (z ** 2) \u002F (4 * n ** 2))) \u002F denom\n    return (max(0.0, center - span), min(1.0, center + span))\n\n\ndef shots_for_epsilon(p: float, epsilon: float = 0.01, z: float = 1.96) -> int:\n    \"\"\"\n    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n    \"\"\"\n    p = min(max(p, 0.0), 1.0)\n    var = p * (1.0 - p)\n    if var == 0.0:\n        return 1\n    n = (z ** 2) * var \u002F (epsilon ** 2)\n    return max(1, int(math.ceil(n)))\n\n\n# ---------- Domain model ----------\n\n@dataclass(frozen=True)\nclass QubitModel:\n    name: str\n    satisfaction_prob: float  # Probability a single shot meets the analytical criterion (bit = 1)\n    description: str\n\n\nDEFAULT_MODELS: List[QubitModel] = [\n    QubitModel(\n        \"Toy working qubit\",\n        0.50,\n        \"Simple demonstrator; about half shots meet the criterion.\"\n    ),\n    QubitModel(\n        \"NISQ qubit\",\n        0.60,\n        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n    ),\n    QubitModel(\n        \"Fault-tolerant (logical)\",\n        0.90,\n        \"Error-corrected logical qubit; high single-shot success probability.\"\n    ),\n    QubitModel(\n        \"Topologically protected\",\n        0.98,\n        \"Intrinsic protection; very high single-shot success probability.\"\n    ),\n    QubitModel(\n        \"Ideal qubit\",\n        1.00,\n        \"Theoretical perfect qubit; always successful in a single shot.\"\n    ),\n]\n\n# ---------- Simulation configuration ----------\n\nCONFIG: Dict = {\n    \"random_seed\": 123456,\n    \"shots_per_model\": 10000,\n    \"epsilons\": [0.02, 0.01, 0.005],  # ±2%, ±1%, ±0.5% targets\n    \"cost_model\": {\n        \"default_cost_per_shot_usd\": 0.00005,\n        \"fixed_overhead_usd\": 0.10\n    },\n    \"timing\": {\n        \"time_per_shot_seconds\": 2e-6\n    }\n}\n\nrng = np.random.default_rng(CONFIG[\"random_seed\"])\n\n\n# ---------- Core simulation ----------\n\ndef simulate_bitstring(p: float, shots: int, rng_: np.random.Generator) -> np.ndarray:\n    \"\"\"\n    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n    \"\"\"\n    return rng_.binomial(1, p, size=shots).astype(int)\n\n\ndef summarize_model(model: QubitModel, bits: np.ndarray, epsilons: List[float]) -> Dict:\n    \"\"\"\n    Calculate summary statistics for a given model and its simulated bitstring.\n    \"\"\"\n    n = int(bits.size)\n    ones = int(bits.sum())\n    zeros = int(n - ones)\n    frac = ones \u002F n if n > 0 else float(\"nan\")\n    lo, hi = wilson_ci(ones, n)\n\n    row = {\n        \"Model\": model.name,\n        \"Description\": model.description,\n        \"Configured p (target)\": model.satisfaction_prob,\n        \"Shots\": n,\n        \"Ones (successes)\": ones,\n        \"Zeros (failures)\": zeros,\n        \"Ones fraction (observed)\": frac,\n        \"Wilson CI 95% low\": lo,\n        \"Wilson CI 95% high\": hi,\n        \"CI width (95%)\": hi - lo,\n    }\n    for eps in epsilons:\n        row[f\"Required shots for ±{int(eps*100)}%\"] = shots_for_epsilon(model.satisfaction_prob, eps)\n    return row\n\n\ndef apply_costs(shots: int, cost_per_shot: float, fixed_overhead: float) -> float:\n    \"\"\"\n    Calculate illustrative cost based on shots and cost model.\n    \"\"\"\n    return fixed_overhead + shots * cost_per_shot\n\n\n# ---------- Run the simulation ----------\n\nall_bitstrings: Dict[str, np.ndarray] = {}\nsummary_rows: List[Dict] = []\n\nprint(\"Running simulation...\")\nfor m in DEFAULT_MODELS:\n    bits = simulate_bitstring(m.satisfaction_prob, CONFIG[\"shots_per_model\"], rng)\n    all_bitstrings[m.name] = bits\n    summary_rows.append(summarize_model(m, bits, CONFIG[\"epsilons\"]))\n\nsummary_df = pd.DataFrame(summary_rows)\n\n# Add cost estimates\ncps = CONFIG[\"cost_model\"][\"default_cost_per_shot_usd\"]\nover = CONFIG[\"cost_model\"][\"fixed_overhead_usd\"]\nif cps > 0 or over > 0:\n    summary_df[\"Illustrative cost (USD)\"] = [\n        apply_costs(int(row[\"Shots\"]), cps, over)\n        for _, row in summary_df.iterrows()\n    ]\n\n# ---------- Save artifacts ----------\n\n# Save raw bitstrings and summary CSV\nfor name, arr in all_bitstrings.items():\n    out_path = f\"bitstrings_{name.replace(' ', '_')}.csv\"\n    pd.DataFrame({\"bit\": arr}).to_csv(out_path, index=False)\n\nsummary_csv_path = \"summary.csv\"\nsummary_df.to_csv(summary_csv_path, index=False)\n\n# Save config JSON\nwith open(\"simulation_config.json\", \"w\") as f:\n    json.dump(CONFIG, f, indent=2)\n\nprint(\"Saved data to CSV and JSON files.\")\n\n# ---------- Visualizations ----------\n\n# Figure 1: Observed ones fraction with Wilson CI\nplt.style.use('seaborn-v0_8-whitegrid')\nplt.figure(figsize=(10, 6))\nx = np.arange(len(summary_df))\ny = summary_df[\"Ones fraction (observed)\"].to_numpy()\n\n# --- FIX APPLIED HERE ---\n# Take the absolute value to prevent tiny negative numbers from floating-point\n# inaccuracies from causing a ValueError in the errorbar plot.\nlower_error = y - summary_df[\"Wilson CI 95% low\"].to_numpy()\nupper_error = summary_df[\"Wilson CI 95% high\"].to_numpy() - y\nyerr = np.vstack([np.abs(lower_error), np.abs(upper_error)])\n# --- END OF FIX ---\n\nplt.bar(x, y, color='skyblue', edgecolor='black')\nplt.errorbar(x, y, yerr=yerr, fmt=\"none\", capsize=5, color=\"black\", elinewidth=1.5)\nplt.xticks(x, summary_df[\"Model\"].tolist(), rotation=25, ha=\"right\")\nplt.ylabel(\"Observed Ones Fraction\")\nplt.title(\"Observed Single-Shot Success Rates with 95% Wilson CI\", fontsize=14)\nplt.tight_layout()\nfig1_path = \"fig_ones_fraction.png\"\nplt.savefig(fig1_path, dpi=200, bbox_inches='tight')\nplt.show()\n\n# Figure 2: Required shots at several ±epsilon targets\nplt.figure(figsize=(10, 6))\nwidth = 0.2\nx = np.arange(len(DEFAULT_MODELS))\ncolors = ['#ff9999','#66b3ff','#99ff99']\n\nfor i, eps in enumerate(CONFIG[\"epsilons\"]):\n    req = [shots_for_epsilon(m.satisfaction_prob, eps) for m in DEFAULT_MODELS]\n    plt.bar(x + i * width, req, width, label=f\"±{eps:.1%}\", color=colors[i], edgecolor='black')\n\nplt.xticks(x + width, [m.name for m in DEFAULT_MODELS], rotation=25, ha=\"right\")\nplt.ylabel(\"Required Shots (Log Scale)\")\nplt.title(\"Approximate Shots to Estimate Success Rate within ±ε\", fontsize=14)\nplt.yscale('log')\nplt.legend(title=\"Error Tolerance\")\nplt.tight_layout()\nfig2_path = \"fig_required_shots.png\"\nplt.savefig(fig2_path, dpi=200, bbox_inches='tight')\nplt.show()\n\nprint(\"Generated and saved plots.\")\n\n# ---------- Display interactive table ----------\n\ndisplay(Markdown(\"##  Qubit Sampling Budget - Summary\"))\ndisplay(summary_df.style.format({\n    \"Configured p (target)\": \"{:.2f}\",\n    \"Ones fraction (observed)\": \"{:.4f}\",\n    \"Wilson CI 95% low\": \"{:.4f}\",\n    \"Wilson CI 95% high\": \"{:.4f}\",\n    \"CI width (95%)\": \"{:.4f}\",\n    \"Illustrative cost (USD)\": \"${:.2f}\"\n}))\n\n\n# ---------- Build a ready-to-run Colab notebook ----------\n\nmd_intro = r\"\"\"\n# Sampling Budget & Bit-String Analogy for Qubit Classes\n\n**High school level (concise):** We treat each measurement (“single shot”) as a bit: `1` if the outcome matches the analytical prediction, `0` if it does not. Different qubit types have different chances of landing a `1`. We simulate many shots, show how often we get `1`s, and estimate how many shots we need for a target accuracy.\n\n**Graduate level (concise):** We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ (noisy intermediate-scale quantum), fault-tolerant (logical), topologically protected, and ideal. We report Wilson score intervals (95%) for observed proportions and estimate the required shot budget $n \\approx z^2\\,p(1-p)\u002F\\epsilon^2$ to bound absolute error by ±$\\epsilon$ at ~95% confidence.\n\"\"\"\n\ncode_config = r\"\"\"\n# @title Configuration\nfrom dataclasses import dataclass\nfrom typing import List, Dict\nimport numpy as np\n\nRANDOM_SEED = 579345  # @param {type:\"number\"}\nSHOTS_PER_MODEL = 10000  # @param {type:\"number\"}\nEPSILONS = [0.02, 0.01, 0.005]  # @param\nCOST_PER_SHOT_USD = 0.00005  # @param {type:\"number\"}\nFIXED_OVERHEAD_USD = 0.10    # @param {type:\"number\"}\n\n@dataclass(frozen=True)\nclass QubitModel:\n    name: str\n    satisfaction_prob: float\n    description: str\n\nMODELS: List[QubitModel] = [\n    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n]\n\nrng = np.random.default_rng(RANDOM_SEED)\n\"\"\"\n\ncode_lib_and_run = r\"\"\"\n# @title Run Simulation & Display Results\nimport math\nfrom typing import Tuple, Dict\nimport pandas as pd\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom IPython.display import display, Markdown\n\ndef wilson_ci(k: int, n: int, z: float = 1.96) -> Tuple[float, float]:\n    if n == 0: return (0.0, 1.0)\n    p_hat, z2_n = k \u002F n, z**2 \u002F n\n    denom = 1.0 + z2_n\n    center = (p_hat + z2_n \u002F 2) \u002F denom\n    span = (z * math.sqrt(p_hat * (1 - p_hat) \u002F n + z**2 \u002F (4 * n**2))) \u002F denom\n    return (max(0.0, center - span), min(1.0, center + span))\n\ndef shots_for_epsilon(p: float, epsilon: float = 0.01, z: float = 1.96) -> int:\n    if p * (1-p) == 0: return 1\n    return max(1, int(math.ceil((z**2 * p * (1-p)) \u002F (epsilon**2))))\n\ndef summarize_model(model, bits: np.ndarray, epsilons) -> Dict:\n    n = len(bits)\n    ones = sum(bits)\n    lo, hi = wilson_ci(ones, n)\n    row = {\n        \"Model\": model.name, \"p (target)\": model.satisfaction_prob, \"Shots\": n,\n        \"Ones\": ones, \"p (observed)\": ones \u002F n if n > 0 else 0,\n        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n    }\n    for eps in epsilons:\n        row[f\"Shots for ±{eps:.1%}\"] = shots_for_epsilon(model.satisfaction_prob, eps)\n    return row\n\n# --- Run Simulation ---\nsummary_rows = [summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\nsummary_df = pd.DataFrame(summary_rows)\nif COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n    summary_df[\"Cost (USD)\"] = FIXED_OVERHEAD_USD + summary_df[\"Shots\"] * COST_PER_SHOT_USD\n\ndisplay(Markdown(\"### Simulation Summary\"))\ndisplay(summary_df.style.format(precision=4).background_gradient(cmap='viridis', subset=['p (observed)', 'CI width']))\n\n# --- Plotting ---\nplt.style.use('seaborn-v0_8-whitegrid')\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6))\n\n# Plot 1\ny = summary_df[\"p (observed)\"].to_numpy()\nlower_error = y - summary_df[\"CI 95% low\"]\nupper_error = summary_df[\"CI 95% high\"] - y\nyerr = np.vstack([np.abs(lower_error), np.abs(upper_error)])\n\nax1.bar(summary_df[\"Model\"], y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black')\nax1.set_title(\"Observed Success Rate (95% Wilson CI)\", fontsize=14)\nax1.tick_params(axis='x', rotation=30, labelsize=10)\n\n# Plot 2\nwidth = 0.2\nx = np.arange(len(MODELS))\ncolors = ['#ff9999','#66b3ff','#99ff99']\nfor i, eps in enumerate(EPSILONS):\n    req = summary_df[f\"Shots for ±{eps:.1%}\"]\n    ax2.bar(x + i * width, req, width, label=f\"±{eps:.1%}\", color=colors[i], edgecolor='black')\n\nax2.set_xticks(x + width, summary_df[\"Model\"])\nax2.set_ylabel(\"Required Shots (Log Scale)\")\nax2.set_yscale('log')\nax2.set_title(\"Shots Needed for ±ε Tolerance\", fontsize=14)\nax2.tick_params(axis='x', rotation=30, labelsize=10)\nax2.legend(title=\"Tolerance\")\n\nplt.tight_layout()\nplt.show()\n\"\"\"\n\nnb = nbf.v4.new_notebook()\nnb[\"cells\"] = [\n    nbf.v4.new_markdown_cell(md_intro),\n    nbf.v4.new_code_cell(code_config),\n    nbf.v4.new_code_cell(code_lib_and_run),\n]\n\nnb_path = \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\nwith open(nb_path, \"w\", encoding=\"utf-8\") as f:\n    nbf.write(nb, f)\n\n# ---------- Final confirmation ----------\nprint(\"\\n\" + \"=\"*50)\nprint(\"SCRIPT COMPLETE\")\nprint(\"=\"*50)\nprint(f\"Google Colab notebook saved to: {nb_path}\")\nprint(f\"Summary data saved to: {summary_csv_path}\")\nprint(f\"Plots saved to: {fig1_path}, {fig2_path}\")\n",[10088,83836,83837,83842,83847,83852,83857,83862,83867,83872,83877,83882,83887,83891,83898,83904,83914,83925,83929,83939,83949,83959,83970,83982,83986,83998,84002,84007,84011,84057,84061,84066,84071,84075,84088,84102,84117,84140,84176,84241,84274,84278,84282,84329,84333,84338,84343,84347,84371,84391,84404,84410,84439,84461,84465,84469,84474,84478,84494,84503,84511,84521,84528,84532,84536,84548,84555,84562,84569,84574,84579,84585,84592,84599,84604,84608,84614,84621,84628,84633,84637,84643,84650,84657,84662,84666,84672,84679,84686,84691,84695,84699,84703,84708,84712,84724,84736,84748,84775,84783,84795,84805,84810,84817,84827,84831,84835,84839,84863,84867,84871,84876,84880,84912,84916,84921,84925,84958,84962,84966,84997,85001,85006,85010,85021,85037,85054,85087,85099,85103,85112,85120,85128,85136,85144,85152,85160,85168,85182,85195,85208,85212,85223,85261,85268,85272,85276,85315,85319,85324,85328,85345,85349,85353,85358,85362,85377,85386,85390,85401,85415,85437,85447,85472,85476,85490,85494,85499,85521,85541,85564,85578,85596,85613,85617,85621,85626,85630,85635,85652,85689,85720,85724,85734,85752,85756,85761,85786,85810,85814,85825,85829,85834,85838,85843,85858,85880,85897,85918,85922,85927,85932,85937,85964,85990,86014,86019,86023,86051,86104,86140,86153,86175,86183,86193,86221,86229,86233,86238,86260,86269,86289,86313,86317,86339,86364,86419,86423,86462,86475,86496,86510,86527,86535,86545,86570,86578,86582,86593,86597,86602,86606,86622,86634,86653,86673,86696,86718,86737,86754,86759,86763,86767,86772,86776,86788,86793,86797,86828,86832,86913,86917,86921,86932,86937,86942,86947,86952,86956,86964,86971,86982,86989,86996,87000,87011,87016,87021,87026,87031,87035,87049,87054,87059,87064,87069,87074,87078,87082,87094,87098,87102,87113,87118,87123,87128,87133,87137,87142,87147,87151,87171,87183,87193,87203,87223,87283,87314,87318,87333,87349,87399,87403,87418,87429,87440,87452,87457,87472,87487,87492,87496,87501,87519,87524,87528,87533,87541,87553,87558,87582,87586,87601,87628,87632,87637,87648,87675,87679,87684,87697,87705,87716,87724,87728,87746,87768,87780,87784,87789,87794,87810,87818,87832,87840,87866,87870,87887,87908,87919,87931,87942,87954,87958,87965,87972,87976,87980,87995,88009,88020,88030,88039,88043,88047,88057,88084,88095,88099,88104,88127,88138,88153,88175,88197],{"__ignoreMap":10090},[40,83838,83839],{"class":10276,"line":10277},[40,83840,83841],{"class":10280},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[40,83843,83844],{"class":10276,"line":10284},[40,83845,83846],{"class":10280},"# generates tables and figures, so you can iterate further.\n",[40,83848,83849],{"class":10276,"line":10291},[40,83850,83851],{"class":10280},"#\n",[40,83853,83854],{"class":10276,"line":10297},[40,83855,83856],{"class":10280},"# Contents created in the current directory:\n",[40,83858,83859],{"class":10276,"line":10304},[40,83860,83861],{"class":10280},"#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n",[40,83863,83864],{"class":10276,"line":10310},[40,83865,83866],{"class":10280},"#  - summary.csv                                 (summary table)\n",[40,83868,83869],{"class":10276,"line":10316},[40,83870,83871],{"class":10280},"#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n",[40,83873,83874],{"class":10276,"line":10322},[40,83875,83876],{"class":10280},"#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n",[40,83878,83879],{"class":10276,"line":10327},[40,83880,83881],{"class":10280},"#  - fig_required_shots.png                      (required shots for ±ε)\n",[40,83883,83884],{"class":10276,"line":10333},[40,83885,83886],{"class":10280},"#  - simulation_config.json                      (run parameters)\n",[40,83888,83889],{"class":10276,"line":10339},[40,83890,10301],{"emptyLinePlaceholder":10300},[40,83892,83893,83895],{"class":10276,"line":10344},[40,83894,10375],{"class":10352},[40,83896,83897],{"class":10363}," json\n",[40,83899,83900,83902],{"class":10276,"line":10349},[40,83901,10375],{"class":10352},[40,83903,21798],{"class":10363},[40,83905,83906,83908,83910,83912],{"class":10276,"line":10367},[40,83907,10353],{"class":10352},[40,83909,10530],{"class":10363},[40,83911,10375],{"class":10352},[40,83913,10535],{"class":10363},[40,83915,83916,83918,83920,83922],{"class":10276,"line":10372},[40,83917,10353],{"class":10352},[40,83919,10554],{"class":10363},[40,83921,10375],{"class":10352},[40,83923,83924],{"class":10363}," List, Dict, Tuple\n",[40,83926,83927],{"class":10276,"line":10387},[40,83928,10301],{"emptyLinePlaceholder":10300},[40,83930,83931,83933,83935,83937],{"class":10276,"line":10400},[40,83932,10375],{"class":10352},[40,83934,10378],{"class":10363},[40,83936,10381],{"class":10352},[40,83938,10384],{"class":10363},[40,83940,83941,83943,83945,83947],{"class":10276,"line":10413},[40,83942,10375],{"class":10352},[40,83944,73473],{"class":10363},[40,83946,10381],{"class":10352},[40,83948,73478],{"class":10363},[40,83950,83951,83953,83955,83957],{"class":10276,"line":10418},[40,83952,10375],{"class":10352},[40,83954,10405],{"class":10363},[40,83956,10381],{"class":10352},[40,83958,10410],{"class":10363},[40,83960,83961,83963,83965,83967],{"class":10276,"line":10424},[40,83962,10353],{"class":10352},[40,83964,10609],{"class":10363},[40,83966,10375],{"class":10352},[40,83968,83969],{"class":10363}," display, Markdown\n",[40,83971,83972,83974,83977,83979],{"class":10276,"line":10437},[40,83973,10375],{"class":10352},[40,83975,83976],{"class":10363}," nbformat ",[40,83978,10381],{"class":10352},[40,83980,83981],{"class":10363}," nbf\n",[40,83983,83984],{"class":10276,"line":10456},[40,83985,10301],{"emptyLinePlaceholder":10300},[40,83987,83988,83990,83992,83994,83996],{"class":10276,"line":10472},[40,83989,27384],{"class":10363},[40,83991,27387],{"class":10287},[40,83993,10741],{"class":10363},[40,83995,162],{"class":10732},[40,83997,27394],{"class":10446},[40,83999,84000],{"class":10276,"line":10496},[40,84001,10301],{"emptyLinePlaceholder":10300},[40,84003,84004],{"class":10276,"line":10510},[40,84005,84006],{"class":10280},"# ---------- Helper math ----------\n",[40,84008,84009],{"class":10276,"line":10696},[40,84010,10301],{"emptyLinePlaceholder":10300},[40,84012,84013,84015,84018,84020,84022,84024,84026,84028,84030,84032,84034,84036,84038,84040,84042,84044,84047,84049,84051,84053,84055],{"class":10276,"line":10702},[40,84014,10965],{"class":10352},[40,84016,84017],{"class":10430}," wilson_ci",[40,84019,167],{"class":10363},[40,84021,13346],{"class":10973},[40,84023,10443],{"class":10363},[40,84025,10821],{"class":10732},[40,84027,10484],{"class":10363},[40,84029,1796],{"class":10973},[40,84031,10443],{"class":10363},[40,84033,10821],{"class":10732},[40,84035,10484],{"class":10363},[40,84037,273],{"class":10973},[40,84039,10443],{"class":10363},[40,84041,10784],{"class":10732},[40,84043,10772],{"class":10732},[40,84045,84046],{"class":10446}," 1.96",[40,84048,12138],{"class":10363},[40,84050,10784],{"class":10732},[40,84052,10484],{"class":10363},[40,84054,10784],{"class":10732},[40,84056,12152],{"class":10363},[40,84058,84059],{"class":10276,"line":10708},[40,84060,10723],{"class":10287},[40,84062,84063],{"class":10276,"line":10714},[40,84064,84065],{"class":10287},"    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[40,84067,84068],{"class":10276,"line":10720},[40,84069,84070],{"class":10287},"    Returns (low, high). Works for edge cases k=0 or k=n.\n",[40,84072,84073],{"class":10276,"line":10726},[40,84074,10723],{"class":10287},[40,84076,84077,84079,84082,84084,84086],{"class":10276,"line":10764},[40,84078,11323],{"class":10352},[40,84080,84081],{"class":10363}," n ",[40,84083,11231],{"class":10732},[40,84085,11234],{"class":10446},[40,84087,10639],{"class":10363},[40,84089,84090,84092,84094,84096,84098,84100],{"class":10276,"line":10778},[40,84091,11339],{"class":10352},[40,84093,920],{"class":10363},[40,84095,10801],{"class":10446},[40,84097,10484],{"class":10363},[40,84099,10810],{"class":10446},[40,84101,10761],{"class":10363},[40,84103,84104,84107,84109,84112,84114],{"class":10276,"line":10815},[40,84105,84106],{"class":10363},"    p_hat ",[40,84108,162],{"class":10732},[40,84110,84111],{"class":10363}," k ",[40,84113,11243],{"class":10732},[40,84115,84116],{"class":10363}," n\n",[40,84118,84119,84121,84123,84125,84127,84130,84132,84134,84136,84138],{"class":10276,"line":10829},[40,84120,24895],{"class":10363},[40,84122,162],{"class":10732},[40,84124,22530],{"class":10446},[40,84126,12042],{"class":10732},[40,84128,84129],{"class":10363}," (z ",[40,84131,22571],{"class":10732},[40,84133,12960],{"class":10446},[40,84135,12055],{"class":10363},[40,84137,11243],{"class":10732},[40,84139,84116],{"class":10363},[40,84141,84142,84145,84147,84150,84152,84154,84156,84158,84160,84162,84164,84166,84168,84171,84173],{"class":10276,"line":10842},[40,84143,84144],{"class":10363},"    center ",[40,84146,162],{"class":10732},[40,84148,84149],{"class":10363}," (p_hat ",[40,84151,1053],{"class":10732},[40,84153,84129],{"class":10363},[40,84155,22571],{"class":10732},[40,84157,12960],{"class":10446},[40,84159,12055],{"class":10363},[40,84161,11243],{"class":10732},[40,84163,920],{"class":10363},[40,84165,1044],{"class":10446},[40,84167,11881],{"class":10732},[40,84169,84170],{"class":10363}," n)) ",[40,84172,11243],{"class":10732},[40,84174,84175],{"class":10363}," denom\n",[40,84177,84178,84181,84183,84185,84187,84189,84191,84194,84196,84198,84200,84202,84205,84207,84210,84212,84214,84216,84218,84220,84222,84224,84226,84228,84230,84232,84234,84237,84239],{"class":10276,"line":10855},[40,84179,84180],{"class":10363},"    span ",[40,84182,162],{"class":10732},[40,84184,84129],{"class":10363},[40,84186,11752],{"class":10732},[40,84188,22358],{"class":10363},[40,84190,2218],{"class":10430},[40,84192,84193],{"class":10363},"((p_hat ",[40,84195,11752],{"class":10732},[40,84197,920],{"class":10363},[40,84199,78],{"class":10446},[40,84201,11764],{"class":10732},[40,84203,84204],{"class":10363}," p_hat) ",[40,84206,11243],{"class":10732},[40,84208,84209],{"class":10363}," n) ",[40,84211,1053],{"class":10732},[40,84213,84129],{"class":10363},[40,84215,22571],{"class":10732},[40,84217,12960],{"class":10446},[40,84219,12055],{"class":10363},[40,84221,11243],{"class":10732},[40,84223,920],{"class":10363},[40,84225,17136],{"class":10446},[40,84227,11881],{"class":10732},[40,84229,84081],{"class":10363},[40,84231,22571],{"class":10732},[40,84233,12960],{"class":10446},[40,84235,84236],{"class":10363},"))) ",[40,84238,11243],{"class":10732},[40,84240,84175],{"class":10363},[40,84242,84243,84245,84247,84249,84251,84253,84256,84258,84261,84263,84265,84267,84269,84271],{"class":10276,"line":10868},[40,84244,11201],{"class":10352},[40,84246,920],{"class":10363},[40,84248,3032],{"class":10732},[40,84250,167],{"class":10363},[40,84252,10801],{"class":10446},[40,84254,84255],{"class":10363},", center ",[40,84257,11056],{"class":10732},[40,84259,84260],{"class":10363}," span), ",[40,84262,3067],{"class":10732},[40,84264,167],{"class":10363},[40,84266,10810],{"class":10446},[40,84268,84255],{"class":10363},[40,84270,1053],{"class":10732},[40,84272,84273],{"class":10363}," span))\n",[40,84275,84276],{"class":10276,"line":10881},[40,84277,10301],{"emptyLinePlaceholder":10300},[40,84279,84280],{"class":10276,"line":10907},[40,84281,10301],{"emptyLinePlaceholder":10300},[40,84283,84284,84286,84289,84291,84293,84295,84297,84299,84302,84304,84306,84308,84311,84313,84315,84317,84319,84321,84323,84325,84327],{"class":10276,"line":10920},[40,84285,10965],{"class":10352},[40,84287,84288],{"class":10430}," shots_for_epsilon",[40,84290,167],{"class":10363},[40,84292,12],{"class":10973},[40,84294,10443],{"class":10363},[40,84296,10784],{"class":10732},[40,84298,10484],{"class":10363},[40,84300,84301],{"class":10973},"epsilon",[40,84303,10443],{"class":10363},[40,84305,10784],{"class":10732},[40,84307,10772],{"class":10732},[40,84309,84310],{"class":10446}," 0.01",[40,84312,10484],{"class":10363},[40,84314,273],{"class":10973},[40,84316,10443],{"class":10363},[40,84318,10784],{"class":10732},[40,84320,10772],{"class":10732},[40,84322,84046],{"class":10446},[40,84324,22117],{"class":10363},[40,84326,10821],{"class":10732},[40,84328,10639],{"class":10363},[40,84330,84331],{"class":10276,"line":10933},[40,84332,10723],{"class":10287},[40,84334,84335],{"class":10276,"line":10946},[40,84336,84337],{"class":10287},"    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[40,84339,84340],{"class":10276,"line":10951},[40,84341,84342],{"class":10287},"    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[40,84344,84345],{"class":10276,"line":10956},[40,84346,10723],{"class":10287},[40,84348,84349,84352,84354,84356,84358,84360,84363,84365,84367,84369],{"class":10276,"line":10962},[40,84350,84351],{"class":10363},"    p ",[40,84353,162],{"class":10732},[40,84355,22453],{"class":10732},[40,84357,167],{"class":10363},[40,84359,3032],{"class":10732},[40,84361,84362],{"class":10363},"(p, ",[40,84364,10801],{"class":10446},[40,84366,11040],{"class":10363},[40,84368,10810],{"class":10446},[40,84370,10761],{"class":10363},[40,84372,84373,84376,84378,84380,84382,84384,84386,84388],{"class":10276,"line":10983},[40,84374,84375],{"class":10363},"    var ",[40,84377,162],{"class":10732},[40,84379,13938],{"class":10363},[40,84381,11752],{"class":10732},[40,84383,920],{"class":10363},[40,84385,10810],{"class":10446},[40,84387,11764],{"class":10732},[40,84389,84390],{"class":10363}," p)\n",[40,84392,84393,84395,84398,84400,84402],{"class":10276,"line":11005},[40,84394,11323],{"class":10352},[40,84396,84397],{"class":10363}," var ",[40,84399,11231],{"class":10732},[40,84401,22435],{"class":10446},[40,84403,10639],{"class":10363},[40,84405,84406,84408],{"class":10276,"line":11011},[40,84407,11339],{"class":10352},[40,84409,27423],{"class":10446},[40,84411,84412,84414,84416,84418,84420,84422,84424,84426,84428,84430,84433,84435,84437],{"class":10276,"line":11022},[40,84413,15978],{"class":10363},[40,84415,162],{"class":10732},[40,84417,84129],{"class":10363},[40,84419,22571],{"class":10732},[40,84421,12960],{"class":10446},[40,84423,12055],{"class":10363},[40,84425,11752],{"class":10732},[40,84427,84397],{"class":10363},[40,84429,11243],{"class":10732},[40,84431,84432],{"class":10363}," (epsilon ",[40,84434,22571],{"class":10732},[40,84436,12960],{"class":10446},[40,84438,10761],{"class":10363},[40,84440,84441,84443,84445,84447,84449,84451,84453,84456,84458],{"class":10276,"line":11064},[40,84442,11201],{"class":10352},[40,84444,22290],{"class":10732},[40,84446,167],{"class":10363},[40,84448,78],{"class":10446},[40,84450,10484],{"class":10363},[40,84452,10821],{"class":10732},[40,84454,84455],{"class":10363},"(math.",[40,84457,81390],{"class":10430},[40,84459,84460],{"class":10363},"(n)))\n",[40,84462,84463],{"class":10276,"line":11103},[40,84464,10301],{"emptyLinePlaceholder":10300},[40,84466,84467],{"class":10276,"line":11142},[40,84468,10301],{"emptyLinePlaceholder":10300},[40,84470,84471],{"class":10276,"line":11148},[40,84472,84473],{"class":10280},"# ---------- Domain model ----------\n",[40,84475,84476],{"class":10276,"line":11181},[40,84477,10301],{"emptyLinePlaceholder":10300},[40,84479,84480,84483,84485,84488,84490,84492],{"class":10276,"line":11198},[40,84481,84482],{"class":10430},"@dataclass",[40,84484,167],{"class":10363},[40,84486,84487],{"class":11903},"frozen",[40,84489,162],{"class":10732},[40,84491,13091],{"class":10446},[40,84493,10761],{"class":10363},[40,84495,84496,84498,84501],{"class":10276,"line":11249},[40,84497,10632],{"class":10352},[40,84499,84500],{"class":10635}," QubitModel",[40,84502,10639],{"class":10363},[40,84504,84505,84508],{"class":10276,"line":11254},[40,84506,84507],{"class":10363},"    name: ",[40,84509,84510],{"class":10732},"str\n",[40,84512,84513,84516,84518],{"class":10276,"line":11259},[40,84514,84515],{"class":10363},"    satisfaction_prob: ",[40,84517,10784],{"class":10732},[40,84519,84520],{"class":10280},"  # Probability a single shot meets the analytical criterion (bit = 1)\n",[40,84522,84523,84526],{"class":10276,"line":11265},[40,84524,84525],{"class":10363},"    description: ",[40,84527,84510],{"class":10732},[40,84529,84530],{"class":10276,"line":11294},[40,84531,10301],{"emptyLinePlaceholder":10300},[40,84533,84534],{"class":10276,"line":11300},[40,84535,10301],{"emptyLinePlaceholder":10300},[40,84537,84538,84541,84544,84546],{"class":10276,"line":11320},[40,84539,84540],{"class":10446},"DEFAULT_MODELS",[40,84542,84543],{"class":10363},": List[QubitModel] ",[40,84545,162],{"class":10732},[40,84547,37422],{"class":10363},[40,84549,84550,84553],{"class":10276,"line":11336},[40,84551,84552],{"class":10430},"    QubitModel",[40,84554,12527],{"class":10363},[40,84556,84557,84560],{"class":10276,"line":11381},[40,84558,84559],{"class":10287},"        \"Toy working qubit\"",[40,84561,10507],{"class":10363},[40,84563,84564,84567],{"class":10276,"line":11395},[40,84565,84566],{"class":10446},"        0.50",[40,84568,10507],{"class":10363},[40,84570,84571],{"class":10276,"line":11438},[40,84572,84573],{"class":10287},"        \"Simple demonstrator; about half shots meet the criterion.\"\n",[40,84575,84576],{"class":10276,"line":11452},[40,84577,84578],{"class":10363},"    ),\n",[40,84580,84581,84583],{"class":10276,"line":11493},[40,84582,84552],{"class":10430},[40,84584,12527],{"class":10363},[40,84586,84587,84590],{"class":10276,"line":11507},[40,84588,84589],{"class":10287},"        \"NISQ qubit\"",[40,84591,10507],{"class":10363},[40,84593,84594,84597],{"class":10276,"line":11512},[40,84595,84596],{"class":10446},"        0.60",[40,84598,10507],{"class":10363},[40,84600,84601],{"class":10276,"line":11517},[40,84602,84603],{"class":10287},"        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[40,84605,84606],{"class":10276,"line":11545},[40,84607,84578],{"class":10363},[40,84609,84610,84612],{"class":10276,"line":11557},[40,84611,84552],{"class":10430},[40,84613,12527],{"class":10363},[40,84615,84616,84619],{"class":10276,"line":11563},[40,84617,84618],{"class":10287},"        \"Fault-tolerant (logical)\"",[40,84620,10507],{"class":10363},[40,84622,84623,84626],{"class":10276,"line":11582},[40,84624,84625],{"class":10446},"        0.90",[40,84627,10507],{"class":10363},[40,84629,84630],{"class":10276,"line":11596},[40,84631,84632],{"class":10287},"        \"Error-corrected logical qubit; high single-shot success probability.\"\n",[40,84634,84635],{"class":10276,"line":11609},[40,84636,84578],{"class":10363},[40,84638,84639,84641],{"class":10276,"line":11622},[40,84640,84552],{"class":10430},[40,84642,12527],{"class":10363},[40,84644,84645,84648],{"class":10276,"line":11637},[40,84646,84647],{"class":10287},"        \"Topologically protected\"",[40,84649,10507],{"class":10363},[40,84651,84652,84655],{"class":10276,"line":11708},[40,84653,84654],{"class":10446},"        0.98",[40,84656,10507],{"class":10363},[40,84658,84659],{"class":10276,"line":11727},[40,84660,84661],{"class":10287},"        \"Intrinsic protection; very high single-shot success probability.\"\n",[40,84663,84664],{"class":10276,"line":11744},[40,84665,84578],{"class":10363},[40,84667,84668,84670],{"class":10276,"line":11790},[40,84669,84552],{"class":10430},[40,84671,12527],{"class":10363},[40,84673,84674,84677],{"class":10276,"line":11795},[40,84675,84676],{"class":10287},"        \"Ideal qubit\"",[40,84678,10507],{"class":10363},[40,84680,84681,84684],{"class":10276,"line":11800},[40,84682,84683],{"class":10446},"        1.00",[40,84685,10507],{"class":10363},[40,84687,84688],{"class":10276,"line":11806},[40,84689,84690],{"class":10287},"        \"Theoretical perfect qubit; always successful in a single shot.\"\n",[40,84692,84693],{"class":10276,"line":11834},[40,84694,84578],{"class":10363},[40,84696,84697],{"class":10276,"line":11840},[40,84698,25466],{"class":10363},[40,84700,84701],{"class":10276,"line":11860},[40,84702,10301],{"emptyLinePlaceholder":10300},[40,84704,84705],{"class":10276,"line":11887},[40,84706,84707],{"class":10280},"# ---------- Simulation configuration ----------\n",[40,84709,84710],{"class":10276,"line":11914},[40,84711,10301],{"emptyLinePlaceholder":10300},[40,84713,84714,84717,84720,84722],{"class":10276,"line":11938},[40,84715,84716],{"class":10446},"CONFIG",[40,84718,84719],{"class":10363},": Dict ",[40,84721,162],{"class":10732},[40,84723,11019],{"class":10363},[40,84725,84726,84729,84731,84734],{"class":10276,"line":11960},[40,84727,84728],{"class":10287},"    \"random_seed\"",[40,84730,10443],{"class":10363},[40,84732,84733],{"class":10446},"123456",[40,84735,10507],{"class":10363},[40,84737,84738,84741,84743,84746],{"class":10276,"line":11975},[40,84739,84740],{"class":10287},"    \"shots_per_model\"",[40,84742,10443],{"class":10363},[40,84744,84745],{"class":10446},"10000",[40,84747,10507],{"class":10363},[40,84749,84750,84753,84756,84759,84761,84764,84766,84769,84772],{"class":10276,"line":11996},[40,84751,84752],{"class":10287},"    \"epsilons\"",[40,84754,84755],{"class":10363},": [",[40,84757,84758],{"class":10446},"0.02",[40,84760,10484],{"class":10363},[40,84762,84763],{"class":10446},"0.01",[40,84765,10484],{"class":10363},[40,84767,84768],{"class":10446},"0.005",[40,84770,84771],{"class":10363},"],  ",[40,84773,84774],{"class":10280},"# ±2%, ±1%, ±0.5% targets\n",[40,84776,84777,84780],{"class":10276,"line":12001},[40,84778,84779],{"class":10287},"    \"cost_model\"",[40,84781,84782],{"class":10363},": {\n",[40,84784,84785,84788,84790,84793],{"class":10276,"line":12006},[40,84786,84787],{"class":10287},"        \"default_cost_per_shot_usd\"",[40,84789,10443],{"class":10363},[40,84791,84792],{"class":10446},"0.00005",[40,84794,10507],{"class":10363},[40,84796,84797,84800,84802],{"class":10276,"line":12028},[40,84798,84799],{"class":10287},"        \"fixed_overhead_usd\"",[40,84801,10443],{"class":10363},[40,84803,84804],{"class":10446},"0.10\n",[40,84806,84807],{"class":10276,"line":12034},[40,84808,84809],{"class":10363},"    },\n",[40,84811,84812,84815],{"class":10276,"line":12105},[40,84813,84814],{"class":10287},"    \"timing\"",[40,84816,84782],{"class":10363},[40,84818,84819,84822,84824],{"class":10276,"line":12110},[40,84820,84821],{"class":10287},"        \"time_per_shot_seconds\"",[40,84823,10443],{"class":10363},[40,84825,84826],{"class":10446},"2e-6\n",[40,84828,84829],{"class":10276,"line":12115},[40,84830,11145],{"class":10363},[40,84832,84833],{"class":10276,"line":12121},[40,84834,74784],{"class":10363},[40,84836,84837],{"class":10276,"line":12155},[40,84838,10301],{"emptyLinePlaceholder":10300},[40,84840,84841,84844,84846,84849,84852,84854,84856,84858,84861],{"class":10276,"line":12161},[40,84842,84843],{"class":10363},"rng ",[40,84845,162],{"class":10732},[40,84847,84848],{"class":10363}," np.random.",[40,84850,84851],{"class":10430},"default_rng",[40,84853,167],{"class":10363},[40,84855,84716],{"class":10446},[40,84857,1654],{"class":10363},[40,84859,84860],{"class":10287},"\"random_seed\"",[40,84862,23390],{"class":10363},[40,84864,84865],{"class":10276,"line":12177},[40,84866,10301],{"emptyLinePlaceholder":10300},[40,84868,84869],{"class":10276,"line":12192},[40,84870,10301],{"emptyLinePlaceholder":10300},[40,84872,84873],{"class":10276,"line":12231},[40,84874,84875],{"class":10280},"# ---------- Core simulation ----------\n",[40,84877,84878],{"class":10276,"line":12245},[40,84879,10301],{"emptyLinePlaceholder":10300},[40,84881,84882,84884,84887,84889,84891,84893,84895,84897,84900,84902,84904,84906,84909],{"class":10276,"line":12280},[40,84883,10965],{"class":10352},[40,84885,84886],{"class":10430}," simulate_bitstring",[40,84888,167],{"class":10363},[40,84890,12],{"class":10973},[40,84892,10443],{"class":10363},[40,84894,10784],{"class":10732},[40,84896,10484],{"class":10363},[40,84898,84899],{"class":10973},"shots",[40,84901,10443],{"class":10363},[40,84903,10821],{"class":10732},[40,84905,10484],{"class":10363},[40,84907,84908],{"class":10973},"rng_",[40,84910,84911],{"class":10363},": np.random.Generator) -> np.ndarray:\n",[40,84913,84914],{"class":10276,"line":12294},[40,84915,10723],{"class":10287},[40,84917,84918],{"class":10276,"line":12329},[40,84919,84920],{"class":10287},"    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[40,84922,84923],{"class":10276,"line":12343},[40,84924,10723],{"class":10287},[40,84926,84927,84929,84932,84935,84937,84939,84942,84945,84947,84950,84952,84954,84956],{"class":10276,"line":12376},[40,84928,11201],{"class":10352},[40,84930,84931],{"class":10363}," rng_.",[40,84933,84934],{"class":10430},"binomial",[40,84936,167],{"class":10363},[40,84938,78],{"class":10446},[40,84940,84941],{"class":10363},", p, ",[40,84943,84944],{"class":11903},"size",[40,84946,162],{"class":10732},[40,84948,84949],{"class":10363},"shots).",[40,84951,82906],{"class":10430},[40,84953,167],{"class":10363},[40,84955,10821],{"class":10732},[40,84957,10761],{"class":10363},[40,84959,84960],{"class":10276,"line":12409},[40,84961,10301],{"emptyLinePlaceholder":10300},[40,84963,84964],{"class":10276,"line":12414},[40,84965,10301],{"emptyLinePlaceholder":10300},[40,84967,84968,84970,84973,84975,84978,84981,84984,84986,84989,84992,84994],{"class":10276,"line":12419},[40,84969,10965],{"class":10352},[40,84971,84972],{"class":10430}," summarize_model",[40,84974,167],{"class":10363},[40,84976,84977],{"class":10973},"model",[40,84979,84980],{"class":10363},": QubitModel, ",[40,84982,84983],{"class":10973},"bits",[40,84985,12019],{"class":10363},[40,84987,84988],{"class":10973},"epsilons",[40,84990,84991],{"class":10363},": List[",[40,84993,10784],{"class":10732},[40,84995,84996],{"class":10363},"]) -> Dict:\n",[40,84998,84999],{"class":10276,"line":12435},[40,85000,10723],{"class":10287},[40,85002,85003],{"class":10276,"line":12447},[40,85004,85005],{"class":10287},"    Calculate summary statistics for a given model and its simulated bitstring.\n",[40,85007,85008],{"class":10276,"line":12463},[40,85009,10723],{"class":10287},[40,85011,85012,85014,85016,85018],{"class":10276,"line":12480},[40,85013,15978],{"class":10363},[40,85015,162],{"class":10732},[40,85017,37310],{"class":10732},[40,85019,85020],{"class":10363},"(bits.size)\n",[40,85022,85023,85026,85028,85030,85033,85035],{"class":10276,"line":12498},[40,85024,85025],{"class":10363},"    ones ",[40,85027,162],{"class":10732},[40,85029,37310],{"class":10732},[40,85031,85032],{"class":10363},"(bits.",[40,85034,12092],{"class":10430},[40,85036,23593],{"class":10363},[40,85038,85039,85042,85044,85046,85049,85051],{"class":10276,"line":12504},[40,85040,85041],{"class":10363},"    zeros ",[40,85043,162],{"class":10732},[40,85045,37310],{"class":10732},[40,85047,85048],{"class":10363},"(n ",[40,85050,11056],{"class":10732},[40,85052,85053],{"class":10363}," ones)\n",[40,85055,85056,85059,85061,85064,85066,85068,85070,85072,85074,85076,85078,85080,85082,85085],{"class":10276,"line":12517},[40,85057,85058],{"class":10363},"    frac ",[40,85060,162],{"class":10732},[40,85062,85063],{"class":10363}," ones ",[40,85065,11243],{"class":10732},[40,85067,84081],{"class":10363},[40,85069,11225],{"class":10352},[40,85071,84081],{"class":10363},[40,85073,22432],{"class":10732},[40,85075,11234],{"class":10446},[40,85077,11237],{"class":10352},[40,85079,75857],{"class":10732},[40,85081,167],{"class":10363},[40,85083,85084],{"class":10287},"\"nan\"",[40,85086,10761],{"class":10363},[40,85088,85089,85092,85094,85096],{"class":10276,"line":12530},[40,85090,85091],{"class":10363},"    lo, hi ",[40,85093,162],{"class":10732},[40,85095,84017],{"class":10430},[40,85097,85098],{"class":10363},"(ones, n)\n",[40,85100,85101],{"class":10276,"line":12542},[40,85102,10301],{"emptyLinePlaceholder":10300},[40,85104,85105,85108,85110],{"class":10276,"line":12571},[40,85106,85107],{"class":10363},"    row ",[40,85109,162],{"class":10732},[40,85111,11019],{"class":10363},[40,85113,85114,85117],{"class":10276,"line":12622},[40,85115,85116],{"class":10287},"        \"Model\"",[40,85118,85119],{"class":10363},": model.name,\n",[40,85121,85122,85125],{"class":10276,"line":12628},[40,85123,85124],{"class":10287},"        \"Description\"",[40,85126,85127],{"class":10363},": model.description,\n",[40,85129,85130,85133],{"class":10276,"line":12634},[40,85131,85132],{"class":10287},"        \"Configured p (target)\"",[40,85134,85135],{"class":10363},": model.satisfaction_prob,\n",[40,85137,85138,85141],{"class":10276,"line":12639},[40,85139,85140],{"class":10287},"        \"Shots\"",[40,85142,85143],{"class":10363},": n,\n",[40,85145,85146,85149],{"class":10276,"line":12644},[40,85147,85148],{"class":10287},"        \"Ones (successes)\"",[40,85150,85151],{"class":10363},": ones,\n",[40,85153,85154,85157],{"class":10276,"line":12667},[40,85155,85156],{"class":10287},"        \"Zeros (failures)\"",[40,85158,85159],{"class":10363},": zeros,\n",[40,85161,85162,85165],{"class":10276,"line":12673},[40,85163,85164],{"class":10287},"        \"Ones fraction (observed)\"",[40,85166,85167],{"class":10363},": frac,\n",[40,85169,85170,85173,85176,85179],{"class":10276,"line":12684},[40,85171,85172],{"class":10287},"        \"Wilson CI 95",[40,85174,85175],{"class":10446},"% lo",[40,85177,85178],{"class":10287},"w\"",[40,85180,85181],{"class":10363},": lo,\n",[40,85183,85184,85186,85189,85192],{"class":10276,"line":12697},[40,85185,85172],{"class":10287},[40,85187,85188],{"class":10446},"% hi",[40,85190,85191],{"class":10287},"gh\"",[40,85193,85194],{"class":10363},": hi,\n",[40,85196,85197,85200,85203,85205],{"class":10276,"line":12723},[40,85198,85199],{"class":10287},"        \"CI width (95%)\"",[40,85201,85202],{"class":10363},": hi ",[40,85204,11056],{"class":10732},[40,85206,85207],{"class":10363}," lo,\n",[40,85209,85210],{"class":10276,"line":12736},[40,85211,11145],{"class":10363},[40,85213,85214,85216,85218,85220],{"class":10276,"line":12777},[40,85215,17425],{"class":10352},[40,85217,22549],{"class":10363},[40,85219,13846],{"class":10352},[40,85221,85222],{"class":10363}," epsilons:\n",[40,85224,85225,85228,85230,85233,85235,85237,85240,85242,85245,85247,85249,85252,85254,85256,85258],{"class":10276,"line":12790},[40,85226,85227],{"class":10363},"        row[",[40,85229,13262],{"class":10352},[40,85231,85232],{"class":10287},"\"Required shots for ±",[40,85234,13268],{"class":10446},[40,85236,10821],{"class":10732},[40,85238,85239],{"class":10363},"(eps",[40,85241,11752],{"class":10732},[40,85243,85244],{"class":10446},"100",[40,85246,207],{"class":10363},[40,85248,13280],{"class":10446},[40,85250,85251],{"class":10287},"%\"",[40,85253,10741],{"class":10363},[40,85255,162],{"class":10732},[40,85257,84288],{"class":10430},[40,85259,85260],{"class":10363},"(model.satisfaction_prob, eps)\n",[40,85262,85263,85265],{"class":10276,"line":16980},[40,85264,11201],{"class":10352},[40,85266,85267],{"class":10363}," row\n",[40,85269,85270],{"class":10276,"line":16991},[40,85271,10301],{"emptyLinePlaceholder":10300},[40,85273,85274],{"class":10276,"line":16996},[40,85275,10301],{"emptyLinePlaceholder":10300},[40,85277,85278,85280,85283,85285,85287,85289,85291,85293,85296,85298,85300,85302,85305,85307,85309,85311,85313],{"class":10276,"line":17001},[40,85279,10965],{"class":10352},[40,85281,85282],{"class":10430}," apply_costs",[40,85284,167],{"class":10363},[40,85286,84899],{"class":10973},[40,85288,10443],{"class":10363},[40,85290,10821],{"class":10732},[40,85292,10484],{"class":10363},[40,85294,85295],{"class":10973},"cost_per_shot",[40,85297,10443],{"class":10363},[40,85299,10784],{"class":10732},[40,85301,10484],{"class":10363},[40,85303,85304],{"class":10973},"fixed_overhead",[40,85306,10443],{"class":10363},[40,85308,10784],{"class":10732},[40,85310,22117],{"class":10363},[40,85312,10784],{"class":10732},[40,85314,10639],{"class":10363},[40,85316,85317],{"class":10276,"line":17019},[40,85318,10723],{"class":10287},[40,85320,85321],{"class":10276,"line":17047},[40,85322,85323],{"class":10287},"    Calculate illustrative cost based on shots and cost model.\n",[40,85325,85326],{"class":10276,"line":17064},[40,85327,10723],{"class":10287},[40,85329,85330,85332,85335,85337,85340,85342],{"class":10276,"line":17077},[40,85331,11201],{"class":10352},[40,85333,85334],{"class":10363}," fixed_overhead ",[40,85336,1053],{"class":10732},[40,85338,85339],{"class":10363}," shots ",[40,85341,11752],{"class":10732},[40,85343,85344],{"class":10363}," cost_per_shot\n",[40,85346,85347],{"class":10276,"line":17110},[40,85348,10301],{"emptyLinePlaceholder":10300},[40,85350,85351],{"class":10276,"line":17145},[40,85352,10301],{"emptyLinePlaceholder":10300},[40,85354,85355],{"class":10276,"line":17194},[40,85356,85357],{"class":10280},"# ---------- Run the simulation ----------\n",[40,85359,85360],{"class":10276,"line":17202},[40,85361,10301],{"emptyLinePlaceholder":10300},[40,85363,85364,85367,85369,85372,85374],{"class":10276,"line":17207},[40,85365,85366],{"class":10363},"all_bitstrings: Dict[",[40,85368,10733],{"class":10732},[40,85370,85371],{"class":10363},", np.ndarray] ",[40,85373,162],{"class":10732},[40,85375,85376],{"class":10363}," {}\n",[40,85378,85379,85382,85384],{"class":10276,"line":17241},[40,85380,85381],{"class":10363},"summary_rows: List[Dict] ",[40,85383,162],{"class":10732},[40,85385,13835],{"class":10363},[40,85387,85388],{"class":10276,"line":17269},[40,85389,10301],{"emptyLinePlaceholder":10300},[40,85391,85392,85394,85396,85399],{"class":10276,"line":17295},[40,85393,14993],{"class":10732},[40,85395,167],{"class":10363},[40,85397,85398],{"class":10287},"\"Running simulation...\"",[40,85400,10761],{"class":10363},[40,85402,85403,85405,85408,85410,85413],{"class":10276,"line":17324},[40,85404,13840],{"class":10352},[40,85406,85407],{"class":10363}," m ",[40,85409,13846],{"class":10352},[40,85411,85412],{"class":10446}," DEFAULT_MODELS",[40,85414,10639],{"class":10363},[40,85416,85417,85420,85422,85424,85427,85429,85431,85434],{"class":10276,"line":17341},[40,85418,85419],{"class":10363},"    bits ",[40,85421,162],{"class":10732},[40,85423,84886],{"class":10430},[40,85425,85426],{"class":10363},"(m.satisfaction_prob, ",[40,85428,84716],{"class":10446},[40,85430,1654],{"class":10363},[40,85432,85433],{"class":10287},"\"shots_per_model\"",[40,85435,85436],{"class":10363},"], rng)\n",[40,85438,85439,85442,85444],{"class":10276,"line":17350},[40,85440,85441],{"class":10363},"    all_bitstrings[m.name] ",[40,85443,162],{"class":10732},[40,85445,85446],{"class":10363}," bits\n",[40,85448,85449,85452,85454,85456,85459,85462,85464,85466,85469],{"class":10276,"line":17358},[40,85450,85451],{"class":10363},"    summary_rows.",[40,85453,13888],{"class":10430},[40,85455,167],{"class":10363},[40,85457,85458],{"class":10430},"summarize_model",[40,85460,85461],{"class":10363},"(m, bits, ",[40,85463,84716],{"class":10446},[40,85465,1654],{"class":10363},[40,85467,85468],{"class":10287},"\"epsilons\"",[40,85470,85471],{"class":10363},"]))\n",[40,85473,85474],{"class":10276,"line":17363},[40,85475,10301],{"emptyLinePlaceholder":10300},[40,85477,85478,85481,85483,85485,85487],{"class":10276,"line":17368},[40,85479,85480],{"class":10363},"summary_df ",[40,85482,162],{"class":10732},[40,85484,74328],{"class":10363},[40,85486,74331],{"class":10430},[40,85488,85489],{"class":10363},"(summary_rows)\n",[40,85491,85492],{"class":10276,"line":17383},[40,85493,10301],{"emptyLinePlaceholder":10300},[40,85495,85496],{"class":10276,"line":17416},[40,85497,85498],{"class":10280},"# Add cost estimates\n",[40,85500,85501,85504,85506,85509,85511,85514,85516,85519],{"class":10276,"line":17422},[40,85502,85503],{"class":10363},"cps ",[40,85505,162],{"class":10732},[40,85507,85508],{"class":10446}," CONFIG",[40,85510,1654],{"class":10363},[40,85512,85513],{"class":10287},"\"cost_model\"",[40,85515,79373],{"class":10363},[40,85517,85518],{"class":10287},"\"default_cost_per_shot_usd\"",[40,85520,25466],{"class":10363},[40,85522,85523,85526,85528,85530,85532,85534,85536,85539],{"class":10276,"line":17435},[40,85524,85525],{"class":10363},"over ",[40,85527,162],{"class":10732},[40,85529,85508],{"class":10446},[40,85531,1654],{"class":10363},[40,85533,85513],{"class":10287},[40,85535,79373],{"class":10363},[40,85537,85538],{"class":10287},"\"fixed_overhead_usd\"",[40,85540,25466],{"class":10363},[40,85542,85543,85545,85548,85550,85552,85555,85558,85560,85562],{"class":10276,"line":17453},[40,85544,11225],{"class":10352},[40,85546,85547],{"class":10363}," cps ",[40,85549,22432],{"class":10732},[40,85551,11234],{"class":10446},[40,85553,85554],{"class":10352}," or",[40,85556,85557],{"class":10363}," over ",[40,85559,22432],{"class":10732},[40,85561,11234],{"class":10446},[40,85563,10639],{"class":10363},[40,85565,85566,85569,85572,85574,85576],{"class":10276,"line":17472},[40,85567,85568],{"class":10363},"    summary_df[",[40,85570,85571],{"class":10287},"\"Illustrative cost (USD)\"",[40,85573,10741],{"class":10363},[40,85575,162],{"class":10732},[40,85577,37422],{"class":10363},[40,85579,85580,85583,85585,85587,85590,85593],{"class":10276,"line":17495},[40,85581,85582],{"class":10430},"        apply_costs",[40,85584,167],{"class":10363},[40,85586,10821],{"class":10732},[40,85588,85589],{"class":10363},"(row[",[40,85591,85592],{"class":10287},"\"Shots\"",[40,85594,85595],{"class":10363},"]), cps, over)\n",[40,85597,85598,85600,85603,85605,85608,85611],{"class":10276,"line":17504},[40,85599,75278],{"class":10352},[40,85601,85602],{"class":10363}," _, row ",[40,85604,13846],{"class":10352},[40,85606,85607],{"class":10363}," summary_df.",[40,85609,85610],{"class":10430},"iterrows",[40,85612,12174],{"class":10363},[40,85614,85615],{"class":10276,"line":17509},[40,85616,75308],{"class":10363},[40,85618,85619],{"class":10276,"line":17514},[40,85620,10301],{"emptyLinePlaceholder":10300},[40,85622,85623],{"class":10276,"line":17520},[40,85624,85625],{"class":10280},"# ---------- Save artifacts ----------\n",[40,85627,85628],{"class":10276,"line":17535},[40,85629,10301],{"emptyLinePlaceholder":10300},[40,85631,85632],{"class":10276,"line":17546},[40,85633,85634],{"class":10280},"# Save raw bitstrings and summary CSV\n",[40,85636,85637,85639,85642,85644,85647,85649],{"class":10276,"line":17583},[40,85638,13840],{"class":10352},[40,85640,85641],{"class":10363}," name, arr ",[40,85643,13846],{"class":10352},[40,85645,85646],{"class":10363}," all_bitstrings.",[40,85648,75289],{"class":10430},[40,85650,85651],{"class":10363},"():\n",[40,85653,85654,85657,85659,85661,85664,85666,85669,85672,85674,85677,85679,85682,85684,85686],{"class":10276,"line":17606},[40,85655,85656],{"class":10363},"    out_path ",[40,85658,162],{"class":10732},[40,85660,13867],{"class":10352},[40,85662,85663],{"class":10287},"\"bitstrings_",[40,85665,13268],{"class":10446},[40,85667,85668],{"class":10363},"name.",[40,85670,85671],{"class":10430},"replace",[40,85673,167],{"class":10363},[40,85675,85676],{"class":10287},"' '",[40,85678,10484],{"class":10363},[40,85680,85681],{"class":10287},"'_'",[40,85683,207],{"class":10363},[40,85685,13280],{"class":10446},[40,85687,85688],{"class":10287},".csv\"\n",[40,85690,85691,85694,85696,85699,85702,85705,85708,85711,85714,85716,85718],{"class":10276,"line":17617},[40,85692,85693],{"class":10363},"    pd.",[40,85695,74331],{"class":10430},[40,85697,85698],{"class":10363},"({",[40,85700,85701],{"class":10287},"\"bit\"",[40,85703,85704],{"class":10363},": arr}).",[40,85706,85707],{"class":10430},"to_csv",[40,85709,85710],{"class":10363},"(out_path, ",[40,85712,85713],{"class":11903},"index",[40,85715,162],{"class":10732},[40,85717,10504],{"class":10446},[40,85719,10761],{"class":10363},[40,85721,85722],{"class":10276,"line":17634},[40,85723,10301],{"emptyLinePlaceholder":10300},[40,85725,85726,85729,85731],{"class":10276,"line":17646},[40,85727,85728],{"class":10363},"summary_csv_path ",[40,85730,162],{"class":10732},[40,85732,85733],{"class":10287}," \"summary.csv\"\n",[40,85735,85736,85739,85741,85744,85746,85748,85750],{"class":10276,"line":17651},[40,85737,85738],{"class":10363},"summary_df.",[40,85740,85707],{"class":10430},[40,85742,85743],{"class":10363},"(summary_csv_path, ",[40,85745,85713],{"class":11903},[40,85747,162],{"class":10732},[40,85749,10504],{"class":10446},[40,85751,10761],{"class":10363},[40,85753,85754],{"class":10276,"line":17656},[40,85755,10301],{"emptyLinePlaceholder":10300},[40,85757,85758],{"class":10276,"line":76297},[40,85759,85760],{"class":10280},"# Save config JSON\n",[40,85762,85763,85766,85769,85771,85774,85776,85779,85781,85783],{"class":10276,"line":76302},[40,85764,85765],{"class":10352},"with",[40,85767,85768],{"class":10732}," open",[40,85770,167],{"class":10363},[40,85772,85773],{"class":10287},"\"simulation_config.json\"",[40,85775,10484],{"class":10363},[40,85777,85778],{"class":10287},"\"w\"",[40,85780,12055],{"class":10363},[40,85782,10381],{"class":10352},[40,85784,85785],{"class":10363}," f:\n",[40,85787,85788,85791,85794,85796,85798,85801,85804,85806,85808],{"class":10276,"line":76313},[40,85789,85790],{"class":10363},"    json.",[40,85792,85793],{"class":10430},"dump",[40,85795,167],{"class":10363},[40,85797,84716],{"class":10446},[40,85799,85800],{"class":10363},", f, ",[40,85802,85803],{"class":11903},"indent",[40,85805,162],{"class":10732},[40,85807,1044],{"class":10446},[40,85809,10761],{"class":10363},[40,85811,85812],{"class":10276,"line":76321},[40,85813,10301],{"emptyLinePlaceholder":10300},[40,85815,85816,85818,85820,85823],{"class":10276,"line":76331},[40,85817,14993],{"class":10732},[40,85819,167],{"class":10363},[40,85821,85822],{"class":10287},"\"Saved data to CSV and JSON files.\"",[40,85824,10761],{"class":10363},[40,85826,85827],{"class":10276,"line":76336},[40,85828,10301],{"emptyLinePlaceholder":10300},[40,85830,85831],{"class":10276,"line":76341},[40,85832,85833],{"class":10280},"# ---------- Visualizations ----------\n",[40,85835,85836],{"class":10276,"line":76346},[40,85837,10301],{"emptyLinePlaceholder":10300},[40,85839,85840],{"class":10276,"line":76353},[40,85841,85842],{"class":10280},"# Figure 1: Observed ones fraction with Wilson CI\n",[40,85844,85845,85848,85851,85853,85856],{"class":10276,"line":76369},[40,85846,85847],{"class":10363},"plt.style.",[40,85849,85850],{"class":10430},"use",[40,85852,167],{"class":10363},[40,85854,85855],{"class":10287},"'seaborn-v0_8-whitegrid'",[40,85857,10761],{"class":10363},[40,85859,85860,85862,85864,85866,85868,85870,85872,85874,85876,85878],{"class":10276,"line":76378},[40,85861,23469],{"class":10363},[40,85863,16373],{"class":10430},[40,85865,167],{"class":10363},[40,85867,13078],{"class":11903},[40,85869,162],{"class":10732},[40,85871,167],{"class":10363},[40,85873,77314],{"class":10446},[40,85875,10484],{"class":10363},[40,85877,12785],{"class":10446},[40,85879,13612],{"class":10363},[40,85881,85882,85884,85886,85888,85890,85892,85894],{"class":10276,"line":76387},[40,85883,25495],{"class":10363},[40,85885,162],{"class":10732},[40,85887,11156],{"class":10363},[40,85889,25502],{"class":10430},[40,85891,167],{"class":10363},[40,85893,25507],{"class":10732},[40,85895,85896],{"class":10363},"(summary_df))\n",[40,85898,85899,85902,85904,85907,85910,85913,85916],{"class":10276,"line":76396},[40,85900,85901],{"class":10363},"y ",[40,85903,162],{"class":10732},[40,85905,85906],{"class":10363}," summary_df[",[40,85908,85909],{"class":10287},"\"Ones fraction (observed)\"",[40,85911,85912],{"class":10363},"].",[40,85914,85915],{"class":10430},"to_numpy",[40,85917,12174],{"class":10363},[40,85919,85920],{"class":10276,"line":76405},[40,85921,10301],{"emptyLinePlaceholder":10300},[40,85923,85924],{"class":10276,"line":76428},[40,85925,85926],{"class":10280},"# --- FIX APPLIED HERE ---\n",[40,85928,85929],{"class":10276,"line":76434},[40,85930,85931],{"class":10280},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[40,85933,85934],{"class":10276,"line":76439},[40,85935,85936],{"class":10280},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[40,85938,85939,85942,85944,85947,85949,85951,85954,85956,85958,85960,85962],{"class":10276,"line":76444},[40,85940,85941],{"class":10363},"lower_error ",[40,85943,162],{"class":10732},[40,85945,85946],{"class":10363}," y ",[40,85948,11056],{"class":10732},[40,85950,85906],{"class":10363},[40,85952,85953],{"class":10287},"\"Wilson CI 95",[40,85955,85175],{"class":10446},[40,85957,85178],{"class":10287},[40,85959,85912],{"class":10363},[40,85961,85915],{"class":10430},[40,85963,12174],{"class":10363},[40,85965,85966,85969,85971,85973,85975,85977,85979,85981,85983,85985,85987],{"class":10276,"line":76449},[40,85967,85968],{"class":10363},"upper_error ",[40,85970,162],{"class":10732},[40,85972,85906],{"class":10363},[40,85974,85953],{"class":10287},[40,85976,85188],{"class":10446},[40,85978,85191],{"class":10287},[40,85980,85912],{"class":10363},[40,85982,85915],{"class":10430},[40,85984,27025],{"class":10363},[40,85986,11056],{"class":10732},[40,85988,85989],{"class":10363}," y\n",[40,85991,85992,85995,85997,85999,86001,86004,86006,86009,86011],{"class":10276,"line":76459},[40,85993,85994],{"class":10363},"yerr ",[40,85996,162],{"class":10732},[40,85998,11156],{"class":10363},[40,86000,76246],{"class":10430},[40,86002,86003],{"class":10363},"([np.",[40,86005,23431],{"class":10430},[40,86007,86008],{"class":10363},"(lower_error), np.",[40,86010,23431],{"class":10430},[40,86012,86013],{"class":10363},"(upper_error)])\n",[40,86015,86016],{"class":10276,"line":76468},[40,86017,86018],{"class":10280},"# --- END OF FIX ---\n",[40,86020,86021],{"class":10276,"line":76477},[40,86022,10301],{"emptyLinePlaceholder":10300},[40,86024,86025,86027,86029,86032,86034,86036,86039,86041,86044,86046,86049],{"class":10276,"line":76482},[40,86026,23469],{"class":10363},[40,86028,25517],{"class":10430},[40,86030,86031],{"class":10363},"(x, y, ",[40,86033,23550],{"class":11903},[40,86035,162],{"class":10732},[40,86037,86038],{"class":10287},"'skyblue'",[40,86040,10484],{"class":10363},[40,86042,86043],{"class":11903},"edgecolor",[40,86045,162],{"class":10732},[40,86047,86048],{"class":10287},"'black'",[40,86050,10761],{"class":10363},[40,86052,86053,86055,86058,86060,86063,86065,86068,86071,86073,86075,86077,86080,86082,86084,86086,86088,86090,86093,86095,86098,86100,86102],{"class":10276,"line":76488},[40,86054,23469],{"class":10363},[40,86056,86057],{"class":10430},"errorbar",[40,86059,86031],{"class":10363},[40,86061,86062],{"class":11903},"yerr",[40,86064,162],{"class":10732},[40,86066,86067],{"class":10363},"yerr, ",[40,86069,86070],{"class":11903},"fmt",[40,86072,162],{"class":10732},[40,86074,12617],{"class":10287},[40,86076,10484],{"class":10363},[40,86078,86079],{"class":11903},"capsize",[40,86081,162],{"class":10732},[40,86083,12566],{"class":10446},[40,86085,10484],{"class":10363},[40,86087,23550],{"class":11903},[40,86089,162],{"class":10732},[40,86091,86092],{"class":10287},"\"black\"",[40,86094,10484],{"class":10363},[40,86096,86097],{"class":11903},"elinewidth",[40,86099,162],{"class":10732},[40,86101,23229],{"class":10446},[40,86103,10761],{"class":10363},[40,86105,86106,86108,86110,86113,86116,86118,86121,86123,86126,86128,86130,86132,86134,86136,86138],{"class":10276,"line":76503},[40,86107,23469],{"class":10363},[40,86109,25527],{"class":10430},[40,86111,86112],{"class":10363},"(x, summary_df[",[40,86114,86115],{"class":10287},"\"Model\"",[40,86117,85912],{"class":10363},[40,86119,86120],{"class":10430},"tolist",[40,86122,13575],{"class":10363},[40,86124,86125],{"class":11903},"rotation",[40,86127,162],{"class":10732},[40,86129,64874],{"class":10446},[40,86131,10484],{"class":10363},[40,86133,12211],{"class":11903},[40,86135,162],{"class":10732},[40,86137,12267],{"class":10287},[40,86139,10761],{"class":10363},[40,86141,86142,86144,86146,86148,86151],{"class":10276,"line":76514},[40,86143,23469],{"class":10363},[40,86145,23626],{"class":10430},[40,86147,167],{"class":10363},[40,86149,86150],{"class":10287},"\"Observed Ones Fraction\"",[40,86152,10761],{"class":10363},[40,86154,86155,86157,86159,86161,86164,86166,86168,86170,86173],{"class":10276,"line":76529},[40,86156,23469],{"class":10363},[40,86158,23656],{"class":10430},[40,86160,167],{"class":10363},[40,86162,86163],{"class":10287},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[40,86165,10484],{"class":10363},[40,86167,12553],{"class":11903},[40,86169,162],{"class":10732},[40,86171,86172],{"class":10446},"14",[40,86174,10761],{"class":10363},[40,86176,86177,86179,86181],{"class":10276,"line":76539},[40,86178,23469],{"class":10363},[40,86180,30425],{"class":10430},[40,86182,12174],{"class":10363},[40,86184,86185,86188,86190],{"class":10276,"line":76547},[40,86186,86187],{"class":10363},"fig1_path ",[40,86189,162],{"class":10732},[40,86191,86192],{"class":10287}," \"fig_ones_fraction.png\"\n",[40,86194,86195,86197,86200,86203,86205,86207,86209,86211,86214,86216,86219],{"class":10276,"line":76556},[40,86196,23469],{"class":10363},[40,86198,86199],{"class":10430},"savefig",[40,86201,86202],{"class":10363},"(fig1_path, ",[40,86204,23491],{"class":11903},[40,86206,162],{"class":10732},[40,86208,10447],{"class":10446},[40,86210,10484],{"class":10363},[40,86212,86213],{"class":11903},"bbox_inches",[40,86215,162],{"class":10732},[40,86217,86218],{"class":10287},"'tight'",[40,86220,10761],{"class":10363},[40,86222,86223,86225,86227],{"class":10276,"line":76561},[40,86224,23469],{"class":10363},[40,86226,23686],{"class":10430},[40,86228,12174],{"class":10363},[40,86230,86231],{"class":10276,"line":76583},[40,86232,10301],{"emptyLinePlaceholder":10300},[40,86234,86235],{"class":10276,"line":76588},[40,86236,86237],{"class":10280},"# Figure 2: Required shots at several ±epsilon targets\n",[40,86239,86240,86242,86244,86246,86248,86250,86252,86254,86256,86258],{"class":10276,"line":76604},[40,86241,23469],{"class":10363},[40,86243,16373],{"class":10430},[40,86245,167],{"class":10363},[40,86247,13078],{"class":11903},[40,86249,162],{"class":10732},[40,86251,167],{"class":10363},[40,86253,77314],{"class":10446},[40,86255,10484],{"class":10363},[40,86257,12785],{"class":10446},[40,86259,13612],{"class":10363},[40,86261,86262,86265,86267],{"class":10276,"line":76621},[40,86263,86264],{"class":10363},"width ",[40,86266,162],{"class":10732},[40,86268,22339],{"class":10446},[40,86270,86271,86273,86275,86277,86279,86281,86283,86285,86287],{"class":10276,"line":76626},[40,86272,25495],{"class":10363},[40,86274,162],{"class":10732},[40,86276,11156],{"class":10363},[40,86278,25502],{"class":10430},[40,86280,167],{"class":10363},[40,86282,25507],{"class":10732},[40,86284,167],{"class":10363},[40,86286,84540],{"class":10446},[40,86288,13612],{"class":10363},[40,86290,86291,86294,86296,86298,86301,86303,86306,86308,86311],{"class":10276,"line":76639},[40,86292,86293],{"class":10363},"colors ",[40,86295,162],{"class":10732},[40,86297,24010],{"class":10363},[40,86299,86300],{"class":10287},"'#ff9999'",[40,86302,73],{"class":10363},[40,86304,86305],{"class":10287},"'#66b3ff'",[40,86307,73],{"class":10363},[40,86309,86310],{"class":10287},"'#99ff99'",[40,86312,25466],{"class":10363},[40,86314,86315],{"class":10276,"line":76652},[40,86316,10301],{"emptyLinePlaceholder":10300},[40,86318,86319,86321,86324,86326,86328,86330,86332,86334,86336],{"class":10276,"line":76657},[40,86320,13840],{"class":10352},[40,86322,86323],{"class":10363}," i, eps ",[40,86325,13846],{"class":10352},[40,86327,23931],{"class":10732},[40,86329,167],{"class":10363},[40,86331,84716],{"class":10446},[40,86333,1654],{"class":10363},[40,86335,85468],{"class":10287},[40,86337,86338],{"class":10363},"]):\n",[40,86340,86341,86344,86346,86348,86351,86354,86356,86358,86360,86362],{"class":10276,"line":76665},[40,86342,86343],{"class":10363},"    req ",[40,86345,162],{"class":10732},[40,86347,24010],{"class":10363},[40,86349,86350],{"class":10430},"shots_for_epsilon",[40,86352,86353],{"class":10363},"(m.satisfaction_prob, eps) ",[40,86355,13840],{"class":10352},[40,86357,85407],{"class":10363},[40,86359,13846],{"class":10352},[40,86361,85412],{"class":10446},[40,86363,25466],{"class":10363},[40,86365,86366,86368,86370,86373,86375,86377,86379,86382,86384,86386,86388,86391,86393,86395,86398,86400,86402,86404,86406,86408,86411,86413,86415,86417],{"class":10276,"line":76670},[40,86367,13617],{"class":10363},[40,86369,25517],{"class":10430},[40,86371,86372],{"class":10363},"(x ",[40,86374,1053],{"class":10732},[40,86376,79692],{"class":10363},[40,86378,11752],{"class":10732},[40,86380,86381],{"class":10363}," width, req, width, ",[40,86383,15415],{"class":11903},[40,86385,162],{"class":10732},[40,86387,13262],{"class":10352},[40,86389,86390],{"class":10287},"\"±",[40,86392,13268],{"class":10446},[40,86394,22105],{"class":10363},[40,86396,86397],{"class":10352},":.1%",[40,86399,13280],{"class":10446},[40,86401,13478],{"class":10287},[40,86403,10484],{"class":10363},[40,86405,23550],{"class":11903},[40,86407,162],{"class":10732},[40,86409,86410],{"class":10363},"colors[i], ",[40,86412,86043],{"class":11903},[40,86414,162],{"class":10732},[40,86416,86048],{"class":10287},[40,86418,10761],{"class":10363},[40,86420,86421],{"class":10276,"line":76675},[40,86422,10301],{"emptyLinePlaceholder":10300},[40,86424,86425,86427,86429,86431,86433,86436,86438,86440,86442,86444,86446,86448,86450,86452,86454,86456,86458,86460],{"class":10276,"line":76685},[40,86426,23469],{"class":10363},[40,86428,25527],{"class":10430},[40,86430,86372],{"class":10363},[40,86432,1053],{"class":10732},[40,86434,86435],{"class":10363}," width, [m.name ",[40,86437,13840],{"class":10352},[40,86439,85407],{"class":10363},[40,86441,13846],{"class":10352},[40,86443,85412],{"class":10446},[40,86445,11672],{"class":10363},[40,86447,86125],{"class":11903},[40,86449,162],{"class":10732},[40,86451,64874],{"class":10446},[40,86453,10484],{"class":10363},[40,86455,12211],{"class":11903},[40,86457,162],{"class":10732},[40,86459,12267],{"class":10287},[40,86461,10761],{"class":10363},[40,86463,86464,86466,86468,86470,86473],{"class":10276,"line":76693},[40,86465,23469],{"class":10363},[40,86467,23626],{"class":10430},[40,86469,167],{"class":10363},[40,86471,86472],{"class":10287},"\"Required Shots (Log Scale)\"",[40,86474,10761],{"class":10363},[40,86476,86477,86479,86481,86483,86486,86488,86490,86492,86494],{"class":10276,"line":76701},[40,86478,23469],{"class":10363},[40,86480,23656],{"class":10430},[40,86482,167],{"class":10363},[40,86484,86485],{"class":10287},"\"Approximate Shots to Estimate Success Rate within ±ε\"",[40,86487,10484],{"class":10363},[40,86489,12553],{"class":11903},[40,86491,162],{"class":10732},[40,86493,86172],{"class":10446},[40,86495,10761],{"class":10363},[40,86497,86498,86500,86503,86505,86508],{"class":10276,"line":76713},[40,86499,23469],{"class":10363},[40,86501,86502],{"class":10430},"yscale",[40,86504,167],{"class":10363},[40,86506,86507],{"class":10287},"'log'",[40,86509,10761],{"class":10363},[40,86511,86512,86514,86516,86518,86520,86522,86525],{"class":10276,"line":76725},[40,86513,23469],{"class":10363},[40,86515,23677],{"class":10430},[40,86517,167],{"class":10363},[40,86519,23656],{"class":11903},[40,86521,162],{"class":10732},[40,86523,86524],{"class":10287},"\"Error Tolerance\"",[40,86526,10761],{"class":10363},[40,86528,86529,86531,86533],{"class":10276,"line":76734},[40,86530,23469],{"class":10363},[40,86532,30425],{"class":10430},[40,86534,12174],{"class":10363},[40,86536,86537,86540,86542],{"class":10276,"line":76740},[40,86538,86539],{"class":10363},"fig2_path ",[40,86541,162],{"class":10732},[40,86543,86544],{"class":10287}," \"fig_required_shots.png\"\n",[40,86546,86547,86549,86551,86554,86556,86558,86560,86562,86564,86566,86568],{"class":10276,"line":76750},[40,86548,23469],{"class":10363},[40,86550,86199],{"class":10430},[40,86552,86553],{"class":10363},"(fig2_path, ",[40,86555,23491],{"class":11903},[40,86557,162],{"class":10732},[40,86559,10447],{"class":10446},[40,86561,10484],{"class":10363},[40,86563,86213],{"class":11903},[40,86565,162],{"class":10732},[40,86567,86218],{"class":10287},[40,86569,10761],{"class":10363},[40,86571,86572,86574,86576],{"class":10276,"line":76760},[40,86573,23469],{"class":10363},[40,86575,23686],{"class":10430},[40,86577,12174],{"class":10363},[40,86579,86580],{"class":10276,"line":76769},[40,86581,10301],{"emptyLinePlaceholder":10300},[40,86583,86584,86586,86588,86591],{"class":10276,"line":76778},[40,86585,14993],{"class":10732},[40,86587,167],{"class":10363},[40,86589,86590],{"class":10287},"\"Generated and saved plots.\"",[40,86592,10761],{"class":10363},[40,86594,86595],{"class":10276,"line":76789},[40,86596,10301],{"emptyLinePlaceholder":10300},[40,86598,86599],{"class":10276,"line":76801},[40,86600,86601],{"class":10280},"# ---------- Display interactive table ----------\n",[40,86603,86604],{"class":10276,"line":76812},[40,86605,10301],{"emptyLinePlaceholder":10300},[40,86607,86608,86610,86612,86615,86617,86620],{"class":10276,"line":76817},[40,86609,14975],{"class":10430},[40,86611,167],{"class":10363},[40,86613,86614],{"class":10430},"Markdown",[40,86616,167],{"class":10363},[40,86618,86619],{"class":10287},"\"##  Qubit Sampling Budget - Summary\"",[40,86621,13612],{"class":10363},[40,86623,86624,86626,86629,86632],{"class":10276,"line":76822},[40,86625,14975],{"class":10430},[40,86627,86628],{"class":10363},"(summary_df.style.",[40,86630,86631],{"class":10430},"format",[40,86633,10434],{"class":10363},[40,86635,86636,86639,86641,86643,86645,86647,86649,86651],{"class":10276,"line":76827},[40,86637,86638],{"class":10287},"    \"Configured p (target)\"",[40,86640,10443],{"class":10363},[40,86642,13478],{"class":10287},[40,86644,13268],{"class":10446},[40,86646,14595],{"class":10352},[40,86648,13280],{"class":10446},[40,86650,13478],{"class":10287},[40,86652,10507],{"class":10363},[40,86654,86655,86658,86660,86662,86664,86667,86669,86671],{"class":10276,"line":76837},[40,86656,86657],{"class":10287},"    \"Ones fraction (observed)\"",[40,86659,10443],{"class":10363},[40,86661,13478],{"class":10287},[40,86663,13268],{"class":10446},[40,86665,86666],{"class":10352},":.4f",[40,86668,13280],{"class":10446},[40,86670,13478],{"class":10287},[40,86672,10507],{"class":10363},[40,86674,86675,86678,86680,86682,86684,86686,86688,86690,86692,86694],{"class":10276,"line":76844},[40,86676,86677],{"class":10287},"    \"Wilson CI 95",[40,86679,85175],{"class":10446},[40,86681,85178],{"class":10287},[40,86683,10443],{"class":10363},[40,86685,13478],{"class":10287},[40,86687,13268],{"class":10446},[40,86689,86666],{"class":10352},[40,86691,13280],{"class":10446},[40,86693,13478],{"class":10287},[40,86695,10507],{"class":10363},[40,86697,86698,86700,86702,86704,86706,86708,86710,86712,86714,86716],{"class":10276,"line":76852},[40,86699,86677],{"class":10287},[40,86701,85188],{"class":10446},[40,86703,85191],{"class":10287},[40,86705,10443],{"class":10363},[40,86707,13478],{"class":10287},[40,86709,13268],{"class":10446},[40,86711,86666],{"class":10352},[40,86713,13280],{"class":10446},[40,86715,13478],{"class":10287},[40,86717,10507],{"class":10363},[40,86719,86720,86723,86725,86727,86729,86731,86733,86735],{"class":10276,"line":76863},[40,86721,86722],{"class":10287},"    \"CI width (95%)\"",[40,86724,10443],{"class":10363},[40,86726,13478],{"class":10287},[40,86728,13268],{"class":10446},[40,86730,86666],{"class":10352},[40,86732,13280],{"class":10446},[40,86734,13478],{"class":10287},[40,86736,10507],{"class":10363},[40,86738,86739,86742,86744,86746,86748,86750,86752],{"class":10276,"line":76874},[40,86740,86741],{"class":10287},"    \"Illustrative cost (USD)\"",[40,86743,10443],{"class":10363},[40,86745,77873],{"class":10287},[40,86747,13268],{"class":10446},[40,86749,14595],{"class":10352},[40,86751,13280],{"class":10446},[40,86753,14632],{"class":10287},[40,86755,86756],{"class":10276,"line":76883},[40,86757,86758],{"class":10363},"}))\n",[40,86760,86761],{"class":10276,"line":76889},[40,86762,10301],{"emptyLinePlaceholder":10300},[40,86764,86765],{"class":10276,"line":76899},[40,86766,10301],{"emptyLinePlaceholder":10300},[40,86768,86769],{"class":10276,"line":76909},[40,86770,86771],{"class":10280},"# ---------- Build a ready-to-run Colab notebook ----------\n",[40,86773,86774],{"class":10276,"line":76918},[40,86775,10301],{"emptyLinePlaceholder":10300},[40,86777,86778,86781,86783,86786],{"class":10276,"line":76927},[40,86779,86780],{"class":10363},"md_intro ",[40,86782,162],{"class":10732},[40,86784,86785],{"class":10352}," r",[40,86787,10288],{"class":10356},[40,86789,86790],{"class":10276,"line":76936},[40,86791,86792],{"class":10280},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[40,86794,86795],{"class":10276,"line":76948},[40,86796,10301],{"emptyLinePlaceholder":10300},[40,86798,86799,86801,86804,86806,86809,86811,86813,86815,86818,86820,86823,86825],{"class":10276,"line":76960},[40,86800,22571],{"class":10446},[40,86802,86803],{"class":10356},"High school level ",[40,86805,167],{"class":10446},[40,86807,86808],{"class":10356},"concise",[40,86810,207],{"class":10446},[40,86812,9910],{"class":10356},[40,86814,22571],{"class":10446},[40,86816,86817],{"class":10356}," We treat each measurement ",[40,86819,167],{"class":10446},[40,86821,86822],{"class":10356},"“single shot”",[40,86824,207],{"class":10446},[40,86826,86827],{"class":10356}," as a bit: `1` if the outcome matches the analytical prediction, `0` if it does not. Different qubit types have different chances of landing a `1`. We simulate many shots, show how often we get `1`s, and estimate how many shots we need for a target accuracy.\n",[40,86829,86830],{"class":10276,"line":76969},[40,86831,10301],{"emptyLinePlaceholder":10300},[40,86833,86834,86836,86839,86841,86843,86845,86847,86849,86852,86854,86857,86859,86862,86864,86867,86869,86872,86874,86877,86879,86882,86885,86888,86891,86893,86895,86898,86900,86902,86905,86908,86910],{"class":10276,"line":76974},[40,86835,22571],{"class":10446},[40,86837,86838],{"class":10356},"Graduate level ",[40,86840,167],{"class":10446},[40,86842,86808],{"class":10356},[40,86844,207],{"class":10446},[40,86846,9910],{"class":10356},[40,86848,22571],{"class":10446},[40,86850,86851],{"class":10356}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[40,86853,167],{"class":10446},[40,86855,86856],{"class":10356},"noisy intermediate-scale quantum",[40,86858,207],{"class":10446},[40,86860,86861],{"class":10356},", fault-tolerant ",[40,86863,167],{"class":10446},[40,86865,86866],{"class":10356},"logical",[40,86868,207],{"class":10446},[40,86870,86871],{"class":10356},", topologically protected, and ideal. We report Wilson score intervals ",[40,86873,167],{"class":10446},[40,86875,86876],{"class":10356},"95%",[40,86878,207],{"class":10446},[40,86880,86881],{"class":10356}," for observed proportions and estimate the required shot budget $n ",[40,86883,86884],{"class":10732},"\\a",[40,86886,86887],{"class":10356},"pprox z^2",[40,86889,86890],{"class":10732},"\\,",[40,86892,12],{"class":10356},[40,86894,167],{"class":10446},[40,86896,86897],{"class":10356},"1-p",[40,86899,207],{"class":10446},[40,86901,11243],{"class":10356},[40,86903,86904],{"class":10732},"\\e",[40,86906,86907],{"class":10356},"psilon^2$ to bound absolute error by ±$",[40,86909,86904],{"class":10732},[40,86911,86912],{"class":10356},"psilon$ at ~95% confidence.\n",[40,86914,86915],{"class":10276,"line":76979},[40,86916,10288],{"class":10356},[40,86918,86919],{"class":10276,"line":76984},[40,86920,10301],{"emptyLinePlaceholder":10300},[40,86922,86923,86926,86928,86930],{"class":10276,"line":76994},[40,86924,86925],{"class":10363},"code_config ",[40,86927,162],{"class":10732},[40,86929,86785],{"class":10352},[40,86931,10288],{"class":10356},[40,86933,86934],{"class":10276,"line":77001},[40,86935,86936],{"class":10280},"# @title Configuration\n",[40,86938,86939],{"class":10276,"line":77008},[40,86940,86941],{"class":10356},"from dataclasses import dataclass\n",[40,86943,86944],{"class":10276,"line":77019},[40,86945,86946],{"class":10356},"from typing import List, Dict\n",[40,86948,86949],{"class":10276,"line":77030},[40,86950,86951],{"class":10356},"import numpy as np\n",[40,86953,86954],{"class":10276,"line":77039},[40,86955,10301],{"emptyLinePlaceholder":10300},[40,86957,86958,86961],{"class":10276,"line":77045},[40,86959,86960],{"class":10356},"RANDOM_SEED = 579345  ",[40,86962,86963],{"class":10280},"# @param {type:\"number\"}\n",[40,86965,86966,86969],{"class":10276,"line":77055},[40,86967,86968],{"class":10356},"SHOTS_PER_MODEL = 10000  ",[40,86970,86963],{"class":10280},[40,86972,86973,86976,86979],{"class":10276,"line":77063},[40,86974,86975],{"class":10356},"EPSILONS = ",[40,86977,86978],{"class":10446},"[0.02, 0.01, 0.005]",[40,86980,86981],{"class":10280},"  # @param\n",[40,86983,86984,86987],{"class":10276,"line":77070},[40,86985,86986],{"class":10356},"COST_PER_SHOT_USD = 0.00005  ",[40,86988,86963],{"class":10280},[40,86990,86991,86994],{"class":10276,"line":77077},[40,86992,86993],{"class":10356},"FIXED_OVERHEAD_USD = 0.10    ",[40,86995,86963],{"class":10280},[40,86997,86998],{"class":10276,"line":77086},[40,86999,10301],{"emptyLinePlaceholder":10300},[40,87001,87002,87004,87006,87009],{"class":10276,"line":77095},[40,87003,84482],{"class":10356},[40,87005,167],{"class":10446},[40,87007,87008],{"class":10356},"frozen=True",[40,87010,10761],{"class":10446},[40,87012,87013],{"class":10276,"line":77104},[40,87014,87015],{"class":10356},"class QubitModel:\n",[40,87017,87018],{"class":10276,"line":77113},[40,87019,87020],{"class":10356},"    name: str\n",[40,87022,87023],{"class":10276,"line":77124},[40,87024,87025],{"class":10356},"    satisfaction_prob: float\n",[40,87027,87028],{"class":10276,"line":77136},[40,87029,87030],{"class":10356},"    description: str\n",[40,87032,87033],{"class":10276,"line":77148},[40,87034,10301],{"emptyLinePlaceholder":10300},[40,87036,87037,87040,87043,87046],{"class":10276,"line":77157},[40,87038,87039],{"class":10356},"MODELS: List",[40,87041,87042],{"class":10446},"[QubitModel]",[40,87044,87045],{"class":10356}," = ",[40,87047,87048],{"class":10446},"[\n",[40,87050,87051],{"class":10276,"line":77162},[40,87052,87053],{"class":10446},"    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[40,87055,87056],{"class":10276,"line":77167},[40,87057,87058],{"class":10446},"    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[40,87060,87061],{"class":10276,"line":77172},[40,87062,87063],{"class":10446},"    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[40,87065,87066],{"class":10276,"line":77182},[40,87067,87068],{"class":10446},"    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[40,87070,87071],{"class":10276,"line":77189},[40,87072,87073],{"class":10446},"    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[40,87075,87076],{"class":10276,"line":77196},[40,87077,25466],{"class":10446},[40,87079,87080],{"class":10276,"line":77208},[40,87081,10301],{"emptyLinePlaceholder":10300},[40,87083,87084,87087,87089,87092],{"class":10276,"line":77219},[40,87085,87086],{"class":10356},"rng = np.random.default_rng",[40,87088,167],{"class":10446},[40,87090,87091],{"class":10356},"RANDOM_SEED",[40,87093,10761],{"class":10446},[40,87095,87096],{"class":10276,"line":77228},[40,87097,10288],{"class":10356},[40,87099,87100],{"class":10276,"line":77234},[40,87101,10301],{"emptyLinePlaceholder":10300},[40,87103,87104,87107,87109,87111],{"class":10276,"line":77250},[40,87105,87106],{"class":10363},"code_lib_and_run ",[40,87108,162],{"class":10732},[40,87110,86785],{"class":10352},[40,87112,10288],{"class":10356},[40,87114,87115],{"class":10276,"line":77255},[40,87116,87117],{"class":10280},"# @title Run Simulation & Display Results\n",[40,87119,87120],{"class":10276,"line":77264},[40,87121,87122],{"class":10356},"import math\n",[40,87124,87125],{"class":10276,"line":77274},[40,87126,87127],{"class":10356},"from typing import Tuple, Dict\n",[40,87129,87130],{"class":10276,"line":77283},[40,87131,87132],{"class":10356},"import pandas as pd\n",[40,87134,87135],{"class":10276,"line":77292},[40,87136,86951],{"class":10356},[40,87138,87139],{"class":10276,"line":77298},[40,87140,87141],{"class":10356},"import matplotlib.pyplot as plt\n",[40,87143,87144],{"class":10276,"line":77307},[40,87145,87146],{"class":10356},"from IPython.display import display, Markdown\n",[40,87148,87149],{"class":10276,"line":77319},[40,87150,10301],{"emptyLinePlaceholder":10300},[40,87152,87153,87156,87158,87161,87163,87166,87169],{"class":10276,"line":77331},[40,87154,87155],{"class":10356},"def wilson_ci",[40,87157,167],{"class":10446},[40,87159,87160],{"class":10356},"k: int, n: int, z: float = 1.96",[40,87162,207],{"class":10446},[40,87164,87165],{"class":10356}," -> Tuple",[40,87167,87168],{"class":10446},"[float, float]",[40,87170,10639],{"class":10356},[40,87172,87173,87176,87178,87181],{"class":10276,"line":77343},[40,87174,87175],{"class":10356},"    if n == 0: return ",[40,87177,167],{"class":10446},[40,87179,87180],{"class":10356},"0.0, 1.0",[40,87182,10761],{"class":10446},[40,87184,87185,87188,87190],{"class":10276,"line":77348},[40,87186,87187],{"class":10356},"    p_hat, z2_n = k \u002F n, z",[40,87189,22571],{"class":10446},[40,87191,87192],{"class":10356},"2 \u002F n\n",[40,87194,87195,87198,87200],{"class":10276,"line":77353},[40,87196,87197],{"class":10356},"    denom = 1.0 ",[40,87199,1053],{"class":10446},[40,87201,87202],{"class":10356}," z2_n\n",[40,87204,87205,87208,87210,87213,87215,87218,87220],{"class":10276,"line":77358},[40,87206,87207],{"class":10356},"    center = ",[40,87209,167],{"class":10446},[40,87211,87212],{"class":10356},"p_hat ",[40,87214,1053],{"class":10446},[40,87216,87217],{"class":10356}," z2_n \u002F 2",[40,87219,207],{"class":10446},[40,87221,87222],{"class":10356}," \u002F denom\n",[40,87224,87225,87228,87230,87233,87235,87238,87240,87242,87244,87246,87249,87251,87254,87256,87259,87261,87264,87266,87269,87271,87274,87276,87278,87281],{"class":10276,"line":77376},[40,87226,87227],{"class":10356},"    span = ",[40,87229,167],{"class":10446},[40,87231,87232],{"class":10356},"z ",[40,87234,11752],{"class":10446},[40,87236,87237],{"class":10356}," math.sqrt",[40,87239,167],{"class":10446},[40,87241,87212],{"class":10356},[40,87243,11752],{"class":10446},[40,87245,920],{"class":10446},[40,87247,87248],{"class":10356},"1 - p_hat",[40,87250,207],{"class":10446},[40,87252,87253],{"class":10356}," \u002F n ",[40,87255,1053],{"class":10446},[40,87257,87258],{"class":10356}," z",[40,87260,22571],{"class":10446},[40,87262,87263],{"class":10356},"2 \u002F ",[40,87265,167],{"class":10446},[40,87267,87268],{"class":10356},"4 ",[40,87270,11752],{"class":10446},[40,87272,87273],{"class":10356}," n",[40,87275,22571],{"class":10446},[40,87277,1044],{"class":10356},[40,87279,87280],{"class":10446},")))",[40,87282,87222],{"class":10356},[40,87284,87285,87288,87290,87292,87294,87297,87299,87302,87304,87307,87309,87312],{"class":10276,"line":77382},[40,87286,87287],{"class":10356},"    return ",[40,87289,167],{"class":10446},[40,87291,3032],{"class":10356},[40,87293,167],{"class":10446},[40,87295,87296],{"class":10356},"0.0, center - span",[40,87298,207],{"class":10446},[40,87300,87301],{"class":10356},", min",[40,87303,167],{"class":10446},[40,87305,87306],{"class":10356},"1.0, center ",[40,87308,1053],{"class":10446},[40,87310,87311],{"class":10356}," span",[40,87313,13612],{"class":10446},[40,87315,87316],{"class":10276,"line":77391},[40,87317,10301],{"emptyLinePlaceholder":10300},[40,87319,87320,87323,87325,87328,87330],{"class":10276,"line":77398},[40,87321,87322],{"class":10356},"def shots_for_epsilon",[40,87324,167],{"class":10446},[40,87326,87327],{"class":10356},"p: float, epsilon: float = 0.01, z: float = 1.96",[40,87329,207],{"class":10446},[40,87331,87332],{"class":10356}," -> int:\n",[40,87334,87335,87338,87340,87342,87344,87346],{"class":10276,"line":77405},[40,87336,87337],{"class":10356},"    if p ",[40,87339,11752],{"class":10446},[40,87341,920],{"class":10446},[40,87343,86897],{"class":10356},[40,87345,207],{"class":10446},[40,87347,87348],{"class":10356}," == 0: return 1\n",[40,87350,87351,87354,87356,87359,87361,87364,87366,87368,87370,87373,87375,87377,87379,87381,87383,87385,87388,87390,87392,87394,87396],{"class":10276,"line":77413},[40,87352,87353],{"class":10356},"    return max",[40,87355,167],{"class":10446},[40,87357,87358],{"class":10356},"1, int",[40,87360,167],{"class":10446},[40,87362,87363],{"class":10356},"math.ceil",[40,87365,16469],{"class":10446},[40,87367,273],{"class":10356},[40,87369,22571],{"class":10446},[40,87371,87372],{"class":10356},"2 ",[40,87374,11752],{"class":10446},[40,87376,13938],{"class":10356},[40,87378,11752],{"class":10446},[40,87380,920],{"class":10446},[40,87382,86897],{"class":10356},[40,87384,2407],{"class":10446},[40,87386,87387],{"class":10356}," \u002F ",[40,87389,167],{"class":10446},[40,87391,84301],{"class":10356},[40,87393,22571],{"class":10446},[40,87395,1044],{"class":10356},[40,87397,87398],{"class":10446},"))))\n",[40,87400,87401],{"class":10276,"line":77424},[40,87402,10301],{"emptyLinePlaceholder":10300},[40,87404,87405,87408,87410,87413,87415],{"class":10276,"line":77435},[40,87406,87407],{"class":10356},"def summarize_model",[40,87409,167],{"class":10446},[40,87411,87412],{"class":10356},"model, bits: np.ndarray, epsilons",[40,87414,207],{"class":10446},[40,87416,87417],{"class":10356}," -> Dict:\n",[40,87419,87420,87423,87425,87427],{"class":10276,"line":77446},[40,87421,87422],{"class":10356},"    n = len",[40,87424,167],{"class":10446},[40,87426,84983],{"class":10356},[40,87428,10761],{"class":10446},[40,87430,87431,87434,87436,87438],{"class":10276,"line":77451},[40,87432,87433],{"class":10356},"    ones = sum",[40,87435,167],{"class":10446},[40,87437,84983],{"class":10356},[40,87439,10761],{"class":10446},[40,87441,87442,87445,87447,87450],{"class":10276,"line":77460},[40,87443,87444],{"class":10356},"    lo, hi = wilson_ci",[40,87446,167],{"class":10446},[40,87448,87449],{"class":10356},"ones, n",[40,87451,10761],{"class":10446},[40,87453,87454],{"class":10276,"line":77467},[40,87455,87456],{"class":10356},"    row = {\n",[40,87458,87459,87462,87464,87467,87469],{"class":10276,"line":77474},[40,87460,87461],{"class":10356},"        \"Model\": model.name, \"p ",[40,87463,167],{"class":10446},[40,87465,87466],{"class":10356},"target",[40,87468,207],{"class":10446},[40,87470,87471],{"class":10356},"\": model.satisfaction_prob, \"Shots\": n,\n",[40,87473,87474,87477,87479,87482,87484],{"class":10276,"line":77482},[40,87475,87476],{"class":10356},"        \"Ones\": ones, \"p ",[40,87478,167],{"class":10446},[40,87480,87481],{"class":10356},"observed",[40,87483,207],{"class":10446},[40,87485,87486],{"class":10356},"\": ones \u002F n if n > 0 else 0,\n",[40,87488,87489],{"class":10276,"line":77503},[40,87490,87491],{"class":10356},"        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[40,87493,87494],{"class":10276,"line":77514},[40,87495,11145],{"class":10356},[40,87497,87498],{"class":10276,"line":77525},[40,87499,87500],{"class":10356},"    for eps in epsilons:\n",[40,87502,87503,87506,87509,87512,87514,87517],{"class":10276,"line":77536},[40,87504,87505],{"class":10356},"        row",[40,87507,87508],{"class":10446},"[f\"Shots for ±{eps:.1%}\"]",[40,87510,87511],{"class":10356}," = shots_for_epsilon",[40,87513,167],{"class":10446},[40,87515,87516],{"class":10356},"model.satisfaction_prob, eps",[40,87518,10761],{"class":10446},[40,87520,87521],{"class":10276,"line":77547},[40,87522,87523],{"class":10356},"    return row\n",[40,87525,87526],{"class":10276,"line":77552},[40,87527,10301],{"emptyLinePlaceholder":10300},[40,87529,87530],{"class":10276,"line":77557},[40,87531,87532],{"class":10280},"# --- Run Simulation ---\n",[40,87534,87535,87538],{"class":10276,"line":77562},[40,87536,87537],{"class":10356},"summary_rows = ",[40,87539,87540],{"class":10446},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[40,87542,87543,87546,87548,87551],{"class":10276,"line":77576},[40,87544,87545],{"class":10356},"summary_df = pd.DataFrame",[40,87547,167],{"class":10446},[40,87549,87550],{"class":10356},"summary_rows",[40,87552,10761],{"class":10446},[40,87554,87555],{"class":10276,"line":77588},[40,87556,87557],{"class":10356},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[40,87559,87560,87563,87566,87569,87571,87574,87577,87579],{"class":10276,"line":77594},[40,87561,87562],{"class":10356},"    summary_df",[40,87564,87565],{"class":10446},"[\"Cost (USD)\"]",[40,87567,87568],{"class":10356}," = FIXED_OVERHEAD_USD ",[40,87570,1053],{"class":10446},[40,87572,87573],{"class":10356}," summary_df",[40,87575,87576],{"class":10446},"[\"Shots\"]",[40,87578,11881],{"class":10446},[40,87580,87581],{"class":10356}," COST_PER_SHOT_USD\n",[40,87583,87584],{"class":10276,"line":77599},[40,87585,10301],{"emptyLinePlaceholder":10300},[40,87587,87588,87590,87592,87594,87596,87598],{"class":10276,"line":77606},[40,87589,14975],{"class":10356},[40,87591,167],{"class":10446},[40,87593,86614],{"class":10356},[40,87595,167],{"class":10446},[40,87597,13478],{"class":10356},[40,87599,87600],{"class":10280},"### Simulation Summary\"))\n",[40,87602,87603,87605,87607,87610,87612,87615,87617,87620,87622,87625],{"class":10276,"line":77616},[40,87604,14975],{"class":10356},[40,87606,167],{"class":10446},[40,87608,87609],{"class":10356},"summary_df.style.format",[40,87611,167],{"class":10446},[40,87613,87614],{"class":10356},"precision=4",[40,87616,207],{"class":10446},[40,87618,87619],{"class":10356},".background_gradient",[40,87621,167],{"class":10446},[40,87623,87624],{"class":10356},"cmap='viridis', subset=",[40,87626,87627],{"class":10446},"['p (observed)', 'CI width']))\n",[40,87629,87630],{"class":10276,"line":77621},[40,87631,10301],{"emptyLinePlaceholder":10300},[40,87633,87634],{"class":10276,"line":77633},[40,87635,87636],{"class":10280},"# --- Plotting ---\n",[40,87638,87639,87642,87644,87646],{"class":10276,"line":77644},[40,87640,87641],{"class":10356},"plt.style.use",[40,87643,167],{"class":10446},[40,87645,85855],{"class":10356},[40,87647,10761],{"class":10446},[40,87649,87650,87653,87655,87658,87660,87663,87665,87668,87670,87673],{"class":10276,"line":77654},[40,87651,87652],{"class":10356},"fig, ",[40,87654,167],{"class":10446},[40,87656,87657],{"class":10356},"ax1, ax2",[40,87659,207],{"class":10446},[40,87661,87662],{"class":10356}," = plt.subplots",[40,87664,167],{"class":10446},[40,87666,87667],{"class":10356},"1, 2, figsize=",[40,87669,167],{"class":10446},[40,87671,87672],{"class":10356},"16, 6",[40,87674,13612],{"class":10446},[40,87676,87677],{"class":10276,"line":77665},[40,87678,10301],{"emptyLinePlaceholder":10300},[40,87680,87681],{"class":10276,"line":77670},[40,87682,87683],{"class":10280},"# Plot 1\n",[40,87685,87686,87689,87692,87695],{"class":10276,"line":77675},[40,87687,87688],{"class":10356},"y = summary_df",[40,87690,87691],{"class":10446},"[\"p (observed)\"]",[40,87693,87694],{"class":10356},".to_numpy",[40,87696,12174],{"class":10446},[40,87698,87699,87702],{"class":10276,"line":77687},[40,87700,87701],{"class":10356},"lower_error = y - summary_df",[40,87703,87704],{"class":10446},"[\"CI 95% low\"]\n",[40,87706,87707,87710,87713],{"class":10276,"line":77698},[40,87708,87709],{"class":10356},"upper_error = summary_df",[40,87711,87712],{"class":10446},"[\"CI 95% high\"]",[40,87714,87715],{"class":10356}," - y\n",[40,87717,87718,87721],{"class":10276,"line":77707},[40,87719,87720],{"class":10356},"yerr = np.vstack",[40,87722,87723],{"class":10446},"([np.abs(lower_error), np.abs(upper_error)])\n",[40,87725,87726],{"class":10276,"line":77718},[40,87727,10301],{"emptyLinePlaceholder":10300},[40,87729,87730,87733,87735,87738,87741,87744],{"class":10276,"line":77723},[40,87731,87732],{"class":10356},"ax1.bar",[40,87734,167],{"class":10446},[40,87736,87737],{"class":10356},"summary_df",[40,87739,87740],{"class":10446},"[\"Model\"]",[40,87742,87743],{"class":10356},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[40,87745,10761],{"class":10446},[40,87747,87748,87751,87753,87756,87758,87761,87763,87766],{"class":10276,"line":77728},[40,87749,87750],{"class":10356},"ax1.set_title",[40,87752,167],{"class":10446},[40,87754,87755],{"class":10356},"\"Observed Success Rate ",[40,87757,167],{"class":10446},[40,87759,87760],{"class":10356},"95% Wilson CI",[40,87762,207],{"class":10446},[40,87764,87765],{"class":10356},"\", fontsize=14",[40,87767,10761],{"class":10446},[40,87769,87770,87773,87775,87778],{"class":10276,"line":77738},[40,87771,87772],{"class":10356},"ax1.tick_params",[40,87774,167],{"class":10446},[40,87776,87777],{"class":10356},"axis='x', rotation=30, labelsize=10",[40,87779,10761],{"class":10446},[40,87781,87782],{"class":10276,"line":77744},[40,87783,10301],{"emptyLinePlaceholder":10300},[40,87785,87786],{"class":10276,"line":77758},[40,87787,87788],{"class":10280},"# Plot 2\n",[40,87790,87791],{"class":10276,"line":77763},[40,87792,87793],{"class":10356},"width = 0.2\n",[40,87795,87796,87799,87801,87803,87805,87808],{"class":10276,"line":77773},[40,87797,87798],{"class":10356},"x = np.arange",[40,87800,167],{"class":10446},[40,87802,25507],{"class":10356},[40,87804,167],{"class":10446},[40,87806,87807],{"class":10356},"MODELS",[40,87809,13612],{"class":10446},[40,87811,87812,87815],{"class":10276,"line":77778},[40,87813,87814],{"class":10356},"colors = ",[40,87816,87817],{"class":10446},"['#ff9999','#66b3ff','#99ff99']\n",[40,87819,87820,87823,87825,87828,87830],{"class":10276,"line":77790},[40,87821,87822],{"class":10356},"for i, eps in enumerate",[40,87824,167],{"class":10446},[40,87826,87827],{"class":10356},"EPSILONS",[40,87829,207],{"class":10446},[40,87831,10639],{"class":10356},[40,87833,87834,87837],{"class":10276,"line":77795},[40,87835,87836],{"class":10356},"    req = summary_df",[40,87838,87839],{"class":10446},"[f\"Shots for ±{eps:.1%}\"]\n",[40,87841,87842,87845,87847,87849,87851,87853,87855,87858,87861,87864],{"class":10276,"line":77800},[40,87843,87844],{"class":10356},"    ax2.bar",[40,87846,167],{"class":10446},[40,87848,25495],{"class":10356},[40,87850,1053],{"class":10446},[40,87852,79692],{"class":10356},[40,87854,11752],{"class":10446},[40,87856,87857],{"class":10356}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[40,87859,87860],{"class":10446},"[i]",[40,87862,87863],{"class":10356},", edgecolor='black'",[40,87865,10761],{"class":10446},[40,87867,87868],{"class":10276,"line":77822},[40,87869,10301],{"emptyLinePlaceholder":10300},[40,87871,87872,87875,87877,87879,87881,87884],{"class":10276,"line":77827},[40,87873,87874],{"class":10356},"ax2.set_xticks",[40,87876,167],{"class":10446},[40,87878,25495],{"class":10356},[40,87880,1053],{"class":10446},[40,87882,87883],{"class":10356}," width, summary_df",[40,87885,87886],{"class":10446},"[\"Model\"])\n",[40,87888,87889,87892,87894,87897,87899,87902,87904,87906],{"class":10276,"line":77850},[40,87890,87891],{"class":10356},"ax2.set_ylabel",[40,87893,167],{"class":10446},[40,87895,87896],{"class":10356},"\"Required Shots ",[40,87898,167],{"class":10446},[40,87900,87901],{"class":10356},"Log Scale",[40,87903,207],{"class":10446},[40,87905,13478],{"class":10356},[40,87907,10761],{"class":10446},[40,87909,87910,87913,87915,87917],{"class":10276,"line":77861},[40,87911,87912],{"class":10356},"ax2.set_yscale",[40,87914,167],{"class":10446},[40,87916,86507],{"class":10356},[40,87918,10761],{"class":10446},[40,87920,87921,87924,87926,87929],{"class":10276,"line":77900},[40,87922,87923],{"class":10356},"ax2.set_title",[40,87925,167],{"class":10446},[40,87927,87928],{"class":10356},"\"Shots Needed for ±ε Tolerance\", fontsize=14",[40,87930,10761],{"class":10446},[40,87932,87933,87936,87938,87940],{"class":10276,"line":77910},[40,87934,87935],{"class":10356},"ax2.tick_params",[40,87937,167],{"class":10446},[40,87939,87777],{"class":10356},[40,87941,10761],{"class":10446},[40,87943,87944,87947,87949,87952],{"class":10276,"line":77920},[40,87945,87946],{"class":10356},"ax2.legend",[40,87948,167],{"class":10446},[40,87950,87951],{"class":10356},"title=\"Tolerance\"",[40,87953,10761],{"class":10446},[40,87955,87956],{"class":10276,"line":77939},[40,87957,10301],{"emptyLinePlaceholder":10300},[40,87959,87960,87963],{"class":10276,"line":77957},[40,87961,87962],{"class":10356},"plt.tight_layout",[40,87964,12174],{"class":10446},[40,87966,87967,87970],{"class":10276,"line":77962},[40,87968,87969],{"class":10356},"plt.show",[40,87971,12174],{"class":10446},[40,87973,87974],{"class":10276,"line":77967},[40,87975,10288],{"class":10356},[40,87977,87978],{"class":10276,"line":77979},[40,87979,10301],{"emptyLinePlaceholder":10300},[40,87981,87982,87985,87987,87990,87993],{"class":10276,"line":77989},[40,87983,87984],{"class":10363},"nb ",[40,87986,162],{"class":10732},[40,87988,87989],{"class":10363}," nbf.v4.",[40,87991,87992],{"class":10430},"new_notebook",[40,87994,12174],{"class":10363},[40,87996,87997,88000,88003,88005,88007],{"class":10276,"line":77999},[40,87998,87999],{"class":10363},"nb[",[40,88001,88002],{"class":10287},"\"cells\"",[40,88004,10741],{"class":10363},[40,88006,162],{"class":10732},[40,88008,37422],{"class":10363},[40,88010,88011,88014,88017],{"class":10276,"line":78004},[40,88012,88013],{"class":10363},"    nbf.v4.",[40,88015,88016],{"class":10430},"new_markdown_cell",[40,88018,88019],{"class":10363},"(md_intro),\n",[40,88021,88022,88024,88027],{"class":10276,"line":78009},[40,88023,88013],{"class":10363},[40,88025,88026],{"class":10430},"new_code_cell",[40,88028,88029],{"class":10363},"(code_config),\n",[40,88031,88032,88034,88036],{"class":10276,"line":78017},[40,88033,88013],{"class":10363},[40,88035,88026],{"class":10430},[40,88037,88038],{"class":10363},"(code_lib_and_run),\n",[40,88040,88041],{"class":10276,"line":78027},[40,88042,25466],{"class":10363},[40,88044,88045],{"class":10276,"line":78042},[40,88046,10301],{"emptyLinePlaceholder":10300},[40,88048,88049,88052,88054],{"class":10276,"line":78053},[40,88050,88051],{"class":10363},"nb_path ",[40,88053,162],{"class":10732},[40,88055,88056],{"class":10287}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[40,88058,88059,88061,88063,88066,88068,88070,88073,88075,88078,88080,88082],{"class":10276,"line":78065},[40,88060,85765],{"class":10352},[40,88062,85768],{"class":10732},[40,88064,88065],{"class":10363},"(nb_path, ",[40,88067,85778],{"class":10287},[40,88069,10484],{"class":10363},[40,88071,88072],{"class":11903},"encoding",[40,88074,162],{"class":10732},[40,88076,88077],{"class":10287},"\"utf-8\"",[40,88079,12055],{"class":10363},[40,88081,10381],{"class":10352},[40,88083,85785],{"class":10363},[40,88085,88086,88089,88092],{"class":10276,"line":78070},[40,88087,88088],{"class":10363},"    nbf.",[40,88090,88091],{"class":10430},"write",[40,88093,88094],{"class":10363},"(nb, f)\n",[40,88096,88097],{"class":10276,"line":78077},[40,88098,10301],{"emptyLinePlaceholder":10300},[40,88100,88101],{"class":10276,"line":78086},[40,88102,88103],{"class":10280},"# ---------- Final confirmation ----------\n",[40,88105,88106,88108,88110,88112,88114,88116,88118,88121,88123,88125],{"class":10276,"line":78100},[40,88107,14993],{"class":10732},[40,88109,167],{"class":10363},[40,88111,13478],{"class":10287},[40,88113,13454],{"class":10732},[40,88115,13478],{"class":10287},[40,88117,12042],{"class":10732},[40,88119,88120],{"class":10287}," \"=\"",[40,88122,11752],{"class":10732},[40,88124,65267],{"class":10446},[40,88126,10761],{"class":10363},[40,88128,88129,88131,88133,88136],{"class":10276,"line":78111},[40,88130,14993],{"class":10732},[40,88132,167],{"class":10363},[40,88134,88135],{"class":10287},"\"SCRIPT COMPLETE\"",[40,88137,10761],{"class":10363},[40,88139,88140,88142,88144,88147,88149,88151],{"class":10276,"line":78123},[40,88141,14993],{"class":10732},[40,88143,167],{"class":10363},[40,88145,88146],{"class":10287},"\"=\"",[40,88148,11752],{"class":10732},[40,88150,65267],{"class":10446},[40,88152,10761],{"class":10363},[40,88154,88155,88157,88159,88161,88164,88166,88169,88171,88173],{"class":10276,"line":78128},[40,88156,14993],{"class":10732},[40,88158,167],{"class":10363},[40,88160,13262],{"class":10352},[40,88162,88163],{"class":10287},"\"Google Colab notebook saved to: ",[40,88165,13268],{"class":10446},[40,88167,88168],{"class":10363},"nb_path",[40,88170,13280],{"class":10446},[40,88172,13478],{"class":10287},[40,88174,10761],{"class":10363},[40,88176,88177,88179,88181,88183,88186,88188,88191,88193,88195],{"class":10276,"line":78133},[40,88178,14993],{"class":10732},[40,88180,167],{"class":10363},[40,88182,13262],{"class":10352},[40,88184,88185],{"class":10287},"\"Summary data saved to: ",[40,88187,13268],{"class":10446},[40,88189,88190],{"class":10363},"summary_csv_path",[40,88192,13280],{"class":10446},[40,88194,13478],{"class":10287},[40,88196,10761],{"class":10363},[40,88198,88199,88201,88203,88205,88208,88210,88213,88215,88217,88219,88222,88224,88226],{"class":10276,"line":78143},[40,88200,14993],{"class":10732},[40,88202,167],{"class":10363},[40,88204,13262],{"class":10352},[40,88206,88207],{"class":10287},"\"Plots saved to: ",[40,88209,13268],{"class":10446},[40,88211,88212],{"class":10363},"fig1_path",[40,88214,13280],{"class":10446},[40,88216,10484],{"class":10287},[40,88218,13268],{"class":10446},[40,88220,88221],{"class":10363},"fig2_path",[40,88223,13280],{"class":10446},[40,88225,13478],{"class":10287},[40,88227,10761],{"class":10363},[10082,88229,88232],{"className":88230,"code":88231,"language":2932,"meta":10090},[10085],"Running simulation...\nSaved data to CSV and JSON files.\n",[10088,88233,88231],{"__ignoreMap":10090},[13997,88235],{"alt":74651,"src":88236},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[13997,88238],{"alt":79479,"src":88239},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[10082,88241,88244],{"className":88242,"code":88243,"language":2932,"meta":10090},[10085],"Generated and saved plots.\n",[10088,88245,88243],{"__ignoreMap":10090},[10082,88247,88250],{"className":88248,"code":88249,"language":2932,"meta":10090},[10085],"==================================================\nSCRIPT COMPLETE\n==================================================\nGoogle Colab notebook saved to: Sampling_Budget_Qubit_Bitstrings.ipynb\nSummary data saved to: summary.csv\nPlots saved to: fig_ones_fraction.png, fig_required_shots.png\n",[10088,88251,88249],{"__ignoreMap":10090},[25,88253],{},[28,88255,21679],{"id":21678},[88257,88258,88259,88266,88273,88280,88287,88294,88301,88308,88315,88322,88329,88336,88343,88354,88361,88368],"ol",{},[9746,88260,88261,88262,88265],{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[15,88263,88264],{},"The American Statistician",", 52(2), pp. 119-126.",[9746,88267,88268,88269,88272],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[15,88270,88271],{},"Statistical Science",", 16(2), pp. 101-133.",[9746,88274,88275,88276,88279],{},"Cochran, W.G. (1977) ",[15,88277,88278],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[9746,88281,88282,88283,88286],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[15,88284,88285],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[9746,88288,88289,88290,88293],{},"Harris, C.R., Millman, K.J., van der Walt, S.J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N.J., Kern, R., Picus, M., Hoyer, S., van Kerkwijk, M.H., Brett, M., Haldane, A., del Río, J.F., Wiebe, M., Peterson, P., Gérard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C. and Oliphant, T.E. (2020) 'Array programming with NumPy', ",[15,88291,88292],{},"Nature",", 585, pp. 357–362.",[9746,88295,88296,88297,88300],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[15,88298,88299],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[9746,88302,88303,88304,88307],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[15,88305,88306],{},"Annals of Physics",", 303(1), pp. 2-30.",[9746,88309,88310,88311,88314],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[15,88312,88313],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[9746,88316,88317,88318,88321],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[15,88319,88320],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[9746,88323,88324,88325,88328],{},"Newman, M. (2013) ",[15,88326,88327],{},"Computational Physics",". CreateSpace Independent Publishing Platform.",[9746,88330,88331,88332,88335],{},"Nielsen, M.A. and Chuang, I.L. (2010) ",[15,88333,88334],{},"Quantum Computation and Quantum Information: 10th Anniversary Edition",". Cambridge: Cambridge University Press.",[9746,88337,88338,88339,88342],{},"Preskill, J. (2018) 'Quantum Computing in the NISQ era and beyond', ",[15,88340,88341],{},"Quantum",", 2, p. 79.",[9746,88344,88345,88346,88349,88350,1123],{},"The pandas development team (2020) ",[15,88347,88348],{},"pandas-dev\u002Fpandas: Pandas",". Zenodo. Available at: ",[19,88351,88353],{"href":88352},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",[9746,88355,88356,88357,88360],{},"Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Millman, K.J., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C.J., Polat, İ., Feng, Y., Moore, E.W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P. and SciPy 1.0 Contributors (2020) 'SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python', ",[15,88358,88359],{},"Nature Methods",", 17, pp. 261–272.",[9746,88362,88363,88364,88367],{},"Wilde, M.M. (2017) ",[15,88365,88366],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[9746,88369,88370,88371,88374],{},"Wilson, E.B. (1927) 'Probable inference, the law of succession, and statistical inference', ",[15,88372,88373],{},"Journal of the American Statistical Association",", 22(158), pp. 209-212.",[28,88376,83769],{"id":83768},[12,88378,83772,88379,83775,88381,1123],{},[19,88380,22],{"href":83831},[19,88382,83779],{"href":88383},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[20069,88385,88386],{},"html pre.shiki code .sV9Aq, html code.shiki .sV9Aq{--shiki-default:#7F848E;--shiki-default-font-style:italic}html pre.shiki code .seHd6, html code.shiki .seHd6{--shiki-default:#C678DD}html pre.shiki code .sn6KH, html code.shiki .sn6KH{--shiki-default:#ABB2BF}html pre.shiki code .subq3, html code.shiki .subq3{--shiki-default:#98C379}html pre.shiki code .sjrmR, html code.shiki .sjrmR{--shiki-default:#56B6C2}html pre.shiki code .sVC51, html code.shiki .sVC51{--shiki-default:#D19A66}html pre.shiki code .sVbv2, html code.shiki .sVbv2{--shiki-default:#61AFEF}html pre.shiki code .sb9H8, html code.shiki .sb9H8{--shiki-default:#D19A66;--shiki-default-font-style:italic}html pre.shiki code .s_ZVi, html code.shiki .s_ZVi{--shiki-default:#E06C75;--shiki-default-font-style:italic}html pre.shiki code .sU0A5, html code.shiki .sU0A5{--shiki-default:#E5C07B}html pre.shiki code .sVyAn, html code.shiki .sVyAn{--shiki-default:#E06C75}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}",{"title":10090,"searchDepth":10284,"depth":10284,"links":88388},[88389,88390],{"id":21678,"depth":10284,"text":21679},{"id":83768,"depth":10284,"text":83769},[20088,20089,20090,88392],"Sampling budget",[88394],{"username":7,"name":20094,"role":20095,"bio":20096,"links":88395},[88396,88397],{"label":20099,"href":20100},{"label":20102,"href":83831},{"username":7,"name":20094,"role":20104},"A Sampling-Budget & Bit-String Analogy for Qubit Classifications. A notebook by Onri Jay Benally, republished with permission.",{},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications","4 min read",[],{"title":83821,"description":88399},"blog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications",[27288,20121],"VYECc6Ag2BqxrKJqgbl-HK6Xs12_7admknxECKyri-c",[],1787180369027]