[{"data":1,"prerenderedAt":81427},["ShallowReactive",2],{"blog-post-expert-notes\u002Fpulse-shapes-and-envelopes":3,"sibling-dives-expert-notes\u002Fpulse-shapes-and-envelopes":12409,"learn-track-expert-notes\u002Fpulse-shapes-and-envelopes":81426},{"id":4,"title":5,"authors":6,"body":8,"breadcrumb":12373,"builders":12378,"byline":12389,"challenge":12391,"courseAuthor":12391,"courseLead":12391,"dek":12391,"description":12392,"draft":12393,"extension":12394,"eyebrow":12395,"featured":12393,"finish":12391,"fork":12391,"hero":12391,"heroAlt":12391,"heroCta":12391,"heroImage":12391,"homepageFeatured":12393,"kind":12396,"lessonCount":12391,"meta":12397,"navigation":111,"newsItems":12391,"next":12391,"ogImage":12398,"order":12391,"outcomes":12391,"path":12399,"publishDate":12400,"readingTime":12401,"related":12402,"relatedProjects":12391,"seo":12403,"stem":12404,"tags":12405,"track":12391,"trackName":12391,"__hash__":12408},"blog\u002Fblog\u002Fexpert-notes\u002Fpulse-shapes-and-envelopes.md","Pulse Shapes and Envelopes, Visualized",[7],"onri-jay-benally",{"type":9,"value":10,"toc":12367},"minimark",[11,24,27,40,64,390,395,666,670,1004,1008,1363,1367,1606,1610,1920,1924,2277,2281,2664,2668,2671,3297,3301,3853,3857,4484,4488,5121,5125,5128,5135,5142,5574,5784,5790,6350,6551,6557,7149,7381,7387,7620,7623,7628,10208,10802,10806,10810,11268,11272,11645,11649,12359,12363],[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\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FPulse_Shapes_and_Envelopes_Visualization.ipynb","the original notebook",". The text, code, and figures below are Onri's, as published.",[12,25,26],{},"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.",[28,29,30,34,37],"ul",{},[31,32,33],"li",{},"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.",[31,35,36],{},"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.",[31,38,39],{},"Optimal control techniques, such as GRAPE or CRAB, can generate tailored pulse shapes that might include sinusoidal modulations.",[41,42,43],"blockquote",{},[28,44,45,52,58],{},[31,46,47,51],{},[48,49,50],"strong",{},"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.",[31,53,54,57],{},[48,55,56],{},"Chopped Random Basis (CRAB)",": A method for quantum optimal control that uses a randomly truncated basis to optimize control pulses.",[31,59,60,63],{},[48,61,62],{},"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.",[65,66,71],"pre",{"className":67,"code":68,"language":69,"meta":70,"style":70},"language-python shiki shiki-themes one-dark-pro","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","python","",[72,73,74,93,106,113,120,126,147,152,158,169,180,220,225,231,284,289,295,319,334,349,364,379],"code",{"__ignoreMap":70},[75,76,79,83,87,90],"span",{"class":77,"line":78},"line",1,[75,80,82],{"class":81},"seHd6","import",[75,84,86],{"class":85},"sn6KH"," numpy ",[75,88,89],{"class":81},"as",[75,91,92],{"class":85}," np\n",[75,94,96,98,101,103],{"class":77,"line":95},2,[75,97,82],{"class":81},[75,99,100],{"class":85}," matplotlib.pyplot ",[75,102,89],{"class":81},[75,104,105],{"class":85}," plt\n",[75,107,109],{"class":77,"line":108},3,[75,110,112],{"emptyLinePlaceholder":111},true,"\n",[75,114,116],{"class":77,"line":115},4,[75,117,119],{"class":118},"sV9Aq","# Onri prefers using rcParams to increase the quality of all plots to be higher than the default settings.\n",[75,121,123],{"class":77,"line":122},5,[75,124,125],{"class":118},"# This only needs to be done once at beginning of every Jupyter Notebook.\n",[75,127,129,132,136,139,143],{"class":77,"line":128},6,[75,130,131],{"class":85},"plt.rcParams[",[75,133,135],{"class":134},"subq3","'figure.dpi'",[75,137,138],{"class":85},"] ",[75,140,142],{"class":141},"sjrmR","=",[75,144,146],{"class":145},"sVC51"," 200\n",[75,148,150],{"class":77,"line":149},7,[75,151,112],{"emptyLinePlaceholder":111},[75,153,155],{"class":77,"line":154},8,[75,156,157],{"class":118},"# Define the parameters for the Gaussian modulation function\n",[75,159,161,164,166],{"class":77,"line":160},9,[75,162,163],{"class":85},"mean ",[75,165,142],{"class":141},[75,167,168],{"class":145}," 0\n",[75,170,172,175,177],{"class":77,"line":171},10,[75,173,174],{"class":85},"std_dev ",[75,176,142],{"class":141},[75,178,179],{"class":145}," 1\n",[75,181,183,186,188,191,195,198,201,204,207,209,211,214,217],{"class":77,"line":182},11,[75,184,185],{"class":85},"time ",[75,187,142],{"class":141},[75,189,190],{"class":85}," np.",[75,192,194],{"class":193},"sVbv2","linspace",[75,196,197],{"class":85},"(",[75,199,200],{"class":141},"-",[75,202,203],{"class":145},"5",[75,205,206],{"class":85},", ",[75,208,203],{"class":145},[75,210,206],{"class":85},[75,212,213],{"class":145},"1000",[75,215,216],{"class":85},")  ",[75,218,219],{"class":118},"# Generate time values from -5 to 5\n",[75,221,223],{"class":77,"line":222},12,[75,224,112],{"emptyLinePlaceholder":111},[75,226,228],{"class":77,"line":227},13,[75,229,230],{"class":118},"# Calculate the Gaussian modulation 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color\n",[75,296,298,301,304,307,311,313,316],{"class":77,"line":297},17,[75,299,300],{"class":85},"plt.",[75,302,303],{"class":193},"plot",[75,305,306],{"class":85},"(time, gaussian_modulation, ",[75,308,310],{"class":309},"s_ZVi","color",[75,312,142],{"class":141},[75,314,315],{"class":134},"'blue'",[75,317,318],{"class":85},")\n",[75,320,322,324,327,329,332],{"class":77,"line":321},18,[75,323,300],{"class":85},[75,325,326],{"class":193},"title",[75,328,197],{"class":85},[75,330,331],{"class":134},"'Gaussian Modulation 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notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-01.webp",[65,396,398],{"className":67,"code":397,"language":69,"meta":70,"style":70},"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",[72,399,400,410,420,424,428,437,445,453,481,485,489,535,539,544,581,593,605,617,641,646,658],{"__ignoreMap":70},[75,401,402,404,406,408],{"class":77,"line":78},[75,403,82],{"class":81},[75,405,86],{"class":85},[75,407,89],{"class":81},[75,409,92],{"class":85},[75,411,412,414,416,418],{"class":77,"line":95},[75,413,82],{"class":81},[75,415,100],{"class":85},[75,417,89],{"class":81},[75,419,105],{"class":85},[75,421,422],{"class":77,"line":108},[75,423,112],{"emptyLinePlaceholder":111},[75,425,426],{"class":77,"line":115},[75,427,157],{"class":118},[75,429,430,433,435],{"class":77,"line":122},[75,431,432],{"class":85},"amplitude ",[75,434,142],{"class":141},[75,436,179],{"class":145},[75,438,439,441,443],{"class":77,"line":128},[75,440,163],{"class":85},[75,442,142],{"class":141},[75,444,168],{"class":145},[75,446,447,449,451],{"class":77,"line":149},[75,448,174],{"class":85},[75,450,142],{"class":141},[75,452,179],{"class":145},[75,454,455,457,459,461,463,465,467,469,471,473,475,477,479],{"class":77,"line":154},[75,456,185],{"class":85},[75,458,142],{"class":141},[75,460,190],{"class":85},[75,462,194],{"class":193},[75,464,197],{"class":85},[75,466,200],{"class":141},[75,468,203],{"class":145},[75,470,206],{"class":85},[75,472,203],{"class":145},[75,474,206],{"class":85},[75,476,213],{"class":145},[75,478,216],{"class":85},[75,480,219],{"class":118},[75,482,483],{"class":77,"line":160},[75,484,112],{"emptyLinePlaceholder":111},[75,486,487],{"class":77,"line":171},[75,488,230],{"class":118},[75,490,491,493,495,498,501,503,505,507,509,511,513,515,517,519,521,523,525,527,529,531,533],{"class":77,"line":182},[75,492,236],{"class":85},[75,494,142],{"class":141},[75,496,497],{"class":85}," amplitude ",[75,499,500],{"class":141},"*",[75,502,190],{"class":85},[75,504,243],{"class":193},[75,506,197],{"class":85},[75,508,200],{"class":141},[75,510,250],{"class":85},[75,512,200],{"class":141},[75,514,255],{"class":85},[75,516,258],{"class":141},[75,518,261],{"class":145},[75,520,264],{"class":141},[75,522,267],{"class":85},[75,524,270],{"class":145},[75,526,273],{"class":141},[75,528,276],{"class":85},[75,530,258],{"class":141},[75,532,261],{"class":145},[75,534,283],{"class":85},[75,536,537],{"class":77,"line":222},[75,538,112],{"emptyLinePlaceholder":111},[75,540,541],{"class":77,"line":227},[75,542,543],{"class":118},"# Plot the Gaussian modulation function\n",[75,545,546,548,550,552,554,556,559,561,564,566,569,571,574,576,579],{"class":77,"line":233},[75,547,300],{"class":85},[75,549,303],{"class":193},[75,551,306],{"class":85},[75,553,310],{"class":309},[75,555,142],{"class":141},[75,557,558],{"class":134},"'red'",[75,560,206],{"class":85},[75,562,563],{"class":309},"linestyle",[75,565,142],{"class":141},[75,567,568],{"class":134},"'dashed'",[75,570,206],{"class":85},[75,572,573],{"class":309},"label",[75,575,142],{"class":141},[75,577,578],{"class":134},"'Gaussian Modulation'",[75,580,318],{"class":85},[75,582,583,585,587,589,591],{"class":77,"line":286},[75,584,300],{"class":85},[75,586,326],{"class":193},[75,588,197],{"class":85},[75,590,331],{"class":134},[75,592,318],{"class":85},[75,594,595,597,599,601,603],{"class":77,"line":291},[75,596,300],{"class":85},[75,598,341],{"class":193},[75,600,197],{"class":85},[75,602,346],{"class":134},[75,604,318],{"class":85},[75,606,607,609,611,613,615],{"class":77,"line":297},[75,608,300],{"class":85},[75,610,356],{"class":193},[75,612,197],{"class":85},[75,614,361],{"class":134},[75,616,318],{"class":85},[75,618,619,621,624,626,628,631,633,636,638],{"class":77,"line":321},[75,620,300],{"class":85},[75,622,623],{"class":193},"ylim",[75,625,197],{"class":85},[75,627,200],{"class":141},[75,629,630],{"class":145},"1.0",[75,632,206],{"class":85},[75,634,635],{"class":145},"1.1",[75,637,216],{"class":85},[75,639,640],{"class":118},"# Force the y-axis to display the range from -1.0 to 1.0\n",[75,642,643],{"class":77,"line":336},[75,644,645],{"class":118},"# plt.legend()\n",[75,647,648,650,652,654,656],{"class":77,"line":351},[75,649,300],{"class":85},[75,651,371],{"class":193},[75,653,197],{"class":85},[75,655,376],{"class":145},[75,657,318],{"class":85},[75,659,660,662,664],{"class":77,"line":366},[75,661,300],{"class":85},[75,663,386],{"class":193},[75,665,389],{"class":85},[391,667],{"alt":668,"src":669},"Output 2 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-02.webp",[65,671,673],{"className":67,"code":672,"language":69,"meta":70,"style":70},"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",[72,674,675,685,695,699,704,712,720,728,758,762,767,797,801,806,850,854,859,885,924,937,949,961,982,995],{"__ignoreMap":70},[75,676,677,679,681,683],{"class":77,"line":78},[75,678,82],{"class":81},[75,680,86],{"class":85},[75,682,89],{"class":81},[75,684,92],{"class":85},[75,686,687,689,691,693],{"class":77,"line":95},[75,688,82],{"class":81},[75,690,100],{"class":85},[75,692,89],{"class":81},[75,694,105],{"class":85},[75,696,697],{"class":77,"line":108},[75,698,112],{"emptyLinePlaceholder":111},[75,700,701],{"class":77,"line":115},[75,702,703],{"class":118},"# Define the parameters for the Gaussian 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function\n",[75,807,808,810,812,814,816,818,820,822,824,826,828,830,832,834,836,838,840,842,844,846,848],{"class":77,"line":233},[75,809,236],{"class":85},[75,811,142],{"class":141},[75,813,497],{"class":85},[75,815,500],{"class":141},[75,817,190],{"class":85},[75,819,243],{"class":193},[75,821,197],{"class":85},[75,823,200],{"class":141},[75,825,250],{"class":85},[75,827,200],{"class":141},[75,829,255],{"class":85},[75,831,258],{"class":141},[75,833,261],{"class":145},[75,835,264],{"class":141},[75,837,267],{"class":85},[75,839,270],{"class":145},[75,841,273],{"class":141},[75,843,276],{"class":85},[75,845,258],{"class":141},[75,847,261],{"class":145},[75,849,283],{"class":85},[75,851,852],{"class":77,"line":286},[75,853,112],{"emptyLinePlaceholder":111},[75,855,856],{"class":77,"line":291},[75,857,858],{"class":118},"# Create the updated lollipop plot with blue color\n",[75,860,861,863,866,868,871,873,875,878,880,883],{"class":77,"line":297},[75,862,300],{"class":85},[75,864,865],{"class":193},"figure",[75,867,197],{"class":85},[75,869,870],{"class":309},"figsize",[75,872,142],{"class":141},[75,874,197],{"class":85},[75,876,877],{"class":145},"8",[75,879,206],{"class":85},[75,881,882],{"class":145},"6",[75,884,283],{"class":85},[75,886,887,889,892,894,897,899,902,904,907,909,912,914,917,919,922],{"class":77,"line":321},[75,888,300],{"class":85},[75,890,891],{"class":193},"stem",[75,893,306],{"class":85},[75,895,896],{"class":309},"linefmt",[75,898,142],{"class":141},[75,900,901],{"class":134},"'b-'",[75,903,206],{"class":85},[75,905,906],{"class":309},"markerfmt",[75,908,142],{"class":141},[75,910,911],{"class":134},"'bo'",[75,913,206],{"class":85},[75,915,916],{"class":309},"basefmt",[75,918,142],{"class":141},[75,920,921],{"class":134},"'k-'",[75,923,318],{"class":85},[75,925,926,928,930,932,935],{"class":77,"line":336},[75,927,300],{"class":85},[75,929,326],{"class":193},[75,931,197],{"class":85},[75,933,934],{"class":134},"'Lollipop Plot of Gaussian Function'",[75,936,318],{"class":85},[75,938,939,941,943,945,947],{"class":77,"line":351},[75,940,300],{"class":85},[75,942,341],{"class":193},[75,944,197],{"class":85},[75,946,346],{"class":134},[75,948,318],{"class":85},[75,950,951,953,955,957,959],{"class":77,"line":366},[75,952,300],{"class":85},[75,954,356],{"class":193},[75,956,197],{"class":85},[75,958,361],{"class":134},[75,960,318],{"class":85},[75,962,963,965,967,969,971,973,975,977,979],{"class":77,"line":381},[75,964,300],{"class":85},[75,966,623],{"class":193},[75,968,197],{"class":85},[75,970,200],{"class":141},[75,972,630],{"class":145},[75,974,206],{"class":85},[75,976,635],{"class":145},[75,978,216],{"class":85},[75,980,981],{"class":118},"# Force the y-axis to display the range from -1.0 to 1.1\n",[75,983,985,987,989,991,993],{"class":77,"line":984},23,[75,986,300],{"class":85},[75,988,371],{"class":193},[75,990,197],{"class":85},[75,992,376],{"class":145},[75,994,318],{"class":85},[75,996,998,1000,1002],{"class":77,"line":997},24,[75,999,300],{"class":85},[75,1001,386],{"class":193},[75,1003,389],{"class":85},[391,1005],{"alt":1006,"src":1007},"Output 3 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-03.webp",[65,1009,1011],{"className":67,"code":1010,"language":69,"meta":70,"style":70},"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",[72,1012,1013,1023,1033,1037,1042,1050,1058,1066,1079,1107,1111,1115,1155,1159,1164,1199,1203,1208,1223,1227,1232,1265,1298,1311,1323,1336,1341,1354],{"__ignoreMap":70},[75,1014,1015,1017,1019,1021],{"class":77,"line":78},[75,1016,82],{"class":81},[75,1018,86],{"class":85},[75,1020,89],{"class":81},[75,1022,92],{"class":85},[75,1024,1025,1027,1029,1031],{"class":77,"line":95},[75,1026,82],{"class":81},[75,1028,100],{"class":85},[75,1030,89],{"class":81},[75,1032,105],{"class":85},[75,1034,1035],{"class":77,"line":108},[75,1036,112],{"emptyLinePlaceholder":111},[75,1038,1039],{"class":77,"line":115},[75,1040,1041],{"class":118},"# Define the parameters for the Gaussian-modulated sine wave\n",[75,1043,1044,1046,1048],{"class":77,"line":122},[75,1045,432],{"class":85},[75,1047,142],{"class":141},[75,1049,179],{"class":145},[75,1051,1052,1054,1056],{"class":77,"line":128},[75,1053,163],{"class":85},[75,1055,142],{"class":141},[75,1057,168],{"class":145},[75,1059,1060,1062,1064],{"class":77,"line":149},[75,1061,174],{"class":85},[75,1063,142],{"class":141},[75,1065,179],{"class":145},[75,1067,1068,1071,1073,1076],{"class":77,"line":154},[75,1069,1070],{"class":85},"frequency ",[75,1072,142],{"class":141},[75,1074,1075],{"class":145}," 5",[75,1077,1078],{"class":118},"  # Frequency of the sine wave\n",[75,1080,1081,1083,1085,1087,1089,1091,1093,1095,1097,1099,1101,1103,1105],{"class":77,"line":160},[75,1082,185],{"class":85},[75,1084,142],{"class":141},[75,1086,190],{"class":85},[75,1088,194],{"class":193},[75,1090,197],{"class":85},[75,1092,200],{"class":141},[75,1094,203],{"class":145},[75,1096,206],{"class":85},[75,1098,203],{"class":145},[75,1100,206],{"class":85},[75,1102,213],{"class":145},[75,1104,216],{"class":85},[75,1106,219],{"class":118},[75,1108,1109],{"class":77,"line":171},[75,1110,112],{"emptyLinePlaceholder":111},[75,1112,1113],{"class":77,"line":182},[75,1114,805],{"class":118},[75,1116,1117,1119,1121,1123,1125,1127,1129,1131,1133,1135,1137,1139,1141,1143,1145,1147,1149,1151,1153],{"class":77,"line":222},[75,1118,236],{"class":85},[75,1120,142],{"class":141},[75,1122,190],{"class":85},[75,1124,243],{"class":193},[75,1126,197],{"class":85},[75,1128,200],{"class":141},[75,1130,250],{"class":85},[75,1132,200],{"class":141},[75,1134,255],{"class":85},[75,1136,258],{"class":141},[75,1138,261],{"class":145},[75,1140,264],{"class":141},[75,1142,267],{"class":85},[75,1144,270],{"class":145},[75,1146,273],{"class":141},[75,1148,276],{"class":85},[75,1150,258],{"class":141},[75,1152,261],{"class":145},[75,1154,283],{"class":85},[75,1156,1157],{"class":77,"line":227},[75,1158,112],{"emptyLinePlaceholder":111},[75,1160,1161],{"class":77,"line":233},[75,1162,1163],{"class":118},"# Calculate the sine wave\n",[75,1165,1166,1169,1171,1173,1175,1177,1180,1182,1184,1186,1189,1191,1194,1196],{"class":77,"line":286},[75,1167,1168],{"class":85},"sine_wave ",[75,1170,142],{"class":141},[75,1172,497],{"class":85},[75,1174,500],{"class":141},[75,1176,190],{"class":85},[75,1178,1179],{"class":193},"sin",[75,1181,197],{"class":85},[75,1183,270],{"class":145},[75,1185,273],{"class":141},[75,1187,1188],{"class":85}," np.pi ",[75,1190,500],{"class":141},[75,1192,1193],{"class":85}," frequency ",[75,1195,500],{"class":141},[75,1197,1198],{"class":85}," time)\n",[75,1200,1201],{"class":77,"line":291},[75,1202,112],{"emptyLinePlaceholder":111},[75,1204,1205],{"class":77,"line":297},[75,1206,1207],{"class":118},"# Modulate the sine wave with the Gaussian function\n",[75,1209,1210,1213,1215,1218,1220],{"class":77,"line":321},[75,1211,1212],{"class":85},"gaussian_modulated_sine_wave ",[75,1214,142],{"class":141},[75,1216,1217],{"class":85}," gaussian_modulation ",[75,1219,500],{"class":141},[75,1221,1222],{"class":85}," sine_wave\n",[75,1224,1225],{"class":77,"line":336},[75,1226,112],{"emptyLinePlaceholder":111},[75,1228,1229],{"class":77,"line":351},[75,1230,1231],{"class":118},"# Plot only the function of the Gaussian-modulated sine wave\n",[75,1233,1234,1236,1238,1240,1242,1244,1246,1248,1250,1252,1254,1256,1258,1260,1263],{"class":77,"line":366},[75,1235,300],{"class":85},[75,1237,303],{"class":193},[75,1239,306],{"class":85},[75,1241,310],{"class":309},[75,1243,142],{"class":141},[75,1245,558],{"class":134},[75,1247,206],{"class":85},[75,1249,563],{"class":309},[75,1251,142],{"class":141},[75,1253,568],{"class":134},[75,1255,206],{"class":85},[75,1257,573],{"class":309},[75,1259,142],{"class":141},[75,1261,1262],{"class":134},"'Modulation'",[75,1264,318],{"class":85},[75,1266,1267,1269,1271,1274,1276,1279,1281,1283,1285,1287,1289,1291,1293,1295],{"class":77,"line":381},[75,1268,300],{"class":85},[75,1270,303],{"class":193},[75,1272,1273],{"class":85},"(time, ",[75,1275,200],{"class":141},[75,1277,1278],{"class":85},"gaussian_modulation, ",[75,1280,310],{"class":309},[75,1282,142],{"class":141},[75,1284,558],{"class":134},[75,1286,206],{"class":85},[75,1288,563],{"class":309},[75,1290,142],{"class":141},[75,1292,568],{"class":134},[75,1294,216],{"class":85},[75,1296,1297],{"class":118},"# Add the negative function for completeness\n",[75,1299,1300,1302,1304,1306,1309],{"class":77,"line":984},[75,1301,300],{"class":85},[75,1303,326],{"class":193},[75,1305,197],{"class":85},[75,1307,1308],{"class":134},"'Upper & Lower Gaussian Modulation Function'",[75,1310,318],{"class":85},[75,1312,1313,1315,1317,1319,1321],{"class":77,"line":997},[75,1314,300],{"class":85},[75,1316,341],{"class":193},[75,1318,197],{"class":85},[75,1320,346],{"class":134},[75,1322,318],{"class":85},[75,1324,1326,1328,1330,1332,1334],{"class":77,"line":1325},25,[75,1327,300],{"class":85},[75,1329,356],{"class":193},[75,1331,197],{"class":85},[75,1333,361],{"class":134},[75,1335,318],{"class":85},[75,1337,1339],{"class":77,"line":1338},26,[75,1340,645],{"class":118},[75,1342,1344,1346,1348,1350,1352],{"class":77,"line":1343},27,[75,1345,300],{"class":85},[75,1347,371],{"class":193},[75,1349,197],{"class":85},[75,1351,376],{"class":145},[75,1353,318],{"class":85},[75,1355,1357,1359,1361],{"class":77,"line":1356},28,[75,1358,300],{"class":85},[75,1360,386],{"class":193},[75,1362,389],{"class":85},[391,1364],{"alt":1365,"src":1366},"Output 4 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-04.webp",[65,1368,1370],{"className":67,"code":1369,"language":69,"meta":70,"style":70},"'''\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",[72,1371,1372,1377,1382,1387,1392,1396,1400,1410,1420,1424,1429,1441,1452,1481,1485,1490,1520,1524,1529,1546,1559,1571,1583,1598],{"__ignoreMap":70},[75,1373,1374],{"class":77,"line":78},[75,1375,1376],{"class":134},"'''\n",[75,1378,1379],{"class":77,"line":95},[75,1380,1381],{"class":134},"This plot represents a sine wave at a chosen frequency, commonly used to match the resonance frequency\n",[75,1383,1384],{"class":77,"line":108},[75,1385,1386],{"class":134},"of a quantum device on a chip. In quantum computing, such signals serve as carrier waves for qubit\n",[75,1388,1389],{"class":77,"line":115},[75,1390,1391],{"class":134},"control pulses before modulation is applied.\n",[75,1393,1394],{"class":77,"line":122},[75,1395,1376],{"class":134},[75,1397,1398],{"class":77,"line":128},[75,1399,112],{"emptyLinePlaceholder":111},[75,1401,1402,1404,1406,1408],{"class":77,"line":149},[75,1403,82],{"class":81},[75,1405,86],{"class":85},[75,1407,89],{"class":81},[75,1409,92],{"class":85},[75,1411,1412,1414,1416,1418],{"class":77,"line":154},[75,1413,82],{"class":81},[75,1415,100],{"class":85},[75,1417,89],{"class":81},[75,1419,105],{"class":85},[75,1421,1422],{"class":77,"line":160},[75,1423,112],{"emptyLinePlaceholder":111},[75,1425,1426],{"class":77,"line":171},[75,1427,1428],{"class":118},"# Define sine wave parameters\n",[75,1430,1431,1433,1435,1438],{"class":77,"line":182},[75,1432,432],{"class":85},[75,1434,142],{"class":141},[75,1436,1437],{"class":145}," 1",[75,1439,1440],{"class":118},"  # Peak amplitude of the wave\n",[75,1442,1443,1445,1447,1449],{"class":77,"line":222},[75,1444,1070],{"class":85},[75,1446,142],{"class":141},[75,1448,1075],{"class":145},[75,1450,1451],{"class":118},"  # Frequency of the sine wave (in arbitrary units, could be GHz for quantum systems)\n",[75,1453,1454,1456,1458,1460,1462,1464,1466,1468,1470,1472,1474,1476,1478],{"class":77,"line":227},[75,1455,185],{"class":85},[75,1457,142],{"class":141},[75,1459,190],{"class":85},[75,1461,194],{"class":193},[75,1463,197],{"class":85},[75,1465,200],{"class":141},[75,1467,203],{"class":145},[75,1469,206],{"class":85},[75,1471,203],{"class":145},[75,1473,206],{"class":85},[75,1475,213],{"class":145},[75,1477,216],{"class":85},[75,1479,1480],{"class":118},"# Generate 1000 evenly spaced time values from -5 to 5\n",[75,1482,1483],{"class":77,"line":233},[75,1484,112],{"emptyLinePlaceholder":111},[75,1486,1487],{"class":77,"line":286},[75,1488,1489],{"class":118},"# Compute the sine wave, representing an unmodulated carrier signal\n",[75,1491,1492,1494,1496,1498,1500,1502,1504,1506,1508,1510,1512,1514,1516,1518],{"class":77,"line":291},[75,1493,1168],{"class":85},[75,1495,142],{"class":141},[75,1497,497],{"class":85},[75,1499,500],{"class":141},[75,1501,190],{"class":85},[75,1503,1179],{"class":193},[75,1505,197],{"class":85},[75,1507,270],{"class":145},[75,1509,273],{"class":141},[75,1511,1188],{"class":85},[75,1513,500],{"class":141},[75,1515,1193],{"class":85},[75,1517,500],{"class":141},[75,1519,1198],{"class":85},[75,1521,1522],{"class":77,"line":297},[75,1523,112],{"emptyLinePlaceholder":111},[75,1525,1526],{"class":77,"line":321},[75,1527,1528],{"class":118},"# Plot the sine wave with blue color\n",[75,1530,1531,1533,1535,1538,1540,1542,1544],{"class":77,"line":336},[75,1532,300],{"class":85},[75,1534,303],{"class":193},[75,1536,1537],{"class":85},"(time, sine_wave, ",[75,1539,310],{"class":309},[75,1541,142],{"class":141},[75,1543,315],{"class":134},[75,1545,318],{"class":85},[75,1547,1548,1550,1552,1554,1557],{"class":77,"line":351},[75,1549,300],{"class":85},[75,1551,326],{"class":193},[75,1553,197],{"class":85},[75,1555,1556],{"class":134},"'Unmodulated Sine Wave (Carrier Signal)'",[75,1558,318],{"class":85},[75,1560,1561,1563,1565,1567,1569],{"class":77,"line":366},[75,1562,300],{"class":85},[75,1564,341],{"class":193},[75,1566,197],{"class":85},[75,1568,346],{"class":134},[75,1570,318],{"class":85},[75,1572,1573,1575,1577,1579,1581],{"class":77,"line":381},[75,1574,300],{"class":85},[75,1576,356],{"class":193},[75,1578,197],{"class":85},[75,1580,361],{"class":134},[75,1582,318],{"class":85},[75,1584,1585,1587,1589,1591,1593,1595],{"class":77,"line":984},[75,1586,300],{"class":85},[75,1588,371],{"class":193},[75,1590,197],{"class":85},[75,1592,376],{"class":145},[75,1594,216],{"class":85},[75,1596,1597],{"class":118},"# No grid for a cleaner look\n",[75,1599,1600,1602,1604],{"class":77,"line":997},[75,1601,300],{"class":85},[75,1603,386],{"class":193},[75,1605,389],{"class":85},[391,1607],{"alt":1608,"src":1609},"Output 5 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-05.webp",[65,1611,1613],{"className":67,"code":1612,"language":69,"meta":70,"style":70},"'''\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",[72,1614,1615,1619,1624,1629,1634,1638,1648,1658,1662,1666,1674,1682,1690,1700,1728,1732,1736,1776,1780,1784,1814,1818,1822,1834,1838,1843,1860,1873,1885,1898,1911],{"__ignoreMap":70},[75,1616,1617],{"class":77,"line":78},[75,1618,1376],{"class":134},[75,1620,1621],{"class":77,"line":95},[75,1622,1623],{"class":134},"When a modulating pulse shape is applied to an unmodulated sine wave tone, then the result will be a\n",[75,1625,1626],{"class":77,"line":108},[75,1627,1628],{"class":134},"modulated sine wave output as shown in the plot below. The example shown is modulation by a\n",[75,1630,1631],{"class":77,"line":115},[75,1632,1633],{"class":134},"Gaussian wave.\n",[75,1635,1636],{"class":77,"line":122},[75,1637,1376],{"class":134},[75,1639,1640,1642,1644,1646],{"class":77,"line":128},[75,1641,82],{"class":81},[75,1643,86],{"class":85},[75,1645,89],{"class":81},[75,1647,92],{"class":85},[75,1649,1650,1652,1654,1656],{"class":77,"line":149},[75,1651,82],{"class":81},[75,1653,100],{"class":85},[75,1655,89],{"class":81},[75,1657,105],{"class":85},[75,1659,1660],{"class":77,"line":154},[75,1661,112],{"emptyLinePlaceholder":111},[75,1663,1664],{"class":77,"line":160},[75,1665,1041],{"class":118},[75,1667,1668,1670,1672],{"class":77,"line":171},[75,1669,432],{"class":85},[75,1671,142],{"class":141},[75,1673,179],{"class":145},[75,1675,1676,1678,1680],{"class":77,"line":182},[75,1677,163],{"class":85},[75,1679,142],{"class":141},[75,1681,168],{"class":145},[75,1683,1684,1686,1688],{"class":77,"line":222},[75,1685,174],{"class":85},[75,1687,142],{"class":141},[75,1689,179],{"class":145},[75,1691,1692,1694,1696,1698],{"class":77,"line":227},[75,1693,1070],{"class":85},[75,1695,142],{"class":141},[75,1697,1075],{"class":145},[75,1699,1078],{"class":118},[75,1701,1702,1704,1706,1708,1710,1712,1714,1716,1718,1720,1722,1724,1726],{"class":77,"line":233},[75,1703,185],{"class":85},[75,1705,142],{"class":141},[75,1707,190],{"class":85},[75,1709,194],{"class":193},[75,1711,197],{"class":85},[75,1713,200],{"class":141},[75,1715,203],{"class":145},[75,1717,206],{"class":85},[75,1719,203],{"class":145},[75,1721,206],{"class":85},[75,1723,213],{"class":145},[75,1725,216],{"class":85},[75,1727,219],{"class":118},[75,1729,1730],{"class":77,"line":286},[75,1731,112],{"emptyLinePlaceholder":111},[75,1733,1734],{"class":77,"line":291},[75,1735,805],{"class":118},[75,1737,1738,1740,1742,1744,1746,1748,1750,1752,1754,1756,1758,1760,1762,1764,1766,1768,1770,1772,1774],{"class":77,"line":297},[75,1739,236],{"class":85},[75,1741,142],{"class":141},[75,1743,190],{"class":85},[75,1745,243],{"class":193},[75,1747,197],{"class":85},[75,1749,200],{"class":141},[75,1751,250],{"class":85},[75,1753,200],{"class":141},[75,1755,255],{"class":85},[75,1757,258],{"class":141},[75,1759,261],{"class":145},[75,1761,264],{"class":141},[75,1763,267],{"class":85},[75,1765,270],{"class":145},[75,1767,273],{"class":141},[75,1769,276],{"class":85},[75,1771,258],{"class":141},[75,1773,261],{"class":145},[75,1775,283],{"class":85},[75,1777,1778],{"class":77,"line":321},[75,1779,112],{"emptyLinePlaceholder":111},[75,1781,1782],{"class":77,"line":336},[75,1783,1163],{"class":118},[75,1785,1786,1788,1790,1792,1794,1796,1798,1800,1802,1804,1806,1808,1810,1812],{"class":77,"line":351},[75,1787,1168],{"class":85},[75,1789,142],{"class":141},[75,1791,497],{"class":85},[75,1793,500],{"class":141},[75,1795,190],{"class":85},[75,1797,1179],{"class":193},[75,1799,197],{"class":85},[75,1801,270],{"class":145},[75,1803,273],{"class":141},[75,1805,1188],{"class":85},[75,1807,500],{"class":141},[75,1809,1193],{"class":85},[75,1811,500],{"class":141},[75,1813,1198],{"class":85},[75,1815,1816],{"class":77,"line":366},[75,1817,112],{"emptyLinePlaceholder":111},[75,1819,1820],{"class":77,"line":381},[75,1821,1207],{"class":118},[75,1823,1824,1826,1828,1830,1832],{"class":77,"line":984},[75,1825,1212],{"class":85},[75,1827,142],{"class":141},[75,1829,1217],{"class":85},[75,1831,500],{"class":141},[75,1833,1222],{"class":85},[75,1835,1836],{"class":77,"line":997},[75,1837,112],{"emptyLinePlaceholder":111},[75,1839,1840],{"class":77,"line":1325},[75,1841,1842],{"class":118},"# Plot the Gaussian-modulated sine wave with blue color\n",[75,1844,1845,1847,1849,1852,1854,1856,1858],{"class":77,"line":1338},[75,1846,300],{"class":85},[75,1848,303],{"class":193},[75,1850,1851],{"class":85},"(time, gaussian_modulated_sine_wave, ",[75,1853,310],{"class":309},[75,1855,142],{"class":141},[75,1857,315],{"class":134},[75,1859,318],{"class":85},[75,1861,1862,1864,1866,1868,1871],{"class":77,"line":1343},[75,1863,300],{"class":85},[75,1865,326],{"class":193},[75,1867,197],{"class":85},[75,1869,1870],{"class":134},"'Gaussian-Modulated Sine Wave Pulse'",[75,1872,318],{"class":85},[75,1874,1875,1877,1879,1881,1883],{"class":77,"line":1356},[75,1876,300],{"class":85},[75,1878,341],{"class":193},[75,1880,197],{"class":85},[75,1882,346],{"class":134},[75,1884,318],{"class":85},[75,1886,1888,1890,1892,1894,1896],{"class":77,"line":1887},29,[75,1889,300],{"class":85},[75,1891,356],{"class":193},[75,1893,197],{"class":85},[75,1895,361],{"class":134},[75,1897,318],{"class":85},[75,1899,1901,1903,1905,1907,1909],{"class":77,"line":1900},30,[75,1902,300],{"class":85},[75,1904,371],{"class":193},[75,1906,197],{"class":85},[75,1908,376],{"class":145},[75,1910,318],{"class":85},[75,1912,1914,1916,1918],{"class":77,"line":1913},31,[75,1915,300],{"class":85},[75,1917,386],{"class":193},[75,1919,389],{"class":85},[391,1921],{"alt":1922,"src":1923},"Output 6 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-06.webp",[65,1925,1927],{"className":67,"code":1926,"language":69,"meta":70,"style":70},"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",[72,1928,1929,1939,1949,1953,1957,1965,1973,1981,1991,2019,2023,2027,2067,2071,2075,2105,2109,2113,2125,2129,2134,2149,2153,2158,2182,2216,2229,2241,2253,2257,2269],{"__ignoreMap":70},[75,1930,1931,1933,1935,1937],{"class":77,"line":78},[75,1932,82],{"class":81},[75,1934,86],{"class":85},[75,1936,89],{"class":81},[75,1938,92],{"class":85},[75,1940,1941,1943,1945,1947],{"class":77,"line":95},[75,1942,82],{"class":81},[75,1944,100],{"class":85},[75,1946,89],{"class":81},[75,1948,105],{"class":85},[75,1950,1951],{"class":77,"line":108},[75,1952,112],{"emptyLinePlaceholder":111},[75,1954,1955],{"class":77,"line":115},[75,1956,1041],{"class":118},[75,1958,1959,1961,1963],{"class":77,"line":122},[75,1960,432],{"class":85},[75,1962,142],{"class":141},[75,1964,179],{"class":145},[75,1966,1967,1969,1971],{"class":77,"line":128},[75,1968,163],{"class":85},[75,1970,142],{"class":141},[75,1972,168],{"class":145},[75,1974,1975,1977,1979],{"class":77,"line":149},[75,1976,174],{"class":85},[75,1978,142],{"class":141},[75,1980,179],{"class":145},[75,1982,1983,1985,1987,1989],{"class":77,"line":154},[75,1984,1070],{"class":85},[75,1986,142],{"class":141},[75,1988,1075],{"class":145},[75,1990,1078],{"class":118},[75,1992,1993,1995,1997,1999,2001,2003,2005,2007,2009,2011,2013,2015,2017],{"class":77,"line":160},[75,1994,185],{"class":85},[75,1996,142],{"class":141},[75,1998,190],{"class":85},[75,2000,194],{"class":193},[75,2002,197],{"class":85},[75,2004,200],{"class":141},[75,2006,203],{"class":145},[75,2008,206],{"class":85},[75,2010,203],{"class":145},[75,2012,206],{"class":85},[75,2014,213],{"class":145},[75,2016,216],{"class":85},[75,2018,219],{"class":118},[75,2020,2021],{"class":77,"line":171},[75,2022,112],{"emptyLinePlaceholder":111},[75,2024,2025],{"class":77,"line":182},[75,2026,805],{"class":118},[75,2028,2029,2031,2033,2035,2037,2039,2041,2043,2045,2047,2049,2051,2053,2055,2057,2059,2061,2063,2065],{"class":77,"line":222},[75,2030,236],{"class":85},[75,2032,142],{"class":141},[75,2034,190],{"class":85},[75,2036,243],{"class":193},[75,2038,197],{"class":85},[75,2040,200],{"class":141},[75,2042,250],{"class":85},[75,2044,200],{"class":141},[75,2046,255],{"class":85},[75,2048,258],{"class":141},[75,2050,261],{"class":145},[75,2052,264],{"class":141},[75,2054,267],{"class":85},[75,2056,270],{"class":145},[75,2058,273],{"class":141},[75,2060,276],{"class":85},[75,2062,258],{"class":141},[75,2064,261],{"class":145},[75,2066,283],{"class":85},[75,2068,2069],{"class":77,"line":227},[75,2070,112],{"emptyLinePlaceholder":111},[75,2072,2073],{"class":77,"line":233},[75,2074,1163],{"class":118},[75,2076,2077,2079,2081,2083,2085,2087,2089,2091,2093,2095,2097,2099,2101,2103],{"class":77,"line":286},[75,2078,1168],{"class":85},[75,2080,142],{"class":141},[75,2082,497],{"class":85},[75,2084,500],{"class":141},[75,2086,190],{"class":85},[75,2088,1179],{"class":193},[75,2090,197],{"class":85},[75,2092,270],{"class":145},[75,2094,273],{"class":141},[75,2096,1188],{"class":85},[75,2098,500],{"class":141},[75,2100,1193],{"class":85},[75,2102,500],{"class":141},[75,2104,1198],{"class":85},[75,2106,2107],{"class":77,"line":291},[75,2108,112],{"emptyLinePlaceholder":111},[75,2110,2111],{"class":77,"line":297},[75,2112,1207],{"class":118},[75,2114,2115,2117,2119,2121,2123],{"class":77,"line":321},[75,2116,1212],{"class":85},[75,2118,142],{"class":141},[75,2120,1217],{"class":85},[75,2122,500],{"class":141},[75,2124,1222],{"class":85},[75,2126,2127],{"class":77,"line":336},[75,2128,112],{"emptyLinePlaceholder":111},[75,2130,2131],{"class":77,"line":351},[75,2132,2133],{"class":118},"# Calculate the upper envelope of the modulated sine wave\n",[75,2135,2136,2139,2141,2143,2146],{"class":77,"line":366},[75,2137,2138],{"class":85},"upper_modulation ",[75,2140,142],{"class":141},[75,2142,190],{"class":85},[75,2144,2145],{"class":193},"abs",[75,2147,2148],{"class":85},"(gaussian_modulation)\n",[75,2150,2151],{"class":77,"line":381},[75,2152,112],{"emptyLinePlaceholder":111},[75,2154,2155],{"class":77,"line":984},[75,2156,2157],{"class":118},"# Plot the Gaussian-modulated sine wave and its upper envelope\n",[75,2159,2160,2162,2164,2166,2168,2170,2172,2174,2176,2178,2180],{"class":77,"line":997},[75,2161,300],{"class":85},[75,2163,303],{"class":193},[75,2165,1851],{"class":85},[75,2167,310],{"class":309},[75,2169,142],{"class":141},[75,2171,315],{"class":134},[75,2173,206],{"class":85},[75,2175,573],{"class":309},[75,2177,142],{"class":141},[75,2179,1870],{"class":134},[75,2181,318],{"class":85},[75,2183,2184,2186,2188,2191,2193,2195,2197,2199,2201,2203,2205,2207,2209,2211,2214],{"class":77,"line":1325},[75,2185,300],{"class":85},[75,2187,303],{"class":193},[75,2189,2190],{"class":85},"(time, upper_modulation, ",[75,2192,310],{"class":309},[75,2194,142],{"class":141},[75,2196,558],{"class":134},[75,2198,206],{"class":85},[75,2200,563],{"class":309},[75,2202,142],{"class":141},[75,2204,568],{"class":134},[75,2206,206],{"class":85},[75,2208,573],{"class":309},[75,2210,142],{"class":141},[75,2212,2213],{"class":134},"'Upper Modulation Function'",[75,2215,318],{"class":85},[75,2217,2218,2220,2222,2224,2227],{"class":77,"line":1338},[75,2219,300],{"class":85},[75,2221,326],{"class":193},[75,2223,197],{"class":85},[75,2225,2226],{"class":134},"'Gaussian-Modulated Sine Wave with Upper Modulation Function'",[75,2228,318],{"class":85},[75,2230,2231,2233,2235,2237,2239],{"class":77,"line":1343},[75,2232,300],{"class":85},[75,2234,341],{"class":193},[75,2236,197],{"class":85},[75,2238,346],{"class":134},[75,2240,318],{"class":85},[75,2242,2243,2245,2247,2249,2251],{"class":77,"line":1356},[75,2244,300],{"class":85},[75,2246,356],{"class":193},[75,2248,197],{"class":85},[75,2250,361],{"class":134},[75,2252,318],{"class":85},[75,2254,2255],{"class":77,"line":1887},[75,2256,645],{"class":118},[75,2258,2259,2261,2263,2265,2267],{"class":77,"line":1900},[75,2260,300],{"class":85},[75,2262,371],{"class":193},[75,2264,197],{"class":85},[75,2266,376],{"class":145},[75,2268,318],{"class":85},[75,2270,2271,2273,2275],{"class":77,"line":1913},[75,2272,300],{"class":85},[75,2274,386],{"class":193},[75,2276,389],{"class":85},[391,2278],{"alt":2279,"src":2280},"Output 7 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-07.webp",[65,2282,2284],{"className":67,"code":2283,"language":69,"meta":70,"style":70},"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",[72,2285,2286,2296,2306,2310,2314,2322,2330,2338,2348,2376,2380,2384,2424,2428,2432,2462,2466,2470,2482,2486,2491,2504,2508,2513,2537,2570,2602,2615,2627,2639,2643,2655],{"__ignoreMap":70},[75,2287,2288,2290,2292,2294],{"class":77,"line":78},[75,2289,82],{"class":81},[75,2291,86],{"class":85},[75,2293,89],{"class":81},[75,2295,92],{"class":85},[75,2297,2298,2300,2302,2304],{"class":77,"line":95},[75,2299,82],{"class":81},[75,2301,100],{"class":85},[75,2303,89],{"class":81},[75,2305,105],{"class":85},[75,2307,2308],{"class":77,"line":108},[75,2309,112],{"emptyLinePlaceholder":111},[75,2311,2312],{"class":77,"line":115},[75,2313,1041],{"class":118},[75,2315,2316,2318,2320],{"class":77,"line":122},[75,2317,432],{"class":85},[75,2319,142],{"class":141},[75,2321,179],{"class":145},[75,2323,2324,2326,2328],{"class":77,"line":128},[75,2325,163],{"class":85},[75,2327,142],{"class":141},[75,2329,168],{"class":145},[75,2331,2332,2334,2336],{"class":77,"line":149},[75,2333,174],{"class":85},[75,2335,142],{"class":141},[75,2337,179],{"class":145},[75,2339,2340,2342,2344,2346],{"class":77,"line":154},[75,2341,1070],{"class":85},[75,2343,142],{"class":141},[75,2345,1075],{"class":145},[75,2347,1078],{"class":118},[75,2349,2350,2352,2354,2356,2358,2360,2362,2364,2366,2368,2370,2372,2374],{"class":77,"line":160},[75,2351,185],{"class":85},[75,2353,142],{"class":141},[75,2355,190],{"class":85},[75,2357,194],{"class":193},[75,2359,197],{"class":85},[75,2361,200],{"class":141},[75,2363,203],{"class":145},[75,2365,206],{"class":85},[75,2367,203],{"class":145},[75,2369,206],{"class":85},[75,2371,213],{"class":145},[75,2373,216],{"class":85},[75,2375,219],{"class":118},[75,2377,2378],{"class":77,"line":171},[75,2379,112],{"emptyLinePlaceholder":111},[75,2381,2382],{"class":77,"line":182},[75,2383,805],{"class":118},[75,2385,2386,2388,2390,2392,2394,2396,2398,2400,2402,2404,2406,2408,2410,2412,2414,2416,2418,2420,2422],{"class":77,"line":222},[75,2387,236],{"class":85},[75,2389,142],{"class":141},[75,2391,190],{"class":85},[75,2393,243],{"class":193},[75,2395,197],{"class":85},[75,2397,200],{"class":141},[75,2399,250],{"class":85},[75,2401,200],{"class":141},[75,2403,255],{"class":85},[75,2405,258],{"class":141},[75,2407,261],{"class":145},[75,2409,264],{"class":141},[75,2411,267],{"class":85},[75,2413,270],{"class":145},[75,2415,273],{"class":141},[75,2417,276],{"class":85},[75,2419,258],{"class":141},[75,2421,261],{"class":145},[75,2423,283],{"class":85},[75,2425,2426],{"class":77,"line":227},[75,2427,112],{"emptyLinePlaceholder":111},[75,2429,2430],{"class":77,"line":233},[75,2431,1163],{"class":118},[75,2433,2434,2436,2438,2440,2442,2444,2446,2448,2450,2452,2454,2456,2458,2460],{"class":77,"line":286},[75,2435,1168],{"class":85},[75,2437,142],{"class":141},[75,2439,497],{"class":85},[75,2441,500],{"class":141},[75,2443,190],{"class":85},[75,2445,1179],{"class":193},[75,2447,197],{"class":85},[75,2449,270],{"class":145},[75,2451,273],{"class":141},[75,2453,1188],{"class":85},[75,2455,500],{"class":141},[75,2457,1193],{"class":85},[75,2459,500],{"class":141},[75,2461,1198],{"class":85},[75,2463,2464],{"class":77,"line":291},[75,2465,112],{"emptyLinePlaceholder":111},[75,2467,2468],{"class":77,"line":297},[75,2469,1207],{"class":118},[75,2471,2472,2474,2476,2478,2480],{"class":77,"line":321},[75,2473,1212],{"class":85},[75,2475,142],{"class":141},[75,2477,1217],{"class":85},[75,2479,500],{"class":141},[75,2481,1222],{"class":85},[75,2483,2484],{"class":77,"line":336},[75,2485,112],{"emptyLinePlaceholder":111},[75,2487,2488],{"class":77,"line":351},[75,2489,2490],{"class":118},"# Calculate the function of the modulated sine wave\n",[75,2492,2493,2496,2498,2500,2502],{"class":77,"line":366},[75,2494,2495],{"class":85},"modulation ",[75,2497,142],{"class":141},[75,2499,190],{"class":85},[75,2501,2145],{"class":193},[75,2503,2148],{"class":85},[75,2505,2506],{"class":77,"line":381},[75,2507,112],{"emptyLinePlaceholder":111},[75,2509,2510],{"class":77,"line":984},[75,2511,2512],{"class":118},"# Plot the Gaussian-modulated sine wave and its modulation function\n",[75,2514,2515,2517,2519,2521,2523,2525,2527,2529,2531,2533,2535],{"class":77,"line":997},[75,2516,300],{"class":85},[75,2518,303],{"class":193},[75,2520,1851],{"class":85},[75,2522,310],{"class":309},[75,2524,142],{"class":141},[75,2526,315],{"class":134},[75,2528,206],{"class":85},[75,2530,573],{"class":309},[75,2532,142],{"class":141},[75,2534,1870],{"class":134},[75,2536,318],{"class":85},[75,2538,2539,2541,2543,2546,2548,2550,2552,2554,2556,2558,2560,2562,2564,2566,2568],{"class":77,"line":1325},[75,2540,300],{"class":85},[75,2542,303],{"class":193},[75,2544,2545],{"class":85},"(time, modulation, ",[75,2547,310],{"class":309},[75,2549,142],{"class":141},[75,2551,558],{"class":134},[75,2553,206],{"class":85},[75,2555,563],{"class":309},[75,2557,142],{"class":141},[75,2559,568],{"class":134},[75,2561,206],{"class":85},[75,2563,573],{"class":309},[75,2565,142],{"class":141},[75,2567,1262],{"class":134},[75,2569,318],{"class":85},[75,2571,2572,2574,2576,2578,2580,2583,2585,2587,2589,2591,2593,2595,2597,2599],{"class":77,"line":1338},[75,2573,300],{"class":85},[75,2575,303],{"class":193},[75,2577,1273],{"class":85},[75,2579,200],{"class":141},[75,2581,2582],{"class":85},"modulation, ",[75,2584,310],{"class":309},[75,2586,142],{"class":141},[75,2588,558],{"class":134},[75,2590,206],{"class":85},[75,2592,563],{"class":309},[75,2594,142],{"class":141},[75,2596,568],{"class":134},[75,2598,216],{"class":85},[75,2600,2601],{"class":118},"# Add the negative modulation function for completeness\n",[75,2603,2604,2606,2608,2610,2613],{"class":77,"line":1343},[75,2605,300],{"class":85},[75,2607,326],{"class":193},[75,2609,197],{"class":85},[75,2611,2612],{"class":134},"'Gaussian-Modulated Sine Wave with Modulation Function'",[75,2614,318],{"class":85},[75,2616,2617,2619,2621,2623,2625],{"class":77,"line":1356},[75,2618,300],{"class":85},[75,2620,341],{"class":193},[75,2622,197],{"class":85},[75,2624,346],{"class":134},[75,2626,318],{"class":85},[75,2628,2629,2631,2633,2635,2637],{"class":77,"line":1887},[75,2630,300],{"class":85},[75,2632,356],{"class":193},[75,2634,197],{"class":85},[75,2636,361],{"class":134},[75,2638,318],{"class":85},[75,2640,2641],{"class":77,"line":1900},[75,2642,645],{"class":118},[75,2644,2645,2647,2649,2651,2653],{"class":77,"line":1913},[75,2646,300],{"class":85},[75,2648,371],{"class":193},[75,2650,197],{"class":85},[75,2652,376],{"class":145},[75,2654,318],{"class":85},[75,2656,2658,2660,2662],{"class":77,"line":2657},32,[75,2659,300],{"class":85},[75,2661,386],{"class":193},[75,2663,389],{"class":85},[391,2665],{"alt":2666,"src":2667},"Output 8 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-08.webp",[12,2669,2670],{},"Below are some common pulse envelope shapes used to modulate a sine wave tone.",[65,2672,2674],{"className":67,"code":2673,"language":69,"meta":70,"style":70},"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",[72,2675,2676,2686,2696,2709,2713,2718,2745,2749,2754,2778,2834,2880,2895,2916,2934,2961,2979,2983,2988,3004,3020,3034,3044,3054,3064,3075,3079,3084,3124,3139,3143,3160,3175,3186,3201,3216,3222,3227,3233,3261,3273,3278,3288],{"__ignoreMap":70},[75,2677,2678,2680,2682,2684],{"class":77,"line":78},[75,2679,82],{"class":81},[75,2681,86],{"class":85},[75,2683,89],{"class":81},[75,2685,92],{"class":85},[75,2687,2688,2690,2692,2694],{"class":77,"line":95},[75,2689,82],{"class":81},[75,2691,100],{"class":85},[75,2693,89],{"class":81},[75,2695,105],{"class":85},[75,2697,2698,2701,2704,2706],{"class":77,"line":108},[75,2699,2700],{"class":81},"from",[75,2702,2703],{"class":85}," scipy.signal.windows ",[75,2705,82],{"class":81},[75,2707,2708],{"class":85}," blackmanharris, tukey\n",[75,2710,2711],{"class":77,"line":115},[75,2712,112],{"emptyLinePlaceholder":111},[75,2714,2715],{"class":77,"line":122},[75,2716,2717],{"class":118},"# Define time vector\n",[75,2719,2720,2723,2725,2727,2729,2731,2733,2735,2737,2739,2741,2743],{"class":77,"line":128},[75,2721,2722],{"class":85},"t ",[75,2724,142],{"class":141},[75,2726,190],{"class":85},[75,2728,194],{"class":193},[75,2730,197],{"class":85},[75,2732,200],{"class":141},[75,2734,270],{"class":145},[75,2736,206],{"class":85},[75,2738,270],{"class":145},[75,2740,206],{"class":85},[75,2742,213],{"class":145},[75,2744,318],{"class":85},[75,2746,2747],{"class":77,"line":149},[75,2748,112],{"emptyLinePlaceholder":111},[75,2750,2751],{"class":77,"line":154},[75,2752,2753],{"class":118},"# Define different pulse envelopes\n",[75,2755,2756,2759,2761,2763,2765,2767,2769,2772,2774,2776],{"class":77,"line":160},[75,2757,2758],{"class":85},"gaussian ",[75,2760,142],{"class":141},[75,2762,190],{"class":85},[75,2764,243],{"class":193},[75,2766,197],{"class":85},[75,2768,200],{"class":141},[75,2770,2771],{"class":85},"t",[75,2773,258],{"class":141},[75,2775,270],{"class":145},[75,2777,318],{"class":85},[75,2779,2780,2783,2785,2787,2789,2792,2794,2797,2799,2802,2805,2808,2811,2814,2817,2820,2822,2824,2826,2829,2831],{"class":77,"line":171},[75,2781,2782],{"class":85},"sin2 ",[75,2784,142],{"class":141},[75,2786,190],{"class":85},[75,2788,1179],{"class":193},[75,2790,2791],{"class":85},"(np.pi ",[75,2793,500],{"class":141},[75,2795,2796],{"class":85}," (t ",[75,2798,200],{"class":141},[75,2800,2801],{"class":85}," t.",[75,2803,2804],{"class":193},"min",[75,2806,2807],{"class":85},"()) ",[75,2809,2810],{"class":141},"\u002F",[75,2812,2813],{"class":85}," (t.",[75,2815,2816],{"class":193},"max",[75,2818,2819],{"class":85},"() ",[75,2821,200],{"class":141},[75,2823,2801],{"class":85},[75,2825,2804],{"class":193},[75,2827,2828],{"class":85},"()))",[75,2830,258],{"class":141},[75,2832,2833],{"class":145},"2\n",[75,2835,2836,2839,2841,2843,2846,2848,2850,2852,2854,2856,2858,2860,2862,2864,2866,2868,2870,2872,2874,2876,2878],{"class":77,"line":182},[75,2837,2838],{"class":85},"cos2 ",[75,2840,142],{"class":141},[75,2842,190],{"class":85},[75,2844,2845],{"class":193},"cos",[75,2847,2791],{"class":85},[75,2849,500],{"class":141},[75,2851,2796],{"class":85},[75,2853,200],{"class":141},[75,2855,2801],{"class":85},[75,2857,2804],{"class":193},[75,2859,2807],{"class":85},[75,2861,2810],{"class":141},[75,2863,2813],{"class":85},[75,2865,2816],{"class":193},[75,2867,2819],{"class":85},[75,2869,200],{"class":141},[75,2871,2801],{"class":85},[75,2873,2804],{"class":193},[75,2875,2828],{"class":85},[75,2877,258],{"class":141},[75,2879,2833],{"class":145},[75,2881,2882,2885,2887,2889,2892],{"class":77,"line":222},[75,2883,2884],{"class":85},"rectangular ",[75,2886,142],{"class":141},[75,2888,190],{"class":85},[75,2890,2891],{"class":193},"ones_like",[75,2893,2894],{"class":85},"(t)\n",[75,2896,2897,2900,2902,2905,2908,2910,2912,2914],{"class":77,"line":227},[75,2898,2899],{"class":85},"rectangular[np.",[75,2901,2145],{"class":193},[75,2903,2904],{"class":85},"(t) ",[75,2906,2907],{"class":141},">",[75,2909,1437],{"class":145},[75,2911,138],{"class":85},[75,2913,142],{"class":141},[75,2915,168],{"class":145},[75,2917,2918,2921,2923,2926,2928,2931],{"class":77,"line":233},[75,2919,2920],{"class":85},"blackman_harris ",[75,2922,142],{"class":141},[75,2924,2925],{"class":193}," blackmanharris",[75,2927,197],{"class":85},[75,2929,2930],{"class":141},"len",[75,2932,2933],{"class":85},"(t))\n",[75,2935,2936,2939,2941,2944,2946,2948,2951,2954,2956,2959],{"class":77,"line":286},[75,2937,2938],{"class":85},"tukey_envelope ",[75,2940,142],{"class":141},[75,2942,2943],{"class":193}," tukey",[75,2945,197],{"class":85},[75,2947,2930],{"class":141},[75,2949,2950],{"class":85},"(t), ",[75,2952,2953],{"class":309},"alpha",[75,2955,142],{"class":141},[75,2957,2958],{"class":145},"0.5",[75,2960,318],{"class":85},[75,2962,2963,2966,2968,2970,2972,2974,2977],{"class":77,"line":291},[75,2964,2965],{"class":85},"sech ",[75,2967,142],{"class":141},[75,2969,1437],{"class":145},[75,2971,264],{"class":141},[75,2973,190],{"class":85},[75,2975,2976],{"class":193},"cosh",[75,2978,2894],{"class":85},[75,2980,2981],{"class":77,"line":297},[75,2982,112],{"emptyLinePlaceholder":111},[75,2984,2985],{"class":77,"line":321},[75,2986,2987],{"class":118},"# List of envelopes and titles\n",[75,2989,2990,2993,2995,2998,3001],{"class":77,"line":336},[75,2991,2992],{"class":85},"envelopes ",[75,2994,142],{"class":141},[75,2996,2997],{"class":85}," [(gaussian, ",[75,2999,3000],{"class":134},"\"Gaussian Modulation Function\"",[75,3002,3003],{"class":85},"),\n",[75,3005,3006,3009,3012,3015,3018],{"class":77,"line":351},[75,3007,3008],{"class":85},"             (sin2, ",[75,3010,3011],{"class":134},"\"$Sin^",[75,3013,3014],{"class":145},"{2}",[75,3016,3017],{"class":134},"$ Modulation Function\"",[75,3019,3003],{"class":85},[75,3021,3022,3025,3028,3030,3032],{"class":77,"line":366},[75,3023,3024],{"class":85},"             (cos2, ",[75,3026,3027],{"class":134},"\"$Cos^",[75,3029,3014],{"class":145},[75,3031,3017],{"class":134},[75,3033,3003],{"class":85},[75,3035,3036,3039,3042],{"class":77,"line":381},[75,3037,3038],{"class":85},"             (rectangular, ",[75,3040,3041],{"class":134},"\"Rectangular Modulation Function\"",[75,3043,3003],{"class":85},[75,3045,3046,3049,3052],{"class":77,"line":984},[75,3047,3048],{"class":85},"             (blackman_harris, ",[75,3050,3051],{"class":134},"\"Blackman-Harris Modulation Function\"",[75,3053,3003],{"class":85},[75,3055,3056,3059,3062],{"class":77,"line":997},[75,3057,3058],{"class":85},"             (tukey_envelope, ",[75,3060,3061],{"class":134},"\"Tukey Modulation Function\"",[75,3063,3003],{"class":85},[75,3065,3066,3069,3072],{"class":77,"line":1325},[75,3067,3068],{"class":85},"             (sech, ",[75,3070,3071],{"class":134},"\"Sech Modulation Function\"",[75,3073,3074],{"class":85},")]\n",[75,3076,3077],{"class":77,"line":1338},[75,3078,112],{"emptyLinePlaceholder":111},[75,3080,3081],{"class":77,"line":1343},[75,3082,3083],{"class":118},"# Create subplots\n",[75,3085,3086,3089,3091,3094,3097,3099,3102,3104,3106,3108,3110,3112,3114,3117,3119,3122],{"class":77,"line":1356},[75,3087,3088],{"class":85},"fig, axes ",[75,3090,142],{"class":141},[75,3092,3093],{"class":85}," plt.",[75,3095,3096],{"class":193},"subplots",[75,3098,197],{"class":85},[75,3100,3101],{"class":145},"3",[75,3103,206],{"class":85},[75,3105,3101],{"class":145},[75,3107,206],{"class":85},[75,3109,870],{"class":309},[75,3111,142],{"class":141},[75,3113,197],{"class":85},[75,3115,3116],{"class":145},"12",[75,3118,206],{"class":85},[75,3120,3121],{"class":145},"9",[75,3123,283],{"class":85},[75,3125,3126,3129,3131,3134,3137],{"class":77,"line":1887},[75,3127,3128],{"class":85},"axes ",[75,3130,142],{"class":141},[75,3132,3133],{"class":85}," axes.",[75,3135,3136],{"class":193},"flatten",[75,3138,389],{"class":85},[75,3140,3141],{"class":77,"line":1900},[75,3142,112],{"emptyLinePlaceholder":111},[75,3144,3145,3148,3151,3154,3157],{"class":77,"line":1913},[75,3146,3147],{"class":81},"for",[75,3149,3150],{"class":85}," i, (envelope, title) ",[75,3152,3153],{"class":81},"in",[75,3155,3156],{"class":141}," enumerate",[75,3158,3159],{"class":85},"(envelopes):\n",[75,3161,3162,3165,3167,3170,3173],{"class":77,"line":2657},[75,3163,3164],{"class":85},"    axes[i].",[75,3166,303],{"class":193},[75,3168,3169],{"class":85},"(t, envelope, ",[75,3171,3172],{"class":134},"'b'",[75,3174,318],{"class":85},[75,3176,3178,3180,3183],{"class":77,"line":3177},33,[75,3179,3164],{"class":85},[75,3181,3182],{"class":193},"set_title",[75,3184,3185],{"class":85},"(title)\n",[75,3187,3189,3191,3194,3196,3199],{"class":77,"line":3188},34,[75,3190,3164],{"class":85},[75,3192,3193],{"class":193},"set_xlabel",[75,3195,197],{"class":85},[75,3197,3198],{"class":134},"\"Time\"",[75,3200,318],{"class":85},[75,3202,3204,3206,3209,3211,3214],{"class":77,"line":3203},35,[75,3205,3164],{"class":85},[75,3207,3208],{"class":193},"set_ylabel",[75,3210,197],{"class":85},[75,3212,3213],{"class":134},"\"Amplitude\"",[75,3215,318],{"class":85},[75,3217,3219],{"class":77,"line":3218},36,[75,3220,3221],{"class":118},"    # axes[i].legend([title])\n",[75,3223,3225],{"class":77,"line":3224},37,[75,3226,112],{"emptyLinePlaceholder":111},[75,3228,3230],{"class":77,"line":3229},38,[75,3231,3232],{"class":118},"# Hide unused subplots\n",[75,3234,3236,3238,3241,3243,3246,3249,3252,3254,3256,3258],{"class":77,"line":3235},39,[75,3237,3147],{"class":81},[75,3239,3240],{"class":85}," j ",[75,3242,3153],{"class":81},[75,3244,3245],{"class":141}," range",[75,3247,3248],{"class":85},"(i ",[75,3250,3251],{"class":141},"+",[75,3253,1437],{"class":145},[75,3255,206],{"class":85},[75,3257,2930],{"class":141},[75,3259,3260],{"class":85},"(axes)):\n",[75,3262,3264,3267,3270],{"class":77,"line":3263},40,[75,3265,3266],{"class":85},"    fig.",[75,3268,3269],{"class":193},"delaxes",[75,3271,3272],{"class":85},"(axes[j])\n",[75,3274,3276],{"class":77,"line":3275},41,[75,3277,112],{"emptyLinePlaceholder":111},[75,3279,3281,3283,3286],{"class":77,"line":3280},42,[75,3282,300],{"class":85},[75,3284,3285],{"class":193},"tight_layout",[75,3287,389],{"class":85},[75,3289,3291,3293,3295],{"class":77,"line":3290},43,[75,3292,300],{"class":85},[75,3294,386],{"class":193},[75,3296,389],{"class":85},[391,3298],{"alt":3299,"src":3300},"Output 9 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-09.webp",[65,3302,3304],{"className":67,"code":3303,"language":69,"meta":70,"style":70},"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",[72,3305,3306,3316,3326,3336,3340,3345,3371,3375,3380,3402,3447,3492,3505,3524,3539,3562,3579,3583,3588,3602,3612,3622,3631,3640,3649,3658,3662,3666,3700,3712,3716,3730,3763,3771,3783,3795,3799,3803,3825,3833,3837,3845],{"__ignoreMap":70},[75,3307,3308,3310,3312,3314],{"class":77,"line":78},[75,3309,82],{"class":81},[75,3311,86],{"class":85},[75,3313,89],{"class":81},[75,3315,92],{"class":85},[75,3317,3318,3320,3322,3324],{"class":77,"line":95},[75,3319,82],{"class":81},[75,3321,100],{"class":85},[75,3323,89],{"class":81},[75,3325,105],{"class":85},[75,3327,3328,3330,3332,3334],{"class":77,"line":108},[75,3329,2700],{"class":81},[75,3331,2703],{"class":85},[75,3333,82],{"class":81},[75,3335,2708],{"class":85},[75,3337,3338],{"class":77,"line":115},[75,3339,112],{"emptyLinePlaceholder":111},[75,3341,3342],{"class":77,"line":122},[75,3343,3344],{"class":118},"# Define time vector with reduced points for lollipop plot clarity\n",[75,3346,3347,3349,3351,3353,3355,3357,3359,3361,3363,3365,3367,3369],{"class":77,"line":128},[75,3348,2722],{"class":85},[75,3350,142],{"class":141},[75,3352,190],{"class":85},[75,3354,194],{"class":193},[75,3356,197],{"class":85},[75,3358,200],{"class":141},[75,3360,270],{"class":145},[75,3362,206],{"class":85},[75,3364,270],{"class":145},[75,3366,206],{"class":85},[75,3368,791],{"class":145},[75,3370,318],{"class":85},[75,3372,3373],{"class":77,"line":149},[75,3374,112],{"emptyLinePlaceholder":111},[75,3376,3377],{"class":77,"line":154},[75,3378,3379],{"class":118},"# Define different modulation functions\n",[75,3381,3382,3384,3386,3388,3390,3392,3394,3396,3398,3400],{"class":77,"line":160},[75,3383,236],{"class":85},[75,3385,142],{"class":141},[75,3387,190],{"class":85},[75,3389,243],{"class":193},[75,3391,197],{"class":85},[75,3393,200],{"class":141},[75,3395,2771],{"class":85},[75,3397,258],{"class":141},[75,3399,270],{"class":145},[75,3401,318],{"class":85},[75,3403,3404,3407,3409,3411,3413,3415,3417,3419,3421,3423,3425,3427,3429,3431,3433,3435,3437,3439,3441,3443,3445],{"class":77,"line":171},[75,3405,3406],{"class":85},"sin2_modulation ",[75,3408,142],{"class":141},[75,3410,190],{"class":85},[75,3412,1179],{"class":193},[75,3414,2791],{"class":85},[75,3416,500],{"class":141},[75,3418,2796],{"class":85},[75,3420,200],{"class":141},[75,3422,2801],{"class":85},[75,3424,2804],{"class":193},[75,3426,2807],{"class":85},[75,3428,2810],{"class":141},[75,3430,2813],{"class":85},[75,3432,2816],{"class":193},[75,3434,2819],{"class":85},[75,3436,200],{"class":141},[75,3438,2801],{"class":85},[75,3440,2804],{"class":193},[75,3442,2828],{"class":85},[75,3444,258],{"class":141},[75,3446,2833],{"class":145},[75,3448,3449,3452,3454,3456,3458,3460,3462,3464,3466,3468,3470,3472,3474,3476,3478,3480,3482,3484,3486,3488,3490],{"class":77,"line":182},[75,3450,3451],{"class":85},"cos2_modulation ",[75,3453,142],{"class":141},[75,3455,190],{"class":85},[75,3457,2845],{"class":193},[75,3459,2791],{"class":85},[75,3461,500],{"class":141},[75,3463,2796],{"class":85},[75,3465,200],{"class":141},[75,3467,2801],{"class":85},[75,3469,2804],{"class":193},[75,3471,2807],{"class":85},[75,3473,2810],{"class":141},[75,3475,2813],{"class":85},[75,3477,2816],{"class":193},[75,3479,2819],{"class":85},[75,3481,200],{"class":141},[75,3483,2801],{"class":85},[75,3485,2804],{"class":193},[75,3487,2828],{"class":85},[75,3489,258],{"class":141},[75,3491,2833],{"class":145},[75,3493,3494,3497,3499,3501,3503],{"class":77,"line":222},[75,3495,3496],{"class":85},"rectangular_modulation ",[75,3498,142],{"class":141},[75,3500,190],{"class":85},[75,3502,2891],{"class":193},[75,3504,2894],{"class":85},[75,3506,3507,3510,3512,3514,3516,3518,3520,3522],{"class":77,"line":227},[75,3508,3509],{"class":85},"rectangular_modulation[np.",[75,3511,2145],{"class":193},[75,3513,2904],{"class":85},[75,3515,2907],{"class":141},[75,3517,1437],{"class":145},[75,3519,138],{"class":85},[75,3521,142],{"class":141},[75,3523,168],{"class":145},[75,3525,3526,3529,3531,3533,3535,3537],{"class":77,"line":233},[75,3527,3528],{"class":85},"blackman_harris_modulation ",[75,3530,142],{"class":141},[75,3532,2925],{"class":193},[75,3534,197],{"class":85},[75,3536,2930],{"class":141},[75,3538,2933],{"class":85},[75,3540,3541,3544,3546,3548,3550,3552,3554,3556,3558,3560],{"class":77,"line":286},[75,3542,3543],{"class":85},"tukey_modulation ",[75,3545,142],{"class":141},[75,3547,2943],{"class":193},[75,3549,197],{"class":85},[75,3551,2930],{"class":141},[75,3553,2950],{"class":85},[75,3555,2953],{"class":309},[75,3557,142],{"class":141},[75,3559,2958],{"class":145},[75,3561,318],{"class":85},[75,3563,3564,3567,3569,3571,3573,3575,3577],{"class":77,"line":291},[75,3565,3566],{"class":85},"sech_modulation ",[75,3568,142],{"class":141},[75,3570,1437],{"class":145},[75,3572,264],{"class":141},[75,3574,190],{"class":85},[75,3576,2976],{"class":193},[75,3578,2894],{"class":85},[75,3580,3581],{"class":77,"line":297},[75,3582,112],{"emptyLinePlaceholder":111},[75,3584,3585],{"class":77,"line":321},[75,3586,3587],{"class":118},"# List of modulation functions and titles\n",[75,3589,3590,3593,3595,3598,3600],{"class":77,"line":336},[75,3591,3592],{"class":85},"modulations ",[75,3594,142],{"class":141},[75,3596,3597],{"class":85}," [(gaussian_modulation, ",[75,3599,3000],{"class":134},[75,3601,3003],{"class":85},[75,3603,3604,3607,3610],{"class":77,"line":351},[75,3605,3606],{"class":85},"               (sin2_modulation, ",[75,3608,3609],{"class":134},"\"Sin² Modulation Function\"",[75,3611,3003],{"class":85},[75,3613,3614,3617,3620],{"class":77,"line":366},[75,3615,3616],{"class":85},"               (cos2_modulation, ",[75,3618,3619],{"class":134},"\"Cos² Modulation Function\"",[75,3621,3003],{"class":85},[75,3623,3624,3627,3629],{"class":77,"line":381},[75,3625,3626],{"class":85},"               (rectangular_modulation, ",[75,3628,3041],{"class":134},[75,3630,3003],{"class":85},[75,3632,3633,3636,3638],{"class":77,"line":984},[75,3634,3635],{"class":85},"               (blackman_harris_modulation, ",[75,3637,3051],{"class":134},[75,3639,3003],{"class":85},[75,3641,3642,3645,3647],{"class":77,"line":997},[75,3643,3644],{"class":85},"               (tukey_modulation, ",[75,3646,3061],{"class":134},[75,3648,3003],{"class":85},[75,3650,3651,3654,3656],{"class":77,"line":1325},[75,3652,3653],{"class":85},"               (sech_modulation, ",[75,3655,3071],{"class":134},[75,3657,3074],{"class":85},[75,3659,3660],{"class":77,"line":1338},[75,3661,112],{"emptyLinePlaceholder":111},[75,3663,3664],{"class":77,"line":1343},[75,3665,3083],{"class":118},[75,3667,3668,3670,3672,3674,3676,3678,3680,3682,3684,3686,3688,3690,3692,3694,3696,3698],{"class":77,"line":1356},[75,3669,3088],{"class":85},[75,3671,142],{"class":141},[75,3673,3093],{"class":85},[75,3675,3096],{"class":193},[75,3677,197],{"class":85},[75,3679,3101],{"class":145},[75,3681,206],{"class":85},[75,3683,3101],{"class":145},[75,3685,206],{"class":85},[75,3687,870],{"class":309},[75,3689,142],{"class":141},[75,3691,197],{"class":85},[75,3693,3116],{"class":145},[75,3695,206],{"class":85},[75,3697,3121],{"class":145},[75,3699,283],{"class":85},[75,3701,3702,3704,3706,3708,3710],{"class":77,"line":1887},[75,3703,3128],{"class":85},[75,3705,142],{"class":141},[75,3707,3133],{"class":85},[75,3709,3136],{"class":193},[75,3711,389],{"class":85},[75,3713,3714],{"class":77,"line":1900},[75,3715,112],{"emptyLinePlaceholder":111},[75,3717,3718,3720,3723,3725,3727],{"class":77,"line":1913},[75,3719,3147],{"class":81},[75,3721,3722],{"class":85}," i, (modulation, title) ",[75,3724,3153],{"class":81},[75,3726,3156],{"class":141},[75,3728,3729],{"class":85},"(modulations):\n",[75,3731,3732,3734,3736,3739,3741,3743,3745,3747,3749,3751,3753,3755,3757,3759,3761],{"class":77,"line":2657},[75,3733,3164],{"class":85},[75,3735,891],{"class":193},[75,3737,3738],{"class":85},"(t, modulation, ",[75,3740,896],{"class":309},[75,3742,142],{"class":141},[75,3744,901],{"class":134},[75,3746,206],{"class":85},[75,3748,906],{"class":309},[75,3750,142],{"class":141},[75,3752,911],{"class":134},[75,3754,206],{"class":85},[75,3756,916],{"class":309},[75,3758,142],{"class":141},[75,3760,921],{"class":134},[75,3762,318],{"class":85},[75,3764,3765,3767,3769],{"class":77,"line":3177},[75,3766,3164],{"class":85},[75,3768,3182],{"class":193},[75,3770,3185],{"class":85},[75,3772,3773,3775,3777,3779,3781],{"class":77,"line":3188},[75,3774,3164],{"class":85},[75,3776,3193],{"class":193},[75,3778,197],{"class":85},[75,3780,3198],{"class":134},[75,3782,318],{"class":85},[75,3784,3785,3787,3789,3791,3793],{"class":77,"line":3203},[75,3786,3164],{"class":85},[75,3788,3208],{"class":193},[75,3790,197],{"class":85},[75,3792,3213],{"class":134},[75,3794,318],{"class":85},[75,3796,3797],{"class":77,"line":3218},[75,3798,112],{"emptyLinePlaceholder":111},[75,3800,3801],{"class":77,"line":3224},[75,3802,3232],{"class":118},[75,3804,3805,3807,3809,3811,3813,3815,3817,3819,3821,3823],{"class":77,"line":3229},[75,3806,3147],{"class":81},[75,3808,3240],{"class":85},[75,3810,3153],{"class":81},[75,3812,3245],{"class":141},[75,3814,3248],{"class":85},[75,3816,3251],{"class":141},[75,3818,1437],{"class":145},[75,3820,206],{"class":85},[75,3822,2930],{"class":141},[75,3824,3260],{"class":85},[75,3826,3827,3829,3831],{"class":77,"line":3235},[75,3828,3266],{"class":85},[75,3830,3269],{"class":193},[75,3832,3272],{"class":85},[75,3834,3835],{"class":77,"line":3263},[75,3836,112],{"emptyLinePlaceholder":111},[75,3838,3839,3841,3843],{"class":77,"line":3275},[75,3840,300],{"class":85},[75,3842,3285],{"class":193},[75,3844,389],{"class":85},[75,3846,3847,3849,3851],{"class":77,"line":3280},[75,3848,300],{"class":85},[75,3850,386],{"class":193},[75,3852,389],{"class":85},[391,3854],{"alt":3855,"src":3856},"Output 10 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-10.webp",[65,3858,3860],{"className":67,"code":3859,"language":69,"meta":70,"style":70},"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",[72,3861,3862,3872,3882,3892,3896,3900,3926,3930,3935,3959,4007,4052,4066,4086,4102,4126,4143,4147,4151,4159,4169,4199,4203,4208,4223,4237,4250,4260,4270,4280,4290,4294,4298,4332,4344,4348,4362,4376,4389,4397,4409,4421,4425,4429,4452,4461,4466,4475],{"__ignoreMap":70},[75,3863,3864,3866,3868,3870],{"class":77,"line":78},[75,3865,82],{"class":81},[75,3867,86],{"class":85},[75,3869,89],{"class":81},[75,3871,92],{"class":85},[75,3873,3874,3876,3878,3880],{"class":77,"line":95},[75,3875,82],{"class":81},[75,3877,100],{"class":85},[75,3879,89],{"class":81},[75,3881,105],{"class":85},[75,3883,3884,3886,3888,3890],{"class":77,"line":108},[75,3885,2700],{"class":81},[75,3887,2703],{"class":85},[75,3889,82],{"class":81},[75,3891,2708],{"class":85},[75,3893,3894],{"class":77,"line":115},[75,3895,112],{"emptyLinePlaceholder":111},[75,3897,3898],{"class":77,"line":122},[75,3899,2717],{"class":118},[75,3901,3902,3904,3906,3908,3910,3912,3914,3916,3918,3920,3922,3924],{"class":77,"line":128},[75,3903,185],{"class":85},[75,3905,142],{"class":141},[75,3907,190],{"class":85},[75,3909,194],{"class":193},[75,3911,197],{"class":85},[75,3913,200],{"class":141},[75,3915,203],{"class":145},[75,3917,206],{"class":85},[75,3919,203],{"class":145},[75,3921,206],{"class":85},[75,3923,213],{"class":145},[75,3925,318],{"class":85},[75,3927,3928],{"class":77,"line":149},[75,3929,112],{"emptyLinePlaceholder":111},[75,3931,3932],{"class":77,"line":154},[75,3933,3934],{"class":118},"# Define different modulated sine waves\n",[75,3936,3937,3940,3942,3944,3946,3948,3950,3953,3955,3957],{"class":77,"line":160},[75,3938,3939],{"class":85},"gaussian_modulated_sine ",[75,3941,142],{"class":141},[75,3943,190],{"class":85},[75,3945,243],{"class":193},[75,3947,197],{"class":85},[75,3949,200],{"class":141},[75,3951,3952],{"class":85},"time",[75,3954,258],{"class":141},[75,3956,270],{"class":145},[75,3958,318],{"class":85},[75,3960,3961,3964,3966,3968,3970,3972,3974,3977,3979,3982,3984,3986,3988,3991,3993,3995,3997,3999,4001,4003,4005],{"class":77,"line":171},[75,3962,3963],{"class":85},"sin2_modulated_sine ",[75,3965,142],{"class":141},[75,3967,190],{"class":85},[75,3969,1179],{"class":193},[75,3971,2791],{"class":85},[75,3973,500],{"class":141},[75,3975,3976],{"class":85}," (time ",[75,3978,200],{"class":141},[75,3980,3981],{"class":85}," time.",[75,3983,2804],{"class":193},[75,3985,2807],{"class":85},[75,3987,2810],{"class":141},[75,3989,3990],{"class":85}," (time.",[75,3992,2816],{"class":193},[75,3994,2819],{"class":85},[75,3996,200],{"class":141},[75,3998,3981],{"class":85},[75,4000,2804],{"class":193},[75,4002,2828],{"class":85},[75,4004,258],{"class":141},[75,4006,2833],{"class":145},[75,4008,4009,4012,4014,4016,4018,4020,4022,4024,4026,4028,4030,4032,4034,4036,4038,4040,4042,4044,4046,4048,4050],{"class":77,"line":182},[75,4010,4011],{"class":85},"cos2_modulated_sine ",[75,4013,142],{"class":141},[75,4015,190],{"class":85},[75,4017,2845],{"class":193},[75,4019,2791],{"class":85},[75,4021,500],{"class":141},[75,4023,3976],{"class":85},[75,4025,200],{"class":141},[75,4027,3981],{"class":85},[75,4029,2804],{"class":193},[75,4031,2807],{"class":85},[75,4033,2810],{"class":141},[75,4035,3990],{"class":85},[75,4037,2816],{"class":193},[75,4039,2819],{"class":85},[75,4041,200],{"class":141},[75,4043,3981],{"class":85},[75,4045,2804],{"class":193},[75,4047,2828],{"class":85},[75,4049,258],{"class":141},[75,4051,2833],{"class":145},[75,4053,4054,4057,4059,4061,4063],{"class":77,"line":222},[75,4055,4056],{"class":85},"rectangular_modulated_sine ",[75,4058,142],{"class":141},[75,4060,190],{"class":85},[75,4062,2891],{"class":193},[75,4064,4065],{"class":85},"(time)\n",[75,4067,4068,4071,4073,4076,4078,4080,4082,4084],{"class":77,"line":227},[75,4069,4070],{"class":85},"rectangular_modulated_sine[np.",[75,4072,2145],{"class":193},[75,4074,4075],{"class":85},"(time) ",[75,4077,2907],{"class":141},[75,4079,261],{"class":145},[75,4081,138],{"class":85},[75,4083,142],{"class":141},[75,4085,168],{"class":145},[75,4087,4088,4091,4093,4095,4097,4099],{"class":77,"line":233},[75,4089,4090],{"class":85},"blackman_harris_modulated_sine ",[75,4092,142],{"class":141},[75,4094,2925],{"class":193},[75,4096,197],{"class":85},[75,4098,2930],{"class":141},[75,4100,4101],{"class":85},"(time))\n",[75,4103,4104,4107,4109,4111,4113,4115,4118,4120,4122,4124],{"class":77,"line":286},[75,4105,4106],{"class":85},"tukey_modulated_sine ",[75,4108,142],{"class":141},[75,4110,2943],{"class":193},[75,4112,197],{"class":85},[75,4114,2930],{"class":141},[75,4116,4117],{"class":85},"(time), ",[75,4119,2953],{"class":309},[75,4121,142],{"class":141},[75,4123,2958],{"class":145},[75,4125,318],{"class":85},[75,4127,4128,4131,4133,4135,4137,4139,4141],{"class":77,"line":291},[75,4129,4130],{"class":85},"sech_modulated_sine ",[75,4132,142],{"class":141},[75,4134,1437],{"class":145},[75,4136,264],{"class":141},[75,4138,190],{"class":85},[75,4140,2976],{"class":193},[75,4142,4065],{"class":85},[75,4144,4145],{"class":77,"line":297},[75,4146,112],{"emptyLinePlaceholder":111},[75,4148,4149],{"class":77,"line":321},[75,4150,1428],{"class":118},[75,4152,4153,4155,4157],{"class":77,"line":336},[75,4154,432],{"class":85},[75,4156,142],{"class":141},[75,4158,179],{"class":145},[75,4160,4161,4163,4165,4167],{"class":77,"line":351},[75,4162,1070],{"class":85},[75,4164,142],{"class":141},[75,4166,1075],{"class":145},[75,4168,1078],{"class":118},[75,4170,4171,4173,4175,4177,4179,4181,4183,4185,4187,4189,4191,4193,4195,4197],{"class":77,"line":366},[75,4172,1168],{"class":85},[75,4174,142],{"class":141},[75,4176,497],{"class":85},[75,4178,500],{"class":141},[75,4180,190],{"class":85},[75,4182,1179],{"class":193},[75,4184,197],{"class":85},[75,4186,270],{"class":145},[75,4188,273],{"class":141},[75,4190,1188],{"class":85},[75,4192,500],{"class":141},[75,4194,1193],{"class":85},[75,4196,500],{"class":141},[75,4198,1198],{"class":85},[75,4200,4201],{"class":77,"line":381},[75,4202,112],{"emptyLinePlaceholder":111},[75,4204,4205],{"class":77,"line":984},[75,4206,4207],{"class":118},"# List of modulated sine waves and titles\n",[75,4209,4210,4213,4215,4218,4221],{"class":77,"line":997},[75,4211,4212],{"class":85},"modulated_sines ",[75,4214,142],{"class":141},[75,4216,4217],{"class":85}," [(gaussian_modulated_sine, ",[75,4219,4220],{"class":134},"\"Gaussian Modulated Sine Wave\"",[75,4222,3003],{"class":85},[75,4224,4225,4228,4230,4232,4235],{"class":77,"line":1325},[75,4226,4227],{"class":85},"                    (sin2_modulated_sine, ",[75,4229,3011],{"class":134},[75,4231,3014],{"class":145},[75,4233,4234],{"class":134},"$ Modulated Sine Wave\"",[75,4236,3003],{"class":85},[75,4238,4239,4242,4244,4246,4248],{"class":77,"line":1338},[75,4240,4241],{"class":85},"                    (cos2_modulated_sine, ",[75,4243,3027],{"class":134},[75,4245,3014],{"class":145},[75,4247,4234],{"class":134},[75,4249,3003],{"class":85},[75,4251,4252,4255,4258],{"class":77,"line":1343},[75,4253,4254],{"class":85},"                    (rectangular_modulated_sine, ",[75,4256,4257],{"class":134},"\"Rectangular Modulated Sine Wave\"",[75,4259,3003],{"class":85},[75,4261,4262,4265,4268],{"class":77,"line":1356},[75,4263,4264],{"class":85},"                    (blackman_harris_modulated_sine, ",[75,4266,4267],{"class":134},"\"Blackman-Harris Modulated Sine Wave\"",[75,4269,3003],{"class":85},[75,4271,4272,4275,4278],{"class":77,"line":1887},[75,4273,4274],{"class":85},"                    (tukey_modulated_sine, ",[75,4276,4277],{"class":134},"\"Tukey Modulated Sine Wave\"",[75,4279,3003],{"class":85},[75,4281,4282,4285,4288],{"class":77,"line":1900},[75,4283,4284],{"class":85},"                    (sech_modulated_sine, ",[75,4286,4287],{"class":134},"\"Sech Modulated Sine Wave\"",[75,4289,3074],{"class":85},[75,4291,4292],{"class":77,"line":1913},[75,4293,112],{"emptyLinePlaceholder":111},[75,4295,4296],{"class":77,"line":2657},[75,4297,3083],{"class":118},[75,4299,4300,4302,4304,4306,4308,4310,4312,4314,4316,4318,4320,4322,4324,4326,4328,4330],{"class":77,"line":3177},[75,4301,3088],{"class":85},[75,4303,142],{"class":141},[75,4305,3093],{"class":85},[75,4307,3096],{"class":193},[75,4309,197],{"class":85},[75,4311,3101],{"class":145},[75,4313,206],{"class":85},[75,4315,3101],{"class":145},[75,4317,206],{"class":85},[75,4319,870],{"class":309},[75,4321,142],{"class":141},[75,4323,197],{"class":85},[75,4325,3116],{"class":145},[75,4327,206],{"class":85},[75,4329,3121],{"class":145},[75,4331,283],{"class":85},[75,4333,4334,4336,4338,4340,4342],{"class":77,"line":3188},[75,4335,3128],{"class":85},[75,4337,142],{"class":141},[75,4339,3133],{"class":85},[75,4341,3136],{"class":193},[75,4343,389],{"class":85},[75,4345,4346],{"class":77,"line":3203},[75,4347,112],{"emptyLinePlaceholder":111},[75,4349,4350,4352,4355,4357,4359],{"class":77,"line":3218},[75,4351,3147],{"class":81},[75,4353,4354],{"class":85}," i, (modulated_sine, title) ",[75,4356,3153],{"class":81},[75,4358,3156],{"class":141},[75,4360,4361],{"class":85},"(modulated_sines):\n",[75,4363,4364,4367,4369,4372,4374],{"class":77,"line":3224},[75,4365,4366],{"class":85},"    modulated_wave ",[75,4368,142],{"class":141},[75,4370,4371],{"class":85}," modulated_sine ",[75,4373,500],{"class":141},[75,4375,1222],{"class":85},[75,4377,4378,4380,4382,4385,4387],{"class":77,"line":3229},[75,4379,3164],{"class":85},[75,4381,303],{"class":193},[75,4383,4384],{"class":85},"(time, modulated_wave, ",[75,4386,3172],{"class":134},[75,4388,318],{"class":85},[75,4390,4391,4393,4395],{"class":77,"line":3235},[75,4392,3164],{"class":85},[75,4394,3182],{"class":193},[75,4396,3185],{"class":85},[75,4398,4399,4401,4403,4405,4407],{"class":77,"line":3263},[75,4400,3164],{"class":85},[75,4402,3193],{"class":193},[75,4404,197],{"class":85},[75,4406,3198],{"class":134},[75,4408,318],{"class":85},[75,4410,4411,4413,4415,4417,4419],{"class":77,"line":3275},[75,4412,3164],{"class":85},[75,4414,3208],{"class":193},[75,4416,197],{"class":85},[75,4418,3213],{"class":134},[75,4420,318],{"class":85},[75,4422,4423],{"class":77,"line":3280},[75,4424,112],{"emptyLinePlaceholder":111},[75,4426,4427],{"class":77,"line":3290},[75,4428,3232],{"class":118},[75,4430,4432,4434,4436,4438,4440,4442,4444,4446,4448,4450],{"class":77,"line":4431},44,[75,4433,3147],{"class":81},[75,4435,3240],{"class":85},[75,4437,3153],{"class":81},[75,4439,3245],{"class":141},[75,4441,3248],{"class":85},[75,4443,3251],{"class":141},[75,4445,1437],{"class":145},[75,4447,206],{"class":85},[75,4449,2930],{"class":141},[75,4451,3260],{"class":85},[75,4453,4455,4457,4459],{"class":77,"line":4454},45,[75,4456,3266],{"class":85},[75,4458,3269],{"class":193},[75,4460,3272],{"class":85},[75,4462,4464],{"class":77,"line":4463},46,[75,4465,112],{"emptyLinePlaceholder":111},[75,4467,4469,4471,4473],{"class":77,"line":4468},47,[75,4470,300],{"class":85},[75,4472,3285],{"class":193},[75,4474,389],{"class":85},[75,4476,4478,4480,4482],{"class":77,"line":4477},48,[75,4479,300],{"class":85},[75,4481,386],{"class":193},[75,4483,389],{"class":85},[391,4485],{"alt":4486,"src":4487},"Output 11 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-11.webp",[65,4489,4491],{"className":67,"code":4490,"language":69,"meta":70,"style":70},"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    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12 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-12.webp",[12,5126,5127],{},"Below are some applied examples in the context of common quantum gates.",[12,5129,5130,5131,5134],{},"Relevant equations for the ",[48,5132,5133],{},"modulated Gaussian pulses"," used in the plots for some common quantum gates:",[5136,5137,5139],"h3",{"id":5138},"_1-gaussian-envelope",[48,5140,5141],{},"1. Gaussian Envelope",[12,5143,5144,5145,5573],{},"The Gaussian envelope is used to smoothly shape the pulse to minimize spectral 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{"className":6215},[5308],[75,6217,6219],{"className":6218,"style":5561},[5312],[75,6220],{},[75,6222],{"className":6223},[5270,5380],[75,6225,6227],{"className":6226,"style":5365},[5270,5364],[75,6228,5175],{"className":6229},[5369,5370],[75,6231],{"className":6232,"style":5349},[5274],[75,6234,2845],{"className":6235},[5353],[75,6237,197],{"className":6238},[5263],[75,6240,270],{"className":6241},[5258],[75,6243,5889],{"className":6244,"style":5416},[5258,5259],[75,6246,6248,6252],{"className":6247},[5258],[75,6249,5894],{"className":6250,"style":6251},[5258,5259],"margin-right:0.1076em;",[75,6253,6255],{"className":6254},[5299],[75,6256,6258,6280],{"className":6257},[5303,5304],[75,6259,6261,6277],{"className":6260},[5308],[75,6262,6265],{"className":6263,"style":6264},[5312],"height:0.1514em;",[75,6266,6268,6271],{"style":6267},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[75,6269],{"className":6270,"style":5321},[5320],[75,6272,6274],{"className":6273},[5325,5326,5327,5328],[75,6275,5897],{"className":6276},[5258,5259,5328],[75,6278,5336],{"className":6279},[5335],[75,6281,6283],{"className":6282},[5308],[75,6284,6286],{"className":6285,"style":5343},[5312],[75,6287],{},[75,6289,2771],{"className":6290},[5258,5259],[75,6292],{"className":6293,"style":6294},[5274],"margin-right:0.2222em;",[75,6296,3251],{"className":6297},[5477],[75,6299],{"className":6300,"style":6294},[5274],[75,6302,6304,6307,6347],{"className":6303},[5249],[75,6305],{"className":6306,"style":5254},[5253],[75,6308,6310,6313],{"className":6309},[5258],[75,6311,5906],{"className":6312},[5258,5259],[75,6314,6316],{"className":6315},[5299],[75,6317,6319,6339],{"className":6318},[5303,5304],[75,6320,6322,6336],{"className":6321},[5308],[75,6323,6325],{"className":6324,"style":5942},[5312],[75,6326,6327,6330],{"style":5316},[75,6328],{"className":6329,"style":5321},[5320],[75,6331,6333],{"className":6332},[5325,5326,5327,5328],[75,6334,5820],{"className":6335,"style":5955},[5258,5259,5328],[75,6337,5336],{"className":6338},[5335],[75,6340,6342],{"className":6341},[5308],[75,6343,6345],{"className":6344,"style":5343},[5312],[75,6346],{},[75,6348,5175],{"className":6349},[5270],[28,6351,6352,6435,6544],{},[31,6353,6354,6434],{},[75,6355,6357,6379],{"className":6356},[5148],[75,6358,6360],{"className":6359},[5152],[5154,6361,6362],{"xmlns":5156},[5158,6363,6364,6376],{},[5161,6365,6366,6368,6374],{},[5168,6367,197],{"stretchy":5170},[5179,6369,6370,6372],{},[5164,6371,5894],{},[5164,6373,5897],{},[5168,6375,5175],{"stretchy":5170},[5238,6377,6378],{"encoding":5240},"( f_c )",[75,6380,6382],{"className":6381,"ariaHidden":5196},[5245],[75,6383,6385,6388,6391,6431],{"className":6384},[5249],[75,6386],{"className":6387,"style":5254},[5253],[75,6389,197],{"className":6390},[5263],[75,6392,6394,6397],{"className":6393},[5258],[75,6395,5894],{"className":6396,"style":6251},[5258,5259],[75,6398,6400],{"className":6399},[5299],[75,6401,6403,6423],{"className":6402},[5303,5304],[75,6404,6406,6420],{"className":6405},[5308],[75,6407,6409],{"className":6408,"style":6264},[5312],[75,6410,6411,6414],{"style":6267},[75,6412],{"className":6413,"style":5321},[5320],[75,6415,6417],{"className":6416},[5325,5326,5327,5328],[75,6418,5897],{"className":6419},[5258,5259,5328],[75,6421,5336],{"className":6422},[5335],[75,6424,6426],{"className":6425},[5308],[75,6427,6429],{"className":6428,"style":5343},[5312],[75,6430],{},[75,6432,5175],{"className":6433},[5270]," is the carrier frequency (typically GHz range),",[31,6436,6437,6539,6540,6543],{},[75,6438,6440,6466],{"className":6439},[5148],[75,6441,6443],{"className":6442},[5152],[5154,6444,6445],{"xmlns":5156},[5158,6446,6447,6463],{},[5161,6448,6449,6451,6457,6459,6461],{},[5168,6450,197],{"stretchy":5170},[5179,6452,6453,6455],{},[5164,6454,5906],{},[5164,6456,5820],{},[5168,6458,142],{},[5184,6460,5186],{},[5168,6462,5175],{"stretchy":5170},[5238,6464,6465],{"encoding":5240},"( \\phi_X = 0 )",[75,6467,6469,6527],{"className":6468,"ariaHidden":5196},[5245],[75,6470,6472,6475,6478,6518,6521,6524],{"className":6471},[5249],[75,6473],{"className":6474,"style":5254},[5253],[75,6476,197],{"className":6477},[5263],[75,6479,6481,6484],{"className":6480},[5258],[75,6482,5906],{"className":6483},[5258,5259],[75,6485,6487],{"className":6486},[5299],[75,6488,6490,6510],{"className":6489},[5303,5304],[75,6491,6493,6507],{"className":6492},[5308],[75,6494,6496],{"className":6495,"style":5942},[5312],[75,6497,6498,6501],{"style":5316},[75,6499],{"className":6500,"style":5321},[5320],[75,6502,6504],{"className":6503},[5325,5326,5327,5328],[75,6505,5820],{"className":6506,"style":5955},[5258,5259,5328],[75,6508,5336],{"className":6509},[5335],[75,6511,6513],{"className":6512},[5308],[75,6514,6516],{"className":6515,"style":5343},[5312],[75,6517],{},[75,6519],{"className":6520,"style":5275},[5274],[75,6522,142],{"className":6523},[5279],[75,6525],{"className":6526,"style":5275},[5274],[75,6528,6530,6533,6536],{"className":6529},[5249],[75,6531],{"className":6532,"style":5254},[5253],[75,6534,5186],{"className":6535},[5258],[75,6537,5175],{"className":6538},[5270]," for an ",[48,6541,6542],{},"X gate",".",[31,6545,6546,6547,6550],{},"The modulation is centered at the qubit frequency with ",[48,6548,6549],{},"zero phase shift",", leading to a cosine-like oscillation.",[5136,6552,6554],{"id":6553},"_3-modulated-pulse-for-pauli-y-gate",[48,6555,6556],{},"3. Modulated Pulse for Pauli-Y Gate",[12,6558,5792,6559,5796,6562,5800,6613,5573],{},[48,6560,6561],{},"Pauli-Y gate",[48,6563,6564,6612],{},[75,6565,6567,6590],{"className":6566},[5148],[75,6568,6570],{"className":6569},[5152],[5154,6571,6572],{"xmlns":5156},[5158,6573,6574,6587],{},[5161,6575,6576,6578,6580,6583,6585],{},[5168,6577,197],{"stretchy":5170},[5164,6579,5889],{},[5164,6581,2810],{"mathvariant":6582},"normal",[5184,6584,270],{},[5168,6586,5175],{"stretchy":5170},[5238,6588,6589],{"encoding":5240},"(\\pi\u002F2)",[75,6591,6593],{"className":6592,"ariaHidden":5196},[5245],[75,6594,6596,6599,6602,6605,6609],{"className":6595},[5249],[75,6597],{"className":6598,"style":5254},[5253],[75,6600,197],{"className":6601},[5263],[75,6603,5889],{"className":6604,"style":5416},[5258,5259],[75,6606,6608],{"className":6607},[5258],"\u002F2",[75,6610,5175],{"className":6611},[5270]," phase-shifted cosine modulation",[75,6614,6616,6721],{"className":6615},[5148],[75,6617,6619],{"className":6618},[5152],[5154,6620,6621],{"xmlns":5156},[5158,6622,6623,6718],{},[5161,6624,6625,6632,6634,6636,6638,6640,6646,6648,6650,6690,6692,6694,6696,6698,6700,6706,6708,6710,6716],{},[5179,6626,6627,6629],{},[5164,6628,5817],{},[5164,6630,6631],{},"Y",[5168,6633,197],{"stretchy":5170},[5164,6635,2771],{},[5168,6637,5175],{"stretchy":5170},[5168,6639,142],{},[5179,6641,6642,6644],{},[5164,6643,5166],{},[5184,6645,5186],{},[5164,6647,243],{},[5168,6649,5191],{},[5161,6651,6652,6654,6656,6688],{},[5168,6653,197],{"fence":5196},[5168,6655,5199],{},[5201,6657,6658,6678],{},[5161,6659,6660,6662,6664,6666,6672],{},[5168,6661,197],{"stretchy":5170},[5164,6663,2771],{},[5168,6665,5199],{},[5179,6667,6668,6670],{},[5164,6669,2771],{},[5184,6671,5186],{},[5218,6673,6674,6676],{},[5168,6675,5175],{"stretchy":5170},[5184,6677,270],{},[5161,6679,6680,6682],{},[5184,6681,270],{},[5218,6683,6684,6686],{},[5164,6685,5232],{},[5184,6687,270],{},[5168,6689,5175],{"fence":5196},[5164,6691,2845],{},[5168,6693,5191],{},[5168,6695,197],{"stretchy":5170},[5184,6697,270],{},[5164,6699,5889],{},[5179,6701,6702,6704],{},[5164,6703,5894],{},[5164,6705,5897],{},[5164,6707,2771],{},[5168,6709,3251],{},[5179,6711,6712,6714],{},[5164,6713,5906],{},[5164,6715,6631],{},[5168,6717,5175],{"stretchy":5170},[5238,6719,6720],{"encoding":5240},"E_Y(t) = A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right) \\cos(2\\pi f_c t + \\phi_Y)",[75,6722,6724,6788,7100],{"className":6723,"ariaHidden":5196},[5245],[75,6725,6727,6730,6770,6773,6776,6779,6782,6785],{"className":6726},[5249],[75,6728],{"className":6729,"style":5254},[5253],[75,6731,6733,6736],{"className":6732},[5258],[75,6734,5817],{"className":6735,"style":5929},[5258,5259],[75,6737,6739],{"className":6738},[5299],[75,6740,6742,6762],{"className":6741},[5303,5304],[75,6743,6745,6759],{"className":6744},[5308],[75,6746,6748],{"className":6747,"style":5942},[5312],[75,6749,6750,6753],{"style":5945},[75,6751],{"className":6752,"style":5321},[5320],[75,6754,6756],{"className":6755},[5325,5326,5327,5328],[75,6757,6631],{"className":6758,"style":6294},[5258,5259,5328],[75,6760,5336],{"className":6761},[5335],[75,6763,6765],{"className":6764},[5308],[75,6766,6768],{"className":6767,"style":5343},[5312],[75,6769],{},[75,6771,197],{"className":6772},[5263],[75,6774,2771],{"className":6775},[5258,5259],[75,6777,5175],{"className":6778},[5270],[75,6780],{"className":6781,"style":5275},[5274],[75,6783,142],{"className":6784},[5279],[75,6786],{"className":6787,"style":5275},[5274],[75,6789,6791,6794,6834,6837,6840,6843,7033,7036,7039,7042,7045,7048,7088,7091,7094,7097],{"className":6790},[5249],[75,6792],{"className":6793,"style":5289},[5253],[75,6795,6797,6800],{"className":6796},[5258],[75,6798,5166],{"className":6799},[5258,5259],[75,6801,6803],{"className":6802},[5299],[75,6804,6806,6826],{"className":6805},[5303,5304],[75,6807,6809,6823],{"className":6808},[5308],[75,6810,6812],{"className":6811,"style":5313},[5312],[75,6813,6814,6817],{"style":5316},[75,6815],{"className":6816,"style":5321},[5320],[75,6818,6820],{"className":6819},[5325,5326,5327,5328],[75,6821,5186],{"className":6822},[5258,5328],[75,6824,5336],{"className":6825},[5335],[75,6827,6829],{"className":6828},[5308],[75,6830,6832],{"className":6831,"style":5343},[5312],[75,6833],{},[75,6835],{"className":6836,"style":5349},[5274],[75,6838,243],{"className":6839},[5353],[75,6841],{"className":6842,"style":5349},[5274],[75,6844,6846,6852,6855,7027],{"className":6845},[5360],[75,6847,6849],{"className":6848,"style":5365},[5263,5364],[75,6850,197],{"className":6851},[5369,5370],[75,6853,5199],{"className":6854},[5258],[75,6856,6858,6861,7024],{"className":6857},[5258],[75,6859],{"className":6860},[5263,5380],[75,6862,6864],{"className":6863},[5201],[75,6865,6867,7016],{"className":6866},[5303,5304],[75,6868,6870,7013],{"className":6869},[5308],[75,6871,6873,6916,6924],{"className":6872,"style":5393},[5312],[75,6874,6875,6878],{"style":5396},[75,6876],{"className":6877,"style":5400},[5320],[75,6879,6881],{"className":6880},[5325,5326,5327,5328],[75,6882,6884,6887],{"className":6883},[5258,5328],[75,6885,270],{"className":6886},[5258,5328],[75,6888,6890,6893],{"className":6889},[5258,5328],[75,6891,5232],{"className":6892,"style":5416},[5258,5259,5328],[75,6894,6896],{"className":6895},[5299],[75,6897,6899],{"className":6898},[5303],[75,6900,6902],{"className":6901},[5308],[75,6903,6905],{"className":6904,"style":5429},[5312],[75,6906,6907,6910],{"style":5432},[75,6908],{"className":6909,"style":5436},[5320],[75,6911,6913],{"className":6912},[5325,5440,5441,5328],[75,6914,270],{"className":6915},[5258,5328],[75,6917,6918,6921],{"style":5447},[75,6919],{"className":6920,"style":5400},[5320],[75,6922],{"className":6923,"style":5455},[5454],[75,6925,6926,6929],{"style":5458},[75,6927],{"className":6928,"style":5400},[5320],[75,6930,6932],{"className":6931},[5325,5326,5327,5328],[75,6933,6935,6938,6941,6944,6984],{"className":6934},[5258,5328],[75,6936,197],{"className":6937},[5263,5328],[75,6939,2771],{"className":6940},[5258,5259,5328],[75,6942,5199],{"className":6943},[5477,5328],[75,6945,6947,6950],{"className":6946},[5258,5328],[75,6948,2771],{"className":6949},[5258,5259,5328],[75,6951,6953],{"className":6952},[5299],[75,6954,6956,6976],{"className":6955},[5303,5304],[75,6957,6959,6973],{"className":6958},[5308],[75,6960,6962],{"className":6961,"style":5496},[5312],[75,6963,6964,6967],{"style":5499},[75,6965],{"className":6966,"style":5436},[5320],[75,6968,6970],{"className":6969},[5325,5440,5441,5328],[75,6971,5186],{"className":6972},[5258,5328],[75,6974,5336],{"className":6975},[5335],[75,6977,6979],{"className":6978},[5308],[75,6980,6982],{"className":6981,"style":5518},[5312],[75,6983],{},[75,6985,6987,6990],{"className":6986},[5270,5328],[75,6988,5175],{"className":6989},[5270,5328],[75,6991,6993],{"className":6992},[5299],[75,6994,6996],{"className":6995},[5303],[75,6997,6999],{"className":6998},[5308],[75,7000,7002],{"className":7001,"style":5539},[5312],[75,7003,7004,7007],{"style":5542},[75,7005],{"className":7006,"style":5436},[5320],[75,7008,7010],{"className":7009},[5325,5440,5441,5328],[75,7011,270],{"className":7012},[5258,5328],[75,7014,5336],{"className":7015},[5335],[75,7017,7019],{"className":7018},[5308],[75,7020,7022],{"className":7021,"style":5561},[5312],[75,7023],{},[75,7025],{"className":7026},[5270,5380],[75,7028,7030],{"className":7029,"style":5365},[5270,5364],[75,7031,5175],{"className":7032},[5369,5370],[75,7034],{"className":7035,"style":5349},[5274],[75,7037,2845],{"className":7038},[5353],[75,7040,197],{"className":7041},[5263],[75,7043,270],{"className":7044},[5258],[75,7046,5889],{"className":7047,"style":5416},[5258,5259],[75,7049,7051,7054],{"className":7050},[5258],[75,7052,5894],{"className":7053,"style":6251},[5258,5259],[75,7055,7057],{"className":7056},[5299],[75,7058,7060,7080],{"className":7059},[5303,5304],[75,7061,7063,7077],{"className":7062},[5308],[75,7064,7066],{"className":7065,"style":6264},[5312],[75,7067,7068,7071],{"style":6267},[75,7069],{"className":7070,"style":5321},[5320],[75,7072,7074],{"className":7073},[5325,5326,5327,5328],[75,7075,5897],{"className":7076},[5258,5259,5328],[75,7078,5336],{"className":7079},[5335],[75,7081,7083],{"className":7082},[5308],[75,7084,7086],{"className":7085,"style":5343},[5312],[75,7087],{},[75,7089,2771],{"className":7090},[5258,5259],[75,7092],{"className":7093,"style":6294},[5274],[75,7095,3251],{"className":7096},[5477],[75,7098],{"className":7099,"style":6294},[5274],[75,7101,7103,7106,7146],{"className":7102},[5249],[75,7104],{"className":7105,"style":5254},[5253],[75,7107,7109,7112],{"className":7108},[5258],[75,7110,5906],{"className":7111},[5258,5259],[75,7113,7115],{"className":7114},[5299],[75,7116,7118,7138],{"className":7117},[5303,5304],[75,7119,7121,7135],{"className":7120},[5308],[75,7122,7124],{"className":7123,"style":5942},[5312],[75,7125,7126,7129],{"style":5316},[75,7127],{"className":7128,"style":5321},[5320],[75,7130,7132],{"className":7131},[5325,5326,5327,5328],[75,7133,6631],{"className":7134,"style":6294},[5258,5259,5328],[75,7136,5336],{"className":7137},[5335],[75,7139,7141],{"className":7140},[5308],[75,7142,7144],{"className":7143,"style":5343},[5312],[75,7145],{},[75,7147,5175],{"className":7148},[5270],[28,7150,7151,7328],{},[31,7152,7153,7327],{},[75,7154,7156,7186],{"className":7155},[5148],[75,7157,7159],{"className":7158},[5152],[5154,7160,7161],{"xmlns":5156},[5158,7162,7163,7183],{},[5161,7164,7165,7167,7173,7175,7181],{},[5168,7166,197],{"stretchy":5170},[5179,7168,7169,7171],{},[5164,7170,5906],{},[5164,7172,6631],{},[5168,7174,142],{},[5201,7176,7177,7179],{},[5164,7178,5889],{},[5184,7180,270],{},[5168,7182,5175],{"stretchy":5170},[5238,7184,7185],{"encoding":5240},"( \\phi_Y = \\frac{\\pi}{2} )",[75,7187,7189,7247],{"className":7188,"ariaHidden":5196},[5245],[75,7190,7192,7195,7198,7238,7241,7244],{"className":7191},[5249],[75,7193],{"className":7194,"style":5254},[5253],[75,7196,197],{"className":7197},[5263],[75,7199,7201,7204],{"className":7200},[5258],[75,7202,5906],{"className":7203},[5258,5259],[75,7205,7207],{"className":7206},[5299],[75,7208,7210,7230],{"className":7209},[5303,5304],[75,7211,7213,7227],{"className":7212},[5308],[75,7214,7216],{"className":7215,"style":5942},[5312],[75,7217,7218,7221],{"style":5316},[75,7219],{"className":7220,"style":5321},[5320],[75,7222,7224],{"className":7223},[5325,5326,5327,5328],[75,7225,6631],{"className":7226,"style":6294},[5258,5259,5328],[75,7228,5336],{"className":7229},[5335],[75,7231,7233],{"className":7232},[5308],[75,7234,7236],{"className":7235,"style":5343},[5312],[75,7237],{},[75,7239],{"className":7240,"style":5275},[5274],[75,7242,142],{"className":7243},[5279],[75,7245],{"className":7246,"style":5275},[5274],[75,7248,7250,7254,7324],{"className":7249},[5249],[75,7251],{"className":7252,"style":7253},[5253],"height:1.095em;vertical-align:-0.345em;",[75,7255,7257,7260,7321],{"className":7256},[5258],[75,7258],{"className":7259},[5263,5380],[75,7261,7263],{"className":7262},[5201],[75,7264,7266,7313],{"className":7265},[5303,5304],[75,7267,7269,7310],{"className":7268},[5308],[75,7270,7273,7287,7295],{"className":7271,"style":7272},[5312],"height:0.6954em;",[75,7274,7275,7278],{"style":5396},[75,7276],{"className":7277,"style":5400},[5320],[75,7279,7281],{"className":7280},[5325,5326,5327,5328],[75,7282,7284],{"className":7283},[5258,5328],[75,7285,270],{"className":7286},[5258,5328],[75,7288,7289,7292],{"style":5447},[75,7290],{"className":7291,"style":5400},[5320],[75,7293],{"className":7294,"style":5455},[5454],[75,7296,7298,7301],{"style":7297},"top:-3.394em;",[75,7299],{"className":7300,"style":5400},[5320],[75,7302,7304],{"className":7303},[5325,5326,5327,5328],[75,7305,7307],{"className":7306},[5258,5328],[75,7308,5889],{"className":7309,"style":5416},[5258,5259,5328],[75,7311,5336],{"className":7312},[5335],[75,7314,7316],{"className":7315},[5308],[75,7317,7319],{"className":7318,"style":5561},[5312],[75,7320],{},[75,7322],{"className":7323},[5270,5380],[75,7325,5175],{"className":7326},[5270]," shifts the phase by 90° (producing a sine-like waveform).",[31,7329,7330,7331,7380],{},"The modulation has a ",[48,7332,7333,7379],{},[75,7334,7336,7358],{"className":7335},[5148],[75,7337,7339],{"className":7338},[5152],[5154,7340,7341],{"xmlns":5156},[5158,7342,7343,7355],{},[5161,7344,7345,7347,7349,7351,7353],{},[5168,7346,197],{"stretchy":5170},[5164,7348,5889],{},[5164,7350,2810],{"mathvariant":6582},[5184,7352,270],{},[5168,7354,5175],{"stretchy":5170},[5238,7356,7357],{"encoding":5240},"( \\pi\u002F2 )",[75,7359,7361],{"className":7360,"ariaHidden":5196},[5245],[75,7362,7364,7367,7370,7373,7376],{"className":7363},[5249],[75,7365],{"className":7366,"style":5254},[5253],[75,7368,197],{"className":7369},[5263],[75,7371,5889],{"className":7372,"style":5416},[5258,5259],[75,7374,6608],{"className":7375},[5258],[75,7377,5175],{"className":7378},[5270]," phase shift",", which shifts the waveform by a quarter cycle, making it a sine-like oscillation.",[5136,7382,7384],{"id":7383},"_4-why-no-pulse-for-pauli-z-gate",[48,7385,7386],{},"4. Why No Pulse for Pauli-Z Gate?",[12,7388,7389,7390,7434,7435,5800,7438,7619],{},"For ",[48,7391,7392,7393],{},"Pauli-Z ",[75,7394,7396,7415],{"className":7395},[5148],[75,7397,7399],{"className":7398},[5152],[5154,7400,7401],{"xmlns":5156},[5158,7402,7403,7412],{},[5161,7404,7405,7407,7410],{},[5168,7406,197],{"stretchy":5170},[5164,7408,7409],{},"Z",[5168,7411,5175],{"stretchy":5170},[5238,7413,7414],{"encoding":5240},"( Z )",[75,7416,7418],{"className":7417,"ariaHidden":5196},[5245],[75,7419,7421,7424,7427,7431],{"className":7420},[5249],[75,7422],{"className":7423,"style":5254},[5253],[75,7425,197],{"className":7426},[5263],[75,7428,7409],{"className":7429,"style":7430},[5258,5259],"margin-right:0.0715em;",[75,7432,5175],{"className":7433},[5270],", no physical pulse is needed because it is implemented as a ",[48,7436,7437],{},"virtual phase shift",[75,7439,7441,7504],{"className":7440},[5148],[75,7442,7444],{"className":7443},[5152],[5154,7445,7446],{"xmlns":5156},[5158,7447,7448,7501],{},[5161,7449,7450,7453,7455,7458,7461,7463,7465,7467,7470,7473,7475,7478,7480,7482,7495,7497,7499],{},[5164,7451,7452],{"mathvariant":6582},"∣",[5184,7454,5186],{},[5168,7456,7457],{"stretchy":5170},"⟩",[5168,7459,7460],{},"→",[5164,7462,7452],{"mathvariant":6582},[5184,7464,5186],{},[5168,7466,7457],{"stretchy":5170},[5168,7468,7469],{"separator":5196},",",[5274,7471],{"width":7472},"1em",[5164,7474,7452],{"mathvariant":6582},[5184,7476,7477],{},"1",[5168,7479,7457],{"stretchy":5170},[5168,7481,7460],{},[5218,7483,7484,7487],{},[5164,7485,7486],{},"e",[5161,7488,7489,7492],{},[5164,7490,7491],{},"i",[5164,7493,7494],{},"θ",[5164,7496,7452],{"mathvariant":6582},[5184,7498,7477],{},[5168,7500,7457],{"stretchy":5170},[5238,7502,7503],{"encoding":5240},"|0\\rangle \\rightarrow |0\\rangle, \\quad |1\\rangle \\rightarrow 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Pulse",[7650,9646,9647],{},[75,9648,9650,9766],{"className":9649},[5148],[75,9651,9653],{"className":9652},[5152],[5154,9654,9655],{"xmlns":5156},[5158,9656,9657,9763],{},[5161,9658,9659,9677,9679,9681,9683,9685,9691,9693,9695,9735,9737,9739,9741,9743,9745,9751,9753,9755,9761],{},[5179,9660,9661,9663],{},[5164,9662,5817],{},[5161,9664,9665,9668,9671,9674],{},[5164,9666,9667],{},"C",[5164,9669,9670],{},"N",[5164,9672,9673],{},"O",[5164,9675,9676],{},"T",[5168,9678,197],{"stretchy":5170},[5164,9680,2771],{},[5168,9682,5175],{"stretchy":5170},[5168,9684,142],{},[5179,9686,9687,9689],{},[5164,9688,5166],{},[5184,9690,5186],{},[5164,9692,243],{},[5168,9694,5191],{},[5161,9696,9697,9699,9701,9733],{},[5168,9698,197],{"fence":5196},[5168,9700,5199],{},[5201,9702,9703,9723],{},[5161,9704,9705,9707,9709,9711,9717],{},[5168,9706,197],{"stretchy":5170},[5164,9708,2771],{},[5168,9710,5199],{},[5179,9712,9713,9715],{},[5164,9714,2771],{},[5184,9716,5186],{},[5218,9718,9719,9721],{},[5168,9720,5175],{"stretchy":5170},[5184,9722,270],{},[5161,9724,9725,9727],{},[5184,9726,270],{},[5218,9728,9729,9731],{},[5164,9730,5232],{},[5184,9732,270],{},[5168,9734,5175],{"fence":5196},[5164,9736,2845],{},[5168,9738,5191],{},[5168,9740,197],{"stretchy":5170},[5184,9742,270],{},[5164,9744,5889],{},[5179,9746,9747,9749],{},[5164,9748,5894],{},[5164,9750,5897],{},[5164,9752,2771],{},[5168,9754,3251],{},[5179,9756,9757,9759],{},[5164,9758,5906],{},[5164,9760,5820],{},[5168,9762,5175],{"stretchy":5170},[5238,9764,9765],{"encoding":5240},"E_{CNOT}(t) = A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right) \\cos(2\\pi f_c t + \\phi_X)",[75,9767,9769,9847,10159],{"className":9768,"ariaHidden":5196},[5245],[75,9770,9772,9775,9829,9832,9835,9838,9841,9844],{"className":9771},[5249],[75,9773],{"className":9774,"style":5254},[5253],[75,9776,9778,9781],{"className":9777},[5258],[75,9779,5817],{"className":9780,"style":5929},[5258,5259],[75,9782,9784],{"className":9783},[5299],[75,9785,9787,9821],{"className":9786},[5303,5304],[75,9788,9790,9818],{"className":9789},[5308],[75,9791,9793],{"className":9792,"style":5942},[5312],[75,9794,9795,9798],{"style":5945},[75,9796],{"className":9797,"style":5321},[5320],[75,9799,9801],{"className":9800},[5325,5326,5327,5328],[75,9802,9804,9807,9811,9814],{"className":9803},[5258,5328],[75,9805,9667],{"className":9806,"style":7430},[5258,5259,5328],[75,9808,9670],{"className":9809,"style":9810},[5258,5259,5328],"margin-right:0.109em;",[75,9812,9673],{"className":9813,"style":7612},[5258,5259,5328],[75,9815,9676],{"className":9816,"style":9817},[5258,5259,5328],"margin-right:0.1389em;",[75,9819,5336],{"className":9820},[5335],[75,9822,9824],{"className":9823},[5308],[75,9825,9827],{"className":9826,"style":5343},[5312],[75,9828],{},[75,9830,197],{"className":9831},[5263],[75,9833,2771],{"className":9834},[5258,5259],[75,9836,5175],{"className":9837},[5270],[75,9839],{"className":9840,"style":5275},[5274],[75,9842,142],{"className":9843},[5279],[75,9845],{"className":9846,"style":5275},[5274],[75,9848,9850,9853,9893,9896,9899,9902,10092,10095,10098,10101,10104,10107,10147,10150,10153,10156],{"className":9849},[5249],[75,9851],{"className":9852,"style":5289},[5253],[75,9854,9856,9859],{"className":9855},[5258],[75,9857,5166],{"className":9858},[5258,5259],[75,9860,9862],{"className":9861},[5299],[75,9863,9865,9885],{"className":9864},[5303,5304],[75,9866,9868,9882],{"className":9867},[5308],[75,9869,9871],{"className":9870,"style":5313},[5312],[75,9872,9873,9876],{"style":5316},[75,9874],{"className":9875,"style":5321},[5320],[75,9877,9879],{"className":9878},[5325,5326,5327,5328],[75,9880,5186],{"className":9881},[5258,5328],[75,9883,5336],{"className":9884},[5335],[75,9886,9888],{"className":9887},[5308],[75,9889,9891],{"className":9890,"style":5343},[5312],[75,9892],{},[75,9894],{"className":9895,"style":5349},[5274],[75,9897,243],{"className":9898},[5353],[75,9900],{"className":9901,"style":5349},[5274],[75,9903,9905,9911,9914,10086],{"className":9904},[5360],[75,9906,9908],{"className":9907,"style":5365},[5263,5364],[75,9909,197],{"className":9910},[5369,5370],[75,9912,5199],{"className":9913},[5258],[75,9915,9917,9920,10083],{"className":9916},[5258],[75,9918],{"className":9919},[5263,5380],[75,9921,9923],{"className":9922},[5201],[75,9924,9926,10075],{"className":9925},[5303,5304],[75,9927,9929,10072],{"className":9928},[5308],[75,9930,9932,9975,9983],{"className":9931,"style":5393},[5312],[75,9933,9934,9937],{"style":5396},[75,9935],{"className":9936,"style":5400},[5320],[75,9938,9940],{"className":9939},[5325,5326,5327,5328],[75,9941,9943,9946],{"className":9942},[5258,5328],[75,9944,270],{"className":9945},[5258,5328],[75,9947,9949,9952],{"className":9948},[5258,5328],[75,9950,5232],{"className":9951,"style":5416},[5258,5259,5328],[75,9953,9955],{"className":9954},[5299],[75,9956,9958],{"className":9957},[5303],[75,9959,9961],{"className":9960},[5308],[75,9962,9964],{"className":9963,"style":5429},[5312],[75,9965,9966,9969],{"style":5432},[75,9967],{"className":9968,"style":5436},[5320],[75,9970,9972],{"className":9971},[5325,5440,5441,5328],[75,9973,270],{"className":9974},[5258,5328],[75,9976,9977,9980],{"style":5447},[75,9978],{"className":9979,"style":5400},[5320],[75,9981],{"className":9982,"style":5455},[5454],[75,9984,9985,9988],{"style":5458},[75,9986],{"className":9987,"style":5400},[5320],[75,9989,9991],{"className":9990},[5325,5326,5327,5328],[75,9992,9994,9997,10000,10003,10043],{"className":9993},[5258,5328],[75,9995,197],{"className":9996},[5263,5328],[75,9998,2771],{"className":9999},[5258,5259,5328],[75,10001,5199],{"className":10002},[5477,5328],[75,10004,10006,10009],{"className":10005},[5258,5328],[75,10007,2771],{"className":10008},[5258,5259,5328],[75,10010,10012],{"className":10011},[5299],[75,10013,10015,10035],{"className":10014},[5303,5304],[75,10016,10018,10032],{"className":10017},[5308],[75,10019,10021],{"className":10020,"style":5496},[5312],[75,10022,10023,10026],{"style":5499},[75,10024],{"className":10025,"style":5436},[5320],[75,10027,10029],{"className":10028},[5325,5440,5441,5328],[75,10030,5186],{"className":10031},[5258,5328],[75,10033,5336],{"className":10034},[5335],[75,10036,10038],{"className":10037},[5308],[75,10039,10041],{"className":10040,"style":5518},[5312],[75,10042],{},[75,10044,10046,10049],{"className":10045},[5270,5328],[75,10047,5175],{"className":10048},[5270,5328],[75,10050,10052],{"className":10051},[5299],[75,10053,10055],{"className":10054},[5303],[75,10056,10058],{"className":10057},[5308],[75,10059,10061],{"className":10060,"style":5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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",[72,10212,10213,10223,10233,10237,10242,10255,10267,10279,10283,10288,10309,10335,10339,10344,10392,10396,10401,10410,10420,10429,10438,10447,10452,10456,10461,10507,10521,10525,10530,10545,10578,10586,10599,10611,10615,10620,10655,10659,10664,10682,10686,10691,10725,10729,10734,10747,10760,10772,10784,10788,10793],{"__ignoreMap":70},[75,10214,10215,10217,10219,10221],{"class":77,"line":78},[75,10216,82],{"class":81},[75,10218,86],{"class":85},[75,10220,89],{"class":81},[75,10222,92],{"class":85},[75,10224,10225,10227,10229,10231],{"class":77,"line":95},[75,10226,82],{"class":81},[75,10228,100],{"class":85},[75,10230,89],{"class":81},[75,10232,105],{"class":85},[75,10234,10235],{"class":77,"line":108},[75,10236,112],{"emptyLinePlaceholder":111},[75,10238,10239],{"class":77,"line":115},[75,10240,10241],{"class":118},"# Define parameters\n",[75,10243,10244,10247,10249,10252],{"class":77,"line":122},[75,10245,10246],{"class":85},"t0 ",[75,10248,142],{"class":141},[75,10250,10251],{"class":145}," 0",[75,10253,10254],{"class":118},"      # Pulse center\n",[75,10256,10257,10260,10262,10264],{"class":77,"line":128},[75,10258,10259],{"class":85},"sigma ",[75,10261,142],{"class":141},[75,10263,1075],{"class":145},[75,10265,10266],{"class":118},"   # Standard deviation of Gaussian modulation function (ns)\n",[75,10268,10269,10272,10274,10276],{"class":77,"line":149},[75,10270,10271],{"class":85},"A0 ",[75,10273,142],{"class":141},[75,10275,1437],{"class":145},[75,10277,10278],{"class":118},"      # Peak amplitude\n",[75,10280,10281],{"class":77,"line":154},[75,10282,112],{"emptyLinePlaceholder":111},[75,10284,10285],{"class":77,"line":160},[75,10286,10287],{"class":118},"# Increase the number of points by 1.5x for denser lollipop visualization\n",[75,10289,10290,10293,10295,10298,10300,10302,10304,10307],{"class":77,"line":171},[75,10291,10292],{"class":85},"num_points ",[75,10294,142],{"class":141},[75,10296,10297],{"class":141}," int",[75,10299,197],{"class":85},[75,10301,791],{"class":145},[75,10303,273],{"class":141},[75,10305,10306],{"class":145}," 1.5",[75,10308,318],{"class":85},[75,10310,10311,10313,10315,10317,10319,10321,10323,10325,10327,10329,10332],{"class":77,"line":182},[75,10312,2722],{"class":85},[75,10314,142],{"class":141},[75,10316,190],{"class":85},[75,10318,194],{"class":193},[75,10320,197],{"class":85},[75,10322,200],{"class":141},[75,10324,752],{"class":145},[75,10326,206],{"class":85},[75,10328,752],{"class":145},[75,10330,10331],{"class":85},", num_points)  ",[75,10333,10334],{"class":118},"# Adjusted time axis\n",[75,10336,10337],{"class":77,"line":222},[75,10338,112],{"emptyLinePlaceholder":111},[75,10340,10341],{"class":77,"line":227},[75,10342,10343],{"class":118},"# Compute Gaussian modulation function\n",[75,10345,10346,10348,10350,10353,10355,10357,10359,10361,10363,10366,10368,10371,10373,10375,10377,10379,10381,10383,10386,10388,10390],{"class":77,"line":233},[75,10347,236],{"class":85},[75,10349,142],{"class":141},[75,10351,10352],{"class":85}," A0 ",[75,10354,500],{"class":141},[75,10356,190],{"class":85},[75,10358,243],{"class":193},[75,10360,197],{"class":85},[75,10362,200],{"class":141},[75,10364,10365],{"class":85},"(t ",[75,10367,200],{"class":141},[75,10369,10370],{"class":85}," t0)",[75,10372,258],{"class":141},[75,10374,270],{"class":145},[75,10376,264],{"class":141},[75,10378,267],{"class":85},[75,10380,270],{"class":145},[75,10382,273],{"class":141},[75,10384,10385],{"class":85}," sigma",[75,10387,258],{"class":141},[75,10389,270],{"class":145},[75,10391,283],{"class":85},[75,10393,10394],{"class":77,"line":286},[75,10395,112],{"emptyLinePlaceholder":111},[75,10397,10398],{"class":77,"line":291},[75,10399,10400],{"class":118},"# Store modulation functions and titles in an array\n",[75,10402,10403,10405,10407],{"class":77,"line":297},[75,10404,3592],{"class":85},[75,10406,142],{"class":141},[75,10408,10409],{"class":85}," [\n",[75,10411,10412,10415,10418],{"class":77,"line":321},[75,10413,10414],{"class":85},"    (gaussian_modulation, ",[75,10416,10417],{"class":134},"\"Gaussian Modulation Function for Pauli-X Gate\"",[75,10419,3003],{"class":85},[75,10421,10422,10424,10427],{"class":77,"line":336},[75,10423,10414],{"class":85},[75,10425,10426],{"class":134},"\"Gaussian Modulation Function for Pauli-Y Gate\"",[75,10428,3003],{"class":85},[75,10430,10431,10433,10436],{"class":77,"line":351},[75,10432,10414],{"class":85},[75,10434,10435],{"class":134},"\"Gaussian Modulation Function for Hadamard Gate\"",[75,10437,3003],{"class":85},[75,10439,10440,10442,10445],{"class":77,"line":366},[75,10441,10414],{"class":85},[75,10443,10444],{"class":134},"\"Gaussian Modulation Function for CNOT Gate Target Qubit\"",[75,10446,318],{"class":85},[75,10448,10449],{"class":77,"line":381},[75,10450,10451],{"class":85},"]\n",[75,10453,10454],{"class":77,"line":984},[75,10455,112],{"emptyLinePlaceholder":111},[75,10457,10458],{"class":77,"line":997},[75,10459,10460],{"class":118},"# Create a figure with constrained layout\n",[75,10462,10463,10466,10468,10470,10472,10474,10476,10478,10480,10482,10484,10486,10488,10490,10492,10494,10497,10500,10502,10505],{"class":77,"line":1325},[75,10464,10465],{"class":85},"fig, axs ",[75,10467,142],{"class":141},[75,10469,3093],{"class":85},[75,10471,3096],{"class":193},[75,10473,197],{"class":85},[75,10475,270],{"class":145},[75,10477,206],{"class":85},[75,10479,270],{"class":145},[75,10481,206],{"class":85},[75,10483,870],{"class":309},[75,10485,142],{"class":141},[75,10487,197],{"class":85},[75,10489,3116],{"class":145},[75,10491,206],{"class":85},[75,10493,877],{"class":145},[75,10495,10496],{"class":85},"), ",[75,10498,10499],{"class":309},"constrained_layout",[75,10501,142],{"class":141},[75,10503,10504],{"class":145},"True",[75,10506,318],{"class":85},[75,10508,10509,10512,10514,10517,10519],{"class":77,"line":1338},[75,10510,10511],{"class":85},"axs ",[75,10513,142],{"class":141},[75,10515,10516],{"class":85}," axs.",[75,10518,3136],{"class":193},[75,10520,389],{"class":85},[75,10522,10523],{"class":77,"line":1343},[75,10524,112],{"emptyLinePlaceholder":111},[75,10526,10527],{"class":77,"line":1356},[75,10528,10529],{"class":118},"# Lollipop plot setup\n",[75,10531,10532,10534,10537,10539,10542],{"class":77,"line":1887},[75,10533,3147],{"class":81},[75,10535,10536],{"class":85}," ax, (modulation, title) ",[75,10538,3153],{"class":81},[75,10540,10541],{"class":141}," zip",[75,10543,10544],{"class":85},"(axs, modulations):\n",[75,10546,10547,10550,10552,10554,10556,10558,10560,10562,10564,10566,10568,10570,10572,10574,10576],{"class":77,"line":1900},[75,10548,10549],{"class":85},"    ax.",[75,10551,891],{"class":193},[75,10553,3738],{"class":85},[75,10555,896],{"class":309},[75,10557,142],{"class":141},[75,10559,901],{"class":134},[75,10561,206],{"class":85},[75,10563,906],{"class":309},[75,10565,142],{"class":141},[75,10567,911],{"class":134},[75,10569,206],{"class":85},[75,10571,916],{"class":309},[75,10573,142],{"class":141},[75,10575,921],{"class":134},[75,10577,318],{"class":85},[75,10579,10580,10582,10584],{"class":77,"line":1913},[75,10581,10549],{"class":85},[75,10583,3182],{"class":193},[75,10585,3185],{"class":85},[75,10587,10588,10590,10592,10594,10597],{"class":77,"line":2657},[75,10589,10549],{"class":85},[75,10591,3193],{"class":193},[75,10593,197],{"class":85},[75,10595,10596],{"class":134},"\"Time (ns)\"",[75,10598,318],{"class":85},[75,10600,10601,10603,10605,10607,10609],{"class":77,"line":3177},[75,10602,10549],{"class":85},[75,10604,3208],{"class":193},[75,10606,197],{"class":85},[75,10608,3213],{"class":134},[75,10610,318],{"class":85},[75,10612,10613],{"class":77,"line":3188},[75,10614,112],{"emptyLinePlaceholder":111},[75,10616,10617],{"class":77,"line":3203},[75,10618,10619],{"class":118},"# Memory Addressing with Lollipop Representation\n",[75,10621,10622,10625,10627,10629,10631,10633,10635,10637,10639,10641,10643,10645,10647,10649,10651,10653],{"class":77,"line":3218},[75,10623,10624],{"class":85},"fig_combined, ax_combined ",[75,10626,142],{"class":141},[75,10628,3093],{"class":85},[75,10630,3096],{"class":193},[75,10632,197],{"class":85},[75,10634,870],{"class":309},[75,10636,142],{"class":141},[75,10638,197],{"class":85},[75,10640,3116],{"class":145},[75,10642,206],{"class":85},[75,10644,882],{"class":145},[75,10646,10496],{"class":85},[75,10648,10499],{"class":309},[75,10650,142],{"class":141},[75,10652,10504],{"class":145},[75,10654,318],{"class":85},[75,10656,10657],{"class":77,"line":3224},[75,10658,112],{"emptyLinePlaceholder":111},[75,10660,10661],{"class":77,"line":3229},[75,10662,10663],{"class":118},"# Memory addresses as an index range\n",[75,10665,10666,10669,10671,10673,10676,10678,10680],{"class":77,"line":3235},[75,10667,10668],{"class":85},"memory_addresses ",[75,10670,142],{"class":141},[75,10672,190],{"class":85},[75,10674,10675],{"class":193},"arange",[75,10677,197],{"class":85},[75,10679,2930],{"class":141},[75,10681,2933],{"class":85},[75,10683,10684],{"class":77,"line":3263},[75,10685,112],{"emptyLinePlaceholder":111},[75,10687,10688],{"class":77,"line":3275},[75,10689,10690],{"class":118},"# Lollipop-style plot for memory addressing\n",[75,10692,10693,10696,10698,10701,10703,10705,10707,10709,10711,10713,10715,10717,10719,10721,10723],{"class":77,"line":3280},[75,10694,10695],{"class":85},"ax_combined.",[75,10697,891],{"class":193},[75,10699,10700],{"class":85},"(memory_addresses, gaussian_modulation, ",[75,10702,896],{"class":309},[75,10704,142],{"class":141},[75,10706,901],{"class":134},[75,10708,206],{"class":85},[75,10710,906],{"class":309},[75,10712,142],{"class":141},[75,10714,911],{"class":134},[75,10716,206],{"class":85},[75,10718,916],{"class":309},[75,10720,142],{"class":141},[75,10722,921],{"class":134},[75,10724,318],{"class":85},[75,10726,10727],{"class":77,"line":3290},[75,10728,112],{"emptyLinePlaceholder":111},[75,10730,10731],{"class":77,"line":4431},[75,10732,10733],{"class":118},"# Labels and title\n",[75,10735,10736,10738,10740,10742,10745],{"class":77,"line":4454},[75,10737,10695],{"class":85},[75,10739,3182],{"class":193},[75,10741,197],{"class":85},[75,10743,10744],{"class":134},"\"Memory Addressing with Lollipop Representation for Quantum Gate Control\"",[75,10746,318],{"class":85},[75,10748,10749,10751,10753,10755,10758],{"class":77,"line":4463},[75,10750,10695],{"class":85},[75,10752,3193],{"class":193},[75,10754,197],{"class":85},[75,10756,10757],{"class":134},"\"Memory Address Index\"",[75,10759,318],{"class":85},[75,10761,10762,10764,10766,10768,10770],{"class":77,"line":4468},[75,10763,10695],{"class":85},[75,10765,3208],{"class":193},[75,10767,197],{"class":85},[75,10769,3213],{"class":134},[75,10771,318],{"class":85},[75,10773,10774,10776,10778,10780,10782],{"class":77,"line":4477},[75,10775,10695],{"class":85},[75,10777,371],{"class":193},[75,10779,197],{"class":85},[75,10781,10504],{"class":145},[75,10783,318],{"class":85},[75,10785,10786],{"class":77,"line":5105},[75,10787,112],{"emptyLinePlaceholder":111},[75,10789,10790],{"class":77,"line":5114},[75,10791,10792],{"class":118},"# Display plots\n",[75,10794,10796,10798,10800],{"class":77,"line":10795},51,[75,10797,300],{"class":85},[75,10799,386],{"class":193},[75,10801,389],{"class":85},[391,10803],{"alt":10804,"src":10805},"Output 13 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-13.webp",[391,10807],{"alt":10808,"src":10809},"Output 14 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-14.webp",[65,10811,10813],{"className":67,"code":10812,"language":69,"meta":70,"style":70},"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",[72,10814,10815,10825,10835,10839,10843,10853,10863,10873,10885,10889,10894,10912,10936,10940,10944,10988,10992,10997,11029,11060,11064,11069,11104,11137,11150,11163,11176,11207,11236,11248,11256,11260],{"__ignoreMap":70},[75,10816,10817,10819,10821,10823],{"class":77,"line":78},[75,10818,82],{"class":81},[75,10820,86],{"class":85},[75,10822,89],{"class":81},[75,10824,92],{"class":85},[75,10826,10827,10829,10831,10833],{"class":77,"line":95},[75,10828,82],{"class":81},[75,10830,100],{"class":85},[75,10832,89],{"class":81},[75,10834,105],{"class":85},[75,10836,10837],{"class":77,"line":108},[75,10838,112],{"emptyLinePlaceholder":111},[75,10840,10841],{"class":77,"line":115},[75,10842,10241],{"class":118},[75,10844,10845,10847,10849,10851],{"class":77,"line":122},[75,10846,10246],{"class":85},[75,10848,142],{"class":141},[75,10850,10251],{"class":145},[75,10852,10254],{"class":118},[75,10854,10855,10857,10859,10861],{"class":77,"line":128},[75,10856,10259],{"class":85},[75,10858,142],{"class":141},[75,10860,1075],{"class":145},[75,10862,10266],{"class":118},[75,10864,10865,10867,10869,10871],{"class":77,"line":149},[75,10866,10271],{"class":85},[75,10868,142],{"class":141},[75,10870,1437],{"class":145},[75,10872,10278],{"class":118},[75,10874,10875,10877,10879,10882],{"class":77,"line":154},[75,10876,1070],{"class":85},[75,10878,142],{"class":141},[75,10880,10881],{"class":145}," 0.2",[75,10883,10884],{"class":118},"  # Frequency of the carrier wave (normalized)\n",[75,10886,10887],{"class":77,"line":160},[75,10888,112],{"emptyLinePlaceholder":111},[75,10890,10891],{"class":77,"line":171},[75,10892,10893],{"class":118},"# Increase the number of points by 1.5x for denser visualization\n",[75,10895,10896,10898,10900,10902,10904,10906,10908,10910],{"class":77,"line":182},[75,10897,10292],{"class":85},[75,10899,142],{"class":141},[75,10901,10297],{"class":141},[75,10903,197],{"class":85},[75,10905,791],{"class":145},[75,10907,273],{"class":141},[75,10909,10306],{"class":145},[75,10911,318],{"class":85},[75,10913,10914,10916,10918,10920,10922,10924,10926,10928,10930,10932,10934],{"class":77,"line":222},[75,10915,2722],{"class":85},[75,10917,142],{"class":141},[75,10919,190],{"class":85},[75,10921,194],{"class":193},[75,10923,197],{"class":85},[75,10925,200],{"class":141},[75,10927,752],{"class":145},[75,10929,206],{"class":85},[75,10931,752],{"class":145},[75,10933,10331],{"class":85},[75,10935,10334],{"class":118},[75,10937,10938],{"class":77,"line":227},[75,10939,112],{"emptyLinePlaceholder":111},[75,10941,10942],{"class":77,"line":233},[75,10943,10343],{"class":118},[75,10945,10946,10948,10950,10952,10954,10956,10958,10960,10962,10964,10966,10968,10970,10972,10974,10976,10978,10980,10982,10984,10986],{"class":77,"line":286},[75,10947,236],{"class":85},[75,10949,142],{"class":141},[75,10951,10352],{"class":85},[75,10953,500],{"class":141},[75,10955,190],{"class":85},[75,10957,243],{"class":193},[75,10959,197],{"class":85},[75,10961,200],{"class":141},[75,10963,10365],{"class":85},[75,10965,200],{"class":141},[75,10967,10370],{"class":85},[75,10969,258],{"class":141},[75,10971,270],{"class":145},[75,10973,264],{"class":141},[75,10975,267],{"class":85},[75,10977,270],{"class":145},[75,10979,273],{"class":141},[75,10981,10385],{"class":85},[75,10983,258],{"class":141},[75,10985,270],{"class":145},[75,10987,283],{"class":85},[75,10989,10990],{"class":77,"line":291},[75,10991,112],{"emptyLinePlaceholder":111},[75,10993,10994],{"class":77,"line":297},[75,10995,10996],{"class":118},"# Generate In-phase (I) and Quadrature (Q) components\n",[75,10998,10999,11002,11004,11006,11008,11010,11012,11014,11016,11018,11020,11022,11024,11026],{"class":77,"line":321},[75,11000,11001],{"class":85},"I ",[75,11003,142],{"class":141},[75,11005,1217],{"class":85},[75,11007,500],{"class":141},[75,11009,190],{"class":85},[75,11011,2845],{"class":193},[75,11013,197],{"class":85},[75,11015,270],{"class":145},[75,11017,273],{"class":141},[75,11019,1188],{"class":85},[75,11021,500],{"class":141},[75,11023,1193],{"class":85},[75,11025,500],{"class":141},[75,11027,11028],{"class":85}," t)\n",[75,11030,11031,11034,11036,11038,11040,11042,11044,11046,11048,11050,11052,11054,11056,11058],{"class":77,"line":336},[75,11032,11033],{"class":85},"Q ",[75,11035,142],{"class":141},[75,11037,1217],{"class":85},[75,11039,500],{"class":141},[75,11041,190],{"class":85},[75,11043,1179],{"class":193},[75,11045,197],{"class":85},[75,11047,270],{"class":145},[75,11049,273],{"class":141},[75,11051,1188],{"class":85},[75,11053,500],{"class":141},[75,11055,1193],{"class":85},[75,11057,500],{"class":141},[75,11059,11028],{"class":85},[75,11061,11062],{"class":77,"line":351},[75,11063,112],{"emptyLinePlaceholder":111},[75,11065,11066],{"class":77,"line":366},[75,11067,11068],{"class":118},"# Plot IQ diagram\n",[75,11070,11071,11074,11076,11078,11080,11082,11084,11086,11088,11090,11092,11094,11096,11098,11100,11102],{"class":77,"line":381},[75,11072,11073],{"class":85},"fig, ax ",[75,11075,142],{"class":141},[75,11077,3093],{"class":85},[75,11079,3096],{"class":193},[75,11081,197],{"class":85},[75,11083,870],{"class":309},[75,11085,142],{"class":141},[75,11087,197],{"class":85},[75,11089,882],{"class":145},[75,11091,206],{"class":85},[75,11093,882],{"class":145},[75,11095,10496],{"class":85},[75,11097,10499],{"class":309},[75,11099,142],{"class":141},[75,11101,10504],{"class":145},[75,11103,318],{"class":85},[75,11105,11106,11109,11111,11114,11117,11119,11121,11123,11126,11128,11130,11132,11135],{"class":77,"line":984},[75,11107,11108],{"class":85},"ax.",[75,11110,303],{"class":193},[75,11112,11113],{"class":85},"(I, Q, ",[75,11115,11116],{"class":134},"'bo-'",[75,11118,206],{"class":85},[75,11120,2953],{"class":309},[75,11122,142],{"class":141},[75,11124,11125],{"class":145},"0.7",[75,11127,206],{"class":85},[75,11129,573],{"class":309},[75,11131,142],{"class":141},[75,11133,11134],{"class":134},"\"IQ Trajectory\"",[75,11136,318],{"class":85},[75,11138,11139,11141,11143,11145,11148],{"class":77,"line":997},[75,11140,11108],{"class":85},[75,11142,3193],{"class":193},[75,11144,197],{"class":85},[75,11146,11147],{"class":134},"\"In-phase (I)\"",[75,11149,318],{"class":85},[75,11151,11152,11154,11156,11158,11161],{"class":77,"line":1325},[75,11153,11108],{"class":85},[75,11155,3208],{"class":193},[75,11157,197],{"class":85},[75,11159,11160],{"class":134},"\"Quadrature (Q)\"",[75,11162,318],{"class":85},[75,11164,11165,11167,11169,11171,11174],{"class":77,"line":1338},[75,11166,11108],{"class":85},[75,11168,3182],{"class":193},[75,11170,197],{"class":85},[75,11172,11173],{"class":134},"\"IQ Diagram for Gaussian Modulated Pulse\"",[75,11175,318],{"class":85},[75,11177,11178,11180,11183,11185,11187,11189,11191,11193,11196,11198,11201,11203,11205],{"class":77,"line":1343},[75,11179,11108],{"class":85},[75,11181,11182],{"class":193},"axhline",[75,11184,197],{"class":85},[75,11186,5186],{"class":145},[75,11188,206],{"class":85},[75,11190,310],{"class":309},[75,11192,142],{"class":141},[75,11194,11195],{"class":134},"'k'",[75,11197,206],{"class":85},[75,11199,11200],{"class":309},"linewidth",[75,11202,142],{"class":141},[75,11204,2958],{"class":145},[75,11206,318],{"class":85},[75,11208,11209,11211,11214,11216,11218,11220,11222,11224,11226,11228,11230,11232,11234],{"class":77,"line":1356},[75,11210,11108],{"class":85},[75,11212,11213],{"class":193},"axvline",[75,11215,197],{"class":85},[75,11217,5186],{"class":145},[75,11219,206],{"class":85},[75,11221,310],{"class":309},[75,11223,142],{"class":141},[75,11225,11195],{"class":134},[75,11227,206],{"class":85},[75,11229,11200],{"class":309},[75,11231,142],{"class":141},[75,11233,2958],{"class":145},[75,11235,318],{"class":85},[75,11237,11238,11240,11242,11244,11246],{"class":77,"line":1887},[75,11239,11108],{"class":85},[75,11241,371],{"class":193},[75,11243,197],{"class":85},[75,11245,10504],{"class":145},[75,11247,318],{"class":85},[75,11249,11250,11252,11254],{"class":77,"line":1900},[75,11251,11108],{"class":85},[75,11253,5058],{"class":193},[75,11255,389],{"class":85},[75,11257,11258],{"class":77,"line":1913},[75,11259,112],{"emptyLinePlaceholder":111},[75,11261,11262,11264,11266],{"class":77,"line":2657},[75,11263,300],{"class":85},[75,11265,386],{"class":193},[75,11267,389],{"class":85},[391,11269],{"alt":11270,"src":11271},"Output 15 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-15.webp",[65,11273,11275],{"className":67,"code":11274,"language":69,"meta":70,"style":70},"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",[72,11276,11277,11287,11297,11301,11305,11315,11325,11335,11339,11343,11361,11385,11389,11393,11437,11441,11445,11453,11461,11469,11477,11485,11489,11493,11498,11540,11552,11556,11568,11601,11609,11621,11633,11637],{"__ignoreMap":70},[75,11278,11279,11281,11283,11285],{"class":77,"line":78},[75,11280,82],{"class":81},[75,11282,86],{"class":85},[75,11284,89],{"class":81},[75,11286,92],{"class":85},[75,11288,11289,11291,11293,11295],{"class":77,"line":95},[75,11290,82],{"class":81},[75,11292,100],{"class":85},[75,11294,89],{"class":81},[75,11296,105],{"class":85},[75,11298,11299],{"class":77,"line":108},[75,11300,112],{"emptyLinePlaceholder":111},[75,11302,11303],{"class":77,"line":115},[75,11304,10241],{"class":118},[75,11306,11307,11309,11311,11313],{"class":77,"line":122},[75,11308,10246],{"class":85},[75,11310,142],{"class":141},[75,11312,10251],{"class":145},[75,11314,10254],{"class":118},[75,11316,11317,11319,11321,11323],{"class":77,"line":128},[75,11318,10259],{"class":85},[75,11320,142],{"class":141},[75,11322,1075],{"class":145},[75,11324,10266],{"class":118},[75,11326,11327,11329,11331,11333],{"class":77,"line":149},[75,11328,10271],{"class":85},[75,11330,142],{"class":141},[75,11332,1437],{"class":145},[75,11334,10278],{"class":118},[75,11336,11337],{"class":77,"line":154},[75,11338,112],{"emptyLinePlaceholder":111},[75,11340,11341],{"class":77,"line":160},[75,11342,10287],{"class":118},[75,11344,11345,11347,11349,11351,11353,11355,11357,11359],{"class":77,"line":171},[75,11346,10292],{"class":85},[75,11348,142],{"class":141},[75,11350,10297],{"class":141},[75,11352,197],{"class":85},[75,11354,791],{"class":145},[75,11356,273],{"class":141},[75,11358,10306],{"class":145},[75,11360,318],{"class":85},[75,11362,11363,11365,11367,11369,11371,11373,11375,11377,11379,11381,11383],{"class":77,"line":182},[75,11364,2722],{"class":85},[75,11366,142],{"class":141},[75,11368,190],{"class":85},[75,11370,194],{"class":193},[75,11372,197],{"class":85},[75,11374,200],{"class":141},[75,11376,752],{"class":145},[75,11378,206],{"class":85},[75,11380,752],{"class":145},[75,11382,10331],{"class":85},[75,11384,10334],{"class":118},[75,11386,11387],{"class":77,"line":222},[75,11388,112],{"emptyLinePlaceholder":111},[75,11390,11391],{"class":77,"line":227},[75,11392,10343],{"class":118},[75,11394,11395,11397,11399,11401,11403,11405,11407,11409,11411,11413,11415,11417,11419,11421,11423,11425,11427,11429,11431,11433,11435],{"class":77,"line":233},[75,11396,236],{"class":85},[75,11398,142],{"class":141},[75,11400,10352],{"class":85},[75,11402,500],{"class":141},[75,11404,190],{"class":85},[75,11406,243],{"class":193},[75,11408,197],{"class":85},[75,11410,200],{"class":141},[75,11412,10365],{"class":85},[75,11414,200],{"class":141},[75,11416,10370],{"class":85},[75,11418,258],{"class":141},[75,11420,270],{"class":145},[75,11422,264],{"class":141},[75,11424,267],{"class":85},[75,11426,270],{"class":145},[75,11428,273],{"class":141},[75,11430,10385],{"class":85},[75,11432,258],{"class":141},[75,11434,270],{"class":145},[75,11436,283],{"class":85},[75,11438,11439],{"class":77,"line":286},[75,11440,112],{"emptyLinePlaceholder":111},[75,11442,11443],{"class":77,"line":291},[75,11444,10400],{"class":118},[75,11446,11447,11449,11451],{"class":77,"line":297},[75,11448,3592],{"class":85},[75,11450,142],{"class":141},[75,11452,10409],{"class":85},[75,11454,11455,11457,11459],{"class":77,"line":321},[75,11456,10414],{"class":85},[75,11458,10417],{"class":134},[75,11460,3003],{"class":85},[75,11462,11463,11465,11467],{"class":77,"line":336},[75,11464,10414],{"class":85},[75,11466,10426],{"class":134},[75,11468,3003],{"class":85},[75,11470,11471,11473,11475],{"class":77,"line":351},[75,11472,10414],{"class":85},[75,11474,10435],{"class":134},[75,11476,3003],{"class":85},[75,11478,11479,11481,11483],{"class":77,"line":366},[75,11480,10414],{"class":85},[75,11482,10444],{"class":134},[75,11484,318],{"class":85},[75,11486,11487],{"class":77,"line":381},[75,11488,10451],{"class":85},[75,11490,11491],{"class":77,"line":984},[75,11492,112],{"emptyLinePlaceholder":111},[75,11494,11495],{"class":77,"line":997},[75,11496,11497],{"class":118},"# Create lollipop plots\n",[75,11499,11500,11502,11504,11506,11508,11510,11512,11514,11516,11518,11520,11522,11524,11526,11528,11530,11532,11534,11536,11538],{"class":77,"line":1325},[75,11501,10465],{"class":85},[75,11503,142],{"class":141},[75,11505,3093],{"class":85},[75,11507,3096],{"class":193},[75,11509,197],{"class":85},[75,11511,270],{"class":145},[75,11513,206],{"class":85},[75,11515,270],{"class":145},[75,11517,206],{"class":85},[75,11519,870],{"class":309},[75,11521,142],{"class":141},[75,11523,197],{"class":85},[75,11525,3116],{"class":145},[75,11527,206],{"class":85},[75,11529,877],{"class":145},[75,11531,10496],{"class":85},[75,11533,10499],{"class":309},[75,11535,142],{"class":141},[75,11537,10504],{"class":145},[75,11539,318],{"class":85},[75,11541,11542,11544,11546,11548,11550],{"class":77,"line":1338},[75,11543,10511],{"class":85},[75,11545,142],{"class":141},[75,11547,10516],{"class":85},[75,11549,3136],{"class":193},[75,11551,389],{"class":85},[75,11553,11554],{"class":77,"line":1343},[75,11555,112],{"emptyLinePlaceholder":111},[75,11557,11558,11560,11562,11564,11566],{"class":77,"line":1356},[75,11559,3147],{"class":81},[75,11561,10536],{"class":85},[75,11563,3153],{"class":81},[75,11565,10541],{"class":141},[75,11567,10544],{"class":85},[75,11569,11570,11572,11574,11577,11579,11581,11583,11585,11587,11589,11591,11593,11595,11597,11599],{"class":77,"line":1887},[75,11571,10549],{"class":85},[75,11573,891],{"class":193},[75,11575,11576],{"class":85},"(t, gaussian_modulation, ",[75,11578,896],{"class":309},[75,11580,142],{"class":141},[75,11582,901],{"class":134},[75,11584,206],{"class":85},[75,11586,906],{"class":309},[75,11588,142],{"class":141},[75,11590,911],{"class":134},[75,11592,206],{"class":85},[75,11594,916],{"class":309},[75,11596,142],{"class":141},[75,11598,921],{"class":134},[75,11600,318],{"class":85},[75,11602,11603,11605,11607],{"class":77,"line":1900},[75,11604,10549],{"class":85},[75,11606,3182],{"class":193},[75,11608,3185],{"class":85},[75,11610,11611,11613,11615,11617,11619],{"class":77,"line":1913},[75,11612,10549],{"class":85},[75,11614,3193],{"class":193},[75,11616,197],{"class":85},[75,11618,10596],{"class":134},[75,11620,318],{"class":85},[75,11622,11623,11625,11627,11629,11631],{"class":77,"line":2657},[75,11624,10549],{"class":85},[75,11626,3208],{"class":193},[75,11628,197],{"class":85},[75,11630,3213],{"class":134},[75,11632,318],{"class":85},[75,11634,11635],{"class":77,"line":3177},[75,11636,112],{"emptyLinePlaceholder":111},[75,11638,11639,11641,11643],{"class":77,"line":3188},[75,11640,300],{"class":85},[75,11642,386],{"class":193},[75,11644,389],{"class":85},[391,11646],{"alt":11647,"src":11648},"Output 16 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-16.webp",[65,11650,11652],{"className":67,"code":11651,"language":69,"meta":70,"style":70},"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",[72,11653,11654,11664,11674,11678,11682,11692,11703,11713,11726,11738,11754,11770,11799,11803,11807,11852,11856,11861,11906,11946,11986,12029,12033,12038,12053,12063,12073,12083,12093,12106,12110,12115,12157,12169,12173,12187,12233,12264,12290,12313,12325,12337,12342,12347,12351],{"__ignoreMap":70},[75,11655,11656,11658,11660,11662],{"class":77,"line":78},[75,11657,82],{"class":81},[75,11659,86],{"class":85},[75,11661,89],{"class":81},[75,11663,92],{"class":85},[75,11665,11666,11668,11670,11672],{"class":77,"line":95},[75,11667,82],{"class":81},[75,11669,100],{"class":85},[75,11671,89],{"class":81},[75,11673,105],{"class":85},[75,11675,11676],{"class":77,"line":108},[75,11677,112],{"emptyLinePlaceholder":111},[75,11679,11680],{"class":77,"line":115},[75,11681,10241],{"class":118},[75,11683,11684,11686,11688,11690],{"class":77,"line":122},[75,11685,10246],{"class":85},[75,11687,142],{"class":141},[75,11689,10251],{"class":145},[75,11691,10254],{"class":118},[75,11693,11694,11696,11698,11700],{"class":77,"line":128},[75,11695,10259],{"class":85},[75,11697,142],{"class":141},[75,11699,1075],{"class":145},[75,11701,11702],{"class":118},"   # Standard deviation of Gaussian modulation (ns)\n",[75,11704,11705,11707,11709,11711],{"class":77,"line":149},[75,11706,10271],{"class":85},[75,11708,142],{"class":141},[75,11710,1437],{"class":145},[75,11712,10278],{"class":118},[75,11714,11715,11718,11720,11723],{"class":77,"line":154},[75,11716,11717],{"class":85},"fc ",[75,11719,142],{"class":141},[75,11721,11722],{"class":145}," 5e9",[75,11724,11725],{"class":118},"    # Carrier frequency (5 GHz, typical for superconducting qubits)\n",[75,11727,11728,11731,11733,11735],{"class":77,"line":160},[75,11729,11730],{"class":85},"phi_X ",[75,11732,142],{"class":141},[75,11734,10251],{"class":145},[75,11736,11737],{"class":118},"           # X gate (Pauli-X) - $\\pi$ rotation around X-axis\n",[75,11739,11740,11743,11745,11747,11749,11751],{"class":77,"line":171},[75,11741,11742],{"class":85},"phi_Y ",[75,11744,142],{"class":141},[75,11746,1188],{"class":85},[75,11748,2810],{"class":141},[75,11750,261],{"class":145},[75,11752,11753],{"class":118},"   # Y gate (Pauli-Y) - $\\pi\u002F2$ rotation around Y-axis\n",[75,11755,11756,11759,11761,11763,11765,11767],{"class":77,"line":182},[75,11757,11758],{"class":85},"phi_H ",[75,11760,142],{"class":141},[75,11762,1188],{"class":85},[75,11764,2810],{"class":141},[75,11766,261],{"class":145},[75,11768,11769],{"class":118},"   # Hadamard gate (H) - $\\pi\u002F2$ rotation around Y-axis\n",[75,11771,11772,11774,11776,11778,11780,11782,11784,11786,11788,11790,11792,11794,11796],{"class":77,"line":222},[75,11773,2722],{"class":85},[75,11775,142],{"class":141},[75,11777,190],{"class":85},[75,11779,194],{"class":193},[75,11781,197],{"class":85},[75,11783,200],{"class":141},[75,11785,752],{"class":145},[75,11787,206],{"class":85},[75,11789,752],{"class":145},[75,11791,206],{"class":85},[75,11793,213],{"class":145},[75,11795,216],{"class":85},[75,11797,11798],{"class":118},"# Time axis in nanoseconds\n",[75,11800,11801],{"class":77,"line":227},[75,11802,112],{"emptyLinePlaceholder":111},[75,11804,11805],{"class":77,"line":233},[75,11806,10343],{"class":118},[75,11808,11809,11812,11814,11816,11818,11820,11822,11824,11826,11828,11830,11832,11834,11836,11838,11840,11842,11844,11846,11848,11850],{"class":77,"line":286},[75,11810,11811],{"class":85},"gaussian_envelope ",[75,11813,142],{"class":141},[75,11815,10352],{"class":85},[75,11817,500],{"class":141},[75,11819,190],{"class":85},[75,11821,243],{"class":193},[75,11823,197],{"class":85},[75,11825,200],{"class":141},[75,11827,10365],{"class":85},[75,11829,200],{"class":141},[75,11831,10370],{"class":85},[75,11833,258],{"class":141},[75,11835,270],{"class":145},[75,11837,264],{"class":141},[75,11839,267],{"class":85},[75,11841,270],{"class":145},[75,11843,273],{"class":141},[75,11845,10385],{"class":85},[75,11847,258],{"class":141},[75,11849,270],{"class":145},[75,11851,283],{"class":85},[75,11853,11854],{"class":77,"line":291},[75,11855,112],{"emptyLinePlaceholder":111},[75,11857,11858],{"class":77,"line":297},[75,11859,11860],{"class":118},"# Compute modulated pulses for X, Y, H, and CNOT gates\n",[75,11862,11863,11866,11868,11871,11873,11875,11877,11879,11881,11883,11885,11887,11890,11892,11895,11897,11900,11903],{"class":77,"line":321},[75,11864,11865],{"class":85},"modulated_pulse_X ",[75,11867,142],{"class":141},[75,11869,11870],{"class":85}," gaussian_envelope ",[75,11872,500],{"class":141},[75,11874,190],{"class":85},[75,11876,2845],{"class":193},[75,11878,197],{"class":85},[75,11880,270],{"class":145},[75,11882,273],{"class":141},[75,11884,1188],{"class":85},[75,11886,500],{"class":141},[75,11888,11889],{"class":85}," fc ",[75,11891,500],{"class":141},[75,11893,11894],{"class":85}," t ",[75,11896,500],{"class":141},[75,11898,11899],{"class":145}," 1e-9",[75,11901,11902],{"class":141}," +",[75,11904,11905],{"class":85}," phi_X)\n",[75,11907,11908,11911,11913,11915,11917,11919,11921,11923,11925,11927,11929,11931,11933,11935,11937,11939,11941,11943],{"class":77,"line":336},[75,11909,11910],{"class":85},"modulated_pulse_Y ",[75,11912,142],{"class":141},[75,11914,11870],{"class":85},[75,11916,500],{"class":141},[75,11918,190],{"class":85},[75,11920,2845],{"class":193},[75,11922,197],{"class":85},[75,11924,270],{"class":145},[75,11926,273],{"class":141},[75,11928,1188],{"class":85},[75,11930,500],{"class":141},[75,11932,11889],{"class":85},[75,11934,500],{"class":141},[75,11936,11894],{"class":85},[75,11938,500],{"class":141},[75,11940,11899],{"class":145},[75,11942,11902],{"class":141},[75,11944,11945],{"class":85}," phi_Y)\n",[75,11947,11948,11951,11953,11955,11957,11959,11961,11963,11965,11967,11969,11971,11973,11975,11977,11979,11981,11983],{"class":77,"line":351},[75,11949,11950],{"class":85},"modulated_pulse_H ",[75,11952,142],{"class":141},[75,11954,11870],{"class":85},[75,11956,500],{"class":141},[75,11958,190],{"class":85},[75,11960,2845],{"class":193},[75,11962,197],{"class":85},[75,11964,270],{"class":145},[75,11966,273],{"class":141},[75,11968,1188],{"class":85},[75,11970,500],{"class":141},[75,11972,11889],{"class":85},[75,11974,500],{"class":141},[75,11976,11894],{"class":85},[75,11978,500],{"class":141},[75,11980,11899],{"class":145},[75,11982,11902],{"class":141},[75,11984,11985],{"class":85}," phi_H)\n",[75,11987,11988,11991,11993,11995,11997,11999,12001,12003,12005,12007,12009,12011,12013,12015,12017,12019,12021,12023,12026],{"class":77,"line":366},[75,11989,11990],{"class":85},"modulated_pulse_CNOT ",[75,11992,142],{"class":141},[75,11994,11870],{"class":85},[75,11996,500],{"class":141},[75,11998,190],{"class":85},[75,12000,2845],{"class":193},[75,12002,197],{"class":85},[75,12004,270],{"class":145},[75,12006,273],{"class":141},[75,12008,1188],{"class":85},[75,12010,500],{"class":141},[75,12012,11889],{"class":85},[75,12014,500],{"class":141},[75,12016,11894],{"class":85},[75,12018,500],{"class":141},[75,12020,11899],{"class":145},[75,12022,11902],{"class":141},[75,12024,12025],{"class":85}," phi_X)  ",[75,12027,12028],{"class":118},"# Similar to X gate\n",[75,12030,12031],{"class":77,"line":381},[75,12032,112],{"emptyLinePlaceholder":111},[75,12034,12035],{"class":77,"line":984},[75,12036,12037],{"class":118},"# Store pulses and titles in an array\n",[75,12039,12040,12043,12045,12047,12050],{"class":77,"line":997},[75,12041,12042],{"class":85},"pulses ",[75,12044,142],{"class":141},[75,12046,190],{"class":85},[75,12048,12049],{"class":193},"array",[75,12051,12052],{"class":85},"([\n",[75,12054,12055,12058,12061],{"class":77,"line":1325},[75,12056,12057],{"class":85},"    (modulated_pulse_X, ",[75,12059,12060],{"class":134},"\"Pauli-X Gate\"",[75,12062,3003],{"class":85},[75,12064,12065,12068,12071],{"class":77,"line":1338},[75,12066,12067],{"class":85},"    (modulated_pulse_Y, ",[75,12069,12070],{"class":134},"\"Pauli-Y Gate\"",[75,12072,3003],{"class":85},[75,12074,12075,12078,12081],{"class":77,"line":1343},[75,12076,12077],{"class":85},"    (modulated_pulse_H, ",[75,12079,12080],{"class":134},"\"Hadamard Gate\"",[75,12082,3003],{"class":85},[75,12084,12085,12088,12091],{"class":77,"line":1356},[75,12086,12087],{"class":85},"    (modulated_pulse_CNOT, ",[75,12089,12090],{"class":134},"\"CNOT Gate Target Qubit\"",[75,12092,318],{"class":85},[75,12094,12095,12098,12101,12104],{"class":77,"line":1887},[75,12096,12097],{"class":85},"], ",[75,12099,12100],{"class":309},"dtype",[75,12102,12103],{"class":141},"=object",[75,12105,318],{"class":85},[75,12107,12108],{"class":77,"line":1900},[75,12109,112],{"emptyLinePlaceholder":111},[75,12111,12112],{"class":77,"line":1913},[75,12113,12114],{"class":118},"# Create plots with blue pulse lines\n",[75,12116,12117,12119,12121,12123,12125,12127,12129,12131,12133,12135,12137,12139,12141,12143,12145,12147,12149,12151,12153,12155],{"class":77,"line":2657},[75,12118,10465],{"class":85},[75,12120,142],{"class":141},[75,12122,3093],{"class":85},[75,12124,3096],{"class":193},[75,12126,197],{"class":85},[75,12128,270],{"class":145},[75,12130,206],{"class":85},[75,12132,270],{"class":145},[75,12134,206],{"class":85},[75,12136,870],{"class":309},[75,12138,142],{"class":141},[75,12140,197],{"class":85},[75,12142,3116],{"class":145},[75,12144,206],{"class":85},[75,12146,877],{"class":145},[75,12148,10496],{"class":85},[75,12150,10499],{"class":309},[75,12152,142],{"class":141},[75,12154,10504],{"class":145},[75,12156,318],{"class":85},[75,12158,12159,12161,12163,12165,12167],{"class":77,"line":3177},[75,12160,10511],{"class":85},[75,12162,142],{"class":141},[75,12164,10516],{"class":85},[75,12166,3136],{"class":193},[75,12168,389],{"class":85},[75,12170,12171],{"class":77,"line":3188},[75,12172,112],{"emptyLinePlaceholder":111},[75,12174,12175,12177,12180,12182,12184],{"class":77,"line":3203},[75,12176,3147],{"class":81},[75,12178,12179],{"class":85}," ax, (pulse, title) ",[75,12181,3153],{"class":81},[75,12183,10541],{"class":141},[75,12185,12186],{"class":85},"(axs, pulses):\n",[75,12188,12189,12191,12193,12196,12198,12200,12202,12204,12206,12208,12210,12212,12214,12217,12220,12222,12225,12228,12230],{"class":77,"line":3218},[75,12190,10549],{"class":85},[75,12192,303],{"class":193},[75,12194,12195],{"class":85},"(t, pulse, ",[75,12197,3172],{"class":134},[75,12199,206],{"class":85},[75,12201,11200],{"class":309},[75,12203,142],{"class":141},[75,12205,2958],{"class":145},[75,12207,206],{"class":85},[75,12209,573],{"class":309},[75,12211,142],{"class":141},[75,12213,5894],{"class":81},[75,12215,12216],{"class":134},"\"",[75,12218,12219],{"class":145},"{",[75,12221,326],{"class":85},[75,12223,12224],{"class":145},"}",[75,12226,12227],{"class":134}," Modulated Pulse\"",[75,12229,216],{"class":85},[75,12231,12232],{"class":118},"# Blue line with reduced thickness\n",[75,12234,12235,12237,12239,12242,12245,12247,12249,12251,12253,12255,12257,12259,12262],{"class":77,"line":3224},[75,12236,10549],{"class":85},[75,12238,303],{"class":193},[75,12240,12241],{"class":85},"(t, gaussian_envelope, ",[75,12243,12244],{"class":134},"'r--'",[75,12246,206],{"class":85},[75,12248,11200],{"class":309},[75,12250,142],{"class":141},[75,12252,630],{"class":145},[75,12254,206],{"class":85},[75,12256,573],{"class":309},[75,12258,142],{"class":141},[75,12260,12261],{"class":134},"\"Gaussian Envelope\"",[75,12263,318],{"class":85},[75,12265,12266,12268,12270,12273,12275,12278,12280,12282,12284,12286,12288],{"class":77,"line":3229},[75,12267,10549],{"class":85},[75,12269,303],{"class":193},[75,12271,12272],{"class":85},"(t, ",[75,12274,200],{"class":141},[75,12276,12277],{"class":85},"gaussian_envelope, ",[75,12279,12244],{"class":134},[75,12281,206],{"class":85},[75,12283,11200],{"class":309},[75,12285,142],{"class":141},[75,12287,630],{"class":145},[75,12289,318],{"class":85},[75,12291,12292,12294,12296,12298,12300,12303,12305,12307,12309,12311],{"class":77,"line":3235},[75,12293,10549],{"class":85},[75,12295,3182],{"class":193},[75,12297,197],{"class":85},[75,12299,5894],{"class":81},[75,12301,12302],{"class":134},"\"Modulated Pulse for ",[75,12304,12219],{"class":145},[75,12306,326],{"class":85},[75,12308,12224],{"class":145},[75,12310,12216],{"class":134},[75,12312,318],{"class":85},[75,12314,12315,12317,12319,12321,12323],{"class":77,"line":3263},[75,12316,10549],{"class":85},[75,12318,3193],{"class":193},[75,12320,197],{"class":85},[75,12322,10596],{"class":134},[75,12324,318],{"class":85},[75,12326,12327,12329,12331,12333,12335],{"class":77,"line":3275},[75,12328,10549],{"class":85},[75,12330,3208],{"class":193},[75,12332,197],{"class":85},[75,12334,3213],{"class":134},[75,12336,318],{"class":85},[75,12338,12339],{"class":77,"line":3280},[75,12340,12341],{"class":118},"    # ax.legend()\n",[75,12343,12344],{"class":77,"line":3290},[75,12345,12346],{"class":118},"    # ax.grid()\n",[75,12348,12349],{"class":77,"line":4431},[75,12350,112],{"emptyLinePlaceholder":111},[75,12352,12353,12355,12357],{"class":77,"line":4454},[75,12354,300],{"class":85},[75,12356,386],{"class":193},[75,12358,389],{"class":85},[391,12360],{"alt":12361,"src":12362},"Output 17 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-17.webp",[12364,12365,12366],"style",{},"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":70,"searchDepth":95,"depth":95,"links":12368},[12369,12370,12371,12372],{"id":5138,"depth":108,"text":5141},{"id":5786,"depth":108,"text":5789},{"id":6553,"depth":108,"text":6556},{"id":7383,"depth":108,"text":7386},[12374,12375,12376,12377],"Qollab","Explore","Expert notes","Pulse shapes",[12379],{"username":7,"name":12380,"role":12381,"bio":12382,"links":12383},"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.",[12384,12387],{"label":12385,"href":12386},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":12388,"href":21},"Original notebook ↗",{"username":7,"name":12380,"role":12390},"Quantum Hardware Engineer, University of Minnesota",null,"How microwave pulses are shaped to control a quantum chip: Gaussian and sinusoidal envelopes, and the modulated pulses behind common gates.",false,"md","Deep dive · Hardware","article",{},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fpulse-shapes-and-envelopes","2026-08-06","16 min read",[],{"title":5,"description":12392},"blog\u002Fexpert-notes\u002Fpulse-shapes-and-envelopes",[12406,12407],"hardware","visualization","lUWuH8SAXjG9DJfLvm20n28sMwmAzXp3sg2F30H98eQ",[12410,19724,31316,50978,76836],{"id":12411,"title":12412,"authors":12413,"body":12414,"breadcrumb":19706,"builders":19708,"byline":19713,"challenge":12391,"courseAuthor":12391,"courseLead":12391,"dek":12391,"description":19714,"draft":12393,"extension":12394,"eyebrow":12395,"featured":12393,"finish":12391,"fork":12391,"hero":12391,"heroAlt":12391,"heroCta":12391,"heroImage":12391,"homepageFeatured":12393,"kind":12396,"lessonCount":12391,"meta":19715,"navigation":111,"newsItems":12391,"next":12391,"ogImage":19716,"order":12391,"outcomes":12391,"path":19717,"publishDate":12400,"readingTime":19718,"related":19719,"relatedProjects":12391,"seo":19720,"stem":19721,"tags":19722,"track":12391,"trackName":12391,"__hash__":19723},"blog\u002Fblog\u002Fexpert-notes\u002Fcross-resonance-gate-visualization.md","Cross-Resonance Gate: Slices and Heatmaps",[7],{"type":9,"value":12415,"toc":19701},[12416,12424,12429,13960,13962,13966,14044,17003,17006,17009,17012,17015,17993,17996,19695,19698],[12,12417,12418],{},[15,12419,17,12420,12423],{},[19,12421,22],{"href":12422},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FCross_Resonance_Gate_Visualization.ipynb"," and shared under CC-BY-4.0. The text, code, and figures below are Onri's, as published.",[12425,12426,12428],"h2",{"id":12427},"cross-resonance-cr-gate","Cross-Resonance (CR) Gate",[7629,12430,12431,12445],{},[7632,12432,12433],{},[7635,12434,12435,12439,12442],{},[7638,12436,12438],{"align":12437},"right","#",[7638,12440,12441],{},"Figure",[7638,12443,12444],{},"What to notice",[7645,12446,12447,12683,12915,13158,13357,13630,13802],{},[7635,12448,12449,12451,12652],{},[7650,12450,7477],{"align":12437},[7650,12452,12453,12536,12537,12620,12621,5175],{},[75,12454,12456,12479],{"className":12455},[5148],[75,12457,12459],{"className":12458},[5152],[5154,12460,12461],{"xmlns":5156},[5158,12462,12463,12476],{},[5161,12464,12465],{},[5179,12466,12467,12470],{},[5164,12468,12469],{},"ω",[5161,12471,12472,12474],{},[5164,12473,7409],{},[5164,12475,5820],{},[5238,12477,12478],{"encoding":5240},"\\omega_{ZX}",[75,12480,12482],{"className":12481,"ariaHidden":5196},[5245],[75,12483,12485,12489],{"className":12484},[5249],[75,12486],{"className":12487,"style":12488},[5253],"height:0.5806em;vertical-align:-0.15em;",[75,12490,12492,12495],{"className":12491},[5258],[75,12493,12469],{"className":12494,"style":5416},[5258,5259],[75,12496,12498],{"className":12497},[5299],[75,12499,12501,12528],{"className":12500},[5303,5304],[75,12502,12504,12525],{"className":12503},[5308],[75,12505,12507],{"className":12506,"style":5942},[5312],[75,12508,12510,12513],{"style":12509},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[75,12511],{"className":12512,"style":5321},[5320],[75,12514,12516],{"className":12515},[5325,5326,5327,5328],[75,12517,12519,12522],{"className":12518},[5258,5328],[75,12520,7409],{"className":12521,"style":7430},[5258,5259,5328],[75,12523,5820],{"className":12524,"style":5955},[5258,5259,5328],[75,12526,5336],{"className":12527},[5335],[75,12529,12531],{"className":12530},[5308],[75,12532,12534],{"className":12533,"style":5343},[5312],[75,12535],{}," vs ",[75,12538,12540,12563],{"className":12539},[5148],[75,12541,12543],{"className":12542},[5152],[5154,12544,12545],{"xmlns":5156},[5158,12546,12547,12560],{},[5161,12548,12549],{},[5179,12550,12551,12554],{},[5164,12552,12553],{"mathvariant":6582},"Δ",[5161,12555,12556,12558],{},[5164,12557,5897],{},[5164,12559,2771],{},[5238,12561,12562],{"encoding":5240},"\\Delta_{ct}",[75,12564,12566],{"className":12565,"ariaHidden":5196},[5245],[75,12567,12569,12573],{"className":12568},[5249],[75,12570],{"className":12571,"style":12572},[5253],"height:0.8333em;vertical-align:-0.15em;",[75,12574,12576,12579],{"className":12575},[5258],[75,12577,12553],{"className":12578},[5258],[75,12580,12582],{"className":12581},[5299],[75,12583,12585,12612],{"className":12584},[5303,5304],[75,12586,12588,12609],{"className":12587},[5308],[75,12589,12592],{"className":12590,"style":12591},[5312],"height:0.2806em;",[75,12593,12594,12597],{"style":5316},[75,12595],{"className":12596,"style":5321},[5320],[75,12598,12600],{"className":12599},[5325,5326,5327,5328],[75,12601,12603,12606],{"className":12602},[5258,5328],[75,12604,5897],{"className":12605},[5258,5259,5328],[75,12607,2771],{"className":12608},[5258,5259,5328],[75,12610,5336],{"className":12611},[5335],[75,12613,12615],{"className":12614},[5308],[75,12616,12618],{"className":12617,"style":5343},[5312],[75,12619],{}," (slice at fixed ",[75,12622,12624,12639],{"className":12623},[5148],[75,12625,12627],{"className":12626},[5152],[5154,12628,12629],{"xmlns":5156},[5158,12630,12631,12636],{},[5161,12632,12633],{},[5164,12634,12635],{"mathvariant":6582},"Ω",[5238,12637,12638],{"encoding":5240},"\\Omega",[75,12640,12642],{"className":12641,"ariaHidden":5196},[5245],[75,12643,12645,12649],{"className":12644},[5249],[75,12646],{"className":12647,"style":12648},[5253],"height:0.6833em;",[75,12650,12635],{"className":12651},[5258],[7650,12653,12654,12655,6543],{},"Entangling-rate sign flips at poles; magnitude peaks between poles; roughly linear in drive amplitude ",[75,12656,12658,12671],{"className":12657},[5148],[75,12659,12661],{"className":12660},[5152],[5154,12662,12663],{"xmlns":5156},[5158,12664,12665,12669],{},[5161,12666,12667],{},[5164,12668,12635],{"mathvariant":6582},[5238,12670,12638],{"encoding":5240},[75,12672,12674],{"className":12673,"ariaHidden":5196},[5245],[75,12675,12677,12680],{"className":12676},[5249],[75,12678],{"className":12679,"style":12648},[5253],[75,12681,12635],{"className":12682},[5258],[7635,12684,12685,12687,12848],{},[7650,12686,270],{"align":12437},[7650,12688,12689,12536,12769],{},[75,12690,12692,12714],{"className":12691},[5148],[75,12693,12695],{"className":12694},[5152],[5154,12696,12697],{"xmlns":5156},[5158,12698,12699,12711],{},[5161,12700,12701],{},[5179,12702,12703,12705],{},[5164,12704,12469],{},[5161,12706,12707,12709],{},[5164,12708,7409],{},[5164,12710,7409],{},[5238,12712,12713],{"encoding":5240},"\\omega_{ZZ}",[75,12715,12717],{"className":12716,"ariaHidden":5196},[5245],[75,12718,12720,12723],{"className":12719},[5249],[75,12721],{"className":12722,"style":12488},[5253],[75,12724,12726,12729],{"className":12725},[5258],[75,12727,12469],{"className":12728,"style":5416},[5258,5259],[75,12730,12732],{"className":12731},[5299],[75,12733,12735,12761],{"className":12734},[5303,5304],[75,12736,12738,12758],{"className":12737},[5308],[75,12739,12741],{"className":12740,"style":5942},[5312],[75,12742,12743,12746],{"style":12509},[75,12744],{"className":12745,"style":5321},[5320],[75,12747,12749],{"className":12748},[5325,5326,5327,5328],[75,12750,12752,12755],{"className":12751},[5258,5328],[75,12753,7409],{"className":12754,"style":7430},[5258,5259,5328],[75,12756,7409],{"className":12757,"style":7430},[5258,5259,5328],[75,12759,5336],{"className":12760},[5335],[75,12762,12764],{"className":12763},[5308],[75,12765,12767],{"className":12766,"style":5343},[5312],[75,12768],{},[75,12770,12772,12793],{"className":12771},[5148],[75,12773,12775],{"className":12774},[5152],[5154,12776,12777],{"xmlns":5156},[5158,12778,12779,12791],{},[5161,12780,12781],{},[5179,12782,12783,12785],{},[5164,12784,12553],{"mathvariant":6582},[5161,12786,12787,12789],{},[5164,12788,5897],{},[5164,12790,2771],{},[5238,12792,12562],{"encoding":5240},[75,12794,12796],{"className":12795,"ariaHidden":5196},[5245],[75,12797,12799,12802],{"className":12798},[5249],[75,12800],{"className":12801,"style":12572},[5253],[75,12803,12805,12808],{"className":12804},[5258],[75,12806,12553],{"className":12807},[5258],[75,12809,12811],{"className":12810},[5299],[75,12812,12814,12840],{"className":12813},[5303,5304],[75,12815,12817,12837],{"className":12816},[5308],[75,12818,12820],{"className":12819,"style":12591},[5312],[75,12821,12822,12825],{"style":5316},[75,12823],{"className":12824,"style":5321},[5320],[75,12826,12828],{"className":12827},[5325,5326,5327,5328],[75,12829,12831,12834],{"className":12830},[5258,5328],[75,12832,5897],{"className":12833},[5258,5259,5328],[75,12835,2771],{"className":12836},[5258,5259,5328],[75,12838,5336],{"className":12839},[5335],[75,12841,12843],{"className":12842},[5308],[75,12844,12846],{"className":12845,"style":5343},[5312],[75,12847],{},[7650,12849,12850,12851,12885,12886,12914],{},"Static ",[75,12852,12854,12870],{"className":12853},[5148],[75,12855,12857],{"className":12856},[5152],[5154,12858,12859],{"xmlns":5156},[5158,12860,12861,12867],{},[5161,12862,12863,12865],{},[5164,12864,7409],{},[5164,12866,7409],{},[5238,12868,12869],{"encoding":5240},"ZZ",[75,12871,12873],{"className":12872,"ariaHidden":5196},[5245],[75,12874,12876,12879,12882],{"className":12875},[5249],[75,12877],{"className":12878,"style":12648},[5253],[75,12880,7409],{"className":12881,"style":7430},[5258,5259],[75,12883,7409],{"className":12884,"style":7430},[5258,5259]," varies strongly near poles; lowest between poles; independent of ",[75,12887,12889,12902],{"className":12888},[5148],[75,12890,12892],{"className":12891},[5152],[5154,12893,12894],{"xmlns":5156},[5158,12895,12896,12900],{},[5161,12897,12898],{},[5164,12899,12635],{"mathvariant":6582},[5238,12901,12638],{"encoding":5240},[75,12903,12905],{"className":12904,"ariaHidden":5196},[5245],[75,12906,12908,12911],{"className":12907},[5249],[75,12909],{"className":12910,"style":12648},[5253],[75,12912,12635],{"className":12913},[5258]," at leading order.",[7635,12916,12917,12919,13151],{},[7650,12918,3101],{"align":12437},[7650,12920,12921,12536,13072],{},[75,12922,12924,12962],{"className":12923},[5148],[75,12925,12927],{"className":12926},[5152],[5154,12928,12929],{"xmlns":5156},[5158,12930,12931,12959],{},[5161,12932,12933,12935,12945,12947,12957],{},[5168,12934,7452],{"stretchy":5170},[5179,12936,12937,12939],{},[5164,12938,12469],{},[5161,12940,12941,12943],{},[5164,12942,7409],{},[5164,12944,5820],{},[5164,12946,2810],{"mathvariant":6582},[5179,12948,12949,12951],{},[5164,12950,12469],{},[5161,12952,12953,12955],{},[5164,12954,7409],{},[5164,12956,7409],{},[5168,12958,7452],{"stretchy":5170},[5238,12960,12961],{"encoding":5240},"\\lvert \\omega_{ZX}\u002F\\omega_{ZZ}\\rvert",[75,12963,12965],{"className":12964,"ariaHidden":5196},[5245],[75,12966,12968,12971,12974,13020,13023,13069],{"className":12967},[5249],[75,12969],{"className":12970,"style":5254},[5253],[75,12972,7452],{"className":12973},[5263],[75,12975,12977,12980],{"className":12976},[5258],[75,12978,12469],{"className":12979,"style":5416},[5258,5259],[75,12981,12983],{"className":12982},[5299],[75,12984,12986,13012],{"className":12985},[5303,5304],[75,12987,12989,13009],{"className":12988},[5308],[75,12990,12992],{"className":12991,"style":5942},[5312],[75,12993,12994,12997],{"style":12509},[75,12995],{"className":12996,"style":5321},[5320],[75,12998,13000],{"className":12999},[5325,5326,5327,5328],[75,13001,13003,13006],{"className":13002},[5258,5328],[75,13004,7409],{"className":13005,"style":7430},[5258,5259,5328],[75,13007,5820],{"className":13008,"style":5955},[5258,5259,5328],[75,13010,5336],{"className":13011},[5335],[75,13013,13015],{"className":13014},[5308],[75,13016,13018],{"className":13017,"style":5343},[5312],[75,13019],{},[75,13021,2810],{"className":13022},[5258],[75,13024,13026,13029],{"className":13025},[5258],[75,13027,12469],{"className":13028,"style":5416},[5258,5259],[75,13030,13032],{"className":13031},[5299],[75,13033,13035,13061],{"className":13034},[5303,5304],[75,13036,13038,13058],{"className":13037},[5308],[75,13039,13041],{"className":13040,"style":5942},[5312],[75,13042,13043,13046],{"style":12509},[75,13044],{"className":13045,"style":5321},[5320],[75,13047,13049],{"className":13048},[5325,5326,5327,5328],[75,13050,13052,13055],{"className":13051},[5258,5328],[75,13053,7409],{"className":13054,"style":7430},[5258,5259,5328],[75,13056,7409],{"className":13057,"style":7430},[5258,5259,5328],[75,13059,5336],{"className":13060},[5335],[75,13062,13064],{"className":13063},[5308],[75,13065,13067],{"className":13066,"style":5343},[5312],[75,13068],{},[75,13070,7452],{"className":13071},[5270],[75,13073,13075,13096],{"className":13074},[5148],[75,13076,13078],{"className":13077},[5152],[5154,13079,13080],{"xmlns":5156},[5158,13081,13082,13094],{},[5161,13083,13084],{},[5179,13085,13086,13088],{},[5164,13087,12553],{"mathvariant":6582},[5161,13089,13090,13092],{},[5164,13091,5897],{},[5164,13093,2771],{},[5238,13095,12562],{"encoding":5240},[75,13097,13099],{"className":13098,"ariaHidden":5196},[5245],[75,13100,13102,13105],{"className":13101},[5249],[75,13103],{"className":13104,"style":12572},[5253],[75,13106,13108,13111],{"className":13107},[5258],[75,13109,12553],{"className":13110},[5258],[75,13112,13114],{"className":13113},[5299],[75,13115,13117,13143],{"className":13116},[5303,5304],[75,13118,13120,13140],{"className":13119},[5308],[75,13121,13123],{"className":13122,"style":12591},[5312],[75,13124,13125,13128],{"style":5316},[75,13126],{"className":13127,"style":5321},[5320],[75,13129,13131],{"className":13130},[5325,5326,5327,5328],[75,13132,13134,13137],{"className":13133},[5258,5328],[75,13135,5897],{"className":13136},[5258,5259,5328],[75,13138,2771],{"className":13139},[5258,5259,5328],[75,13141,5336],{"className":13142},[5335],[75,13144,13146],{"className":13145},[5308],[75,13147,13149],{"className":13148,"style":5343},[5312],[75,13150],{},[7650,13152,13153,13154,13157],{},"Heuristic ",[48,13155,13156],{},"speed-vs-static"," figure; tall regions mark friendlier operating points (away from poles).",[7635,13159,13160,13163,13325],{},[7650,13161,13162],{"align":12437},"4",[7650,13164,13165,13324],{},[75,13166,13168,13208],{"className":13167},[5148],[75,13169,13171],{"className":13170},[5152],[5154,13172,13173],{"xmlns":5156},[5158,13174,13175,13205],{},[5161,13176,13177,13187,13189,13199,13201,13203],{},[5179,13178,13179,13181],{},[5164,13180,12469],{},[5161,13182,13183,13185],{},[5164,13184,7409],{},[5164,13186,5820],{},[5168,13188,197],{"stretchy":5170},[5179,13190,13191,13193],{},[5164,13192,12553],{"mathvariant":6582},[5161,13194,13195,13197],{},[5164,13196,5897],{},[5164,13198,2771],{},[5168,13200,7469],{"separator":5196},[5164,13202,12635],{"mathvariant":6582},[5168,13204,5175],{"stretchy":5170},[5238,13206,13207],{"encoding":5240},"\\omega_{ZX}(\\Delta_{ct}, \\Omega)",[75,13209,13211],{"className":13210,"ariaHidden":5196},[5245],[75,13212,13214,13217,13263,13266,13312,13315,13318,13321],{"className":13213},[5249],[75,13215],{"className":13216,"style":5254},[5253],[75,13218,13220,13223],{"className":13219},[5258],[75,13221,12469],{"className":13222,"style":5416},[5258,5259],[75,13224,13226],{"className":13225},[5299],[75,13227,13229,13255],{"className":13228},[5303,5304],[75,13230,13232,13252],{"className":13231},[5308],[75,13233,13235],{"className":13234,"style":5942},[5312],[75,13236,13237,13240],{"style":12509},[75,13238],{"className":13239,"style":5321},[5320],[75,13241,13243],{"className":13242},[5325,5326,5327,5328],[75,13244,13246,13249],{"className":13245},[5258,5328],[75,13247,7409],{"className":13248,"style":7430},[5258,5259,5328],[75,13250,5820],{"className":13251,"style":5955},[5258,5259,5328],[75,13253,5336],{"className":13254},[5335],[75,13256,13258],{"className":13257},[5308],[75,13259,13261],{"className":13260,"style":5343},[5312],[75,13262],{},[75,13264,197],{"className":13265},[5263],[75,13267,13269,13272],{"className":13268},[5258],[75,13270,12553],{"className":13271},[5258],[75,13273,13275],{"className":13274},[5299],[75,13276,13278,13304],{"className":13277},[5303,5304],[75,13279,13281,13301],{"className":13280},[5308],[75,13282,13284],{"className":13283,"style":12591},[5312],[75,13285,13286,13289],{"style":5316},[75,13287],{"className":13288,"style":5321},[5320],[75,13290,13292],{"className":13291},[5325,5326,5327,5328],[75,13293,13295,13298],{"className":13294},[5258,5328],[75,13296,5897],{"className":13297},[5258,5259,5328],[75,13299,2771],{"className":13300},[5258,5259,5328],[75,13302,5336],{"className":13303},[5335],[75,13305,13307],{"className":13306},[5308],[75,13308,13310],{"className":13309,"style":5343},[5312],[75,13311],{},[75,13313,7469],{"className":13314},[7544],[75,13316],{"className":13317,"style":5349},[5274],[75,13319,12635],{"className":13320},[5258],[75,13322,5175],{"className":13323},[5270]," heatmap",[7650,13326,13327,13328,13356],{},"Linear growth with ",[75,13329,13331,13344],{"className":13330},[5148],[75,13332,13334],{"className":13333},[5152],[5154,13335,13336],{"xmlns":5156},[5158,13337,13338,13342],{},[5161,13339,13340],{},[5164,13341,12635],{"mathvariant":6582},[5238,13343,12638],{"encoding":5240},[75,13345,13347],{"className":13346,"ariaHidden":5196},[5245],[75,13348,13350,13353],{"className":13349},[5249],[75,13351],{"className":13352,"style":12648},[5253],[75,13354,12635],{"className":13355},[5258],"; detuning structure set by anharmonicities; region boundaries visible as sharp changes.",[7635,13358,13359,13361,13593],{},[7650,13360,203],{"align":12437},[7650,13362,13363,13324],{},[75,13364,13366,13422],{"className":13365},[5148],[75,13367,13369],{"className":13368},[5152],[5154,13370,13371],{"xmlns":5156},[5158,13372,13373,13419],{},[5161,13374,13375,13377,13387,13389,13399,13401,13403,13413,13415,13417],{},[5168,13376,7452],{"stretchy":5170},[5179,13378,13379,13381],{},[5164,13380,12469],{},[5161,13382,13383,13385],{},[5164,13384,7409],{},[5164,13386,5820],{},[5164,13388,2810],{"mathvariant":6582},[5179,13390,13391,13393],{},[5164,13392,12469],{},[5161,13394,13395,13397],{},[5164,13396,7409],{},[5164,13398,7409],{},[5168,13400,7452],{"stretchy":5170},[5168,13402,197],{"stretchy":5170},[5179,13404,13405,13407],{},[5164,13406,12553],{"mathvariant":6582},[5161,13408,13409,13411],{},[5164,13410,5897],{},[5164,13412,2771],{},[5168,13414,7469],{"separator":5196},[5164,13416,12635],{"mathvariant":6582},[5168,13418,5175],{"stretchy":5170},[5238,13420,13421],{"encoding":5240},"\\lvert \\omega_{ZX}\u002F\\omega_{ZZ}\\rvert(\\Delta_{ct}, \\Omega)",[75,13423,13425],{"className":13424,"ariaHidden":5196},[5245],[75,13426,13428,13431,13434,13480,13483,13529,13532,13535,13581,13584,13587,13590],{"className":13427},[5249],[75,13429],{"className":13430,"style":5254},[5253],[75,13432,7452],{"className":13433},[5263],[75,13435,13437,13440],{"className":13436},[5258],[75,13438,12469],{"className":13439,"style":5416},[5258,5259],[75,13441,13443],{"className":13442},[5299],[75,13444,13446,13472],{"className":13445},[5303,5304],[75,13447,13449,13469],{"className":13448},[5308],[75,13450,13452],{"className":13451,"style":5942},[5312],[75,13453,13454,13457],{"style":12509},[75,13455],{"className":13456,"style":5321},[5320],[75,13458,13460],{"className":13459},[5325,5326,5327,5328],[75,13461,13463,13466],{"className":13462},[5258,5328],[75,13464,7409],{"className":13465,"style":7430},[5258,5259,5328],[75,13467,5820],{"className":13468,"style":5955},[5258,5259,5328],[75,13470,5336],{"className":13471},[5335],[75,13473,13475],{"className":13474},[5308],[75,13476,13478],{"className":13477,"style":5343},[5312],[75,13479],{},[75,13481,2810],{"className":13482},[5258],[75,13484,13486,13489],{"className":13485},[5258],[75,13487,12469],{"className":13488,"style":5416},[5258,5259],[75,13490,13492],{"className":13491},[5299],[75,13493,13495,13521],{"className":13494},[5303,5304],[75,13496,13498,13518],{"className":13497},[5308],[75,13499,13501],{"className":13500,"style":5942},[5312],[75,13502,13503,13506],{"style":12509},[75,13504],{"className":13505,"style":5321},[5320],[75,13507,13509],{"className":13508},[5325,5326,5327,5328],[75,13510,13512,13515],{"className":13511},[5258,5328],[75,13513,7409],{"className":13514,"style":7430},[5258,5259,5328],[75,13516,7409],{"className":13517,"style":7430},[5258,5259,5328],[75,13519,5336],{"className":13520},[5335],[75,13522,13524],{"className":13523},[5308],[75,13525,13527],{"className":13526,"style":5343},[5312],[75,13528],{},[75,13530,7452],{"className":13531},[5270],[75,13533,197],{"className":13534},[5263],[75,13536,13538,13541],{"className":13537},[5258],[75,13539,12553],{"className":13540},[5258],[75,13542,13544],{"className":13543},[5299],[75,13545,13547,13573],{"className":13546},[5303,5304],[75,13548,13550,13570],{"className":13549},[5308],[75,13551,13553],{"className":13552,"style":12591},[5312],[75,13554,13555,13558],{"style":5316},[75,13556],{"className":13557,"style":5321},[5320],[75,13559,13561],{"className":13560},[5325,5326,5327,5328],[75,13562,13564,13567],{"className":13563},[5258,5328],[75,13565,5897],{"className":13566},[5258,5259,5328],[75,13568,2771],{"className":13569},[5258,5259,5328],[75,13571,5336],{"className":13572},[5335],[75,13574,13576],{"className":13575},[5308],[75,13577,13579],{"className":13578,"style":5343},[5312],[75,13580],{},[75,13582,7469],{"className":13583},[7544],[75,13585],{"className":13586,"style":5349},[5274],[75,13588,12635],{"className":13589},[5258],[75,13591,5175],{"className":13592},[5270],[7650,13594,13595,13596,13629],{},"Bright bands away from poles suggest high entangling-rate with low static ",[75,13597,13599,13614],{"className":13598},[5148],[75,13600,13602],{"className":13601},[5152],[5154,13603,13604],{"xmlns":5156},[5158,13605,13606,13612],{},[5161,13607,13608,13610],{},[5164,13609,7409],{},[5164,13611,7409],{},[5238,13613,12869],{"encoding":5240},[75,13615,13617],{"className":13616,"ariaHidden":5196},[5245],[75,13618,13620,13623,13626],{"className":13619},[5249],[75,13621],{"className":13622,"style":12648},[5253],[75,13624,7409],{"className":13625,"style":7430},[5258,5259],[75,13627,7409],{"className":13628,"style":7430},[5258,5259]," (good for echoed-CR).",[7635,13631,13632,13634,13637],{},[7650,13633,882],{"align":12437},[7650,13635,13636],{},"Echoed-CR coefficients (two-segment echo)",[7650,13638,13639,13640,13668,13669,13703,13704,13732,13733,206,13768,13801],{},"Echo preserves odd-in-",[75,13641,13643,13656],{"className":13642},[5148],[75,13644,13646],{"className":13645},[5152],[5154,13647,13648],{"xmlns":5156},[5158,13649,13650,13654],{},[5161,13651,13652],{},[5164,13653,12635],{"mathvariant":6582},[5238,13655,12638],{"encoding":5240},[75,13657,13659],{"className":13658,"ariaHidden":5196},[5245],[75,13660,13662,13665],{"className":13661},[5249],[75,13663],{"className":13664,"style":12648},[5253],[75,13666,12635],{"className":13667},[5258]," ",[75,13670,13672,13688],{"className":13671},[5148],[75,13673,13675],{"className":13674},[5152],[5154,13676,13677],{"xmlns":5156},[5158,13678,13679,13685],{},[5161,13680,13681,13683],{},[5164,13682,7409],{},[5164,13684,5820],{},[5238,13686,13687],{"encoding":5240},"ZX",[75,13689,13691],{"className":13690,"ariaHidden":5196},[5245],[75,13692,13694,13697,13700],{"className":13693},[5249],[75,13695],{"className":13696,"style":12648},[5253],[75,13698,7409],{"className":13699,"style":7430},[5258,5259],[75,13701,5820],{"className":13702,"style":5955},[5258,5259]," and suppresses even-in-",[75,13705,13707,13720],{"className":13706},[5148],[75,13708,13710],{"className":13709},[5152],[5154,13711,13712],{"xmlns":5156},[5158,13713,13714,13718],{},[5161,13715,13716],{},[5164,13717,12635],{"mathvariant":6582},[5238,13719,12638],{"encoding":5240},[75,13721,13723],{"className":13722,"ariaHidden":5196},[5245],[75,13724,13726,13729],{"className":13725},[5249],[75,13727],{"className":13728,"style":12648},[5253],[75,13730,12635],{"className":13731},[5258]," terms (",[75,13734,13736,13753],{"className":13735},[5148],[75,13737,13739],{"className":13738},[5152],[5154,13740,13741],{"xmlns":5156},[5158,13742,13743,13750],{},[5161,13744,13745,13748],{},[5164,13746,13747],{},"I",[5164,13749,7409],{},[5238,13751,13752],{"encoding":5240},"IZ",[75,13754,13756],{"className":13755,"ariaHidden":5196},[5245],[75,13757,13759,13762,13765],{"className":13758},[5249],[75,13760],{"className":13761,"style":12648},[5253],[75,13763,13747],{"className":13764,"style":5955},[5258,5259],[75,13766,7409],{"className":13767,"style":7430},[5258,5259],[75,13769,13771,13786],{"className":13770},[5148],[75,13772,13774],{"className":13773},[5152],[5154,13775,13776],{"xmlns":5156},[5158,13777,13778,13784],{},[5161,13779,13780,13782],{},[5164,13781,7409],{},[5164,13783,7409],{},[5238,13785,12869],{"encoding":5240},[75,13787,13789],{"className":13788,"ariaHidden":5196},[5245],[75,13790,13792,13795,13798],{"className":13791},[5249],[75,13793],{"className":13794,"style":12648},[5253],[75,13796,7409],{"className":13797,"style":7430},[5258,5259],[75,13799,7409],{"className":13800,"style":7430},[5258,5259],") to first order.",[7635,13803,13804,13807,13888],{},[7650,13805,13806],{"align":12437},"7",[7650,13808,13809,13810,12536,13849,5175],{},"Conditional target Rabi (control ",[75,13811,13813,13831],{"className":13812},[5148],[75,13814,13816],{"className":13815},[5152],[5154,13817,13818],{"xmlns":5156},[5158,13819,13820,13828],{},[5161,13821,13822,13824,13826],{},[5168,13823,7452],{"stretchy":5170},[5184,13825,5186],{},[5168,13827,7457],{"stretchy":5170},[5238,13829,13830],{"encoding":5240},"\\lvert0\\rangle",[75,13832,13834],{"className":13833,"ariaHidden":5196},[5245],[75,13835,13837,13840,13843,13846],{"className":13836},[5249],[75,13838],{"className":13839,"style":5254},[5253],[75,13841,7452],{"className":13842},[5263],[75,13844,5186],{"className":13845},[5258],[75,13847,7457],{"className":13848},[5270],[75,13850,13852,13870],{"className":13851},[5148],[75,13853,13855],{"className":13854},[5152],[5154,13856,13857],{"xmlns":5156},[5158,13858,13859,13867],{},[5161,13860,13861,13863,13865],{},[5168,13862,7452],{"stretchy":5170},[5184,13864,7477],{},[5168,13866,7457],{"stretchy":5170},[5238,13868,13869],{"encoding":5240},"\\lvert1\\rangle",[75,13871,13873],{"className":13872,"ariaHidden":5196},[5245],[75,13874,13876,13879,13882,13885],{"className":13875},[5249],[75,13877],{"className":13878,"style":5254},[5253],[75,13880,7452],{"className":13881},[5263],[75,13883,7477],{"className":13884},[5258],[75,13886,7457],{"className":13887},[5270],[7650,13889,13890,13891,13924,13925,13959],{},"Target oscillation frequency\u002Fphase depends on control state due to the ",[75,13892,13894,13909],{"className":13893},[5148],[75,13895,13897],{"className":13896},[5152],[5154,13898,13899],{"xmlns":5156},[5158,13900,13901,13907],{},[5161,13902,13903,13905],{},[5164,13904,7409],{},[5164,13906,5820],{},[5238,13908,13687],{"encoding":5240},[75,13910,13912],{"className":13911,"ariaHidden":5196},[5245],[75,13913,13915,13918,13921],{"className":13914},[5249],[75,13916],{"className":13917,"style":12648},[5253],[75,13919,7409],{"className":13920,"style":7430},[5258,5259],[75,13922,5820],{"className":13923,"style":5955},[5258,5259]," term (and any residual ",[75,13926,13928,13944],{"className":13927},[5148],[75,13929,13931],{"className":13930},[5152],[5154,13932,13933],{"xmlns":5156},[5158,13934,13935,13941],{},[5161,13936,13937,13939],{},[5164,13938,13747],{},[5164,13940,5820],{},[5238,13942,13943],{"encoding":5240},"IX",[75,13945,13947],{"className":13946,"ariaHidden":5196},[5245],[75,13948,13950,13953,13956],{"className":13949},[5249],[75,13951],{"className":13952,"style":12648},[5253],[75,13954,13747],{"className":13955,"style":5955},[5258,5259],[75,13957,5820],{"className":13958,"style":5955},[5258,5259],").",[7621,13961],{},[5136,13963,13965],{"id":13964},"references","References",[28,13967,13968,14015,14030],{},[31,13969,13970,13971,13974,13975,14008,14009,14012],{},"The Quantum Aviary — ",[15,13972,13973],{},"How the Cross-Resonance Gate Works"," (overview of ",[75,13976,13978,13993],{"className":13977},[5148],[75,13979,13981],{"className":13980},[5152],[5154,13982,13983],{"xmlns":5156},[5158,13984,13985,13991],{},[5161,13986,13987,13989],{},[5164,13988,7409],{},[5164,13990,5820],{},[5238,13992,13687],{"encoding":5240},[75,13994,13996],{"className":13995,"ariaHidden":5196},[5245],[75,13997,13999,14002,14005],{"className":13998},[5249],[75,14000],{"className":14001,"style":12648},[5253],[75,14003,7409],{"className":14004,"style":7430},[5258,5259],[75,14006,5820],{"className":14007,"style":5955},[5258,5259],", echo, and intuition)",[14010,14011],"br",{},[19,14013,14014],{"href":14014},"https:\u002F\u002Fthequantumaviary.blogspot.com\u002F2021\u002F07\u002Fhow-cross-resonance-gate-works.html",[31,14016,14017,14018,206,14021,14024,14025,14027],{},"Malekakhlagh, Magesan & McKay — ",[15,14019,14020],{},"First-principles analysis of cross-resonance gate operation",[48,14022,14023],{},"Phys. Rev. A"," 102, 042605 (2020)",[14010,14026],{},[19,14028,14029],{"href":14029},"https:\u002F\u002Flink.aps.org\u002Fdoi\u002F10.1103\u002FPhysRevA.102.042605",[31,14031,14032,14033,206,14036,14038,14039,14041],{},"Magesan & Gambetta — ",[15,14034,14035],{},"Effective Hamiltonian models of the cross-resonance gate",[48,14037,14023],{}," 101, 052308 (2020)",[14010,14040],{},[19,14042,14043],{"href":14043},"https:\u002F\u002Flink.aps.org\u002Fdoi\u002F10.1103\u002FPhysRevA.101.052308",[65,14045,14047],{"className":67,"code":14046,"language":69,"meta":70,"style":70},"#@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",[72,14048,14049,14054,14058,14070,14082,14089,14099,14109,14113,14118,14123,14127,14132,14143,14160,14175,14193,14209,14224,14239,14255,14259,14264,14292,14305,14332,14344,14348,14359,14363,14367,14372,14376,14424,14429,14446,14463,14476,14480,14511,14516,14531,14553,14573,14588,14624,14640,14647,14684,14715,14756,14776,14781,14817,14823,14836,14891,14949,14957,14962,14981,14987,15010,15015,15023,15036,15049,15056,15067,15078,15083,15088,15094,15099,15149,15158,15204,15233,15286,15292,15309,15314,15361,15369,15417,15440,15480,15518,15523,15538,15543,15549,15590,15606,15622,15650,15655,15660,15666,15671,15696,15721,15726,15760,15790,15827,15832,15838,15871,15929,15951,15966,15994,16024,16045,16054,16063,16068,16074,16105,16151,16172,16183,16206,16233,16254,16263,16272,16277,16283,16309,16327,16345,16358,16363,16394,16439,16488,16520,16549,16572,16616,16625,16630,16636,16652,16669,16682,16699,16711,16746,16751,16777,16798,16803,16834,16867,16904,16931,16954,16994],{"__ignoreMap":70},[75,14050,14051],{"class":77,"line":78},[75,14052,14053],{"class":118},"#@title Cross-Resonance (CR) gate — slices & heatmaps (PRA 2020 formulas)\n",[75,14055,14056],{"class":77,"line":95},[75,14057,112],{"emptyLinePlaceholder":111},[75,14059,14060,14062,14065,14067],{"class":77,"line":108},[75,14061,2700],{"class":81},[75,14063,14064],{"class":85}," dataclasses ",[75,14066,82],{"class":81},[75,14068,14069],{"class":85}," dataclass\n",[75,14071,14072,14074,14077,14079],{"class":77,"line":115},[75,14073,2700],{"class":81},[75,14075,14076],{"class":85}," typing ",[75,14078,82],{"class":81},[75,14080,14081],{"class":85}," Tuple\n",[75,14083,14084,14086],{"class":77,"line":122},[75,14085,82],{"class":81},[75,14087,14088],{"class":85}," math\n",[75,14090,14091,14093,14095,14097],{"class":77,"line":128},[75,14092,82],{"class":81},[75,14094,86],{"class":85},[75,14096,89],{"class":81},[75,14098,92],{"class":85},[75,14100,14101,14103,14105,14107],{"class":77,"line":149},[75,14102,82],{"class":81},[75,14104,100],{"class":85},[75,14106,89],{"class":81},[75,14108,105],{"class":85},[75,14110,14111],{"class":77,"line":154},[75,14112,112],{"emptyLinePlaceholder":111},[75,14114,14115],{"class":77,"line":160},[75,14116,14117],{"class":118},"# -----------------------------\n",[75,14119,14120],{"class":77,"line":171},[75,14121,14122],{"class":118},"# Control knobs (MHz)\n",[75,14124,14125],{"class":77,"line":182},[75,14126,14117],{"class":118},[75,14128,14129],{"class":77,"line":222},[75,14130,14131],{"class":193},"@dataclass\n",[75,14133,14134,14137,14141],{"class":77,"line":227},[75,14135,14136],{"class":81},"class",[75,14138,14140],{"class":14139},"sU0A5"," CRKnobs",[75,14142,5800],{"class":85},[75,14144,14145,14148,14151,14154,14157],{"class":77,"line":233},[75,14146,14147],{"class":85},"    f_c: ",[75,14149,14150],{"class":141},"float",[75,14152,14153],{"class":141}," =",[75,14155,14156],{"class":145}," 5114.0",[75,14158,14159],{"class":118},"        # control frequency\n",[75,14161,14162,14165,14167,14169,14172],{"class":77,"line":286},[75,14163,14164],{"class":85},"    f_t: ",[75,14166,14150],{"class":141},[75,14168,14153],{"class":141},[75,14170,14171],{"class":145}," 4914.0",[75,14173,14174],{"class":118},"        # target frequency\n",[75,14176,14177,14180,14182,14184,14187,14190],{"class":77,"line":291},[75,14178,14179],{"class":85},"    alpha_c: ",[75,14181,14150],{"class":141},[75,14183,14153],{"class":141},[75,14185,14186],{"class":141}," -",[75,14188,14189],{"class":145},"330.0",[75,14191,14192],{"class":118},"    # control anharmonicity (negative)\n",[75,14194,14195,14198,14200,14202,14204,14206],{"class":77,"line":297},[75,14196,14197],{"class":85},"    alpha_t: ",[75,14199,14150],{"class":141},[75,14201,14153],{"class":141},[75,14203,14186],{"class":141},[75,14205,14189],{"class":145},[75,14207,14208],{"class":118},"    # target anharmonicity\n",[75,14210,14211,14214,14216,14218,14221],{"class":77,"line":321},[75,14212,14213],{"class":85},"    J: ",[75,14215,14150],{"class":141},[75,14217,14153],{"class":141},[75,14219,14220],{"class":145}," 3.8",[75,14222,14223],{"class":118},"             # exchange coupling\n",[75,14225,14226,14229,14231,14233,14236],{"class":77,"line":336},[75,14227,14228],{"class":85},"    Omega0: ",[75,14230,14150],{"class":141},[75,14232,14153],{"class":141},[75,14234,14235],{"class":145}," 30.0",[75,14237,14238],{"class":118},"       # default drive for Δ-slices\n",[75,14240,14241,14244,14247,14249,14252],{"class":77,"line":351},[75,14242,14243],{"class":85},"    use_energy_basis: ",[75,14245,14246],{"class":141},"bool",[75,14248,14153],{"class":141},[75,14250,14251],{"class":145}," False",[75,14253,14254],{"class":118},"  # toggle ν-corrections (~10–15%)\n",[75,14256,14257],{"class":77,"line":366},[75,14258,112],{"emptyLinePlaceholder":111},[75,14260,14261],{"class":77,"line":381},[75,14262,14263],{"class":118},"    # Sweep grids\n",[75,14265,14266,14269,14271,14273,14275,14277,14279,14281,14283,14286,14288,14290],{"class":77,"line":984},[75,14267,14268],{"class":85},"    delta_span: Tuple[",[75,14270,14150],{"class":141},[75,14272,206],{"class":85},[75,14274,14150],{"class":141},[75,14276,138],{"class":85},[75,14278,142],{"class":141},[75,14280,267],{"class":85},[75,14282,200],{"class":141},[75,14284,14285],{"class":145},"500.0",[75,14287,206],{"class":85},[75,14289,14285],{"class":145},[75,14291,318],{"class":85},[75,14293,14294,14297,14300,14302],{"class":77,"line":997},[75,14295,14296],{"class":85},"    delta_points: ",[75,14298,14299],{"class":141},"int",[75,14301,14153],{"class":141},[75,14303,14304],{"class":145}," 1801\n",[75,14306,14307,14310,14312,14314,14316,14318,14320,14322,14325,14327,14330],{"class":77,"line":1325},[75,14308,14309],{"class":85},"    omega_span: Tuple[",[75,14311,14150],{"class":141},[75,14313,206],{"class":85},[75,14315,14150],{"class":141},[75,14317,138],{"class":85},[75,14319,142],{"class":141},[75,14321,267],{"class":85},[75,14323,14324],{"class":145},"0.0",[75,14326,206],{"class":85},[75,14328,14329],{"class":145},"60.0",[75,14331,318],{"class":85},[75,14333,14334,14337,14339,14341],{"class":77,"line":1338},[75,14335,14336],{"class":85},"    omega_points: ",[75,14338,14299],{"class":141},[75,14340,14153],{"class":141},[75,14342,14343],{"class":145}," 361\n",[75,14345,14346],{"class":77,"line":1343},[75,14347,112],{"emptyLinePlaceholder":111},[75,14349,14350,14353,14355,14357],{"class":77,"line":1356},[75,14351,14352],{"class":85},"K ",[75,14354,142],{"class":141},[75,14356,14140],{"class":193},[75,14358,389],{"class":85},[75,14360,14361],{"class":77,"line":1887},[75,14362,112],{"emptyLinePlaceholder":111},[75,14364,14365],{"class":77,"line":1900},[75,14366,14117],{"class":118},[75,14368,14369],{"class":77,"line":1913},[75,14370,14371],{"class":118},"# Helpers & ν-coefficients\n",[75,14373,14374],{"class":77,"line":2657},[75,14375,14117],{"class":118},[75,14377,14378,14381,14384,14386,14389,14392,14394,14396,14399,14401,14403,14405,14408,14410,14412,14414,14417,14420,14422],{"class":77,"line":3177},[75,14379,14380],{"class":81},"def",[75,14382,14383],{"class":193}," _safe_div",[75,14385,197],{"class":85},[75,14387,19],{"class":14388},"sb9H8",[75,14390,14391],{"class":85},": ",[75,14393,14150],{"class":141},[75,14395,206],{"class":85},[75,14397,14398],{"class":14388},"b",[75,14400,14391],{"class":85},[75,14402,14150],{"class":141},[75,14404,206],{"class":85},[75,14406,14407],{"class":14388},"eps",[75,14409,14391],{"class":85},[75,14411,14150],{"class":141},[75,14413,14153],{"class":141},[75,14415,14416],{"class":145}," 1e-12",[75,14418,14419],{"class":85},") -> ",[75,14421,14150],{"class":141},[75,14423,5800],{"class":85},[75,14425,14426],{"class":77,"line":3188},[75,14427,14428],{"class":134},"    \"\"\"Return a\u002Fb with a small guard to avoid blow-ups at poles.\"\"\"\n",[75,14430,14431,14434,14437,14440,14443],{"class":77,"line":3203},[75,14432,14433],{"class":81},"    if",[75,14435,14436],{"class":141}," abs",[75,14438,14439],{"class":85},"(b) ",[75,14441,14442],{"class":141},"\u003C",[75,14444,14445],{"class":85}," eps:\n",[75,14447,14448,14451,14453,14456,14458,14460],{"class":77,"line":3218},[75,14449,14450],{"class":81},"        return",[75,14452,190],{"class":85},[75,14454,14455],{"class":193},"sign",[75,14457,14439],{"class":85},[75,14459,500],{"class":141},[75,14461,14462],{"class":145}," 1e12\n",[75,14464,14465,14468,14471,14473],{"class":77,"line":3224},[75,14466,14467],{"class":81},"    return",[75,14469,14470],{"class":85}," a ",[75,14472,2810],{"class":141},[75,14474,14475],{"class":85}," b\n",[75,14477,14478],{"class":77,"line":3229},[75,14479,112],{"emptyLinePlaceholder":111},[75,14481,14482,14484,14487,14489,14492,14494,14496,14498,14501,14503,14505,14507,14509],{"class":77,"line":3235},[75,14483,14380],{"class":81},[75,14485,14486],{"class":193}," _epsilon_from_alpha_over_omega",[75,14488,197],{"class":85},[75,14490,14491],{"class":14388},"alpha_mhz",[75,14493,14391],{"class":85},[75,14495,14150],{"class":141},[75,14497,206],{"class":85},[75,14499,14500],{"class":14388},"omega_mhz",[75,14502,14391],{"class":85},[75,14504,14150],{"class":141},[75,14506,14419],{"class":85},[75,14508,14150],{"class":141},[75,14510,5800],{"class":85},[75,14512,14513],{"class":77,"line":3263},[75,14514,14515],{"class":134},"    \"\"\"Solve ε from (9 − 4r) ε² + 16(1 − r) ε + 64 r = 0, with r = α\u002Fω.\"\"\"\n",[75,14517,14518,14521,14523,14526,14528],{"class":77,"line":3275},[75,14519,14520],{"class":85},"    r ",[75,14522,142],{"class":141},[75,14524,14525],{"class":85}," alpha_mhz ",[75,14527,2810],{"class":141},[75,14529,14530],{"class":85}," omega_mhz\n",[75,14532,14533,14536,14538,14540,14543,14545,14548,14550],{"class":77,"line":3280},[75,14534,14535],{"class":85},"    A ",[75,14537,142],{"class":141},[75,14539,267],{"class":85},[75,14541,14542],{"class":145},"9.0",[75,14544,14186],{"class":141},[75,14546,14547],{"class":145}," 4.0",[75,14549,273],{"class":141},[75,14551,14552],{"class":85}," r)\n",[75,14554,14555,14558,14560,14563,14565,14567,14569,14571],{"class":77,"line":3290},[75,14556,14557],{"class":85},"    B ",[75,14559,142],{"class":141},[75,14561,14562],{"class":145}," 16.0",[75,14564,273],{"class":141},[75,14566,267],{"class":85},[75,14568,630],{"class":145},[75,14570,14186],{"class":141},[75,14572,14552],{"class":85},[75,14574,14575,14578,14580,14583,14585],{"class":77,"line":4431},[75,14576,14577],{"class":85},"    C ",[75,14579,142],{"class":141},[75,14581,14582],{"class":145}," 64.0",[75,14584,273],{"class":141},[75,14586,14587],{"class":85}," r\n",[75,14589,14590,14593,14595,14598,14601,14603,14606,14608,14610,14612,14615,14617,14620,14622],{"class":77,"line":4454},[75,14591,14592],{"class":85},"    disc ",[75,14594,142],{"class":141},[75,14596,14597],{"class":141}," max",[75,14599,14600],{"class":85},"(B ",[75,14602,500],{"class":141},[75,14604,14605],{"class":85}," B ",[75,14607,200],{"class":141},[75,14609,14547],{"class":145},[75,14611,273],{"class":141},[75,14613,14614],{"class":85}," A ",[75,14616,500],{"class":141},[75,14618,14619],{"class":85}," C, ",[75,14621,14324],{"class":145},[75,14623,318],{"class":85},[75,14625,14626,14628,14630,14633,14635,14638],{"class":77,"line":4463},[75,14627,14433],{"class":81},[75,14629,14436],{"class":141},[75,14631,14632],{"class":85},"(A) ",[75,14634,14442],{"class":141},[75,14636,14637],{"class":145}," 1e-15",[75,14639,5800],{"class":85},[75,14641,14642,14644],{"class":77,"line":4468},[75,14643,14450],{"class":81},[75,14645,14646],{"class":145}," 0.2\n",[75,14648,14649,14652,14654,14656,14658,14661,14663,14666,14669,14672,14674,14676,14679,14681],{"class":77,"line":4477},[75,14650,14651],{"class":85},"    e1 ",[75,14653,142],{"class":141},[75,14655,267],{"class":85},[75,14657,200],{"class":141},[75,14659,14660],{"class":85},"B ",[75,14662,3251],{"class":141},[75,14664,14665],{"class":85}," math.",[75,14667,14668],{"class":193},"sqrt",[75,14670,14671],{"class":85},"(disc)) ",[75,14673,2810],{"class":141},[75,14675,267],{"class":85},[75,14677,14678],{"class":145},"2.0",[75,14680,273],{"class":141},[75,14682,14683],{"class":85}," A)\n",[75,14685,14686,14689,14691,14693,14695,14697,14699,14701,14703,14705,14707,14709,14711,14713],{"class":77,"line":5105},[75,14687,14688],{"class":85},"    e2 ",[75,14690,142],{"class":141},[75,14692,267],{"class":85},[75,14694,200],{"class":141},[75,14696,14660],{"class":85},[75,14698,200],{"class":141},[75,14700,14665],{"class":85},[75,14702,14668],{"class":193},[75,14704,14671],{"class":85},[75,14706,2810],{"class":141},[75,14708,267],{"class":85},[75,14710,14678],{"class":145},[75,14712,273],{"class":141},[75,14714,14683],{"class":85},[75,14716,14717,14720,14722,14725,14727,14730,14732,14735,14738,14740,14742,14745,14748,14750,14753],{"class":77,"line":5114},[75,14718,14719],{"class":85},"    roots ",[75,14721,142],{"class":141},[75,14723,14724],{"class":85}," [e ",[75,14726,3147],{"class":81},[75,14728,14729],{"class":85}," e ",[75,14731,3153],{"class":81},[75,14733,14734],{"class":85}," (e1, e2) ",[75,14736,14737],{"class":81},"if",[75,14739,14729],{"class":85},[75,14741,2907],{"class":141},[75,14743,14744],{"class":145}," 0.0",[75,14746,14747],{"class":81}," and",[75,14749,190],{"class":85},[75,14751,14752],{"class":193},"isfinite",[75,14754,14755],{"class":85},"(e)]\n",[75,14757,14758,14760,14763,14766,14768,14771,14774],{"class":77,"line":10795},[75,14759,14467],{"class":81},[75,14761,14762],{"class":141}," min",[75,14764,14765],{"class":85},"(roots) ",[75,14767,14737],{"class":81},[75,14769,14770],{"class":85}," roots ",[75,14772,14773],{"class":81},"else",[75,14775,14646],{"class":145},[75,14777,14779],{"class":77,"line":14778},52,[75,14780,112],{"emptyLinePlaceholder":111},[75,14782,14784,14786,14789,14791,14793,14795,14797,14799,14801,14803,14805,14808,14810,14812,14814],{"class":77,"line":14783},53,[75,14785,14380],{"class":81},[75,14787,14788],{"class":193}," _nu_energy_basis",[75,14790,197],{"class":85},[75,14792,14491],{"class":14388},[75,14794,14391],{"class":85},[75,14796,14150],{"class":141},[75,14798,206],{"class":85},[75,14800,14500],{"class":14388},[75,14802,14391],{"class":85},[75,14804,14150],{"class":141},[75,14806,14807],{"class":85},") -> Tuple[",[75,14809,14150],{"class":141},[75,14811,206],{"class":85},[75,14813,14150],{"class":141},[75,14815,14816],{"class":85},"]:\n",[75,14818,14820],{"class":77,"line":14819},54,[75,14821,14822],{"class":134},"    \"\"\"Energy-basis ν_01, ν_12 up to O(ε²).\"\"\"\n",[75,14824,14826,14829,14831,14833],{"class":77,"line":14825},55,[75,14827,14828],{"class":85},"    eps ",[75,14830,142],{"class":141},[75,14832,14486],{"class":193},[75,14834,14835],{"class":85},"(alpha_mhz, omega_mhz)\n",[75,14837,14839,14842,14844,14847,14849,14851,14853,14855,14858,14861,14863,14866,14868,14870,14873,14875,14878,14880,14882,14885,14887,14889],{"class":77,"line":14838},56,[75,14840,14841],{"class":85},"    nu01 ",[75,14843,142],{"class":141},[75,14845,14846],{"class":145}," 1.0",[75,14848,14186],{"class":141},[75,14850,267],{"class":85},[75,14852,630],{"class":145},[75,14854,264],{"class":141},[75,14856,14857],{"class":145}," 8.0",[75,14859,14860],{"class":85},") ",[75,14862,500],{"class":141},[75,14864,14865],{"class":85}," eps ",[75,14867,200],{"class":141},[75,14869,267],{"class":85},[75,14871,14872],{"class":145},"11.0",[75,14874,264],{"class":141},[75,14876,14877],{"class":145}," 256.0",[75,14879,14860],{"class":85},[75,14881,500],{"class":141},[75,14883,14884],{"class":85}," (eps ",[75,14886,258],{"class":141},[75,14888,261],{"class":145},[75,14890,318],{"class":85},[75,14892,14894,14897,14899,14901,14903,14905,14908,14910,14912,14914,14916,14919,14921,14924,14926,14928,14930,14932,14934,14937,14939,14941,14943,14945,14947],{"class":77,"line":14893},57,[75,14895,14896],{"class":85},"    nu12 ",[75,14898,142],{"class":141},[75,14900,267],{"class":85},[75,14902,630],{"class":145},[75,14904,14186],{"class":141},[75,14906,14907],{"class":145}," 0.25",[75,14909,273],{"class":141},[75,14911,14865],{"class":85},[75,14913,200],{"class":141},[75,14915,267],{"class":85},[75,14917,14918],{"class":145},"73.0",[75,14920,264],{"class":141},[75,14922,14923],{"class":145}," 512.0",[75,14925,14860],{"class":85},[75,14927,500],{"class":141},[75,14929,14884],{"class":85},[75,14931,258],{"class":141},[75,14933,261],{"class":145},[75,14935,14936],{"class":85},")) ",[75,14938,2810],{"class":141},[75,14940,14665],{"class":85},[75,14942,14668],{"class":193},[75,14944,197],{"class":85},[75,14946,14678],{"class":145},[75,14948,318],{"class":85},[75,14950,14952,14954],{"class":77,"line":14951},58,[75,14953,14467],{"class":81},[75,14955,14956],{"class":85}," nu01, nu12\n",[75,14958,14960],{"class":77,"line":14959},59,[75,14961,112],{"emptyLinePlaceholder":111},[75,14963,14965,14967,14970,14973,14975,14977,14979],{"class":77,"line":14964},60,[75,14966,14380],{"class":81},[75,14968,14969],{"class":193}," _nu_kerr",[75,14971,14972],{"class":85},"() -> Tuple[",[75,14974,14150],{"class":141},[75,14976,206],{"class":85},[75,14978,14150],{"class":141},[75,14980,14816],{"class":85},[75,14982,14984],{"class":77,"line":14983},61,[75,14985,14986],{"class":134},"    \"\"\"Kerr-limit charge matrix elements.\"\"\"\n",[75,14988,14990,14992,14994,14996,14998,15000,15002,15004,15006,15008],{"class":77,"line":14989},62,[75,14991,14467],{"class":81},[75,14993,14846],{"class":145},[75,14995,206],{"class":85},[75,14997,630],{"class":145},[75,14999,264],{"class":141},[75,15001,14665],{"class":85},[75,15003,14668],{"class":193},[75,15005,197],{"class":85},[75,15007,14678],{"class":145},[75,15009,318],{"class":85},[75,15011,15013],{"class":77,"line":15012},63,[75,15014,112],{"emptyLinePlaceholder":111},[75,15016,15018,15020],{"class":77,"line":15017},64,[75,15019,14737],{"class":81},[75,15021,15022],{"class":85}," K.use_energy_basis:\n",[75,15024,15026,15029,15031,15033],{"class":77,"line":15025},65,[75,15027,15028],{"class":85},"    nu_c01, nu_c12 ",[75,15030,142],{"class":141},[75,15032,14788],{"class":193},[75,15034,15035],{"class":85},"(K.alpha_c, K.f_c)\n",[75,15037,15039,15042,15044,15046],{"class":77,"line":15038},66,[75,15040,15041],{"class":85},"    nu_t01, nu_t12 ",[75,15043,142],{"class":141},[75,15045,14788],{"class":193},[75,15047,15048],{"class":85},"(K.alpha_t, K.f_t)\n",[75,15050,15052,15054],{"class":77,"line":15051},67,[75,15053,14773],{"class":81},[75,15055,5800],{"class":85},[75,15057,15059,15061,15063,15065],{"class":77,"line":15058},68,[75,15060,15028],{"class":85},[75,15062,142],{"class":141},[75,15064,14969],{"class":193},[75,15066,389],{"class":85},[75,15068,15070,15072,15074,15076],{"class":77,"line":15069},69,[75,15071,15041],{"class":85},[75,15073,142],{"class":141},[75,15075,14969],{"class":193},[75,15077,389],{"class":85},[75,15079,15081],{"class":77,"line":15080},70,[75,15082,112],{"emptyLinePlaceholder":111},[75,15084,15086],{"class":77,"line":15085},71,[75,15087,14117],{"class":118},[75,15089,15091],{"class":77,"line":15090},72,[75,15092,15093],{"class":118},"# Leading-order rates (MHz)\n",[75,15095,15097],{"class":77,"line":15096},73,[75,15098,14117],{"class":118},[75,15100,15102,15104,15107,15109,15112,15114,15116,15118,15121,15123,15125,15127,15130,15132,15134,15136,15139,15141,15143,15145,15147],{"class":77,"line":15101},74,[75,15103,14380],{"class":81},[75,15105,15106],{"class":193}," omega_zx_mhz",[75,15108,197],{"class":85},[75,15110,15111],{"class":14388},"delta_ct",[75,15113,14391],{"class":85},[75,15115,14150],{"class":141},[75,15117,206],{"class":85},[75,15119,15120],{"class":14388},"J",[75,15122,14391],{"class":85},[75,15124,14150],{"class":141},[75,15126,206],{"class":85},[75,15128,15129],{"class":14388},"Omega",[75,15131,14391],{"class":85},[75,15133,14150],{"class":141},[75,15135,206],{"class":85},[75,15137,15138],{"class":14388},"alpha_c",[75,15140,14391],{"class":85},[75,15142,14150],{"class":141},[75,15144,14419],{"class":85},[75,15146,14150],{"class":141},[75,15148,5800],{"class":85},[75,15150,15152,15155],{"class":77,"line":15151},75,[75,15153,15154],{"class":81},"    r",[75,15156,15157],{"class":134},"\"\"\"Entangling rate ω_ZX ≈ JΩ[(Δ+α_c)^{-1} − Δ^{-1}] with small ν-correction.\"\"\"\n",[75,15159,15161,15164,15166,15169,15171,15174,15176,15178,15181,15183,15185,15188,15190,15193,15195,15197,15199,15201],{"class":77,"line":15160},76,[75,15162,15163],{"class":85},"    kerr ",[75,15165,142],{"class":141},[75,15167,15168],{"class":85}," J ",[75,15170,500],{"class":141},[75,15172,15173],{"class":85}," Omega ",[75,15175,500],{"class":141},[75,15177,267],{"class":85},[75,15179,15180],{"class":193},"_safe_div",[75,15182,197],{"class":85},[75,15184,630],{"class":145},[75,15186,15187],{"class":85},", delta_ct ",[75,15189,3251],{"class":141},[75,15191,15192],{"class":85}," alpha_c) ",[75,15194,200],{"class":141},[75,15196,14383],{"class":193},[75,15198,197],{"class":85},[75,15200,630],{"class":145},[75,15202,15203],{"class":85},", delta_ct))\n",[75,15205,15207,15210,15212,15215,15217,15219,15221,15223,15225,15228,15230],{"class":77,"line":15206},77,[75,15208,15209],{"class":85},"    eb ",[75,15211,142],{"class":141},[75,15213,15214],{"class":145}," 0.5",[75,15216,273],{"class":141},[75,15218,15168],{"class":85},[75,15220,500],{"class":141},[75,15222,15173],{"class":85},[75,15224,500],{"class":141},[75,15226,15227],{"class":85}," nu_t01 ",[75,15229,500],{"class":141},[75,15231,15232],{"class":85}," (\n",[75,15234,15236,15239,15241,15243,15245,15247,15249,15251,15253,15255,15257,15259,15261,15264,15266,15269,15271,15273,15275,15277,15279,15281,15283],{"class":77,"line":15235},78,[75,15237,15238],{"class":85},"        (nu_c12 ",[75,15240,258],{"class":141},[75,15242,261],{"class":145},[75,15244,14860],{"class":85},[75,15246,500],{"class":141},[75,15248,14383],{"class":193},[75,15250,197],{"class":85},[75,15252,630],{"class":145},[75,15254,15187],{"class":85},[75,15256,3251],{"class":141},[75,15258,15192],{"class":85},[75,15260,200],{"class":141},[75,15262,15263],{"class":145}," 2.0",[75,15265,273],{"class":141},[75,15267,15268],{"class":85}," (nu_c01 ",[75,15270,258],{"class":141},[75,15272,261],{"class":145},[75,15274,14860],{"class":85},[75,15276,500],{"class":141},[75,15278,14383],{"class":193},[75,15280,197],{"class":85},[75,15282,630],{"class":145},[75,15284,15285],{"class":85},", delta_ct)\n",[75,15287,15289],{"class":77,"line":15288},79,[75,15290,15291],{"class":85},"    )\n",[75,15293,15295,15297,15299,15301,15304,15306],{"class":77,"line":15294},80,[75,15296,14467],{"class":81},[75,15298,15214],{"class":145},[75,15300,273],{"class":141},[75,15302,15303],{"class":85}," (kerr ",[75,15305,3251],{"class":141},[75,15307,15308],{"class":85}," eb)\n",[75,15310,15312],{"class":77,"line":15311},81,[75,15313,112],{"emptyLinePlaceholder":111},[75,15315,15317,15319,15322,15324,15326,15328,15330,15332,15334,15336,15338,15340,15342,15344,15346,15348,15351,15353,15355,15357,15359],{"class":77,"line":15316},82,[75,15318,14380],{"class":81},[75,15320,15321],{"class":193}," omega_zz_mhz",[75,15323,197],{"class":85},[75,15325,15111],{"class":14388},[75,15327,14391],{"class":85},[75,15329,14150],{"class":141},[75,15331,206],{"class":85},[75,15333,15120],{"class":14388},[75,15335,14391],{"class":85},[75,15337,14150],{"class":141},[75,15339,206],{"class":85},[75,15341,15138],{"class":14388},[75,15343,14391],{"class":85},[75,15345,14150],{"class":141},[75,15347,206],{"class":85},[75,15349,15350],{"class":14388},"alpha_t",[75,15352,14391],{"class":85},[75,15354,14150],{"class":141},[75,15356,14419],{"class":85},[75,15358,14150],{"class":141},[75,15360,5800],{"class":85},[75,15362,15364,15366],{"class":77,"line":15363},83,[75,15365,15154],{"class":81},[75,15367,15368],{"class":134},"\"\"\"Static ZZ ≈ J²[(Δ−α_t)^{-1} − (Δ+α_c)^{-1}] with small ν-correction.\"\"\"\n",[75,15370,15372,15374,15376,15379,15381,15383,15385,15387,15389,15391,15393,15395,15397,15399,15402,15404,15406,15408,15410,15412,15414],{"class":77,"line":15371},84,[75,15373,15163],{"class":85},[75,15375,142],{"class":141},[75,15377,15378],{"class":85}," (J ",[75,15380,258],{"class":141},[75,15382,261],{"class":145},[75,15384,14860],{"class":85},[75,15386,500],{"class":141},[75,15388,267],{"class":85},[75,15390,15180],{"class":193},[75,15392,197],{"class":85},[75,15394,630],{"class":145},[75,15396,15187],{"class":85},[75,15398,200],{"class":141},[75,15400,15401],{"class":85}," alpha_t) ",[75,15403,200],{"class":141},[75,15405,14383],{"class":193},[75,15407,197],{"class":85},[75,15409,630],{"class":145},[75,15411,15187],{"class":85},[75,15413,3251],{"class":141},[75,15415,15416],{"class":85}," alpha_c))\n",[75,15418,15420,15422,15424,15426,15428,15430,15432,15434,15436,15438],{"class":77,"line":15419},85,[75,15421,15209],{"class":85},[75,15423,142],{"class":141},[75,15425,15214],{"class":145},[75,15427,273],{"class":141},[75,15429,15378],{"class":85},[75,15431,258],{"class":141},[75,15433,261],{"class":145},[75,15435,14860],{"class":85},[75,15437,500],{"class":141},[75,15439,15232],{"class":85},[75,15441,15443,15446,15448,15450,15452,15454,15457,15459,15461,15463,15465,15467,15469,15471,15473,15475,15477],{"class":77,"line":15442},86,[75,15444,15445],{"class":85},"        (nu_c01 ",[75,15447,258],{"class":141},[75,15449,261],{"class":145},[75,15451,14860],{"class":85},[75,15453,500],{"class":141},[75,15455,15456],{"class":85}," (nu_t12 ",[75,15458,258],{"class":141},[75,15460,261],{"class":145},[75,15462,14860],{"class":85},[75,15464,500],{"class":141},[75,15466,14383],{"class":193},[75,15468,197],{"class":85},[75,15470,630],{"class":145},[75,15472,15187],{"class":85},[75,15474,200],{"class":141},[75,15476,15401],{"class":85},[75,15478,15479],{"class":141},"-\n",[75,15481,15483,15486,15488,15490,15492,15494,15497,15499,15501,15503,15505,15507,15509,15511,15513,15515],{"class":77,"line":15482},87,[75,15484,15485],{"class":85},"        (nu_t01 ",[75,15487,258],{"class":141},[75,15489,261],{"class":145},[75,15491,14860],{"class":85},[75,15493,500],{"class":141},[75,15495,15496],{"class":85}," (nu_c12 ",[75,15498,258],{"class":141},[75,15500,261],{"class":145},[75,15502,14860],{"class":85},[75,15504,500],{"class":141},[75,15506,14383],{"class":193},[75,15508,197],{"class":85},[75,15510,630],{"class":145},[75,15512,15187],{"class":85},[75,15514,3251],{"class":141},[75,15516,15517],{"class":85}," alpha_c)\n",[75,15519,15521],{"class":77,"line":15520},88,[75,15522,15291],{"class":85},[75,15524,15526,15528,15530,15532,15534,15536],{"class":77,"line":15525},89,[75,15527,14467],{"class":81},[75,15529,15214],{"class":145},[75,15531,273],{"class":141},[75,15533,15303],{"class":85},[75,15535,3251],{"class":141},[75,15537,15308],{"class":85},[75,15539,15541],{"class":77,"line":15540},90,[75,15542,112],{"emptyLinePlaceholder":111},[75,15544,15546],{"class":77,"line":15545},91,[75,15547,15548],{"class":118},"# Region boundaries (poles) for I–V:\n",[75,15550,15552,15555,15557,15560,15562,15564,15566,15569,15571,15573,15575,15577,15580,15582,15585,15587],{"class":77,"line":15551},92,[75,15553,15554],{"class":145},"REGION_LINES",[75,15556,14153],{"class":141},[75,15558,15559],{"class":85}," (K.alpha_t, ",[75,15561,14324],{"class":145},[75,15563,206],{"class":85},[75,15565,200],{"class":141},[75,15567,15568],{"class":85},"K.alpha_c ",[75,15570,2810],{"class":141},[75,15572,15263],{"class":145},[75,15574,206],{"class":85},[75,15576,200],{"class":141},[75,15578,15579],{"class":85},"K.alpha_c, ",[75,15581,200],{"class":141},[75,15583,15584],{"class":145},"1.5",[75,15586,273],{"class":141},[75,15588,15589],{"class":85}," K.alpha_c)\n",[75,15591,15593,15595,15598,15600,15603],{"class":77,"line":15592},93,[75,15594,14380],{"class":81},[75,15596,15597],{"class":193}," _annotate_regions",[75,15599,197],{"class":85},[75,15601,15602],{"class":14388},"ax",[75,15604,15605],{"class":85},"):\n",[75,15607,15609,15612,15615,15617,15620],{"class":77,"line":15608},94,[75,15610,15611],{"class":81},"    for",[75,15613,15614],{"class":85}," x ",[75,15616,3153],{"class":81},[75,15618,15619],{"class":145}," REGION_LINES",[75,15621,5800],{"class":85},[75,15623,15625,15628,15630,15633,15635,15637,15640,15642,15644,15646,15648],{"class":77,"line":15624},95,[75,15626,15627],{"class":85},"        ax.",[75,15629,11213],{"class":193},[75,15631,15632],{"class":85},"(x, ",[75,15634,563],{"class":309},[75,15636,142],{"class":141},[75,15638,15639],{"class":134},"\"--\"",[75,15641,206],{"class":85},[75,15643,11200],{"class":309},[75,15645,142],{"class":141},[75,15647,630],{"class":145},[75,15649,318],{"class":85},[75,15651,15653],{"class":77,"line":15652},96,[75,15654,112],{"emptyLinePlaceholder":111},[75,15656,15658],{"class":77,"line":15657},97,[75,15659,14117],{"class":118},[75,15661,15663],{"class":77,"line":15662},98,[75,15664,15665],{"class":118},"# Slices\n",[75,15667,15669],{"class":77,"line":15668},99,[75,15670,14117],{"class":118},[75,15672,15674,15677,15679,15681,15683,15686,15688,15691,15693],{"class":77,"line":15673},100,[75,15675,15676],{"class":145},"DELTA",[75,15678,14153],{"class":141},[75,15680,190],{"class":85},[75,15682,194],{"class":193},[75,15684,15685],{"class":85},"(K.delta_span[",[75,15687,5186],{"class":145},[75,15689,15690],{"class":85},"], K.delta_span[",[75,15692,7477],{"class":145},[75,15694,15695],{"class":85},"], K.delta_points)\n",[75,15697,15699,15702,15704,15706,15708,15711,15713,15716,15718],{"class":77,"line":15698},101,[75,15700,15701],{"class":145},"OMEGA",[75,15703,14153],{"class":141},[75,15705,190],{"class":85},[75,15707,194],{"class":193},[75,15709,15710],{"class":85},"(K.omega_span[",[75,15712,5186],{"class":145},[75,15714,15715],{"class":85},"], K.omega_span[",[75,15717,7477],{"class":145},[75,15719,15720],{"class":85},"], K.omega_points)\n",[75,15722,15724],{"class":77,"line":15723},102,[75,15725,112],{"emptyLinePlaceholder":111},[75,15727,15729,15732,15734,15736,15738,15741,15744,15747,15749,15752,15754,15757],{"class":77,"line":15728},103,[75,15730,15731],{"class":85},"wzx_slice ",[75,15733,142],{"class":141},[75,15735,190],{"class":85},[75,15737,12049],{"class":193},[75,15739,15740],{"class":85},"([",[75,15742,15743],{"class":193},"omega_zx_mhz",[75,15745,15746],{"class":85},"(d, K.J, K.Omega0, K.alpha_c) ",[75,15748,3147],{"class":81},[75,15750,15751],{"class":85}," d ",[75,15753,3153],{"class":81},[75,15755,15756],{"class":145}," DELTA",[75,15758,15759],{"class":85},"])\n",[75,15761,15763,15766,15768,15770,15772,15774,15777,15780,15782,15784,15786,15788],{"class":77,"line":15762},104,[75,15764,15765],{"class":85},"wzz_slice ",[75,15767,142],{"class":141},[75,15769,190],{"class":85},[75,15771,12049],{"class":193},[75,15773,15740],{"class":85},[75,15775,15776],{"class":193},"omega_zz_mhz",[75,15778,15779],{"class":85},"(d, K.J, K.alpha_c, K.alpha_t) ",[75,15781,3147],{"class":81},[75,15783,15751],{"class":85},[75,15785,3153],{"class":81},[75,15787,15756],{"class":145},[75,15789,15759],{"class":85},[75,15791,15793,15796,15798,15800,15802,15805,15808,15810,15812,15815,15817,15819,15822,15824],{"class":77,"line":15792},105,[75,15794,15795],{"class":85},"ratio_slice ",[75,15797,142],{"class":141},[75,15799,190],{"class":85},[75,15801,2145],{"class":193},[75,15803,15804],{"class":85},"(np.",[75,15806,15807],{"class":193},"where",[75,15809,15804],{"class":85},[75,15811,2145],{"class":193},[75,15813,15814],{"class":85},"(wzz_slice) ",[75,15816,2907],{"class":141},[75,15818,14416],{"class":145},[75,15820,15821],{"class":85},", wzx_slice ",[75,15823,2810],{"class":141},[75,15825,15826],{"class":85}," wzz_slice, np.nan))\n",[75,15828,15830],{"class":77,"line":15829},106,[75,15831,112],{"emptyLinePlaceholder":111},[75,15833,15835],{"class":77,"line":15834},107,[75,15836,15837],{"class":118},"# Plot 1: ω_ZX vs Δ_ct\n",[75,15839,15841,15843,15845,15847,15849,15851,15853,15855,15857,15859,15861,15864,15866,15869],{"class":77,"line":15840},108,[75,15842,300],{"class":85},[75,15844,865],{"class":193},[75,15846,197],{"class":85},[75,15848,870],{"class":309},[75,15850,142],{"class":141},[75,15852,197],{"class":85},[75,15854,877],{"class":145},[75,15856,206],{"class":85},[75,15858,203],{"class":145},[75,15860,10496],{"class":85},[75,15862,15863],{"class":309},"dpi",[75,15865,142],{"class":141},[75,15867,15868],{"class":145},"150",[75,15870,318],{"class":85},[75,15872,15874,15876,15878,15880,15882,15885,15887,15889,15892,15895,15898,15900,15903,15906,15908,15911,15914,15916,15919,15921,15923,15925,15927],{"class":77,"line":15873},109,[75,15875,300],{"class":85},[75,15877,303],{"class":193},[75,15879,197],{"class":85},[75,15881,15676],{"class":145},[75,15883,15884],{"class":85},", wzx_slice, ",[75,15886,573],{"class":309},[75,15888,142],{"class":141},[75,15890,15891],{"class":81},"rf",[75,15893,15894],{"class":134},"'$\\omega_",[75,15896,15897],{"class":141},"{{",[75,15899,13687],{"class":134},[75,15901,15902],{"class":141},"}}",[75,15904,15905],{"class":134},"$ (Ω=",[75,15907,12219],{"class":145},[75,15909,15910],{"class":85},"K.Omega0",[75,15912,15913],{"class":81},":.1f",[75,15915,12224],{"class":145},[75,15917,15918],{"class":134}," MHz)'",[75,15920,206],{"class":85},[75,15922,310],{"class":309},[75,15924,142],{"class":141},[75,15926,315],{"class":134},[75,15928,318],{"class":85},[75,15930,15932,15934,15936,15938,15940,15942,15944,15946,15949],{"class":77,"line":15931},110,[75,15933,300],{"class":85},[75,15935,11182],{"class":193},[75,15937,197],{"class":85},[75,15939,14324],{"class":145},[75,15941,206],{"class":85},[75,15943,563],{"class":309},[75,15945,142],{"class":141},[75,15947,15948],{"class":134},"\":\"",[75,15950,318],{"class":85},[75,15952,15954,15957,15960,15963],{"class":77,"line":15953},111,[75,15955,15956],{"class":193},"_annotate_regions",[75,15958,15959],{"class":85},"(plt.",[75,15961,15962],{"class":193},"gca",[75,15964,15965],{"class":85},"())\n",[75,15967,15969,15971,15973,15975,15978,15982,15984,15987,15989,15992],{"class":77,"line":15968},112,[75,15970,300],{"class":85},[75,15972,341],{"class":193},[75,15974,197],{"class":85},[75,15976,15977],{"class":81},"r",[75,15979,15981],{"class":15980},"sVyAn","'$\\Delta_{ct}$ ",[75,15983,197],{"class":145},[75,15985,15986],{"class":15980},"MHz",[75,15988,5175],{"class":145},[75,15990,15991],{"class":15980},"'",[75,15993,318],{"class":85},[75,15995,15997,15999,16001,16003,16005,16008,16011,16014,16016,16018,16020,16022],{"class":77,"line":15996},113,[75,15998,300],{"class":85},[75,16000,356],{"class":193},[75,16002,197],{"class":85},[75,16004,15977],{"class":81},[75,16006,16007],{"class":15980},"'$",[75,16009,16010],{"class":141},"\\o",[75,16012,16013],{"class":15980},"mega_{ZX}$ ",[75,16015,197],{"class":145},[75,16017,15986],{"class":15980},[75,16019,5175],{"class":145},[75,16021,15991],{"class":15980},[75,16023,318],{"class":85},[75,16025,16027,16029,16031,16033,16035,16038,16040,16043],{"class":77,"line":16026},114,[75,16028,300],{"class":85},[75,16030,326],{"class":193},[75,16032,197],{"class":85},[75,16034,15977],{"class":81},[75,16036,16037],{"class":15980},"'Leading-order $",[75,16039,16010],{"class":141},[75,16041,16042],{"class":15980},"mega_{ZX}$ vs detuning'",[75,16044,318],{"class":85},[75,16046,16048,16050,16052],{"class":77,"line":16047},115,[75,16049,300],{"class":85},[75,16051,5058],{"class":193},[75,16053,389],{"class":85},[75,16055,16057,16059,16061],{"class":77,"line":16056},116,[75,16058,300],{"class":85},[75,16060,386],{"class":193},[75,16062,389],{"class":85},[75,16064,16066],{"class":77,"line":16065},117,[75,16067,112],{"emptyLinePlaceholder":111},[75,16069,16071],{"class":77,"line":16070},118,[75,16072,16073],{"class":118},"# Plot 2: ω_ZZ vs Δ_ct\n",[75,16075,16077,16079,16081,16083,16085,16087,16089,16091,16093,16095,16097,16099,16101,16103],{"class":77,"line":16076},119,[75,16078,300],{"class":85},[75,16080,865],{"class":193},[75,16082,197],{"class":85},[75,16084,870],{"class":309},[75,16086,142],{"class":141},[75,16088,197],{"class":85},[75,16090,877],{"class":145},[75,16092,206],{"class":85},[75,16094,203],{"class":145},[75,16096,10496],{"class":85},[75,16098,15863],{"class":309},[75,16100,142],{"class":141},[75,16102,15868],{"class":145},[75,16104,318],{"class":85},[75,16106,16108,16110,16112,16114,16116,16119,16121,16123,16125,16127,16129,16132,16134,16137,16139,16141,16143,16145,16147,16149],{"class":77,"line":16107},120,[75,16109,300],{"class":85},[75,16111,303],{"class":193},[75,16113,197],{"class":85},[75,16115,15676],{"class":145},[75,16117,16118],{"class":85},", wzz_slice, ",[75,16120,573],{"class":309},[75,16122,142],{"class":141},[75,16124,15977],{"class":81},[75,16126,16007],{"class":15980},[75,16128,16010],{"class":141},[75,16130,16131],{"class":15980},"mega_{ZZ}$ ",[75,16133,197],{"class":145},[75,16135,16136],{"class":15980},"static; lowest order",[75,16138,5175],{"class":145},[75,16140,15991],{"class":15980},[75,16142,206],{"class":85},[75,16144,310],{"class":309},[75,16146,142],{"class":141},[75,16148,315],{"class":134},[75,16150,318],{"class":85},[75,16152,16154,16156,16158,16160,16162,16164,16166,16168,16170],{"class":77,"line":16153},121,[75,16155,300],{"class":85},[75,16157,11182],{"class":193},[75,16159,197],{"class":85},[75,16161,14324],{"class":145},[75,16163,206],{"class":85},[75,16165,563],{"class":309},[75,16167,142],{"class":141},[75,16169,15948],{"class":134},[75,16171,318],{"class":85},[75,16173,16175,16177,16179,16181],{"class":77,"line":16174},122,[75,16176,15956],{"class":193},[75,16178,15959],{"class":85},[75,16180,15962],{"class":193},[75,16182,15965],{"class":85},[75,16184,16186,16188,16190,16192,16194,16196,16198,16200,16202,16204],{"class":77,"line":16185},123,[75,16187,300],{"class":85},[75,16189,341],{"class":193},[75,16191,197],{"class":85},[75,16193,15977],{"class":81},[75,16195,15981],{"class":15980},[75,16197,197],{"class":145},[75,16199,15986],{"class":15980},[75,16201,5175],{"class":145},[75,16203,15991],{"class":15980},[75,16205,318],{"class":85},[75,16207,16209,16211,16213,16215,16217,16219,16221,16223,16225,16227,16229,16231],{"class":77,"line":16208},124,[75,16210,300],{"class":85},[75,16212,356],{"class":193},[75,16214,197],{"class":85},[75,16216,15977],{"class":81},[75,16218,16007],{"class":15980},[75,16220,16010],{"class":141},[75,16222,16131],{"class":15980},[75,16224,197],{"class":145},[75,16226,15986],{"class":15980},[75,16228,5175],{"class":145},[75,16230,15991],{"class":15980},[75,16232,318],{"class":85},[75,16234,16236,16238,16240,16242,16244,16247,16249,16252],{"class":77,"line":16235},125,[75,16237,300],{"class":85},[75,16239,326],{"class":193},[75,16241,197],{"class":85},[75,16243,15977],{"class":81},[75,16245,16246],{"class":15980},"'Lowest-order $",[75,16248,16010],{"class":141},[75,16250,16251],{"class":15980},"mega_{ZZ}$ vs detuning'",[75,16253,318],{"class":85},[75,16255,16257,16259,16261],{"class":77,"line":16256},126,[75,16258,300],{"class":85},[75,16260,5058],{"class":193},[75,16262,389],{"class":85},[75,16264,16266,16268,16270],{"class":77,"line":16265},127,[75,16267,300],{"class":85},[75,16269,386],{"class":193},[75,16271,389],{"class":85},[75,16273,16275],{"class":77,"line":16274},128,[75,16276,112],{"emptyLinePlaceholder":111},[75,16278,16280],{"class":77,"line":16279},129,[75,16281,16282],{"class":118},"# Heatmap 1: ω_ZX(Δ, Ω)\n",[75,16284,16286,16289,16291,16293,16296,16299,16301,16304,16306],{"class":77,"line":16285},130,[75,16287,16288],{"class":85},"wzx_map ",[75,16290,142],{"class":141},[75,16292,190],{"class":85},[75,16294,16295],{"class":193},"empty",[75,16297,16298],{"class":85},"((",[75,16300,15676],{"class":145},[75,16302,16303],{"class":85},".size, ",[75,16305,15701],{"class":145},[75,16307,16308],{"class":85},".size))\n",[75,16310,16312,16314,16317,16319,16321,16323,16325],{"class":77,"line":16311},131,[75,16313,3147],{"class":81},[75,16315,16316],{"class":85}," i, d ",[75,16318,3153],{"class":81},[75,16320,3156],{"class":141},[75,16322,197],{"class":85},[75,16324,15676],{"class":145},[75,16326,15605],{"class":85},[75,16328,16330,16332,16335,16337,16339,16341,16343],{"class":77,"line":16329},132,[75,16331,15611],{"class":81},[75,16333,16334],{"class":85}," j, om ",[75,16336,3153],{"class":81},[75,16338,3156],{"class":141},[75,16340,197],{"class":85},[75,16342,15701],{"class":145},[75,16344,15605],{"class":85},[75,16346,16348,16351,16353,16355],{"class":77,"line":16347},133,[75,16349,16350],{"class":85},"        wzx_map[i, j] ",[75,16352,142],{"class":141},[75,16354,15106],{"class":193},[75,16356,16357],{"class":85},"(d, K.J, om, K.alpha_c)\n",[75,16359,16361],{"class":77,"line":16360},134,[75,16362,112],{"emptyLinePlaceholder":111},[75,16364,16366,16368,16370,16372,16374,16376,16378,16380,16382,16384,16386,16388,16390,16392],{"class":77,"line":16365},135,[75,16367,300],{"class":85},[75,16369,865],{"class":193},[75,16371,197],{"class":85},[75,16373,870],{"class":309},[75,16375,142],{"class":141},[75,16377,197],{"class":85},[75,16379,877],{"class":145},[75,16381,206],{"class":85},[75,16383,882],{"class":145},[75,16385,10496],{"class":85},[75,16387,15863],{"class":309},[75,16389,142],{"class":141},[75,16391,15868],{"class":145},[75,16393,318],{"class":85},[75,16395,16397,16400,16402,16405,16407,16409,16411,16414,16416,16418,16420,16422,16424,16426,16428,16430,16432,16434,16436],{"class":77,"line":16396},136,[75,16398,16399],{"class":85},"extent ",[75,16401,142],{"class":141},[75,16403,16404],{"class":85}," [",[75,16406,15701],{"class":145},[75,16408,6543],{"class":85},[75,16410,2804],{"class":193},[75,16412,16413],{"class":85},"(), ",[75,16415,15701],{"class":145},[75,16417,6543],{"class":85},[75,16419,2816],{"class":193},[75,16421,16413],{"class":85},[75,16423,15676],{"class":145},[75,16425,6543],{"class":85},[75,16427,2804],{"class":193},[75,16429,16413],{"class":85},[75,16431,15676],{"class":145},[75,16433,6543],{"class":85},[75,16435,2816],{"class":193},[75,16437,16438],{"class":85},"()]\n",[75,16440,16442,16444,16447,16450,16453,16455,16458,16460,16463,16465,16468,16470,16473,16475,16478,16481,16483,16486],{"class":77,"line":16441},137,[75,16443,300],{"class":85},[75,16445,16446],{"class":193},"imshow",[75,16448,16449],{"class":85},"(wzx_map, ",[75,16451,16452],{"class":309},"origin",[75,16454,142],{"class":141},[75,16456,16457],{"class":134},"\"lower\"",[75,16459,206],{"class":85},[75,16461,16462],{"class":309},"aspect",[75,16464,142],{"class":141},[75,16466,16467],{"class":134},"\"auto\"",[75,16469,206],{"class":85},[75,16471,16472],{"class":309},"extent",[75,16474,142],{"class":141},[75,16476,16477],{"class":85},"extent, ",[75,16479,16480],{"class":309},"cmap",[75,16482,142],{"class":141},[75,16484,16485],{"class":134},"\"magma\"",[75,16487,318],{"class":85},[75,16489,16491,16493,16496,16498,16500,16502,16504,16506,16508,16510,16512,16514,16516,16518],{"class":77,"line":16490},138,[75,16492,300],{"class":85},[75,16494,16495],{"class":193},"colorbar",[75,16497,197],{"class":85},[75,16499,573],{"class":309},[75,16501,142],{"class":141},[75,16503,15977],{"class":81},[75,16505,16007],{"class":15980},[75,16507,16010],{"class":141},[75,16509,16013],{"class":15980},[75,16511,197],{"class":145},[75,16513,15986],{"class":15980},[75,16515,5175],{"class":145},[75,16517,15991],{"class":15980},[75,16519,318],{"class":85},[75,16521,16523,16525,16527,16529,16531,16533,16536,16539,16541,16543,16545,16547],{"class":77,"line":16522},139,[75,16524,300],{"class":85},[75,16526,341],{"class":193},[75,16528,197],{"class":85},[75,16530,15977],{"class":81},[75,16532,16007],{"class":15980},[75,16534,16535],{"class":141},"\\O",[75,16537,16538],{"class":15980},"mega$ ",[75,16540,197],{"class":145},[75,16542,15986],{"class":15980},[75,16544,5175],{"class":145},[75,16546,15991],{"class":15980},[75,16548,318],{"class":85},[75,16550,16552,16554,16556,16558,16560,16562,16564,16566,16568,16570],{"class":77,"line":16551},140,[75,16553,300],{"class":85},[75,16555,356],{"class":193},[75,16557,197],{"class":85},[75,16559,15977],{"class":81},[75,16561,15981],{"class":15980},[75,16563,197],{"class":145},[75,16565,15986],{"class":15980},[75,16567,5175],{"class":145},[75,16569,15991],{"class":15980},[75,16571,318],{"class":85},[75,16573,16575,16577,16579,16581,16583,16585,16587,16590,16592,16595,16597,16600,16602,16605,16607,16610,16612,16614],{"class":77,"line":16574},141,[75,16576,300],{"class":85},[75,16578,326],{"class":193},[75,16580,197],{"class":85},[75,16582,15977],{"class":81},[75,16584,16007],{"class":15980},[75,16586,16010],{"class":141},[75,16588,16589],{"class":15980},"mega_{ZX}",[75,16591,197],{"class":145},[75,16593,16594],{"class":15980},"\\Delta_{ct}, ",[75,16596,16535],{"class":141},[75,16598,16599],{"class":15980},"mega",[75,16601,5175],{"class":145},[75,16603,16604],{"class":15980},"$ ",[75,16606,197],{"class":145},[75,16608,16609],{"class":15980},"leading-order",[75,16611,5175],{"class":145},[75,16613,15991],{"class":15980},[75,16615,318],{"class":85},[75,16617,16619,16621,16623],{"class":77,"line":16618},142,[75,16620,300],{"class":85},[75,16622,386],{"class":193},[75,16624,389],{"class":85},[75,16626,16628],{"class":77,"line":16627},143,[75,16629,112],{"emptyLinePlaceholder":111},[75,16631,16633],{"class":77,"line":16632},144,[75,16634,16635],{"class":118},"# Heatmap 2: |ω_ZX\u002Fω_ZZ|(Δ, Ω) (clipped for visibility)\n",[75,16637,16639,16642,16644,16646,16649],{"class":77,"line":16638},145,[75,16640,16641],{"class":85},"ratio_map ",[75,16643,142],{"class":141},[75,16645,190],{"class":85},[75,16647,16648],{"class":193},"empty_like",[75,16650,16651],{"class":85},"(wzx_map)\n",[75,16653,16655,16657,16659,16661,16663,16665,16667],{"class":77,"line":16654},146,[75,16656,3147],{"class":81},[75,16658,16316],{"class":85},[75,16660,3153],{"class":81},[75,16662,3156],{"class":141},[75,16664,197],{"class":85},[75,16666,15676],{"class":145},[75,16668,15605],{"class":85},[75,16670,16672,16675,16677,16679],{"class":77,"line":16671},147,[75,16673,16674],{"class":85},"    zz ",[75,16676,142],{"class":141},[75,16678,15321],{"class":193},[75,16680,16681],{"class":85},"(d, K.J, K.alpha_c, K.alpha_t)\n",[75,16683,16685,16687,16689,16691,16693,16695,16697],{"class":77,"line":16684},148,[75,16686,15611],{"class":81},[75,16688,16334],{"class":85},[75,16690,3153],{"class":81},[75,16692,3156],{"class":141},[75,16694,197],{"class":85},[75,16696,15701],{"class":145},[75,16698,15605],{"class":85},[75,16700,16702,16705,16707,16709],{"class":77,"line":16701},149,[75,16703,16704],{"class":85},"        zx ",[75,16706,142],{"class":141},[75,16708,15106],{"class":193},[75,16710,16357],{"class":85},[75,16712,16714,16717,16719,16721,16724,16726,16729,16731,16733,16736,16738,16740,16743],{"class":77,"line":16713},150,[75,16715,16716],{"class":85},"        ratio_map[i, j] ",[75,16718,142],{"class":141},[75,16720,14436],{"class":141},[75,16722,16723],{"class":85},"(zx ",[75,16725,2810],{"class":141},[75,16727,16728],{"class":85}," zz) ",[75,16730,14737],{"class":81},[75,16732,14436],{"class":141},[75,16734,16735],{"class":85},"(zz) ",[75,16737,2907],{"class":141},[75,16739,14416],{"class":145},[75,16741,16742],{"class":81}," else",[75,16744,16745],{"class":85}," np.nan\n",[75,16747,16749],{"class":77,"line":16748},151,[75,16750,112],{"emptyLinePlaceholder":111},[75,16752,16754,16757,16759,16761,16764,16767,16769,16772,16775],{"class":77,"line":16753},152,[75,16755,16756],{"class":85},"clip_max ",[75,16758,142],{"class":141},[75,16760,190],{"class":85},[75,16762,16763],{"class":193},"nanpercentile",[75,16765,16766],{"class":85},"(ratio_map[np.",[75,16768,14752],{"class":193},[75,16770,16771],{"class":85},"(ratio_map)], ",[75,16773,16774],{"class":145},"99.5",[75,16776,318],{"class":85},[75,16778,16780,16783,16785,16787,16790,16793,16795],{"class":77,"line":16779},153,[75,16781,16782],{"class":85},"ratio_map_vis ",[75,16784,142],{"class":141},[75,16786,190],{"class":85},[75,16788,16789],{"class":193},"clip",[75,16791,16792],{"class":85},"(ratio_map, ",[75,16794,14324],{"class":145},[75,16796,16797],{"class":85},", clip_max)\n",[75,16799,16801],{"class":77,"line":16800},154,[75,16802,112],{"emptyLinePlaceholder":111},[75,16804,16806,16808,16810,16812,16814,16816,16818,16820,16822,16824,16826,16828,16830,16832],{"class":77,"line":16805},155,[75,16807,300],{"class":85},[75,16809,865],{"class":193},[75,16811,197],{"class":85},[75,16813,870],{"class":309},[75,16815,142],{"class":141},[75,16817,197],{"class":85},[75,16819,877],{"class":145},[75,16821,206],{"class":85},[75,16823,882],{"class":145},[75,16825,10496],{"class":85},[75,16827,15863],{"class":309},[75,16829,142],{"class":141},[75,16831,15868],{"class":145},[75,16833,318],{"class":85},[75,16835,16837,16839,16841,16844,16846,16848,16850,16852,16854,16856,16858,16860,16862,16864],{"class":77,"line":16836},156,[75,16838,300],{"class":85},[75,16840,16446],{"class":193},[75,16842,16843],{"class":85},"(ratio_map_vis, ",[75,16845,16452],{"class":309},[75,16847,142],{"class":141},[75,16849,16457],{"class":134},[75,16851,206],{"class":85},[75,16853,16462],{"class":309},[75,16855,142],{"class":141},[75,16857,16467],{"class":134},[75,16859,206],{"class":85},[75,16861,16472],{"class":309},[75,16863,142],{"class":141},[75,16865,16866],{"class":85},"extent)\n",[75,16868,16870,16872,16874,16876,16878,16880,16882,16884,16887,16889,16892,16894,16897,16899,16902],{"class":77,"line":16869},157,[75,16871,300],{"class":85},[75,16873,16495],{"class":193},[75,16875,197],{"class":85},[75,16877,573],{"class":309},[75,16879,142],{"class":141},[75,16881,15977],{"class":81},[75,16883,16007],{"class":15980},[75,16885,16886],{"class":85},"|",[75,16888,16010],{"class":141},[75,16890,16891],{"class":15980},"mega_{ZX}\u002F",[75,16893,16010],{"class":141},[75,16895,16896],{"class":15980},"mega_{ZZ}",[75,16898,16886],{"class":85},[75,16900,16901],{"class":15980},"$'",[75,16903,318],{"class":85},[75,16905,16907,16909,16911,16913,16915,16917,16919,16921,16923,16925,16927,16929],{"class":77,"line":16906},158,[75,16908,300],{"class":85},[75,16910,341],{"class":193},[75,16912,197],{"class":85},[75,16914,15977],{"class":81},[75,16916,16007],{"class":15980},[75,16918,16535],{"class":141},[75,16920,16538],{"class":15980},[75,16922,197],{"class":145},[75,16924,15986],{"class":15980},[75,16926,5175],{"class":145},[75,16928,15991],{"class":15980},[75,16930,318],{"class":85},[75,16932,16934,16936,16938,16940,16942,16944,16946,16948,16950,16952],{"class":77,"line":16933},159,[75,16935,300],{"class":85},[75,16937,356],{"class":193},[75,16939,197],{"class":85},[75,16941,15977],{"class":81},[75,16943,15981],{"class":15980},[75,16945,197],{"class":145},[75,16947,15986],{"class":15980},[75,16949,5175],{"class":145},[75,16951,15991],{"class":15980},[75,16953,318],{"class":85},[75,16955,16957,16959,16961,16963,16965,16968,16970,16972,16974,16976,16978,16980,16983,16985,16988,16990,16992],{"class":77,"line":16956},160,[75,16958,300],{"class":85},[75,16960,326],{"class":193},[75,16962,197],{"class":85},[75,16964,15977],{"class":81},[75,16966,16967],{"class":15980},"'Heuristic $",[75,16969,16886],{"class":85},[75,16971,16010],{"class":141},[75,16973,16891],{"class":15980},[75,16975,16010],{"class":141},[75,16977,16896],{"class":15980},[75,16979,16886],{"class":85},[75,16981,16982],{"class":15980},"$ map ",[75,16984,197],{"class":145},[75,16986,16987],{"class":15980},"clipped at 99.5th pct",[75,16989,5175],{"class":145},[75,16991,15991],{"class":15980},[75,16993,318],{"class":85},[75,16995,16997,16999,17001],{"class":77,"line":16996},161,[75,16998,300],{"class":85},[75,17000,386],{"class":193},[75,17002,389],{"class":85},[391,17004],{"alt":393,"src":17005},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-01.webp",[391,17007],{"alt":668,"src":17008},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-02.webp",[391,17010],{"alt":1006,"src":17011},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-03.webp",[391,17013],{"alt":1365,"src":17014},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-04.webp",[65,17016,17018],{"className":67,"code":17017,"language":69,"meta":70,"style":70},"#@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",[72,17019,17020,17025,17029,17035,17045,17055,17059,17113,17118,17137,17167,17206,17241,17281,17319,17341,17409,17455,17532,17603,17610,17614,17619,17633,17645,17659,17674,17678,17702,17743,17747,17759,17781,17791,17795,17807,17811,17843,17897,17921,17939,17949,17959,17972,17985],{"__ignoreMap":70},[75,17021,17022],{"class":77,"line":78},[75,17023,17024],{"class":118},"#@title Echoed-CR demonstration (two-segment echo, Eqs. 28a–d in PRA 2020)\n",[75,17026,17027],{"class":77,"line":95},[75,17028,112],{"emptyLinePlaceholder":111},[75,17030,17031,17033],{"class":77,"line":108},[75,17032,82],{"class":81},[75,17034,14088],{"class":85},[75,17036,17037,17039,17041,17043],{"class":77,"line":115},[75,17038,82],{"class":81},[75,17040,86],{"class":85},[75,17042,89],{"class":81},[75,17044,92],{"class":85},[75,17046,17047,17049,17051,17053],{"class":77,"line":122},[75,17048,82],{"class":81},[75,17050,100],{"class":85},[75,17052,89],{"class":81},[75,17054,105],{"class":85},[75,17056,17057],{"class":77,"line":128},[75,17058,112],{"emptyLinePlaceholder":111},[75,17060,17061,17063,17066,17068,17071,17073,17075,17077,17080,17082,17084,17086,17089,17091,17093,17095,17098,17100,17102,17104,17107,17109,17111],{"class":77,"line":149},[75,17062,14380],{"class":81},[75,17064,17065],{"class":193}," 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conv ",[75,17124,142],{"class":141},[75,17126,15263],{"class":145},[75,17128,273],{"class":141},[75,17130,17131],{"class":85}," math.pi ",[75,17133,2810],{"class":141},[75,17135,17136],{"class":145}," 1000.0\n",[75,17138,17139,17142,17144,17147,17149,17152,17154,17157,17159,17162,17164],{"class":77,"line":171},[75,17140,17141],{"class":85},"    zx, ix, iz, zz ",[75,17143,142],{"class":141},[75,17145,17146],{"class":85}," zx_mhz",[75,17148,500],{"class":141},[75,17150,17151],{"class":85},"conv, ix_mhz",[75,17153,500],{"class":141},[75,17155,17156],{"class":85},"conv, iz_mhz",[75,17158,500],{"class":141},[75,17160,17161],{"class":85},"conv, zz_mhz",[75,17163,500],{"class":141},[75,17165,17166],{"class":85},"conv\n",[75,17168,17169,17172,17174,17176,17178,17181,17183,17186,17188,17190,17192,17195,17197,17200,17202,17204],{"class":77,"line":182},[75,17170,17171],{"class":85},"    w_plus  ",[75,17173,142],{"class":141},[75,17175,14665],{"class":85},[75,17177,14668],{"class":193},[75,17179,17180],{"class":85},"((zx ",[75,17182,3251],{"class":141},[75,17184,17185],{"class":85}," ix)",[75,17187,258],{"class":141},[75,17189,270],{"class":145},[75,17191,11902],{"class":141},[75,17193,17194],{"class":85}," (iz ",[75,17196,3251],{"class":141},[75,17198,17199],{"class":85}," zz)",[75,17201,258],{"class":141},[75,17203,270],{"class":145},[75,17205,318],{"class":85},[75,17207,17208,17211,17213,17215,17217,17219,17221,17223,17225,17227,17229,17231,17233,17235,17237,17239],{"class":77,"line":222},[75,17209,17210],{"class":85},"    w_minus ",[75,17212,142],{"class":141},[75,17214,14665],{"class":85},[75,17216,14668],{"class":193},[75,17218,17180],{"class":85},[75,17220,200],{"class":141},[75,17222,17185],{"class":85},[75,17224,258],{"class":141},[75,17226,270],{"class":145},[75,17228,11902],{"class":141},[75,17230,17194],{"class":85},[75,17232,200],{"class":141},[75,17234,17199],{"class":85},[75,17236,258],{"class":141},[75,17238,270],{"class":145},[75,17240,318],{"class":85},[75,17242,17243,17246,17248,17250,17252,17254,17256,17258,17261,17263,17266,17268,17270,17272,17274,17276,17278],{"class":77,"line":227},[75,17244,17245],{"class":85},"    c_plus, s_plus ",[75,17247,142],{"class":141},[75,17249,14665],{"class":85},[75,17251,2845],{"class":193},[75,17253,197],{"class":85},[75,17255,2958],{"class":145},[75,17257,500],{"class":141},[75,17259,17260],{"class":85},"w_plus",[75,17262,500],{"class":141},[75,17264,17265],{"class":85},"tau_ns), math.",[75,17267,1179],{"class":193},[75,17269,197],{"class":85},[75,17271,2958],{"class":145},[75,17273,500],{"class":141},[75,17275,17260],{"class":85},[75,17277,500],{"class":141},[75,17279,17280],{"class":85},"tau_ns)\n",[75,17282,17283,17286,17288,17290,17292,17294,17296,17298,17301,17303,17305,17307,17309,17311,17313,17315,17317],{"class":77,"line":233},[75,17284,17285],{"class":85},"    c_minus, s_minus ",[75,17287,142],{"class":141},[75,17289,14665],{"class":85},[75,17291,2845],{"class":193},[75,17293,197],{"class":85},[75,17295,2958],{"class":145},[75,17297,500],{"class":141},[75,17299,17300],{"class":85},"w_minus",[75,17302,500],{"class":141},[75,17304,17265],{"class":85},[75,17306,1179],{"class":193},[75,17308,197],{"class":85},[75,17310,2958],{"class":145},[75,17312,500],{"class":141},[75,17314,17300],{"class":85},[75,17316,500],{"class":141},[75,17318,17280],{"class":85},[75,17320,17321,17324,17326,17329,17331,17334,17336,17339],{"class":77,"line":286},[75,17322,17323],{"class":85},"    denom ",[75,17325,142],{"class":141},[75,17327,17328],{"class":85}," (w_plus",[75,17330,500],{"class":141},[75,17332,17333],{"class":85},"w_minus ",[75,17335,3251],{"class":141},[75,17337,17338],{"class":145}," 1e-30",[75,17340,318],{"class":85},[75,17342,17343,17346,17348,17351,17353,17356,17358,17361,17363,17365,17367,17370,17372,17374,17376,17379,17381,17383,17385,17388,17390,17392,17394,17396,17399,17401,17404,17406],{"class":77,"line":291},[75,17344,17345],{"class":85},"    u_ii ",[75,17347,142],{"class":141},[75,17349,17350],{"class":85}," c_plus",[75,17352,500],{"class":141},[75,17354,17355],{"class":85},"c_minus ",[75,17357,3251],{"class":141},[75,17359,17360],{"class":85}," ((ix",[75,17362,258],{"class":141},[75,17364,270],{"class":145},[75,17366,14186],{"class":141},[75,17368,17369],{"class":85}," iz",[75,17371,258],{"class":141},[75,17373,270],{"class":145},[75,17375,14186],{"class":141},[75,17377,17378],{"class":85}," zx",[75,17380,258],{"class":141},[75,17382,270],{"class":145},[75,17384,11902],{"class":141},[75,17386,17387],{"class":85}," zz",[75,17389,258],{"class":141},[75,17391,270],{"class":145},[75,17393,5175],{"class":85},[75,17395,2810],{"class":141},[75,17397,17398],{"class":85},"denom)",[75,17400,500],{"class":141},[75,17402,17403],{"class":85},"s_plus",[75,17405,500],{"class":141},[75,17407,17408],{"class":85},"s_minus\n",[75,17410,17411,17414,17416,17418,17421,17423,17426,17428,17431,17433,17436,17438,17441,17443,17446,17448,17451,17453],{"class":77,"line":297},[75,17412,17413],{"class":85},"    u_iy ",[75,17415,142],{"class":141},[75,17417,261],{"class":145},[75,17419,17420],{"class":81},"j",[75,17422,500],{"class":141},[75,17424,17425],{"class":85},"(zx",[75,17427,500],{"class":141},[75,17429,17430],{"class":85},"zz ",[75,17432,200],{"class":141},[75,17434,17435],{"class":85}," ix",[75,17437,500],{"class":141},[75,17439,17440],{"class":85},"iz)",[75,17442,2810],{"class":141},[75,17444,17445],{"class":85},"denom ",[75,17447,500],{"class":141},[75,17449,17450],{"class":85}," s_plus",[75,17452,500],{"class":141},[75,17454,17408],{"class":85},[75,17456,17457,17460,17462,17464,17466,17468,17471,17473,17476,17478,17481,17483,17485,17488,17490,17493,17495,17498,17500,17502,17504,17506,17508,17510,17512,17514,17517,17519,17521,17523,17525,17527,17529],{"class":77,"line":321},[75,17458,17459],{"class":85},"    u_iz ",[75,17461,142],{"class":141},[75,17463,1437],{"class":145},[75,17465,17420],{"class":81},[75,17467,500],{"class":141},[75,17469,17470],{"class":85},"((zz ",[75,17472,200],{"class":141},[75,17474,17475],{"class":85}," iz)",[75,17477,2810],{"class":141},[75,17479,17480],{"class":85},"(w_minus ",[75,17482,3251],{"class":141},[75,17484,17338],{"class":145},[75,17486,17487],{"class":85},"))",[75,17489,500],{"class":141},[75,17491,17492],{"class":85},"c_plus",[75,17494,500],{"class":141},[75,17496,17497],{"class":85},"s_minus ",[75,17499,200],{"class":141},[75,17501,1437],{"class":145},[75,17503,17420],{"class":81},[75,17505,500],{"class":141},[75,17507,17470],{"class":85},[75,17509,3251],{"class":141},[75,17511,17475],{"class":85},[75,17513,2810],{"class":141},[75,17515,17516],{"class":85},"(w_plus ",[75,17518,3251],{"class":141},[75,17520,17338],{"class":145},[75,17522,17487],{"class":85},[75,17524,500],{"class":141},[75,17526,17403],{"class":85},[75,17528,500],{"class":141},[75,17530,17531],{"class":85},"c_minus\n",[75,17533,17534,17537,17539,17541,17543,17545,17548,17550,17553,17555,17557,17559,17561,17563,17565,17567,17569,17571,17573,17575,17577,17579,17581,17583,17585,17587,17589,17591,17593,17595,17597,17599,17601],{"class":77,"line":336},[75,17535,17536],{"class":85},"    u_zx ",[75,17538,142],{"class":141},[75,17540,1437],{"class":145},[75,17542,17420],{"class":81},[75,17544,500],{"class":141},[75,17546,17547],{"class":85},"((ix ",[75,17549,200],{"class":141},[75,17551,17552],{"class":85}," zx)",[75,17554,2810],{"class":141},[75,17556,17480],{"class":85},[75,17558,3251],{"class":141},[75,17560,17338],{"class":145},[75,17562,17487],{"class":85},[75,17564,500],{"class":141},[75,17566,17492],{"class":85},[75,17568,500],{"class":141},[75,17570,17497],{"class":85},[75,17572,200],{"class":141},[75,17574,1437],{"class":145},[75,17576,17420],{"class":81},[75,17578,500],{"class":141},[75,17580,17547],{"class":85},[75,17582,3251],{"class":141},[75,17584,17552],{"class":85},[75,17586,2810],{"class":141},[75,17588,17516],{"class":85},[75,17590,3251],{"class":141},[75,17592,17338],{"class":145},[75,17594,17487],{"class":85},[75,17596,500],{"class":141},[75,17598,17403],{"class":85},[75,17600,500],{"class":141},[75,17602,17531],{"class":85},[75,17604,17605,17607],{"class":77,"line":351},[75,17606,14467],{"class":81},[75,17608,17609],{"class":85}," u_ii, u_iy, u_iz, u_zx\n",[75,17611,17612],{"class":77,"line":366},[75,17613,112],{"emptyLinePlaceholder":111},[75,17615,17616],{"class":77,"line":381},[75,17617,17618],{"class":118},"# Representative point near Region III speed-up:\n",[75,17620,17621,17624,17626,17628,17630],{"class":77,"line":984},[75,17622,17623],{"class":85},"f_c, f_t ",[75,17625,142],{"class":141},[75,17627,14156],{"class":145},[75,17629,206],{"class":85},[75,17631,17632],{"class":145},"4914.0\n",[75,17634,17635,17638,17640,17642],{"class":77,"line":997},[75,17636,17637],{"class":85},"alpha_c ",[75,17639,142],{"class":141},[75,17641,14186],{"class":141},[75,17643,17644],{"class":145},"330.0\n",[75,17646,17647,17650,17652,17654,17656],{"class":77,"line":1325},[75,17648,17649],{"class":85},"J, Omega ",[75,17651,142],{"class":141},[75,17653,14220],{"class":145},[75,17655,206],{"class":85},[75,17657,17658],{"class":145},"30.0\n",[75,17660,17661,17664,17666,17669,17671],{"class":77,"line":1338},[75,17662,17663],{"class":85},"delta0 ",[75,17665,142],{"class":141},[75,17667,17668],{"class":85}," f_c ",[75,17670,200],{"class":141},[75,17672,17673],{"class":85}," f_t\n",[75,17675,17676],{"class":77,"line":1343},[75,17677,112],{"emptyLinePlaceholder":111},[75,17679,17680,17682,17684,17686,17688,17690,17692,17694,17696,17698,17700],{"class":77,"line":1356},[75,17681,14380],{"class":81},[75,17683,15106],{"class":193},[75,17685,197],{"class":85},[75,17687,15111],{"class":14388},[75,17689,206],{"class":85},[75,17691,15120],{"class":14388},[75,17693,206],{"class":85},[75,17695,15129],{"class":14388},[75,17697,206],{"class":85},[75,17699,15138],{"class":14388},[75,17701,15605],{"class":85},[75,17703,17704,17706,17709,17711,17713,17715,17717,17719,17721,17724,17726,17728,17730,17732,17735,17738,17741],{"class":77,"line":1887},[75,17705,14467],{"class":81},[75,17707,17708],{"class":85}," J",[75,17710,500],{"class":141},[75,17712,15129],{"class":85},[75,17714,500],{"class":141},[75,17716,197],{"class":85},[75,17718,630],{"class":145},[75,17720,2810],{"class":141},[75,17722,17723],{"class":85},"(delta_ct ",[75,17725,3251],{"class":141},[75,17727,15192],{"class":85},[75,17729,200],{"class":141},[75,17731,14846],{"class":145},[75,17733,17734],{"class":141},"\u002Fmax",[75,17736,17737],{"class":85},"(delta_ct, ",[75,17739,17740],{"class":145},"1e-12",[75,17742,283],{"class":85},[75,17744,17745],{"class":77,"line":1900},[75,17746,112],{"emptyLinePlaceholder":111},[75,17748,17749,17752,17754,17756],{"class":77,"line":1913},[75,17750,17751],{"class":85},"w_zx ",[75,17753,142],{"class":141},[75,17755,15106],{"class":193},[75,17757,17758],{"class":85},"(delta0, J, Omega, alpha_c)\n",[75,17760,17761,17764,17766,17768,17770,17773,17775,17778],{"class":77,"line":2657},[75,17762,17763],{"class":85},"w_ix, w_iz, w_zz ",[75,17765,142],{"class":141},[75,17767,14846],{"class":145},[75,17769,206],{"class":85},[75,17771,17772],{"class":145},"10.0",[75,17774,206],{"class":85},[75,17776,17777],{"class":145},"0.1",[75,17779,17780],{"class":118},"   # illustrative single-qubit terms \u002F static ZZ\n",[75,17782,17783,17786,17788],{"class":77,"line":3177},[75,17784,17785],{"class":85},"tau_ns ",[75,17787,142],{"class":141},[75,17789,17790],{"class":145}," 100.0\n",[75,17792,17793],{"class":77,"line":3188},[75,17794,112],{"emptyLinePlaceholder":111},[75,17796,17797,17800,17802,17804],{"class":77,"line":3203},[75,17798,17799],{"class":85},"u_ii, u_iy, u_iz, u_zx ",[75,17801,142],{"class":141},[75,17803,17065],{"class":193},[75,17805,17806],{"class":85},"(w_zx, w_ix, w_iz, w_zz, tau_ns)\n",[75,17808,17809],{"class":77,"line":3218},[75,17810,112],{"emptyLinePlaceholder":111},[75,17812,17813,17815,17817,17819,17821,17823,17825,17828,17830,17833,17835,17837,17839,17841],{"class":77,"line":3224},[75,17814,300],{"class":85},[75,17816,865],{"class":193},[75,17818,197],{"class":85},[75,17820,870],{"class":309},[75,17822,142],{"class":141},[75,17824,197],{"class":85},[75,17826,17827],{"class":145},"7.5",[75,17829,206],{"class":85},[75,17831,17832],{"class":145},"4.8",[75,17834,10496],{"class":85},[75,17836,15863],{"class":309},[75,17838,142],{"class":141},[75,17840,15868],{"class":145},[75,17842,318],{"class":85},[75,17844,17845,17848,17850,17852,17854,17856,17858,17861,17863,17865,17867,17869,17871,17873,17876,17878,17880,17882,17884,17886,17888,17891,17893,17895],{"class":77,"line":3229},[75,17846,17847],{"class":85},"labels ",[75,17849,142],{"class":141},[75,17851,16404],{"class":85},[75,17853,15977],{"class":81},[75,17855,16007],{"class":15980},[75,17857,16886],{"class":85},[75,17859,17860],{"class":15980},"u_{ZX}",[75,17862,16886],{"class":85},[75,17864,16901],{"class":15980},[75,17866,206],{"class":85},[75,17868,15977],{"class":81},[75,17870,16007],{"class":15980},[75,17872,16886],{"class":85},[75,17874,17875],{"class":15980},"u_{IY}",[75,17877,16886],{"class":85},[75,17879,16901],{"class":15980},[75,17881,206],{"class":85},[75,17883,15977],{"class":81},[75,17885,16007],{"class":15980},[75,17887,16886],{"class":85},[75,17889,17890],{"class":15980},"u_{IZ}",[75,17892,16886],{"class":85},[75,17894,16901],{"class":15980},[75,17896,10451],{"class":85},[75,17898,17899,17902,17904,17906,17908,17911,17913,17916,17918],{"class":77,"line":3235},[75,17900,17901],{"class":85},"vals ",[75,17903,142],{"class":141},[75,17905,16404],{"class":85},[75,17907,2145],{"class":141},[75,17909,17910],{"class":85},"(u_zx), ",[75,17912,2145],{"class":141},[75,17914,17915],{"class":85},"(u_iy), ",[75,17917,2145],{"class":141},[75,17919,17920],{"class":85},"(u_iz)]\n",[75,17922,17923,17926,17928,17930,17932,17934,17936],{"class":77,"line":3263},[75,17924,17925],{"class":85},"x ",[75,17927,142],{"class":141},[75,17929,190],{"class":85},[75,17931,10675],{"class":193},[75,17933,197],{"class":85},[75,17935,2930],{"class":141},[75,17937,17938],{"class":85},"(labels))\n",[75,17940,17941,17943,17946],{"class":77,"line":3275},[75,17942,300],{"class":85},[75,17944,17945],{"class":193},"bar",[75,17947,17948],{"class":85},"(x, vals)\n",[75,17950,17951,17953,17956],{"class":77,"line":3280},[75,17952,300],{"class":85},[75,17954,17955],{"class":193},"xticks",[75,17957,17958],{"class":85},"(x, labels)\n",[75,17960,17961,17963,17965,17967,17970],{"class":77,"line":3290},[75,17962,300],{"class":85},[75,17964,356],{"class":193},[75,17966,197],{"class":85},[75,17968,17969],{"class":134},"'Coefficient magnitude (unitless)'",[75,17971,318],{"class":85},[75,17973,17974,17976,17978,17980,17983],{"class":77,"line":4431},[75,17975,300],{"class":85},[75,17977,326],{"class":193},[75,17979,197],{"class":85},[75,17981,17982],{"class":134},"'Echoed-CR: effective coefficients after two-segment echo'",[75,17984,318],{"class":85},[75,17986,17987,17989,17991],{"class":77,"line":4454},[75,17988,300],{"class":85},[75,17990,386],{"class":193},[75,17992,389],{"class":85},[391,17994],{"alt":1608,"src":17995},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-05.webp",[65,17997,17999],{"className":67,"code":17998,"language":69,"meta":70,"style":70},"#@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",[72,18000,18001,18006,18011,18015,18025,18031,18041,18051,18055,18059,18064,18068,18072,18081,18094,18107,18122,18137,18150,18164,18168,18173,18187,18202,18217,18232,18236,18241,18253,18265,18269,18279,18283,18287,18292,18296,18336,18371,18375,18397,18417,18421,18443,18459,18463,18485,18501,18505,18510,18515,18559,18595,18599,18643,18685,18689,18693,18698,18702,18720,18743,18757,18778,18792,18796,18825,18829,18844,18859,18875,18898,18910,18922,18934,18957,18977,18997,19017,19035,19039,19043,19048,19052,19077,19093,19109,19125,19146,19160,19200,19207,19211,19216,19251,19286,19321,19356,19360,19364,19369,19373,19385,19403,19407,19412,19424,19436,19440,19445,19493,19539,19543,19547,19552,19556,19587,19613,19640,19653,19666,19679,19687],{"__ignoreMap":70},[75,18002,18003],{"class":77,"line":78},[75,18004,18005],{"class":118},"#@title Conditional target Rabi under CR effective Hamiltonian\n",[75,18007,18008],{"class":77,"line":95},[75,18009,18010],{"class":118},"# we diagonalize once and propagate.\n",[75,18012,18013],{"class":77,"line":108},[75,18014,112],{"emptyLinePlaceholder":111},[75,18016,18017,18019,18021,18023],{"class":77,"line":115},[75,18018,2700],{"class":81},[75,18020,14064],{"class":85},[75,18022,82],{"class":81},[75,18024,14069],{"class":85},[75,18026,18027,18029],{"class":77,"line":122},[75,18028,82],{"class":81},[75,18030,14088],{"class":85},[75,18032,18033,18035,18037,18039],{"class":77,"line":128},[75,18034,82],{"class":81},[75,18036,86],{"class":85},[75,18038,89],{"class":81},[75,18040,92],{"class":85},[75,18042,18043,18045,18047,18049],{"class":77,"line":149},[75,18044,82],{"class":81},[75,18046,100],{"class":85},[75,18048,89],{"class":81},[75,18050,105],{"class":85},[75,18052,18053],{"class":77,"line":154},[75,18054,112],{"emptyLinePlaceholder":111},[75,18056,18057],{"class":77,"line":160},[75,18058,14117],{"class":118},[75,18060,18061],{"class":77,"line":171},[75,18062,18063],{"class":118},"# Control knobs (MHz unless noted)\n",[75,18065,18066],{"class":77,"line":182},[75,18067,14117],{"class":118},[75,18069,18070],{"class":77,"line":222},[75,18071,14131],{"class":193},[75,18073,18074,18076,18079],{"class":77,"line":227},[75,18075,14136],{"class":81},[75,18077,18078],{"class":14139}," RabiKnobs",[75,18080,5800],{"class":85},[75,18082,18083,18085,18087,18089,18091],{"class":77,"line":233},[75,18084,14147],{"class":85},[75,18086,14150],{"class":141},[75,18088,14153],{"class":141},[75,18090,14156],{"class":145},[75,18092,18093],{"class":118},"          # control frequency\n",[75,18095,18096,18098,18100,18102,18104],{"class":77,"line":286},[75,18097,14164],{"class":85},[75,18099,14150],{"class":141},[75,18101,14153],{"class":141},[75,18103,14171],{"class":145},[75,18105,18106],{"class":118},"          # target frequency\n",[75,18108,18109,18111,18113,18115,18117,18119],{"class":77,"line":291},[75,18110,14179],{"class":85},[75,18112,14150],{"class":141},[75,18114,14153],{"class":141},[75,18116,14186],{"class":141},[75,18118,14189],{"class":145},[75,18120,18121],{"class":118},"      # control anharmonicity\n",[75,18123,18124,18126,18128,18130,18132,18134],{"class":77,"line":297},[75,18125,14197],{"class":85},[75,18127,14150],{"class":141},[75,18129,14153],{"class":141},[75,18131,14186],{"class":141},[75,18133,14189],{"class":145},[75,18135,18136],{"class":118},"      # target anharmonicity\n",[75,18138,18139,18141,18143,18145,18147],{"class":77,"line":321},[75,18140,14213],{"class":85},[75,18142,14150],{"class":141},[75,18144,14153],{"class":141},[75,18146,14220],{"class":145},[75,18148,18149],{"class":118},"               # exchange coupling\n",[75,18151,18152,18155,18157,18159,18161],{"class":77,"line":336},[75,18153,18154],{"class":85},"    Omega: ",[75,18156,14150],{"class":141},[75,18158,14153],{"class":141},[75,18160,14235],{"class":145},[75,18162,18163],{"class":118},"          # CR drive amplitude on control\n",[75,18165,18166],{"class":77,"line":351},[75,18167,112],{"emptyLinePlaceholder":111},[75,18169,18170],{"class":77,"line":366},[75,18171,18172],{"class":118},"    # Residuals (typical orders, tweak for your device)\n",[75,18174,18175,18178,18180,18182,18184],{"class":77,"line":381},[75,18176,18177],{"class":85},"    wIX_mhz: ",[75,18179,14150],{"class":141},[75,18181,14153],{"class":141},[75,18183,14846],{"class":145},[75,18185,18186],{"class":118},"         # IX (drive crosstalk on target)\n",[75,18188,18189,18192,18194,18196,18199],{"class":77,"line":984},[75,18190,18191],{"class":85},"    wIZ_khz: ",[75,18193,14150],{"class":141},[75,18195,14153],{"class":141},[75,18197,18198],{"class":145}," 20.0",[75,18200,18201],{"class":118},"        # IZ (AC Stark on target)\n",[75,18203,18204,18207,18209,18211,18214],{"class":77,"line":997},[75,18205,18206],{"class":85},"    wZI_mhz: ",[75,18208,14150],{"class":141},[75,18210,14153],{"class":141},[75,18212,18213],{"class":145}," 10.0",[75,18215,18216],{"class":118},"        # ZI (AC Stark on control)\n",[75,18218,18219,18222,18224,18226,18229],{"class":77,"line":1325},[75,18220,18221],{"class":85},"    wZZ_khz: ",[75,18223,14150],{"class":141},[75,18225,14153],{"class":141},[75,18227,18228],{"class":145}," 100.0",[75,18230,18231],{"class":118},"       # ZZ (static coupling residual)\n",[75,18233,18234],{"class":77,"line":1338},[75,18235,112],{"emptyLinePlaceholder":111},[75,18237,18238],{"class":77,"line":1343},[75,18239,18240],{"class":118},"    # Time axis\n",[75,18242,18243,18246,18248,18250],{"class":77,"line":1356},[75,18244,18245],{"class":85},"    t_end_ns: ",[75,18247,14150],{"class":141},[75,18249,14153],{"class":141},[75,18251,18252],{"class":145}," 400.0\n",[75,18254,18255,18258,18260,18262],{"class":77,"line":1887},[75,18256,18257],{"class":85},"    n_steps: ",[75,18259,14299],{"class":141},[75,18261,14153],{"class":141},[75,18263,18264],{"class":145}," 2001\n",[75,18266,18267],{"class":77,"line":1900},[75,18268,112],{"emptyLinePlaceholder":111},[75,18270,18271,18273,18275,18277],{"class":77,"line":1913},[75,18272,14352],{"class":85},[75,18274,142],{"class":141},[75,18276,18078],{"class":193},[75,18278,389],{"class":85},[75,18280,18281],{"class":77,"line":2657},[75,18282,112],{"emptyLinePlaceholder":111},[75,18284,18285],{"class":77,"line":3177},[75,18286,14117],{"class":118},[75,18288,18289],{"class":77,"line":3188},[75,18290,18291],{"class":118},"# Helpers and literature-rate formulas\n",[75,18293,18294],{"class":77,"line":3203},[75,18295,14117],{"class":118},[75,18297,18298,18300,18302,18304,18306,18308,18310,18312,18314,18316,18318,18320,18322,18324,18326,18328,18330,18332,18334],{"class":77,"line":3218},[75,18299,14380],{"class":81},[75,18301,14383],{"class":193},[75,18303,197],{"class":85},[75,18305,19],{"class":14388},[75,18307,14391],{"class":85},[75,18309,14150],{"class":141},[75,18311,206],{"class":85},[75,18313,14398],{"class":14388},[75,18315,14391],{"class":85},[75,18317,14150],{"class":141},[75,18319,206],{"class":85},[75,18321,14407],{"class":14388},[75,18323,14391],{"class":85},[75,18325,14150],{"class":141},[75,18327,14153],{"class":141},[75,18329,14416],{"class":145},[75,18331,14419],{"class":85},[75,18333,14150],{"class":141},[75,18335,5800],{"class":85},[75,18337,18338,18340,18342,18344,18347,18349,18351,18353,18356,18358,18360,18362,18364,18366,18368],{"class":77,"line":3224},[75,18339,14467],{"class":81},[75,18341,14470],{"class":85},[75,18343,2810],{"class":141},[75,18345,18346],{"class":85}," (b ",[75,18348,14737],{"class":81},[75,18350,14436],{"class":141},[75,18352,14439],{"class":85},[75,18354,18355],{"class":141},">=",[75,18357,14865],{"class":85},[75,18359,14773],{"class":81},[75,18361,190],{"class":85},[75,18363,14455],{"class":193},[75,18365,14439],{"class":85},[75,18367,500],{"class":141},[75,18369,18370],{"class":85}," eps)\n",[75,18372,18373],{"class":77,"line":3229},[75,18374,112],{"emptyLinePlaceholder":111},[75,18376,18377,18379,18382,18384,18387,18389,18391,18393,18395],{"class":77,"line":3235},[75,18378,14380],{"class":81},[75,18380,18381],{"class":193}," hz_to_rad_per_ns",[75,18383,197],{"class":85},[75,18385,18386],{"class":14388},"hz",[75,18388,14391],{"class":85},[75,18390,14150],{"class":141},[75,18392,14419],{"class":85},[75,18394,14150],{"class":141},[75,18396,5800],{"class":85},[75,18398,18399,18401,18403,18405,18407,18409,18412,18414],{"class":77,"line":3263},[75,18400,14467],{"class":81},[75,18402,15263],{"class":145},[75,18404,273],{"class":141},[75,18406,17131],{"class":85},[75,18408,500],{"class":141},[75,18410,18411],{"class":85}," hz ",[75,18413,500],{"class":141},[75,18415,18416],{"class":145}," 1e-9\n",[75,18418,18419],{"class":77,"line":3275},[75,18420,112],{"emptyLinePlaceholder":111},[75,18422,18423,18425,18428,18430,18433,18435,18437,18439,18441],{"class":77,"line":3280},[75,18424,14380],{"class":81},[75,18426,18427],{"class":193}," mhz_to_rad_per_ns",[75,18429,197],{"class":85},[75,18431,18432],{"class":14388},"mhz",[75,18434,14391],{"class":85},[75,18436,14150],{"class":141},[75,18438,14419],{"class":85},[75,18440,14150],{"class":141},[75,18442,5800],{"class":85},[75,18444,18445,18447,18449,18452,18454,18457],{"class":77,"line":3290},[75,18446,14467],{"class":81},[75,18448,18381],{"class":193},[75,18450,18451],{"class":85},"(mhz ",[75,18453,500],{"class":141},[75,18455,18456],{"class":145}," 1e6",[75,18458,318],{"class":85},[75,18460,18461],{"class":77,"line":4431},[75,18462,112],{"emptyLinePlaceholder":111},[75,18464,18465,18467,18470,18472,18475,18477,18479,18481,18483],{"class":77,"line":4454},[75,18466,14380],{"class":81},[75,18468,18469],{"class":193}," khz_to_rad_per_ns",[75,18471,197],{"class":85},[75,18473,18474],{"class":14388},"khz",[75,18476,14391],{"class":85},[75,18478,14150],{"class":141},[75,18480,14419],{"class":85},[75,18482,14150],{"class":141},[75,18484,5800],{"class":85},[75,18486,18487,18489,18491,18494,18496,18499],{"class":77,"line":4463},[75,18488,14467],{"class":81},[75,18490,18381],{"class":193},[75,18492,18493],{"class":85},"(khz ",[75,18495,500],{"class":141},[75,18497,18498],{"class":145}," 1e3",[75,18500,318],{"class":85},[75,18502,18503],{"class":77,"line":4468},[75,18504,112],{"emptyLinePlaceholder":111},[75,18506,18507],{"class":77,"line":4477},[75,18508,18509],{"class":118},"# Leading-order CR analytics (weak drive, effective model)\n",[75,18511,18512],{"class":77,"line":5105},[75,18513,18514],{"class":118},"# 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",[75,18516,18517,18519,18521,18523,18525,18527,18529,18531,18533,18535,18537,18539,18541,18543,18545,18547,18549,18551,18553,18555,18557],{"class":77,"line":5114},[75,18518,14380],{"class":81},[75,18520,15106],{"class":193},[75,18522,197],{"class":85},[75,18524,15111],{"class":14388},[75,18526,14391],{"class":85},[75,18528,14150],{"class":141},[75,18530,206],{"class":85},[75,18532,15120],{"class":14388},[75,18534,14391],{"class":85},[75,18536,14150],{"class":141},[75,18538,206],{"class":85},[75,18540,15129],{"class":14388},[75,18542,14391],{"class":85},[75,18544,14150],{"class":141},[75,18546,206],{"class":85},[75,18548,15138],{"class":14388},[75,18550,14391],{"class":85},[75,18552,14150],{"class":141},[75,18554,14419],{"class":85},[75,18556,14150],{"class":141},[75,18558,5800],{"class":85},[75,18560,18561,18563,18565,18567,18569,18571,18573,18575,18577,18579,18581,18583,18585,18587,18589,18591,18593],{"class":77,"line":10795},[75,18562,14467],{"class":81},[75,18564,15168],{"class":85},[75,18566,500],{"class":141},[75,18568,15173],{"class":85},[75,18570,500],{"class":141},[75,18572,267],{"class":85},[75,18574,15180],{"class":193},[75,18576,197],{"class":85},[75,18578,630],{"class":145},[75,18580,15187],{"class":85},[75,18582,3251],{"class":141},[75,18584,15192],{"class":85},[75,18586,200],{"class":141},[75,18588,14383],{"class":193},[75,18590,197],{"class":85},[75,18592,630],{"class":145},[75,18594,15203],{"class":85},[75,18596,18597],{"class":77,"line":14778},[75,18598,112],{"emptyLinePlaceholder":111},[75,18600,18601,18603,18605,18607,18609,18611,18613,18615,18617,18619,18621,18623,18625,18627,18629,18631,18633,18635,18637,18639,18641],{"class":77,"line":14783},[75,18602,14380],{"class":81},[75,18604,15321],{"class":193},[75,18606,197],{"class":85},[75,18608,15111],{"class":14388},[75,18610,14391],{"class":85},[75,18612,14150],{"class":141},[75,18614,206],{"class":85},[75,18616,15120],{"class":14388},[75,18618,14391],{"class":85},[75,18620,14150],{"class":141},[75,18622,206],{"class":85},[75,18624,15138],{"class":14388},[75,18626,14391],{"class":85},[75,18628,14150],{"class":141},[75,18630,206],{"class":85},[75,18632,15350],{"class":14388},[75,18634,14391],{"class":85},[75,18636,14150],{"class":141},[75,18638,14419],{"class":85},[75,18640,14150],{"class":141},[75,18642,5800],{"class":85},[75,18644,18645,18647,18649,18651,18653,18655,18657,18659,18661,18663,18665,18667,18669,18671,18673,18675,18677,18679,18681,18683],{"class":77,"line":14819},[75,18646,14467],{"class":81},[75,18648,15378],{"class":85},[75,18650,258],{"class":141},[75,18652,261],{"class":145},[75,18654,14860],{"class":85},[75,18656,500],{"class":141},[75,18658,267],{"class":85},[75,18660,15180],{"class":193},[75,18662,197],{"class":85},[75,18664,630],{"class":145},[75,18666,15187],{"class":85},[75,18668,200],{"class":141},[75,18670,15401],{"class":85},[75,18672,200],{"class":141},[75,18674,14383],{"class":193},[75,18676,197],{"class":85},[75,18678,630],{"class":145},[75,18680,15187],{"class":85},[75,18682,3251],{"class":141},[75,18684,15416],{"class":85},[75,18686,18687],{"class":77,"line":14825},[75,18688,112],{"emptyLinePlaceholder":111},[75,18690,18691],{"class":77,"line":14838},[75,18692,14117],{"class":118},[75,18694,18695],{"class":77,"line":14893},[75,18696,18697],{"class":118},"# Build H_eff (4×4) in rad\u002Fns\n",[75,18699,18700],{"class":77,"line":14951},[75,18701,14117],{"class":118},[75,18703,18704,18707,18709,18711,18714,18716,18718],{"class":77,"line":14959},[75,18705,18706],{"class":85},"I2 ",[75,18708,142],{"class":141},[75,18710,190],{"class":85},[75,18712,18713],{"class":193},"eye",[75,18715,197],{"class":85},[75,18717,270],{"class":145},[75,18719,318],{"class":85},[75,18721,18722,18725,18727,18729,18731,18734,18736,18738,18740],{"class":77,"line":14964},[75,18723,18724],{"class":85},"X ",[75,18726,142],{"class":141},[75,18728,190],{"class":85},[75,18730,12049],{"class":193},[75,18732,18733],{"class":85},"([[",[75,18735,14324],{"class":145},[75,18737,206],{"class":85},[75,18739,630],{"class":145},[75,18741,18742],{"class":85},"],\n",[75,18744,18745,18748,18750,18752,18754],{"class":77,"line":14983},[75,18746,18747],{"class":85},"              [",[75,18749,630],{"class":145},[75,18751,206],{"class":85},[75,18753,14324],{"class":145},[75,18755,18756],{"class":85},"]])\n",[75,18758,18759,18762,18764,18766,18768,18770,18772,18774,18776],{"class":77,"line":14989},[75,18760,18761],{"class":85},"Z ",[75,18763,142],{"class":141},[75,18765,190],{"class":85},[75,18767,12049],{"class":193},[75,18769,18733],{"class":85},[75,18771,630],{"class":145},[75,18773,206],{"class":85},[75,18775,14324],{"class":145},[75,18777,18742],{"class":85},[75,18779,18780,18782,18784,18786,18788,18790],{"class":77,"line":15012},[75,18781,18747],{"class":85},[75,18783,14324],{"class":145},[75,18785,206],{"class":85},[75,18787,200],{"class":141},[75,18789,630],{"class":145},[75,18791,18756],{"class":85},[75,18793,18794],{"class":77,"line":15017},[75,18795,112],{"emptyLinePlaceholder":111},[75,18797,18798,18800,18803,18805,18807,18809,18811,18814,18817,18819,18822],{"class":77,"line":15025},[75,18799,14380],{"class":81},[75,18801,18802],{"class":193}," kron2",[75,18804,197],{"class":85},[75,18806,19],{"class":14388},[75,18808,206],{"class":85},[75,18810,14398],{"class":14388},[75,18812,18813],{"class":85},"): ",[75,18815,18816],{"class":81},"return",[75,18818,190],{"class":85},[75,18820,18821],{"class":193},"kron",[75,18823,18824],{"class":85},"(a, b)\n",[75,18826,18827],{"class":77,"line":15038},[75,18828,112],{"emptyLinePlaceholder":111},[75,18830,18831,18833,18836,18838,18841],{"class":77,"line":15051},[75,18832,14380],{"class":81},[75,18834,18835],{"class":193}," build_heff_radns",[75,18837,197],{"class":85},[75,18839,18840],{"class":14388},"K",[75,18842,18843],{"class":85},": RabiKnobs) -> np.ndarray:\n",[75,18845,18846,18849,18851,18854,18856],{"class":77,"line":15058},[75,18847,18848],{"class":85},"    Δ ",[75,18850,142],{"class":141},[75,18852,18853],{"class":85}," K.f_c ",[75,18855,200],{"class":141},[75,18857,18858],{"class":85}," K.f_t\n",[75,18860,18861,18864,18866,18868,18870,18872],{"class":77,"line":15069},[75,18862,18863],{"class":85},"    w_zx ",[75,18865,142],{"class":141},[75,18867,18427],{"class":193},[75,18869,197],{"class":85},[75,18871,15743],{"class":193},[75,18873,18874],{"class":85},"(Δ, K.J, K.Omega, K.alpha_c))\n",[75,18876,18877,18880,18882,18884,18886,18888,18891,18893,18895],{"class":77,"line":15080},[75,18878,18879],{"class":85},"    w_zz ",[75,18881,142],{"class":141},[75,18883,18427],{"class":193},[75,18885,197],{"class":85},[75,18887,15776],{"class":193},[75,18889,18890],{"class":85},"(Δ, K.J, K.alpha_c, K.alpha_t)) ",[75,18892,3251],{"class":141},[75,18894,18469],{"class":193},[75,18896,18897],{"class":85},"(K.wZZ_khz)\n",[75,18899,18900,18903,18905,18907],{"class":77,"line":15085},[75,18901,18902],{"class":85},"    w_ix ",[75,18904,142],{"class":141},[75,18906,18427],{"class":193},[75,18908,18909],{"class":85},"(K.wIX_mhz)\n",[75,18911,18912,18915,18917,18919],{"class":77,"line":15090},[75,18913,18914],{"class":85},"    w_iz ",[75,18916,142],{"class":141},[75,18918,18469],{"class":193},[75,18920,18921],{"class":85},"(K.wIZ_khz)\n",[75,18923,18924,18927,18929,18931],{"class":77,"line":15096},[75,18925,18926],{"class":85},"    w_zi ",[75,18928,142],{"class":141},[75,18930,18427],{"class":193},[75,18932,18933],{"class":85},"(K.wZI_mhz)\n",[75,18935,18936,18938,18941,18943,18945,18947,18949,18951,18954],{"class":77,"line":15101},[75,18937,14467],{"class":81},[75,18939,18940],{"class":85}," ((w_ix",[75,18942,2810],{"class":141},[75,18944,14678],{"class":145},[75,18946,14860],{"class":85},[75,18948,500],{"class":141},[75,18950,18802],{"class":193},[75,18952,18953],{"class":85},"(I2, X) ",[75,18955,18956],{"class":141},"+\n",[75,18958,18959,18962,18964,18966,18968,18970,18972,18975],{"class":77,"line":15151},[75,18960,18961],{"class":85},"            (w_iz",[75,18963,2810],{"class":141},[75,18965,14678],{"class":145},[75,18967,14860],{"class":85},[75,18969,500],{"class":141},[75,18971,18802],{"class":193},[75,18973,18974],{"class":85},"(I2, Z) ",[75,18976,18956],{"class":141},[75,18978,18979,18982,18984,18986,18988,18990,18992,18995],{"class":77,"line":15160},[75,18980,18981],{"class":85},"            (w_zi",[75,18983,2810],{"class":141},[75,18985,14678],{"class":145},[75,18987,14860],{"class":85},[75,18989,500],{"class":141},[75,18991,18802],{"class":193},[75,18993,18994],{"class":85},"(Z,  I2) ",[75,18996,18956],{"class":141},[75,18998,18999,19002,19004,19006,19008,19010,19012,19015],{"class":77,"line":15206},[75,19000,19001],{"class":85},"            (w_zx",[75,19003,2810],{"class":141},[75,19005,14678],{"class":145},[75,19007,14860],{"class":85},[75,19009,500],{"class":141},[75,19011,18802],{"class":193},[75,19013,19014],{"class":85},"(Z,  X) ",[75,19016,18956],{"class":141},[75,19018,19019,19022,19024,19026,19028,19030,19032],{"class":77,"line":15235},[75,19020,19021],{"class":85},"            (w_zz",[75,19023,2810],{"class":141},[75,19025,14678],{"class":145},[75,19027,14860],{"class":85},[75,19029,500],{"class":141},[75,19031,18802],{"class":193},[75,19033,19034],{"class":85},"(Z,  Z))\n",[75,19036,19037],{"class":77,"line":15288},[75,19038,112],{"emptyLinePlaceholder":111},[75,19040,19041],{"class":77,"line":15294},[75,19042,14117],{"class":118},[75,19044,19045],{"class":77,"line":15311},[75,19046,19047],{"class":118},"# Time evolution via diagonalization\n",[75,19049,19050],{"class":77,"line":15316},[75,19051,14117],{"class":118},[75,19053,19054,19056,19059,19061,19063,19066,19069,19071,19074],{"class":77,"line":15363},[75,19055,14380],{"class":81},[75,19057,19058],{"class":193}," evolve_state_constH",[75,19060,197],{"class":85},[75,19062,9119],{"class":14388},[75,19064,19065],{"class":85},": np.ndarray, ",[75,19067,19068],{"class":14388},"psi0",[75,19070,19065],{"class":85},[75,19072,19073],{"class":14388},"t_ns",[75,19075,19076],{"class":85},": np.ndarray) -> np.ndarray:\n",[75,19078,19079,19082,19084,19087,19090],{"class":77,"line":15371},[75,19080,19081],{"class":85},"    E, V ",[75,19083,142],{"class":141},[75,19085,19086],{"class":85}," np.linalg.",[75,19088,19089],{"class":193},"eigh",[75,19091,19092],{"class":85},"(H)\n",[75,19094,19095,19098,19100,19103,19106],{"class":77,"line":15419},[75,19096,19097],{"class":85},"    Vinv ",[75,19099,142],{"class":141},[75,19101,19102],{"class":85}," V.",[75,19104,19105],{"class":193},"conj",[75,19107,19108],{"class":85},"().T\n",[75,19110,19111,19114,19116,19119,19122],{"class":77,"line":15442},[75,19112,19113],{"class":85},"    amps0 ",[75,19115,142],{"class":141},[75,19117,19118],{"class":85}," Vinv ",[75,19120,19121],{"class":141},"@",[75,19123,19124],{"class":85}," psi0\n",[75,19126,19127,19130,19132,19134,19136,19139,19141,19144],{"class":77,"line":15482},[75,19128,19129],{"class":85},"    out ",[75,19131,142],{"class":141},[75,19133,190],{"class":85},[75,19135,16295],{"class":193},[75,19137,19138],{"class":85},"((psi0.size, t_ns.size), ",[75,19140,12100],{"class":309},[75,19142,19143],{"class":141},"=complex",[75,19145,318],{"class":85},[75,19147,19148,19150,19153,19155,19157],{"class":77,"line":15520},[75,19149,15611],{"class":81},[75,19151,19152],{"class":85}," k, t ",[75,19154,3153],{"class":81},[75,19156,3156],{"class":141},[75,19158,19159],{"class":85},"(t_ns):\n",[75,19161,19162,19165,19167,19170,19172,19175,19177,19179,19181,19183,19185,19187,19190,19192,19195,19197],{"class":77,"line":15525},[75,19163,19164],{"class":85},"        out[:, k] ",[75,19166,142],{"class":141},[75,19168,19169],{"class":85}," V ",[75,19171,19121],{"class":141},[75,19173,19174],{"class":85}," (np.",[75,19176,243],{"class":193},[75,19178,197],{"class":85},[75,19180,200],{"class":141},[75,19182,7477],{"class":145},[75,19184,17420],{"class":81},[75,19186,273],{"class":141},[75,19188,19189],{"class":85}," E ",[75,19191,500],{"class":141},[75,19193,19194],{"class":85}," t) ",[75,19196,500],{"class":141},[75,19198,19199],{"class":85}," amps0)\n",[75,19201,19202,19204],{"class":77,"line":15540},[75,19203,14467],{"class":81},[75,19205,19206],{"class":85}," out\n",[75,19208,19209],{"class":77,"line":15545},[75,19210,112],{"emptyLinePlaceholder":111},[75,19212,19213],{"class":77,"line":15551},[75,19214,19215],{"class":118},"# Basis |00>, |01>, |10>, |11>\n",[75,19217,19218,19221,19223,19225,19227,19229,19231,19233,19235,19237,19239,19241,19243,19245,19247,19249],{"class":77,"line":15592},[75,19219,19220],{"class":85},"e00 ",[75,19222,142],{"class":141},[75,19224,190],{"class":85},[75,19226,12049],{"class":193},[75,19228,15740],{"class":85},[75,19230,7477],{"class":145},[75,19232,7469],{"class":85},[75,19234,5186],{"class":145},[75,19236,7469],{"class":85},[75,19238,5186],{"class":145},[75,19240,7469],{"class":85},[75,19242,5186],{"class":145},[75,19244,12097],{"class":85},[75,19246,12100],{"class":309},[75,19248,19143],{"class":141},[75,19250,318],{"class":85},[75,19252,19253,19256,19258,19260,19262,19264,19266,19268,19270,19272,19274,19276,19278,19280,19282,19284],{"class":77,"line":15608},[75,19254,19255],{"class":85},"e01 ",[75,19257,142],{"class":141},[75,19259,190],{"class":85},[75,19261,12049],{"class":193},[75,19263,15740],{"class":85},[75,19265,5186],{"class":145},[75,19267,7469],{"class":85},[75,19269,7477],{"class":145},[75,19271,7469],{"class":85},[75,19273,5186],{"class":145},[75,19275,7469],{"class":85},[75,19277,5186],{"class":145},[75,19279,12097],{"class":85},[75,19281,12100],{"class":309},[75,19283,19143],{"class":141},[75,19285,318],{"class":85},[75,19287,19288,19291,19293,19295,19297,19299,19301,19303,19305,19307,19309,19311,19313,19315,19317,19319],{"class":77,"line":15624},[75,19289,19290],{"class":85},"e10 ",[75,19292,142],{"class":141},[75,19294,190],{"class":85},[75,19296,12049],{"class":193},[75,19298,15740],{"class":85},[75,19300,5186],{"class":145},[75,19302,7469],{"class":85},[75,19304,5186],{"class":145},[75,19306,7469],{"class":85},[75,19308,7477],{"class":145},[75,19310,7469],{"class":85},[75,19312,5186],{"class":145},[75,19314,12097],{"class":85},[75,19316,12100],{"class":309},[75,19318,19143],{"class":141},[75,19320,318],{"class":85},[75,19322,19323,19326,19328,19330,19332,19334,19336,19338,19340,19342,19344,19346,19348,19350,19352,19354],{"class":77,"line":15652},[75,19324,19325],{"class":85},"e11 ",[75,19327,142],{"class":141},[75,19329,190],{"class":85},[75,19331,12049],{"class":193},[75,19333,15740],{"class":85},[75,19335,5186],{"class":145},[75,19337,7469],{"class":85},[75,19339,5186],{"class":145},[75,19341,7469],{"class":85},[75,19343,5186],{"class":145},[75,19345,7469],{"class":85},[75,19347,7477],{"class":145},[75,19349,12097],{"class":85},[75,19351,12100],{"class":309},[75,19353,19143],{"class":141},[75,19355,318],{"class":85},[75,19357,19358],{"class":77,"line":15657},[75,19359,112],{"emptyLinePlaceholder":111},[75,19361,19362],{"class":77,"line":15662},[75,19363,14117],{"class":118},[75,19365,19366],{"class":77,"line":15668},[75,19367,19368],{"class":118},"# Conditional Rabi simulation\n",[75,19370,19371],{"class":77,"line":15673},[75,19372,14117],{"class":118},[75,19374,19375,19378,19380,19382],{"class":77,"line":15698},[75,19376,19377],{"class":85},"H ",[75,19379,142],{"class":141},[75,19381,18835],{"class":193},[75,19383,19384],{"class":85},"(K)\n",[75,19386,19387,19390,19392,19394,19396,19398,19400],{"class":77,"line":15723},[75,19388,19389],{"class":85},"t_ns ",[75,19391,142],{"class":141},[75,19393,190],{"class":85},[75,19395,194],{"class":193},[75,19397,197],{"class":85},[75,19399,14324],{"class":145},[75,19401,19402],{"class":85},", K.t_end_ns, K.n_steps)\n",[75,19404,19405],{"class":77,"line":15728},[75,19406,112],{"emptyLinePlaceholder":111},[75,19408,19409],{"class":77,"line":15762},[75,19410,19411],{"class":118},"# Prepare control in |0> (|00>) vs |1> (|10>), target initially |0>\n",[75,19413,19414,19417,19419,19421],{"class":77,"line":15792},[75,19415,19416],{"class":85},"ψ_c0 ",[75,19418,142],{"class":141},[75,19420,19058],{"class":193},[75,19422,19423],{"class":85},"(H, e00, t_ns)\n",[75,19425,19426,19429,19431,19433],{"class":77,"line":15829},[75,19427,19428],{"class":85},"ψ_c1 ",[75,19430,142],{"class":141},[75,19432,19058],{"class":193},[75,19434,19435],{"class":85},"(H, e10, t_ns)\n",[75,19437,19438],{"class":77,"line":15834},[75,19439,112],{"emptyLinePlaceholder":111},[75,19441,19442],{"class":77,"line":15840},[75,19443,19444],{"class":118},"# Target |1> probability = P(|01>) + P(|11>)\n",[75,19446,19447,19450,19452,19454,19456,19459,19461,19463,19465,19468,19470,19472,19474,19476,19478,19481,19483,19485,19487,19489,19491],{"class":77,"line":15873},[75,19448,19449],{"class":85},"p_t1_c0 ",[75,19451,142],{"class":141},[75,19453,190],{"class":85},[75,19455,2145],{"class":193},[75,19457,19458],{"class":85},"(e01.",[75,19460,19105],{"class":193},[75,19462,2819],{"class":85},[75,19464,19121],{"class":141},[75,19466,19467],{"class":85}," ψ_c0)",[75,19469,258],{"class":141},[75,19471,270],{"class":145},[75,19473,11902],{"class":141},[75,19475,190],{"class":85},[75,19477,2145],{"class":193},[75,19479,19480],{"class":85},"(e11.",[75,19482,19105],{"class":193},[75,19484,2819],{"class":85},[75,19486,19121],{"class":141},[75,19488,19467],{"class":85},[75,19490,258],{"class":141},[75,19492,2833],{"class":145},[75,19494,19495,19498,19500,19502,19504,19506,19508,19510,19512,19515,19517,19519,19521,19523,19525,19527,19529,19531,19533,19535,19537],{"class":77,"line":15931},[75,19496,19497],{"class":85},"p_t1_c1 ",[75,19499,142],{"class":141},[75,19501,190],{"class":85},[75,19503,2145],{"class":193},[75,19505,19458],{"class":85},[75,19507,19105],{"class":193},[75,19509,2819],{"class":85},[75,19511,19121],{"class":141},[75,19513,19514],{"class":85}," ψ_c1)",[75,19516,258],{"class":141},[75,19518,270],{"class":145},[75,19520,11902],{"class":141},[75,19522,190],{"class":85},[75,19524,2145],{"class":193},[75,19526,19480],{"class":85},[75,19528,19105],{"class":193},[75,19530,2819],{"class":85},[75,19532,19121],{"class":141},[75,19534,19514],{"class":85},[75,19536,258],{"class":141},[75,19538,2833],{"class":145},[75,19540,19541],{"class":77,"line":15953},[75,19542,112],{"emptyLinePlaceholder":111},[75,19544,19545],{"class":77,"line":15968},[75,19546,14117],{"class":118},[75,19548,19549],{"class":77,"line":15996},[75,19550,19551],{"class":118},"# Plot\n",[75,19553,19554],{"class":77,"line":16026},[75,19555,14117],{"class":118},[75,19557,19558,19560,19562,19564,19566,19568,19570,19573,19575,19577,19579,19581,19583,19585],{"class":77,"line":16047},[75,19559,300],{"class":85},[75,19561,865],{"class":193},[75,19563,197],{"class":85},[75,19565,870],{"class":309},[75,19567,142],{"class":141},[75,19569,197],{"class":85},[75,19571,19572],{"class":145},"8.2",[75,19574,206],{"class":85},[75,19576,17832],{"class":145},[75,19578,10496],{"class":85},[75,19580,15863],{"class":309},[75,19582,142],{"class":141},[75,19584,15868],{"class":145},[75,19586,318],{"class":85},[75,19588,19589,19591,19593,19596,19598,19600,19603,19605,19607,19609,19611],{"class":77,"line":16056},[75,19590,300],{"class":85},[75,19592,303],{"class":193},[75,19594,19595],{"class":85},"(t_ns, p_t1_c0, ",[75,19597,573],{"class":309},[75,19599,142],{"class":141},[75,19601,19602],{"class":134},"'Control |0⟩, init |00⟩'",[75,19604,206],{"class":85},[75,19606,310],{"class":309},[75,19608,142],{"class":141},[75,19610,315],{"class":134},[75,19612,318],{"class":85},[75,19614,19615,19617,19619,19622,19624,19626,19629,19631,19633,19635,19638],{"class":77,"line":16065},[75,19616,300],{"class":85},[75,19618,303],{"class":193},[75,19620,19621],{"class":85},"(t_ns, p_t1_c1, ",[75,19623,573],{"class":309},[75,19625,142],{"class":141},[75,19627,19628],{"class":134},"'Control |1⟩, init |10⟩'",[75,19630,206],{"class":85},[75,19632,310],{"class":309},[75,19634,142],{"class":141},[75,19636,19637],{"class":134},"'orange'",[75,19639,318],{"class":85},[75,19641,19642,19644,19646,19648,19651],{"class":77,"line":16070},[75,19643,300],{"class":85},[75,19645,341],{"class":193},[75,19647,197],{"class":85},[75,19649,19650],{"class":134},"'Time (ns)'",[75,19652,318],{"class":85},[75,19654,19655,19657,19659,19661,19664],{"class":77,"line":16076},[75,19656,300],{"class":85},[75,19658,356],{"class":193},[75,19660,197],{"class":85},[75,19662,19663],{"class":134},"'Target |1⟩ probability'",[75,19665,318],{"class":85},[75,19667,19668,19670,19672,19674,19677],{"class":77,"line":16107},[75,19669,300],{"class":85},[75,19671,326],{"class":193},[75,19673,197],{"class":85},[75,19675,19676],{"class":134},"'Conditional target Rabi under CR effective Hamiltonian'",[75,19678,318],{"class":85},[75,19680,19681,19683,19685],{"class":77,"line":16153},[75,19682,300],{"class":85},[75,19684,5058],{"class":193},[75,19686,389],{"class":85},[75,19688,19689,19691,19693],{"class":77,"line":16174},[75,19690,300],{"class":85},[75,19692,386],{"class":193},[75,19694,389],{"class":85},[391,19696],{"alt":1922,"src":19697},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-06.webp",[12364,19699,19700],{},"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":70,"searchDepth":95,"depth":95,"links":19702},[19703],{"id":12427,"depth":95,"text":12428,"children":19704},[19705],{"id":13964,"depth":108,"text":13965},[12374,12375,12376,19707],"Cross-resonance gate",[19709],{"username":7,"name":12380,"role":12381,"bio":12382,"links":19710},[19711,19712],{"label":12385,"href":12386},{"label":12388,"href":12422},{"username":7,"name":12380,"role":12390},"How the ZX entangling rate and static ZZ vary with detuning and drive amplitude, plotted from the PRA 2020 cross-resonance formulas.",{},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fcross-resonance-gate-visualization","9 min 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1",[75,32804,32806,32830],{"className":32805,"ariaHidden":5196},[5245],[75,32807,32809,32812,32815,32818,32821,32824,32827],{"className":32808},[5249],[75,32810],{"className":32811,"style":5254},[5253],[75,32813,32791],{"className":32814},[5263],[75,32816,15977],{"className":32817},[5258,32200],[75,32819,32791],{"className":32820},[5270],[75,32822],{"className":32823,"style":5275},[5274],[75,32825,142],{"className":32826},[5279],[75,32828],{"className":32829,"style":5275},[5274],[75,32831,32833,32837],{"className":32832},[5249],[75,32834],{"className":32835,"style":32836},[5253],"height:0.6444em;",[75,32838,7477],{"className":32839},[5258],[12,32841,32842,32845,32846,32986],{},[48,32843,32844],{},"Unit vector on the sphere"," (spherical angles 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(rigid rotation):"," for any unitary (U),",[75,34854,34856],{"className":34855},[31410],[75,34857,34859,34954],{"className":34858},[5148],[75,34860,34862],{"className":34861},[5152],[5154,34863,34864],{"xmlns":5156,"display":31419},[5158,34865,34866,34951],{},[31438,34867,34868],{"width":33623},[31443,34869,34870,34872,34944,34946],{},[31446,34871],{"width":33628},[31446,34873,34874],{},[5161,34875,34876,34895,34897,34899,34901,34903,34905,34907,34913,34916,34918,34928,34930,34932,34934,34936,34938,34940,34942],{},[5179,34877,34878,34880],{},[5164,34879,33637],{},[5161,34881,34882,34885,34887],{},[5164,34883,34884],{},"U",[5164,34886,32265],{},[5218,34888,34889,34891],{},[5164,34890,34884],{},[5168,34892,34894],{"lspace":34893,"rspace":34893},"0em","†",[5168,34896,197],{"stretchy":5170},[5164,34898,7494],{},[5168,34900,7469],{"separator":5196},[5164,34902,5906],{},[5168,34904,5175],{"stretchy":5170},[5168,34906,142],{},[5179,34908,34909,34911],{},[5164,34910,33637],{},[5164,34912,32265],{},[5168,34914,34915],{"stretchy":5170},"!",[5168,34917,197],{"fence":5170,"stretchy":5196,"minsize":32278,"maxsize":32278},[5218,34919,34920,34922],{},[5164,34921,32181],{},[5161,34923,34924,34926],{},[5168,34925,5199],{},[5184,34927,7477],{},[5164,34929,33004],{"mathvariant":32172},[5168,34931,197],{"stretchy":5170},[5164,34933,7494],{},[5168,34935,7469],{"separator":5196},[5164,34937,5906],{},[5168,34939,5175],{"stretchy":5170},[5168,34941,5175],{"fence":5170,"stretchy":5196,"minsize":32278,"maxsize":32278},[5168,34943,7469],{"separator":5196},[31446,34945],{"width":33628},[31446,34947,34948],{},[32328,34949,34950],{},"(Covariance)",[5238,34952,34953],{"encoding":5240},"W_{U\\rho U^{\\dagger}}(\\theta,\\phi) = W_{\\rho}!\\big(R^{-1}\\mathbf n(\\theta,\\phi)\\big),\n\\tag{Covariance}",[75,34955,34957,35073,35200],{"className":34956,"ariaHidden":5196},[5245],[75,34958,34960,34964,35046,35049,35052,35055,35058,35061,35064,35067,35070],{"className":34959},[5249],[75,34961],{"className":34962,"style":34963},[5253],"height:1.0887em;vertical-align:-0.3387em;",[75,34965,34967,34970],{"className":34966},[5258],[75,34968,33637],{"className":34969,"style":9817},[5258,5259],[75,34971,34973],{"className":34972},[5299],[75,34974,34976,35037],{"className":34975},[5303,5304],[75,34977,34979,35034],{"className":34978},[5308],[75,34980,34983],{"className":34981,"style":34982},[5312],"height:0.3448em;",[75,34984,34986,34989],{"style":34985},"top:-2.4974em;margin-left:-0.1389em;margin-right:0.05em;",[75,34987],{"className":34988,"style":5321},[5320],[75,34990,34992],{"className":34991},[5325,5326,5327,5328],[75,34993,34995,34998,35001],{"className":34994},[5258,5328],[75,34996,34884],{"className":34997,"style":9810},[5258,5259,5328],[75,34999,32265],{"className":35000},[5258,5259,5328],[75,35002,35004,35007],{"className":35003},[5258,5328],[75,35005,34884],{"className":35006,"style":9810},[5258,5259,5328],[75,35008,35010],{"className":35009},[5299],[75,35011,35013],{"className":35012},[5303],[75,35014,35016],{"className":35015},[5308],[75,35017,35020],{"className":35018,"style":35019},[5312],"height:0.782em;",[75,35021,35022,35025],{"style":5432},[75,35023],{"className":35024,"style":5436},[5320],[75,35026,35028],{"className":35027},[5325,5440,5441,5328],[75,35029,35031],{"className":35030},[5258,5328],[75,35032,34894],{"className":35033},[5258,5328],[75,35035,5336],{"className":35036},[5335],[75,35038,35040],{"className":35039},[5308],[75,35041,35044],{"className":35042,"style":35043},[5312],"height:0.3387em;",[75,35045],{},[75,35047,197],{"className":35048},[5263],[75,35050,7494],{"className":35051,"style":7612},[5258,5259],[75,35053,7469],{"className":35054},[7544],[75,35056],{"className":35057,"style":5349},[5274],[75,35059,5906],{"className":35060},[5258,5259],[75,35062,5175],{"className":35063},[5270],[75,35065],{"className":35066,"style":5275},[5274],[75,35068,142],{"className":35069},[5279],[75,35071],{"className":35072,"style":5275},[5274],[75,35074,35076,35080,35123,35126,35132,35170,35173,35176,35179,35182,35185,35188,35191,35197],{"className":35075},[5249],[75,35077],{"className":35078,"style":35079},[5253],"height:1.2141em;vertical-align:-0.35em;",[75,35081,35083,35086],{"className":35082},[5258],[75,35084,33637],{"className":35085,"style":9817},[5258,5259],[75,35087,35089],{"className":35088},[5299],[75,35090,35092,35115],{"className":35091},[5303,5304],[75,35093,35095,35112],{"className":35094},[5308],[75,35096,35098],{"className":35097,"style":6264},[5312],[75,35099,35100,35103],{"style":33755},[75,35101],{"className":35102,"style":5321},[5320],[75,35104,35106],{"className":35105},[5325,5326,5327,5328],[75,35107,35109],{"className":35108},[5258,5328],[75,35110,32265],{"className":35111},[5258,5259,5328],[75,35113,5336],{"className":35114},[5335],[75,35116,35118],{"className":35117},[5308],[75,35119,35121],{"className":35120,"style":31823},[5312],[75,35122],{},[75,35124,34915],{"className":35125},[5270],[75,35127,35129],{"className":35128},[5258],[75,35130,197],{"className":35131},[5369,5441],[75,35133,35135,35139],{"className":35134},[5258],[75,35136,32181],{"className":35137,"style":35138},[5258,5259],"margin-right:0.0077em;",[75,35140,35142],{"className":35141},[5299],[75,35143,35145],{"className":35144},[5303],[75,35146,35148],{"className":35147},[5308],[75,35149,35152],{"className":35150,"style":35151},[5312],"height:0.8641em;",[75,35153,35155,35158],{"style":35154},"top:-3.113em;margin-right:0.05em;",[75,35156],{"className":35157,"style":5321},[5320],[75,35159,35161],{"className":35160},[5325,5326,5327,5328],[75,35162,35164,35167],{"className":35163},[5258,5328],[75,35165,5199],{"className":35166},[5258,5328],[75,35168,7477],{"className":35169},[5258,5328],[75,35171,33004],{"className":35172},[5258,32200],[75,35174,197],{"className":35175},[5263],[75,35177,7494],{"className":35178,"style":7612},[5258,5259],[75,35180,7469],{"className":35181},[7544],[75,35183],{"className":35184,"style":5349},[5274],[75,35186,5906],{"className":35187},[5258,5259],[75,35189,5175],{"className":35190},[5270],[75,35192,35194],{"className":35193},[5258],[75,35195,5175],{"className":35196},[5369,5441],[75,35198,7469],{"className":35199},[7544],[75,35201,35203,35206],{"className":35202},[34124],[75,35204],{"className":35205,"style":35079},[5253],[75,35207,35209,35212,35219],{"className":35208},[5258,34131],[75,35210,197],{"className":35211},[5258],[75,35213,35215],{"className":35214},[5258],[75,35216,35218],{"className":35217},[5258],"Covariance",[75,35220,5175],{"className":35221},[5258],[12,35223,35224,35225,35300],{},"where (",[75,35226,35228,35256],{"className":35227},[5148],[75,35229,35231],{"className":35230},[5152],[5154,35232,35233],{"xmlns":5156},[5158,35234,35235,35253],{},[5161,35236,35237,35239,35241,35247,35249,35251],{},[5164,35238,32181],{},[5168,35240,32175],{},[5161,35242,35243,35245],{},[5164,35244,33564],{"mathvariant":6582},[5164,35246,9673],{"mathvariant":6582},[5168,35248,197],{"stretchy":5170},[5184,35250,3101],{},[5168,35252,5175],{"stretchy":5170},[5238,35254,35255],{"encoding":5240},"R\\in\\mathrm{SO}(3)",[75,35257,35259,35278],{"className":35258,"ariaHidden":5196},[5245],[75,35260,35262,35266,35269,35272,35275],{"className":35261},[5249],[75,35263],{"className":35264,"style":35265},[5253],"height:0.7224em;vertical-align:-0.0391em;",[75,35267,32181],{"className":35268,"style":35138},[5258,5259],[75,35270],{"className":35271,"style":5275},[5274],[75,35273,32175],{"className":35274},[5279],[75,35276],{"className":35277,"style":5275},[5274],[75,35279,35281,35284,35291,35294,35297],{"className":35280},[5249],[75,35282],{"className":35283,"style":5254},[5253],[75,35285,35287],{"className":35286},[5258],[75,35288,35290],{"className":35289},[5258,32585],"SO",[75,35292,197],{"className":35293},[5263],[75,35295,3101],{"className":35296},[5258],[75,35298,5175],{"className":35299},[5270],") is the 3D rotation associated with (U) (see §3–§4).",[7621,35302],{},[12425,35304,35306,35307,35337],{"id":35305},"_3-pauli-gates-as-πpiπ-rotations","3. Pauli Gates as (",[75,35308,35310,35324],{"className":35309},[5148],[75,35311,35313],{"className":35312},[5152],[5154,35314,35315],{"xmlns":5156},[5158,35316,35317,35321],{},[5161,35318,35319],{},[5164,35320,5889],{},[5238,35322,35323],{"encoding":5240},"\\pi",[75,35325,35327],{"className":35326,"ariaHidden":5196},[5245],[75,35328,35330,35334],{"className":35329},[5249],[75,35331],{"className":35332,"style":35333},[5253],"height:0.4306em;",[75,35335,5889],{"className":35336,"style":5416},[5258,5259],")-Rotations",[12,35339,35340,35341,35369],{},"The single‑qubit Pauli gates are equivalent (up to a global phase) to (",[75,35342,35344,35357],{"className":35343},[5148],[75,35345,35347],{"className":35346},[5152],[5154,35348,35349],{"xmlns":5156},[5158,35350,35351,35355],{},[5161,35352,35353],{},[5164,35354,5889],{},[5238,35356,35323],{"encoding":5240},[75,35358,35360],{"className":35359,"ariaHidden":5196},[5245],[75,35361,35363,35366],{"className":35362},[5249],[75,35364],{"className":35365,"style":35333},[5253],[75,35367,5889],{"className":35368,"style":5416},[5258,5259],")-rotations about Cartesian axes:",[75,35371,35373],{"className":35372},[31410],[75,35374,35376,35482],{"className":35375},[5148],[75,35377,35379],{"className":35378},[5152],[5154,35380,35381],{"xmlns":5156,"display":31419},[5158,35382,35383,35479],{},[5161,35384,35385,35387,35389,35413,35415,35417,35419,35421,35445,35447,35449,35451,35453,35477],{},[5164,35386,5820],{},[5168,35388,142],{},[5218,35390,35391,35393],{},[5164,35392,7486],{},[5161,35394,35395,35397,35399,35405,35407],{},[5168,35396,5199],{},[5164,35398,7491],{},[5201,35400,35401,35403],{},[5164,35402,5889],{},[5184,35404,270],{},[5168,35406,7469],{"separator":5196},[5179,35408,35409,35411],{},[5164,35410,5232],{},[5164,35412,31430],{},[5168,35414,7469],{"separator":5196},[5274,35416],{"width":32297},[5164,35418,6631],{},[5168,35420,142],{},[5218,35422,35423,35425],{},[5164,35424,7486],{},[5161,35426,35427,35429,35431,35437,35439],{},[5168,35428,5199],{},[5164,35430,7491],{},[5201,35432,35433,35435],{},[5164,35434,5889],{},[5184,35436,270],{},[5168,35438,7469],{"separator":5196},[5179,35440,35441,35443],{},[5164,35442,5232],{},[5164,35444,31485],{},[5168,35446,7469],{"separator":5196},[5274,35448],{"width":32297},[5164,35450,7409],{},[5168,35452,142],{},[5218,35454,35455,35457],{},[5164,35456,7486],{},[5161,35458,35459,35461,35463,35469,35471],{},[5168,35460,5199],{},[5164,35462,7491],{},[5201,35464,35465,35467],{},[5164,35466,5889],{},[5184,35468,270],{},[5168,35470,7469],{"separator":5196},[5179,35472,35473,35475],{},[5164,35474,5232],{},[5164,35476,31538],{},[5164,35478,6543],{"mathvariant":6582},[5238,35480,35481],{"encoding":5240},"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}.",[75,35483,35485,35503,35688,35862],{"className":35484,"ariaHidden":5196},[5245],[75,35486,35488,35491,35494,35497,35500],{"className":35487},[5249],[75,35489],{"className":35490,"style":12648},[5253],[75,35492,5820],{"className":35493,"style":5955},[5258,5259],[75,35495],{"className":35496,"style":5275},[5274],[75,35498,142],{"className":35499},[5279],[75,35501],{"className":35502,"style":5275},[5274],[75,35504,35506,35510,35667,35670,35673,35676,35679,35682,35685],{"className":35505},[5249],[75,35507],{"className":35508,"style":35509},[5253],"height:1.0915em;vertical-align:-0.1944em;",[75,35511,35513,35516],{"className":35512},[5258],[75,35514,7486],{"className":35515},[5258,5259],[75,35517,35519],{"className":35518},[5299],[75,35520,35522],{"className":35521},[5303],[75,35523,35525],{"className":35524},[5308],[75,35526,35529],{"className":35527,"style":35528},[5312],"height:0.8971em;",[75,35530,35532,35535],{"style":35531},"top:-3.413em;margin-right:0.05em;",[75,35533],{"className":35534,"style":5400},[5320],[75,35536,35538],{"className":35537},[5325,5326,5327,5328],[75,35539,35541,35544,35547,35622,35625],{"className":35540},[5258,5328],[75,35542,5199],{"className":35543},[5258,5328],[75,35545,7491],{"className":35546},[5258,5259,5328],[75,35548,35550,35554,35619],{"className":35549},[5258,5328],[75,35551],{"className":35552},[5263,5380,5325,5440,35553],"size6",[75,35555,35557],{"className":35556},[5201],[75,35558,35560,35610],{"className":35559},[5303,5304],[75,35561,35563,35607],{"className":35562},[5308],[75,35564,35567,35582,35592],{"className":35565,"style":35566},[5312],"height:0.6915em;",[75,35568,35570,35573],{"style":35569},"top:-2.656em;",[75,35571],{"className":35572,"style":5400},[5320],[75,35574,35576],{"className":35575},[5325,5440,5441,5328],[75,35577,35579],{"className":35578},[5258,5328],[75,35580,270],{"className":35581},[5258,5328],[75,35583,35585,35588],{"style":35584},"top:-3.2255em;",[75,35586],{"className":35587,"style":5400},[5320],[75,35589],{"className":35590,"style":35591},[5454,5328],"border-bottom-width:0.049em;",[75,35593,35595,35598],{"style":35594},"top:-3.384em;",[75,35596],{"className":35597,"style":5400},[5320],[75,35599,35601],{"className":35600},[5325,5440,5441,5328],[75,35602,35604],{"className":35603},[5258,5328],[75,35605,5889],{"className":35606,"style":5416},[5258,5259,5328],[75,35608,5336],{"className":35609},[5335],[75,35611,35613],{"className":35612},[5308],[75,35614,35617],{"className":35615,"style":35616},[5312],"height:0.344em;",[75,35618],{},[75,35620],{"className":35621},[5270,5380,5325,5440,35553],[75,35623,7469],{"className":35624},[7544,5328],[75,35626,35628,35631],{"className":35627},[5258,5328],[75,35629,5232],{"className":35630,"style":5416},[5258,5259,5328],[75,35632,35634],{"className":35633},[5299],[75,35635,35637,35659],{"className":35636},[5303,5304],[75,35638,35640,35656],{"className":35639},[5308],[75,35641,35644],{"className":35642,"style":35643},[5312],"height:0.1645em;",[75,35645,35647,35650],{"style":35646},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[75,35648],{"className":35649,"style":5436},[5320],[75,35651,35653],{"className":35652},[5325,5440,5441,5328],[75,35654,31430],{"className":35655},[5258,5259,5328],[75,35657,5336],{"className":35658},[5335],[75,35660,35662],{"className":35661},[5308],[75,35663,35665],{"className":35664,"style":5518},[5312],[75,35666],{},[75,35668,7469],{"className":35669},[7544],[75,35671],{"className":35672,"style":32516},[5274],[75,35674],{"className":35675,"style":5349},[5274],[75,35677,6631],{"className":35678,"style":6294},[5258,5259],[75,35680],{"className":35681,"style":5275},[5274],[75,35683,142],{"className":35684},[5279],[75,35686],{"className":35687,"style":5275},[5274],[75,35689,35691,35694,35841,35844,35847,35850,35853,35856,35859],{"className":35690},[5249],[75,35692],{"className":35693,"style":35509},[5253],[75,35695,35697,35700],{"className":35696},[5258],[75,35698,7486],{"className":35699},[5258,5259],[75,35701,35703],{"className":35702},[5299],[75,35704,35706],{"className":35705},[5303],[75,35707,35709],{"className":35708},[5308],[75,35710,35712],{"className":35711,"style":35528},[5312],[75,35713,35714,35717],{"style":35531},[75,35715],{"className":35716,"style":5400},[5320],[75,35718,35720],{"className":35719},[5325,5326,5327,5328],[75,35721,35723,35726,35729,35797,35800],{"className":35722},[5258,5328],[75,35724,5199],{"className":35725},[5258,5328],[75,35727,7491],{"className":35728},[5258,5259,5328],[75,35730,35732,35735,35794],{"className":35731},[5258,5328],[75,35733],{"className":35734},[5263,5380,5325,5440,35553],[75,35736,35738],{"className":35737},[5201],[75,35739,35741,35786],{"className":35740},[5303,5304],[75,35742,35744,35783],{"className":35743},[5308],[75,35745,35747,35761,35769],{"className":35746,"style":35566},[5312],[75,35748,35749,35752],{"style":35569},[75,35750],{"className":35751,"style":5400},[5320],[75,35753,35755],{"className":35754},[5325,5440,5441,5328],[75,35756,35758],{"className":35757},[5258,5328],[75,35759,270],{"className":35760},[5258,5328],[75,35762,35763,35766],{"style":35584},[75,35764],{"className":35765,"style":5400},[5320],[75,35767],{"className":35768,"style":35591},[5454,5328],[75,35770,35771,35774],{"style":35594},[75,35772],{"className":35773,"style":5400},[5320],[75,35775,35777],{"className":35776},[5325,5440,5441,5328],[75,35778,35780],{"className":35779},[5258,5328],[75,35781,5889],{"className":35782,"style":5416},[5258,5259,5328],[75,35784,5336],{"className":35785},[5335],[75,35787,35789],{"className":35788},[5308],[75,35790,35792],{"className":35791,"style":35616},[5312],[75,35793],{},[75,35795],{"className":35796},[5270,5380,5325,5440,35553],[75,35798,7469],{"className":35799},[7544,5328],[75,35801,35803,35806],{"className":35802},[5258,5328],[75,35804,5232],{"className":35805,"style":5416},[5258,5259,5328],[75,35807,35809],{"className":35808},[5299],[75,35810,35812,35832],{"className":35811},[5303,5304],[75,35813,35815,35829],{"className":35814},[5308],[75,35816,35818],{"className":35817,"style":35643},[5312],[75,35819,35820,35823],{"style":35646},[75,35821],{"className":35822,"style":5436},[5320],[75,35824,35826],{"className":35825},[5325,5440,5441,5328],[75,35827,31485],{"className":35828,"style":5416},[5258,5259,5328],[75,35830,5336],{"className":35831},[5335],[75,35833,35835],{"className":35834},[5308],[75,35836,35839],{"className":35837,"style":35838},[5312],"height:0.2819em;",[75,35840],{},[75,35842,7469],{"className":35843},[7544],[75,35845],{"className":35846,"style":32516},[5274],[75,35848],{"className":35849,"style":5349},[5274],[75,35851,7409],{"className":35852,"style":7430},[5258,5259],[75,35854],{"className":35855,"style":5275},[5274],[75,35857,142],{"className":35858},[5279],[75,35860],{"className":35861,"style":5275},[5274],[75,35863,35865,35868,36014],{"className":35864},[5249],[75,35866],{"className":35867,"style":35528},[5253],[75,35869,35871,35874],{"className":35870},[5258],[75,35872,7486],{"className":35873},[5258,5259],[75,35875,35877],{"className":35876},[5299],[75,35878,35880],{"className":35879},[5303],[75,35881,35883],{"className":35882},[5308],[75,35884,35886],{"className":35885,"style":35528},[5312],[75,35887,35888,35891],{"style":35531},[75,35889],{"className":35890,"style":5400},[5320],[75,35892,35894],{"className":35893},[5325,5326,5327,5328],[75,35895,35897,35900,35903,35971,35974],{"className":35896},[5258,5328],[75,35898,5199],{"className":35899},[5258,5328],[75,35901,7491],{"className":35902},[5258,5259,5328],[75,35904,35906,35909,35968],{"className":35905},[5258,5328],[75,35907],{"className":35908},[5263,5380,5325,5440,35553],[75,35910,35912],{"className":35911},[5201],[75,35913,35915,35960],{"className":35914},[5303,5304],[75,35916,35918,35957],{"className":35917},[5308],[75,35919,35921,35935,35943],{"className":35920,"style":35566},[5312],[75,35922,35923,35926],{"style":35569},[75,35924],{"className":35925,"style":5400},[5320],[75,35927,35929],{"className":35928},[5325,5440,5441,5328],[75,35930,35932],{"className":35931},[5258,5328],[75,35933,270],{"className":35934},[5258,5328],[75,35936,35937,35940],{"style":35584},[75,35938],{"className":35939,"style":5400},[5320],[75,35941],{"className":35942,"style":35591},[5454,5328],[75,35944,35945,35948],{"style":35594},[75,35946],{"className":35947,"style":5400},[5320],[75,35949,35951],{"className":35950},[5325,5440,5441,5328],[75,35952,35954],{"className":35953},[5258,5328],[75,35955,5889],{"className":35956,"style":5416},[5258,5259,5328],[75,35958,5336],{"className":35959},[5335],[75,35961,35963],{"className":35962},[5308],[75,35964,35966],{"className":35965,"style":35616},[5312],[75,35967],{},[75,35969],{"className":35970},[5270,5380,5325,5440,35553],[75,35972,7469],{"className":35973},[7544,5328],[75,35975,35977,35980],{"className":35976},[5258,5328],[75,35978,5232],{"className":35979,"style":5416},[5258,5259,5328],[75,35981,35983],{"className":35982},[5299],[75,35984,35986,36006],{"className":35985},[5303,5304],[75,35987,35989,36003],{"className":35988},[5308],[75,35990,35992],{"className":35991,"style":35643},[5312],[75,35993,35994,35997],{"style":35646},[75,35995],{"className":35996,"style":5436},[5320],[75,35998,36000],{"className":35999},[5325,5440,5441,5328],[75,36001,31538],{"className":36002,"style":32002},[5258,5259,5328],[75,36004,5336],{"className":36005},[5335],[75,36007,36009],{"className":36008},[5308],[75,36010,36012],{"className":36011,"style":5518},[5312],[75,36013],{},[75,36015,6543],{"className":36016},[5258],[12,36018,36019,36020,36244,36245,36247,36248,13959],{},"They act on the Bloch vector by the corresponding rotation matrices (",[75,36021,36023,36075],{"className":36022},[5148],[75,36024,36026],{"className":36025},[5152],[5154,36027,36028],{"xmlns":5156},[5158,36029,36030,36072],{},[5161,36031,36032,36038,36040,36042,36044,36046,36052,36054,36056,36058,36060,36066,36068,36070],{},[5179,36033,36034,36036],{},[5164,36035,32181],{},[5164,36037,31430],{},[5168,36039,197],{"stretchy":5170},[5164,36041,5889],{},[5168,36043,5175],{"stretchy":5170},[5168,36045,7469],{"separator":5196},[5179,36047,36048,36050],{},[5164,36049,32181],{},[5164,36051,31485],{},[5168,36053,197],{"stretchy":5170},[5164,36055,5889],{},[5168,36057,5175],{"stretchy":5170},[5168,36059,7469],{"separator":5196},[5179,36061,36062,36064],{},[5164,36063,32181],{},[5164,36065,31538],{},[5168,36067,197],{"stretchy":5170},[5164,36069,5889],{},[5168,36071,5175],{"stretchy":5170},[5238,36073,36074],{"encoding":5240},"R_x(\\pi),R_y(\\pi),R_z(\\pi)",[75,36076,36078],{"className":36077,"ariaHidden":5196},[5245],[75,36079,36081,36084,36125,36128,36131,36134,36137,36140,36180,36183,36186,36189,36192,36195,36235,36238,36241],{"className":36080},[5249],[75,36082],{"className":36083,"style":32578},[5253],[75,36085,36087,36090],{"className":36086},[5258],[75,36088,32181],{"className":36089,"style":35138},[5258,5259],[75,36091,36093],{"className":36092},[5299],[75,36094,36096,36117],{"className":36095},[5303,5304],[75,36097,36099,36114],{"className":36098},[5308],[75,36100,36102],{"className":36101,"style":6264},[5312],[75,36103,36105,36108],{"style":36104},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[75,36106],{"className":36107,"style":5321},[5320],[75,36109,36111],{"className":36110},[5325,5326,5327,5328],[75,36112,31430],{"className":36113},[5258,5259,5328],[75,36115,5336],{"className":36116},[5335],[75,36118,36120],{"className":36119},[5308],[75,36121,36123],{"className":36122,"style":5343},[5312],[75,36124],{},[75,36126,197],{"className":36127},[5263],[75,36129,5889],{"className":36130,"style":5416},[5258,5259],[75,36132,5175],{"className":36133},[5270],[75,36135,7469],{"className":36136},[7544],[75,36138],{"className":36139,"style":5349},[5274],[75,36141,36143,36146],{"className":36142},[5258],[75,36144,32181],{"className":36145,"style":35138},[5258,5259],[75,36147,36149],{"className":36148},[5299],[75,36150,36152,36172],{"className":36151},[5303,5304],[75,36153,36155,36169],{"className":36154},[5308],[75,36156,36158],{"className":36157,"style":6264},[5312],[75,36159,36160,36163],{"style":36104},[75,36161],{"className":36162,"style":5321},[5320],[75,36164,36166],{"className":36165},[5325,5326,5327,5328],[75,36167,31485],{"className":36168,"style":5416},[5258,5259,5328],[75,36170,5336],{"className":36171},[5335],[75,36173,36175],{"className":36174},[5308],[75,36176,36178],{"className":36177,"style":31823},[5312],[75,36179],{},[75,36181,197],{"className":36182},[5263],[75,36184,5889],{"className":36185,"style":5416},[5258,5259],[75,36187,5175],{"className":36188},[5270],[75,36190,7469],{"className":36191},[7544],[75,36193],{"className":36194,"style":5349},[5274],[75,36196,36198,36201],{"className":36197},[5258],[75,36199,32181],{"className":36200,"style":35138},[5258,5259],[75,36202,36204],{"className":36203},[5299],[75,36205,36207,36227],{"className":36206},[5303,5304],[75,36208,36210,36224],{"className":36209},[5308],[75,36211,36213],{"className":36212,"style":6264},[5312],[75,36214,36215,36218],{"style":36104},[75,36216],{"className":36217,"style":5321},[5320],[75,36219,36221],{"className":36220},[5325,5326,5327,5328],[75,36222,31538],{"className":36223,"style":32002},[5258,5259,5328],[75,36225,5336],{"className":36226},[5335],[75,36228,36230],{"className":36229},[5308],[75,36231,36233],{"className":36232,"style":5343},[5312],[75,36234],{},[75,36236,197],{"className":36237},[5263],[75,36239,5889],{"className":36240,"style":5416},[5258,5259],[75,36242,5175],{"className":36243},[5270],"), so ",[48,36246,34950],{}," implies the Wigner pattern simply rotates rigidly on (",[75,36249,36251,36268],{"className":36250},[5148],[75,36252,36254],{"className":36253},[5152],[5154,36255,36256],{"xmlns":5156},[5158,36257,36258,36266],{},[5161,36259,36260],{},[5218,36261,36262,36264],{},[5164,36263,33564],{},[5184,36265,270],{},[5238,36267,33569],{"encoding":5240},[75,36269,36271],{"className":36270,"ariaHidden":5196},[5245],[75,36272,36274,36277],{"className":36273},[5249],[75,36275],{"className":36276,"style":32216},[5253],[75,36278,36280,36283],{"className":36279},[5258],[75,36281,33564],{"className":36282,"style":5929},[5258,5259],[75,36284,36286],{"className":36285},[5299],[75,36287,36289],{"className":36288},[5303],[75,36290,36292],{"className":36291},[5308],[75,36293,36295],{"className":36294,"style":32216},[5312],[75,36296,36297,36300],{"style":7596},[75,36298],{"className":36299,"style":5321},[5320],[75,36301,36303],{"className":36302},[5325,5326,5327,5328],[75,36304,270],{"className":36305},[5258,5328],[7621,36307],{},[12425,36309,36311],{"id":36310},"_4-general-su2-rotation-and-the-so3-map","4. General SU(2) Rotation and the SO(3) Map",[12,36313,36314,36315,36384,36385,36416],{},"A general single‑qubit rotation about a unit axis (",[75,36316,36318,36339],{"className":36317},[5148],[75,36319,36321],{"className":36320},[5152],[5154,36322,36323],{"xmlns":5156},[5158,36324,36325,36336],{},[5161,36326,36327],{},[36328,36329,36330,36333],"mover",{"accent":5196},[5164,36331,36332],{"mathvariant":32172},"u",[5168,36334,36335],{},"^",[5238,36337,36338],{"encoding":5240},"\\hat{\\mathbf 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(",[75,36386,36388,36403],{"className":36387},[5148],[75,36389,36391],{"className":36390},[5152],[5154,36392,36393],{"xmlns":5156},[5158,36394,36395,36400],{},[5161,36396,36397],{},[5164,36398,36399],{},"α",[5238,36401,36402],{"encoding":5240},"\\alpha",[75,36404,36406],{"className":36405,"ariaHidden":5196},[5245],[75,36407,36409,36412],{"className":36408},[5249],[75,36410],{"className":36411,"style":35333},[5253],[75,36413,36399],{"className":36414,"style":36415},[5258,5259],"margin-right:0.0037em;",") 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center",[31443,37432,37433,37439,37453,37473,37479,37503,37513],{},[31446,37434,37435],{},[31449,37436,37437],{"scriptlevel":5186,"displaystyle":5170},[5184,37438,5186],{},[31446,37440,37441],{},[31449,37442,37443],{"scriptlevel":5186,"displaystyle":5170},[5161,37444,37445,37447],{},[5168,37446,5199],{},[5179,37448,37449,37451],{},[5164,37450,36332],{},[5164,37452,31538],{},[31446,37454,37455],{},[31449,37456,37457],{"scriptlevel":5186,"displaystyle":5170},[5161,37458,37459,37465,37467],{},[5179,37460,37461,37463],{},[5164,37462,36332],{},[5164,37464,31485],{},[32328,37466,32330],{},[5179,37468,37469,37471],{},[5164,37470,36332],{},[5164,37472,31538],{},[31446,37474,37475],{},[31449,37476,37477],{"scriptlevel":5186,"displaystyle":5170},[5184,37478,5186],{},[31446,37480,37481],{},[31449,37482,37483],{"scriptlevel":5186,"displaystyle":5170},[5161,37484,37485,37487,37493,37495,37497],{},[5168,37486,5199],{},[5179,37488,37489,37491],{},[5164,37490,36332],{},[5164,37492,31430],{},[32328,37494,32330],{},[5168,37496,5199],{},[5179,37498,37499,37501],{},[5164,37500,36332],{},[5164,37502,31485],{},[31446,37504,37505],{},[31449,37506,37507],{"scriptlevel":5186,"displaystyle":5170},[5179,37508,37509,37511],{},[5164,37510,36332],{},[5164,37512,31430],{},[31446,37514,37515],{},[31449,37516,37517],{"scriptlevel":5186,"displaystyle":5170},[5184,37518,5186],{},[5168,37520,5175],{"fence":5196},[5164,37522,6543],{"mathvariant":6582},[5238,37524,37525],{"encoding":5240},"K(\\hat{\\mathbf 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0\n\\end{pmatrix}.",[75,37527,37529,37584],{"className":37528,"ariaHidden":5196},[5245],[75,37530,37532,37535,37538,37541,37572,37575,37578,37581],{"className":37531},[5249],[75,37533],{"className":37534,"style":5254},[5253],[75,37536,18840],{"className":37537,"style":7430},[5258,5259],[75,37539,197],{"className":37540},[5263],[75,37542,37544],{"className":37543},[5258,36352],[75,37545,37547],{"className":37546},[5303],[75,37548,37550],{"className":37549},[5308],[75,37551,37553,37561],{"className":37552,"style":36348},[5312],[75,37554,37555,37558],{"style":33435},[75,37556],{"className":37557,"style":5400},[5320],[75,37559,36332],{"className":37560},[5258,32200],[75,37562,37563,37566],{"style":36372},[75,37564],{"className":37565,"style":5400},[5320],[75,37567,37569],{"className":37568,"style":36380},[36379],[75,37570,36335],{"className":37571},[5258],[75,37573,5175],{"className":37574},[5270],[75,37576],{"className":37577,"style":5275},[5274],[75,37579,142],{"className":37580},[5279],[75,37582],{"className":37583,"style":5275},[5274],[75,37585,37587,37590,38137,38140],{"className":37586},[5249],[75,37588],{"className":37589,"style":32380},[5253],[75,37591,37593,37599,38131],{"className":37592},[5360],[75,37594,37596],{"className":37595,"style":5365},[5263,5364],[75,37597,197],{"className":37598},[5369,5441],[75,37600,37602],{"className":37601},[5258],[75,37603,37605,37642,37645,37648,37722,37725,37728,37842,37845,37848,37882,37885,37888,38014,38017,38020,38091,38094,38097],{"className":37604},[31438],[75,37606,37608],{"className":37607},[31667],[75,37609,37611,37633],{"className":37610},[5303,5304],[75,37612,37614,37630],{"className":37613},[5308],[75,37615,37618],{"className":37616,"style":37617},[5312],"height:0.85em;",[75,37619,37621,37624],{"style":37620},"top:-3.01em;",[75,37622],{"className":37623,"style":5400},[5320],[75,37625,37627],{"className":37626},[5258],[75,37628,5186],{"className":37629},[5258],[75,37631,5336],{"className":37632},[5335],[75,37634,37636],{"className":37635},[5308],[75,37637,37640],{"className":37638,"style":37639},[5312],"height:0.35em;",[75,37641],{},[75,37643],{"className":37644,"style":31718},[31717],[75,37646],{"className":37647,"style":31718},[31717],[75,37649,37651],{"className":37650},[31667],[75,37652,37654,37714],{"className":37653},[5303,5304],[75,37655,37657,37711],{"className":37656},[5308],[75,37658,37660],{"className":37659,"style":37617},[5312],[75,37661,37662,37665],{"style":37620},[75,37663],{"className":37664,"style":5400},[5320],[75,37666,37668,37671],{"className":37667},[5258],[75,37669,5199],{"className":37670},[5258],[75,37672,37674,37677],{"className":37673},[5258],[75,37675,36332],{"className":37676},[5258,5259],[75,37678,37680],{"className":37679},[5299],[75,37681,37683,37703],{"className":37682},[5303,5304],[75,37684,37686,37700],{"className":37685},[5308],[75,37687,37689],{"className":37688,"style":6264},[5312],[75,37690,37691,37694],{"style":5316},[75,37692],{"className":37693,"style":5321},[5320],[75,37695,37697],{"className":37696},[5325,5326,5327,5328],[75,37698,31538],{"className":37699,"style":32002},[5258,5259,5328],[75,37701,5336],{"className":37702},[5335],[75,37704,37706],{"className":37705},[5308],[75,37707,37709],{"className":37708,"style":5343},[5312],[75,37710],{},[75,37712,5336],{"className":37713},[5335],[75,37715,37717],{"className":37716},[5308],[75,37718,37720],{"className":37719,"style":37639},[5312],[75,37721],{},[75,37723],{"className":37724,"style":31718},[31717],[75,37726],{"className":37727,"style":31718},[31717],[75,37729,37731],{"className":37730},[31667],[75,37732,37734,37834],{"className":37733},[5303,5304],[75,37735,37737,37831],{"className":37736},[5308],[75,37738,37740],{"className":37739,"style":37617},[5312],[75,37741,37742,37745],{"style":37620},[75,37743],{"className":37744,"style":5400},[5320],[75,37746,37748,37788,37791],{"className":37747},[5258],[75,37749,37751,37754],{"className":37750},[5258],[75,37752,36332],{"className":37753},[5258,5259],[75,37755,37757],{"className":37756},[5299],[75,37758,37760,37780],{"className":37759},[5303,5304],[75,37761,37763,37777],{"className":37762},[5308],[75,37764,37766],{"className":37765,"style":6264},[5312],[75,37767,37768,37771],{"style":5316},[75,37769],{"className":37770,"style":5321},[5320],[75,37772,37774],{"className":37773},[5325,5326,5327,5328],[75,37775,31485],{"className":37776,"style":5416},[5258,5259,5328],[75,37778,5336],{"className":37779},[5335],[75,37781,37783],{"className":37782},[5308],[75,37784,37786],{"className":37785,"style":31823},[5312],[75,37787],{},[75,37789,32330],{"className":37790},[5274],[75,37792,37794,37797],{"className":37793},[5258],[75,37795,36332],{"className":37796},[5258,5259],[75,37798,37800],{"className":37799},[5299],[75,37801,37803,37823],{"className":37802},[5303,5304],[75,37804,37806,37820],{"className":37805},[5308],[75,37807,37809],{"className":37808,"style":6264},[5312],[75,37810,37811,37814],{"style":5316},[75,37812],{"classN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the corresponding rotation matrix 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Optional: Discrete (",[75,39258,39260,39279],{"className":39259},[5148],[75,39261,39263],{"className":39262},[5152],[5154,39264,39265],{"xmlns":5156},[5158,39266,39267,39276],{},[5161,39268,39269,39271,39274],{},[5184,39270,270],{},[5168,39272,39273],{},"×",[5184,39275,270],{},[5238,39277,39278],{"encoding":5240},"2\\times2",[75,39280,39282,39301],{"className":39281,"ariaHidden":5196},[5245],[75,39283,39285,39289,39292,39295,39298],{"className":39284},[5249],[75,39286],{"className":39287,"style":39288},[5253],"height:0.7278em;vertical-align:-0.0833em;",[75,39290,270],{"className":39291},[5258],[75,39293],{"className":39294,"style":6294},[5274],[75,39296,39273],{"className":39297},[5477],[75,39299],{"className":39300,"style":6294},[5274],[75,39302,39304,39307],{"className":39303},[5249],[75,39305],{"className":39306,"style":32836},[5253],[75,39308,270],{"className":39309},[5258],") (Finite‑State) Wigner",[12,39312,39313,39314,39394,39395,39465],{},"On a 4‑point discrete phase space (",[75,39315,39317,39346],{"className":39316},[5148],[75,39318,39320],{"className":39319},[5152],[5154,39321,39322],{"xmlns":5156},[5158,39323,39324,39343],{},[5161,39325,39326,39328,39330],{},[5164,39327,36399],{},[5168,39329,32175],{},[5161,39331,39332,39334,39337,39339,39341],{},[5168,39333,197],{"stretchy":5170},[5164,39335,39336],{},"q",[5168,39338,7469],{"separator":5196},[5164,39340,12],{},[5168,39342,5175],{"stretchy":5170},[5238,39344,39345],{"encoding":5240},"\\alpha\\in{(q,p)}",[75,39347,39349,39367],{"className":39348,"ariaHidden":5196},[5245],[75,39350,39352,39355,39358,39361,39364],{"className":39351},[5249],[75,39353],{"className":39354,"style":32196},[5253],[75,39356,36399],{"className":39357,"style":36415},[5258,5259],[75,39359],{"className":39360,"style":5275},[5274],[75,39362,32175],{"className":39363},[5279],[75,39365],{"className":39366,"style":5275},[5274],[75,39368,39370,39373],{"className":39369},[5249],[75,39371],{"className":39372,"style":5254},[5253],[75,39374,39376,39379,39382,39385,39388,39391],{"className":39375},[5258],[75,39377,197],{"className":39378},[5263],[75,39380,39336],{"className":39381,"style":5416},[5258,5259],[75,39383,7469],{"className":39384},[7544],[75,39386],{"className":39387,"style":5349},[5274],[75,39389,12],{"className":39390},[5258,5259],[75,39392,5175],{"className":39393},[5270],"), define phase‑point operators (",[75,39396,39398,39416],{"className":39397},[5148],[75,39399,39401],{"className":39400},[5152],[5154,39402,39403],{"xmlns":5156},[5158,39404,39405,39413],{},[5161,39406,39407],{},[5179,39408,39409,39411],{},[5164,39410,5166],{},[5164,39412,36399],{},[5238,39414,39415],{"encoding":5240},"A_\\alpha",[75,39417,39419],{"className":39418,"ariaHidden":5196},[5245],[75,39420,39422,39425],{"className":39421},[5249],[75,39423],{"className":39424,"style":12572},[5253],[75,39426,39428,39431],{"className":39427},[5258],[75,39429,5166],{"className":39430},[5258,5259],[75,39432,39434],{"className":39433},[5299],[75,39435,39437,39457],{"className":39436},[5303,5304],[75,39438,39440,39454],{"className":39439},[5308],[75,39441,39443],{"className":39442,"style":6264},[5312],[75,39444,39445,39448],{"style":5316},[75,39446],{"className":39447,"style":5321},[5320],[75,39449,39451],{"className":39450},[5325,5326,5327,5328],[75,39452,36399],{"className":39453,"style":36415},[5258,5259,5328],[75,39455,5336],{"className":39456},[5335],[75,39458,39460],{"className":39459},[5308],[75,39461,39463],{"className":39462,"style":5343},[5312],[75,39464],{},") with",[75,39467,39469],{"className":39468},[31410],[75,39470,39472,39549],{"className":39471},[5148],[75,39473,39475],{"className":39474},[5152],[5154,39476,39477],{"xmlns":5156,"display":31419},[5158,39478,39479,39546],{},[5161,39480,39481,39489,39495,39497,39499,39501,39503,39505,39507,39509,39511,39517,39519,39521,39523,39525,39527,39533,39535,39544],{},[39482,39483,39484,39487],"munder",{},[5168,39485,39486],{},"∑",[5164,39488,36399],{},[5179,39490,39491,39493],{},[5164,39492,5166],{},[5164,39494,36399],{},[5168,39496,142],{},[5184,39498,270],{},[5164,39500,13747],{},[5168,39502,7469],{"separator":5196},[5274,39504],{"width":7472},[5164,39506,32308],{"mathvariant":6582},[5168,39508,5191],{},[5168,39510,197],{"stretchy":5170},[5179,39512,39513,39515],{},[5164,39514,5166],{},[5164,39516,36399],{},[5168,39518,5175],{"stretchy":5170},[5168,39520,142],{},[5184,39522,7477],{},[5168,39524,7469],{"separator":5196},[5274,39526],{"width":7472},[5179,39528,39529,39531],{},[5164,39530,5166],{},[5164,39532,36399],{},[5168,39534,142],{},[39536,39537,39538,39540,39542],"msubsup",{},[5164,39539,5166],{},[5164,39541,36399],{},[5168,39543,34894],{"lspace":34893,"rspace":34893},[5164,39545,6543],{"mathvariant":6582},[5238,39547,39548],{"encoding":5240},"\\sum_{\\alpha} A_\\alpha = 2I,\\quad \\operatorname{Tr}(A_\\alpha)=1,\\quad A_\\alpha = A_\\alpha^{\\dagger}.",[75,39550,39552,39666,39748,39815],{"className":39551,"ariaHidden":5196},[5245],[75,39553,39555,39559,39614,39617,39657,39660,39663],{"className":39554},[5249],[75,39556],{"className":39557,"style":39558},[5253],"height:2.3em;vertical-align:-1.25em;",[75,39560,39563],{"className":39561},[5353,39562],"op-limits",[75,39564,39566,39605],{"className":39565},[5303,5304],[75,39567,39569,39602],{"className":39568},[5308],[75,39570,39573,39589],{"className":39571,"style":39572},[5312],"height:1.05em;",[75,39574,39576,39580],{"style":39575},"top:-1.9em;margin-left:0em;",[75,39577],{"className":39578,"style":39579},[5320],"height:3.05em;",[75,39581,39583],{"className":39582},[5325,5326,5327,5328],[75,39584,39586],{"className":39585},[5258,5328],[75,39587,36399],{"className":39588,"style":36415},[5258,5259,5328],[75,39590,39592,39595],{"style":39591},"top:-3.05em;",[75,39593],{"className":39594,"style":39579},[5320],[75,39596,39597],{},[75,39598,39486],{"className":39599},[5353,39600,39601],"op-symbol","large-op",[75,39603,5336],{"className":39604},[5335],[75,39606,39608],{"className":39607},[5308],[75,39609,39612],{"className":39610,"style":39611},[5312],"height:1.25em;",[75,39613],{},[75,39615],{"className":39616,"style":5349},[5274],[75,39618,39620,39623],{"className":39619},[5258],[75,39621,5166],{"className":39622},[5258,5259],[75,39624,39626],{"className":39625},[5299],[75,39627,39629,39649],{"className":39628},[5303,5304],[75,39630,39632,39646],{"className":39631},[5308],[75,39633,39635],{"className":39634,"style":6264},[5312],[75,39636,39637,39640],{"style":5316},[75,39638],{"className":39639,"style":5321},[5320],[75,39641,39643],{"className":39642},[5325,5326,5327,5328],[75,39644,36399],{"className":39645,"style":36415},[5258,5259,5328],[75,39647,5336],{"className":39648},[5335],[75,39650,39652],{"className":39651},[5308],[75,39653,39655],{"className":39654,"style":5343},[5312],[75,39656],{},[75,39658],{"className":39659,"style":5275},[5274],[75,39661,142],{"className":39662},[5279],[75,39664],{"className":39665,"style":5275},[5274],[75,39667,39669,39672,39675,39678,39681,39684,39687,39693,39696,39736,39739,39742,39745],{"className":39668},[5249],[75,39670],{"className":39671,"style":5254},[5253],[75,39673,270],{"className":39674},[5258],[75,39676,13747],{"className":39677,"style":5955},[5258,5259],[75,39679,7469],{"className":39680},[7544],[75,39682],{"className":39683,"style":7548},[5274],[75,39685],{"className":39686,"style":5349},[5274],[75,39688,39690],{"className":39689},[5353],[75,39691,32308],{"className":39692},[5258,32585],[75,39694,197],{"className":39695},[5263],[75,39697,39699,39702],{"className":39698},[5258],[75,39700,5166],{"className":39701},[5258,5259],[75,39703,39705],{"className":39704},[5299],[75,39706,39708,39728],{"className":39707},[5303,5304],[75,39709,39711,39725],{"className":39710},[5308],[75,39712,39714],{"className":39713,"style":6264},[5312],[75,39715,39716,39719],{"style":5316},[75,39717],{"className":39718,"style":5321},[5320],[75,39720,39722],{"className":39721},[5325,5326,5327,5328],[75,39723,36399],{"className":39724,"style":36415},[5258,5259,5328],[75,39726,5336],{"className":39727},[5335],[75,39729,39731],{"className":39730},[5308],[75,39732,39734],{"className":39733,"style":5343},[5312],[75,39735],{},[75,39737,5175],{"className":39738},[5270],[75,39740],{"className":39741,"style":5275},[5274],[75,39743,142],{"className":39744},[5279],[75,39746],{"className":39747,"style":5275},[5274],[75,39749,39751,39754,39757,39760,39763,39766,39806,39809,39812],{"className":39750},[5249],[75,39752],{"className":39753,"style":38345},[5253],[75,39755,7477],{"className":39756},[5258],[75,39758,7469],{"className":39759},[7544],[75,39761],{"className":39762,"style":7548},[5274],[75,39764],{"className":39765,"style":5349},[5274],[75,39767,39769,39772],{"className":39768},[5258],[75,39770,5166],{"className":39771},[5258,5259],[75,39773,39775],{"className":39774},[5299],[75,39776,39778,39798],{"className":39777},[5303,5304],[75,39779,39781,39795],{"className":39780},[5308],[75,39782,39784],{"className":39783,"style":6264},[5312],[75,39785,39786,39789],{"style":5316},[75,39787],{"className":39788,"style":5321},[5320],[75,39790,39792],{"className":39791},[5325,5326,5327,5328],[75,39793,36399],{"className":39794,"style":36415},[5258,5259,5328],[75,39796,5336],{"className":39797},[5335],[75,39799,39801],{"className":39800},[5308],[75,39802,39804],{"className":39803,"style":5343},[5312],[75,39805],{},[75,39807],{"className":39808,"style":5275},[5274],[75,39810,142],{"className":39811},[5279],[75,39813],{"className":39814,"style":5275},[5274],[75,39816,39818,39822,39878],{"className":39817},[5249],[75,39819],{"className":39820,"style":39821},[5253],"height:1.1461em;vertical-align:-0.247em;",[75,39823,39825,39828],{"className":39824},[5258],[75,39826,5166],{"className":39827},[5258,5259],[75,39829,39831],{"className":39830},[5299],[75,39832,39834,39869],{"className":39833},[53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constants are useful for fixing a consistent color scale across frames.",[7621,40912],{},[12425,40914,40916],{"id":40915},"_8-acronymnotation-glossary","8. Acronym\u002FNotation Glossary",[28,40918,40919,40925,40931,40937,40943,41061],{},[31,40920,40921,40924],{},[48,40922,40923],{},"SW"," — Stratonovich–Weyl (kernel\u002Fcorrespondence).",[31,40926,40927,40930],{},[48,40928,40929],{},"SU(2)"," — Special Unitary group of degree two (spin‑(1\u002F2) rotations).",[31,40932,40933,40936],{},[48,40934,40935],{},"SO(3)"," — Special Orthogonal group in 3D (real rotation matrices).",[31,40938,40939,40942],{},[48,40940,40941],{},"WF"," — Wigner function.",[31,40944,40945,40986,40987,41028,41029,41060],{},[48,40946,197,40947,5175],{},[75,40948,40950,40968],{"className":40949},[5148],[75,40951,40953],{"className":40952},[5152],[5154,40954,40955],{"xmlns":5156},[5158,40956,40957,40965],{},[5161,40958,40959,40961,40963],{},[5168,40960,7452],{"stretchy":5170},[5164,40962,32718],{},[5168,40964,7457],{"stretchy":5170},[5238,40966,40967],{"encoding":5240},"\\lvert\\psi\\rangle",[75,40969,40971],{"className":40970,"ariaHidden":5196},[5245],[75,40972,40974,40977,40980,40983],{"className":40973},[5249],[75,40975],{"className":40976,"style":5254},[5253],[75,40978,7452],{"className":40979},[5263],[75,40981,32718],{"className":40982,"style":5416},[5258,5259],[75,40984,7457],{"className":40985},[5270]," — State vector (ket); ",[48,40988,197,40989,5175],{},[75,40990,40992,41010],{"className":40991},[5148],[75,40993,40995],{"className":40994},[5152],[5154,40996,40997],{"xmlns":5156},[5158,40998,40999,41007],{},[5161,41000,41001,41003,41005],{},[5168,41002,32723],{"stretchy":5170},[5164,41004,32718],{},[5168,41006,7452],{"stretchy":5170},[5238,41008,41009],{"encoding":5240},"\\langle\\psi\\rvert",[75,41011,41013],{"className":41012,"ariaHidden":5196},[5245],[75,41014,41016,41019,41022,41025],{"className":41015},[5249],[75,41017],{"className":41018,"style":5254},[5253],[75,41020,32723],{"className":41021},[5263],[75,41023,32718],{"className":41024,"style":5416},[5258,5259],[75,41026,7452],{"className":41027},[5270]," — bra; ",[48,41030,197,41031,5175],{},[75,41032,41034,41048],{"className":41033},[5148],[75,41035,41037],{"className":41036},[5152],[5154,41038,41039],{"xmlns":5156},[5158,41040,41041,41045],{},[5161,41042,41043],{},[5164,41044,32265],{},[5238,41046,41047],{"encoding":5240},"\\rho",[75,41049,41051],{"className":41050,"ariaHidden":5196},[5245],[75,41052,41054,41057],{"className":41053},[5249],[75,41055],{"className":41056,"style":32361},[5253],[75,41058,32265],{"className":41059},[5258,5259]," — density operator.",[31,41062,41063,41066,41067,41186,41187,13959],{},[48,41064,41065],{},"Adjoint map"," — Conjugation action (",[75,41068,41070,41102],{"className":41069},[5148],[75,41071,41073],{"className":41072},[5152],[5154,41074,41075],{"xmlns":5156},[5158,41076,41077,41099],{},[5161,41078,41079,41081,41084,41086,41089,41091,41093],{},[5164,41080,34884],{},[5168,41082,41083],{},":",[5164,41085,32265],{},[5168,41087,41088],{},"↦",[5164,41090,34884],{},[5164,41092,32265],{},[5218,41094,41095,41097],{},[5164,41096,34884],{},[5168,41098,34894],{"lspace":34893,"rspace":34893},[5238,41100,41101],{"encoding":5240},"U: \\rho\\mapsto U\\rho U^{\\dagger}",[75,41103,41105,41123,41141],{"className":41104,"ariaHidden":5196},[5245],[75,41106,41108,41111,41114,41117,41120],{"className":41107},[5249],[75,41109],{"className":41110,"style":12648},[5253],[75,41112,34884],{"className":41113,"style":9810},[5258,5259],[75,41115],{"className":41116,"style":5275},[5274],[75,41118,41083],{"className":41119},[5279],[75,41121],{"className":41122,"style":5275},[5274],[75,41124,41126,41129,41132,41135,41138],{"className":41125},[5249],[75,41127],{"className":41128,"style":36954},[5253],[75,41130,32265],{"className":41131},[5258,5259],[75,41133],{"className":41134,"style":5275},[5274],[75,41136,41088],{"className":41137},[5279],[75,41139],{"className":41140,"style":5275},[5274],[75,41142,41144,41148,41151,41154],{"className":41143},[5249],[75,41145],{"className":41146,"style":41147},[5253],"height:1.0435em;vertical-align:-0.1944em;",[75,41149,34884],{"className":41150,"style":9810},[5258,5259],[75,41152,32265],{"className":41153},[5258,5259],[75,41155,41157,41160],{"className":41156},[5258],[75,41158,34884],{"className":41159,"style":9810},[5258,5259],[75,41161,41163],{"className":41162},[5299],[75,41164,41166],{"className":41165},[5303],[75,41167,41169],{"className":41168},[5308],[75,41170,41172],{"className":41171,"style":7593},[5312],[75,41173,41174,41177],{"style":7596},[75,41175],{"className":41176,"style":5321},[5320],[75,41178,41180],{"className":41179},[5325,5326,5327,5328],[75,41181,41183],{"className":41182},[5258,5328],[75,41184,34894],{"className":41185},[5258,5328],") inducing (",[75,41188,41190,41210],{"className":41189},[5148],[75,41191,41193],{"className":41192},[5152],[5154,41194,41195],{"xmlns":5156},[5158,41196,41197,41207],{},[5161,41198,41199,41201,41203,41205],{},[5164,41200,15977],{"mathvariant":32172},[5168,41202,41088],{},[5164,41204,32181],{},[5164,41206,15977],{"mathvariant":32172},[5238,41208,41209],{"encoding":5240},"\\mathbf r\\mapsto R\\mathbf r",[75,41211,41213,41232],{"className":41212,"ariaHidden":5196},[5245],[75,41214,41216,41220,41223,41226,41229],{"className":41215},[5249],[75,41217],{"className":41218,"style":41219},[5253],"height:0.522em;vertical-align:-0.011em;",[75,41221,15977],{"className":41222},[5258,32200],[75,41224],{"className":41225,"style":5275},[5274],[75,41227,41088],{"className":41228},[5279],[75,41230],{"className":41231,"style":5275},[5274],[75,41233,41235,41238,41241],{"className":41234},[5249],[75,41236],{"className":41237,"style":12648},[5253],[75,41239,32181],{"className":41240,"style":35138},[5258,5259],[75,41242,15977],{"className":41243},[5258,32200],[7621,41245],{},[12425,41247,41249],{"id":41248},"_9-mindmap","9. Mind‑Map",[12,41251,41252],{},"Wigner for Qubits",[65,41254,41258],{"className":41255,"code":41257,"language":34131},[41256],"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",[72,41259,41257],{"__ignoreMap":70},[7621,41261],{},[12425,41263,41265],{"id":41264},"_10-minimal-substitutions-if-variations-are-needed","10. Minimal Substitutions (if variations are needed)",[28,41267,41268,41330],{},[31,41269,41270,41271,41300,41301,41329],{},"If a different SW kernel normalization is chosen, the prefactors in (",[75,41272,41274,41288],{"className":41273},[5148],[75,41275,41277],{"className":41276},[5152],[5154,41278,41279],{"xmlns":5156},[5158,41280,41281,41285],{},[5161,41282,41283],{},[5164,41284,12553],{"mathvariant":6582},[5238,41286,41287],{"encoding":5240},"\\Delta",[75,41289,41291],{"className":41290,"ariaHidden":5196},[5245],[75,41292,41294,41297],{"className":41293},[5249],[75,41295],{"className":41296,"style":12648},[5253],[75,41298,12553],{"className":41299},[5258],") and hence (",[75,41302,41304,41317],{"className":41303},[5148],[75,41305,41307],{"className":41306},[5152],[5154,41308,41309],{"xmlns":5156},[5158,41310,41311,41315],{},[5161,41312,41313],{},[5164,41314,33637],{},[5238,41316,33637],{"encoding":5240},[75,41318,41320],{"className":41319,"ariaHidden":5196},[5245],[75,41321,41323,41326],{"className":41322},[5249],[75,41324],{"className":41325,"style":12648},[5253],[75,41327,33637],{"className":41328,"style":9817},[5258,5259],") adjust accordingly, while covariance and linearity remain.",[31,41331,41332,41333,41394,41395,41434],{},"Mixed states use the same formulas with (",[75,41334,41336,41358],{"className":41335},[5148],[75,41337,41339],{"className":41338},[5152],[5154,41340,41341],{"xmlns":5156},[5158,41342,41343,41355],{},[5161,41344,41345,41347,41349,41351,41353],{},[5168,41346,32791],{"stretchy":5170},[5164,41348,15977],{"mathvariant":32172},[5168,41350,32791],{"stretchy":5170},[5168,41352,14442],{},[5184,41354,7477],{},[5238,41356,41357],{"encoding":5240},"\\lVert\\mathbf r\\rVert\u003C1",[75,41359,41361,41385],{"className":41360,"ariaHidden":5196},[5245],[75,41362,41364,41367,41370,41373,41376,41379,41382],{"className":41363},[5249],[75,41365],{"className":41366,"style":5254},[5253],[75,41368,32791],{"className":41369},[5263],[75,41371,15977],{"className":41372},[5258,32200],[75,41374,32791],{"className":41375},[5270],[75,41377],{"className":41378,"style":5275},[5274],[75,41380,14442],{"className":41381},[5279],[75,41383],{"className":41384,"style":5275},[5274],[75,41386,41388,41391],{"className":41387},[5249],[75,41389],{"className":41390,"style":32836},[5253],[75,41392,7477],{"className":41393},[5258],") (the extrema shrink linearly with (",[75,41396,41398,41416],{"className":41397},[5148],[75,41399,41401],{"className":41400},[5152],[5154,41402,41403],{"xmlns":5156},[5158,41404,41405,41413],{},[5161,41406,41407,41409,41411],{},[5168,41408,32791],{"stretchy":5170},[5164,41410,15977],{"mathvariant":32172},[5168,41412,32791],{"stretchy":5170},[5238,41414,41415],{"encoding":5240},"\\lVert\\mathbf r\\rVert",[75,41417,41419],{"className":41418,"ariaHidden":5196},[5245],[75,41420,41422,41425,41428,41431],{"className":41421},[5249],[75,41423],{"className":41424,"style":5254},[5253],[75,41426,32791],{"className":41427},[5263],[75,41429,15977],{"className":41430},[5258,32200],[75,41432,32791],{"className":41433},[5270],")).",[65,41436,41438],{"className":67,"code":41437,"language":69,"meta":70,"style":70},"# @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",[72,41439,41440,41445,41450,41455,41459,41464,41469,41474,41478,41483,41488,41492,41496,41509,41513,41523,41535,41545,41549,41554,41565,41581,41596,41618,41630],{"__ignoreMap":70},[75,41441,41442],{"class":77,"line":78},[75,41443,41444],{"class":118},"# @title Cell 1 — Imports & global plotting config (DPI=200)\n",[75,41446,41447],{"class":77,"line":95},[75,41448,41449],{"class":134},"\"\"\"\n",[75,41451,41452],{"class":77,"line":108},[75,41453,41454],{"class":134},"Spin (SU(2)) Wigner-function animations for single-qubit gates.\n",[75,41456,41457],{"class":77,"line":115},[75,41458,112],{"emptyLinePlaceholder":111},[75,41460,41461],{"class":77,"line":122},[75,41462,41463],{"class":134},"This notebook renders the qubit Wigner function\n",[75,41465,41466],{"class":77,"line":128},[75,41467,41468],{"class":134},"    W(θ, φ) = 1\u002F2 + (√3\u002F2) * r · n(θ, φ)\n",[75,41470,41471],{"class":77,"line":149},[75,41472,41473],{"class":134},"and animates its rigid rotation under Pauli and custom axis–angle gates.\n",[75,41475,41476],{"class":77,"line":154},[75,41477,112],{"emptyLinePlaceholder":111},[75,41479,41480],{"class":77,"line":160},[75,41481,41482],{"class":134},"All titles and dynamic text are drawn *inside* the plotted region so they\n",[75,41484,41485],{"class":77,"line":171},[75,41486,41487],{"class":134},"never clip in the saved GIFs. Use Cells 3 and 4 to generate animations.\n",[75,41489,41490],{"class":77,"line":182},[75,41491,41449],{"class":134},[75,41493,41494],{"class":77,"line":222},[75,41495,112],{"emptyLinePlaceholder":111},[75,41497,41498,41500,41503,41506],{"class":77,"line":227},[75,41499,2700],{"class":81},[75,41501,41502],{"class":15980}," __future__",[75,41504,41505],{"class":81}," import",[75,41507,41508],{"class":85}," annotations\n",[75,41510,41511],{"class":77,"line":233},[75,41512,112],{"emptyLinePlaceholder":111},[75,41514,41515,41517,41519,41521],{"class":77,"line":286},[75,41516,82],{"class":81},[75,41518,86],{"class":85},[75,41520,89],{"class":81},[75,41522,92],{"class":85},[75,41524,41525,41527,41530,41532],{"class":77,"line":291},[75,41526,82],{"class":81},[75,41528,41529],{"class":85}," matplotlib ",[75,41531,89],{"class":81},[75,41533,41534],{"class":85}," mpl\n",[75,41536,41537,41539,41541,41543],{"class":77,"line":297},[75,41538,82],{"class":81},[75,41540,100],{"class":85},[75,41542,89],{"class":81},[75,41544,105],{"class":85},[75,41546,41547],{"class":77,"line":321},[75,41548,112],{"emptyLinePlaceholder":111},[75,41550,41551],{"class":77,"line":336},[75,41552,41553],{"class":118},"# Global Matplotlib configuration: DPI = 200 everywhere\n",[75,41555,41556,41559,41562],{"class":77,"line":351},[75,41557,41558],{"class":85},"mpl.rcParams.",[75,41560,41561],{"class":193},"update",[75,41563,41564],{"class":85},"({\n",[75,41566,41567,41570,41572,41575,41578],{"class":77,"line":366},[75,41568,41569],{"class":134},"    \"figure.dpi\"",[75,41571,14391],{"class":85},[75,41573,41574],{"class":145},"200",[75,41576,41577],{"class":85},",       ",[75,41579,41580],{"class":118},"# on-screen DPI\n",[75,41582,41583,41586,41588,41590,41593],{"class":77,"line":381},[75,41584,41585],{"class":134},"    \"savefig.dpi\"",[75,41587,14391],{"class":85},[75,41589,41574],{"class":145},[75,41591,41592],{"class":85},",      ",[75,41594,41595],{"class":118},"# saved file DPI\n",[75,41597,41598,41601,41604,41607,41609,41612,41615],{"class":77,"line":984},[75,41599,41600],{"class":134},"    \"figure.figsize\"",[75,41602,41603],{"class":85},": (",[75,41605,41606],{"class":145},"6.4",[75,41608,206],{"class":85},[75,41610,41611],{"class":145},"3.6",[75,41613,41614],{"class":85},"),  ",[75,41616,41617],{"class":118},"# 1280×720 at DPI=200\n",[75,41619,41620,41623,41625,41627],{"class":77,"line":997},[75,41621,41622],{"class":134},"    \"axes.grid\"",[75,41624,14391],{"class":85},[75,41626,376],{"class":145},[75,41628,41629],{"class":85},",\n",[75,41631,41632],{"class":77,"line":1325},[75,41633,41634],{"class":85},"})\n",[65,41636,41638],{"className":67,"code":41637,"language":69,"meta":70,"style":70},"# @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",[72,41639,41640,41645,41655,41667,41678,41685,41689,41699,41709,41721,41733,41737,41741,41745,41754,41759,41763,41768,41773,41778,41783,41788,41793,41798,41803,41808,41813,41818,41823,41828,41833,41867,41879,41912,41924,41936,41948,41960,41972,41997,42009,42021,42033,42037,42041,42046,42064,42085,42090,42100,42138,42176,42214,42219,42249,42264,42307,42311,42315,42320,42348,42353,42372,42386,42428,42441,42483,42496,42536,42549,42553,42557,42583,42594,42599,42617,42630,42642,42654,42668,42737,42754,42770,42813,42817,42821,42826,42853,42858,42876,42900,42925,42948,42969,42983,43003,43007,43011,43030,43035,43100,43104,43108,43113,43144,43149,43163,43177,43215,43228,43262,43275,43309,43322,43354,43386,43390,43394,43408,43419,43434,43450,43467,43472,43484,43496,43507,43534,43583,43588,43592,43596,43600,43622,43627,43637,43649,43674,43686,43726,43737],{"__ignoreMap":70},[75,41641,41642],{"class":77,"line":78},[75,41643,41644],{"class":118},"# @title Cell 2 — Math & utilities (PEP 8 \u002F PEP 257)\n",[75,41646,41647,41649,41651,41653],{"class":77,"line":95},[75,41648,2700],{"class":81},[75,41650,14064],{"class":85},[75,41652,82],{"class":81},[75,41654,14069],{"class":85},[75,41656,41657,41659,41662,41664],{"class":77,"line":108},[75,41658,2700],{"class":81},[75,41660,41661],{"class":85}," pathlib ",[75,41663,82],{"class":81},[75,41665,41666],{"class":85}," Path\n",[75,41668,41669,41671,41673,41675],{"class":77,"line":115},[75,41670,2700],{"class":81},[75,41672,14076],{"class":85},[75,41674,82],{"class":81},[75,41676,41677],{"class":85}," Tuple, Iterable\n",[75,41679,41680,41682],{"class":77,"line":122},[75,41681,82],{"class":81},[75,41683,41684],{"class":85}," textwrap\n",[75,41686,41687],{"class":77,"line":128},[75,41688,112],{"emptyLinePlaceholder":111},[75,41690,41691,41693,41695,41697],{"class":77,"line":149},[75,41692,82],{"class":81},[75,41694,86],{"class":85},[75,41696,89],{"class":81},[75,41698,92],{"class":85},[75,41700,41701,41703,41705,41707],{"class":77,"line":154},[75,41702,82],{"class":81},[75,41704,100],{"class":85},[75,41706,89],{"class":81},[75,41708,105],{"class":85},[75,41710,41711,41713,41716,41718],{"class":77,"line":160},[75,41712,2700],{"class":81},[75,41714,41715],{"class":85}," matplotlib.animation ",[75,41717,82],{"class":81},[75,41719,41720],{"class":85}," FuncAnimation, PillowWriter\n",[75,41722,41723,41725,41728,41730],{"class":77,"line":171},[75,41724,2700],{"class":81},[75,41726,41727],{"class":85}," IPython.display ",[75,41729,82],{"class":81},[75,41731,41732],{"class":85}," Image, display\n",[75,41734,41735],{"class":77,"line":182},[75,41736,112],{"emptyLinePlaceholder":111},[75,41738,41739],{"class":77,"line":222},[75,41740,112],{"emptyLinePlaceholder":111},[75,41742,41743],{"class":77,"line":227},[75,41744,14131],{"class":193},[75,41746,41747,41749,41752],{"class":77,"line":233},[75,41748,14136],{"class":81},[75,41750,41751],{"class":14139}," Knobs",[75,41753,5800],{"class":85},[75,41755,41756],{"class":77,"line":286},[75,41757,41758],{"class":134},"    \"\"\"User controls for Wigner animations.\n",[75,41760,41761],{"class":77,"line":291},[75,41762,112],{"emptyLinePlaceholder":111},[75,41764,41765],{"class":77,"line":297},[75,41766,41767],{"class":134},"    Attributes:\n",[75,41769,41770],{"class":77,"line":321},[75,41771,41772],{"class":134},"        gate_axes: Pauli axes to animate, each in {\"x\", \"y\", \"z\"}.\n",[75,41774,41775],{"class":77,"line":336},[75,41776,41777],{"class":134},"        initial_state: Starting Bloch eigenstate, or \"custom\".\n",[75,41779,41780],{"class":77,"line":351},[75,41781,41782],{"class":134},"        custom_r: Custom Bloch vector if initial_state == \"custom\".\n",[75,41784,41785],{"class":77,"line":366},[75,41786,41787],{"class":134},"        frames: Number of frames (≥ 2).\n",[75,41789,41790],{"class":77,"line":381},[75,41791,41792],{"class":134},"        n_theta: Polar samples (θ ∈ [0, π]).\n",[75,41794,41795],{"class":77,"line":984},[75,41796,41797],{"class":134},"        n_phi: Azimuth samples (φ ∈ [0, 2π]).\n",[75,41799,41800],{"class":77,"line":997},[75,41801,41802],{"class":134},"        fps: GIF framerate.\n",[75,41804,41805],{"class":77,"line":1325},[75,41806,41807],{"class":134},"        out_dir: Output directory (e.g., \"\u002Fcontent\" in Colab).\n",[75,41809,41810],{"class":77,"line":1338},[75,41811,41812],{"class":134},"        figsize: Figure size in inches (DPI set globally).\n",[75,41814,41815],{"class":77,"line":1343},[75,41816,41817],{"class":134},"        info_loc: In-axes anchor (\"ul\", \"ur\", \"ll\", \"lr\").\n",[75,41819,41820],{"class":77,"line":1356},[75,41821,41822],{"class":134},"        info_fontsize: Font size for in-axes info text.\n",[75,41824,41825],{"class":77,"line":1887},[75,41826,41827],{"class":134},"        wrap_chars: Manual wrap width for info text.\n",[75,41829,41830],{"class":77,"line":1900},[75,41831,41832],{"class":134},"    \"\"\"\n",[75,41834,41835,41838,41841,41843,41846,41848,41850,41852,41855,41857,41860,41862,41865],{"class":77,"line":1913},[75,41836,41837],{"class":85},"    gate_axes: Tuple[",[75,41839,41840],{"class":141},"str",[75,41842,206],{"class":85},[75,41844,41845],{"class":145},"...",[75,41847,138],{"class":85},[75,41849,142],{"class":141},[75,41851,267],{"class":85},[75,41853,41854],{"class":134},"\"x\"",[75,41856,206],{"class":85},[75,41858,41859],{"class":134},"\"y\"",[75,41861,206],{"class":85},[75,41863,41864],{"class":134},"\"z\"",[75,41866,318],{"class":85},[75,41868,41869,41872,41874,41876],{"class":77,"line":2657},[75,41870,41871],{"class":85},"    initial_state: ",[75,41873,41840],{"class":141},[75,41875,14153],{"class":141},[75,41877,41878],{"class":134}," \"+z\"\n",[75,41880,41881,41884,41886,41888,41890,41892,41894,41896,41898,41900,41902,41904,41906,41908,41910],{"class":77,"line":3177},[75,41882,41883],{"class":85},"    custom_r: Tuple[",[75,41885,14150],{"class":141},[75,41887,206],{"class":85},[75,41889,14150],{"class":141},[75,41891,206],{"class":85},[75,41893,14150],{"class":141},[75,41895,138],{"class":85},[75,41897,142],{"class":141},[75,41899,267],{"class":85},[75,41901,14324],{"class":145},[75,41903,206],{"class":85},[75,41905,14324],{"class":145},[75,41907,206],{"class":85},[75,41909,630],{"class":145},[75,41911,318],{"class":85},[75,41913,41914,41917,41919,41921],{"class":77,"line":3188},[75,41915,41916],{"class":85},"    frames: ",[75,41918,14299],{"class":141},[75,41920,14153],{"class":141},[75,41922,41923],{"class":145}," 24\n",[75,41925,41926,41929,41931,41933],{"class":77,"line":3203},[75,41927,41928],{"class":85},"    n_theta: ",[75,41930,14299],{"class":141},[75,41932,14153],{"class":141},[75,41934,41935],{"class":145}," 80\n",[75,41937,41938,41941,41943,41945],{"class":77,"line":3218},[75,41939,41940],{"class":85},"    n_phi: ",[75,41942,14299],{"class":141},[75,41944,14153],{"class":141},[75,41946,41947],{"class":145}," 160\n",[75,41949,41950,41953,41955,41957],{"class":77,"line":3224},[75,41951,41952],{"class":85},"    fps: ",[75,41954,14299],{"class":141},[75,41956,14153],{"class":141},[75,41958,41959],{"class":145}," 20\n",[75,41961,41962,41965,41967,41969],{"class":77,"line":3229},[75,41963,41964],{"class":85},"    out_dir: ",[75,41966,41840],{"class":141},[75,41968,14153],{"class":141},[75,41970,41971],{"class":134}," \"\u002Fcontent\"\n",[75,41973,41974,41977,41979,41981,41983,41985,41987,41989,41991,41993,41995],{"class":77,"line":3235},[75,41975,41976],{"class":85},"    figsize: Tuple[",[75,41978,14150],{"class":141},[75,41980,206],{"class":85},[75,41982,14150],{"class":141},[75,41984,138],{"class":85},[75,41986,142],{"class":141},[75,41988,267],{"class":85},[75,41990,41606],{"class":145},[75,41992,206],{"class":85},[75,41994,41611],{"class":145},[75,41996,318],{"class":85},[75,41998,41999,42002,42004,42006],{"class":77,"line":3263},[75,42000,42001],{"class":85},"    info_loc: ",[75,42003,41840],{"class":141},[75,42005,14153],{"class":141},[75,42007,42008],{"class":134}," \"ul\"\n",[75,42010,42011,42014,42016,42018],{"class":77,"line":3275},[75,42012,42013],{"class":85},"    info_fontsize: ",[75,42015,14299],{"class":141},[75,42017,14153],{"class":141},[75,42019,42020],{"class":145}," 9\n",[75,42022,42023,42026,42028,42030],{"class":77,"line":3280},[75,42024,42025],{"class":85},"    wrap_chars: ",[75,42027,14299],{"class":141},[75,42029,14153],{"class":141},[75,42031,42032],{"class":145}," 42\n",[75,42034,42035],{"class":77,"line":3290},[75,42036,112],{"emptyLinePlaceholder":111},[75,42038,42039],{"class":77,"line":4431},[75,42040,112],{"emptyLinePlaceholder":111},[75,42042,42043],{"class":77,"line":4454},[75,42044,42045],{"class":118},"# ------------------------------ Bloch helpers ---------------------------------\n",[75,42047,42048,42050,42053,42055,42058,42060,42062],{"class":77,"line":4463},[75,42049,14380],{"class":81},[75,42051,42052],{"class":193}," bloch_vector",[75,42054,197],{"class":85},[75,42056,42057],{"class":14388},"initial_state",[75,42059,14391],{"class":85},[75,42061,41840],{"class":141},[75,42063,41629],{"class":85},[75,42065,42066,42069,42072,42074,42076,42078,42080,42082],{"class":77,"line":4468},[75,42067,42068],{"class":14388},"                 custom_r",[75,42070,42071],{"class":85},": Tuple[",[75,42073,14150],{"class":141},[75,42075,206],{"class":85},[75,42077,14150],{"class":141},[75,42079,206],{"class":85},[75,42081,14150],{"class":141},[75,42083,42084],{"class":85},"]) -> np.ndarray:\n",[75,42086,42087],{"class":77,"line":4477},[75,42088,42089],{"class":134},"    \"\"\"Return a unit Bloch vector for a named eigenstate or custom input.\"\"\"\n",[75,42091,42092,42095,42097],{"class":77,"line":5105},[75,42093,42094],{"class":85},"    mapping ",[75,42096,142],{"class":141},[75,42098,42099],{"class":85}," {\n",[75,42101,42102,42105,42107,42109,42111,42113,42115,42117,42119,42122,42124,42126,42128,42130,42132,42134,42136],{"class":77,"line":5114},[75,42103,42104],{"class":134},"        \"+z\"",[75,42106,41603],{"class":85},[75,42108,14324],{"class":145},[75,42110,206],{"class":85},[75,42112,14324],{"class":145},[75,42114,206],{"class":85},[75,42116,630],{"class":145},[75,42118,10496],{"class":85},[75,42120,42121],{"class":134},"\"-z\"",[75,42123,41603],{"class":85},[75,42125,14324],{"class":145},[75,42127,206],{"class":85},[75,42129,14324],{"class":145},[75,42131,206],{"class":85},[75,42133,200],{"class":141},[75,42135,630],{"class":145},[75,42137,3003],{"class":85},[75,42139,42140,42143,42145,42147,42149,42151,42153,42155,42157,42160,42162,42164,42166,42168,42170,42172,42174],{"class":77,"line":10795},[75,42141,42142],{"class":134},"        \"+x\"",[75,42144,41603],{"class":85},[75,42146,630],{"class":145},[75,42148,206],{"class":85},[75,42150,14324],{"class":145},[75,42152,206],{"class":85},[75,42154,14324],{"class":145},[75,42156,10496],{"class":85},[75,42158,42159],{"class":134},"\"-x\"",[75,42161,41603],{"class":85},[75,42163,200],{"class":141},[75,42165,630],{"class":145},[75,42167,206],{"class":85},[75,42169,14324],{"class":145},[75,42171,206],{"class":85},[75,42173,14324],{"class":145},[75,42175,3003],{"class":85},[75,42177,42178,42181,42183,42185,42187,42189,42191,42193,42195,42198,42200,42202,42204,42206,42208,42210,42212],{"class":77,"line":14778},[75,42179,42180],{"class":134},"        \"+y\"",[75,42182,41603],{"class":85},[75,42184,14324],{"class":145},[75,42186,206],{"class":85},[75,42188,630],{"class":145},[75,42190,206],{"class":85},[75,42192,14324],{"class":145},[75,42194,10496],{"class":85},[75,42196,42197],{"class":134},"\"-y\"",[75,42199,41603],{"class":85},[75,42201,14324],{"class":145},[75,42203,206],{"class":85},[75,42205,200],{"class":141},[75,42207,630],{"class":145},[75,42209,206],{"class":85},[75,42211,14324],{"class":145},[75,42213,3003],{"class":85},[75,42215,42216],{"class":77,"line":14783},[75,42217,42218],{"class":85},"    }\n",[75,42220,42221,42223,42225,42227,42230,42233,42236,42239,42242,42245,42247],{"class":77,"line":14819},[75,42222,14520],{"class":85},[75,42224,142],{"class":141},[75,42226,190],{"class":85},[75,42228,42229],{"class":193},"asarray",[75,42231,42232],{"class":85},"(mapping.",[75,42234,42235],{"class":193},"get",[75,42237,42238],{"class":85},"(initial_state.",[75,42240,42241],{"class":193},"lower",[75,42243,42244],{"class":85},"(), custom_r), ",[75,42246,14150],{"class":141},[75,42248,318],{"class":85},[75,42250,42251,42254,42256,42258,42261],{"class":77,"line":14825},[75,42252,42253],{"class":85},"    nrm ",[75,42255,142],{"class":141},[75,42257,19086],{"class":85},[75,42259,42260],{"class":193},"norm",[75,42262,42263],{"class":85},"(r)\n",[75,42265,42266,42268,42270,42272,42274,42276,42278,42280,42282,42284,42287,42289,42292,42295,42297,42299,42302,42304],{"class":77,"line":14838},[75,42267,14467],{"class":81},[75,42269,190],{"class":85},[75,42271,12049],{"class":193},[75,42273,15740],{"class":85},[75,42275,14324],{"class":145},[75,42277,206],{"class":85},[75,42279,14324],{"class":145},[75,42281,206],{"class":85},[75,42283,630],{"class":145},[75,42285,42286],{"class":85},"]) ",[75,42288,14737],{"class":81},[75,42290,42291],{"class":85}," nrm ",[75,42293,42294],{"class":141},"==",[75,42296,10251],{"class":145},[75,42298,16742],{"class":81},[75,42300,42301],{"class":85}," r ",[75,42303,2810],{"class":141},[75,42305,42306],{"class":85}," nrm\n",[75,42308,42309],{"class":77,"line":14893},[75,42310,112],{"emptyLinePlaceholder":111},[75,42312,42313],{"class":77,"line":14951},[75,42314,112],{"emptyLinePlaceholder":111},[75,42316,42317],{"class":77,"line":14959},[75,42318,42319],{"class":118},"# ---------------------------- Axis–angle rotations ----------------------------\n",[75,42321,42322,42324,42327,42329,42332,42334,42336,42338,42341,42343,42345],{"class":77,"line":14964},[75,42323,14380],{"class":81},[75,42325,42326],{"class":193}," rotation_matrix",[75,42328,197],{"class":85},[75,42330,42331],{"class":14388},"axis",[75,42333,14391],{"class":85},[75,42335,41840],{"class":141},[75,42337,206],{"class":85},[75,42339,42340],{"class":14388},"angle",[75,42342,14391],{"class":85},[75,42344,14150],{"class":141},[75,42346,42347],{"class":85},") -> np.ndarray:\n",[75,42349,42350],{"class":77,"line":14983},[75,42351,42352],{"class":134},"    \"\"\"Rodrigues rotation matrix for axis in {'x','y','z'} and angle (radians).\"\"\"\n",[75,42354,42355,42358,42360,42362,42364,42367,42369],{"class":77,"line":14989},[75,42356,42357],{"class":85},"    c, s ",[75,42359,142],{"class":141},[75,42361,190],{"class":85},[75,42363,2845],{"class":193},[75,42365,42366],{"class":85},"(angle), np.",[75,42368,1179],{"class":193},[75,42370,42371],{"class":85},"(angle)\n",[75,42373,42374,42376,42379,42381,42384],{"class":77,"line":15012},[75,42375,14433],{"class":81},[75,42377,42378],{"class":85}," axis ",[75,42380,42294],{"class":141},[75,42382,42383],{"class":134}," \"x\"",[75,42385,5800],{"class":85},[75,42387,42388,42390,42392,42394,42396,42398,42400,42402,42404,42406,42409,42411,42414,42416,42419,42421,42424,42426],{"class":77,"line":15017},[75,42389,14450],{"class":81},[75,42391,190],{"class":85},[75,42393,12049],{"class":193},[75,42395,18733],{"class":85},[75,42397,7477],{"class":145},[75,42399,206],{"class":85},[75,42401,5186],{"class":145},[75,42403,206],{"class":85},[75,42405,5186],{"class":145},[75,42407,42408],{"class":85},"], [",[75,42410,5186],{"class":145},[75,42412,42413],{"class":85},", c, ",[75,42415,200],{"class":141},[75,42417,42418],{"class":85},"s], [",[75,42420,5186],{"class":145},[75,42422,42423],{"class":85},", s, c]], ",[75,42425,14150],{"class":141},[75,42427,318],{"class":85},[75,42429,42430,42432,42434,42436,42439],{"class":77,"line":15025},[75,42431,14433],{"class":81},[75,42433,42378],{"class":85},[75,42435,42294],{"class":141},[75,42437,42438],{"class":134}," \"y\"",[75,42440,5800],{"class":85},[75,42442,42443,42445,42447,42449,42452,42454,42457,42459,42461,42463,42465,42467,42469,42471,42474,42476,42479,42481],{"class":77,"line":15038},[75,42444,14450],{"class":81},[75,42446,190],{"class":85},[75,42448,12049],{"class":193},[75,42450,42451],{"class":85},"([[c, ",[75,42453,5186],{"class":145},[75,42455,42456],{"class":85},", s], [",[75,42458,5186],{"class":145},[75,42460,206],{"class":85},[75,42462,7477],{"class":145},[75,42464,206],{"class":85},[75,42466,5186],{"class":145},[75,42468,42408],{"class":85},[75,42470,200],{"class":141},[75,42472,42473],{"class":85},"s, ",[75,42475,5186],{"class":145},[75,42477,42478],{"class":85},", c]], ",[75,42480,14150],{"class":141},[75,42482,318],{"class":85},[75,42484,42485,42487,42489,42491,42494],{"class":77,"line":15051},[75,42486,14433],{"class":81},[75,42488,42378],{"class":85},[75,42490,42294],{"class":141},[75,42492,42493],{"class":134}," \"z\"",[75,42495,5800],{"class":85},[75,42497,42498,42500,42502,42504,42506,42508,42510,42512,42515,42517,42519,42521,42523,42525,42527,42529,42532,42534],{"class":77,"line":15058},[75,42499,14450],{"class":81},[75,42501,190],{"class":85},[75,42503,12049],{"class":193},[75,42505,42451],{"class":85},[75,42507,200],{"class":141},[75,42509,42473],{"class":85},[75,42511,5186],{"class":145},[75,42513,42514],{"class":85},"], [s, c, ",[75,42516,5186],{"class":145},[75,42518,42408],{"class":85},[75,42520,5186],{"class":145},[75,42522,206],{"class":85},[75,42524,5186],{"class":145},[75,42526,206],{"class":85},[75,42528,7477],{"class":145},[75,42530,42531],{"class":85},"]], ",[75,42533,14150],{"class":141},[75,42535,318],{"class":85},[75,42537,42538,42541,42544,42547],{"class":77,"line":15069},[75,42539,42540],{"class":81},"    raise",[75,42542,42543],{"class":85}," ValueError(",[75,42545,42546],{"class":134},"\"axis must be one of {'x','y','z'}\"",[75,42548,318],{"class":85},[75,42550,42551],{"class":77,"line":15080},[75,42552,112],{"emptyLinePlaceholder":111},[75,42554,42555],{"class":77,"line":15085},[75,42556,112],{"emptyLinePlaceholder":111},[75,42558,42559,42561,42564,42566,42569,42571,42573,42575,42577,42579,42581],{"class":77,"line":15090},[75,42560,14380],{"class":81},[75,42562,42563],{"class":193}," rotation_matrix_axis",[75,42565,197],{"class":85},[75,42567,42568],{"class":14388},"axis_vec",[75,42570,42071],{"class":85},[75,42572,14150],{"class":141},[75,42574,206],{"class":85},[75,42576,14150],{"class":141},[75,42578,206],{"class":85},[75,42580,14150],{"class":141},[75,42582,18742],{"class":85},[75,42584,42585,42588,42590,42592],{"class":77,"line":15096},[75,42586,42587],{"class":14388},"                         angle",[75,42589,14391],{"class":85},[75,42591,14150],{"class":141},[75,42593,42347],{"class":85},[75,42595,42596],{"class":77,"line":15101},[75,42597,42598],{"class":134},"    \"\"\"Rodrigues matrix for arbitrary axis vector and angle (radians).\"\"\"\n",[75,42600,42601,42604,42606,42608,42610,42613,42615],{"class":77,"line":15151},[75,42602,42603],{"class":85},"    v ",[75,42605,142],{"class":141},[75,42607,190],{"class":85},[75,42609,42229],{"class":193},[75,42611,42612],{"class":85},"(axis_vec, ",[75,42614,14150],{"class":141},[75,42616,318],{"class":85},[75,42618,42619,42621,42623,42625,42627],{"class":77,"line":15160},[75,42620,42253],{"class":85},[75,42622,142],{"class":141},[75,42624,19086],{"class":85},[75,42626,42260],{"class":193},[75,42628,42629],{"class":85},"(v)\n",[75,42631,42632,42634,42636,42638,42640],{"class":77,"line":15206},[75,42633,14433],{"class":81},[75,42635,42291],{"class":85},[75,42637,42294],{"class":141},[75,42639,10251],{"class":145},[75,42641,5800],{"class":85},[75,42643,42644,42647,42649,42652],{"class":77,"line":15235},[75,42645,42646],{"class":81},"        raise",[75,42648,42543],{"class":85},[75,42650,42651],{"class":134},"\"axis_vec must be nonzero\"",[75,42653,318],{"class":85},[75,42655,42656,42659,42661,42664,42666],{"class":77,"line":15288},[75,42657,42658],{"class":85},"    k ",[75,42660,142],{"class":141},[75,42662,42663],{"class":85}," v ",[75,42665,2810],{"class":141},[75,42667,42306],{"class":85},[75,42669,42670,42673,42675,42677,42679,42681,42683,42685,42687,42690,42692,42695,42697,42700,42702,42704,42706,42708,42710,42712,42714,42717,42719,42721,42723,42725,42727,42729,42731,42733,42735],{"class":77,"line":15294},[75,42671,42672],{"class":85},"    K ",[75,42674,142],{"class":141},[75,42676,190],{"class":85},[75,42678,12049],{"class":193},[75,42680,18733],{"class":85},[75,42682,14324],{"class":145},[75,42684,206],{"class":85},[75,42686,200],{"class":141},[75,42688,42689],{"class":85},"k[",[75,42691,270],{"class":145},[75,42693,42694],{"class":85},"], k[",[75,42696,7477],{"class":145},[75,42698,42699],{"class":85},"]], [k[",[75,42701,270],{"class":145},[75,42703,12097],{"class":85},[75,42705,14324],{"class":145},[75,42707,206],{"class":85},[75,42709,200],{"class":141},[75,42711,42689],{"class":85},[75,42713,5186],{"class":145},[75,42715,42716],{"class":85},"]], [",[75,42718,200],{"class":141},[75,42720,42689],{"class":85},[75,42722,7477],{"class":145},[75,42724,42694],{"class":85},[75,42726,5186],{"class":145},[75,42728,12097],{"class":85},[75,42730,14324],{"class":145},[75,42732,42531],{"class":85},[75,42734,14150],{"class":141},[75,42736,318],{"class":85},[75,42738,42739,42742,42744,42746,42748,42750,42752],{"class":77,"line":15311},[75,42740,42741],{"class":85},"    I ",[75,42743,142],{"class":141},[75,42745,190],{"class":85},[75,42747,18713],{"class":193},[75,42749,197],{"class":85},[75,42751,3101],{"class":145},[75,42753,318],{"class":85},[75,42755,42756,42758,42760,42762,42764,42766,42768],{"class":77,"line":15316},[75,42757,42357],{"class":85},[75,42759,142],{"class":141},[75,42761,190],{"class":85},[75,42763,2845],{"class":193},[75,42765,42366],{"class":85},[75,42767,1179],{"class":193},[75,42769,42371],{"class":85},[75,42771,42772,42774,42777,42779,42782,42784,42786,42788,42790,42793,42795,42797,42800,42803,42805,42808,42810],{"class":77,"line":15363},[75,42773,14467],{"class":81},[75,42775,42776],{"class":85}," c ",[75,42778,500],{"class":141},[75,42780,42781],{"class":85}," I ",[75,42783,3251],{"class":141},[75,42785,267],{"class":85},[75,42787,7477],{"class":145},[75,42789,14186],{"class":141},[75,42791,42792],{"class":85}," c) ",[75,42794,500],{"class":141},[75,42796,190],{"class":85},[75,42798,42799],{"class":193},"outer",[75,42801,42802],{"class":85},"(k, k) ",[75,42804,3251],{"class":141},[75,42806,42807],{"class":85}," s ",[75,42809,500],{"class":141},[75,42811,42812],{"class":85}," K\n",[75,42814,42815],{"class":77,"line":15371},[75,42816,112],{"emptyLinePlaceholder":111},[75,42818,42819],{"class":77,"line":15419},[75,42820,112],{"emptyLinePlaceholder":111},[75,42822,42823],{"class":77,"line":15442},[75,42824,42825],{"class":118},"# ------------------------------ Wigner on S^2 ---------------------------------\n",[75,42827,42828,42830,42833,42835,42838,42840,42842,42844,42847,42849,42851],{"class":77,"line":15482},[75,42829,14380],{"class":81},[75,42831,42832],{"class":193}," n_grid",[75,42834,197],{"class":85},[75,42836,42837],{"class":14388},"n_theta",[75,42839,14391],{"class":85},[75,42841,14299],{"class":141},[75,42843,206],{"class":85},[75,42845,42846],{"class":14388},"n_phi",[75,42848,14391],{"class":85},[75,42850,14299],{"class":141},[75,42852,42347],{"class":85},[75,42854,42855],{"class":77,"line":15520},[75,42856,42857],{"class":134},"    \"\"\"Return unit-vector grid n(θ,φ) stacked as (3, n_theta, n_phi).\"\"\"\n",[75,42859,42860,42863,42865,42867,42869,42871,42873],{"class":77,"line":15525},[75,42861,42862],{"class":85},"    theta ",[75,42864,142],{"class":141},[75,42866,190],{"class":85},[75,42868,194],{"class":193},[75,42870,197],{"class":85},[75,42872,14324],{"class":145},[75,42874,42875],{"class":85},", np.pi, n_theta)\n",[75,42877,42878,42881,42883,42885,42887,42889,42891,42893,42895,42897],{"class":77,"line":15540},[75,42879,42880],{"class":85},"    phi ",[75,42882,142],{"class":141},[75,42884,190],{"class":85},[75,42886,194],{"class":193},[75,42888,197],{"class":85},[75,42890,14324],{"class":145},[75,42892,206],{"class":85},[75,42894,14678],{"class":145},[75,42896,273],{"class":141},[75,42898,42899],{"class":85}," np.pi, n_phi)\n",[75,42901,42902,42905,42907,42909,42912,42915,42918,42920,42923],{"class":77,"line":15545},[75,42903,42904],{"class":85},"    th, ph ",[75,42906,142],{"class":141},[75,42908,190],{"class":85},[75,42910,42911],{"class":193},"meshgrid",[75,42913,42914],{"class":85},"(theta, phi, ",[75,42916,42917],{"class":309},"indexing",[75,42919,142],{"class":141},[75,42921,42922],{"class":134},"\"ij\"",[75,42924,318],{"class":85},[75,42926,42927,42930,42932,42934,42936,42939,42941,42943,42945],{"class":77,"line":15551},[75,42928,42929],{"class":85},"    nx ",[75,42931,142],{"class":141},[75,42933,190],{"class":85},[75,42935,1179],{"class":193},[75,42937,42938],{"class":85},"(th) ",[75,42940,500],{"class":141},[75,42942,190],{"class":85},[75,42944,2845],{"class":193},[75,42946,42947],{"class":85},"(ph)\n",[75,42949,42950,42953,42955,42957,42959,42961,42963,42965,42967],{"class":77,"line":15592},[75,42951,42952],{"class":85},"    ny ",[75,42954,142],{"class":141},[75,42956,190],{"class":85},[75,42958,1179],{"class":193},[75,42960,42938],{"class":85},[75,42962,500],{"class":141},[75,42964,190],{"class":85},[75,42966,1179],{"class":193},[75,42968,42947],{"class":85},[75,42970,42971,42974,42976,42978,42980],{"class":77,"line":15608},[75,42972,42973],{"class":85},"    nz ",[75,42975,142],{"class":141},[75,42977,190],{"class":85},[75,42979,2845],{"class":193},[75,42981,42982],{"class":85},"(th)\n",[75,42984,42985,42987,42989,42992,42995,42997,42999,43001],{"class":77,"line":15624},[75,42986,14467],{"class":81},[75,42988,190],{"class":85},[75,42990,42991],{"class":193},"stack",[75,42993,42994],{"class":85},"((nx, ny, nz), ",[75,42996,42331],{"class":309},[75,42998,142],{"class":141},[75,43000,5186],{"class":145},[75,43002,318],{"class":85},[75,43004,43005],{"class":77,"line":15652},[75,43006,112],{"emptyLinePlaceholder":111},[75,43008,43009],{"class":77,"line":15657},[75,43010,112],{"emptyLinePlaceholder":111},[75,43012,43013,43015,43018,43020,43023,43025,43028],{"class":77,"line":15662},[75,43014,14380],{"class":81},[75,43016,43017],{"class":193}," spin_wigner_qubit",[75,43019,197],{"class":85},[75,43021,43022],{"class":14388},"r_vec",[75,43024,19065],{"class":85},[75,43026,43027],{"class":14388},"ngrid",[75,43029,19076],{"class":85},[75,43031,43032],{"class":77,"line":15668},[75,43033,43034],{"class":134},"    \"\"\"Spin SW Wigner for a qubit: W = 1\u002F2 + (√3\u002F2) * r·n on the (θ,φ) grid.\"\"\"\n",[75,43036,43037,43039,43041,43043,43045,43047,43049,43052,43054,43056,43058,43060,43062,43065,43068,43070,43072,43074,43076,43078,43080,43082,43084,43087,43090,43092,43094,43096,43098],{"class":77,"line":15673},[75,43038,14467],{"class":81},[75,43040,15214],{"class":145},[75,43042,11902],{"class":141},[75,43044,19174],{"class":85},[75,43046,14668],{"class":193},[75,43048,197],{"class":85},[75,43050,43051],{"class":145},"3.0",[75,43053,14860],{"class":85},[75,43055,2810],{"class":141},[75,43057,15263],{"class":145},[75,43059,14860],{"class":85},[75,43061,500],{"class":141},[75,43063,43064],{"class":85}," (r_vec.",[75,43066,43067],{"class":193},"reshape",[75,43069,197],{"class":85},[75,43071,3101],{"class":145},[75,43073,206],{"class":85},[75,43075,7477],{"class":145},[75,43077,206],{"class":85},[75,43079,7477],{"class":145},[75,43081,14860],{"class":85},[75,43083,500],{"class":141},[75,43085,43086],{"class":85}," ngrid).",[75,43088,43089],{"class":193},"sum",[75,43091,197],{"class":85},[75,43093,42331],{"class":309},[75,43095,142],{"class":141},[75,43097,5186],{"class":145},[75,43099,318],{"class":85},[75,43101,43102],{"class":77,"line":15698},[75,43103,112],{"emptyLinePlaceholder":111},[75,43105,43106],{"class":77,"line":15723},[75,43107,112],{"emptyLinePlaceholder":111},[75,43109,43110],{"class":77,"line":15728},[75,43111,43112],{"class":118},"# ----------------------- In-axes title & info placement -----------------------\n",[75,43114,43115,43117,43120,43122,43125,43127,43129,43131,43133,43135,43137,43139,43142],{"class":77,"line":15762},[75,43116,14380],{"class":81},[75,43118,43119],{"class":193}," _info_xy",[75,43121,197],{"class":85},[75,43123,43124],{"class":14388},"loc",[75,43126,14391],{"class":85},[75,43128,41840],{"class":141},[75,43130,14807],{"class":85},[75,43132,14150],{"class":141},[75,43134,206],{"class":85},[75,43136,14150],{"class":141},[75,43138,206],{"class":85},[75,43140,43141],{"class":141},"dict",[75,43143,14816],{"class":85},[75,43145,43146],{"class":77,"line":15792},[75,43147,43148],{"class":134},"    \"\"\"Map location code to (x, y, kwargs) in axes-fraction coordinates.\"\"\"\n",[75,43150,43151,43154,43156,43159,43161],{"class":77,"line":15829},[75,43152,43153],{"class":85},"    loc ",[75,43155,142],{"class":141},[75,43157,43158],{"class":85}," loc.",[75,43160,42241],{"class":193},[75,43162,389],{"class":85},[75,43164,43165,43167,43170,43172,43175],{"class":77,"line":15834},[75,43166,14433],{"class":81},[75,43168,43169],{"class":85}," loc ",[75,43171,42294],{"class":141},[75,43173,43174],{"class":134}," \"ul\"",[75,43176,5800],{"class":85},[75,43178,43179,43181,43184,43186,43189,43191,43193,43195,43198,43200,43203,43205,43208,43210,43213],{"class":77,"line":15840},[75,43180,14450],{"class":81},[75,43182,43183],{"class":145}," 0.015",[75,43185,206],{"class":85},[75,43187,43188],{"class":145},"0.985",[75,43190,206],{"class":85},[75,43192,43141],{"class":141},[75,43194,197],{"class":85},[75,43196,43197],{"class":309},"ha",[75,43199,142],{"class":141},[75,43201,43202],{"class":134},"\"left\"",[75,43204,206],{"class":85},[75,43206,43207],{"class":309},"va",[75,43209,142],{"class":141},[75,43211,43212],{"class":134},"\"top\"",[75,43214,318],{"class":85},[75,43216,43217,43219,43221,43223,43226],{"class":77,"line":15873},[75,43218,14433],{"class":81},[75,43220,43169],{"class":85},[75,43222,42294],{"class":141},[75,43224,43225],{"class":134}," \"ur\"",[75,43227,5800],{"class":85},[75,43229,43230,43232,43235,43237,43239,43241,43243,43245,43247,43249,43252,43254,43256,43258,43260],{"class":77,"line":15931},[75,43231,14450],{"class":81},[75,43233,43234],{"class":145}," 0.985",[75,43236,206],{"class":85},[75,43238,43188],{"class":145},[75,43240,206],{"class":85},[75,43242,43141],{"class":141},[75,43244,197],{"class":85},[75,43246,43197],{"class":309},[75,43248,142],{"class":141},[75,43250,43251],{"class":134},"\"right\"",[75,43253,206],{"class":85},[75,43255,43207],{"class":309},[75,43257,142],{"class":141},[75,43259,43212],{"class":134},[75,43261,318],{"class":85},[75,43263,43264,43266,43268,43270,43273],{"class":77,"line":15953},[75,43265,14433],{"class":81},[75,43267,43169],{"class":85},[75,43269,42294],{"class":141},[75,43271,43272],{"class":134}," \"ll\"",[75,43274,5800],{"class":85},[75,43276,43277,43279,43281,43283,43286,43288,43290,43292,43294,43296,43298,43300,43302,43304,43307],{"class":77,"line":15968},[75,43278,14450],{"class":81},[75,43280,43183],{"class":145},[75,43282,206],{"class":85},[75,43284,43285],{"class":145},"0.015",[75,43287,206],{"class":85},[75,43289,43141],{"class":141},[75,43291,197],{"class":85},[75,43293,43197],{"class":309},[75,43295,142],{"class":141},[75,43297,43202],{"class":134},[75,43299,206],{"class":85},[75,43301,43207],{"class":309},[75,43303,142],{"class":141},[75,43305,43306],{"class":134},"\"bottom\"",[75,43308,318],{"class":85},[75,43310,43311,43313,43315,43317,43320],{"class":77,"line":15996},[75,43312,14433],{"class":81},[75,43314,43169],{"class":85},[75,43316,42294],{"class":141},[75,43318,43319],{"class":134}," \"lr\"",[75,43321,5800],{"class":85},[75,43323,43324,43326,43328,43330,43332,43334,43336,43338,43340,43342,43344,43346,43348,43350,43352],{"class":77,"line":16026},[75,43325,14450],{"class":81},[75,43327,43234],{"class":145},[75,43329,206],{"class":85},[75,43331,43285],{"class":145},[75,43333,206],{"class":85},[75,43335,43141],{"class":141},[75,43337,197],{"class":85},[75,43339,43197],{"class":309},[75,43341,142],{"class":141},[75,43343,43251],{"class":134},[75,43345,206],{"class":85},[75,43347,43207],{"class":309},[75,43349,142],{"class":141},[75,43351,43306],{"class":134},[75,43353,318],{"class":85},[75,43355,43356,43358,43360,43362,43364,43366,43368,43370,43372,43374,43376,43378,43380,43382,43384],{"class":77,"line":16047},[75,43357,14467],{"class":81},[75,43359,43183],{"class":145},[75,43361,206],{"class":85},[75,43363,43188],{"class":145},[75,43365,206],{"class":85},[75,43367,43141],{"class":141},[75,43369,197],{"class":85},[75,43371,43197],{"class":309},[75,43373,142],{"class":141},[75,43375,43202],{"class":134},[75,43377,206],{"class":85},[75,43379,43207],{"class":309},[75,43381,142],{"class":141},[75,43383,43212],{"class":134},[75,43385,318],{"class":85},[75,43387,43388],{"class":77,"line":16056},[75,43389,112],{"emptyLinePlaceholder":111},[75,43391,43392],{"class":77,"line":16065},[75,43393,112],{"emptyLinePlaceholder":111},[75,43395,43396,43398,43401,43403,43405],{"class":77,"line":16070},[75,43397,14380],{"class":81},[75,43399,43400],{"class":193}," add_info_box",[75,43402,197],{"class":85},[75,43404,15602],{"class":14388},[75,43406,43407],{"class":85},": plt.Axes,\n",[75,43409,43410,43413,43415,43417],{"class":77,"line":16076},[75,43411,43412],{"class":14388},"                 text",[75,43414,14391],{"class":85},[75,43416,41840],{"class":141},[75,43418,41629],{"class":85},[75,43420,43421,43424,43426,43428,43430,43432],{"class":77,"line":16107},[75,43422,43423],{"class":14388},"                 loc",[75,43425,14391],{"class":85},[75,43427,41840],{"class":141},[75,43429,14153],{"class":141},[75,43431,43174],{"class":134},[75,43433,41629],{"class":85},[75,43435,43436,43439,43441,43443,43445,43448],{"class":77,"line":16153},[75,43437,43438],{"class":14388},"                 fontsize",[75,43440,14391],{"class":85},[75,43442,14299],{"class":141},[75,43444,14153],{"class":141},[75,43446,43447],{"class":145}," 9",[75,43449,41629],{"class":85},[75,43451,43452,43455,43457,43459,43461,43464],{"class":77,"line":16174},[75,43453,43454],{"class":14388},"                 wrap",[75,43456,14391],{"class":85},[75,43458,14299],{"class":141},[75,43460,14153],{"class":141},[75,43462,43463],{"class":145}," 42",[75,43465,43466],{"class":85},") -> plt.Text:\n",[75,43468,43469],{"class":77,"line":16185},[75,43470,43471],{"class":134},"    \"\"\"Create a wrapped, padded text box inside the axes (never clips).\"\"\"\n",[75,43473,43474,43477,43479,43481],{"class":77,"line":16208},[75,43475,43476],{"class":85},"    x, y, kw ",[75,43478,142],{"class":141},[75,43480,43119],{"class":193},[75,43482,43483],{"class":85},"(loc)\n",[75,43485,43486,43488,43491,43493],{"class":77,"line":16235},[75,43487,14467],{"class":81},[75,43489,43490],{"class":85}," ax.",[75,43492,34131],{"class":193},[75,43494,43495],{"class":85},"(\n",[75,43497,43498,43501,43504],{"class":77,"line":16256},[75,43499,43500],{"class":85},"        x, y, textwrap.",[75,43502,43503],{"class":193},"fill",[75,43505,43506],{"class":85},"(text, wrap),\n",[75,43508,43509,43512,43514,43517,43520,43522,43525,43528,43530,43532],{"class":77,"line":16265},[75,43510,43511],{"class":309},"        transform",[75,43513,142],{"class":141},[75,43515,43516],{"class":85},"ax.transAxes, ",[75,43518,43519],{"class":309},"fontsize",[75,43521,142],{"class":141},[75,43523,43524],{"class":85},"fontsize, ",[75,43526,43527],{"class":309},"zorder",[75,43529,142],{"class":141},[75,43531,203],{"class":145},[75,43533,41629],{"class":85},[75,43535,43536,43539,43542,43544,43547,43549,43552,43554,43557,43559,43562,43564,43566,43568,43571,43573,43576,43578,43581],{"class":77,"line":16274},[75,43537,43538],{"class":309},"        bbox",[75,43540,43541],{"class":141},"=dict",[75,43543,197],{"class":85},[75,43545,43546],{"class":309},"boxstyle",[75,43548,142],{"class":141},[75,43550,43551],{"class":134},"\"round,pad=0.3\"",[75,43553,206],{"class":85},[75,43555,43556],{"class":309},"fc",[75,43558,142],{"class":141},[75,43560,43561],{"class":134},"\"white\"",[75,43563,206],{"class":85},[75,43565,2953],{"class":309},[75,43567,142],{"class":141},[75,43569,43570],{"class":145},"0.75",[75,43572,206],{"class":85},[75,43574,43575],{"class":309},"ec",[75,43577,142],{"class":141},[75,43579,43580],{"class":134},"\"none\"",[75,43582,3003],{"class":85},[75,43584,43585],{"class":77,"line":16279},[75,43586,43587],{"class":85},"        **kw,\n",[75,43589,43590],{"class":77,"line":16285},[75,43591,15291],{"class":85},[75,43593,43594],{"class":77,"line":16311},[75,43595,112],{"emptyLinePlaceholder":111},[75,43597,43598],{"class":77,"line":16329},[75,43599,112],{"emptyLinePlaceholder":111},[75,43601,43602,43604,43607,43609,43611,43614,43616,43618,43620],{"class":77,"line":16347},[75,43603,14380],{"class":81},[75,43605,43606],{"class":193}," add_title_inside",[75,43608,197],{"class":85},[75,43610,15602],{"class":14388},[75,43612,43613],{"class":85},": plt.Axes, ",[75,43615,34131],{"class":14388},[75,43617,14391],{"class":85},[75,43619,41840],{"class":141},[75,43621,43466],{"class":85},[75,43623,43624],{"class":77,"line":16360},[75,43625,43626],{"class":134},"    \"\"\"Draw a title inside the Axes at the very top center (never clips).\"\"\"\n",[75,43628,43629,43631,43633,43635],{"class":77,"line":16365},[75,43630,14467],{"class":81},[75,43632,43490],{"class":85},[75,43634,34131],{"class":193},[75,43636,43495],{"class":85},[75,43638,43639,43642,43644,43646],{"class":77,"line":16396},[75,43640,43641],{"class":145},"        0.5",[75,43643,206],{"class":85},[75,43645,43188],{"class":145},[75,43647,43648],{"class":85},", text,\n",[75,43650,43651,43653,43655,43657,43659,43661,43664,43666,43668,43670,43672],{"class":77,"line":16441},[75,43652,43511],{"class":309},[75,43654,142],{"class":141},[75,43656,43516],{"class":85},[75,43658,43197],{"class":309},[75,43660,142],{"class":141},[75,43662,43663],{"class":134},"\"center\"",[75,43665,206],{"class":85},[75,43667,43207],{"class":309},[75,43669,142],{"class":141},[75,43671,43212],{"class":134},[75,43673,41629],{"class":85},[75,43675,43676,43679,43681,43684],{"class":77,"line":16490},[75,43677,43678],{"class":309},"        fontsize",[75,43680,142],{"class":141},[75,43682,43683],{"class":145},"11",[75,43685,41629],{"class":85},[75,43687,43688,43690,43692,43694,43696,43698,43700,43702,43704,43706,43708,43710,43712,43714,43716,43718,43720,43722,43724],{"class":77,"line":16522},[75,43689,43538],{"class":309},[75,43691,43541],{"class":141},[75,43693,197],{"class":85},[75,43695,43546],{"class":309},[75,43697,142],{"class":141},[75,43699,43551],{"class":134},[75,43701,206],{"class":85},[75,43703,43556],{"class":309},[75,43705,142],{"class":141},[75,43707,43561],{"class":134},[75,43709,206],{"class":85},[75,43711,2953],{"class":309},[75,43713,142],{"class":141},[75,43715,43570],{"class":145},[75,43717,206],{"class":85},[75,43719,43575],{"class":309},[75,43721,142],{"class":141},[75,43723,43580],{"class":134},[75,43725,3003],{"class":85},[75,43727,43728,43731,43733,43735],{"class":77,"line":16551},[75,43729,43730],{"class":309},"        zorder",[75,43732,142],{"class":141},[75,43734,882],{"class":145},[75,43736,41629],{"class":85},[75,43738,43739],{"class":77,"line":16574},[75,43740,15291],{"class":85},[65,43742,43744],{"className":67,"code":43743,"language":69,"meta":70,"style":70},"# @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",[72,43745,43746,43751,43761,43765,43782,43790,43801,43812,43823,43834,43849,43860,43871,43882,43890,43895,43912,43924,43951,43978,44003,44007,44035,44069,44095,44137,44152,44156,44169,44182,44213,44217,44247,44257,44261,44280,44309,44331,44347,44403,44410,44414,44453,44481,44515,44533,44544,44551,44555,44559,44564,44579,44600,44612,44628,44653,44664,44676,44693,44724,44728,44732,44744,44754,44766,44797,44807,44815,44820,44825,44830,44835,44839,44843,44848,44860,44881],{"__ignoreMap":70},[75,43747,43748],{"class":77,"line":78},[75,43749,43750],{"class":118},"# @title Cell 3 — Pauli X\u002FY\u002FZ Wigner animations (titles *inside* the plot)\n",[75,43752,43753,43755,43757,43759],{"class":77,"line":95},[75,43754,2700],{"class":81},[75,43756,41661],{"class":85},[75,43758,82],{"class":81},[75,43760,41666],{"class":85},[75,43762,43763],{"class":77,"line":108},[75,43764,112],{"emptyLinePlaceholder":111},[75,43766,43767,43769,43772,43774,43776,43778,43780],{"class":77,"line":115},[75,43768,14380],{"class":81},[75,43770,43771],{"class":193}," animate_wigner_pauli",[75,43773,197],{"class":85},[75,43775,42331],{"class":14388},[75,43777,14391],{"class":85},[75,43779,41840],{"class":141},[75,43781,41629],{"class":85},[75,43783,43784,43787],{"class":77,"line":122},[75,43785,43786],{"class":14388},"                         r0",[75,43788,43789],{"class":85},": np.ndarray,\n",[75,43791,43792,43795,43797,43799],{"class":77,"line":128},[75,43793,43794],{"class":14388},"                         frames",[75,43796,14391],{"class":85},[75,43798,14299],{"class":141},[75,43800,41629],{"class":85},[75,43802,43803,43806,43808,43810],{"class":77,"line":149},[75,43804,43805],{"class":14388},"                         n_theta",[75,43807,14391],{"class":85},[75,43809,14299],{"class":141},[75,43811,41629],{"class":85},[75,43813,43814,43817,43819,43821],{"class":77,"line":154},[75,43815,43816],{"class":14388},"                         n_phi",[75,43818,14391],{"class":85},[75,43820,14299],{"class":141},[75,43822,41629],{"class":85},[75,43824,43825,43828,43830,43832],{"class":77,"line":160},[75,43826,43827],{"class":14388},"                         fps",[75,43829,14391],{"class":85},[75,43831,14299],{"class":141},[75,43833,41629],{"class":85},[75,43835,43836,43839,43841,43843,43845,43847],{"class":77,"line":171},[75,43837,43838],{"class":14388},"                         figsize",[75,43840,42071],{"class":85},[75,43842,14150],{"class":141},[75,43844,206],{"class":85},[75,43846,14150],{"class":141},[75,43848,18742],{"class":85},[75,43850,43851,43854,43856,43858],{"class":77,"line":182},[75,43852,43853],{"class":14388},"                         info_loc",[75,43855,14391],{"class":85},[75,43857,41840],{"class":141},[75,43859,41629],{"class":85},[75,43861,43862,43865,43867,43869],{"class":77,"line":222},[75,43863,43864],{"class":14388},"                         info_fontsize",[75,43866,14391],{"class":85},[75,43868,14299],{"class":141},[75,43870,41629],{"class":85},[75,43872,43873,43876,43878,43880],{"class":77,"line":227},[75,43874,43875],{"class":14388},"                         wrap_chars",[75,43877,14391],{"class":85},[75,43879,14299],{"class":141},[75,43881,41629],{"class":85},[75,43883,43884,43887],{"class":77,"line":233},[75,43885,43886],{"class":14388},"                         out_path",[75,43888,43889],{"class":85},": Path) -> Path:\n",[75,43891,43892],{"class":77,"line":286},[75,43893,43894],{"class":134},"    \"\"\"Animate Wigner under a Pauli rotation U = exp(-i σ_axis * π * t \u002F 2).\"\"\"\n",[75,43896,43897,43900,43903,43905,43907,43909],{"class":77,"line":291},[75,43898,43899],{"class":81},"    assert",[75,43901,43902],{"class":85}," frames ",[75,43904,18355],{"class":141},[75,43906,261],{"class":145},[75,43908,206],{"class":85},[75,43910,43911],{"class":134},"\"frames must be ≥ 2\"\n",[75,43913,43914,43917,43919,43921],{"class":77,"line":297},[75,43915,43916],{"class":85},"    ngrid ",[75,43918,142],{"class":141},[75,43920,42832],{"class":193},[75,43922,43923],{"class":85},"(n_theta, n_phi)\n",[75,43925,43926,43929,43931,43933,43935,43937,43939,43941,43943,43945,43947,43949],{"class":77,"line":321},[75,43927,43928],{"class":85},"    w_min ",[75,43930,142],{"class":141},[75,43932,15214],{"class":145},[75,43934,14186],{"class":141},[75,43936,19174],{"class":85},[75,43938,14668],{"class":193},[75,43940,197],{"class":85},[75,43942,43051],{"class":145},[75,43944,14860],{"class":85},[75,43946,2810],{"class":141},[75,43948,15263],{"class":145},[75,43950,318],{"class":85},[75,43952,43953,43956,43958,43960,43962,43964,43966,43968,43970,43972,43974,43976],{"class":77,"line":336},[75,43954,43955],{"class":85},"    w_max ",[75,43957,142],{"class":141},[75,43959,15214],{"class":145},[75,43961,11902],{"class":141},[75,43963,19174],{"class":85},[75,43965,14668],{"class":193},[75,43967,197],{"class":85},[75,43969,43051],{"class":145},[75,43971,14860],{"class":85},[75,43973,2810],{"class":141},[75,43975,15263],{"class":145},[75,43977,318],{"class":85},[75,43979,43980,43983,43985,43987,43989,43991,43993,43995,43998,44000],{"class":77,"line":351},[75,43981,43982],{"class":85},"    extent ",[75,43984,142],{"class":141},[75,43986,267],{"class":85},[75,43988,14324],{"class":145},[75,43990,206],{"class":85},[75,43992,14678],{"class":145},[75,43994,273],{"class":141},[75,43996,43997],{"class":85}," np.pi, ",[75,43999,14324],{"class":145},[75,44001,44002],{"class":85},", np.pi)\n",[75,44004,44005],{"class":77,"line":366},[75,44006,112],{"emptyLinePlaceholder":111},[75,44008,44009,44012,44014,44016,44018,44020,44022,44024,44027,44029,44031,44033],{"class":77,"line":381},[75,44010,44011],{"class":85},"    fig, ax ",[75,44013,142],{"class":141},[75,44015,3093],{"class":85},[75,44017,3096],{"class":193},[75,44019,197],{"class":85},[75,44021,870],{"class":309},[75,44023,142],{"class":141},[75,44025,44026],{"class":85},"figsize, 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wrap",[75,44253,142],{"class":141},[75,44255,44256],{"class":85},"wrap_chars)\n",[75,44258,44259],{"class":77,"line":3188},[75,44260,112],{"emptyLinePlaceholder":111},[75,44262,44263,44266,44269,44271,44274,44276,44278],{"class":77,"line":3203},[75,44264,44265],{"class":81},"    def",[75,44267,44268],{"class":193}," update",[75,44270,197],{"class":85},[75,44272,44273],{"class":14388},"k",[75,44275,14391],{"class":85},[75,44277,14299],{"class":141},[75,44279,15605],{"class":85},[75,44281,44282,44285,44287,44290,44292,44295,44297,44299,44301,44303,44306],{"class":77,"line":3218},[75,44283,44284],{"class":85},"        ang ",[75,44286,142],{"class":141},[75,44288,44289],{"class":85}," (k ",[75,44291,2810],{"class":141},[75,44293,44294],{"class":85}," (frames ",[75,44296,200],{"class":141},[75,44298,1437],{"class":145},[75,44300,14936],{"class":85},[75,44302,500],{"class":141},[75,44304,44305],{"class":85}," np.pi  ",[75,44307,44308],{"class":118},"# 0 → π\n",[75,44310,44311,44314,44316,44318,44321,44323,44326,44328],{"class":77,"line":3224},[75,44312,44313],{"class":85},"        r ",[75,44315,142],{"class":141},[75,44317,42326],{"class":193},[75,44319,44320],{"class":85},"(axis.",[75,44322,42241],{"class":193},[75,44324,44325],{"class":85},"(), ang) ",[75,44327,19121],{"class":141},[75,44329,44330],{"class":85}," r0\n",[75,44332,44333,44336,44339,44341,44344],{"class":77,"line":3229},[75,44334,44335],{"class":85},"        im.",[75,44337,44338],{"class":193},"set_data",[75,44340,197],{"class":85},[75,44342,44343],{"class":193},"spin_wigner_qubit",[75,44345,44346],{"class":85},"(r, ngrid))\n",[75,44348,44349,44352,44355,44357,44359,44362,44364,44367,44370,44372,44375,44378,44381,44383,44385,44387,44390,44392,44394,44397,44399,44401],{"class":77,"line":3235},[75,44350,44351],{"class":85},"        info.",[75,44353,44354],{"class":193},"set_text",[75,44356,197],{"class":85},[75,44358,5894],{"class":81},[75,44360,44361],{"class":134},"\"θ = 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out_path.parent.",[75,44459,44460],{"class":193},"mkdir",[75,44462,197],{"class":85},[75,44464,44465],{"class":309},"parents",[75,44467,142],{"class":141},[75,44469,10504],{"class":145},[75,44471,206],{"class":85},[75,44473,44474],{"class":309},"exist_ok",[75,44476,142],{"class":141},[75,44478,10504],{"class":145},[75,44480,318],{"class":85},[75,44482,44483,44486,44489,44492,44495,44497,44500,44502,44505,44507,44510,44512],{"class":77,"line":4431},[75,44484,44485],{"class":85},"    anim.",[75,44487,44488],{"class":193},"save",[75,44490,44491],{"class":85},"(out_path.",[75,44493,44494],{"class":193},"as_posix",[75,44496,16413],{"class":85},[75,44498,44499],{"class":309},"writer",[75,44501,142],{"class":141},[75,44503,44504],{"class":193},"PillowWriter",[75,44506,197],{"class":85},[75,44508,44509],{"class":309},"fps",[75,44511,142],{"class":141},[75,44513,44514],{"class":85},"fps),\n",[75,44516,44517,44520,44522,44524,44527,44529,44531],{"class":77,"line":4454},[75,44518,44519],{"class":309},"              savefig_kwargs",[75,44521,43541],{"class":141},[75,44523,197],{"class":85},[75,44525,44526],{"class":309},"facecolor",[75,44528,142],{"class":141},[75,44530,43561],{"class":134},[75,44532,283],{"class":85},[75,44534,44535,44538,44541],{"class":77,"line":4463},[75,44536,44537],{"class":85},"    plt.",[75,44539,44540],{"class":193},"close",[75,44542,44543],{"class":85},"(fig)\n",[75,44545,44546,44548],{"class":77,"line":4468},[75,44547,14467],{"class":81},[75,44549,44550],{"class":85}," out_path\n",[75,44552,44553],{"class":77,"line":4477},[75,44554,112],{"emptyLinePlaceholder":111},[75,44556,44557],{"class":77,"line":5105},[75,44558,112],{"emptyLinePlaceholder":111},[75,44560,44561],{"class":77,"line":5114},[75,44562,44563],{"class":118},"# ---------------------------- Run X, Y, Z animations --------------------------\n",[75,44565,44566,44569,44571,44573,44576],{"class":77,"line":10795},[75,44567,44568],{"class":85},"knobs ",[75,44570,142],{"class":141},[75,44572,41751],{"class":193},[75,44574,44575],{"class":85},"(  ",[75,44577,44578],{"class":118},"# tweak if desired\n",[75,44580,44581,44584,44586,44588,44590,44592,44594,44596,44598],{"class":77,"line":14778},[75,44582,44583],{"class":309},"    gate_axes",[75,44585,142],{"class":141},[75,44587,197],{"class":85},[75,44589,41854],{"class":134},[75,44591,206],{"class":85},[75,44593,41859],{"class":134},[75,44595,206],{"class":85},[75,44597,41864],{"class":134},[75,44599,3003],{"class":85},[75,44601,44602,44605,44607,44610],{"class":77,"line":14783},[75,44603,44604],{"class":309},"    initial_state",[75,44606,142],{"class":141},[75,44608,44609],{"class":134},"\"+z\"",[75,44611,41629],{"class":85},[75,44613,44614,44617,44619,44622,44625],{"class":77,"line":14819},[75,44615,44616],{"class":309},"    frames",[75,44618,142],{"class":141},[75,44620,44621],{"class":145},"24",[75,44623,44624],{"class":85},",                ",[75,44626,44627],{"class":118},"# raise to 48 for smoother motion\n",[75,44629,44630,44633,44635,44638,44640,44642,44644,44647,44650],{"class":77,"line":14825},[75,44631,44632],{"class":309},"    n_theta",[75,44634,142],{"class":141},[75,44636,44637],{"class":145},"80",[75,44639,206],{"class":85},[75,44641,42846],{"class":309},[75,44643,142],{"class":141},[75,44645,44646],{"class":145},"160",[75,44648,44649],{"class":85},",    ",[75,44651,44652],{"class":118},"# raise to 120×240 for more detail\n",[75,44654,44655,44658,44660,44662],{"class":77,"line":14838},[75,44656,44657],{"class":309},"    fps",[75,44659,142],{"class":141},[75,44661,752],{"class":145},[75,44663,41629],{"class":85},[75,44665,44666,44669,44671,44674],{"class":77,"line":14893},[75,44667,44668],{"class":309},"    out_dir",[75,44670,142],{"class":141},[75,44672,44673],{"class":134},"\"\u002Fcontent\"",[75,44675,41629],{"class":85},[75,44677,44678,44681,44683,44685,44687,44689,44691],{"class":77,"line":14951},[75,44679,44680],{"class":309},"    figsize",[75,44682,142],{"class":141},[75,44684,197],{"class":85},[75,44686,41606],{"class":145},[75,44688,206],{"class":85},[75,44690,41611],{"class":145},[75,44692,3003],{"class":85},[75,44694,44695,44698,44700,44703,44705,44708,44710,44712,44714,44717,44719,44722],{"class":77,"line":14959},[75,44696,44697],{"class":309},"    info_loc",[75,44699,142],{"class":141},[75,44701,44702],{"class":134},"\"ul\"",[75,44704,206],{"class":85},[75,44706,44707],{"class":309},"info_fontsize",[75,44709,142],{"class":141},[75,44711,3121],{"class":145},[75,44713,206],{"class":85},[75,44715,44716],{"class":309},"wrap_chars",[75,44718,142],{"class":141},[75,44720,44721],{"class":145},"42",[75,44723,41629],{"class":85},[75,44725,44726],{"class":77,"line":14964},[75,44727,318],{"class":85},[75,44729,44730],{"class":77,"line":14983},[75,44731,112],{"emptyLinePlaceholder":111},[75,44733,44734,44737,44739,44741],{"class":77,"line":14989},[75,44735,44736],{"class":85},"r0 ",[75,44738,142],{"class":141},[75,44740,42052],{"class":193},[75,44742,44743],{"class":85},"(knobs.initial_state, knobs.custom_r)\n",[75,44745,44746,44749,44751],{"class":77,"line":15012},[75,44747,44748],{"class":85},"outs ",[75,44750,142],{"class":141},[75,44752,44753],{"class":85}," []\n",[75,44755,44756,44758,44761,44763],{"class":77,"line":15017},[75,44757,3147],{"class":81},[75,44759,44760],{"class":85}," axname ",[75,44762,3153],{"class":81},[75,44764,44765],{"class":85}," knobs.gate_axes:\n",[75,44767,44768,44771,44773,44776,44779,44781,44784,44787,44789,44792,44794],{"class":77,"line":15025},[75,44769,44770],{"class":85},"    path ",[75,44772,142],{"class":141},[75,44774,44775],{"class":193}," Path",[75,44777,44778],{"class":85},"(knobs.out_dir) ",[75,44780,2810],{"class":141},[75,44782,44783],{"class":81}," f",[75,44785,44786],{"class":134},"\"wigner_pauli_",[75,44788,12219],{"class":145},[75,44790,44791],{"class":85},"axname",[75,44793,12224],{"class":145},[75,44795,44796],{"class":134},".gif\"\n",[75,44798,44799,44802,44805],{"class":77,"line":15038},[75,44800,44801],{"class":85},"    outs.",[75,44803,44804],{"class":193},"append",[75,44806,43495],{"class":85},[75,44808,44809,44812],{"class":77,"line":15051},[75,44810,44811],{"class":193},"        animate_wigner_pauli",[75,44813,44814],{"class":85},"(axname, r0, knobs.frames,\n",[75,44816,44817],{"class":77,"line":15058},[75,44818,44819],{"class":85},"                             knobs.n_theta, knobs.n_phi, knobs.fps,\n",[75,44821,44822],{"class":77,"line":15069},[75,44823,44824],{"class":85},"                             knobs.figsize, knobs.info_loc,\n",[75,44826,44827],{"class":77,"line":15080},[75,44828,44829],{"class":85},"                             knobs.info_fontsize, knobs.wrap_chars,\n",[75,44831,44832],{"class":77,"line":15085},[75,44833,44834],{"class":85},"                             path)\n",[75,44836,44837],{"class":77,"line":15090},[75,44838,15291],{"class":85},[75,44840,44841],{"class":77,"line":15096},[75,44842,112],{"emptyLinePlaceholder":111},[75,44844,44845],{"class":77,"line":15101},[75,44846,44847],{"class":118},"# Display inline\n",[75,44849,44850,44852,44855,44857],{"class":77,"line":15151},[75,44851,3147],{"class":81},[75,44853,44854],{"class":85}," p ",[75,44856,3153],{"class":81},[75,44858,44859],{"class":85}," outs:\n",[75,44861,44862,44865,44867,44870,44872,44875,44878],{"class":77,"line":15160},[75,44863,44864],{"class":193},"    display",[75,44866,197],{"class":85},[75,44868,44869],{"class":193},"Image",[75,44871,197],{"class":85},[75,44873,44874],{"class":309},"filename",[75,44876,44877],{"class":141},"=str",[75,44879,44880],{"class":85},"(p)))\n",[75,44882,44883,44886,44888,44891],{"class":77,"line":15206},[75,44884,44885],{"class":141},"    print",[75,44887,197],{"class":85},[75,44889,44890],{"class":134},"\"Saved:\"",[75,44892,44893],{"class":85},", p)\n",[65,44895,44898],{"className":44896,"code":44897,"language":34131,"meta":70},[41256],"Saved: \u002Fcontent\u002Fwigner_pauli_x.gif\n",[72,44899,44897],{"__ignoreMap":70},[65,44901,44904],{"className":44902,"code":44903,"language":34131,"meta":70},[41256],"Saved: \u002Fcontent\u002Fwigner_pauli_y.gif\n",[72,44905,44903],{"__ignoreMap":70},[65,44907,44910],{"className":44908,"code":44909,"language":34131,"meta":70},[41256],"Saved: \u002Fcontent\u002Fwigner_pauli_z.gif\n",[72,44911,44909],{"__ignoreMap":70},[391,44913],{"alt":393,"src":44914},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-01.webp",[391,44916],{"alt":668,"src":44917},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-02.webp",[391,44919],{"alt":1006,"src":44920},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-03.webp",[65,44922,44924],{"className":67,"code":44923,"language":69,"meta":70,"style":70},"# @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",[72,44925,44926,44931,44941,44945,44970,44981,44988,44999,45010,45021,45032,45047,45058,45069,45080,45087,45092,45106,45116,45142,45168,45190,45194,45220,45250,45272,45306,45318,45322,45334,45346,45357,45361,45387,45395,45399,45415,45443,45455,45468,45480,45488,45545,45581,45586,45592,45596,45628,45652,45678,45694,45702,45708,45712,45716,45721,45737,45744,45765,45777,45787,45801,45819,45829,45838,45854,45864,45874,45878,45882,45900],{"__ignoreMap":70},[75,44927,44928],{"class":77,"line":78},[75,44929,44930],{"class":118},"# @title Cell 4 — Custom axis–angle gate (example: 45° about y)\n",[75,44932,44933,44935,44937,44939],{"class":77,"line":95},[75,44934,2700],{"class":81},[75,44936,41661],{"class":85},[75,44938,82],{"class":81},[75,44940,41666],{"class":85},[75,44942,44943],{"class":77,"line":108},[75,44944,112],{"emptyLinePlaceholder":111},[75,44946,44947,44949,44952,44954,44956,44958,44960,44962,44964,44966,44968],{"class":77,"line":115},[75,44948,14380],{"class":81},[75,44950,44951],{"class":193}," animate_wigner_custom",[75,44953,197],{"class":85},[75,44955,42568],{"class":14388},[75,44957,42071],{"class":85},[75,44959,14150],{"class":141},[75,44961,206],{"class":85},[75,44963,14150],{"class":141},[75,44965,206],{"class":85},[75,44967,14150],{"class":141},[75,44969,18742],{"class":85},[75,44971,44972,44975,44977,44979],{"class":77,"line":122},[75,44973,44974],{"class":14388},"                          angle_deg",[75,44976,14391],{"class":85},[75,44978,14150],{"class":141},[75,44980,41629],{"class":85},[75,44982,44983,44986],{"class":77,"line":128},[75,44984,44985],{"class":14388},"                          r0",[75,44987,43789],{"class":85},[75,44989,44990,44993,44995,44997],{"class":77,"line":149},[75,44991,44992],{"class":14388},"                          frames",[75,44994,14391],{"class":85},[75,44996,14299],{"class":141},[75,44998,41629],{"class":85},[75,45000,45001,45004,45006,45008],{"class":77,"line":154},[75,45002,45003],{"class":14388},"                          n_theta",[75,45005,14391],{"class":85},[75,45007,14299],{"class":141},[75,45009,41629],{"class":85},[75,45011,45012,45015,45017,45019],{"class":77,"line":160},[75,45013,45014],{"class":14388},"                          n_phi",[75,45016,14391],{"class":85},[75,45018,14299],{"class":141},[75,45020,41629],{"class":85},[75,45022,45023,45026,45028,45030],{"class":77,"line":171},[75,45024,45025],{"class":14388},"                          fps",[75,45027,14391],{"class":85},[75,45029,14299],{"class":141},[75,45031,41629],{"class":85},[75,45033,45034,45037,45039,45041,45043,45045],{"class":77,"line":182},[75,45035,45036],{"class":14388},"                          figsize",[75,45038,42071],{"class":85},[75,45040,14150],{"class":141},[75,45042,206],{"class":85},[75,45044,14150],{"class":141},[75,45046,18742],{"class":85},[75,45048,45049,45052,45054,45056],{"class":77,"line":222},[75,45050,45051],{"class":14388},"                          info_loc",[75,45053,14391],{"class":85},[75,45055,41840],{"class":141},[75,45057,41629],{"class":85},[75,45059,45060,45063,45065,45067],{"class":77,"line":227},[75,45061,45062],{"class":14388},"                          info_fontsize",[75,45064,14391],{"class":85},[75,45066,14299],{"class":141},[75,45068,41629],{"class":85},[75,45070,45071,45074,45076,45078],{"class":77,"line":233},[75,45072,45073],{"class":14388},"                          wrap_chars",[75,45075,14391],{"class":85},[75,45077,14299],{"class":141},[75,45079,41629],{"class":85},[75,45081,45082,45085],{"class":77,"line":286},[75,45083,45084],{"class":14388},"                          out_path",[75,45086,43889],{"class":85},[75,45088,45089],{"class":77,"line":291},[75,45090,45091],{"class":134},"    \"\"\"Animate Wigner under an arbitrary axis–angle rotation 0 → angle_deg.\"\"\"\n",[75,45093,45094,45096,45098,45100,45102,45104],{"class":77,"line":297},[75,45095,43899],{"class":81},[75,45097,43902],{"class":85},[75,45099,18355],{"class":141},[75,45101,261],{"class":145},[75,45103,206],{"class":85},[75,45105,43911],{"class":134},[75,45107,45108,45110,45112,45114],{"class":77,"line":321},[75,45109,43916],{"class":85},[75,45111,142],{"class":141},[75,45113,42832],{"class":193},[75,45115,43923],{"class":85},[75,45117,45118,45120,45122,45124,45126,45128,45130,45132,45134,45136,45138,45140],{"class":77,"line":336},[75,45119,43928],{"class":85},[75,45121,142],{"class":141},[75,45123,15214],{"class":145},[75,45125,14186],{"class":141},[75,45127,19174],{"class":85},[75,45129,14668],{"class":193},[75,45131,197],{"class":85},[75,45133,43051],{"class":145},[75,45135,14860],{"class":85},[75,45137,2810],{"class":141},[75,45139,15263],{"class":145},[75,45141,318],{"class":85},[75,45143,45144,45146,45148,45150,45152,45154,45156,45158,45160,45162,45164,45166],{"class":77,"line":351},[75,45145,43955],{"class":85},[75,45147,142],{"class":141},[75,45149,15214],{"class":145},[75,45151,11902],{"class":141},[75,45153,19174],{"class":85},[75,45155,14668],{"class":193},[75,45157,197],{"class":85},[75,45159,43051],{"class":145},[75,45161,14860],{"class":85},[75,45163,2810],{"class":141},[75,45165,15263],{"class":145},[75,45167,318],{"class":85},[75,45169,45170,45172,45174,45176,45178,45180,45182,45184,45186,45188],{"class":77,"line":366},[75,45171,43982],{"class":85},[75,45173,142],{"class":141},[75,45175,267],{"class":85},[75,45177,14324],{"class":145},[75,45179,206],{"class":85},[75,45181,14678],{"class":145},[75,45183,273],{"class":141},[75,45185,43997],{"class":85},[75,45187,14324],{"class":145},[75,45189,44002],{"class":85},[75,45191,45192],{"class":77,"line":381},[75,45193,112],{"emptyLinePlaceholder":111},[75,45195,45196,45198,45200,45202,45204,45206,45208,45210,45212,45214,45216,45218],{"class":77,"line":984},[75,45197,44011],{"class":85},[75,45199,142],{"class":141},[75,45201,3093],{"class":85},[75,45203,3096],{"class":193},[75,45205,197],{"class":85},[75,45207,870],{"class":309},[75,45209,142],{"class":141},[75,45211,44026],{"class":85},[75,45213,10499],{"class":309},[75,45215,142],{"class":141},[75,45217,10504],{"class":145},[75,45219,318],{"class":85},[75,45221,45222,45224,45226,45228,45230,45232,45234,45236,45238,45240,45242,45244,45246,45248],{"class":77,"line":997},[75,45223,44039],{"class":85},[75,45225,142],{"class":141},[75,45227,43490],{"class":85},[75,45229,16446],{"class":193},[75,45231,15804],{"class":85},[75,45233,44050],{"class":193},[75,45235,44053],{"class":85},[75,45237,16452],{"class":309},[75,45239,142],{"class":141},[75,45241,16457],{"class":134},[75,45243,206],{"class":85},[75,45245,16472],{"class":309},[75,45247,142],{"class":141},[75,45249,44068],{"class":85},[75,45251,45252,45254,45256,45258,45260,45262,45264,45266,45268,45270],{"class":77,"line":1325},[75,45253,44073],{"class":309},[75,45255,142],{"class":141},[75,45257,44078],{"class":85},[75,45259,44081],{"class":309},[75,45261,142],{"class":141},[75,45263,44086],{"class":85},[75,45265,16462],{"class":309},[75,45267,142],{"class":141},[75,45269,16467],{"class":134},[75,45271,318],{"class":85},[75,45273,45274,45276,45278,45280,45282,45284,45286,45288,45290,45292,45294,45296,45298,45300,45302,45304],{"class":77,"line":1338},[75,45275,44099],{"class":85},[75,45277,142],{"class":141},[75,45279,44104],{"class":85},[75,45281,16495],{"class":193},[75,45283,44109],{"class":85},[75,45285,15602],{"class":309},[75,45287,142],{"class":141},[75,45289,44116],{"class":85},[75,45291,44119],{"class":309},[75,45293,142],{"class":141},[75,45295,44124],{"class":145},[75,45297,206],{"class":85},[75,45299,44129],{"class":309},[75,45301,142],{"class":141},[75,45303,44134],{"class":145},[75,45305,318],{"class":85},[75,45307,45308,45310,45312,45314,45316],{"class":77,"line":1343},[75,45309,44141],{"class":85},[75,45311,44144],{"class":193},[75,45313,197],{"class":85},[75,45315,44149],{"class":134},[75,45317,318],{"class":85},[75,45319,45320],{"class":77,"line":1356},[75,45321,112],{"emptyLinePlaceholder":111},[75,45323,45324,45326,45328,45330,45332],{"class":77,"line":1887},[75,45325,10549],{"class":85},[75,45327,3193],{"class":193},[75,45329,197],{"class":85},[75,45331,44166],{"class":134},[75,45333,318],{"class":85},[75,45335,45336,45338,45340,45342,45344],{"class":77,"line":1900},[75,45337,10549],{"class":85},[75,45339,3208],{"class":193},[75,45341,197],{"class":85},[75,45343,44179],{"class":134},[75,45345,318],{"class":85},[75,45347,45348,45350,45352,45355],{"class":77,"line":1913},[75,45349,44186],{"class":193},[75,45351,44189],{"class":85},[75,45353,45354],{"class":134},"\"Custom rotation\"",[75,45356,318],{"class":85},[75,45358,45359],{"class":77,"line":2657},[75,45360,112],{"emptyLinePlaceholder":111},[75,45362,45363,45365,45367,45369,45371,45373,45375,45377,45379,45381,45383,45385],{"class":77,"line":3177},[75,45364,44221],{"class":85},[75,45366,142],{"class":141},[75,45368,43400],{"class":193},[75,45370,44189],{"class":85},[75,45372,44230],{"class":134},[75,45374,206],{"class":85},[75,45376,43124],{"class":309},[75,45378,142],{"class":141},[75,45380,44239],{"class":85},[75,45382,43519],{"class":309},[75,45384,142],{"class":141},[75,45386,44246],{"class":85},[75,45388,45389,45391,45393],{"class":77,"line":3188},[75,45390,44251],{"class":309},[75,45392,142],{"class":141},[75,45394,44256],{"class":85},[75,45396,45397],{"class":77,"line":3203},[75,45398,112],{"emptyLinePlaceholder":111},[75,45400,45401,45403,45405,45407,45409,45411,45413],{"class":77,"line":3218},[75,45402,44265],{"class":81},[75,45404,44268],{"class":193},[75,45406,197],{"class":85},[75,45408,44273],{"class":14388},[75,45410,14391],{"class":85},[75,45412,14299],{"class":141},[75,45414,15605],{"class":85},[75,45416,45417,45419,45421,45423,45425,45427,45429,45431,45433,45435,45437,45440],{"class":77,"line":3224},[75,45418,44284],{"class":85},[75,45420,142],{"class":141},[75,45422,44289],{"class":85},[75,45424,2810],{"class":141},[75,45426,44294],{"class":85},[75,45428,200],{"class":141},[75,45430,1437],{"class":145},[75,45432,14936],{"class":85},[75,45434,500],{"class":141},[75,45436,190],{"class":85},[75,45438,45439],{"class":193},"deg2rad",[75,45441,45442],{"class":85},"(angle_deg)\n",[75,45444,45445,45448,45450,45452],{"class":77,"line":3229},[75,45446,45447],{"class":85},"        R ",[75,45449,142],{"class":141},[75,45451,42563],{"class":193},[75,45453,45454],{"class":85},"(axis_vec, ang)\n",[75,45456,45457,45459,45461,45464,45466],{"class":77,"line":3235},[75,45458,44313],{"class":85},[75,45460,142],{"class":141},[75,45462,45463],{"class":85}," R ",[75,45465,19121],{"class":141},[75,45467,44330],{"class":85},[75,45469,45470,45472,45474,45476,45478],{"class":77,"line":3263},[75,45471,44335],{"class":85},[75,45473,44338],{"class":193},[75,45475,197],{"class":85},[75,45477,44343],{"class":193},[75,45479,44346],{"class":85},[75,45481,45482,45484,45486],{"class":77,"line":3275},[75,45483,44351],{"class":85},[75,45485,44354],{"class":193},[75,45487,43495],{"class":85},[75,45489,45490,45493,45496,45498,45501,45503,45505,45508,45510,45512,45514,45516,45518,45520,45522,45524,45526,45528,45530,45532,45534,45536,45538,45540,45542],{"class":77,"line":3280},[75,45491,45492],{"class":81},"            f",[75,45494,45495],{"class":134},"\"axis = (",[75,45497,12219],{"class":145},[75,45499,45500],{"class":85},"axis_vec[",[75,45502,5186],{"class":145},[75,45504,32873],{"class":85},[75,45506,45507],{"class":81},":.2f",[75,45509,12224],{"class":145},[75,45511,206],{"class":134},[75,45513,12219],{"class":145},[75,45515,45500],{"class":85},[75,45517,7477],{"class":145},[75,45519,32873],{"class":85},[75,45521,45507],{"class":81},[75,45523,12224],{"class":145},[75,45525,206],{"class":134},[75,45527,12219],{"class":145},[75,45529,45500],{"class":85},[75,45531,270],{"class":145},[75,45533,32873],{"class":85},[75,45535,45507],{"class":81},[75,45537,12224],{"class":145},[75,45539,5175],{"class":134},[75,45541,44377],{"class":141},[75,45543,45544],{"class":134},"\"\n",[75,45546,45547,45549,45551,45553,45556,45559,45562,45564,45566,45569,45571,45574,45576,45578],{"class":77,"line":3290},[75,45548,45492],{"class":81},[75,45550,44361],{"class":134},[75,45552,12219],{"class":145},[75,45554,45555],{"class":85},"np.",[75,45557,45558],{"class":193},"rad2deg",[75,45560,45561],{"class":85},"(ang)",[75,45563,15913],{"class":81},[75,45565,12224],{"class":145},[75,45567,45568],{"class":134},"° \u002F ",[75,45570,12219],{"class":145},[75,45572,45573],{"class":85},"angle_deg",[75,45575,15913],{"class":81},[75,45577,12224],{"class":145},[75,45579,45580],{"class":134},"°\"\n",[75,45582,45583],{"class":77,"line":4431},[75,45584,45585],{"class":85},"        )\n",[75,45587,45588,45590],{"class":77,"line":4454},[75,45589,14450],{"class":81},[75,45591,44409],{"class":85},[75,45593,45594],{"class":77,"line":4463},[75,45595,112],{"emptyLinePlaceholder":111},[75,45597,45598,45600,45602,45604,45606,45608,45610,45612,45614,45616,45618,45620,45622,45624,45626],{"class":77,"line":4468},[75,45599,44418],{"class":85},[75,45601,142],{"class":141},[75,45603,44423],{"class":193},[75,45605,44426],{"class":85},[75,45607,44396],{"class":309},[75,45609,142],{"class":141},[75,45611,44433],{"class":85},[75,45613,44436],{"class":309},[75,45615,142],{"class":141},[75,45617,44441],{"class":145},[75,45619,206],{"class":85},[75,45621,44446],{"class":309},[75,45623,142],{"class":141},[75,45625,376],{"class":145},[75,45627,318],{"class":85},[75,45629,45630,45632,45634,45636,45638,45640,45642,45644,45646,45648,45650],{"class":77,"line":4477},[75,45631,44457],{"class":85},[75,45633,44460],{"class":193},[75,45635,197],{"class":85},[75,45637,44465],{"class":309},[75,45639,142],{"class":141},[75,45641,10504],{"class":145},[75,45643,206],{"class":85},[75,45645,44474],{"class":309},[75,45647,142],{"class":141},[75,45649,10504],{"class":145},[75,45651,318],{"class":85},[75,45653,45654,45656,45658,45660,45662,45664,45666,45668,45670,45672,45674,45676],{"class":77,"line":5105},[75,45655,44485],{"class":85},[75,45657,44488],{"class":193},[75,45659,44491],{"class":85},[75,45661,44494],{"class":193},[75,45663,16413],{"class":85},[75,45665,44499],{"class":309},[75,45667,142],{"class":141},[75,45669,44504],{"class":193},[75,45671,197],{"class":85},[75,45673,44509],{"class":309},[75,45675,142],{"class":141},[75,45677,44514],{"class":85},[75,45679,45680,45682,45684,45686,45688,45690,45692],{"class":77,"line":5114},[75,45681,44519],{"class":309},[75,45683,43541],{"class":141},[75,45685,197],{"class":85},[75,45687,44526],{"class":309},[75,45689,142],{"class":141},[75,45691,43561],{"class":134},[75,45693,283],{"class":85},[75,45695,45696,45698,45700],{"class":77,"line":10795},[75,45697,44537],{"class":85},[75,45699,44540],{"class":193},[75,45701,44543],{"class":85},[75,45703,45704,45706],{"class":77,"line":14778},[75,45705,14467],{"class":81},[75,45707,44550],{"class":85},[75,45709,45710],{"class":77,"line":14783},[75,45711,112],{"emptyLinePlaceholder":111},[75,45713,45714],{"class":77,"line":14819},[75,45715,112],{"emptyLinePlaceholder":111},[75,45717,45718],{"class":77,"line":14825},[75,45719,45720],{"class":118},"# Example: 45° about the y-axis, starting from |+z⟩\n",[75,45722,45723,45726,45728,45730,45732,45734],{"class":77,"line":14838},[75,45724,45725],{"class":85},"custom_out ",[75,45727,142],{"class":141},[75,45729,44775],{"class":193},[75,45731,44778],{"class":85},[75,45733,2810],{"class":141},[75,45735,45736],{"class":134}," \"wigner_custom_y45.gif\"\n",[75,45738,45739,45742],{"class":77,"line":14893},[75,45740,45741],{"class":193},"animate_wigner_custom",[75,45743,43495],{"class":85},[75,45745,45746,45749,45751,45753,45755,45757,45759,45761,45763],{"class":77,"line":14951},[75,45747,45748],{"class":309},"    axis_vec",[75,45750,142],{"class":141},[75,45752,197],{"class":85},[75,45754,14324],{"class":145},[75,45756,206],{"class":85},[75,45758,630],{"class":145},[75,45760,206],{"class":85},[75,45762,14324],{"class":145},[75,45764,3003],{"class":85},[75,45766,45767,45770,45772,45775],{"class":77,"line":14959},[75,45768,45769],{"class":309},"    angle_deg",[75,45771,142],{"class":141},[75,45773,45774],{"class":145},"45.0",[75,45776,41629],{"class":85},[75,45778,45779,45782,45784],{"class":77,"line":14964},[75,45780,45781],{"class":309},"    r0",[75,45783,142],{"class":141},[75,45785,45786],{"class":85},"r0,\n",[75,45788,45789,45791,45793,45795,45798],{"class":77,"line":14983},[75,45790,44616],{"class":309},[75,45792,142],{"class":141},[75,45794,44621],{"class":145},[75,45796,45797],{"class":85},",                 ",[75,45799,45800],{"class":118},"# raise to 48 if you prefer\n",[75,45802,45803,45805,45807,45809,45811,45813,45815,45817],{"class":77,"line":14989},[75,45804,44632],{"class":309},[75,45806,142],{"class":141},[75,45808,44637],{"class":145},[75,45810,206],{"class":85},[75,45812,42846],{"class":309},[75,45814,142],{"class":141},[75,45816,44646],{"class":145},[75,45818,41629],{"class":85},[75,45820,45821,45823,45825,45827],{"class":77,"line":15012},[75,45822,44657],{"class":309},[75,45824,142],{"class":141},[75,45826,752],{"class":145},[75,45828,41629],{"class":85},[75,45830,45831,45833,45835],{"class":77,"line":15017},[75,45832,44680],{"class":309},[75,45834,142],{"class":141},[75,45836,45837],{"class":85},"knobs.figsize,\n",[75,45839,45840,45842,45844,45847,45849,45851],{"class":77,"line":15025},[75,45841,44697],{"class":309},[75,45843,142],{"class":141},[75,45845,45846],{"class":85},"knobs.info_loc, ",[75,45848,44707],{"class":309},[75,45850,142],{"class":141},[75,45852,45853],{"class":85},"knobs.info_fontsize,\n",[75,45855,45856,45859,45861],{"class":77,"line":15038},[75,45857,45858],{"class":309},"    wrap_chars",[75,45860,142],{"class":141},[75,45862,45863],{"class":85},"knobs.wrap_chars,\n",[75,45865,45866,45869,45871],{"class":77,"line":15051},[75,45867,45868],{"class":309},"    out_path",[75,45870,142],{"class":141},[75,45872,45873],{"class":85},"custom_out,\n",[75,45875,45876],{"class":77,"line":15058},[75,45877,318],{"class":85},[75,45879,45880],{"class":77,"line":15069},[75,45881,112],{"emptyLinePlaceholder":111},[75,45883,45884,45887,45889,45891,45893,45895,45897],{"class":77,"line":15080},[75,45885,45886],{"class":193},"display",[75,45888,197],{"class":85},[75,45890,44869],{"class":193},[75,45892,197],{"class":85},[75,45894,44874],{"class":309},[75,45896,44877],{"class":141},[75,45898,45899],{"class":85},"(custom_out)))\n",[75,45901,45902,45905,45907,45909],{"class":77,"line":15085},[75,45903,45904],{"class":141},"print",[75,45906,197],{"class":85},[75,45908,44890],{"class":134},[75,45910,45911],{"class":85},", custom_out)\n",[65,45913,45916],{"className":45914,"code":45915,"language":34131,"meta":70},[41256],"Saved: \u002Fcontent\u002Fwigner_custom_y45.gif\n",[72,45917,45915],{"__ignoreMap":70},[391,45919],{"alt":1365,"src":45920},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-04.webp",[12,45922,45923],{},"The next following scripts are for the 3D Wigner plots.",[65,45925,45927],{"className":67,"code":45926,"language":69,"meta":70,"style":70},"# @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",[72,45928,45929,45934,45939,45943,45953,45957,45967,45977,45987,45991,46001,46011,46021,46033,46045,46055,46059,46063,46067,46076,46081,46085,46089,46094,46098,46103,46108,46113,46117,46122,46127,46132,46137,46142,46146,46156,46188,46198,46209,46220,46230,46242,46256,46266,46290,46300,46304,46308,46313,46349,46357,46393,46429,46465,46469,46495,46514,46524,46528,46532,46557,46573,46585,46623,46635,46673,46685,46723,46734,46738,46742,46767,46772,46788,46810,46830,46850,46870,46882,46903,46910,46914,46918,46935,46940,47001,47005,47009,47031,47036,47048,47060,47064,47069,47090,47111,47124,47128,47133,47159,47185,47211,47225,47251,47265,47269,47297,47326,47351,47365,47386,47390,47395,47421,47455,47459,47473,47478,47490,47510,47529,47544,47548,47552,47557,47592,47604,47608,47613,47625,47637,47650,47664,47694,47726,47766,47776,47780,47793,47832,47841,47881,47891,47895,47899,47916,47943,47959,47971,48003,48036,48084,48091,48095,48128,48154,48179,48207,48223,48231,48237,48241,48246,48261,48294,48300,48312,48330,48349,48372,48381,48386,48391,48397,48412,48423,48460,48482,48493,48510,48522,48527,48532],{"__ignoreMap":70},[75,45930,45931],{"class":77,"line":78},[75,45932,45933],{"class":118},"# @title Cell 5 — 3D Wigner animations for ALL Pauli gates (X, Y, Z)\n",[75,45935,45936],{"class":77,"line":95},[75,45937,45938],{"class":118},"# PEP 8\u002F257 compliant. Stable: updates facecolors in place; no deleting artists.\n",[75,45940,45941],{"class":77,"line":108},[75,45942,112],{"emptyLinePlaceholder":111},[75,45944,45945,45947,45949,45951],{"class":77,"line":115},[75,45946,2700],{"class":81},[75,45948,41502],{"class":15980},[75,45950,41505],{"class":81},[75,45952,41508],{"class":85},[75,45954,45955],{"class":77,"line":122},[75,45956,112],{"emptyLinePlaceholder":111},[75,45958,45959,45961,45963,45965],{"class":77,"line":128},[75,45960,2700],{"class":81},[75,45962,14064],{"class":85},[75,45964,82],{"class":81},[75,45966,14069],{"class":85},[75,45968,45969,45971,45973,45975],{"class":77,"line":149},[75,45970,2700],{"class":81},[75,45972,41661],{"class":85},[75,45974,82],{"class":81},[75,45976,41666],{"class":85},[75,45978,45979,45981,45983,45985],{"class":77,"line":154},[75,45980,2700],{"class":81},[75,45982,14076],{"class":85},[75,45984,82],{"class":81},[75,45986,41677],{"class":85},[75,45988,45989],{"class":77,"line":160},[75,45990,112],{"emptyLinePlaceholder":111},[75,45992,45993,45995,45997,45999],{"class":77,"line":171},[75,45994,82],{"class":81},[75,45996,86],{"class":85},[75,45998,89],{"class":81},[75,46000,92],{"class":85},[75,46002,46003,46005,46007,46009],{"class":77,"line":182},[75,46004,82],{"class":81},[75,46006,100],{"class":85},[75,46008,89],{"class":81},[75,46010,105],{"class":85},[75,46012,46013,46015,46017,46019],{"class":77,"line":222},[75,46014,2700],{"class":81},[75,46016,41715],{"class":85},[75,46018,82],{"class":81},[75,46020,41720],{"class":85},[75,46022,46023,46025,46028,46030],{"class":77,"line":227},[75,46024,2700],{"class":81},[75,46026,46027],{"class":85}," matplotlib.colors ",[75,46029,82],{"class":81},[75,46031,46032],{"class":85}," Normalize\n",[75,46034,46035,46037,46040,46042],{"class":77,"line":233},[75,46036,2700],{"class":81},[75,46038,46039],{"class":85}," matplotlib.cm ",[75,46041,82],{"class":81},[75,46043,46044],{"class":85}," ScalarMappable\n",[75,46046,46047,46049,46051,46053],{"class":77,"line":286},[75,46048,2700],{"class":81},[75,46050,41727],{"class":85},[75,46052,82],{"class":81},[75,46054,41732],{"class":85},[75,46056,46057],{"class":77,"line":291},[75,46058,112],{"emptyLinePlaceholder":111},[75,46060,46061],{"class":77,"line":297},[75,46062,112],{"emptyLinePlaceholder":111},[75,46064,46065],{"class":77,"line":321},[75,46066,14131],{"class":193},[75,46068,46069,46071,46074],{"class":77,"line":336},[75,46070,14136],{"class":81},[75,46072,46073],{"class":14139}," Knobs3D",[75,46075,5800],{"class":85},[75,46077,46078],{"class":77,"line":351},[75,46079,46080],{"class":134},"    \"\"\"User controls for 3D Pauli-gate Wigner 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degrees.\n",[75,46123,46124],{"class":77,"line":1900},[75,46125,46126],{"class":134},"        azim: Camera azimuth in degrees.\n",[75,46128,46129],{"class":77,"line":1913},[75,46130,46131],{"class":134},"        out_dir: Output directory (e.g., \"\u002Fcontent\" on Colab).\n",[75,46133,46134],{"class":77,"line":2657},[75,46135,46136],{"class":134},"        figsize: Figure size in inches (DPI set globally by rcParams).\n",[75,46138,46139],{"class":77,"line":3177},[75,46140,46141],{"class":134},"        info_fontsize: In-axes info text font size.\n",[75,46143,46144],{"class":77,"line":3188},[75,46145,41832],{"class":134},[75,46147,46148,46150,46152,46154],{"class":77,"line":3203},[75,46149,41871],{"class":85},[75,46151,41840],{"class":141},[75,46153,14153],{"class":141},[75,46155,41878],{"class":134},[75,46157,46158,46160,46162,46164,46166,46168,46170,46172,46174,46176,46178,46180,46182,46184,46186],{"class":77,"line":3218},[75,46159,41883],{"class":85},[75,46161,14150],{"class":141},[75,46163,206],{"class":85},[75,46165,14150],{"class":141},[75,46167,206],{"class":85},[75,46169,14150],{"class":141},[75,46171,138],{"class":85},[75,46173,142],{"class":141},[75,46175,267],{"class":85},[75,46177,14324],{"class":145},[75,46179,206],{"class":85},[75,46181,14324],{"class":145},[75,46183,206],{"class":85},[75,46185,630],{"class":145},[75,46187,318],{"class":85},[75,46189,46190,46192,46194,46196],{"class":77,"line":3224},[75,46191,41916],{"class":85},[75,46193,14299],{"class":141},[75,46195,14153],{"class":141},[75,46197,41923],{"class":145},[75,46199,46200,46202,46204,46206],{"class":77,"line":3229},[75,46201,41928],{"class":85},[75,46203,14299],{"class":141},[75,46205,14153],{"class":141},[75,46207,46208],{"class":145}," 64\n",[75,46210,46211,46213,46215,46217],{"class":77,"line":3235},[75,46212,41940],{"class":85},[75,46214,14299],{"class":141},[75,46216,14153],{"class":141},[75,46218,46219],{"class":145}," 128\n",[75,46221,46222,46224,46226,46228],{"class":77,"line":3263},[75,46223,41952],{"class":85},[75,46225,14299],{"class":141},[75,46227,14153],{"class":141},[75,46229,41959],{"class":145},[75,46231,46232,46235,46237,46239],{"class":77,"line":3275},[75,46233,46234],{"class":85},"    elev: ",[75,46236,14150],{"class":141},[75,46238,14153],{"class":141},[75,46240,46241],{"class":145}," 25.0\n",[75,46243,46244,46247,46249,46251,46253],{"class":77,"line":3280},[75,46245,46246],{"class":85},"    azim: ",[75,46248,14150],{"class":141},[75,46250,14153],{"class":141},[75,46252,14186],{"class":141},[75,46254,46255],{"class":145},"60.0\n",[75,46257,46258,46260,46262,46264],{"class":77,"line":3290},[75,46259,41964],{"class":85},[75,46261,41840],{"class":141},[75,46263,14153],{"class":141},[75,46265,41971],{"class":134},[75,46267,46268,46270,46272,46274,46276,46278,46280,46282,46284,46286,46288],{"class":77,"line":4431},[75,46269,41976],{"class":85},[75,46271,14150],{"class":141},[75,46273,206],{"class":85},[75,46275,14150],{"class":141},[75,46277,138],{"class":85},[75,46279,142],{"class":141},[75,46281,267],{"class":85},[75,46283,41606],{"class":145},[75,46285,206],{"class":85},[75,46287,17832],{"class":145},[75,46289,318],{"class":85},[75,46291,46292,46294,46296,46298],{"class":77,"line":4454},[75,46293,42013],{"class":85},[75,46295,14299],{"class":141},[75,46297,14153],{"class":141},[75,46299,42020],{"class":145},[75,46301,46302],{"class":77,"line":4463},[75,46303,112],{"emptyLinePlaceholder":111},[75,46305,46306],{"class":77,"line":4468},[75,46307,112],{"emptyLinePlaceholder":111},[75,46309,46310],{"class":77,"line":4477},[75,46311,46312],{"class":118},"# ----------------------------- Helpers (self-contained) -----------------------------\n",[75,46314,46315,46317,46320,46322,46324,46326,46328,46330,46333,46336,46338,46340,46342,46344,46346],{"class":77,"line":5105},[75,46316,14380],{"class":81},[75,46318,46319],{"class":193}," _bloch_vector",[75,46321,197],{"class":85},[75,46323,573],{"class":14388},[75,46325,14391],{"class":85},[75,46327,41840],{"class":141},[75,46329,206],{"class":85},[75,46331,46332],{"class":14388},"custom",[75,46334,46335],{"class":85},"=(",[75,46337,14324],{"class":145},[75,46339,206],{"class":85},[75,46341,14324],{"class":145},[75,46343,206],{"class":85},[75,46345,630],{"class":145},[75,46347,46348],{"class":85},")) -> np.ndarray:\n",[75,46350,46351,46353,46355],{"class":77,"line":5114},[75,46352,42094],{"class":85},[75,46354,142],{"class":141},[75,46356,42099],{"class":85},[75,46358,46359,46361,46363,46365,46367,46369,46371,46373,46375,46377,46379,46381,46383,46385,46387,46389,46391],{"class":77,"line":10795},[75,46360,42104],{"class":134},[75,46362,41603],{"class":85},[75,46364,14324],{"class":145},[75,46366,206],{"class":85},[75,46368,14324],{"class":145},[75,46370,206],{"class":85},[75,46372,630],{"class":145},[75,46374,10496],{"class":85},[75,46376,42121],{"class":134},[75,46378,41603],{"class":85},[75,46380,14324],{"class":145},[75,46382,206],{"class":85},[75,46384,14324],{"class":145},[75,46386,206],{"class":85},[75,46388,200],{"class":141},[75,46390,630],{"class":145},[75,46392,3003],{"class":85},[75,46394,46395,46397,46399,46401,46403,46405,46407,46409,46411,46413,46415,46417,46419,46421,46423,46425,46427],{"class":77,"line":14778},[75,46396,42142],{"class":134},[75,46398,41603],{"class":85},[75,46400,630],{"class":145},[75,46402,206],{"class":85},[75,46404,14324],{"class":145},[75,46406,206],{"class":85},[75,46408,14324],{"class":145},[75,46410,10496],{"class":85},[75,46412,42159],{"class":134},[75,46414,41603],{"class":85},[75,46416,200],{"class":141},[75,46418,630],{"class":145},[75,46420,206],{"class":85},[75,46422,14324],{"class":145},[75,46424,206],{"class":85},[75,46426,14324],{"class":145},[75,46428,3003],{"class":85},[75,46430,46431,46433,46435,46437,46439,46441,46443,46445,46447,46449,46451,46453,46455,46457,46459,46461,46463],{"class":77,"line":14783},[75,46432,42180],{"class":134},[75,46434,41603],{"class":85},[75,46436,14324],{"class":145},[75,46438,206],{"class":85},[75,46440,630],{"class":145},[75,46442,206],{"class":85},[75,46444,14324],{"class":145},[75,46446,10496],{"class":85},[75,46448,42197],{"class":134},[75,46450,41603],{"class":85},[75,46452,14324],{"class":145},[75,46454,206],{"class":85},[75,46456,200],{"class":141},[75,46458,630],{"class":145},[75,46460,206],{"class":85},[75,46462,14324],{"class":145},[75,46464,3003],{"class":85},[75,46466,46467],{"class":77,"line":14819},[75,46468,42218],{"class":85},[75,46470,46471,46473,46475,46477,46479,46481,46483,46486,46488,46491,46493],{"class":77,"line":14825},[75,46472,14520],{"class":85},[75,46474,142],{"class":141},[75,46476,190],{"class":85},[75,46478,42229],{"class":193},[75,46480,42232],{"class":85},[75,46482,42235],{"class":193},[75,46484,46485],{"class":85},"(label.",[75,46487,42241],{"class":193},[75,46489,46490],{"class":85},"(), 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must be 'x', 'y', or 'z'\"",[75,46733,318],{"class":85},[75,46735,46736],{"class":77,"line":15069},[75,46737,112],{"emptyLinePlaceholder":111},[75,46739,46740],{"class":77,"line":15080},[75,46741,112],{"emptyLinePlaceholder":111},[75,46743,46744,46746,46749,46751,46753,46755,46757,46759,46761,46763,46765],{"class":77,"line":15085},[75,46745,14380],{"class":81},[75,46747,46748],{"class":193}," _n_grid",[75,46750,197],{"class":85},[75,46752,42837],{"class":14388},[75,46754,14391],{"class":85},[75,46756,14299],{"class":141},[75,46758,206],{"class":85},[75,46760,42846],{"class":14388},[75,46762,14391],{"class":85},[75,46764,14299],{"class":141},[75,46766,15605],{"class":85},[75,46768,46769],{"class":77,"line":15090},[75,46770,46771],{"class":134},"    \"\"\"Return (θ, φ, n) with n stacked as (3, n_theta, n_phi).\"\"\"\n",[75,46773,46774,46776,46778,46780,46782,46784,46786],{"class":77,"line":15096},[75,46775,42862],{"class":85},[75,46777,142],{"class":141},[75,46779,190],{"class":85},[75,46781,194],{"class":193},[75,46783,197],{"class":85},[75,46785,14324],{"class":145},[75,46787,42875],{"class":85},[75,46789,46790,46792,46794,46796,46798,46800,46802,46804,46806,46808],{"class":77,"line":15101},[75,46791,42880],{"class":85},[75,46793,142],{"class":141},[75,46795,190],{"class":85},[75,46797,194],{"class":193},[75,46799,197],{"class":85},[75,46801,14324],{"class":145},[75,46803,206],{"class":85},[75,46805,14678],{"class":145},[75,46807,273],{"class":141},[75,46809,42899],{"class":85},[75,46811,46812,46814,46816,46818,46820,46822,46824,46826,46828],{"class":77,"line":15151},[75,46813,42904],{"class":85},[75,46815,142],{"class":141},[75,46817,190],{"class":85},[75,46819,42911],{"class":193},[75,46821,42914],{"class":85},[75,46823,42917],{"class":309},[75,46825,142],{"class":141},[75,46827,42922],{"class":134},[75,46829,318],{"class":85},[75,46831,46832,46834,46836,46838,46840,46842,46844,46846,46848],{"class":77,"line":15160},[75,46833,42929],{"class":85},[75,46835,142],{"class":141},[75,46837,190],{"class":85},[75,46839,1179],{"class":193},[75,46841,42938],{"class":85},[75,46843,500],{"class":141},[75,46845,190],{"class":85},[75,46847,2845],{"class":193},[75,46849,42947],{"class":85},[75,46851,46852,46854,46856,46858,46860,46862,46864,46866,46868],{"class":77,"line":15206},[75,46853,42952],{"class":85},[75,46855,142],{"class":141},[75,46857,190],{"class":85},[75,46859,1179],{"class":193},[75,46861,42938],{"class":85},[75,46863,500],{"class":141},[75,46865,190],{"class":85},[75,46867,1179],{"class":193},[75,46869,42947],{"class":85},[75,46871,46872,46874,46876,46878,46880],{"class":77,"line":15235},[75,46873,42973],{"class":85},[75,46875,142],{"class":141},[75,46877,190],{"class":85},[75,46879,2845],{"class":193},[75,46881,42982],{"class":85},[75,46883,46884,46887,46889,46891,46893,46895,46897,46899,46901],{"class":77,"line":15288},[75,46885,46886],{"class":85},"    n ",[75,46888,142],{"class":141},[75,46890,190],{"class":85},[75,46892,42991],{"class":193},[75,46894,42994],{"class":85},[75,46896,42331],{"class":309},[75,46898,142],{"class":141},[75,46900,5186],{"class":145},[75,46902,318],{"class":85},[75,46904,46905,46907],{"class":77,"line":15294},[75,46906,14467],{"class":81},[75,46908,46909],{"class":85}," th, ph, n\n",[75,46911,46912],{"class":77,"line":15311},[75,46913,112],{"emptyLinePlaceholder":111},[75,46915,46916],{"class":77,"line":15316},[75,46917,112],{"emptyLinePlaceholder":111},[75,46919,46920,46922,46925,46927,46929,46931,46933],{"class":77,"line":15363},[75,46921,14380],{"class":81},[75,46923,46924],{"class":193}," _wigner_qubit",[75,46926,197],{"class":85},[75,46928,43022],{"class":14388},[75,46930,19065],{"class":85},[75,46932,33004],{"class":14388},[75,46934,19076],{"class":85},[75,46936,46937],{"class":77,"line":15371},[75,46938,46939],{"class":134},"    \"\"\"W(θ, φ) = 1\u002F2 + (√3\u002F2) * r·n on the (θ, φ) grid.\"\"\"\n",[75,46941,46942,46944,46946,46948,46950,46952,46954,46956,46958,46960,46962,46964,46966,46968,46970,46972,46974,46976,46978,46980,46982,46984,46986,46989,46991,46993,46995,46997,46999],{"class":77,"line":15419},[75,46943,14467],{"class":81},[75,46945,15214],{"class":145},[75,46947,11902],{"class":141},[75,46949,19174],{"class":85},[75,46951,14668],{"class":193},[75,46953,197],{"class":85},[75,46955,43051],{"class":145},[75,46957,14860],{"class":85},[75,46959,2810],{"class":141},[75,46961,15263],{"class":145},[75,46963,14860],{"class":85},[75,46965,500],{"class":141},[75,46967,43064],{"class":85},[75,46969,43067],{"class":193},[75,46971,197],{"class":85},[75,46973,3101],{"class":145},[75,46975,206],{"class":85},[75,46977,7477],{"class":145},[75,46979,206],{"class":85},[75,46981,7477],{"class":145},[75,46983,14860],{"class":85},[75,46985,500],{"class":141},[75,46987,46988],{"class":85}," n).",[75,46990,43089],{"class":193},[75,46992,197],{"class":85},[75,46994,42331],{"class":309},[75,46996,142],{"class":141},[75,46998,5186],{"class":145},[75,47000,318],{"class":85},[75,47002,47003],{"class":77,"line":15442},[75,47004,112],{"emptyLinePlaceholder":111},[75,47006,47007],{"class":77,"line":15482},[75,47008,112],{"emptyLinePlaceholder":111},[75,47010,47011,47013,47016,47018,47020,47022,47024,47026,47028],{"class":77,"line":15520},[75,47012,14380],{"class":81},[75,47014,47015],{"class":193}," _animate_one_pauli",[75,47017,197],{"class":85},[75,47019,42331],{"class":14388},[75,47021,14391],{"class":85},[75,47023,41840],{"class":141},[75,47025,206],{"class":85},[75,47027,44273],{"class":14388},[75,47029,47030],{"class":85},": Knobs3D) -> Path:\n",[75,47032,47033],{"class":77,"line":15525},[75,47034,47035],{"class":134},"    \"\"\"Render one 3D Wigner animation for a chosen Pauli axis; return output path.\"\"\"\n",[75,47037,47038,47041,47043,47045],{"class":77,"line":15540},[75,47039,47040],{"class":85},"    r0 ",[75,47042,142],{"class":141},[75,47044,46319],{"class":193},[75,47046,47047],{"class":85},"(k.initial_state, k.custom_r)\n",[75,47049,47050,47053,47055,47057],{"class":77,"line":15545},[75,47051,47052],{"class":85},"    th, ph, n ",[75,47054,142],{"class":141},[75,47056,46748],{"class":193},[75,47058,47059],{"class":85},"(k.n_theta, k.n_phi)\n",[75,47061,47062],{"class":77,"line":15551},[75,47063,112],{"emptyLinePlaceholder":111},[75,47065,47066],{"class":77,"line":15592},[75,47067,47068],{"class":118},"    # Unit sphere geometry\n",[75,47070,47071,47074,47076,47078,47080,47082,47084,47086,47088],{"class":77,"line":15608},[75,47072,47073],{"class":85},"    X ",[75,47075,142],{"class":141},[75,47077,190],{"class":85},[75,47079,1179],{"class":193},[75,47081,42938],{"class":85},[75,47083,500],{"class":141},[75,47085,190],{"class":85},[75,47087,2845],{"class":193},[75,47089,42947],{"class":85},[75,47091,47092,47095,47097,47099,47101,47103,47105,47107,47109],{"class":77,"line":15624},[75,47093,47094],{"class":85},"    Y ",[75,47096,142],{"class":141},[75,47098,190],{"class":85},[75,47100,1179],{"class":193},[75,47102,42938],{"class":85},[75,47104,500],{"class":141},[75,47106,190],{"class":85},[75,47108,1179],{"class":193},[75,47110,42947],{"class":85},[75,47112,47113,47116,47118,47120,47122],{"class":77,"line":15652},[75,47114,47115],{"class":85},"    Z ",[75,47117,142],{"class":141},[75,47119,190],{"class":85},[75,47121,2845],{"class":193},[75,47123,42982],{"class":85},[75,47125,47126],{"class":77,"line":15657},[75,47127,112],{"emptyLinePlaceholder":111},[75,47129,47130],{"class":77,"line":15662},[75,47131,47132],{"class":118},"    # Fixed normalization (pure-state bounds) for constant color scale\n",[75,47134,47135,47137,47139,47141,47143,47145,47147,47149,47151,47153,47155,47157],{"class":77,"line":15668},[75,47136,43928],{"class":85},[75,47138,142],{"class":141},[75,47140,15214],{"class":145},[75,47142,14186],{"class":141},[75,47144,19174],{"class":85},[75,47146,14668],{"class":193},[75,47148,197],{"class":85},[75,47150,43051],{"class":145},[75,47152,14860],{"class":85},[75,47154,2810],{"class":141},[75,47156,15263],{"class":145},[75,47158,318],{"class":85},[75,47160,47161,47163,47165,47167,47169,47171,47173,47175,47177,47179,47181,47183],{"class":77,"line":15673},[75,47162,43955],{"class":85},[75,47164,142],{"class":141},[75,47166,15214],{"class":145},[75,47168,11902],{"class":141},[75,47170,19174],{"class":85},[75,47172,14668],{"class":193},[75,47174,197],{"class":85},[75,47176,43051],{"class":145},[75,47178,14860],{"class":85},[75,47180,2810],{"class":141},[75,47182,15263],{"class":145},[75,47184,318],{"class":85},[75,47186,47187,47190,47192,47195,47197,47200,47202,47204,47206,47208],{"class":77,"line":15698},[75,47188,47189],{"class":85},"    norm ",[75,47191,142],{"class":141},[75,47193,47194],{"class":193}," Normalize",[75,47196,197],{"class":85},[75,47198,47199],{"class":309},"vmin",[75,47201,142],{"class":141},[75,47203,44078],{"class":85},[75,47205,44081],{"class":309},[75,47207,142],{"class":141},[75,47209,47210],{"class":85},"w_max)\n",[75,47212,47213,47216,47218,47220,47223],{"class":77,"line":15723},[75,47214,47215],{"class":85},"    cmap ",[75,47217,142],{"class":141},[75,47219,3093],{"class":85},[75,47221,47222],{"class":193},"get_cmap",[75,47224,389],{"class":85},[75,47226,47227,47230,47232,47235,47237,47239,47241,47244,47246,47248],{"class":77,"line":15728},[75,47228,47229],{"class":85},"    sm ",[75,47231,142],{"class":141},[75,47233,47234],{"class":193}," ScalarMappable",[75,47236,197],{"class":85},[75,47238,42260],{"class":309},[75,47240,142],{"class":141},[75,47242,47243],{"class":85},"norm, ",[75,47245,16480],{"class":309},[75,47247,142],{"class":141},[75,47249,47250],{"class":85},"cmap)\n",[75,47252,47253,47256,47259,47262],{"class":77,"line":15762},[75,47254,47255],{"class":85},"    sm.",[75,47257,47258],{"class":193},"set_array",[75,47260,47261],{"class":85},"([])  ",[75,47263,47264],{"class":118},"# required for colorbar from a ScalarMappable\n",[75,47266,47267],{"class":77,"line":15792},[75,47268,112],{"emptyLinePlaceholder":111},[75,47270,47271,47274,47276,47278,47280,47282,47284,47286,47289,47291,47293,47295],{"class":77,"line":15829},[75,47272,47273],{"class":85},"    fig ",[75,47275,142],{"class":141},[75,47277,3093],{"class":85},[75,47279,865],{"class":193},[75,47281,197],{"class":85},[75,47283,870],{"class":309},[75,47285,142],{"class":141},[75,47287,47288],{"class":85},"k.figsize, ",[75,47290,10499],{"class":309},[75,47292,142],{"class":141},[75,47294,10504],{"class":145},[75,47296,318],{"class":85},[75,47298,47299,47302,47304,47306,47309,47311,47314,47316,47319,47321,47324],{"class":77,"line":15834},[75,47300,47301],{"class":85},"    ax ",[75,47303,142],{"class":141},[75,47305,44104],{"class":85},[75,47307,47308],{"class":193},"add_subplot",[75,47310,197],{"class":85},[75,47312,47313],{"class":145},"111",[75,47315,206],{"class":85},[75,47317,47318],{"class":309},"projection",[75,47320,142],{"class":141},[75,47322,47323],{"class":134},"\"3d\"",[75,47325,318],{"class":85},[75,47327,47328,47330,47333,47335,47338,47340,47343,47346,47348],{"class":77,"line":15840},[75,47329,10549],{"class":85},[75,47331,47332],{"class":193},"view_init",[75,47334,197],{"class":85},[75,47336,47337],{"class":309},"elev",[75,47339,142],{"class":141},[75,47341,47342],{"class":85},"k.elev, ",[75,47344,47345],{"class":309},"azim",[75,47347,142],{"class":141},[75,47349,47350],{"class":85},"k.azim)\n",[75,47352,47353,47355,47358,47360,47363],{"class":77,"line":15873},[75,47354,14433],{"class":81},[75,47356,47357],{"class":141}," hasattr",[75,47359,44189],{"class":85},[75,47361,47362],{"class":134},"\"set_box_aspect\"",[75,47364,15605],{"class":85},[75,47366,47367,47369,47372,47374,47376,47378,47380,47382,47384],{"class":77,"line":15931},[75,47368,15627],{"class":85},[75,47370,47371],{"class":193},"set_box_aspect",[75,47373,16298],{"class":85},[75,47375,7477],{"class":145},[75,47377,206],{"class":85},[75,47379,7477],{"class":145},[75,47381,206],{"class":85},[75,47383,7477],{"class":145},[75,47385,283],{"class":85},[75,47387,47388],{"class":77,"line":15953},[75,47389,112],{"emptyLinePlaceholder":111},[75,47391,47392],{"class":77,"line":15968},[75,47393,47394],{"class":118},"    # Initial facecolors (use cell-centered (M-1, N-1) colors for quads)\n",[75,47396,47397,47400,47402,47404,47406,47409,47412,47414,47416,47418],{"class":77,"line":15996},[75,47398,47399],{"class":85},"    W0 ",[75,47401,142],{"class":141},[75,47403,46924],{"class":193},[75,47405,197],{"class":85},[75,47407,47408],{"class":193},"_rotation_matrix",[75,47410,47411],{"class":85},"(axis, ",[75,47413,14324],{"class":145},[75,47415,14860],{"class":85},[75,47417,19121],{"class":141},[75,47419,47420],{"class":85}," r0, n)\n",[75,47422,47423,47426,47428,47431,47433,47435,47438,47440,47442,47445,47447,47449,47452],{"class":77,"line":16026},[75,47424,47425],{"class":145},"    FC0",[75,47427,14153],{"class":141},[75,47429,47430],{"class":193}," cmap",[75,47432,197],{"class":85},[75,47434,42260],{"class":193},[75,47436,47437],{"class":85},"(W0[:",[75,47439,200],{"class":141},[75,47441,7477],{"class":145},[75,47443,47444],{"class":85},", :",[75,47446,200],{"class":141},[75,47448,7477],{"class":145},[75,47450,47451],{"class":85},"]))  ",[75,47453,47454],{"class":118},"# shape: (n_theta-1, n_phi-1, 4)\n",[75,47456,47457],{"class":77,"line":16047},[75,47458,112],{"emptyLinePlaceholder":111},[75,47460,47461,47464,47466,47468,47471],{"class":77,"line":16056},[75,47462,47463],{"class":85},"    surf ",[75,47465,142],{"class":141},[75,47467,43490],{"class":85},[75,47469,47470],{"class":193},"plot_surface",[75,47472,43495],{"class":85},[75,47474,47475],{"class":77,"line":16065},[75,47476,47477],{"class":85},"        X, Y, Z,\n",[75,47479,47480,47483,47485,47488],{"class":77,"line":16070},[75,47481,47482],{"class":309},"        facecolors",[75,47484,142],{"class":141},[75,47486,47487],{"class":145},"FC0",[75,47489,41629],{"class":85},[75,47491,47492,47495,47497,47499,47501,47504,47506,47508],{"class":77,"line":16076},[75,47493,47494],{"class":309},"        rstride",[75,47496,142],{"class":141},[75,47498,7477],{"class":145},[75,47500,206],{"class":85},[75,47502,47503],{"class":309},"cstride",[75,47505,142],{"class":141},[75,47507,7477],{"class":145},[75,47509,41629],{"class":85},[75,47511,47512,47515,47517,47519,47521,47523,47525,47527],{"class":77,"line":16107},[75,47513,47514],{"class":309},"        antialiased",[75,47516,142],{"class":141},[75,47518,376],{"class":145},[75,47520,206],{"class":85},[75,47522,11200],{"class":309},[75,47524,142],{"class":141},[75,47526,5186],{"class":145},[75,47528,41629],{"class":85},[75,47530,47531,47534,47536,47538,47541],{"class":77,"line":16153},[75,47532,47533],{"class":309},"        shade",[75,47535,142],{"class":141},[75,47537,376],{"class":145},[75,47539,47540],{"class":85},",  ",[75,47542,47543],{"class":118},"# use given facecolors directly\n",[75,47545,47546],{"class":77,"line":16174},[75,47547,15291],{"class":85},[75,47549,47550],{"class":77,"line":16185},[75,47551,112],{"emptyLinePlaceholder":111},[75,47553,47554],{"class":77,"line":16208},[75,47555,47556],{"class":118},"    # Colorbar (bound to ScalarMappable using same norm+cmap)\n",[75,47558,47559,47561,47563,47565,47567,47570,47572,47574,47576,47578,47580,47582,47584,47586,47588,47590],{"class":77,"line":16235},[75,47560,44099],{"class":85},[75,47562,142],{"class":141},[75,47564,44104],{"class":85},[75,47566,16495],{"class":193},[75,47568,47569],{"class":85},"(sm, ",[75,47571,15602],{"class":309},[75,47573,142],{"class":141},[75,47575,44116],{"class":85},[75,47577,44119],{"class":309},[75,47579,142],{"class":141},[75,47581,44124],{"class":145},[75,47583,206],{"class":85},[75,47585,44129],{"class":309},[75,47587,142],{"class":141},[75,47589,44134],{"class":145},[75,47591,318],{"class":85},[75,47593,47594,47596,47598,47600,47602],{"class":77,"line":16256},[75,47595,44141],{"class":85},[75,47597,44144],{"class":193},[75,47599,197],{"class":85},[75,47601,44149],{"class":134},[75,47603,318],{"class":85},[75,47605,47606],{"class":77,"line":16265},[75,47607,112],{"emptyLinePlaceholder":111},[75,47609,47610],{"class":77,"line":16274},[75,47611,47612],{"class":118},"    # Labels and in-axes title and info\n",[75,47614,47615,47617,47619,47621,47623],{"class":77,"line":16279},[75,47616,10549],{"class":85},[75,47618,3193],{"class":193},[75,47620,197],{"class":85},[75,47622,41854],{"class":134},[75,47624,318],{"class":85},[75,47626,47627,47629,47631,47633,47635],{"class":77,"line":16285},[75,47628,10549],{"class":85},[75,47630,3208],{"class":193},[75,47632,197],{"class":85},[75,47634,41859],{"class":134},[75,47636,318],{"class":85},[75,47638,47639,47641,47644,47646,47648],{"class":77,"line":16311},[75,47640,10549],{"class":85},[75,47642,47643],{"class":193},"set_zlabel",[75,47645,197],{"class":85},[75,47647,41864],{"class":134},[75,47649,318],{"class":85},[75,47651,47652,47655,47657,47659,47662],{"class":77,"line":16329},[75,47653,47654],{"class":85},"    title_txt ",[75,47656,142],{"class":141},[75,47658,43490],{"class":85},[75,47660,47661],{"class":193},"text2D",[75,47663,43495],{"class":85},[75,47665,47666,47668,47670,47673,47675,47677,47679,47681,47683,47685,47687,47689,47692],{"class":77,"line":16347},[75,47667,43641],{"class":145},[75,47669,206],{"class":85},[75,47671,47672],{"class":145},"0.98",[75,47674,206],{"class":85},[75,47676,5894],{"class":81},[75,47678,44194],{"class":134},[75,47680,12219],{"class":145},[75,47682,44199],{"class":85},[75,47684,44202],{"class":193},[75,47686,44205],{"class":85},[75,47688,12224],{"class":145},[75,47690,47691],{"class":134}," rotation (3D 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   info_txt ",[75,47786,142],{"class":141},[75,47788,43490],{"class":85},[75,47790,47661],{"class":193},[75,47792,43495],{"class":85},[75,47794,47795,47798,47800,47803,47805,47807,47809,47812,47814,47816,47818,47820,47822,47824,47826,47828,47830],{"class":77,"line":16522},[75,47796,47797],{"class":145},"        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       angle ",[75,47922,142],{"class":141},[75,47924,47925],{"class":85}," (frame ",[75,47927,2810],{"class":141},[75,47929,47930],{"class":85}," (k.frames ",[75,47932,200],{"class":141},[75,47934,1437],{"class":145},[75,47936,14936],{"class":85},[75,47938,500],{"class":141},[75,47940,44305],{"class":85},[75,47942,44308],{"class":118},[75,47944,47945,47948,47950,47952,47955,47957],{"class":77,"line":16671},[75,47946,47947],{"class":85},"        r_now ",[75,47949,142],{"class":141},[75,47951,46538],{"class":193},[75,47953,47954],{"class":85},"(axis, angle) ",[75,47956,19121],{"class":141},[75,47958,44330],{"class":85},[75,47960,47961,47964,47966,47968],{"class":77,"line":16684},[75,47962,47963],{"class":85},"        W_now ",[75,47965,142],{"class":141},[75,47967,46924],{"class":193},[75,47969,47970],{"class":85},"(r_now, n)\n",[75,47972,47973,47976,47978,47980,47982,47984,47987,47989,47991,47993,47995,47997,48000],{"class":77,"line":16701},[75,47974,47975],{"class":145},"        FC_now",[75,47977,14153],{"class":141},[75,47979,47430],{"class":193},[75,47981,197],{"class":85},[75,47983,42260],{"class":193},[75,47985,47986],{"class":85},"(W_now[:",[75,47988,200],{"class":141},[75,47990,7477],{"class":145},[75,47992,47444],{"class":85},[75,47994,200],{"class":141},[75,47996,7477],{"class":145},[75,47998,47999],{"class":85},"]))        ",[75,48001,48002],{"class":118},"# (M-1, N-1, 4)\n",[75,48004,48005,48008,48011,48013,48016,48018,48020,48022,48024,48026,48028,48030,48033],{"class":77,"line":16713},[75,48006,48007],{"class":85},"        surf.",[75,48009,48010],{"class":193},"set_facecolors",[75,48012,197],{"class":85},[75,48014,48015],{"class":145},"FC_now",[75,48017,6543],{"class":85},[75,48019,43067],{"class":193},[75,48021,197],{"class":85},[75,48023,200],{"class":141},[75,48025,7477],{"class":145},[75,48027,206],{"class":85},[75,48029,13162],{"class":145},[75,48031,48032],{"class":85},"))  ",[75,48034,48035],{"class":118},"# update in place\n",[75,48037,48038,48041,48043,48045,48047,48049,48051,48053,48055,48057,48059,48061,48063,48065,48067,48069,48071,48073,48075,48078,48080,48082],{"class":77,"line":16748},[75,48039,48040],{"class":85},"        info_txt.",[75,48042,44354],{"class":193},[75,48044,197],{"class":85},[75,48046,5894],{"class":81},[75,48048,44361],{"class":134},[75,48050,12219],{"class":145},[75,48052,42340],{"class":85},[75,48054,44369],{"class":81},[75,48056,12224],{"class":145},[75,48058,44374],{"class":134},[75,48060,44377],{"class":141},[75,48062,44380],{"class":134},[75,48064,12219],{"class":145},[75,48066,47909],{"class":85},[75,48068,3251],{"class":141},[75,48070,44389],{"class":145},[75,48072,2810],{"class":134},[75,48074,12219],{"class":145},[75,48076,48077],{"class":85},"k.frames",[75,48079,12224],{"class":145},[75,48081,12216],{"class":134},[75,48083,318],{"class":85},[75,48085,48086,48088],{"class":77,"line":16753},[75,48087,14450],{"class":81},[75,48089,48090],{"class":85}," (surf, info_txt, title_txt)\n",[75,48092,48093],{"class":77,"line":16779},[75,48094,112],{"emptyLinePlaceholder":111},[75,48096,48097,48099,48101,48103,48105,48107,48109,48112,48114,48116,48118,48120,48122,48124,48126],{"class":77,"line":16800},[75,48098,44418],{"class":85},[75,48100,142],{"class":141},[75,48102,44423],{"class":193},[75,48104,44426],{"class":85},[75,48106,44396],{"class":309},[75,48108,142],{"class":141},[75,48110,48111],{"class":85},"k.frames, ",[75,48113,44436],{"class":309},[75,48115,142],{"class":141},[75,48117,44441],{"class":145},[75,48119,206],{"class":85},[75,48121,44446],{"class":309},[75,48123,142],{"class":141},[75,48125,376],{"class":145},[75,48127,318],{"class":85},[75,48129,48130,48132,48134,48136,48139,48141,48143,48146,48148,48150,48152],{"class":77,"line":16805},[75,48131,19129],{"class":85},[75,48133,142],{"class":141},[75,48135,44775],{"class":193},[75,48137,48138],{"class":85},"(k.out_dir) ",[75,48140,2810],{"class":141},[75,48142,44783],{"class":81},[75,48144,48145],{"class":134},"\"wigner3d_pauli_",[75,48147,12219],{"class":145},[75,48149,42331],{"class":85},[75,48151,12224],{"class":145},[75,48153,44796],{"class":134},[75,48155,48156,48159,48161,48163,48165,48167,48169,48171,48173,48175,48177],{"class":77,"line":16836},[75,48157,48158],{"class":85},"    out.parent.",[75,48160,44460],{"class":193},[75,48162,197],{"class":85},[75,48164,44465],{"class":309},[75,48166,142],{"class":141},[75,48168,10504],{"class":145},[75,48170,206],{"class":85},[75,48172,44474],{"class":309},[75,48174,142],{"class":141},[75,48176,10504],{"class":145},[75,48178,318],{"class":85},[75,48180,48181,48183,48185,48188,48190,48192,48194,48196,48198,48200,48202,48204],{"class":77,"line":16869},[75,48182,44485],{"class":85},[75,48184,44488],{"class":193},[75,48186,48187],{"class":85},"(out.",[75,48189,44494],{"class":193},[75,48191,16413],{"class":85},[75,48193,44499],{"class":309},[75,48195,142],{"class":141},[75,48197,44504],{"class":193},[75,48199,197],{"class":85},[75,48201,44509],{"class":309},[75,48203,142],{"class":141},[75,48205,48206],{"class":85},"k.fps),\n",[75,48208,48209,48211,48213,48215,48217,48219,48221],{"class":77,"line":16906},[75,48210,44519],{"class":309},[75,48212,43541],{"class":141},[75,48214,197],{"class":85},[75,48216,44526],{"class":309},[75,48218,142],{"class":141},[75,48220,43561],{"class":134},[75,48222,283],{"class":85},[75,48224,48225,48227,48229],{"class":77,"line":16933},[75,48226,44537],{"class":85},[75,48228,44540],{"class":193},[75,48230,44543],{"class":85},[75,48232,48233,48235],{"class":77,"line":16956},[75,48234,14467],{"class":81},[75,48236,19206],{"class":85},[75,48238,48239],{"class":77,"line":16996},[75,48240,112],{"emptyLinePlaceholder":111},[75,48242,48244],{"class":77,"line":48243},162,[75,48245,112],{"emptyLinePlaceholder":111},[75,48247,48249,48251,48254,48256,48258],{"class":77,"line":48248},163,[75,48250,14380],{"class":81},[75,48252,48253],{"class":193}," animate_all_paulis_3d",[75,48255,197],{"class":85},[75,48257,44273],{"class":14388},[75,48259,48260],{"class":85},": Knobs3D,\n",[75,48262,48264,48267,48269,48271,48273,48275,48277,48279,48281,48283,48285,48287,48289,48291],{"class":77,"line":48263},164,[75,48265,48266],{"class":14388},"                          axes",[75,48268,42071],{"class":85},[75,48270,41840],{"class":141},[75,48272,206],{"class":85},[75,48274,41845],{"class":145},[75,48276,138],{"class":85},[75,48278,142],{"class":141},[75,48280,267],{"class":85},[75,48282,41854],{"class":134},[75,48284,206],{"class":85},[75,48286,41859],{"class":134},[75,48288,206],{"class":85},[75,48290,41864],{"class":134},[75,48292,48293],{"class":85},")) -> Iterable[Path]:\n",[75,48295,48297],{"class":77,"line":48296},165,[75,48298,48299],{"class":134},"    \"\"\"Generate and display GIFs for all requested Pauli axes; yield file paths.\"\"\"\n",[75,48301,48303,48305,48307,48309],{"class":77,"line":48302},166,[75,48304,15611],{"class":81},[75,48306,44760],{"class":85},[75,48308,3153],{"class":81},[75,48310,48311],{"class":85}," axes:\n",[75,48313,48315,48318,48320,48322,48325,48327],{"class":77,"line":48314},167,[75,48316,48317],{"class":85},"        path ",[75,48319,142],{"class":141},[75,48321,47015],{"class":193},[75,48323,48324],{"class":85},"(axname.",[75,48326,42241],{"class":193},[75,48328,48329],{"class":85},"(), k)\n",[75,48331,48333,48336,48338,48340,48342,48344,48346],{"class":77,"line":48332},168,[75,48334,48335],{"class":193},"        display",[75,48337,197],{"class":85},[75,48339,44869],{"class":193},[75,48341,197],{"class":85},[75,48343,44874],{"class":309},[75,48345,44877],{"class":141},[75,48347,48348],{"class":85},"(path)))\n",[75,48350,48352,48355,48357,48359,48362,48364,48366,48368,48370],{"class":77,"line":48351},169,[75,48353,48354],{"class":141},"        print",[75,48356,197],{"class":85},[75,48358,5894],{"class":81},[75,48360,48361],{"class":134},"\"Saved: ",[75,48363,12219],{"class":145},[75,48365,33466],{"class":85},[75,48367,12224],{"class":145},[75,48369,12216],{"class":134},[75,48371,318],{"class":85},[75,48373,48375,48378],{"class":77,"line":48374},170,[75,48376,48377],{"class":81},"        yield",[75,48379,48380],{"class":85}," path\n",[75,48382,48384],{"class":77,"line":48383},171,[75,48385,112],{"emptyLinePlaceholder":111},[75,48387,48389],{"class":77,"line":48388},172,[75,48390,112],{"emptyLinePlaceholder":111},[75,48392,48394],{"class":77,"line":48393},173,[75,48395,48396],{"class":118},"# ------------------------------ Run all three gates ------------------------------\n",[75,48398,48400,48403,48405,48407,48409],{"class":77,"line":48399},174,[75,48401,48402],{"class":145},"K3D",[75,48404,14153],{"class":141},[75,48406,46073],{"class":193},[75,48408,44575],{"class":85},[75,48410,48411],{"class":118},"# adjust as desired\n",[75,48413,48415,48417,48419,48421],{"class":77,"line":48414},175,[75,48416,44604],{"class":309},[75,48418,142],{"class":141},[75,48420,44609],{"class":134},[75,48422,41629],{"class":85},[75,48424,48426,48428,48430,48432,48434,48436,48438,48441,48443,48445,48447,48450,48452,48454,48456,48458],{"class":77,"line":48425},176,[75,48427,44616],{"class":309},[75,48429,142],{"class":141},[75,48431,44621],{"class":145},[75,48433,206],{"class":85},[75,48435,42837],{"class":309},[75,48437,142],{"class":141},[75,48439,48440],{"class":145},"64",[75,48442,206],{"class":85},[75,48444,42846],{"class":309},[75,48446,142],{"class":141},[75,48448,48449],{"class":145},"128",[75,48451,206],{"class":85},[75,48453,44509],{"class":309},[75,48455,142],{"class":141},[75,48457,752],{"class":145},[75,48459,41629],{"class":85},[75,48461,48463,48466,48468,48471,48473,48475,48478,48480],{"class":77,"line":48462},177,[75,48464,48465],{"class":309},"    elev",[75,48467,142],{"class":141},[75,48469,48470],{"class":145},"25.0",[75,48472,206],{"class":85},[75,48474,47345],{"class":309},[75,48476,48477],{"class":141},"=-",[75,48479,14329],{"class":145},[75,48481,41629],{"class":85},[75,48483,48485,48487,48489,48491],{"class":77,"line":48484},178,[75,48486,44668],{"class":309},[75,48488,142],{"class":141},[75,48490,44673],{"class":134},[75,48492,41629],{"class":85},[75,48494,48496,48498,48500,48502,48504,48506,48508],{"class":77,"line":48495},179,[75,48497,44680],{"class":309},[75,48499,142],{"class":141},[75,48501,197],{"class":85},[75,48503,41606],{"class":145},[75,48505,206],{"class":85},[75,48507,17832],{"class":145},[75,48509,3003],{"class":85},[75,48511,48513,48516,48518,48520],{"class":77,"line":48512},180,[75,48514,48515],{"class":309},"    info_fontsize",[75,48517,142],{"class":141},[75,48519,3121],{"class":145},[75,48521,41629],{"class":85},[75,48523,48525],{"class":77,"line":48524},181,[75,48526,318],{"class":85},[75,48528,48530],{"class":77,"line":48529},182,[75,48531,112],{"emptyLinePlaceholder":111},[75,48533,48535,48538,48540,48543,48545,48547,48549,48552,48554,48556,48558,48560,48562,48564,48566],{"class":77,"line":48534},183,[75,48536,48537],{"class":141},"list",[75,48539,197],{"class":85},[75,48541,48542],{"class":193},"animate_all_paulis_3d",[75,48544,197],{"class":85},[75,48546,48402],{"class":145},[75,48548,206],{"class":85},[75,48550,48551],{"class":309},"axes",[75,48553,142],{"class":141},[75,48555,197],{"class":85},[75,48557,41854],{"class":134},[75,48559,206],{"class":85},[75,48561,41859],{"class":134},[75,48563,206],{"class":85},[75,48565,41864],{"class":134},[75,48567,48568],{"class":85},")))\n",[65,48570,48573],{"className":48571,"code":48572,"language":34131,"meta":70},[41256],"Saved: \u002Fcontent\u002Fwigner3d_pauli_x.gif\n",[72,48574,48572],{"__ignoreMap":70},[65,48576,48579],{"className":48577,"code":48578,"language":34131,"meta":70},[41256],"Saved: \u002Fcontent\u002Fwigner3d_pauli_y.gif\n",[72,48580,48578],{"__ignoreMap":70},[65,48582,48585],{"className":48583,"code":48584,"language":34131,"meta":70},[41256],"Saved: \u002Fcontent\u002Fwigner3d_pauli_z.gif\n",[72,48586,48584],{"__ignoreMap":70},[391,48588],{"alt":1608,"src":48589},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-05.webp",[391,48591],{"alt":1922,"src":48592},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-06.webp",[391,48594],{"alt":2279,"src":48595},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-07.webp",[65,48597,48599],{"className":67,"code":48598,"language":69,"meta":70,"style":70},"# @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",[72,48600,48601,48606,48611,48615,48625,48629,48639,48649,48659,48663,48673,48683,48693,48703,48713,48723,48727,48731,48735,48744,48749,48785,48797,48807,48839,48849,48859,48869,48879,48889,48901,48913,48937,48947,48951,48955,48960,48992,49000,49036,49072,49108,49112,49136,49152,49162,49166,49170,49203,49219,49231,49243,49253,49265,49329,49345,49361,49397,49401,49405,49429,49445,49467,49487,49507,49527,49539,49559,49565,49569,49573,49589,49649,49653,49657,49671,49676,49686,49696,49700,49720,49740,49752,49756,49782,49808,49830,49842,49864,49873,49877,49903,49927,49947,49959,49979,49983,49988,49998,50014,50025,50054,50058,50070,50074,50084,50102,50120,50130,50134,50138,50172,50184,50188,50200,50212,50224,50236,50251,50283,50323,50333,50337,50349,50385,50393,50433,50443,50447,50451,50467,50494,50509,50519,50548,50578,50586,50639,50670,50674,50680,50684,50716,50727,50751,50777,50793,50801,50807,50811,50815,50820,50831,50859,50879,50883,50898,50920],{"__ignoreMap":70},[75,48602,48603],{"class":77,"line":78},[75,48604,48605],{"class":118},"# @title Cell 6 — 3D Wigner animation (custom axis–angle; colors-only, no geometry edits)\n",[75,48607,48608],{"class":77,"line":95},[75,48609,48610],{"class":118},"# PEP 8\u002F257 compliant. Rotates Bloch vector about an arbitrary axis by a target angle.\n",[75,48612,48613],{"class":77,"line":108},[75,48614,112],{"emptyLinePlaceholder":111},[75,48616,48617,48619,48621,48623],{"class":77,"line":115},[75,48618,2700],{"class":81},[75,48620,41502],{"class":15980},[75,48622,41505],{"class":81},[75,48624,41508],{"class":85},[75,48626,48627],{"class":77,"line":122},[75,48628,112],{"emptyLinePlaceholder":111},[75,48630,48631,48633,48635,48637],{"class":77,"line":128},[75,48632,2700],{"class":81},[75,48634,14064],{"class":85},[75,48636,82],{"class":81},[75,48638,14069],{"class":85},[75,48640,48641,48643,48645,48647],{"class":77,"line":149},[75,48642,2700],{"class":81},[75,48644,41661],{"class":85},[75,48646,82],{"class":81},[75,48648,41666],{"class":85},[75,48650,48651,48653,48655,48657],{"class":77,"line":154},[75,48652,2700],{"class":81},[75,48654,14076],{"class":85},[75,48656,82],{"class":81},[75,48658,14081],{"class":85},[75,48660,48661],{"class":77,"line":160},[75,48662,112],{"emptyLinePlaceholder":111},[75,48664,48665,48667,48669,48671],{"class":77,"line":171},[75,48666,82],{"class":81},[75,48668,86],{"class":85},[75,48670,89],{"class":81},[75,48672,92],{"class":85},[75,48674,48675,48677,48679,48681],{"class":77,"line":182},[75,48676,82],{"class":81},[75,48678,100],{"class":85},[75,48680,89],{"class":81},[75,48682,105],{"class":85},[75,48684,48685,48687,48689,48691],{"class":77,"line":222},[75,48686,2700],{"class":81},[75,48688,41715],{"class":85},[75,48690,82],{"class":81},[75,48692,41720],{"class":85},[75,48694,48695,48697,48699,48701],{"class":77,"line":227},[75,48696,2700],{"class":81},[75,48698,46027],{"class":85},[75,48700,82],{"class":81},[75,48702,46032],{"class":85},[75,48704,48705,48707,48709,48711],{"class":77,"line":233},[75,48706,2700],{"class":81},[75,48708,46039],{"class":85},[75,48710,82],{"class":81},[75,48712,46044],{"class":85},[75,48714,48715,48717,48719,48721],{"class":77,"line":286},[75,48716,2700],{"class":81},[75,48718,41727],{"class":85},[75,48720,82],{"class":81},[75,48722,41732],{"class":85},[75,48724,48725],{"class":77,"line":291},[75,48726,112],{"emptyLinePlaceholder":111},[75,48728,48729],{"class":77,"line":297},[75,48730,112],{"emptyLinePlaceholder":111},[75,48732,48733],{"class":77,"line":321},[75,48734,14131],{"class":193},[75,48736,48737,48739,48742],{"class":77,"line":336},[75,48738,14136],{"class":81},[75,48740,48741],{"class":14139}," Knobs3DCustom",[75,48743,5800],{"class":85},[75,48745,48746],{"class":77,"line":351},[75,48747,48748],{"class":134},"    \"\"\"User controls for 3D custom axis–angle Wigner animation.\"\"\"\n",[75,48750,48751,48754,48756,48758,48760,48762,48764,48766,48768,48770,48772,48774,48776,48778,48780,48782],{"class":77,"line":366},[75,48752,48753],{"class":85},"    axis_vec: Tuple[",[75,48755,14150],{"class":141},[75,48757,206],{"class":85},[75,48759,14150],{"class":141},[75,48761,206],{"class":85},[75,48763,14150],{"class":141},[75,48765,138],{"class":85},[75,48767,142],{"class":141},[75,48769,267],{"class":85},[75,48771,14324],{"class":145},[75,48773,206],{"class":85},[75,48775,630],{"class":145},[75,48777,206],{"class":85},[75,48779,14324],{"class":145},[75,48781,216],{"class":85},[75,48783,48784],{"class":118},"# default: y-axis\n",[75,48786,48787,48790,48792,48794],{"class":77,"line":381},[75,48788,48789],{"class":85},"    angle_deg: ",[75,48791,14150],{"class":141},[75,48793,14153],{"class":141},[75,48795,48796],{"class":145}," 45.0\n",[75,48798,48799,48801,48803,48805],{"class":77,"line":984},[75,48800,41871],{"class":85},[75,48802,41840],{"class":141},[75,48804,14153],{"class":141},[75,48806,41878],{"class":134},[75,48808,48809,48811,48813,48815,48817,48819,48821,48823,48825,48827,48829,48831,48833,48835,48837],{"class":77,"line":997},[75,48810,41883],{"class":85},[75,48812,14150],{"class":141},[75,48814,206],{"class":85},[75,48816,14150],{"class":141},[75,48818,206],{"class":85},[75,48820,14150],{"class":141},[75,48822,138],{"class":85},[75,48824,142],{"class":141},[75,48826,267],{"class":85},[75,48828,14324],{"class":145},[75,48830,206],{"class":85},[75,48832,14324],{"class":145},[75,48834,206],{"class":85},[75,48836,630],{"class":145},[75,48838,318],{"class":85},[75,48840,48841,48843,48845,48847],{"class":77,"line":1325},[75,48842,41916],{"class":85},[75,48844,14299],{"class":141},[75,48846,14153],{"class":141},[75,48848,41923],{"class":145},[75,48850,48851,48853,48855,48857],{"class":77,"line":1338},[75,48852,41928],{"class":85},[75,48854,14299],{"class":141},[75,48856,14153],{"class":141},[75,48858,46208],{"class":145},[75,48860,48861,48863,48865,48867],{"class":77,"line":1343},[75,48862,41940],{"class":85},[75,48864,14299],{"class":141},[75,48866,14153],{"class":141},[75,48868,46219],{"class":145},[75,48870,48871,48873,48875,48877],{"class":77,"line":1356},[75,48872,41952],{"class":85},[75,48874,14299],{"class":141},[75,48876,14153],{"class":141},[75,48878,41959],{"class":145},[75,48880,48881,48883,48885,48887],{"class":77,"line":1887},[75,48882,46234],{"class":85},[75,48884,14150],{"class":141},[75,48886,14153],{"class":141},[75,48888,46241],{"class":145},[75,48890,48891,48893,48895,48897,48899],{"class":77,"line":1900},[75,48892,46246],{"class":85},[75,48894,14150],{"class":141},[75,48896,14153],{"class":141},[75,48898,14186],{"class":141},[75,48900,46255],{"class":145},[75,48902,48903,48906,48908,48910],{"class":77,"line":1913},[75,48904,48905],{"class":85},"    out_path: ",[75,48907,41840],{"class":141},[75,48909,14153],{"class":141},[75,48911,48912],{"class":134}," \"\u002Fcontent\u002Fwigner3d_custom_y45.gif\"\n",[75,48914,48915,48917,48919,48921,48923,48925,48927,48929,48931,48933,48935],{"class":77,"line":2657},[75,48916,41976],{"class":85},[75,48918,14150],{"class":141},[75,48920,206],{"class":85},[75,48922,14150],{"class":141},[75,48924,138],{"class":85},[75,48926,142],{"class":141},[75,48928,267],{"class":85},[75,48930,41606],{"class":145},[75,48932,206],{"class":85},[75,48934,17832],{"class":145},[75,48936,318],{"class":85},[75,48938,48939,48941,48943,48945],{"class":77,"line":3177},[75,48940,42013],{"class":85},[75,48942,14299],{"class":141},[75,48944,14153],{"class":141},[75,48946,42020],{"class":145},[75,48948,48949],{"class":77,"line":3188},[75,48950,112],{"emptyLinePlaceholder":111},[75,48952,48953],{"class":77,"line":3203},[75,48954,112],{"emptyLinePlaceholder":111},[75,48956,48957],{"class":77,"line":3218},[75,48958,48959],{"class":118},"# ------------- helpers (self-contained; safe for this cell alone) 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animate_wigner_custom_3d",[75,49665,197],{"class":85},[75,49667,44273],{"class":14388},[75,49669,49670],{"class":85},": Knobs3DCustom) -> Path:\n",[75,49672,49673],{"class":77,"line":15160},[75,49674,49675],{"class":134},"    \"\"\"Animate a 3D sphere with colors driven by Wigner under axis–angle rotation.\"\"\"\n",[75,49677,49678,49680,49682,49684],{"class":77,"line":15206},[75,49679,47040],{"class":85},[75,49681,142],{"class":141},[75,49683,46319],{"class":193},[75,49685,47047],{"class":85},[75,49687,49688,49690,49692,49694],{"class":77,"line":15235},[75,49689,47052],{"class":85},[75,49691,142],{"class":141},[75,49693,46748],{"class":193},[75,49695,47059],{"class":85},[75,49697,49698],{"class":77,"line":15288},[75,49699,112],{"emptyLinePlaceholder":111},[75,49701,49702,49704,49706,49708,49710,49712,49714,49716,49718],{"class":77,"line":15294},[75,49703,47073],{"class":85},[75,49705,142],{"class":141},[75,49707,190],{"class":85},[75,49709,1179],{"class":193},[75,49711,42938],{"class":85},[75,49713,500],{"class":141},[75,49715,190],{"class":85},[75,49717,2845],{"class":193},[75,49719,42947],{"class":85},[75,49721,49722,49724,49726,49728,49730,49732,49734,49736,49738],{"class":77,"line":15311},[75,49723,47094],{"class":85},[75,49725,142],{"class":141},[75,49727,190],{"class":85},[75,49729,1179],{"class":193},[75,49731,42938],{"class":85},[75,49733,500],{"class":141},[75,49735,190],{"class":85},[75,49737,1179],{"class":193},[75,49739,42947],{"class":85},[75,49741,49742,49744,49746,49748,49750],{"class":77,"line":15316},[75,49743,47115],{"class":85},[75,49745,142],{"class":141},[75,49747,190],{"class":85},[75,49749,2845],{"class":193},[75,49751,42982],{"class":85},[75,49753,49754],{"class":77,"line":15363},[75,49755,112],{"emptyLinePlaceholder":111},[75,49757,49758,49760,49762,49764,49766,49768,49770,49772,49774,49776,49778,49780],{"class":77,"line":15371},[75,49759,43928],{"class":85},[75,49761,142],{"class":141},[75,49763,15214],{"class":145},[75,49765,14186],{"class":141},[75,49767,19174],{"class":85},[75,49769,14668],{"class":193},[75,49771,197],{"class":85},[75,49773,43051],{"class":145},[75,49775,14860],{"class":85},[75,49777,2810],{"class":141},[75,49779,15263],{"class":145},[75,49781,318],{"class":85},[75,49783,49784,49786,49788,49790,49792,49794,49796,49798,49800,49802,49804,49806],{"class":77,"line":15419},[75,49785,43955],{"class":85},[75,49787,142],{"class":141},[75,49789,15214],{"class":145},[75,49791,11902],{"class":141},[75,49793,19174],{"class":85},[75,49795,14668],{"class":193},[75,49797,197],{"class":85},[75,49799,43051],{"class":145},[75,49801,14860],{"class":85},[75,49803,2810],{"class":141},[75,49805,15263],{"class":145},[75,49807,318],{"class":85},[75,49809,49810,49812,49814,49816,49818,49820,49822,49824,49826,49828],{"class":77,"line":15442},[75,49811,47189],{"class":85},[75,49813,142],{"class":141},[75,49815,47194],{"class":193},[75,49817,197],{"class":85},[75,49819,47199],{"class":309},[75,49821,142],{"class":141},[75,49823,44078],{"class":85},[75,49825,44081],{"class":309},[75,49827,142],{"class":141},[75,49829,47210],{"class":85},[75,49831,49832,49834,49836,49838,49840],{"class":77,"line":15482},[75,49833,47215],{"class":85},[75,49835,142],{"class":141},[75,49837,3093],{"class":85},[75,49839,47222],{"class":193},[75,49841,389],{"class":85},[75,49843,49844,49846,49848,49850,49852,49854,49856,49858,49860,49862],{"class":77,"line":15520},[75,49845,47229],{"class":85},[75,49847,142],{"class":141},[75,49849,47234],{"class":193},[75,49851,197],{"class":85},[75,49853,42260],{"class":309},[75,49855,142],{"class":141},[75,49857,47243],{"class":85},[75,49859,16480],{"class":309},[75,49861,142],{"class":141},[75,49863,47250],{"class":85},[75,49865,49866,49868,49870],{"class":77,"line":15525},[75,49867,47255],{"class":85},[75,49869,47258],{"class":193},[75,49871,49872],{"class":85},"([])\n",[75,49874,49875],{"class":77,"line":15540},[75,49876,112],{"emptyLinePlaceholder":111},[75,49878,49879,49881,49883,49885,49887,49889,49891,49893,49895,49897,49899,49901],{"class":77,"line":15545},[75,49880,47273],{"class":85},[75,49882,142],{"class":141},[75,49884,3093],{"class":85},[75,49886,865],{"class":193},[75,49888,197],{"class":85},[75,49890,870],{"class":309},[75,49892,142],{"class":141},[75,49894,47288],{"class":85},[75,49896,10499],{"class":309},[75,49898,142],{"class":141},[75,49900,10504],{"class":145},[75,49902,318],{"class":85},[75,49904,49905,49907,49909,49911,49913,49915,49917,49919,49921,49923,49925],{"class":77,"line":15551},[75,49906,47301],{"class":85},[75,49908,142],{"class":141},[75,49910,44104],{"class":85},[75,49912,47308],{"class":193},[75,49914,197],{"class":85},[75,49916,47313],{"class":145},[75,49918,206],{"class":85},[75,49920,47318],{"class":309},[75,49922,142],{"class":141},[75,49924,47323],{"class":134},[75,49926,318],{"class":85},[75,49928,49929,49931,49933,49935,49937,49939,49941,49943,49945],{"class":77,"line":15592},[75,49930,10549],{"class":85},[75,49932,47332],{"class":193},[75,49934,197],{"class":85},[75,49936,47337],{"class":309},[75,49938,142],{"class":141},[75,49940,47342],{"class":85},[75,49942,47345],{"class":309},[75,49944,142],{"class":141},[75,49946,47350],{"class":85},[75,49948,49949,49951,49953,49955,49957],{"class":77,"line":15608},[75,49950,14433],{"class":81},[75,49952,47357],{"class":141},[75,49954,44189],{"class":85},[75,49956,47362],{"class":134},[75,49958,15605],{"class":85},[75,49960,49961,49963,49965,49967,49969,49971,49973,49975,49977],{"class":77,"line":15624},[75,49962,15627],{"class":85},[75,49964,47371],{"class":193},[75,49966,16298],{"class":85},[75,49968,7477],{"class":145},[75,49970,206],{"class":85},[75,49972,7477],{"class":145},[75,49974,206],{"class":85},[75,49976,7477],{"class":145},[75,49978,283],{"class":85},[75,49980,49981],{"class":77,"line":15652},[75,49982,112],{"emptyLinePlaceholder":111},[75,49984,49985],{"class":77,"line":15657},[75,49986,49987],{"class":118},"    # Initial facecolors\n",[75,49989,49990,49993,49995],{"class":77,"line":15662},[75,49991,49992],{"class":85},"    angle0 ",[75,49994,142],{"class":141},[75,49996,49997],{"class":145}," 0.0\n",[75,49999,50000,50003,50005,50007,50010,50012],{"class":77,"line":15668},[75,50001,50002],{"class":85},"    r_init ",[75,50004,142],{"class":141},[75,50006,49176],{"class":193},[75,50008,50009],{"class":85},"(k.axis_vec, angle0) ",[75,50011,19121],{"class":141},[75,50013,44330],{"class":85},[75,50015,50016,50018,50020,50022],{"class":77,"line":15673},[75,50017,47399],{"class":85},[75,50019,142],{"class":141},[75,50021,46924],{"class":193},[75,50023,50024],{"class":85},"(r_init, n)\n",[75,50026,50027,50029,50031,50033,50035,50037,50039,50041,50043,50045,50047,50049,50051],{"class":77,"line":15698},[75,50028,47425],{"class":145},[75,50030,14153],{"class":141},[75,50032,47430],{"class":193},[75,50034,197],{"class":85},[75,50036,42260],{"class":193},[75,50038,47437],{"class":85},[75,50040,200],{"class":141},[75,50042,7477],{"class":145},[75,50044,47444],{"class":85},[75,50046,200],{"class":141},[75,50048,7477],{"class":145},[75,50050,47451],{"class":85},[75,50052,50053],{"class":118},"# (M-1,N-1,4)\n",[75,50055,50056],{"class":77,"line":15723},[75,50057,112],{"emptyLinePlaceholder":111},[75,50059,50060,50062,50064,50066,50068],{"class":77,"line":15728},[75,50061,47463],{"class":85},[75,50063,142],{"class":141},[75,50065,43490],{"class":85},[75,50067,47470],{"class":193},[75,50069,43495],{"class":85},[75,50071,50072],{"class":77,"line":15762},[75,50073,47477],{"class":85},[75,50075,50076,50078,50080,50082],{"class":77,"line":15792},[75,50077,47482],{"class":309},[75,50079,142],{"class":141},[75,50081,47487],{"class":145},[75,50083,41629],{"class":85},[75,50085,50086,50088,50090,50092,50094,50096,50098,50100],{"class":77,"line":15829},[75,50087,47494],{"class":309},[75,50089,142],{"class":141},[75,50091,7477],{"class":145},[75,50093,206],{"class":85},[75,50095,47503],{"class":309},[75,50097,142],{"class":141},[75,50099,7477],{"class":145},[75,50101,41629],{"class":85},[75,50103,50104,50106,50108,50110,50112,50114,50116,50118],{"class":77,"line":15834},[75,50105,47514],{"class":309},[75,50107,142],{"class":141},[75,50109,376],{"class":145},[75,50111,206],{"class":85},[75,50113,11200],{"class":309},[75,50115,142],{"class":141},[75,50117,5186],{"class":145},[75,50119,41629],{"class":85},[75,50121,50122,50124,50126,50128],{"class":77,"line":15840},[75,50123,47533],{"class":309},[75,50125,142],{"class":141},[75,50127,376],{"class":145},[75,50129,41629],{"class":85},[75,50131,50132],{"class":77,"line":15873},[75,50133,15291],{"class":85},[75,50135,50136],{"class":77,"line":15931},[75,50137,112],{"emptyLinePlaceholder":111},[75,50139,50140,50142,50144,50146,50148,50150,50152,50154,50156,50158,50160,50162,50164,50166,50168,50170],{"class":77,"line":15953},[75,50141,44099],{"class":85},[75,50143,142],{"class":141},[75,50145,44104],{"class":85},[75,50147,16495],{"class":193},[75,50149,47569],{"class":85},[75,50151,15602],{"class":309},[75,50153,142],{"class":141},[75,50155,44116],{"class":85},[75,50157,44119],{"class":309},[75,50159,142],{"class":141},[75,50161,44124],{"class":145},[75,50163,206],{"class":85},[75,50165,44129],{"class":309},[75,50167,142],{"class":141},[75,50169,44134],{"class":145},[75,50171,318],{"class":85},[75,50173,50174,50176,50178,50180,50182],{"class":77,"line":15968},[75,50175,44141],{"class":85},[75,50177,44144],{"class":193},[75,50179,197],{"class":85},[75,50181,44149],{"class":134},[75,50183,318],{"class":85},[75,50185,50186],{"class":77,"line":15996},[75,50187,112],{"emptyLinePlaceholder":111},[75,50189,50190,50192,50194,50196,50198],{"class":77,"line":16026},[75,50191,10549],{"class":85},[75,50193,3193],{"class":193},[75,50195,197],{"class":85},[75,50197,41854],{"class":134},[75,50199,318],{"class":85},[75,50201,50202,50204,50206,50208,50210],{"class":77,"line":16047},[75,50203,10549],{"class":85},[75,50205,3208],{"class":193},[75,50207,197],{"class":85},[75,50209,41859],{"class":134},[75,50211,318],{"class":85},[75,50213,50214,50216,50218,50220,50222],{"class":77,"line":16056},[75,50215,10549],{"class":85},[75,50217,47643],{"class":193},[75,50219,197],{"class":85},[75,50221,41864],{"class":134},[75,50223,318],{"class":85},[75,50225,50226,50228,50230,50232,50234],{"class":77,"line":16065},[75,50227,47654],{"class":85},[75,50229,142],{"class":141},[75,50231,43490],{"class":85},[75,50233,47661],{"class":193},[75,50235,43495],{"class":85},[75,50237,50238,50240,50242,50244,50246,50249],{"class":77,"line":16070},[75,50239,43641],{"class":145},[75,50241,206],{"class":85},[75,50243,47672],{"class":145},[75,50245,206],{"class":85},[75,50247,50248],{"class":134},"\"Wigner - custom 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ang) ",[75,50506,19121],{"class":141},[75,50508,44330],{"class":85},[75,50510,50511,50513,50515,50517],{"class":77,"line":16347},[75,50512,47963],{"class":85},[75,50514,142],{"class":141},[75,50516,46924],{"class":193},[75,50518,47970],{"class":85},[75,50520,50521,50523,50525,50527,50529,50531,50533,50535,50537,50539,50541,50543,50546],{"class":77,"line":16360},[75,50522,47975],{"class":145},[75,50524,14153],{"class":141},[75,50526,47430],{"class":193},[75,50528,197],{"class":85},[75,50530,42260],{"class":193},[75,50532,47986],{"class":85},[75,50534,200],{"class":141},[75,50536,7477],{"class":145},[75,50538,47444],{"class":85},[75,50540,200],{"class":141},[75,50542,7477],{"class":145},[75,50544,50545],{"class":85},"]))          ",[75,50547,50053],{"class":118},[75,50549,50550,50552,50554,50556,50558,50560,50562,50564,50566,50568,50570,50572,50575],{"class":77,"line":16365},[75,50551,48007],{"class":85},[75,50553,48010],{"class":193},[75,50555,197],{"class":85},[75,50557,48015],{"class":145},[75,50559,6543],{"class":85},[75,50561,43067],{"class":193},[75,50563,197],{"class":85},[75,50565,200],{"class":141},[75,50567,7477],{"class":145},[75,50569,206],{"class":85},[75,50571,13162],{"class":145},[75,50573,50574],{"class":85},"))    ",[75,50576,50577],{"class":118},"# in-place update\n",[75,50579,50580,50582,50584],{"class":77,"line":16396},[75,50581,48040],{"class":85},[75,50583,44354],{"class":193},[75,50585,43495],{"class":85},[75,50587,50588,50590,50592,50594,50597,50599,50601,50603,50605,50607,50609,50611,50613,50615,50617,50619,50621,50623,50625,50627,50629,50631,50633,50635,50637],{"class":77,"line":16441},[75,50589,45492],{"class":81},[75,50591,45495],{"class":134},[75,50593,12219],{"class":145},[75,50595,50596],{"class":85},"k.axis_vec[",[75,50598,5186],{"class":145},[75,50600,32873],{"class":85},[75,50602,45507],{"class":81},[75,50604,12224],{"class":145},[75,50606,206],{"class":134},[75,50608,12219],{"class":145},[75,50610,50596],{"class":85},[75,50612,7477],{"class":145},[75,50614,32873],{"class":85},[75,50616,45507],{"class":81},[75,50618,12224],{"class":145},[75,50620,206],{"class":134},[75,50622,12219],{"class":145},[75,50624,50596],{"class":85},[75,50626,270],{"class":145},[75,50628,32873],{"class":85},[75,50630,45507],{"class":81},[75,50632,12224],{"class":145},[75,50634,5175],{"class":134},[75,50636,44377],{"class":141},[75,50638,45544],{"class":134},[75,50640,50641,50643,50645,50647,50649,50651,50653,50655,50657,50659,50661,50664,50666,50668],{"class":77,"line":16490},[75,50642,45492],{"class":81},[75,50644,44361],{"class":134},[75,50646,12219],{"class":145},[75,50648,45555],{"class":85},[75,50650,45558],{"class":193},[75,50652,45561],{"class":85},[75,50654,15913],{"class":81},[75,50656,12224],{"class":145},[75,50658,45568],{"class":134},[75,50660,12219],{"class":145},[75,50662,50663],{"class":85},"k.angle_deg",[75,50665,15913],{"class":81},[75,50667,12224],{"class":145},[75,50669,45580],{"class":134},[75,50671,50672],{"class":77,"line":16522},[75,50673,45585],{"class":85},[75,50675,50676,50678],{"class":77,"line":16551},[75,50677,14450],{"class":81},[75,50679,48090],{"class":85},[75,50681,50682],{"class":77,"line":16574},[75,50683,112],{"emptyLinePlaceholder":111},[75,50685,50686,50688,50690,50692,50694,50696,50698,50700,50702,50704,50706,50708,50710,50712,50714],{"class":77,"line":16618},[75,50687,44418],{"class":85},[75,50689,142],{"class":141},[75,50691,44423],{"class":193},[75,50693,44426],{"class":85},[75,50695,44396],{"class":309},[75,50697,142],{"class":141},[75,50699,48111],{"class":85},[75,50701,44436],{"class":309},[75,50703,142],{"class":141},[75,50705,44441],{"class":145},[75,50707,206],{"class":85},[75,50709,44446],{"class":309},[75,50711,142],{"class":141},[75,50713,376],{"class":145},[75,50715,318],{"class":85},[75,50717,50718,50720,50722,50724],{"class":77,"line":16627},[75,50719,19129],{"class":85},[75,50721,142],{"class":141},[75,50723,44775],{"class":193},[75,50725,50726],{"class":85},"(k.out_path)\n",[75,50728,50729,50731,50733,50735,50737,50739,50741,50743,50745,50747,50749],{"class":77,"line":16632},[75,50730,48158],{"class":85},[75,50732,44460],{"class":193},[75,50734,197],{"class":85},[75,50736,44465],{"class":309},[75,50738,142],{"class":141},[75,50740,10504],{"class":145},[75,50742,206],{"class":85},[75,50744,44474],{"class":309},[75,50746,142],{"class":141},[75,50748,10504],{"class":145},[75,50750,318],{"class":85},[75,50752,50753,50755,50757,50759,50761,50763,50765,50767,50769,50771,50773,50775],{"class":77,"line":16638},[75,50754,44485],{"class":85},[75,50756,44488],{"class":193},[75,50758,48187],{"class":85},[75,50760,44494],{"class":193},[75,50762,16413],{"class":85},[75,50764,44499],{"class":309},[75,50766,142],{"class":141},[75,50768,44504],{"class":193},[75,50770,197],{"class":85},[75,50772,44509],{"class":309},[75,50774,142],{"class":141},[75,50776,48206],{"class":85},[75,50778,50779,50781,50783,50785,50787,50789,50791],{"class":77,"line":16654},[75,50780,44519],{"class":309},[75,50782,43541],{"class":141},[75,50784,197],{"class":85},[75,50786,44526],{"class":309},[75,50788,142],{"class":141},[75,50790,43561],{"class":134},[75,50792,283],{"class":85},[75,50794,50795,50797,50799],{"class":77,"line":16671},[75,50796,44537],{"class":85},[75,50798,44540],{"class":193},[75,50800,44543],{"class":85},[75,50802,50803,50805],{"class":77,"line":16684},[75,50804,14467],{"class":81},[75,50806,19206],{"class":85},[75,50808,50809],{"class":77,"line":16701},[75,50810,112],{"emptyLinePlaceholder":111},[75,50812,50813],{"class":77,"line":16713},[75,50814,112],{"emptyLinePlaceholder":111},[75,50816,50817],{"class":77,"line":16748},[75,50818,50819],{"class":118},"# ----------------------------- Example run ------------------------------------\n",[75,50821,50822,50825,50827,50829],{"class":77,"line":16753},[75,50823,50824],{"class":145},"K3C",[75,50826,14153],{"class":141},[75,50828,48741],{"class":193},[75,50830,43495],{"class":85},[75,50832,50833,50835,50837,50839,50841,50843,50845,50847,50849,50851,50853,50855,50857],{"class":77,"line":16779},[75,50834,45748],{"class":309},[75,50836,142],{"class":141},[75,50838,197],{"class":85},[75,50840,14324],{"class":145},[75,50842,206],{"class":85},[75,50844,630],{"class":145},[75,50846,206],{"class":85},[75,50848,14324],{"class":145},[75,50850,10496],{"class":85},[75,50852,45573],{"class":309},[75,50854,142],{"class":141},[75,50856,45774],{"class":145},[75,50858,41629],{"class":85},[75,50860,50861,50863,50865,50867,50869,50872,50874,50877],{"class":77,"line":16800},[75,50862,44604],{"class":309},[75,50864,142],{"class":141},[75,50866,44609],{"class":134},[75,50868,206],{"class":85},[75,50870,50871],{"class":309},"out_path",[75,50873,142],{"class":141},[75,50875,50876],{"class":134},"\"\u002Fcontent\u002Fwigner3d_custom_y45.gif\"",[75,50878,41629],{"class":85},[75,50880,50881],{"class":77,"line":16805},[75,50882,318],{"class":85},[75,50884,50885,50888,50890,50892,50894,50896],{"class":77,"line":16836},[75,50886,50887],{"class":85},"saved_custom 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Pauli Gates as (π\\piπ)-Rotations",{"id":36310,"depth":95,"text":36311},{"id":37116,"depth":95,"text":50951},"5. Rodrigues’ Formula (Axis–Angle (→\\to→) Rotation Matrix)",{"id":39255,"depth":95,"text":50953},"6. Optional: Discrete (2×22\\times22×2) (Finite‑State) Wigner",{"id":40167,"depth":95,"text":40168},{"id":40915,"depth":95,"text":40916},{"id":41248,"depth":95,"text":41249},{"id":41264,"depth":95,"text":41265},[12374,12375,12376,50959],"Wigner functions",[50961],{"username":7,"name":12380,"role":12381,"bio":12382,"links":50962},[50963,50964],{"label":12385,"href":12386},{"label":12388,"href":31328},{"username":7,"name":12380,"role":12390},"Pauli rotations rendered as Wigner functions, animated in two and three dimensions, with the density-operator and Bloch-vector groundwork.","Deep dive · Visualization",{},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fwigner-functions-single-qubit-gates","17 min read",[],{"title":31318,"description":50966},"blog\u002Fexpert-notes\u002Fwigner-functions-single-qubit-gates",[12407,50976],"quantum","F21gsgnqd_6pE29RCUocKz6jo6vSs6rJlQM17vqwdBU",{"id":50979,"title":50980,"authors":50981,"body":50982,"breadcrumb":76817,"builders":76819,"byline":76824,"challenge":12391,"courseAuthor":12391,"courseLead":12391,"dek":12391,"description":76825,"draft":12393,"extension":12394,"eyebrow":12395,"featured":111,"finish":12391,"fork":12391,"hero":12391,"heroAlt":12391,"heroCta":12391,"heroImage":12391,"homepageFeatured":12393,"kind":12396,"lessonCount":12391,"meta":76826,"navigation":111,"newsItems":12391,"next":12391,"ogImage":76827,"order":12391,"outcomes":12391,"path":76828,"publishDate":76829,"readingTime":76830,"related":76831,"relatedProjects":12391,"seo":76832,"stem":76833,"tags":76834,"track":12391,"trackName":12391,"__hash__":76835},"blog\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization.md","Rabi Oscillations: Visualizing Excited-State Probability",[7],{"type":9,"value":50983,"toc":76800},[50984,50991,50994,51103,51166,51170,51173,52638,52642,52723,52935,52938,53089,53092,53215,53324,53424,53427,53431,53434,53555,53615,53773,53776,53998,54001,54730,54767,54771,54774,55108,55238,55548,55551,55778,55896,55900,55970,56440,56442,56674,56677,56856,56911,57112,57115,57118,57369,57401,57681,57684,57834,57866,57986,57989,58256,58259,58377,58381,58384,58750,58753,58887,58894,58898,58901,59043,59046,59588,59591,59817,59993,59997,60071,60342,60344,60409,60441,60608,60677,60902,60905,60908,61269,61271,61595,61598,61602,61605,61712,61744,61928,61971,61975,62106,62352,62355,62748,62751,62962,62965,62969,63442,63902,64021,64140,64143,64147,64248,64251,64350,64433,64618,64621,64821,64824,64925,65322,65326,65329,65814,65817,65821,65824,66083,66086,66299,66302,66404,66407,66437,66748,66754,67234,67325,67331,67507,67513,67666,67670,72494,72498,72504,73206,73210,73856,73860,73864,75019,75025,75225,75506,75510,76315,76319,76775,76779,76783,76787,76797],[12,50985,50986],{},[15,50987,17,50988,12423],{},[19,50989,22],{"href":50990},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb",[12,50992,50993],{},"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",[75,50995,50997],{"className":50996},[31410],[75,50998,51000,51032],{"className":50999},[5148],[75,51001,51003],{"className":51002},[5152],[5154,51004,51005],{"xmlns":5156,"display":31419},[5158,51006,51007,51029],{},[5161,51008,51009,51016,51018,51020,51022,51025,51027],{},[5179,51010,51011,51014],{},[5164,51012,51013],{},"P",[5164,51015,7486],{},[5168,51017,197],{"stretchy":5170},[5164,51019,5894],{},[5168,51021,7469],{"separator":5196},[5164,51023,51024],{},"τ",[5168,51026,5175],{"stretchy":5170},[5168,51028,7469],{"separator":5196},[5238,51030,51031],{"encoding":5240},"P_e(f,\\tau),",[75,51033,51035],{"className":51034,"ariaHidden":5196},[5245],[75,51036,51038,51041,51081,51084,51087,51090,51093,51097,51100],{"className":51037},[5249],[75,51039],{"className":51040,"style":5254},[5253],[75,51042,51044,51047],{"className":51043},[5258],[75,51045,51013],{"className":51046,"style":9817},[5258,5259],[75,51048,51050],{"className":51049},[5299],[75,51051,51053,51073],{"className":51052},[5303,5304],[75,51054,51056,51070],{"className":51055},[5308],[75,51057,51059],{"className":51058,"style":6264},[5312],[75,51060,51061,51064],{"style":33755},[75,51062],{"className":51063,"style":5321},[5320],[75,51065,51067],{"className":51066},[5325,5326,5327,5328],[75,51068,7486],{"className":51069},[5258,5259,5328],[75,51071,5336],{"className":51072},[5335],[75,51074,51076],{"className":51075},[5308],[75,51077,51079],{"className":51078,"style":5343},[5312],[75,51080],{},[75,51082,197],{"className":51083},[5263],[75,51085,5894],{"className":51086,"style":6251},[5258,5259],[75,51088,7469],{"className":51089},[7544],[75,51091],{"className":51092,"style":5349},[5274],[75,51094,51024],{"className":51095,"style":51096},[5258,5259],"margin-right:0.1132em;",[75,51098,5175],{"className":51099},[5270],[75,51101,7469],{"className":51102},[7544],[12,51104,51105,51106,51135,51136,51165],{},"where ",[75,51107,51109,51122],{"className":51108},[5148],[75,51110,51112],{"className":51111},[5152],[5154,51113,51114],{"xmlns":5156},[5158,51115,51116,51120],{},[5161,51117,51118],{},[5164,51119,5894],{},[5238,51121,5894],{"encoding":5240},[75,51123,51125],{"className":51124,"ariaHidden":5196},[5245],[75,51126,51128,51132],{"className":51127},[5249],[75,51129],{"className":51130,"style":51131},[5253],"height:0.8889em;vertical-align:-0.1944em;",[75,51133,5894],{"className":51134,"style":6251},[5258,5259]," denotes the applied drive frequency and ",[75,51137,51139,51153],{"className":51138},[5148],[75,51140,51142],{"className":51141},[5152],[5154,51143,51144],{"xmlns":5156},[5158,51145,51146,51150],{},[5161,51147,51148],{},[5164,51149,51024],{},[5238,51151,51152],{"encoding":5240},"\\tau",[75,51154,51156],{"className":51155,"ariaHidden":5196},[5245],[75,51157,51159,51162],{"className":51158},[5249],[75,51160],{"className":51161,"style":35333},[5253],[75,51163,51024],{"className":51164,"style":51096},[5258,5259]," 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.",[12425,51167,51169],{"id":51168},"nomenclature-and-definitions","Nomenclature and Definitions",[12,51171,51172],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[7629,51174,51175,51185],{},[7632,51176,51177],{},[7635,51178,51179,51182],{},[7638,51180,51181],{},"Symbol or Term",[7638,51183,51184],{},"Meaning",[7645,51186,51187,51234,51280,51326,51453,51530,51607,51642,51862,51959,51994,52040,52190,52267,52492,52527,52622,52630],{},[7635,51188,51189,51231],{},[7650,51190,51191],{},[75,51192,51194,51213],{"className":51193},[5148],[75,51195,51197],{"className":51196},[5152],[5154,51198,51199],{"xmlns":5156},[5158,51200,51201,51210],{},[5161,51202,51203,51205,51208],{},[5168,51204,7452],{"stretchy":5170},[5164,51206,51207],{},"g",[5168,51209,7457],{"stretchy":5170},[5238,51211,51212],{"encoding":5240},"\\lvert g\\rangle",[75,51214,51216],{"className":51215,"ariaHidden":5196},[5245],[75,51217,51219,51222,51225,51228],{"className":51218},[5249],[75,51220],{"className":51221,"style":5254},[5253],[75,51223,7452],{"className":51224},[5263],[75,51226,51207],{"className":51227,"style":5416},[5258,5259],[75,51229,7457],{"className":51230},[5270],[7650,51232,51233],{},"Ground state of the qubit",[7635,51235,51236,51277],{},[7650,51237,51238],{},[75,51239,51241,51259],{"className":51240},[5148],[75,51242,51244],{"className":51243},[5152],[5154,51245,51246],{"xmlns":5156},[5158,51247,51248,51256],{},[5161,51249,51250,51252,51254],{},[5168,51251,7452],{"stretchy":5170},[5164,51253,7486],{},[5168,51255,7457],{"stretchy":5170},[5238,51257,51258],{"encoding":5240},"\\lvert e\\rangle",[75,51260,51262],{"className":51261,"ariaHidden":5196},[5245],[75,51263,51265,51268,51271,51274],{"className":51264},[5249],[75,51266],{"className":51267,"style":5254},[5253],[75,51269,7452],{"className":51270},[5263],[75,51272,7486],{"className":51273},[5258,5259],[75,51275,7457],{"className":51276},[5270],[7650,51278,51279],{},"Excited state of the qubit",[7635,51281,51282,51323],{},[7650,51283,51284],{},[75,51285,51287,51305],{"className":51286},[5148],[75,51288,51290],{"className":51289},[5152],[5154,51291,51292],{"xmlns":5156},[5158,51293,51294,51302],{},[5161,51295,51296,51298,51300],{},[5168,51297,7452],{"stretchy":5170},[5164,51299,32718],{},[5168,51301,7457],{"stretchy":5170},[5238,51303,51304],{"encoding":5240},"\\lvert \\psi\\rangle",[75,51306,51308],{"className":51307,"ariaHidden":5196},[5245],[75,51309,51311,51314,51317,51320],{"className":51310},[5249],[75,51312],{"className":51313,"style":5254},[5253],[75,51315,7452],{"className":51316},[5263],[75,51318,32718],{"className":51319,"style":5416},[5258,5259],[75,51321,7457],{"className":51322},[5270],[7650,51324,51325],{},"Quantum state vector in a two-dimensional Hilbert space",[7635,51327,51328,51374],{},[7650,51329,51330],{},[75,51331,51333,51352],{"className":51332},[5148],[75,51334,51336],{"className":51335},[5152],[5154,51337,51338],{"xmlns":5156},[5158,51339,51340,51349],{},[5161,51341,51342,51344,51346],{},[5164,51343,36399],{},[5168,51345,7469],{"separator":5196},[5164,51347,51348],{},"β",[5238,51350,51351],{"encoding":5240},"\\alpha,\\beta",[75,51353,51355],{"className":51354,"ariaHidden":5196},[5245],[75,51356,51358,51361,51364,51367,51370],{"className":51357},[5249],[75,51359],{"className":51360,"style":51131},[5253],[75,51362,36399],{"className":51363,"style":36415},[5258,5259],[75,51365,7469],{"className":51366},[7544],[75,51368],{"className":51369,"style":5349},[5274],[75,51371,51348],{"className":51372,"style":51373},[5258,5259],"margin-right:0.0528em;",[7650,51375,51376,51377,32154,51415],{},"Complex probability amplitudes for 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of measuring the qubit in the excited state",[7635,51531,51532,51604],{},[7650,51533,51534],{},[75,51535,51537,51555],{"className":51536},[5148],[75,51538,51540],{"className":51539},[5152],[5154,51541,51542],{"xmlns":5156},[5158,51543,51544,51552],{},[5161,51545,51546],{},[5179,51547,51548,51550],{},[5164,51549,5894],{},[5184,51551,5186],{},[5238,51553,51554],{"encoding":5240},"f_0",[75,51556,51558],{"className":51557,"ariaHidden":5196},[5245],[75,51559,51561,51564],{"className":51560},[5249],[75,51562],{"className":51563,"style":51131},[5253],[75,51565,51567,51570],{"className":51566},[5258],[75,51568,5894],{"className":51569,"style":6251},[5258,5259],[75,51571,51573],{"className":51572},[5299],[75,51574,51576,51596],{"className":51575},[5303,5304],[75,51577,51579,51593],{"className":51578},[5308],[75,51580,51582],{"className":51581,"style":5313},[5312],[75,51583,51584,51587],{"style":6267},[75,51585],{"className":51586,"style":5321},[5320],[75,51588,51590],{"className":51589},[5325,5326,5327,5328],[75,51591,5186],{"className":51592},[5258,5328],[75,51594,5336],{"className":51595},[5335],[75,51597,51599],{"className":51598},[5308],[75,51600,51602],{"className":51601,"style":5343},[5312],[75,51603],{},[7650,51605,51606],{},"Natural transition frequency of the qubit",[7635,51608,51609,51639],{},[7650,51610,51611],{},[75,51612,51614,51627],{"className":51613},[5148],[75,51615,51617],{"className":51616},[5152],[5154,51618,51619],{"xmlns":5156},[5158,51620,51621,51625],{},[5161,51622,51623],{},[5164,51624,5894],{},[5238,51626,5894],{"encoding":5240},[75,51628,51630],{"className":51629,"ariaHidden":5196},[5245],[75,51631,51633,51636],{"className":51632},[5249],[75,51634],{"className":51635,"style":51131},[5253],[75,51637,5894],{"className":51638,"style":6251},[5258,5259],[7650,51640,51641],{},"Applied drive frequency",[7635,51643,51644,51716],{},[7650,51645,51646],{},[75,51647,51649,51667],{"className":51648},[5148],[75,51650,51652],{"className":51651},[5152],[5154,51653,51654],{"xmlns":5156},[5158,51655,51656,51664],{},[5161,51657,51658],{},[5179,51659,51660,51662],{},[5164,51661,12469],{},[5184,51663,5186],{},[5238,51665,51666],{"encoding":5240},"\\omega_0",[75,51668,51670],{"className":51669,"ariaHidden":5196},[5245],[75,51671,51673,51676],{"className":51672},[5249],[75,51674],{"className":51675,"style":12488},[5253],[75,51677,51679,51682],{"className":51678},[5258],[75,51680,12469],{"className":51681,"style":5416},[5258,5259],[75,51683,51685],{"className":51684},[5299],[75,51686,51688,51708],{"className":51687},[5303,5304],[75,51689,51691,51705],{"className":51690},[5308],[75,51692,51694],{"className":51693,"style":5313},[5312],[75,51695,51696,51699],{"style":12509},[75,51697],{"className":51698,"style":5321},[5320],[75,51700,51702],{"className":51701},[5325,5326,5327,5328],[75,51703,5186],{"className":51704},[5258,5328],[75,51706,5336],{"className":51707},[5335],[75,51709,51711],{"className":51710},[5308],[75,51712,51714],{"className":51713,"style":5343},[5312],[75,51715],{},[7650,51717,51718,51719],{},"Angular transition frequency, ",[75,51720,51722,51752],{"className":51721},[5148],[75,51723,51725],{"className":51724},[5152],[5154,51726,51727],{"xmlns":5156},[5158,51728,51729,51749],{},[5161,51730,51731,51737,51739,51741,51743],{},[5179,51732,51733,51735],{},[5164,51734,12469],{},[5184,51736,5186],{},[5168,51738,142],{},[5184,51740,270],{},[5164,51742,5889],{},[5179,51744,51745,51747],{},[5164,51746,5894],{},[5184,51748,5186],{},[5238,51750,51751],{"encoding":5240},"\\omega_0=2\\pi f_0",[75,51753,51755,51810],{"className":51754,"ariaHidden":5196},[5245],[75,51756,51758,51761,51801,51804,51807],{"className":51757},[5249],[75,51759],{"className":51760,"style":12488},[5253],[75,51762,51764,51767],{"className":51763},[5258],[75,51765,12469],{"className":51766,"style":5416},[5258,5259],[75,51768,51770],{"className":51769},[5299],[75,51771,51773,51793],{"className":51772},[5303,5304],[75,51774,51776,51790],{"className":51775},[5308],[75,51777,51779],{"className":51778,"style":5313},[5312],[75,51780,51781,51784],{"style":12509},[75,51782],{"className":51783,"style":5321},[5320],[75,51785,51787],{"className":51786},[5325,5326,5327,5328],[75,51788,5186],{"className":51789},[5258,5328],[75,51791,5336],{"className":51792},[5335],[75,51794,51796],{"className":51795},[5308],[75,51797,51799],{"className":51798,"style":5343},[5312],[75,51800],{},[75,51802],{"className":51803,"style":5275},[5274],[75,51805,142],{"className":51806},[5279],[75,51808],{"className":51809,"style":5275},[5274],[75,51811,51813,51816,51819,51822],{"className":51812},[5249],[75,51814],{"className":51815,"style":51131},[5253],[75,51817,270],{"className":51818},[5258],[75,51820,5889],{"className":51821,"style":5416},[5258,5259],[75,51823,51825,51828],{"className":51824},[5258],[75,51826,5894],{"className":51827,"style":6251},[5258,5259],[75,51829,51831],{"className":51830},[5299],[75,51832,51834,51854],{"className":51833},[5303,5304],[75,51835,51837,51851],{"className":51836},[5308],[75,51838,51840],{"className":51839,"style":5313},[5312],[75,51841,51842,51845],{"style":6267},[75,51843],{"className":51844,"style":5321},[5320],[75,51846,51848],{"className":51847},[5325,5326,5327,5328],[75,51849,5186],{"className":51850},[5258,5328],[75,51852,5336],{"className":51853},[5335],[75,51855,51857],{"className":51856},[5308],[75,51858,51860],{"className":51859,"style":5343},[5312],[75,51861],{},[7635,51863,51864,51895],{},[7650,51865,51866],{},[75,51867,51869,51883],{"className":51868},[5148],[75,51870,51872],{"className":51871},[5152],[5154,51873,51874],{"xmlns":5156},[5158,51875,51876,51880],{},[5161,51877,51878],{},[5164,51879,12469],{},[5238,51881,51882],{"encoding":5240},"\\omega",[75,51884,51886],{"className":51885,"ariaHidden":5196},[5245],[75,51887,51889,51892],{"className":51888},[5249],[75,51890],{"className":51891,"style":35333},[5253],[75,51893,12469],{"className":51894,"style":5416},[5258,5259],[7650,51896,51897,51898],{},"Angular drive frequency, ",[75,51899,51901,51923],{"className":51900},[5148],[75,51902,51904],{"className":51903},[5152],[5154,51905,51906],{"xmlns":5156},[5158,51907,51908,51920],{},[5161,51909,51910,51912,51914,51916,51918],{},[5164,51911,12469],{},[5168,51913,142],{},[5184,51915,270],{},[5164,51917,5889],{},[5164,51919,5894],{},[5238,51921,51922],{"encoding":5240},"\\omega=2\\pi f",[75,51924,51926,51944],{"className":51925,"ariaHidden":5196},[5245],[75,51927,51929,51932,51935,51938,51941],{"className":51928},[5249],[75,51930],{"className":51931,"style":35333},[5253],[75,51933,12469],{"className":51934,"style":5416},[5258,5259],[75,51936],{"className":51937,"style":5275},[5274],[75,51939,142],{"className":51940},[5279],[75,51942],{"className":51943,"style":5275},[5274],[75,51945,51947,51950,51953,51956],{"className":51946},[5249],[75,51948],{"className":51949,"style":51131},[5253],[75,51951,270],{"className":51952},[5258],[75,51954,5889],{"className":51955,"style":5416},[5258,5259],[75,51957,5894],{"className":51958,"style":6251},[5258,5259],[7635,51960,51961,51991],{},[7650,51962,51963],{},[75,51964,51966,51979],{"className":51965},[5148],[75,51967,51969],{"className":51968},[5152],[5154,51970,51971],{"xmlns":5156},[5158,51972,51973,51977],{},[5161,51974,51975],{},[5164,51976,12635],{"mathvariant":6582},[5238,51978,12638],{"encoding":5240},[75,51980,51982],{"className":51981,"ariaHidden":5196},[5245],[75,51983,51985,51988],{"className":51984},[5249],[75,51986],{"className":51987,"style":12648},[5253],[75,51989,12635],{"className":51990},[5258],[7650,51992,51993],{},"On-resonance Rabi angular frequency",[7635,51995,51996,52037],{},[7650,51997,51998],{},[75,51999,52001,52021],{"className":52000},[5148],[75,52002,52004],{"className":52003},[5152],[5154,52005,52006],{"xmlns":5156},[5158,52007,52008,52018],{},[5161,52009,52010,52012,52014,52016],{},[5164,52011,12635],{"mathvariant":6582},[5164,52013,2810],{"mathvariant":6582},[5184,52015,270],{},[5164,52017,5889],{},[5238,52019,52020],{"encoding":5240},"\\Omega\u002F2\\pi",[75,52022,52024],{"className":52023,"ariaHidden":5196},[5245],[75,52025,52027,52030,52034],{"className":52026},[5249],[75,52028],{"className":52029,"style":5254},[5253],[75,52031,52033],{"className":52032},[5258],"Ω\u002F2",[75,52035,5889],{"className":52036,"style":5416},[5258,5259],[7650,52038,52039],{},"On-resonance Rabi frequency in ordinary frequency units",[7635,52041,52042,52072],{},[7650,52043,52044],{},[75,52045,52047,52060],{"className":52046},[5148],[75,52048,52050],{"className":52049},[5152],[5154,52051,52052],{"xmlns":5156},[5158,52053,52054,52058],{},[5161,52055,52056],{},[5164,52057,12553],{"mathvariant":6582},[5238,52059,41287],{"encoding":5240},[75,52061,52063],{"className":52062,"ariaHidden":5196},[5245],[75,52064,52066,52069],{"className":52065},[5249],[75,52067],{"className":52068,"style":12648},[5253],[75,52070,12553],{"className":52071},[5258],[7650,52073,52074,52075],{},"Drive detuning, ",[75,52076,52078,52104],{"className":52077},[5148],[75,52079,52081],{"className":52080},[5152],[5154,52082,52083],{"xmlns":5156},[5158,52084,52085,52101],{},[5161,52086,52087,52089,52091,52093,52095],{},[5164,52088,12553],{"mathvariant":6582},[5168,52090,142],{},[5164,52092,12469],{},[5168,52094,5199],{},[5179,52096,52097,52099],{},[5164,52098,12469],{},[5184,52100,5186],{},[5238,52102,52103],{"encoding":5240},"\\Delta=\\omega-\\omega_0",[75,52105,52107,52125,52144],{"className":52106,"ariaHidden":5196},[5245],[75,52108,52110,52113,52116,52119,52122],{"className":52109},[5249],[75,52111],{"className":52112,"style":12648},[5253],[75,52114,12553],{"className":52115},[5258],[75,52117],{"className":52118,"style":5275},[5274],[75,52120,142],{"className":52121},[5279],[75,52123],{"className":52124,"style":5275},[5274],[75,52126,52128,52132,52135,52138,52141],{"className":52127},[5249],[75,52129],{"className":52130,"style":52131},[5253],"height:0.6667em;vertical-align:-0.0833em;",[75,52133,12469],{"className":52134,"style":5416},[5258,5259],[75,52136],{"className":52137,"style":6294},[5274],[75,52139,5199],{"className":52140},[5477],[75,52142],{"className":52143,"style":6294},[5274],[75,52145,52147,52150],{"className":52146},[5249],[75,52148],{"className":52149,"style":12488},[5253],[75,52151,52153,52156],{"className":52152},[5258],[75,52154,12469],{"className":52155,"style":5416},[5258,5259],[75,52157,52159],{"className":52158},[5299],[75,52160,52162,52182],{"className":52161},[5303,5304],[75,52163,52165,52179],{"className":52164},[5308],[75,52166,52168],{"className":52167,"style":5313},[5312],[75,52169,52170,52173],{"style":12509},[75,52171],{"className":52172,"style":5321},[5320],[75,52174,52176],{"className":52175},[5325,5326,5327,5328],[75,52177,5186],{"className":52178},[5258,5328],[75,52180,5336],{"className":52181},[5335],[75,52183,52185],{"className":52184},[5308],[75,52186,52188],{"className":52187,"style":5343},[5312],[75,52189],{},[7635,52191,52192,52264],{},[7650,52193,52194],{},[75,52195,52197,52215],{"className":52196},[5148],[75,52198,52200],{"className":52199},[5152],[5154,52201,52202],{"xmlns":5156},[5158,52203,52204,52212],{},[5161,52205,52206],{},[5179,52207,52208,52210],{},[5164,52209,12635],{"mathvariant":6582},[5164,52211,32181],{},[5238,52213,52214],{"encoding":5240},"\\Omega_R",[75,52216,52218],{"className":52217,"ariaHidden":5196},[5245],[75,52219,52221,52224],{"className":52220},[5249],[75,52222],{"className":52223,"style":12572},[5253],[75,52225,52227,52230],{"className":52226},[5258],[75,52228,12635],{"className":52229},[5258],[75,52231,52233],{"className":52232},[5299],[75,52234,52236,52256],{"className":52235},[5303,5304],[75,52237,52239,52253],{"className":52238},[5308],[75,52240,52242],{"className":52241,"style":5942},[5312],[75,52243,52244,52247],{"style":5316},[75,52245],{"className":52246,"style":5321},[5320],[75,52248,52250],{"className":52249},[5325,5326,5327,5328],[75,52251,32181],{"className":52252,"style":35138},[5258,5259,5328],[75,52254,5336],{"className":52255},[5335],[75,52257,52259],{"className":52258},[5308],[75,52260,52262],{"className":52261,"style":5343},[5312],[75,52263],{},[7650,52265,52266],{},"Generalized Rabi angular frequency",[7635,52268,52269,52344],{},[7650,52270,52271],{},[75,52272,52274,52293],{"className":52273},[5148],[75,52275,52277],{"className":52276},[5152],[5154,52278,52279],{"xmlns":5156},[5158,52280,52281,52290],{},[5161,52282,52283],{},[5179,52284,52285,52288],{},[5164,52286,52287],{},"ν",[5164,52289,32181],{},[5238,52291,52292],{"encoding":5240},"\\nu_R",[75,52294,52296],{"className":52295,"ariaHidden":5196},[5245],[75,52297,52299,52302],{"className":52298},[5249],[75,52300],{"className":52301,"style":12488},[5253],[75,52303,52305,52309],{"className":52304},[5258],[75,52306,52287],{"className":52307,"style":52308},[5258,5259],"margin-right:0.0637em;",[75,52310,52312],{"className":52311},[5299],[75,52313,52315,52336],{"className":52314},[5303,5304],[75,52316,52318,52333],{"className":52317},[5308],[75,52319,52321],{"className":52320,"style":5942},[5312],[75,52322,52324,52327],{"style":52323},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[75,52325],{"className":52326,"style":5321},[5320],[75,52328,52330],{"className":52329},[5325,5326,5327,5328],[75,52331,32181],{"className":52332,"style":35138},[5258,5259,5328],[75,52334,5336],{"className":52335},[5335],[75,52337,52339],{"className":52338},[5308],[75,52340,52342],{"className":52341,"style":5343},[5312],[75,52343],{},[7650,52345,52346,52347],{},"Generalized Rabi frequency, ",[75,52348,52350,52382],{"className":52349},[5148],[75,52351,52353],{"className":52352},[5152],[5154,52354,52355],{"xmlns":5156},[5158,52356,52357,52379],{},[5161,52358,52359,52365,52367,52373,52375,52377],{},[5179,52360,52361,52363],{},[5164,52362,52287],{},[5164,52364,32181],{},[5168,52366,142],{},[5179,52368,52369,52371],{},[5164,52370,12635],{"mathvariant":6582},[5164,52372,32181],{},[5164,52374,2810],{"mathvariant":6582},[5184,52376,270],{},[5164,52378,5889],{},[5238,52380,52381],{"encoding":5240},"\\nu_R=\\Omega_R\u002F2\\pi",[75,52383,52385,52440],{"className":52384,"ariaHidden":5196},[5245],[75,52386,52388,52391,52431,52434,52437],{"className":52387},[5249],[75,52389],{"className":52390,"style":12488},[5253],[75,52392,52394,52397],{"className":52393},[5258],[75,52395,52287],{"className":52396,"style":52308},[5258,5259],[75,52398,52400],{"className":52399},[5299],[75,52401,52403,52423],{"className":52402},[5303,5304],[75,52404,52406,52420],{"className":52405},[5308],[75,52407,52409],{"className":52408,"style":5942},[5312],[75,52410,52411,52414],{"style":52323},[75,52412],{"className":52413,"style":5321},[5320],[75,52415,52417],{"className":52416},[5325,5326,5327,5328],[75,52418,32181],{"className":52419,"style":35138},[5258,5259,5328],[75,52421,5336],{"className":52422},[5335],[75,52424,52426],{"className":52425},[5308],[75,52427,52429],{"className":52428,"style":5343},[5312],[75,52430],{},[75,52432],{"className":52433,"style":5275},[5274],[75,52435,142],{"className":52436},[5279],[75,52438],{"className":52439,"style":5275},[5274],[75,52441,52443,52446,52486,52489],{"className":52442},[5249],[75,52444],{"className":52445,"style":5254},[5253],[75,52447,52449,52452],{"className":52448},[5258],[75,52450,12635],{"className":52451},[5258],[75,52453,52455],{"className":52454},[5299],[75,52456,52458,52478],{"className":52457},[5303,5304],[75,52459,52461,52475],{"className":52460},[5308],[75,52462,52464],{"className":52463,"style":5942},[5312],[75,52465,52466,52469],{"style":5316},[75,52467],{"className":52468,"style":5321},[5320],[75,52470,52472],{"className":52471},[5325,5326,5327,5328],[75,52473,32181],{"className":52474,"style":35138},[5258,5259,5328],[75,52476,5336],{"className":52477},[5335],[75,52479,52481],{"className":52480},[5308],[75,52482,52484],{"className":52483,"style":5343},[5312],[75,52485],{},[75,52487,6608],{"className":52488},[5258],[75,52490,5889],{"className":52491,"style":5416},[5258,5259],[7635,52493,52494,52524],{},[7650,52495,52496],{},[75,52497,52499,52512],{"className":52498},[5148],[75,52500,52502],{"className":52501},[5152],[5154,52503,52504],{"xmlns":5156},[5158,52505,52506,52510],{},[5161,52507,52508],{},[5164,52509,51024],{},[5238,52511,51152],{"encoding":5240},[75,52513,52515],{"className":52514,"ariaHidden":5196},[5245],[75,52516,52518,52521],{"className":52517},[5249],[75,52519],{"className":52520,"style":35333},[5253],[75,52522,51024],{"className":52523,"style":51096},[5258,5259],[7650,52525,52526],{},"Pulse duration",[7635,52528,52529,52619],{},[7650,52530,52531],{},[75,52532,52534,52555],{"className":52533},[5148],[75,52535,52537],{"className":52536},[5152],[5154,52538,52539],{"xmlns":5156},[5158,52540,52541,52552],{},[5161,52542,52543],{},[39536,52544,52545,52547,52549],{},[5164,52546,9676],{},[5184,52548,270],{},[5168,52550,52551],{},"∗",[5238,52553,52554],{"encoding":5240},"T_2^\\ast",[75,52556,52558],{"className":52557,"ariaHidden":5196},[5245],[75,52559,52561,52565],{"className":52560},[5249],[75,52562],{"className":52563,"style":52564},[5253],"height:0.9368em;vertical-align:-0.2481em;",[75,52566,52568,52571],{"className":52567},[5258],[75,52569,9676],{"className":52570,"style":9817},[5258,5259],[75,52572,52574],{"className":52573},[5299],[75,52575,52577,52610],{"className":52576},[5303,5304],[75,52578,52580,52607],{"className":52579},[5308],[75,52581,52584,52596],{"className":52582,"style":52583},[5312],"height:0.6887em;",[75,52585,52587,52590],{"style":52586},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[75,52588],{"className":52589,"style":5321},[5320],[75,52591,52593],{"className":52592},[5325,5326,5327,5328],[75,52594,270],{"className":52595},[5258,5328],[75,52597,52598,52601],{"style":7596},[75,52599],{"className":52600,"style":5321},[5320],[75,52602,52604],{"className":52603},[5325,5326,5327,5328],[75,52605,52551],{"className":52606},[5477,5328],[75,52608,5336],{"className":52609},[5335],[75,52611,52613],{"className":52612},[5308],[75,52614,52617],{"className":52615,"style":52616},[5312],"height:0.2481em;",[75,52618],{},[7650,52620,52621],{},"Phenomenological dephasing time used as an optional damping parameter",[7635,52623,52624,52627],{},[7650,52625,52626],{},"RWA",[7650,52628,52629],{},"Rotating-wave approximation",[7635,52631,52632,52635],{},[7650,52633,52634],{},"FFT",[7650,52636,52637],{},"Fast Fourier transform",[12425,52639,52641],{"id":52640},"physical-model","Physical Model",[12,52643,52644,52645,52683,52684,52722],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[75,52646,52648,52665],{"className":52647},[5148],[75,52649,52651],{"className":52650},[5152],[5154,52652,52653],{"xmlns":5156},[5158,52654,52655,52663],{},[5161,52656,52657,52659,52661],{},[5168,52658,7452],{"stretchy":5170},[5164,52660,51207],{},[5168,52662,7457],{"stretchy":5170},[5238,52664,51212],{"encoding":5240},[75,52666,52668],{"className":52667,"ariaHidden":5196},[5245],[75,52669,52671,52674,52677,52680],{"className":52670},[5249],[75,52672],{"className":52673,"style":5254},[5253],[75,52675,7452],{"className":52676},[5263],[75,52678,51207],{"className":52679,"style":5416},[5258,5259],[75,52681,7457],{"className":52682},[5270]," and an excited state ",[75,52685,52687,52704],{"className":52686},[5148],[75,52688,52690],{"className":52689},[5152],[5154,52691,52692],{"xmlns":5156},[5158,52693,52694,52702],{},[5161,52695,52696,52698,52700],{},[5168,52697,7452],{"stretchy":5170},[5164,52699,7486],{},[5168,52701,7457],{"stretchy":5170},[5238,52703,51258],{"encoding":5240},[75,52705,52707],{"className":52706,"ariaHidden":5196},[5245],[75,52708,52710,52713,52716,52719],{"className":52709},[5249],[75,52711],{"className":52712,"style":5254},[5253],[75,52714,7452],{"className":52715},[5263],[75,52717,7486],{"className":52718},[5258,5259],[75,52720,7457],{"className":52721},[5270],". In the absence of a microwave drive, the energy splitting is",[75,52724,52726],{"className":52725},[31410],[75,52727,52729,52768],{"className":52728},[5148],[75,52730,52732],{"className":52731},[5152],[5154,52733,52734],{"xmlns":5156,"display":31419},[5158,52735,52736,52765],{},[5161,52737,52738,52744,52746,52752,52754,52757,52763],{},[5179,52739,52740,52742],{},[5164,52741,5817],{},[5164,52743,7486],{},[5168,52745,5199],{},[5179,52747,52748,52750],{},[5164,52749,5817],{},[5164,52751,51207],{},[5168,52753,142],{},[5164,52755,52756],{"mathvariant":6582},"ℏ",[5179,52758,52759,52761],{},[5164,52760,12469],{},[5184,52762,5186],{},[5168,52764,7469],{"separator":5196},[5238,52766,52767],{"encoding":5240},"E_e-E_g=\\hbar\\omega_0,",[75,52769,52771,52826,52882],{"className":52770,"ariaHidden":5196},[5245],[75,52772,52774,52777,52817,52820,52823],{"className":52773},[5249],[75,52775],{"className":52776,"style":12572},[5253],[75,52778,52780,52783],{"className":52779},[5258],[75,52781,5817],{"className":52782,"style":5929},[5258,5259],[75,52784,52786],{"className":52785},[5299],[75,52787,52789,52809],{"className":52788},[5303,5304],[75,52790,52792,52806],{"className":52791},[5308],[75,52793,52795],{"className":52794,"style":6264},[5312],[75,52796,52797,52800],{"style":5945},[75,52798],{"className":52799,"style":5321},[5320],[75,52801,52803],{"className":52802},[5325,5326,5327,5328],[75,52804,7486],{"className":52805},[5258,5259,5328],[75,52807,5336],{"className":52808},[5335],[75,52810,52812],{"className":52811},[5308],[75,52813,52815],{"className":52814,"style":5343},[5312],[75,52816],{},[75,52818],{"className":52819,"style":6294},[5274],[75,52821,5199],{"className":52822},[5477],[75,52824],{"className":52825,"style":6294},[5274],[75,52827,52829,52833,52873,52876,52879],{"className":52828},[5249],[75,52830],{"className":52831,"style":52832},[5253],"height:0.9694em;vertical-align:-0.2861em;",[75,52834,52836,52839],{"className":52835},[5258],[75,52837,5817],{"className":52838,"style":5929},[5258,5259],[75,52840,52842],{"className":52841},[5299],[75,52843,52845,52865],{"className":52844},[5303,5304],[75,52846,52848,52862],{"className":52847},[5308],[75,52849,52851],{"className":52850,"style":6264},[5312],[75,52852,52853,52856],{"style":5945},[75,52854],{"className":52855,"style":5321},[5320],[75,52857,52859],{"className":52858},[5325,5326,5327,5328],[75,52860,51207],{"className":52861,"style":5416},[5258,5259,5328],[75,52863,5336],{"className":52864},[5335],[75,52866,52868],{"className":52867},[5308],[75,52869,52871],{"className":52870,"style":31823},[5312],[75,52872],{},[75,52874],{"className":52875,"style":5275},[5274],[75,52877,142],{"className":52878},[5279],[75,52880],{"className":52881,"style":5275},[5274],[75,52883,52885,52889,52892,52932],{"className":52884},[5249],[75,52886],{"className":52887,"style":52888},[5253],"height:0.8833em;vertical-align:-0.1944em;",[75,52890,52756],{"className":52891},[5258],[75,52893,52895,52898],{"className":52894},[5258],[75,52896,12469],{"className":52897,"style":5416},[5258,5259],[75,52899,52901],{"className":52900},[5299],[75,52902,52904,52924],{"className":52903},[5303,5304],[75,52905,52907,52921],{"className":52906},[5308],[75,52908,52910],{"className":52909,"style":5313},[5312],[75,52911,52912,52915],{"style":12509},[75,52913],{"className":52914,"style":5321},[5320],[75,52916,52918],{"className":52917},[5325,5326,5327,5328],[75,52919,5186],{"className":52920},[5258,5328],[75,52922,5336],{"className":52923},[5335],[75,52925,52927],{"className":52926},[5308],[75,52928,52930],{"className":52929,"style":5343},[5312],[75,52931],{},[75,52933,7469],{"className":52934},[7544],[12,52936,52937],{},"where the angular transition frequency is",[75,52939,52941],{"className":52940},[31410],[75,52942,52944,52976],{"className":52943},[5148],[75,52945,52947],{"className":52946},[5152],[5154,52948,52949],{"xmlns":5156,"display":31419},[5158,52950,52951,52973],{},[5161,52952,52953,52959,52961,52963,52965,52971],{},[5179,52954,52955,52957],{},[5164,52956,12469],{},[5184,52958,5186],{},[5168,52960,142],{},[5184,52962,270],{},[5164,52964,5889],{},[5179,52966,52967,52969],{},[5164,52968,5894],{},[5184,52970,5186],{},[5164,52972,6543],{"mathvariant":6582},[5238,52974,52975],{"encoding":5240},"\\omega_0=2\\pi f_0.",[75,52977,52979,53034],{"className":52978,"ariaHidden":5196},[5245],[75,52980,52982,52985,53025,53028,53031],{"className":52981},[5249],[75,52983],{"className":52984,"style":12488},[5253],[75,52986,52988,52991],{"className":52987},[5258],[75,52989,12469],{"className":52990,"style":5416},[5258,5259],[75,52992,52994],{"className":52993},[5299],[75,52995,52997,53017],{"className":52996},[5303,5304],[75,52998,53000,53014],{"className":52999},[5308],[75,53001,53003],{"className":53002,"style":5313},[5312],[75,53004,53005,53008],{"style":12509},[75,53006],{"className":53007,"style":5321},[5320],[75,53009,53011],{"className":53010},[5325,5326,5327,5328],[75,53012,5186],{"className":53013},[5258,5328],[75,53015,5336],{"className":53016},[5335],[75,53018,53020],{"className":53019},[5308],[75,53021,53023],{"className":53022,"style":5343},[5312],[75,53024],{},[75,53026],{"className":53027,"style":5275},[5274],[75,53029,142],{"className":53030},[5279],[75,53032],{"className":53033,"style":5275},[5274],[75,53035,53037,53040,53043,53046,53086],{"className":53036},[5249],[75,53038],{"className":53039,"style":51131},[5253],[75,53041,270],{"className":53042},[5258],[75,53044,5889],{"className":53045,"style":5416},[5258,5259],[75,53047,53049,53052],{"className":53048},[5258],[75,53050,5894],{"className":53051,"style":6251},[5258,5259],[75,53053,53055],{"className":53054},[5299],[75,53056,53058,53078],{"className":53057},[5303,5304],[75,53059,53061,53075],{"className":53060},[5308],[75,53062,53064],{"className":53063,"style":5313},[5312],[75,53065,53066,53069],{"style":6267},[75,53067],{"className":53068,"style":5321},[5320],[75,53070,53072],{"className":53071},[5325,5326,5327,5328],[75,53073,5186],{"className":53074},[5258,5328],[75,53076,5336],{"className":53077},[5335],[75,53079,53081],{"className":53080},[5308],[75,53082,53084],{"className":53083,"style":5343},[5312],[75,53085],{},[75,53087,6543],{"className":53088},[5258],[12,53090,53091],{},"The default configuration uses a transition frequency of",[75,53093,53095],{"className":53094},[31410],[75,53096,53098,53134],{"className":53097},[5148],[75,53099,53101],{"className":53100},[5152],[5154,53102,53103],{"xmlns":5156,"display":31419},[5158,53104,53105,53131],{},[5161,53106,53107,53113,53115,53118,53120,53129],{},[5179,53108,53109,53111],{},[5164,53110,5894],{},[5184,53112,5186],{},[5168,53114,142],{},[5184,53116,53117],{},"5.000",[32328,53119,32330],{},[5161,53121,53122,53125,53127],{},[5164,53123,53124],{"mathvariant":6582},"G",[5164,53126,9119],{"mathvariant":6582},[5164,53128,31538],{"mathvariant":6582},[5164,53130,6543],{"mathvariant":6582},[5238,53132,53133],{"encoding":5240},"f_0=5.000~\\mathrm{GHz}.",[75,53135,53137,53192],{"className":53136,"ariaHidden":5196},[5245],[75,53138,53140,53143,53183,53186,53189],{"className":53139},[5249],[75,53141],{"className":53142,"style":51131},[5253],[75,53144,53146,53149],{"className":53145},[5258],[75,53147,5894],{"className":53148,"style":6251},[5258,5259],[75,53150,53152],{"className":53151},[5299],[75,53153,53155,53175],{"className":53154},[5303,5304],[75,53156,53158,53172],{"className":53157},[5308],[75,53159,53161],{"className":53160,"style":5313},[5312],[75,53162,53163,53166],{"style":6267},[75,53164],{"className":53165,"style":5321},[5320],[75,53167,53169],{"className":53168},[5325,5326,5327,5328],[75,53170,5186],{"className":53171},[5258,5328],[75,53173,5336],{"className":53174},[5335],[75,53176,53178],{"className":53177},[5308],[75,53179,53181],{"className":53180,"style":5343},[5312],[75,53182],{},[75,53184],{"className":53185,"style":5275},[5274],[75,53187,142],{"className":53188},[5279],[75,53190],{"className":53191,"style":5275},[5274],[75,53193,53195,53198,53201,53205,53212],{"className":53194},[5249],[75,53196],{"className":53197,"style":12648},[5253],[75,53199,53117],{"className":53200},[5258],[75,53202,32330],{"className":53203},[5274,53204],"nobreak",[75,53206,53208],{"className":53207},[5258],[75,53209,53211],{"className":53210},[5258,32585],"GHz",[75,53213,6543],{"className":53214},[5258],[12,53216,53217,53218,53246,53247,32154,53285,53323],{},"Applying a microwave drive with frequency ",[75,53219,53221,53234],{"className":53220},[5148],[75,53222,53224],{"className":53223},[5152],[5154,53225,53226],{"xmlns":5156},[5158,53227,53228,53232],{},[5161,53229,53230],{},[5164,53231,5894],{},[5238,53233,5894],{"encoding":5240},[75,53235,53237],{"className":53236,"ariaHidden":5196},[5245],[75,53238,53240,53243],{"className":53239},[5249],[75,53241],{"className":53242,"style":51131},[5253],[75,53244,5894],{"className":53245,"style":6251},[5258,5259]," induces coherent rotations between ",[75,53248,53250,53267],{"className":53249},[5148],[75,53251,53253],{"className":53252},[5152],[5154,53254,53255],{"xmlns":5156},[5158,53256,53257,53265],{},[5161,53258,53259,53261,53263],{},[5168,53260,7452],{"stretchy":5170},[5164,53262,51207],{},[5168,53264,7457],{"stretchy":5170},[5238,53266,51212],{"encoding":5240},[75,53268,53270],{"className":53269,"ariaHidden":5196},[5245],[75,53271,53273,53276,53279,53282],{"className":53272},[5249],[75,53274],{"className":53275,"style":5254},[5253],[75,53277,7452],{"className":53278},[5263],[75,53280,51207],{"className":53281,"style":5416},[5258,5259],[75,53283,7457],{"className":53284},[5270],[75,53286,53288,53305],{"className":53287},[5148],[75,53289,53291],{"className":53290},[5152],[5154,53292,53293],{"xmlns":5156},[5158,53294,53295,53303],{},[5161,53296,53297,53299,53301],{},[5168,53298,7452],{"stretchy":5170},[5164,53300,7486],{},[5168,53302,7457],{"stretchy":5170},[5238,53304,51258],{"encoding":5240},[75,53306,53308],{"className":53307,"ariaHidden":5196},[5245],[75,53309,53311,53314,53317,53320],{"className":53310},[5249],[75,53312],{"className":53313,"style":5254},[5253],[75,53315,7452],{"className":53316},[5263],[75,53318,7486],{"className":53319},[5258,5259],[75,53321,7457],{"className":53322},[5270],". Exact resonance occurs when",[75,53325,53327],{"className":53326},[31410],[75,53328,53330,53354],{"className":53329},[5148],[75,53331,53333],{"className":53332},[5152],[5154,53334,53335],{"xmlns":5156,"display":31419},[5158,53336,53337,53351],{},[5161,53338,53339,53341,53343,53349],{},[5164,53340,5894],{},[5168,53342,142],{},[5179,53344,53345,53347],{},[5164,53346,5894],{},[5184,53348,5186],{},[5164,53350,6543],{"mathvariant":6582},[5238,53352,53353],{"encoding":5240},"f=f_0.",[75,53355,53357,53375],{"className":53356,"ariaHidden":5196},[5245],[75,53358,53360,53363,53366,53369,53372],{"className":53359},[5249],[75,53361],{"className":53362,"style":51131},[5253],[75,53364,5894],{"className":53365,"style":6251},[5258,5259],[75,53367],{"className":53368,"style":5275},[5274],[75,53370,142],{"className":53371},[5279],[75,53373],{"className":53374,"style":5275},[5274],[75,53376,53378,53381,53421],{"className":53377},[5249],[75,53379],{"className":53380,"style":51131},[5253],[75,53382,53384,53387],{"className":53383},[5258],[75,53385,5894],{"className":53386,"style":6251},[5258,5259],[75,53388,53390],{"className":53389},[5299],[75,53391,53393,53413],{"className":53392},[5303,5304],[75,53394,53396,53410],{"className":53395},[5308],[75,53397,53399],{"className":53398,"style":5313},[5312],[75,53400,53401,53404],{"style":6267},[75,53402],{"className":53403,"style":5321},[5320],[75,53405,53407],{"className":53406},[5325,5326,5327,5328],[75,53408,5186],{"className":53409},[5258,5328],[75,53411,5336],{"className":53412},[5335],[75,53414,53416],{"className":53415},[5308],[75,53417,53419],{"className":53418,"style":5343},[5312],[75,53420],{},[75,53422,6543],{"className":53423},[5258],[12,53425,53426],{},"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.",[12425,53428,53430],{"id":53429},"mathematical-foundation","Mathematical Foundation",[12,53432,53433],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[75,53435,53437],{"className":53436},[31410],[75,53438,53440,53480],{"className":53439},[5148],[75,53441,53443],{"className":53442},[5152],[5154,53444,53445],{"xmlns":5156,"display":31419},[5158,53446,53447,53477],{},[5161,53448,53449,53451,53453,53455,53457,53459,53461,53463,53465,53467,53469,53471,53473,53475],{},[5168,53450,7452],{"stretchy":5170},[5164,53452,32718],{},[5168,53454,7457],{"stretchy":5170},[5168,53456,142],{},[5164,53458,36399],{},[5168,53460,7452],{"stretchy":5170},[5164,53462,51207],{},[5168,53464,7457],{"stretchy":5170},[5168,53466,3251],{},[5164,53468,51348],{},[5168,53470,7452],{"stretchy":5170},[5164,53472,7486],{},[5168,53474,7457],{"stretchy":5170},[5168,53476,7469],{"separator":5196},[5238,53478,53479],{"encoding":5240},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[75,53481,53483,53507,53534],{"className":53482,"ariaHidden":5196},[5245],[75,53484,53486,53489,53492,53495,53498,53501,53504],{"className":53485},[5249],[75,53487],{"className":53488,"style":5254},[5253],[75,53490,7452],{"className":53491},[5263],[75,53493,32718],{"className":53494,"style":5416},[5258,5259],[75,53496,7457],{"className":53497},[5270],[75,53499],{"className":53500,"style":5275},[5274],[75,53502,142],{"className":53503},[5279],[75,53505],{"className":53506,"style":5275},[5274],[75,53508,53510,53513,53516,53519,53522,53525,53528,53531],{"className":53509},[5249],[75,53511],{"className":53512,"style":5254},[5253],[75,53514,36399],{"className":53515,"style":36415},[5258,5259],[75,53517,7452],{"className":53518},[5263],[75,53520,51207],{"className":53521,"style":5416},[5258,5259],[75,53523,7457],{"className":53524},[5270],[75,53526],{"className":53527,"style":6294},[5274],[75,53529,3251],{"className":53530},[5477],[75,53532],{"className":53533,"style":6294},[5274],[75,53535,53537,53540,53543,53546,53549,53552],{"className":53536},[5249],[75,53538],{"className":53539,"style":5254},[5253],[75,53541,51348],{"className":53542,"style":51373},[5258,5259],[75,53544,7452],{"className":53545},[5263],[75,53547,7486],{"className":53548},[5258,5259],[75,53550,7457],{"className":53551},[5270],[75,53553,7469],{"className":53554},[7544],[12,53556,51105,53557,32154,53585,53614],{},[75,53558,53560,53573],{"className":53559},[5148],[75,53561,53563],{"className":53562},[5152],[5154,53564,53565],{"xmlns":5156},[5158,53566,53567,53571],{},[5161,53568,53569],{},[5164,53570,36399],{},[5238,53572,36402],{"encoding":5240},[75,53574,53576],{"className":53575,"ariaHidden":5196},[5245],[75,53577,53579,53582],{"className":53578},[5249],[75,53580],{"className":53581,"style":35333},[5253],[75,53583,36399],{"className":53584,"style":36415},[5258,5259],[75,53586,53588,53602],{"className":53587},[5148],[75,53589,53591],{"className":53590},[5152],[5154,53592,53593],{"xmlns":5156},[5158,53594,53595,53599],{},[5161,53596,53597],{},[5164,53598,51348],{},[5238,53600,53601],{"encoding":5240},"\\beta",[75,53603,53605],{"className":53604,"ariaHidden":5196},[5245],[75,53606,53608,53611],{"className":53607},[5249],[75,53609],{"className":53610,"style":51131},[5253],[75,53612,51348],{"className":53613,"style":51373},[5258,5259]," are complex probability amplitudes. Normalization requires",[75,53616,53618],{"className":53617},[31410],[75,53619,53621,53660],{"className":53620},[5148],[75,53622,53624],{"className":53623},[5152],[5154,53625,53626],{"xmlns":5156,"display":31419},[5158,53627,53628,53657],{},[5161,53629,53630,53632,53634,53640,53642,53644,53646,53652,53654],{},[5164,53631,7452],{"mathvariant":6582},[5164,53633,36399],{},[5218,53635,53636,53638],{},[5164,53637,7452],{"mathvariant":6582},[5184,53639,270],{},[5168,53641,3251],{},[5164,53643,7452],{"mathvariant":6582},[5164,53645,51348],{},[5218,53647,53648,53650],{},[5164,53649,7452],{"mathvariant":6582},[5184,53651,270],{},[5168,53653,142],{},[5184,53655,53656],{},"1.",[5238,53658,53659],{"encoding":5240},"|\\alpha|^2+|\\beta|^2=1.",[75,53661,53663,53714,53764],{"className":53662,"ariaHidden":5196},[5245],[75,53664,53666,53670,53673,53676,53705,53708,53711],{"className":53665},[5249],[75,53667],{"className":53668,"style":53669},[5253],"height:1.1141em;vertical-align:-0.25em;",[75,53671,7452],{"className":53672},[5258],[75,53674,36399],{"className":53675,"style":36415},[5258,5259],[75,53677,53679,53682],{"className":53678},[5258],[75,53680,7452],{"className":53681},[5258],[75,53683,53685],{"className":53684},[5299],[75,53686,53688],{"className":53687},[5303],[75,53689,53691],{"className":53690},[5308],[75,53692,53694],{"className":53693,"style":35151},[5312],[75,53695,53696,53699],{"style":35154},[75,53697],{"className":53698,"style":5321},[5320],[75,53700,53702],{"className":53701},[5325,5326,5327,5328],[75,53703,270],{"className":53704},[5258,5328],[75,53706],{"className":53707,"style":6294},[5274],[75,53709,3251],{"className":53710},[5477],[75,53712],{"className":53713,"style":6294},[5274],[75,53715,53717,53720,53723,53726,53755,53758,53761],{"className":53716},[5249],[75,53718],{"className":53719,"style":53669},[5253],[75,53721,7452],{"className":53722},[5258],[75,53724,51348],{"className":53725,"style":51373},[5258,5259],[75,53727,53729,53732],{"className":53728},[5258],[75,53730,7452],{"className":53731},[5258],[75,53733,53735],{"className":53734},[5299],[75,53736,53738],{"className":53737},[5303],[75,53739,53741],{"className":53740},[5308],[75,53742,53744],{"className":53743,"style":35151},[5312],[75,53745,53746,53749],{"style":35154},[75,53747],{"className":53748,"style":5321},[5320],[75,53750,53752],{"className":53751},[5325,5326,5327,5328],[75,53753,270],{"className":53754},[5258,5328],[75,53756],{"className":53757,"style":5275},[5274],[75,53759,142],{"className":53760},[5279],[75,53762],{"className":53763,"style":5275},[5274],[75,53765,53767,53770],{"className":53766},[5249],[75,53768],{"className":53769,"style":32836},[5253],[75,53771,53656],{"className":53772},[5258],[12,53774,53775],{},"The Born rule gives the probability of measuring the excited state,",[75,53777,53779],{"className":53778},[31410],[75,53780,53782,53834],{"className":53781},[5148],[75,53783,53785],{"className":53784},[5152],[5154,53786,53787],{"xmlns":5156,"display":31419},[5158,53788,53789,53831],{},[5161,53790,53791,53797,53799,53801,53803,53805,53807,53809,53811,53817,53819,53821,53823,53829],{},[5179,53792,53793,53795],{},[5164,53794,51013],{},[5164,53796,7486],{},[5168,53798,142],{},[5164,53800,7452],{"mathvariant":6582},[5168,53802,32723],{"stretchy":5170},[5164,53804,7486],{},[5168,53806,7452],{"stretchy":5170},[5164,53808,32718],{},[5168,53810,7457],{"stretchy":5170},[5218,53812,53813,53815],{},[5164,53814,7452],{"mathvariant":6582},[5184,53816,270],{},[5168,53818,142],{},[5164,53820,7452],{"mathvariant":6582},[5164,53822,51348],{},[5218,53824,53825,53827],{},[5164,53826,7452],{"mathvariant":6582},[5184,53828,270],{},[5164,53830,6543],{"mathvariant":6582},[5238,53832,53833],{"encoding":5240},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[75,53835,53837,53892,53954],{"className":53836,"ariaHidden":5196},[5245],[75,53838,53840,53843,53883,53886,53889],{"className":53839},[5249],[75,53841],{"className":53842,"style":12572},[5253],[75,53844,53846,53849],{"className":53845},[5258],[75,53847,51013],{"className":53848,"style":9817},[5258,5259],[75,53850,53852],{"className":53851},[5299],[75,53853,53855,53875],{"className":53854},[5303,5304],[75,53856,53858,53872],{"className":53857},[5308],[75,53859,53861],{"className":53860,"style":6264},[5312],[75,53862,53863,53866],{"style":33755},[75,53864],{"className":53865,"style":5321},[5320],[75,53867,53869],{"className":53868},[5325,5326,5327,5328],[75,53870,7486],{"className":53871},[5258,5259,5328],[75,53873,5336],{"className":53874},[5335],[75,53876,53878],{"className":53877},[5308],[75,53879,53881],{"className":53880,"style":5343},[5312],[75,53882],{},[75,53884],{"className":53885,"style":5275},[5274],[75,53887,142],{"className":53888},[5279],[75,53890],{"className":53891,"style":5275},[5274],[75,53893,53895,53898,53901,53904,53907,53910,53913,53916,53945,53948,53951],{"className":53894},[5249],[75,53896],{"className":53897,"style":53669},[5253],[75,53899,7452],{"className":53900},[5258],[75,53902,32723],{"className":53903},[5263],[75,53905,7486],{"className":53906},[5258,5259],[75,53908,7452],{"className":53909},[5263],[75,53911,32718],{"className":53912,"style":5416},[5258,5259],[75,53914,7457],{"className":53915},[5270],[75,53917,53919,53922],{"className":53918},[5258],[75,53920,7452],{"className":53921},[5258],[75,53923,53925],{"className":53924},[5299],[75,53926,53928],{"className":53927},[5303],[75,53929,53931],{"className":53930},[5308],[75,53932,53934],{"className":53933,"style":35151},[5312],[75,53935,53936,53939],{"style":35154},[75,53937],{"className":53938,"style":5321},[5320],[75,53940,53942],{"className":53941},[5325,5326,5327,5328],[75,53943,270],{"className":53944},[5258,5328],[75,53946],{"className":53947,"style":5275},[5274],[75,53949,142],{"className":53950},[5279],[75,53952],{"className":53953,"style":5275},[5274],[75,53955,53957,53960,53963,53966,53995],{"className":53956},[5249],[75,53958],{"className":53959,"style":53669},[5253],[75,53961,7452],{"className":53962},[5258],[75,53964,51348],{"className":53965,"style":51373},[5258,5259],[75,53967,53969,53972],{"className":53968},[5258],[75,53970,7452],{"className":53971},[5258],[75,53973,53975],{"className":53974},[5299],[75,53976,53978],{"className":53977},[5303],[75,53979,53981],{"className":53980},[5308],[75,53982,53984],{"className":53983,"style":35151},[5312],[75,53985,53986,53989],{"style":35154},[75,53987],{"className":53988,"style":5321},[5320],[75,53990,53992],{"className":53991},[5325,5326,5327,5328],[75,53993,270],{"className":53994},[5258,5328],[75,53996,6543],{"className":53997},[5258],[12,53999,54000],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[75,54002,54004],{"className":54003},[31410],[75,54005,54007,54169],{"className":54006},[5148],[75,54008,54010],{"className":54009},[5152],[5154,54011,54012],{"xmlns":5156,"display":31419},[5158,54013,54014,54166],{},[5161,54015,54016,54022,54024,54060,54062,54064,54070,54072,54112,54114,54116,54122,54124,54164],{},[5179,54017,54018,54020],{},[5164,54019,5232],{},[5164,54021,31430],{},[5168,54023,142],{},[5161,54025,54026,54028,54058],{},[5168,54027,32864],{"fence":5196},[31438,54029,54030,54044],{"rowspacing":31440,"columnalign":31441,"columnspacing":7472},[31443,54031,54032,54038],{},[31446,54033,54034],{},[31449,54035,54036],{"scriptlevel":5186,"displaystyle":5170},[5184,54037,5186],{},[31446,54039,54040],{},[31449,54041,54042],{"scriptlevel":5186,"displaystyle":5170},[5184,54043,7477],{},[31443,54045,54046,54052],{},[31446,54047,54048],{},[31449,54049,54050],{"scriptlevel":5186,"displaystyle":5170},[5184,54051,7477],{},[31446,54053,54054],{},[31449,54055,54056],{"scriptlevel":5186,"displaystyle":5170},[5184,54057,5186],{},[5168,54059,32873],{"fence":5196},[5168,54061,7469],{"separator":5196},[5274,54063],{"width":32297},[5179,54065,54066,54068],{},[5164,54067,5232],{},[5164,54069,31485],{},[5168,54071,142],{},[5161,54073,54074,54076,54110],{},[5168,54075,32864],{"fence":5196},[31438,54077,54078,54096],{"rowspacing":31440,"columnalign":31441,"columnspacing":7472},[31443,54079,54080,54086],{},[31446,54081,54082],{},[31449,54083,54084],{"scriptlevel":5186,"displaystyle":5170},[5184,54085,5186],{},[31446,54087,54088],{},[31449,54089,54090],{"scriptlevel":5186,"displaystyle":5170},[5161,54091,54092,54094],{},[5168,54093,5199],{},[5164,54095,7491],{},[31443,54097,54098,54104],{},[31446,54099,54100],{},[31449,54101,54102],{"scriptlevel":5186,"displaystyle":5170},[5164,54103,7491],{},[31446,54105,54106],{},[31449,54107,54108],{"scriptlevel":5186,"displaystyle":5170},[5184,54109,5186],{},[5168,54111,32873],{"fence":5196},[5168,54113,7469],{"separator":5196},[5274,54115],{"width":32297},[5179,54117,54118,54120],{},[5164,54119,5232],{},[5164,54121,31538],{},[5168,54123,142],{},[5161,54125,54126,54128,54162],{},[5168,54127,32864],{"fence":5196},[31438,54129,54130,54144],{"rowspacing":31440,"columnalign":31441,"columnspacing":7472},[31443,54131,54132,54138],{},[31446,54133,54134],{},[31449,54135,54136],{"scriptlevel":5186,"displaystyle":5170},[5184,54137,7477],{},[31446,54139,54140],{},[31449,54141,54142],{"scriptlevel":5186,"displaystyle":5170},[5184,54143,5186],{},[31443,54145,54146,54152],{},[31446,54147,54148],{},[31449,54149,54150],{"scriptlevel":5186,"displaystyle":5170},[5184,54151,5186],{},[31446,54153,54154],{},[31449,54155,54156],{"scriptlevel":5186,"displaystyle":5170},[5161,54157,54158,54160],{},[5168,54159,5199],{},[5184,54161,7477],{},[5168,54163,32873],{"fence":5196},[5164,54165,6543],{"mathvariant":6582},[5238,54167,54168],{"encoding":5240},"\\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}.",[75,54170,54172,54227,54411,54598],{"className":54171,"ariaHidden":5196},[5245],[75,54173,54175,54178,54218,54221,54224],{"className":54174},[5249],[75,54176],{"className":54177,"style":12488},[5253],[75,54179,54181,54184],{"className":54180},[5258],[75,54182,5232],{"className":54183,"style":5416},[5258,5259],[75,54185,54187],{"className":54186},[5299],[75,54188,54190,54210],{"className":54189},[5303,5304],[75,54191,54193,54207],{"className":54192},[5308],[75,54194,54196],{"className":54195,"style":6264},[5312],[75,54197,54198,54201],{"style":12509},[75,54199],{"className":54200,"style":5321},[5320],[75,54202,54204],{"className":54203},[5325,5326,5327,5328],[75,54205,31430],{"className":54206},[5258,5259,5328],[75,54208,5336],{"className":54209},[5335],[75,54211,54213],{"className":54212},[5308],[75,54214,54216],{"className":54215,"style":5343},[5312],[75,54217],{},[75,54219],{"className":54220,"style":5275},[5274],[75,54222,142],{"className":54223},[5279],[75,54225],{"className":54226,"style":5275},[5274],[75,54228,54230,54233,54350,54353,54356,54359,54362,54402,54405,54408],{"className":54229},[5249],[75,54231],{"className":54232,"style":31648},[5253],[75,54234,54236,54242,54344],{"className":54235},[5360],[75,54237,54239],{"className":54238,"style":5365},[5263,5364],[75,54240,32864],{"className":54241},[5369,5327],[75,54243,54245],{"className":54244},[5258],[75,54246,54248,54293,54296,54299],{"className":54247},[31438],[75,54249,54251],{"className":54250},[31667],[75,54252,54254,54285],{"className":54253},[5303,5304],[75,54255,54257,54282],{"className":54256},[5308],[75,54258,54260,54271],{"className":54259,"style":31677},[5312],[75,54261,54262,54265],{"style":31680},[75,54263],{"className":54264,"style":5400},[5320],[75,54266,54268],{"className":54267},[5258],[75,54269,5186],{"className":54270},[5258],[75,54272,54273,54276],{"style":31692},[75,54274],{"className":54275,"style":5400},[5320],[75,54277,54279],{"className":54278},[5258],[75,54280,7477],{"className":54281},[5258],[75,54283,5336],{"className":54284},[5335],[75,54286,54288],{"className":54287},[5308],[75,54289,54291],{"className":54290,"style":31711},[5312],[75,54292],{},[75,54294],{"className":54295,"style":31718},[31717],[75,54297],{"className":54298,"style":31718},[31717],[75,54300,54302],{"className":54301},[31667],[75,54303,54305,54336],{"className":54304},[5303,5304],[75,54306,54308,54333],{"className":54307},[5308],[75,54309,54311,54322],{"className":54310,"style":31677},[5312],[75,54312,54313,54316],{"style":31680},[75,54314],{"className":54315,"style":5400},[5320],[75,54317,54319],{"className":54318},[5258],[75,54320,7477],{"className":54321},[5258],[75,54323,54324,54327],{"style":31692},[75,54325],{"className":54326,"style":5400},[5320],[75,54328,54330],{"className":54329},[5258],[75,54331,5186],{"className":54332},[5258],[75,54334,5336],{"className":54335},[5335],[75,54337,54339],{"className":54338},[5308],[75,54340,54342],{"className":54341,"style":31711},[5312],[75,54343],{},[75,54345,54347],{"className":54346,"style":5365},[5270,5364],[75,54348,32873],{"className":54349},[5369,5327],[75,54351],{"className":54352,"style":5349},[5274],[75,54354,7469],{"className":54355},[7544],[75,54357],{"className":54358,"style":32516},[5274],[75,54360],{"className":54361,"style":5349},[5274],[75,54363,54365,54368],{"className":54364},[5258],[75,54366,5232],{"className":54367,"style":5416},[5258,5259],[75,54369,54371],{"className":54370},[5299],[75,54372,54374,54394],{"className":54373},[5303,5304],[75,54375,54377,54391],{"className":54376},[5308],[75,54378,54380],{"className":54379,"style":6264},[5312],[75,54381,54382,54385],{"style":12509},[75,54383],{"className":54384,"style":5321},[5320],[75,54386,54388],{"className":54387},[5325,5326,5327,5328],[75,54389,31485],{"className":54390,"style":5416},[5258,5259,5328],[75,54392,5336],{"className":54393},[5335],[75,54395,54397],{"className":54396},[5308],[75,54398,54400],{"className":54399,"style":31823},[5312],[75,54401],{},[75,54403],{"className":54404,"style":5275},[5274],[75,54406,142],{"className":54407},[5279],[75,54409],{"className":54410,"style":5275},[5274],[75,54412,54414,54417,54537,54540,54543,54546,54549,54589,54592,54595],{"className":54413},[5249],[75,54415],{"className":54416,"style":31648},[5253],[75,54418,54420,54426,54531],{"className":54419},[5360],[75,54421,54423],{"className":54422,"style":5365},[5263,5364],[75,54424,32864],{"className":54425},[5369,5327],[75,54427,54429],{"className":54428},[5258],[75,54430,54432,54477,54480,54483],{"className":54431},[31438],[75,54433,54435],{"className":54434},[31667],[75,54436,54438,54469],{"className":54437},[5303,5304],[75,54439,54441,54466],{"className":54440},[5308],[75,54442,54444,54455],{"className":54443,"style":31677},[5312],[75,54445,54446,54449],{"style":31680},[75,54447],{"className":54448,"style":5400},[5320],[75,54450,54452],{"className":54451},[5258],[75,54453,5186],{"className":54454},[5258],[75,54456,54457,54460],{"style":31692},[75,54458],{"className":54459,"style":5400},[5320],[75,54461,54463],{"className":54462},[5258],[75,54464,7491],{"className":54465},[5258,5259],[75,54467,5336],{"className":54468},[5335],[75,54470,54472],{"className":54471},[5308],[75,54473,54475],{"className":54474,"style":31711},[5312],[75,54476],{},[75,54478],{"className":54479,"style":31718},[31717],[75,54481],{"className":54482,"style":31718},[31717],[75,54484,54486],{"className":54485},[31667],[75,54487,54489,54523],{"className":54488},[5303,5304],[75,54490,54492,54520],{"className":54491},[5308],[75,54493,54495,54509],{"className":54494,"style":31677},[5312],[75,54496,54497,54500],{"style":31680},[75,54498],{"className":54499,"style":5400},[5320],[75,54501,54503,54506],{"className":54502},[5258],[75,54504,5199],{"className":54505},[5258],[75,54507,7491],{"className":54508},[5258,5259],[75,54510,54511,54514],{"style":31692},[75,54512],{"className":54513,"style":5400},[5320],[75,54515,54517],{"className":54516},[5258],[75,54518,5186],{"className":54519},[5258],[75,54521,5336],{"className":54522},[5335],[75,54524,54526],{"className":54525},[5308],[75,54527,54529],{"className":54528,"style":31711},[5312],[75,54530],{},[75,54532,54534],{"className":54533,"style":5365},[5270,5364],[75,54535,32873],{"className":54536},[5369,5327],[75,54538],{"className":54539,"style":5349},[5274],[75,54541,7469],{"className":54542},[7544],[75,54544],{"className":54545,"style":32516},[5274],[75,54547],{"className":54548,"style":5349},[5274],[75,54550,54552,54555],{"className":54551},[5258],[75,54553,5232],{"className":54554,"style":5416},[5258,5259],[75,54556,54558],{"className":54557},[5299],[75,54559,54561,54581],{"className":54560},[5303,5304],[75,54562,54564,54578],{"className":54563},[5308],[75,54565,54567],{"className":54566,"style":6264},[5312],[75,54568,54569,54572],{"style":12509},[75,54570],{"className":54571,"style":5321},[5320],[75,54573,54575],{"className":54574},[5325,5326,5327,5328],[75,54576,31538],{"className":54577,"style":32002},[5258,5259,5328],[75,54579,5336],{"className":54580},[5335],[75,54582,54584],{"className":54583},[5308],[75,54585,54587],{"className":54586,"style":5343},[5312],[75,54588],{},[75,54590],{"className":54591,"style":5275},[5274],[75,54593,142],{"className":54594},[5279],[75,54596],{"className":54597,"style":5275},[5274],[75,54599,54601,54604,54724,54727],{"className":54600},[5249],[75,54602],{"className":54603,"style":31648},[5253],[75,54605,54607,54613,54718],{"className":54606},[5360],[75,54608,54610],{"className":54609,"style":5365},[5263,5364],[75,54611,32864],{"className":54612},[5369,5327],[75,54614,54616],{"className":54615},[5258],[75,54617,54619,54664,54667,54670],{"className":54618},[31438],[75,54620,54622],{"className":54621},[31667],[75,54623,54625,54656],{"className":54624},[5303,5304],[75,54626,54628,54653],{"className":54627},[5308],[75,54629,54631,54642],{"className":54630,"style":31677},[5312],[75,54632,54633,54636],{"style":31680},[75,54634],{"className":54635,"style":5400},[5320],[75,54637,54639],{"className":54638},[5258],[75,54640,7477],{"className":54641},[5258],[75,54643,54644,54647],{"style":31692},[75,54645],{"className":54646,"style":5400},[5320],[75,54648,54650],{"className":54649},[5258],[75,54651,5186],{"className":54652},[5258],[75,54654,5336],{"className":54655},[5335],[75,54657,54659],{"className":54658},[5308],[75,54660,54662],{"className":54661,"style":31711},[5312],[75,54663],{},[75,54665],{"className":54666,"style":31718},[31717],[75,54668],{"className":54669,"style":31718},[31717],[75,54671,54673],{"className":54672},[31667],[75,54674,54676,54710],{"className":54675},[5303,5304],[75,54677,54679,54707],{"className":54678},[5308],[75,54680,54682,54693],{"className":54681,"style":31677},[5312],[75,54683,54684,54687],{"style":31680},[75,54685],{"className":54686,"style":5400},[5320],[75,54688,54690],{"className":54689},[5258],[75,54691,5186],{"className":54692},[5258],[75,54694,54695,54698],{"style":31692},[75,54696],{"className":54697,"style":5400},[5320],[75,54699,54701,54704],{"className":54700},[5258],[75,54702,5199],{"className":54703},[5258],[75,54705,7477],{"className":54706},[5258],[75,54708,5336],{"className":54709},[5335],[75,54711,54713],{"className":54712},[5308],[75,54714,54716],{"className":54715,"style":31711},[5312],[75,54717],{},[75,54719,54721],{"className":54720,"style":5365},[5270,5364],[75,54722,32873],{"className":54723},[5369,5327],[75,54725],{"className":54726,"style":5349},[5274],[75,54728,6543],{"className":54729},[5258],[12,54731,54732,54733,54766],{},"Within this representation, a driven two-level system behaves like a spin-",[75,54734,54736,54754],{"className":54735},[5148],[75,54737,54739],{"className":54738},[5152],[5154,54740,54741],{"xmlns":5156},[5158,54742,54743,54751],{},[5161,54744,54745,54747,54749],{},[5184,54746,7477],{},[5164,54748,2810],{"mathvariant":6582},[5184,54750,270],{},[5238,54752,54753],{"encoding":5240},"1\u002F2",[75,54755,54757],{"className":54756,"ariaHidden":5196},[5245],[75,54758,54760,54763],{"className":54759},[5249],[75,54761],{"className":54762,"style":5254},[5253],[75,54764,54753],{"className":54765},[5258]," particle precessing around an effective magnetic field. The drive strength supplies the transverse control component. Applied detuning supplies the longitudinal control component.",[12425,54768,54770],{"id":54769},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[12,54772,54773],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit is",[75,54775,54777],{"className":54776},[31410],[75,54778,54780,54848],{"className":54779},[5148],[75,54781,54783],{"className":54782},[5152],[5154,54784,54785],{"xmlns":5156,"display":31419},[5158,54786,54787,54845],{},[5161,54788,54789,54791,54793,54795,54797,54799,54813,54819,54821,54823,54825,54827,54829,54831,54833,54835,54837,54843],{},[5164,54790,9119],{},[5168,54792,197],{"stretchy":5170},[5164,54794,2771],{},[5168,54796,5175],{"stretchy":5170},[5168,54798,142],{},[5201,54800,54801,54811],{},[5161,54802,54803,54805],{},[5164,54804,52756],{"mathvariant":6582},[5179,54806,54807,54809],{},[5164,54808,12469],{},[5184,54810,5186],{},[5184,54812,270],{},[5179,54814,54815,54817],{},[5164,54816,5232],{},[5164,54818,31538],{},[5168,54820,3251],{},[5164,54822,52756],{"mathvariant":6582},[5164,54824,12635],{"mathvariant":6582},[5164,54826,2845],{},[5168,54828,5191],{},[5168,54830,197],{"stretchy":5170},[5164,54832,12469],{},[5164,54834,2771],{},[5168,54836,5175],{"stretchy":5170},[5179,54838,54839,54841],{},[5164,54840,5232],{},[5164,54842,31430],{},[5168,54844,7469],{"separator":5196},[5238,54846,54847],{"encoding":5240},"H(t)=\\frac{\\hbar\\omega_0}{2}\\sigma_z+\\hbar\\Omega\\cos(\\omega t)\\sigma_x,",[75,54849,54851,54878,55037],{"className":54850,"ariaHidden":5196},[5245],[75,54852,54854,54857,54860,54863,54866,54869,54872,54875],{"className":54853},[5249],[75,54855],{"className":54856,"style":5254},[5253],[75,54858,9119],{"className":54859,"style":9247},[5258,5259],[75,54861,197],{"className":54862},[5263],[75,54864,2771],{"className":54865},[5258,5259],[75,54867,5175],{"className":54868},[5270],[75,54870],{"className":54871,"style":5275},[5274],[75,54873,142],{"className":54874},[5279],[75,54876],{"className":54877,"style":5275},[5274],[75,54879,54881,54885,54988,55028,55031,55034],{"className":54880},[5249],[75,54882],{"className":54883,"style":54884},[5253],"height:2.0519em;vertical-align:-0.686em;",[75,54886,54888,54891,54985],{"className":54887},[5258],[75,54889],{"className":54890},[5263,5380],[75,54892,54894],{"className":54893},[5201],[75,54895,54897,54977],{"className":54896},[5303,5304],[75,54898,54900,54974],{"className":54899},[5308],[75,54901,54904,54915,54923],{"className":54902,"style":54903},[5312],"height:1.3659em;",[75,54905,54906,54909],{"style":33981},[75,54907],{"className":54908,"style":5400},[5320],[75,54910,54912],{"className":54911},[5258],[75,54913,270],{"className":54914},[5258],[75,54916,54917,54920],{"style":5447},[75,54918],{"className":54919,"style":5400},[5320],[75,54921],{"className":54922,"style":5455},[5454],[75,54924,54925,54928],{"style":34001},[75,54926],{"className":54927,"style":5400},[5320],[75,54929,54931,54934],{"className":54930},[5258],[75,54932,52756],{"className":54933},[5258],[75,54935,54937,54940],{"className":54936},[5258],[75,54938,12469],{"className":54939,"style":5416},[5258,5259],[75,54941,54943],{"className":54942},[5299],[75,54944,54946,54966],{"className":54945},[5303,5304],[75,54947,54949,54963],{"className":54948},[5308],[75,54950,54952],{"className":54951,"style":5313},[5312],[75,54953,54954,54957],{"style":12509},[75,54955],{"className":54956,"style":5321},[5320],[75,54958,54960],{"className":54959},[5325,5326,5327,5328],[75,54961,5186],{"className":54962},[5258,5328],[75,54964,5336],{"className":54965},[5335],[75,54967,54969],{"className":54968},[5308],[75,54970,54972],{"className":54971,"style":5343},[5312],[75,54973],{},[75,54975,5336],{"className":54976},[5335],[75,54978,54980],{"className":54979},[5308],[75,54981,54983],{"className":54982,"style":34067},[5312],[75,54984],{},[75,54986],{"className":54987},[5270,5380],[75,54989,54991,54994],{"className":54990},[5258],[75,54992,5232],{"className":54993,"style":5416},[5258,5259],[75,54995,54997],{"className":54996},[5299],[75,54998,55000,55020],{"className":54999},[5303,5304],[75,55001,55003,55017],{"className":55002},[5308],[75,55004,55006],{"className":55005,"style":6264},[5312],[75,55007,55008,55011],{"style":12509},[75,55009],{"className":55010,"style":5321},[5320],[75,55012,55014],{"className":55013},[5325,5326,5327,5328],[75,55015,31538],{"className":55016,"style":32002},[5258,5259,5328],[75,55018,5336],{"className":55019},[5335],[75,55021,55023],{"className":55022},[5308],[75,55024,55026],{"className":55025,"style":5343},[5312],[75,55027],{},[75,55029],{"className":55030,"style":6294},[5274],[75,55032,3251],{"className":55033},[5477],[75,55035],{"className":55036,"style":6294},[5274],[75,55038,55040,55043,55047,55050,55053,55056,55059,55062,55065,55105],{"className":55039},[5249],[75,55041],{"className":55042,"style":5254},[5253],[75,55044,55046],{"className":55045},[5258],"ℏΩ",[75,55048],{"className":55049,"style":5349},[5274],[75,55051,2845],{"className":55052},[5353],[75,55054,197],{"className":55055},[5263],[75,55057,12469],{"className":55058,"style":5416},[5258,5259],[75,55060,2771],{"className":55061},[5258,5259],[75,55063,5175],{"className":55064},[5270],[75,55066,55068,55071],{"className":55067},[5258],[75,55069,5232],{"className":55070,"style":5416},[5258,5259],[75,55072,55074],{"className":55073},[5299],[75,55075,55077,55097],{"className":55076},[5303,5304],[75,55078,55080,55094],{"className":55079},[5308],[75,55081,55083],{"className":55082,"style":6264},[5312],[75,55084,55085,55088],{"style":12509},[75,55086],{"className":55087,"style":5321},[5320],[75,55089,55091],{"className":55090},[5325,5326,5327,5328],[75,55092,31430],{"className":55093},[5258,5259,5328],[75,55095,5336],{"className":55096},[5335],[75,55098,55100],{"className":55099},[5308],[75,55101,55103],{"className":55102,"style":5343},[5312],[75,55104],{},[75,55106,7469],{"className":55107},[7544],[12,55109,51105,55110,55170,55171,55199,55200,55237],{},[75,55111,55113,55134],{"className":55112},[5148],[75,55114,55116],{"className":55115},[5152],[5154,55117,55118],{"xmlns":5156},[5158,55119,55120,55132],{},[5161,55121,55122,55124,55126,55128,55130],{},[5164,55123,12469],{},[5168,55125,142],{},[5184,55127,270],{},[5164,55129,5889],{},[5164,55131,5894],{},[5238,55133,51922],{"encoding":5240},[75,55135,55137,55155],{"className":55136,"ariaHidden":5196},[5245],[75,55138,55140,55143,55146,55149,55152],{"className":55139},[5249],[75,55141],{"className":55142,"style":35333},[5253],[75,55144,12469],{"className":55145,"style":5416},[5258,5259],[75,55147],{"className":55148,"style":5275},[5274],[75,55150,142],{"className":55151},[5279],[75,55153],{"className":55154,"style":5275},[5274],[75,55156,55158,55161,55164,55167],{"className":55157},[5249],[75,55159],{"className":55160,"style":51131},[5253],[75,55162,270],{"className":55163},[5258],[75,55165,5889],{"className":55166,"style":5416},[5258,5259],[75,55168,5894],{"className":55169,"style":6251},[5258,5259]," is the drive angular frequency and ",[75,55172,55174,55187],{"className":55173},[5148],[75,55175,55177],{"className":55176},[5152],[5154,55178,55179],{"xmlns":5156},[5158,55180,55181,55185],{},[5161,55182,55183],{},[5164,55184,12635],{"mathvariant":6582},[5238,55186,12638],{"encoding":5240},[75,55188,55190],{"className":55189,"ariaHidden":5196},[5245],[75,55191,55193,55196],{"className":55192},[5249],[75,55194],{"className":55195,"style":12648},[5253],[75,55197,12635],{"className":55198},[5258]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[75,55201,55203,55222],{"className":55202},[5148],[75,55204,55206],{"className":55205},[5152],[5154,55207,55208],{"xmlns":5156},[5158,55209,55210,55220],{},[5161,55211,55212,55214,55216,55218],{},[5164,55213,12635],{"mathvariant":6582},[5164,55215,2810],{"mathvariant":6582},[5184,55217,270],{},[5164,55219,5889],{},[5238,55221,52020],{"encoding":5240},[75,55223,55225],{"className":55224,"ariaHidden":5196},[5245],[75,55226,55228,55231,55234],{"className":55227},[5249],[75,55229],{"className":55230,"style":5254},[5253],[75,55232,52033],{"className":55233},[5258],[75,55235,5889],{"className":55236,"style":5416},[5258,5259]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[75,55239,55241],{"className":55240},[31410],[75,55242,55244,55302],{"className":55243},[5148],[75,55245,55247],{"className":55246},[5152],[5154,55248,55249],{"xmlns":5156,"display":31419},[5158,55250,55251,55299],{},[5161,55252,55253,55265,55267,55273,55297],{},[5179,55254,55255,55257],{},[5164,55256,9119],{},[5161,55258,55259,55261,55263],{},[5164,55260,32181],{"mathvariant":6582},[5164,55262,33637],{"mathvariant":6582},[5164,55264,5166],{"mathvariant":6582},[5168,55266,142],{},[5201,55268,55269,55271],{},[5164,55270,52756],{"mathvariant":6582},[5184,55272,270],{},[5161,55274,55275,55277,55279,55285,55287,55289,55295],{},[5168,55276,197],{"fence":5196},[5164,55278,12553],{"mathvariant":6582},[5179,55280,55281,55283],{},[5164,55282,5232],{},[5164,55284,31538],{},[5168,55286,3251],{},[5164,55288,12635],{"mathvariant":6582},[5179,55290,55291,55293],{},[5164,55292,5232],{},[5164,55294,31430],{},[5168,55296,5175],{"fence":5196},[5168,55298,7469],{"separator":5196},[5238,55300,55301],{"encoding":5240},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[75,55303,55305,55367],{"className":55304,"ariaHidden":5196},[5245],[75,55306,55308,55311,55358,55361,55364],{"className":55307},[5249],[75,55309],{"className":55310,"style":12572},[5253],[75,55312,55314,55317],{"className":55313},[5258],[75,55315,9119],{"className":55316,"style":9247},[5258,5259],[75,55318,55320],{"className":55319},[5299],[75,55321,55323,55350],{"className":55322},[5303,5304],[75,55324,55326,55347],{"className":55325},[5308],[75,55327,55329],{"className":55328,"style":5942},[5312],[75,55330,55332,55335],{"style":55331},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[75,55333],{"className":55334,"style":5321},[5320],[75,55336,55338],{"className":55337},[5325,5326,5327,5328],[75,55339,55341],{"className":55340},[5258,5328],[75,55342,55344],{"className":55343},[5258,5328],[75,55345,52626],{"className":55346},[5258,32585,5328],[75,55348,5336],{"className":55349},[5335],[75,55351,55353],{"className":55352},[5308],[75,55354,55356],{"className":55355,"style":5343},[5312],[75,55357],{},[75,55359],{"className":55360,"style":5275},[5274],[75,55362,142],{"className":55363},[5279],[75,55365],{"className":55366,"style":5275},[5274],[75,55368,55370,55373,55435,55438,55542,55545],{"className":55369},[5249],[75,55371],{"className":55372,"style":54884},[5253],[75,55374,55376,55379,55432],{"className":55375},[5258],[75,55377],{"className":55378},[5263,5380],[75,55380,55382],{"className":55381},[5201],[75,55383,55385,55424],{"className":55384},[5303,5304],[75,55386,55388,55421],{"className":55387},[5308],[75,55389,55391,55402,55410],{"className":55390,"style":54903},[5312],[75,55392,55393,55396],{"style":33981},[75,55394],{"className":55395,"style":5400},[5320],[75,55397,55399],{"className":55398},[5258],[75,55400,270],{"className":55401},[5258],[75,55403,55404,55407],{"style":5447},[75,55405],{"className":55406,"style":5400},[5320],[75,55408],{"className":55409,"style":5455},[5454],[75,55411,55412,55415],{"style":34001},[75,55413],{"className":55414,"style":5400},[5320],[75,55416,55418],{"className":55417},[5258],[75,55419,52756],{"className":55420},[5258],[75,55422,5336],{"className":55423},[5335],[75,55425,55427],{"className":55426},[5308],[75,55428,55430],{"className":55429,"style":34067},[5312],[75,55431],{},[75,55433],{"className":55434},[5270,5380],[75,55436],{"className":55437,"style":5349},[5274],[75,55439,55441,55444,55447,55487,55490,55493,55496,55499,55539],{"className":55440},[5360],[75,55442,197],{"className":55443,"style":5365},[5263,5364],[75,55445,12553],{"className":55446},[5258],[75,55448,55450,55453],{"className":55449},[5258],[75,55451,5232],{"className":55452,"style":5416},[5258,5259],[75,55454,55456],{"className":55455},[5299],[75,55457,55459,55479],{"className":55458},[5303,5304],[75,55460,55462,55476],{"className":55461},[5308],[75,55463,55465],{"className":55464,"style":6264},[5312],[75,55466,55467,55470],{"style":12509},[75,55468],{"className":55469,"style":5321},[5320],[75,55471,55473],{"className":55472},[5325,5326,5327,5328],[75,55474,31538],{"className":55475,"style":32002},[5258,5259,5328],[75,55477,5336],{"className":55478},[5335],[75,55480,55482],{"className":55481},[5308],[75,55483,55485],{"className":55484,"style":5343},[5312],[75,55486],{},[75,55488],{"className":55489,"style":6294},[5274],[75,55491,3251],{"className":55492},[5477],[75,55494],{"className":55495,"style":6294},[5274],[75,55497,12635],{"className":55498},[5258],[75,55500,55502,55505],{"className":55501},[5258],[75,55503,5232],{"className":55504,"style":5416},[5258,5259],[75,55506,55508],{"className":55507},[5299],[75,55509,55511,55531],{"className":55510},[5303,5304],[75,55512,55514,55528],{"className":55513},[5308],[75,55515,55517],{"className":55516,"style":6264},[5312],[75,55518,55519,55522],{"style":12509},[75,55520],{"className":55521,"style":5321},[5320],[75,55523,55525],{"className":55524},[5325,5326,5327,5328],[75,55526,31430],{"className":55527},[5258,5259,5328],[75,55529,5336],{"className":55530},[5335],[75,55532,55534],{"className":55533},[5308],[75,55535,55537],{"className":55536,"style":5343},[5312],[75,55538],{},[75,55540,5175],{"className":55541,"style":5365},[5270,5364],[75,55543],{"className":55544,"style":5349},[5274],[75,55546,7469],{"className":55547},[7544],[12,55549,55550],{},"with detuning",[75,55552,55554],{"className":55553},[31410],[75,55555,55557,55605],{"className":55556},[5148],[75,55558,55560],{"className":55559},[5152],[5154,55561,55562],{"xmlns":5156,"display":31419},[5158,55563,55564,55602],{},[5161,55565,55566,55568,55570,55572,55574,55580,55582,55584,55586,55588,55590,55592,55598,55600],{},[5164,55567,12553],{"mathvariant":6582},[5168,55569,142],{},[5164,55571,12469],{},[5168,55573,5199],{},[5179,55575,55576,55578],{},[5164,55577,12469],{},[5184,55579,5186],{},[5168,55581,142],{},[5184,55583,270],{},[5164,55585,5889],{},[5168,55587,197],{"stretchy":5170},[5164,55589,5894],{},[5168,55591,5199],{},[5179,55593,55594,55596],{},[5164,55595,5894],{},[5184,55597,5186],{},[5168,55599,5175],{"stretchy":5170},[5164,55601,6543],{"mathvariant":6582},[5238,55603,55604],{"encoding":5240},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[75,55606,55608,55626,55644,55699,55726],{"className":55607,"ariaHidden":5196},[5245],[75,55609,55611,55614,55617,55620,55623],{"className":55610},[5249],[75,55612],{"className":55613,"style":12648},[5253],[75,55615,12553],{"className":55616},[5258],[75,55618],{"className":55619,"style":5275},[5274],[75,55621,142],{"className":55622},[5279],[75,55624],{"className":55625,"style":5275},[5274],[75,55627,55629,55632,55635,55638,55641],{"className":55628},[5249],[75,55630],{"className":55631,"style":52131},[5253],[75,55633,12469],{"className":55634,"style":5416},[5258,5259],[75,55636],{"className":55637,"style":6294},[5274],[75,55639,5199],{"className":55640},[5477],[75,55642],{"className":55643,"style":6294},[5274],[75,55645,55647,55650,55690,55693,55696],{"className":55646},[5249],[75,55648],{"className":55649,"style":12488},[5253],[75,55651,55653,55656],{"className":55652},[5258],[75,55654,12469],{"className":55655,"style":5416},[5258,5259],[75,55657,55659],{"className":55658},[5299],[75,55660,55662,55682],{"className":55661},[5303,5304],[75,55663,55665,55679],{"className":55664},[5308],[75,55666,55668],{"className":55667,"style":5313},[5312],[75,55669,55670,55673],{"style":12509},[75,55671],{"className":55672,"style":5321},[5320],[75,55674,55676],{"className":55675},[5325,5326,5327,5328],[75,55677,5186],{"className":55678},[5258,5328],[75,55680,5336],{"className":55681},[5335],[75,55683,55685],{"className":55684},[5308],[75,55686,55688],{"className":55687,"style":5343},[5312],[75,55689],{},[75,55691],{"className":55692,"style":5275},[5274],[75,55694,142],{"className":55695},[5279],[75,55697],{"className":55698,"style":5275},[5274],[75,55700,55702,55705,55708,55711,55714,55717,55720,55723],{"className":55701},[5249],[75,55703],{"className":55704,"style":5254},[5253],[75,55706,270],{"className":55707},[5258],[75,55709,5889],{"className":55710,"style":5416},[5258,5259],[75,55712,197],{"className":55713},[5263],[75,55715,5894],{"className":55716,"style":6251},[5258,5259],[75,55718],{"className":55719,"style":6294},[5274],[75,55721,5199],{"className":55722},[5477],[75,55724],{"className":55725,"style":6294},[5274],[75,55727,55729,55732,55772,55775],{"className":55728},[5249],[75,55730],{"className":55731,"style":5254},[5253],[75,55733,55735,55738],{"className":55734},[5258],[75,55736,5894],{"className":55737,"style":6251},[5258,5259],[75,55739,55741],{"className":55740},[5299],[75,55742,55744,55764],{"className":55743},[5303,5304],[75,55745,55747,55761],{"className":55746},[5308],[75,55748,55750],{"className":55749,"style":5313},[5312],[75,55751,55752,55755],{"style":6267},[75,55753],{"className":55754,"style":5321},[5320],[75,55756,55758],{"className":55757},[5325,5326,5327,5328],[75,55759,5186],{"className":55760},[5258,5328],[75,55762,5336],{"className":55763},[5335],[75,55765,55767],{"className":55766},[5308],[75,55768,55770],{"className":55769,"style":5343},[5312],[75,55771],{},[75,55773,5175],{"className":55774},[5270],[75,55776,6543],{"className":55777},[5258],[12,55779,55780,55781,32154,55809,55837,55838,55866,55867,55895],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[75,55782,55784,55797],{"className":55783},[5148],[75,55785,55787],{"className":55786},[5152],[5154,55788,55789],{"xmlns":5156},[5158,55790,55791,55795],{},[5161,55792,55793],{},[5164,55794,12635],{"mathvariant":6582},[5238,55796,12638],{"encoding":5240},[75,55798,55800],{"className":55799,"ariaHidden":5196},[5245],[75,55801,55803,55806],{"className":55802},[5249],[75,55804],{"className":55805,"style":12648},[5253],[75,55807,12635],{"className":55808},[5258],[75,55810,55812,55825],{"className":55811},[5148],[75,55813,55815],{"className":55814},[5152],[5154,55816,55817],{"xmlns":5156},[5158,55818,55819,55823],{},[5161,55820,55821],{},[5164,55822,12553],{"mathvariant":6582},[5238,55824,41287],{"encoding":5240},[75,55826,55828],{"className":55827,"ariaHidden":5196},[5245],[75,55829,55831,55834],{"className":55830},[5249],[75,55832],{"className":55833,"style":12648},[5253],[75,55835,12553],{"className":55836},[5258],". The transverse component ",[75,55839,55841,55854],{"className":55840},[5148],[75,55842,55844],{"className":55843},[5152],[5154,55845,55846],{"xmlns":5156},[5158,55847,55848,55852],{},[5161,55849,55850],{},[5164,55851,12635],{"mathvariant":6582},[5238,55853,12638],{"encoding":5240},[75,55855,55857],{"className":55856,"ariaHidden":5196},[5245],[75,55858,55860,55863],{"className":55859},[5249],[75,55861],{"className":55862,"style":12648},[5253],[75,55864,12635],{"className":55865},[5258]," drives population transfer. A longitudinal component ",[75,55868,55870,55883],{"className":55869},[5148],[75,55871,55873],{"className":55872},[5152],[5154,55874,55875],{"xmlns":5156},[5158,55876,55877,55881],{},[5161,55878,55879],{},[5164,55880,12553],{"mathvariant":6582},[5238,55882,41287],{"encoding":5240},[75,55884,55886],{"className":55885,"ariaHidden":5196},[5245],[75,55887,55889,55892],{"className":55888},[5249],[75,55890],{"className":55891,"style":12648},[5253],[75,55893,12553],{"className":55894},[5258]," tilts the rotation axis away from the resonant direction.",[12425,55897,55899],{"id":55898},"excited-state-probability","Excited-State Probability",[12,55901,55902,55903,55941,55942,39886],{},"The simulation assumes that the qubit begins in the ground state ",[75,55904,55906,55923],{"className":55905},[5148],[75,55907,55909],{"className":55908},[5152],[5154,55910,55911],{"xmlns":5156},[5158,55912,55913,55921],{},[5161,55914,55915,55917,55919],{},[5168,55916,7452],{"stretchy":5170},[5164,55918,51207],{},[5168,55920,7457],{"stretchy":5170},[5238,55922,51212],{"encoding":5240},[75,55924,55926],{"className":55925,"ariaHidden":5196},[5245],[75,55927,55929,55932,55935,55938],{"className":55928},[5249],[75,55930],{"className":55931,"style":5254},[5253],[75,55933,7452],{"className":55934},[5263],[75,55936,51207],{"className":55937,"style":5416},[5258,5259],[75,55939,7457],{"className":55940},[5270],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration 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0 400000 1296",[33466,56660],{"d":56661},"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",[75,56663,5336],{"className":56664},[5335],[75,56666,56668],{"className":56667},[5308],[75,56669,56672],{"className":56670,"style":56671},[5312],"height:0.1777em;",[75,56673],{},[12,56675,56676],{},"is the generalized Rabi angular frequency. The amplitude coefficient",[75,56678,56680],{"className":56679},[31410],[75,56681,56683,56711],{"className":56682},[5148],[75,56684,56686],{"className":56685},[5152],[5154,56687,56688],{"xmlns":5156,"display":31419},[5158,56689,56690,56708],{},[5161,56691,56692],{},[5201,56693,56694,56700],{},[5218,56695,56696,56698],{},[5164,56697,12635],{"mathvariant":6582},[5184,56699,270],{},[39536,56701,56702,56704,56706],{},[5164,56703,12635],{"mathvariant":6582},[5164,56705,32181],{},[5184,56707,270],{},[5238,56709,56710],{"encoding":5240},"\\frac{\\Omega^2}{\\Omega_R^2}",[75,56712,56714],{"className":56713,"ariaHidden":5196},[5245],[75,56715,56717,56720],{"className":56716},[5249],[75,56718],{"className":56719,"style":56136},[5253],[75,56721,56723,56726,56853],{"className":56722},[5258],[75,56724],{"className":56725},[5263,5380],[75,56727,56729],{"className":56728},[5201],[75,56730,56732,56845],{"className":56731},[5303,5304],[75,56733,56735,56842],{"className":56734},[5308],[75,56736,56738,56797,56805],{"className":56737,"style":56155},[5312],[75,56739,56740,56743],{"style":33981},[75,56741],{"className":56742,"style":5400},[5320],[75,56744,56746],{"className":56745},[5258],[75,56747,56749,56752],{"className":56748},[5258],[75,56750,12635],{"className":56751},[5258],[75,56753,56755],{"className":56754},[5299],[75,56756,56758,56789],{"className":56757},[5303,5304],[75,56759,56761,56786],{"className":56760},[5308],[75,56762,56764,56775],{"className":56763,"style":56182},[5312],[75,56765,56766,56769],{"style":56185},[75,56767],{"className":56768,"style":5321},[5320],[75,56770,56772],{"className":56771},[5325,5326,5327,5328],[75,56773,32181],{"className":56774,"style":35138},[5258,5259,5328],[75,56776,56777,56780],{"style":56197},[75,56778],{"className":56779,"style":5321},[5320],[75,56781,56783],{"className":56782},[5325,5326,5327,5328],[75,56784,270],{"className":56785},[5258,5328],[75,56787,5336],{"className":56788},[5335],[75,56790,56792],{"className":56791},[5308],[75,56793,56795],{"className":56794,"style":56216},[5312],[75,56796],{},[75,56798,56799,56802],{"style":5447},[75,56800],{"className":56801,"style":5400},[5320],[75,56803],{"className":56804,"style":5455},[5454],[75,56806,56807,56810],{"style":34001},[75,56808],{"className":56809,"style":5400},[5320],[75,56811,56813],{"className":56812},[5258],[75,56814,56816,56819],{"className":56815},[5258],[75,56817,12635],{"className":56818},[5258],[75,56820,56822],{"className":56821},[5299],[75,56823,56825],{"className":56824},[5303],[75,56826,56828],{"className":56827},[5308],[75,56829,56831],{"className":56830,"style":32216},[5312],[75,56832,56833,56836],{"style":7596},[75,56834],{"className":56835,"style":5321},[5320],[75,56837,56839],{"className":56838},[5325,5326,5327,5328],[75,56840,270],{"className":56841},[5258,5328],[75,56843,5336],{"className":56844},[5335],[75,56846,56848],{"className":56847},[5308],[75,56849,56851],{"className":56850,"style":56273},[5312],[75,56852],{},[75,56854],{"className":56855},[5270,5380],[12,56857,56858,56859,56910],{},"sets the maximum possible population transfer. At exact resonance, ",[75,56860,56862,56880],{"className":56861},[5148],[75,56863,56865],{"className":56864},[5152],[5154,56866,56867],{"xmlns":5156},[5158,56868,56869,56877],{},[5161,56870,56871,56873,56875],{},[5164,56872,12553],{"mathvariant":6582},[5168,56874,142],{},[5184,56876,5186],{},[5238,56878,56879],{"encoding":5240},"\\Delta=0",[75,56881,56883,56901],{"className":56882,"ariaHidden":5196},[5245],[75,56884,56886,56889,56892,56895,56898],{"className":56885},[5249],[75,56887],{"className":56888,"style":12648},[5253],[75,56890,12553],{"className":56891},[5258],[75,56893],{"className":56894,"style":5275},[5274],[75,56896,142],{"className":56897},[5279],[75,56899],{"className":56900,"style":5275},[5274],[75,56902,56904,56907],{"className":56903},[5249],[75,56905],{"className":56906,"style":32836},[5253],[75,56908,5186],{"className":56909},[5258],", so",[75,56912,56914],{"className":56913},[31410],[75,56915,56917,56949],{"className":56916},[5148],[75,56918,56920],{"className":56919},[5152],[5154,56921,56922],{"xmlns":5156,"display":31419},[5158,56923,56924,56946],{},[5161,56925,56926,56942,56944],{},[5201,56927,56928,56934],{},[5218,56929,56930,56932],{},[5164,56931,12635],{"mathvariant":6582},[5184,56933,270],{},[39536,56935,56936,56938,56940],{},[5164,56937,12635],{"mathvariant":6582},[5164,56939,32181],{},[5184,56941,270],{},[5168,56943,142],{},[5184,56945,53656],{},[5238,56947,56948],{"encoding":5240},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[75,56950,56952,57103],{"className":56951,"ariaHidden":5196},[5245],[75,56953,56955,56958,57094,57097,57100],{"className":56954},[5249],[75,56956],{"className":56957,"style":56136},[5253],[75,56959,56961,56964,57091],{"className":56960},[5258],[75,56962],{"className":56963},[5263,5380],[75,56965,56967],{"className":56966},[5201],[75,56968,56970,57083],{"className":56969},[5303,5304],[75,56971,56973,57080],{"className":56972},[5308],[75,56974,56976,57035,57043],{"className":56975,"style":56155},[5312],[75,56977,56978,56981],{"style":33981},[75,56979],{"className":56980,"style":5400},[5320],[75,56982,56984],{"className":56983},[5258],[75,56985,56987,56990],{"className":56986},[5258],[75,56988,12635],{"className":56989},[5258],[75,56991,56993],{"className":56992},[5299],[75,56994,56996,57027],{"className":56995},[5303,5304],[75,56997,56999,57024],{"className":56998},[5308],[75,57000,57002,57013],{"className":57001,"style":56182},[5312],[75,57003,57004,57007],{"style":56185},[75,57005],{"className":57006,"style":5321},[5320],[75,57008,57010],{"className":57009},[5325,5326,5327,5328],[75,57011,32181],{"className":57012,"style":35138},[5258,5259,5328],[75,57014,57015,57018],{"style":56197},[75,57016],{"className":57017,"style":5321},[5320],[75,57019,57021],{"className":57020},[5325,5326,5327,5328],[75,57022,270],{"className":57023},[5258,5328],[75,57025,5336],{"className":57026},[5335],[75,57028,57030],{"className":57029},[5308],[75,57031,57033],{"className":57032,"style":56216},[5312],[75,57034],{},[75,57036,57037,57040],{"style":5447},[75,57038],{"className":57039,"style":5400},[5320],[75,57041],{"className":57042,"style":5455},[5454],[75,57044,57045,57048],{"style":34001},[75,57046],{"className":57047,"style":5400},[5320],[75,57049,57051],{"className":57050},[5258],[75,57052,57054,57057],{"className":57053},[5258],[75,57055,12635],{"className":57056},[5258],[75,57058,57060],{"className":57059},[5299],[75,57061,57063],{"className":57062},[5303],[75,57064,57066],{"className":57065},[5308],[75,57067,57069],{"className":57068,"style":32216},[5312],[75,57070,57071,57074],{"style":7596},[75,57072],{"className":57073,"style":5321},[5320],[75,57075,57077],{"className":57076},[5325,5326,5327,5328],[75,57078,270],{"className":57079},[5258,5328],[75,57081,5336],{"className":57082},[5335],[75,57084,57086],{"className":57085},[5308],[75,57087,57089],{"className":57088,"style":56273},[5312],[75,57090],{},[75,57092],{"className":57093},[5270,5380],[75,57095],{"className":57096,"style":5275},[5274],[75,57098,142],{"className":57099},[5279],[75,57101],{"className":57102,"style":5275},[5274],[75,57104,57106,57109],{"className":57105},[5249],[75,57107],{"className":57108,"style":32836},[5253],[75,57110,53656],{"className":57111},[5258],[12,57113,57114],{},"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.",[12,57116,57117],{},"At resonance, the excited-state probability simplifies to",[75,57119,57121],{"className":57120},[31410],[75,57122,57124,57178],{"className":57123},[5148],[75,57125,57127],{"className":57126},[5152],[5154,57128,57129],{"xmlns":5156,"display":31419},[5158,57130,57131,57175],{},[5161,57132,57133,57139,57141,57143,57145,57147,57157,57173],{},[5179,57134,57135,57137],{},[5164,57136,51013],{},[5164,57138,7486],{},[5168,57140,197],{"stretchy":5170},[5164,57142,51024],{},[5168,57144,5175],{"stretchy":5170},[5168,57146,142],{},[5218,57148,57149,57155],{},[5161,57150,57151,57153],{},[5164,57152,1179],{},[5168,57154,5191],{},[5184,57156,270],{},[5161,57158,57159,57161,57171],{},[5168,57160,197],{"fence":5196},[5201,57162,57163,57169],{},[5161,57164,57165,57167],{},[5164,57166,12635],{"mathvariant":6582},[5164,57168,51024],{},[5184,57170,270],{},[5168,57172,5175],{"fence":5196},[5164,57174,6543],{"mathvariant":6582},[5238,57176,57177],{"encoding":5240},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[75,57179,57181,57245],{"className":57180,"ariaHidden":5196},[5245],[75,57182,57184,57187,57227,57230,57233,57236,57239,57242],{"className":57183},[5249],[75,57185],{"className":57186,"style":5254},[5253],[75,57188,57190,57193],{"className":57189},[5258],[75,57191,51013],{"className":57192,"style":9817},[5258,5259],[75,57194,57196],{"className":57195},[5299],[75,57197,57199,57219],{"className":57198},[5303,5304],[75,57200,57202,57216],{"className":57201},[5308],[75,57203,57205],{"className":57204,"style":6264},[5312],[75,57206,57207,57210],{"style":33755},[75,57208],{"className":57209,"style":5321},[5320],[75,57211,57213],{"className":57212},[5325,5326,5327,5328],[75,57214,7486],{"className":57215},[5258,5259,5328],[75,57217,5336],{"className":57218},[5335],[75,57220,57222],{"className":57221},[5308],[75,57223,57225],{"className":57224,"style":5343},[5312],[75,57226],{},[75,57228,197],{"className":57229},[5263],[75,57231,51024],{"className":57232,"style":51096},[5258,5259],[75,57234,5175],{"className":57235},[5270],[75,57237],{"className":57238,"style":5275},[5274],[75,57240,142],{"className":57241},[5279],[75,57243],{"className":57244,"style":5275},[5274],[75,57246,57248,57251,57280,57283,57363,57366],{"className":57247},[5249],[75,57249],{"className":57250,"style":31648},[5253],[75,57252,57254,57257],{"className":57253},[5353],[75,57255,1179],{"className":57256},[5353],[75,57258,57260],{"className":57259},[5299],[75,57261,57263],{"className":57262},[5303],[75,57264,57266],{"className":57265},[5308],[75,57267,57269],{"className":57268,"style":56300},[5312],[75,57270,57271,57274],{"style":56303},[75,57272],{"className":57273,"style":5321},[5320],[75,57275,57277],{"className":57276},[5325,5326,5327,5328],[75,57278,270],{"className":57279},[5258,5328],[75,57281],{"className":57282,"style":5349},[5274],[75,57284,57286,57292,57357],{"className":57285},[5360],[75,57287,57289],{"className":57288,"style":5365},[5263,5364],[75,57290,197],{"className":57291},[5369,5327],[75,57293,57295,57298,57354],{"className":57294},[5258],[75,57296],{"className":57297},[5263,5380],[75,57299,57301],{"className":57300},[5201],[75,57302,57304,57346],{"className":57303},[5303,5304],[75,57305,57307,57343],{"className":57306},[5308],[75,57308,57310,57321,57329],{"className":57309,"style":56343},[5312],[75,57311,57312,57315],{"style":33981},[75,57313],{"className":57314,"style":5400},[5320],[75,57316,57318],{"className":57317},[5258],[75,57319,270],{"className":57320},[5258],[75,57322,57323,57326],{"style":5447},[75,57324],{"className":57325,"style":5400},[5320],[75,57327],{"className":57328,"style":5455},[5454],[75,57330,57331,57334],{"style":34001},[75,57332],{"className":57333,"style":5400},[5320],[75,57335,57337,57340],{"className":57336},[5258],[75,57338,12635],{"className":57339},[5258],[75,57341,51024],{"className":57342,"style":51096},[5258,5259],[75,57344,5336],{"className":57345},[5335],[75,57347,57349],{"className":57348},[5308],[75,57350,57352],{"className":57351,"style":34067},[5312],[75,57353],{},[75,57355],{"className":57356},[5270,5380],[75,57358,57360],{"className":57359,"style":5365},[5270,5364],[75,57361,5175],{"className":57362},[5369,5327],[75,57364],{"className":57365,"style":5349},[5274],[75,57367,6543],{"className":57368},[5258],[12,57370,57371,57372,57400],{},"The pulse duration required for ideal population inversion defines the standard ",[75,57373,57375,57388],{"className":57374},[5148],[75,57376,57378],{"className":57377},[5152],[5154,57379,57380],{"xmlns":5156},[5158,57381,57382,57386],{},[5161,57383,57384],{},[5164,57385,5889],{},[5238,57387,35323],{"encoding":5240},[75,57389,57391],{"className":57390,"ariaHidden":5196},[5245],[75,57392,57394,57397],{"className":57393},[5249],[75,57395],{"className":57396,"style":35333},[5253],[75,57398,5889],{"className":57399,"style":5416},[5258,5259],"-pulse time,",[75,57402,57404],{"className":57403},[31410],[75,57405,57407,57457],{"className":57406},[5148],[75,57408,57410],{"className":57409},[5152],[5154,57411,57412],{"xmlns":5156,"display":31419},[5158,57413,57414,57454],{},[5161,57415,57416,57422,57424,57430,57432,57452],{},[5179,57417,57418,57420],{},[5164,57419,51024],{},[5164,57421,5889],{},[5168,57423,142],{},[5201,57425,57426,57428],{},[5164,57427,5889],{},[5164,57429,12635],{"mathvariant":6582},[5168,57431,142],{},[5201,57433,57434,57436],{},[5184,57435,7477],{},[5161,57437,57438,57440,57442,57444,57446,57448,57450],{},[5184,57439,270],{},[5168,57441,197],{"stretchy":5170},[5164,57443,12635],{"mathvariant":6582},[5164,57445,2810],{"mathvariant":6582},[5184,57447,270],{},[5164,57449,5889],{},[5168,57451,5175],{"stretchy":5170},[5164,57453,6543],{"mathvariant":6582},[5238,57455,57456],{"encoding":5240},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[75,57458,57460,57516,57595],{"className":57459,"ariaHidden":5196},[5245],[75,57461,57463,57466,57507,57510,57513],{"className":57462},[5249],[75,57464],{"className":57465,"style":12488},[5253],[75,57467,57469,57472],{"className":57468},[5258],[75,57470,51024],{"className":57471,"style":51096},[5258,5259],[75,57473,57475],{"className":57474},[5299],[75,57476,57478,57499],{"className":57477},[5303,5304],[75,57479,57481,57496],{"className":57480},[5308],[75,57482,57484],{"className":57483,"style":6264},[5312],[75,57485,57487,57490],{"style":57486},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[75,57488],{"className":57489,"style":5321},[5320],[75,57491,57493],{"className":57492},[5325,5326,5327,5328],[75,57494,5889],{"className":57495,"style":5416},[5258,5259,5328],[75,57497,5336],{"className":57498},[5335],[75,57500,57502],{"className":57501},[5308],[75,57503,57505],{"className":57504,"style":5343},[5312],[75,57506],{},[75,57508],{"className":57509,"style":5275},[5274],[75,57511,142],{"className":57512},[5279],[75,57514],{"className":57515,"style":5275},[5274],[75,57517,57519,57523,57586,57589,57592],{"className":57518},[5249],[75,57520],{"className":57521,"style":57522},[5253],"height:1.7936em;vertical-align:-0.686em;",[75,57524,57526,57529,57583],{"className":57525},[5258],[75,57527],{"className":57528},[5263,5380],[75,57530,57532],{"className":57531},[5201],[75,57533,57535,57575],{"className":57534},[5303,5304],[75,57536,57538,57572],{"className":57537},[5308],[75,57539,57542,57553,57561],{"className":57540,"style":57541},[5312],"height:1.1076em;",[75,57543,57544,57547],{"style":33981},[75,57545],{"className":57546,"style":5400},[5320],[75,57548,57550],{"className":57549},[5258],[75,57551,12635],{"className":57552},[5258],[75,57554,57555,57558],{"style":5447},[75,57556],{"className":57557,"style":5400},[5320],[75,57559],{"className":57560,"style":5455},[5454],[75,57562,57563,57566],{"style":34001},[75,57564],{"className":57565,"style":5400},[5320],[75,57567,57569],{"className":57568},[5258],[75,57570,5889],{"className":57571,"style":5416},[5258,5259],[75,57573,5336],{"className":57574},[5335],[75,57576,57578],{"className":57577},[5308],[75,57579,57581],{"className":57580,"style":34067},[5312],[75,57582],{},[75,57584],{"className":57585},[5270,5380],[75,57587],{"className":57588,"style":5275},[5274],[75,57590,142],{"className":57591},[5279],[75,57593],{"className":57594,"style":5275},[5274],[75,57596,57598,57602,57678],{"className":57597},[5249],[75,57599],{"className":57600,"style":57601},[5253],"height:2.2574em;vertical-align:-0.936em;",[75,57603,57605,57608,57675],{"className":57604},[5258],[75,57606],{"className":57607},[5263,5380],[75,57609,57611],{"className":57610},[5201],[75,57612,57614,57666],{"className":57613},[5303,5304],[75,57615,57617,57663],{"className":57616},[5308],[75,57618,57621,57644,57652],{"className":57619,"style":57620},[5312],"height:1.3214em;",[75,57622,57623,57626],{"style":33981},[75,57624],{"className":57625,"style":5400},[5320],[75,57627,57629,57632,57635,57638,57641],{"className":57628},[5258],[75,57630,270],{"className":57631},[5258],[75,57633,197],{"className":57634},[5263],[75,57636,52033],{"className":57637},[5258],[75,57639,5889],{"className":57640,"style":5416},[5258,5259],[75,57642,5175],{"className":57643},[5270],[75,57645,57646,57649],{"style":5447},[75,57647],{"className":57648,"style":5400},[5320],[75,57650],{"className":57651,"style":5455},[5454],[75,57653,57654,57657],{"style":34001},[75,57655],{"className":57656,"style":5400},[5320],[75,57658,57660],{"className":57659},[5258],[75,57661,7477],{"className":57662},[5258],[75,57664,5336],{"className":57665},[5335],[75,57667,57669],{"className":57668},[5308],[75,57670,57673],{"className":57671,"style":57672},[5312],"height:0.936em;",[75,57674],{},[75,57676],{"className":57677},[5270,5380],[75,57679,6543],{"className":57680},[5258],[12,57682,57683],{},"Given the default drive strength",[75,57685,57687],{"className":57686},[31410],[75,57688,57690,57729],{"className":57689},[5148],[75,57691,57693],{"className":57692},[5152],[5154,57694,57695],{"xmlns":5156,"display":31419},[5158,57696,57697,57726],{},[5161,57698,57699,57709,57711,57713,57715,57724],{},[5201,57700,57701,57703],{},[5164,57702,12635],{"mathvariant":6582},[5161,57704,57705,57707],{},[5184,57706,270],{},[5164,57708,5889],{},[5168,57710,142],{},[5184,57712,752],{},[32328,57714,32330],{},[5161,57716,57717,57720,57722],{},[5164,57718,57719],{"mathvariant":6582},"M",[5164,57721,9119],{"mathvariant":6582},[5164,57723,31538],{"mathvariant":6582},[5168,57725,7469],{"separator":5196},[5238,57727,57728],{"encoding":5240},"\\frac{\\Omega}{2\\pi}=20~\\mathrm{MHz},",[75,57730,57732,57813],{"className":57731,"ariaHidden":5196},[5245],[75,57733,57735,57739,57804,57807,57810],{"className":57734},[5249],[75,57736],{"className":57737,"style":57738},[5253],"height:2.0463em;vertical-align:-0.686em;",[75,57740,57742,57745,57801],{"className":57741},[5258],[75,57743],{"className":57744},[5263,5380],[75,57746,57748],{"className":57747},[5201],[75,57749,57751,57793],{"className":57750},[5303,5304],[75,57752,57754,57790],{"className":57753},[5308],[75,57755,57757,57771,57779],{"className":57756,"style":56343},[5312],[75,57758,57759,57762],{"style":33981},[75,57760],{"className":57761,"style":5400},[5320],[75,57763,57765,57768],{"className":57764},[5258],[75,57766,270],{"className":57767},[5258],[75,57769,5889],{"className":57770,"style":5416},[5258,5259],[75,57772,57773,57776],{"style":5447},[75,57774],{"className":57775,"style":5400},[5320],[75,57777],{"className":57778,"style":5455},[5454],[75,57780,57781,57784],{"style":34001},[75,57782],{"className":57783,"style":5400},[5320],[75,57785,57787],{"className":57786},[5258],[75,57788,12635],{"className":57789},[5258],[75,57791,5336],{"className":57792},[5335],[75,57794,57796],{"className":57795},[5308],[75,57797,57799],{"className":57798,"style":34067},[5312],[75,57800],{},[75,57802],{"className":57803},[5270,5380],[75,57805],{"className":57806,"style":5275},[5274],[75,57808,142],{"className":57809},[5279],[75,57811],{"className":57812,"style":5275},[5274],[75,57814,57816,57819,57822,57825,57831],{"className":57815},[5249],[75,57817],{"className":57818,"style":38345},[5253],[75,57820,752],{"className":57821},[5258],[75,57823,32330],{"className":57824},[5274,53204],[75,57826,57828],{"className":57827},[5258],[75,57829,15986],{"className":57830},[5258,32585],[75,57832,7469],{"className":57833},[7544],[12,57835,57836,57837,57865],{},"the ideal ",[75,57838,57840,57853],{"className":57839},[5148],[75,57841,57843],{"className":57842},[5152],[5154,57844,57845],{"xmlns":5156},[5158,57846,57847,57851],{},[5161,57848,57849],{},[5164,57850,5889],{},[5238,57852,35323],{"encoding":5240},[75,57854,57856],{"className":57855,"ariaHidden":5196},[5245],[75,57857,57859,57862],{"className":57858},[5249],[75,57860],{"className":57861,"style":35333},[5253],[75,57863,5889],{"className":57864,"style":5416},[5258,5259],"-pulse time is",[75,57867,57869],{"className":57868},[31410],[75,57870,57872,57906],{"className":57871},[5148],[75,57873,57875],{"className":57874},[5152],[5154,57876,57877],{"xmlns":5156,"display":31419},[5158,57878,57879,57903],{},[5161,57880,57881,57887,57889,57892,57894,57901],{},[5179,57882,57883,57885],{},[5164,57884,51024],{},[5164,57886,5889],{},[5168,57888,142],{},[5184,57890,57891],{},"25",[32328,57893,32330],{},[5161,57895,57896,57898],{},[5164,57897,33004],{"mathvariant":6582},[5164,57899,57900],{"mathvariant":6582},"s",[5164,57902,6543],{"mathvariant":6582},[5238,57904,57905],{"encoding":5240},"\\tau_\\pi=25~\\mathrm{ns}.",[75,57907,57909,57964],{"className":57908,"ariaHidden":5196},[5245],[75,57910,57912,57915,57955,57958,57961],{"className":57911},[5249],[75,57913],{"className":57914,"style":12488},[5253],[75,57916,57918,57921],{"className":57917},[5258],[75,57919,51024],{"className":57920,"style":51096},[5258,5259],[75,57922,57924],{"className":57923},[5299],[75,57925,57927,57947],{"className":57926},[5303,5304],[75,57928,57930,57944],{"className":57929},[5308],[75,57931,57933],{"className":57932,"style":6264},[5312],[75,57934,57935,57938],{"style":57486},[75,57936],{"className":57937,"style":5321},[5320],[75,57939,57941],{"className":57940},[5325,5326,5327,5328],[75,57942,5889],{"className":57943,"style":5416},[5258,5259,5328],[75,57945,5336],{"className":57946},[5335],[75,57948,57950],{"className":57949},[5308],[75,57951,57953],{"className":57952,"style":5343},[5312],[75,57954],{},[75,57956],{"className":57957,"style":5275},[5274],[75,57959,142],{"className":57960},[5279],[75,57962],{"className":57963,"style":5275},[5274],[75,57965,57967,57970,57973,57976,57983],{"className":57966},[5249],[75,57968],{"className":57969,"style":32836},[5253],[75,57971,57891],{"className":57972},[5258],[75,57974,32330],{"className":57975},[5274,53204],[75,57977,57979],{"className":57978},[5258],[75,57980,57982],{"className":57981},[5258,32585],"ns",[75,57984,6543],{"className":57985},[5258],[12,57987,57988],{},"A full period of the resonant excited-state probability oscillation is",[75,57990,57992],{"className":57991},[31410],[75,57993,57995,58043],{"className":57994},[5148],[75,57996,57998],{"className":57997},[5152],[5154,57999,58000],{"xmlns":5156,"display":31419},[5158,58001,58002,58040],{},[5161,58003,58004,58010,58012,58022,58024,58038],{},[5179,58005,58006,58008],{},[5164,58007,9676],{},[5164,58009,32181],{},[5168,58011,142],{},[5201,58013,58014,58020],{},[5161,58015,58016,58018],{},[5184,58017,270],{},[5164,58019,5889],{},[5164,58021,12635],{"mathvariant":6582},[5168,58023,142],{},[5201,58025,58026,58028],{},[5184,58027,7477],{},[5161,58029,58030,58032,58034,58036],{},[5164,58031,12635],{"mathvariant":6582},[5164,58033,2810],{"mathvariant":6582},[5184,58035,270],{},[5164,58037,5889],{},[5164,58039,6543],{"mathvariant":6582},[5238,58041,58042],{"encoding":5240},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[75,58044,58046,58101,58182],{"className":58045,"ariaHidden":5196},[5245],[75,58047,58049,58052,58092,58095,58098],{"className":58048},[5249],[75,58050],{"className":58051,"style":12572},[5253],[75,58053,58055,58058],{"className":58054},[5258],[75,58056,9676],{"className":58057,"style":9817},[5258,5259],[75,58059,58061],{"className":58060},[5299],[75,58062,58064,58084],{"className":58063},[5303,5304],[75,58065,58067,58081],{"className":58066},[5308],[75,58068,58070],{"className":58069,"style":5942},[5312],[75,58071,58072,58075],{"style":33755},[75,58073],{"className":58074,"style":5321},[5320],[75,58076,58078],{"className":58077},[5325,5326,5327,5328],[75,58079,32181],{"className":58080,"style":35138},[5258,5259,5328],[75,58082,5336],{"className":58083},[5335],[75,58085,58087],{"className":58086},[5308],[75,58088,58090],{"className":58089,"style":5343},[5312],[75,58091],{},[75,58093],{"className":58094,"style":5275},[5274],[75,58096,142],{"className":58097},[5279],[75,58099],{"className":58100,"style":5275},[5274],[75,58102,58104,58108,58173,58176,58179],{"className":58103},[5249],[75,58105],{"className":58106,"style":58107},[5253],"height:2.0074em;vertical-align:-0.686em;",[75,58109,58111,58114,58170],{"className":58110},[5258],[75,58112],{"className":58113},[5263,5380],[75,58115,58117],{"className":58116},[5201],[75,58118,58120,58162],{"className":58119},[5303,5304],[75,58121,58123,58159],{"className":58122},[5308],[75,58124,58126,58137,58145],{"className":58125,"style":57620},[5312],[75,58127,58128,58131],{"style":33981},[75,58129],{"className":58130,"style":5400},[5320],[75,58132,58134],{"className":58133},[5258],[75,58135,12635],{"className":58136},[5258],[75,58138,58139,58142],{"style":5447},[75,58140],{"className":58141,"style":5400},[5320],[75,58143],{"className":58144,"style":5455},[5454],[75,58146,58147,58150],{"style":34001},[75,58148],{"className":58149,"style":5400},[5320],[75,58151,58153,58156],{"className":58152},[5258],[75,58154,270],{"className":58155},[5258],[75,58157,5889],{"className":58158,"style":5416},[5258,5259],[75,58160,5336],{"className":58161},[5335],[75,58163,58165],{"className":58164},[5308],[75,58166,58168],{"className":58167,"style":34067},[5312],[75,58169],{},[75,58171],{"className":58172},[5270,5380],[75,58174],{"className":58175,"style":5275},[5274],[75,58177,142],{"className":58178},[5279],[75,58180],{"className":58181,"style":5275},[5274],[75,58183,58185,58188,58253],{"className":58184},[5249],[75,58186],{"className":58187,"style":57601},[5253],[75,58189,58191,58194,58250],{"className":58190},[5258],[75,58192],{"className":58193},[5263,5380],[75,58195,58197],{"className":58196},[5201],[75,58198,58200,58242],{"className":58199},[5303,5304],[75,58201,58203,58239],{"className":58202},[5308],[75,58204,58206,58220,58228],{"className":58205,"style":57620},[5312],[75,58207,58208,58211],{"style":33981},[75,58209],{"className":58210,"style":5400},[5320],[75,58212,58214,58217],{"className":58213},[5258],[75,58215,52033],{"className":58216},[5258],[75,58218,5889],{"className":58219,"style":5416},[5258,5259],[75,58221,58222,58225],{"style":5447},[75,58223],{"className":58224,"style":5400},[5320],[75,58226],{"className":58227,"style":5455},[5454],[75,58229,58230,58233],{"style":34001},[75,58231],{"className":58232,"style":5400},[5320],[75,58234,58236],{"className":58235},[5258],[75,58237,7477],{"className":58238},[5258],[75,58240,5336],{"className":58241},[5335],[75,58243,58245],{"className":58244},[5308],[75,58246,58248],{"className":58247,"style":57672},[5312],[75,58249],{},[75,58251],{"className":58252},[5270,5380],[75,58254,6543],{"className":58255},[5258],[12,58257,58258],{},"For the same default drive strength, the full temporal period is",[75,58260,58262],{"className":58261},[31410],[75,58263,58265,58298],{"className":58264},[5148],[75,58266,58268],{"className":58267},[5152],[5154,58269,58270],{"xmlns":5156,"display":31419},[5158,58271,58272,58295],{},[5161,58273,58274,58280,58282,58285,58287,58293],{},[5179,58275,58276,58278],{},[5164,58277,9676],{},[5164,58279,32181],{},[5168,58281,142],{},[5184,58283,58284],{},"50",[32328,58286,32330],{},[5161,58288,58289,58291],{},[5164,58290,33004],{"mathvariant":6582},[5164,58292,57900],{"mathvariant":6582},[5164,58294,6543],{"mathvariant":6582},[5238,58296,58297],{"encoding":5240},"T_R=50~\\mathrm{ns}.",[75,58299,58301,58356],{"className":58300,"ariaHidden":5196},[5245],[75,58302,58304,58307,58347,58350,58353],{"className":58303},[5249],[75,58305],{"className":58306,"style":12572},[5253],[75,58308,58310,58313],{"className":58309},[5258],[75,58311,9676],{"className":58312,"style":9817},[5258,5259],[75,58314,58316],{"className":58315},[5299],[75,58317,58319,58339],{"className":58318},[5303,5304],[75,58320,58322,58336],{"className":58321},[5308],[75,58323,58325],{"className":58324,"style":5942},[5312],[75,58326,58327,58330],{"style":33755},[75,58328],{"className":58329,"style":5321},[5320],[75,58331,58333],{"className":58332},[5325,5326,5327,5328],[75,58334,32181],{"className":58335,"style":35138},[5258,5259,5328],[75,58337,5336],{"className":58338},[5335],[75,58340,58342],{"className":58341},[5308],[75,58343,58345],{"className":58344,"style":5343},[5312],[75,58346],{},[75,58348],{"className":58349,"style":5275},[5274],[75,58351,142],{"className":58352},[5279],[75,58354],{"className":58355,"style":5275},[5274],[75,58357,58359,58362,58365,58368,58374],{"className":58358},[5249],[75,58360],{"className":58361,"style":32836},[5253],[75,58363,58284],{"className":58364},[5258],[75,58366,32330],{"className":58367},[5274,53204],[75,58369,58371],{"className":58370},[5258],[75,58372,57982],{"className":58373},[5258,32585],[75,58375,6543],{"className":58376},[5258],[12425,58378,58380],{"id":58379},"phenomenological-dephasing","Phenomenological Dephasing",[12,58382,58383],{},"The computational model includes an optional phenomenological damping factor,",[75,58385,58387],{"className":58386},[31410],[75,58388,58390,58462],{"className":58389},[5148],[75,58391,58393],{"className":58392},[5152],[5154,58394,58395],{"xmlns":5156,"display":31419},[5158,58396,58397,58459],{},[5161,58398,58399,58405,58407,58409,58411,58413,58415,58417,58423,58425,58427,58429,58431,58433,58435,58437,58457],{},[5179,58400,58401,58403],{},[5164,58402,51013],{},[5164,58404,7486],{},[5168,58406,197],{"stretchy":5170},[5164,58408,5894],{},[5168,58410,7469],{"separator":5196},[5164,58412,51024],{},[5168,58414,5175],{"stretchy":5170},[5168,58416,7460],{},[5179,58418,58419,58421],{},[5164,58420,51013],{},[5164,58422,7486],{},[5168,58424,197],{"stretchy":5170},[5164,58426,5894],{},[5168,58428,7469],{"separator":5196},[5164,58430,51024],{},[5168,58432,5175],{"stretchy":5170},[5164,58434,243],{},[5168,58436,5191],{},[5161,58438,58439,58441,58443,58455],{},[5168,58440,197],{"fence":5196},[5168,58442,5199],{},[5201,58444,58445,58447],{},[5164,58446,51024],{},[39536,58448,58449,58451,58453],{},[5164,58450,9676],{},[5184,58452,270],{},[5168,58454,52551],{},[5168,58456,5175],{"fence":5196},[5164,58458,6543],{"mathvariant":6582},[5238,58460,58461],{"encoding":5240},"P_e(f,\\tau)\\rightarrow 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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 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in ordinary frequency units, the generalized Rabi frequency is",[75,59047,59049],{"className":59048},[31410],[75,59050,59052,59140],{"className":59051},[5148],[75,59053,59055],{"className":59054},[5152],[5154,59056,59057],{"xmlns":5156,"display":31419},[5158,59058,59059,59137],{},[5161,59060,59061,59067,59069,59071,59073,59075,59089,59091,59135],{},[5179,59062,59063,59065],{},[5164,59064,52287],{},[5164,59066,32181],{},[5168,59068,197],{"stretchy":5170},[5164,59070,5894],{},[5168,59072,5175],{"stretchy":5170},[5168,59074,142],{},[5201,59076,59077,59083],{},[5179,59078,59079,59081],{},[5164,59080,12635],{"mathvariant":6582},[5164,59082,32181],{},[5161,59084,59085,59087],{},[5184,59086,270],{},[5164,59088,5889],{},[5168,59090,142],{},[33250,59092,59093],{},[5161,59094,59095,59115,59117,59119,59121,59123,59129],{},[5218,59096,59097,59113],{},[5161,59098,59099,59101,59111],{},[5168,59100,197],{"fence":5196},[5201,59102,59103,59105],{},[5164,59104,12635],{"mathvariant":6582},[5161,59106,59107,59109],{},[5184,59108,270],{},[5164,59110,5889],{},[5168,59112,5175],{"fence":5196},[5184,59114,270],{},[5168,59116,3251],{},[5168,59118,197],{"stretchy":5170},[5164,59120,5894],{},[5168,59122,5199],{},[5179,59124,59125,59127],{},[5164,59126,5894],{},[5184,59128,5186],{},[5218,59130,59131,59133],{},[5168,59132,5175],{"stretchy":5170},[5184,59134,270],{},[5164,59136,6543],{"mathvariant":6582},[5238,59138,59139],{"encoding":5240},"\\nu_R(f)=\\frac{\\Omega_R}{2\\pi}=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[75,59141,59143,59207,59324],{"className":59142,"ariaHidden":5196},[5245],[75,59144,59146,59149,59189,59192,59195,59198,59201,59204],{"className":59145},[5249],[75,59147],{"className":59148,"style":5254},[5253],[75,59150,59152,59155],{"className":59151},[5258],[75,59153,52287],{"className":59154,"style":52308},[5258,5259],[75,59156,59158],{"className":59157},[5299],[75,59159,59161,59181],{"className":59160},[5303,5304],[75,59162,59164,59178],{"className":59163},[5308],[75,59165,59167],{"className":59166,"style":5942},[5312],[75,59168,59169,59172],{"style":52323},[75,59170],{"className":59171,"style":5321},[5320],[75,59173,59175],{"className":59174},[5325,5326,5327,5328],[75,59176,32181],{"className":59177,"style":35138},[5258,5259,5328],[75,59179,5336],{"className":59180},[5335],[75,59182,59184],{"className":59183},[5308],[75,59185,59187],{"className":59186,"style":5343},[5312],[75,59188],{},[75,59190,197],{"className":59191},[5263],[75,59193,5894],{"className":59194,"style":6251},[5258,5259],[75,59196,5175],{"className":59197},[5270],[75,59199],{"className":59200,"style":5275},[5274],[75,59202,142],{"className":59203},[5279],[75,59205],{"className":59206,"style":5275},[5274],[75,59208,59210,59213,59315,59318,59321],{"className":59209},[5249],[75,59211],{"className":59212,"style":57738},[5253],[75,59214,59216,59219,59312],{"className":59215},[5258],[75,59217],{"className":59218},[5263,5380],[75,59220,59222],{"className":59221},[5201],[75,59223,59225,59304],{"className":59224},[5303,5304],[75,59226,59228,59301],{"className":59227},[5308],[75,59229,59231,59245,59253],{"className":59230,"style":56343},[5312],[75,59232,59233,59236],{"style":33981},[75,59234],{"className":59235,"style":5400},[5320],[75,59237,59239,59242],{"className":59238},[5258],[75,59240,270],{"className":59241},[5258],[75,59243,5889],{"className":59244,"style":5416},[5258,5259],[75,59246,59247,59250],{"style":5447},[75,59248],{"className":59249,"style":5400},[5320],[75,59251],{"className":59252,"style":5455},[5454],[75,59254,59255,59258],{"style":34001},[75,59256],{"className":59257,"style":5400},[5320],[75,59259,59261],{"className":59260},[5258],[75,59262,59264,59267],{"className":59263},[5258],[75,59265,12635],{"className":59266},[5258],[75,59268,59270],{"className":59269},[5299],[75,59271,59273,59293],{"className":59272},[5303,5304],[75,59274,59276,59290],{"className":59275},[5308],[75,59277,59279],{"className":59278,"style":5942},[5312],[75,59280,59281,59284],{"style":5316},[75,59282],{"className":59283,"style":5321},[5320],[75,59285,59287],{"className":59286},[5325,5326,5327,5328],[75,59288,32181],{"className":59289,"style":35138},[5258,5259,5328],[75,59291,5336],{"className":59292},[5335],[75,59294,59296],{"className":59295},[5308],[75,59297,59299],{"className":59298,"style":5343},[5312],[75,59300],{},[75,59302,5336],{"className":59303},[5335],[75,59305,59307],{"className":59306},[5308],[75,59308,59310],{"className":59309,"style":34067},[5312],[75,59311],{},[75,59313],{"className":59314},[5270,5380],[75,59316],{"className":59317,"style":5275},[5274],[75,59319,142],{"className":59320},[5279],[75,59322],{"className":59323,"style":5275},[5274],[75,59325,59327,59331,59585],{"className":59326},[5249],[75,59328],{"className":59329,"style":59330},[5253],"height:3.04em;vertical-align:-1.0541em;",[75,59332,59334],{"className":59333},[5258,14668],[75,59335,59337,59576],{"className":59336},[5303,5304],[75,59338,59340,59573],{"className":59339},[5308],[75,59341,59344,59556],{"className":59342,"style":59343},[5312],"height:1.9859em;",[75,59345,59348,59352],{"className":59346,"style":59347},[33434],"top:-5em;",[75,59349],{"className":59350,"style":59351},[5320],"height:5em;",[75,59353,59355,59463,59466,59469,59472,59475,59478,59481,59484,59487,59527],{"className":59354,"style":56575},[5258],[75,59356,59358,59438],{"className":59357},[5360],[75,59359,59361,59367,59432],{"className":59360},[5360],[75,59362,59364],{"className":59363,"style":5365},[5263,5364],[75,59365,197],{"className":59366},[5369,5327],[75,59368,59370,59373,59429],{"className":59369},[5258],[75,59371],{"className":59372},[5263,5380],[75,59374,59376],{"className":59375},[5201],[75,59377,59379,59421],{"className":59378},[5303,5304],[75,59380,59382,59418],{"className":59381},[5308],[75,59383,59385,59399,59407],{"className":59384,"style":56343},[5312],[75,59386,59387,59390],{"style":33981},[75,59388],{"className":59389,"style":5400},[5320],[75,59391,59393,59396],{"className":59392},[5258],[75,59394,270],{"className":59395},[5258],[75,59397,5889],{"className":59398,"style":5416},[5258,5259],[75,59400,59401,59404],{"style":5447},[75,59402],{"className":59403,"style":5400},[5320],[75,59405],{"className":59406,"style":5455},[5454],[75,59408,59409,59412],{"style":34001},[75,59410],{"className":59411,"style":5400},[5320],[75,59413,59415],{"className":59414},[5258],[75,59416,12635],{"className":59417},[5258],[75,59419,5336],{"className":59420},[5335],[75,59422,59424],{"className":59423},[5308],[75,59425,59427],{"className":59426,"style":34067},[5312],[75,59428],{},[75,59430],{"className":59431},[5270,5380],[75,59433,59435],{"className":59434,"style":5365},[5270,5364],[75,59436,5175],{"className":59437},[5369,5327],[75,59439,59441],{"className":59440},[5299],[75,59442,59444],{"className":59443},[5303],[75,59445,59447],{"className":59446},[5308],[75,59448,59451],{"className":59449,"style":59450},[5312],"height:1.654em;",[75,59452,59454,59457],{"style":59453},"top:-3.9029em;margin-right:0.05em;",[75,59455],{"className":59456,"style":5321},[5320],[75,59458,59460],{"className":59459},[5325,5326,5327,5328],[75,59461,270],{"className":59462},[5258,5328],[75,59464],{"className":59465,"style":6294},[5274],[75,59467,3251],{"className":59468},[5477],[75,59470],{"className":59471,"style":6294},[5274],[75,59473,197],{"className":59474},[5263],[75,59476,5894],{"className":59477,"style":6251},[5258,5259],[75,59479],{"className":59480,"style":6294},[5274],[75,59482,5199],{"className":59483},[5477],[75,59485],{"className":59486,"style":6294},[5274],[75,59488,59490,59493],{"className":59489},[5258],[75,59491,5894],{"className":59492,"style":6251},[5258,5259],[75,59494,59496],{"className":59495},[5299],[75,59497,59499,59519],{"className":59498},[5303,5304],[75,59500,59502,59516],{"className":59501},[5308],[75,59503,59505],{"className":59504,"style":5313},[5312],[75,59506,59507,59510],{"style":6267},[75,59508],{"className":59509,"style":5321},[5320],[75,59511,59513],{"className":59512},[5325,5326,5327,5328],[75,59514,5186],{"className":59515},[5258,5328],[75,59517,5336],{"className":59518},[5335],[75,59520,59522],{"className":59521},[5308],[75,59523,59525],{"className":59524,"style":5343},[5312],[75,59526],{},[75,59528,59530,59533],{"className":59529},[5270],[75,59531,5175],{"className":59532},[5270],[75,59534,59536],{"className":59535},[5299],[75,59537,59539],{"className":59538},[5303],[75,59540,59542],{"className":59541},[5308],[75,59543,59545],{"className":59544,"style":56594},[5312],[75,59546,59547,59550],{"style":56597},[75,59548],{"className":59549,"style":5321},[5320],[75,59551,59553],{"className":59552},[5325,5326,5327,5328],[75,59554,270],{"className":59555},[5258,5328],[75,59557,59559,59562],{"style":59558},"top:-3.9459em;",[75,59560],{"className":59561,"style":59351},[5320],[75,59563,59566],{"className":59564,"style":59565},[33455],"min-width:1.02em;height:3.08em;",[33458,59567,59570],{"xmlns":33460,"width":33461,"height":59568,"viewBox":59569,"preserveAspectRatio":33464},"3.08em","0 0 400000 3240",[33466,59571],{"d":59572},"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",[75,59574,5336],{"className":59575},[5335],[75,59577,59579],{"className":59578},[5308],[75,59580,59583],{"className":59581,"style":59582},[5312],"height:1.0541em;",[75,59584],{},[75,59586,6543],{"className":59587},[5258],[12,59589,59590],{},"This expression explains the ridge visible in the Fourier-domain heatmap. At resonance, the generalized frequency equals the applied drive strength,",[75,59592,59594],{"className":59593},[31410],[75,59595,59597,59639],{"className":59596},[5148],[75,59598,59600],{"className":59599},[5152],[5154,59601,59602],{"xmlns":5156,"display":31419},[5158,59603,59604,59636],{},[5161,59605,59606,59612,59614,59620,59622,59624,59634],{},[5179,59607,59608,59610],{},[5164,59609,52287],{},[5164,59611,32181],{},[5168,59613,197],{"stretchy":5170},[5179,59615,59616,59618],{},[5164,59617,5894],{},[5184,59619,5186],{},[5168,59621,5175],{"stretchy":5170},[5168,59623,142],{},[5201,59625,59626,59628],{},[5164,59627,12635],{"mathvariant":6582},[5161,59629,59630,59632],{},[5184,59631,270],{},[5164,59633,5889],{},[5164,59635,6543],{"mathvariant":6582},[5238,59637,59638],{"encoding":5240},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[75,59640,59642,59743],{"className":59641,"ariaHidden":5196},[5245],[75,59643,59645,59648,59688,59691,59731,59734,59737,59740],{"className":59644},[5249],[75,59646],{"className":59647,"style":5254},[5253],[75,59649,59651,59654],{"className":59650},[5258],[75,59652,52287],{"className":59653,"style":52308},[5258,5259],[75,59655,59657],{"className":59656},[5299],[75,59658,59660,59680],{"className":59659},[5303,5304],[75,59661,59663,59677],{"className":59662},[5308],[75,59664,59666],{"className":59665,"style":5942},[5312],[75,59667,59668,59671],{"style":52323},[75,59669],{"className":59670,"style":5321},[5320],[75,59672,59674],{"className":59673},[5325,5326,5327,5328],[75,59675,32181],{"className":59676,"style":35138},[5258,5259,5328],[75,59678,5336],{"className":59679},[5335],[75,59681,59683],{"className":59682},[5308],[75,59684,59686],{"className":59685,"style":5343},[5312],[75,59687],{},[75,59689,197],{"className":59690},[5263],[75,59692,59694,59697],{"className":59693},[5258],[75,59695,5894],{"className":59696,"style":6251},[5258,5259],[75,59698,59700],{"className":59699},[5299],[75,59701,59703,59723],{"className":59702},[5303,5304],[75,59704,59706,59720],{"className":59705},[5308],[75,59707,59709],{"className":59708,"style":5313},[5312],[75,59710,59711,59714],{"style":6267},[75,59712],{"className":59713,"style":5321},[5320],[75,59715,59717],{"className":59716},[5325,5326,5327,5328],[75,59718,5186],{"className":59719},[5258,5328],[75,59721,5336],{"className":59722},[5335],[75,59724,59726],{"className":59725},[5308],[75,59727,59729],{"className":59728,"style":5343},[5312],[75,59730],{},[75,59732,5175],{"className":59733},[5270],[75,59735],{"className":59736,"style":5275},[5274],[75,59738,142],{"className":59739},[5279],[75,59741],{"className":59742,"style":5275},[5274],[75,59744,59746,59749,59814],{"className":59745},[5249],[75,59747],{"className":59748,"style":57738},[5253],[75,59750,59752,59755,59811],{"className":59751},[5258],[75,59753],{"className":59754},[5263,5380],[75,59756,59758],{"className":59757},[5201],[75,59759,59761,59803],{"className":59760},[5303,5304],[75,59762,59764,59800],{"className":59763},[5308],[75,59765,59767,59781,59789],{"className":59766,"style":56343},[5312],[75,59768,59769,59772],{"style":33981},[75,59770],{"className":59771,"style":5400},[5320],[75,59773,59775,59778],{"className":59774},[5258],[75,59776,270],{"className":59777},[5258],[75,59779,5889],{"className":59780,"style":5416},[5258,5259],[75,59782,59783,59786],{"style":5447},[75,59784],{"className":59785,"style":5400},[5320],[75,59787],{"className":59788,"style":5455},[5454],[75,59790,59791,59794],{"style":34001},[75,59792],{"className":59793,"style":5400},[5320],[75,59795,59797],{"className":59796},[5258],[75,59798,12635],{"className":59799},[5258],[75,59801,5336],{"className":59802},[5335],[75,59804,59806],{"className":59805},[5308],[75,59807,59809],{"className":59808,"style":34067},[5312],[75,59810],{},[75,59812],{"className":59813},[5270,5380],[75,59815,6543],{"className":59816},[5258],[12,59818,59819,59820,59889,59890,59992],{},"As the system moves away from resonance, ",[75,59821,59823,59840],{"className":59822},[5148],[75,59824,59826],{"className":59825},[5152],[5154,59827,59828],{"xmlns":5156},[5158,59829,59830,59838],{},[5161,59831,59832],{},[5179,59833,59834,59836],{},[5164,59835,52287],{},[5164,59837,32181],{},[5238,59839,52292],{"encoding":5240},[75,59841,59843],{"className":59842,"ariaHidden":5196},[5245],[75,59844,59846,59849],{"className":59845},[5249],[75,59847],{"className":59848,"style":12488},[5253],[75,59850,59852,59855],{"className":59851},[5258],[75,59853,52287],{"className":59854,"style":52308},[5258,5259],[75,59856,59858],{"className":59857},[5299],[75,59859,59861,59881],{"className":59860},[5303,5304],[75,59862,59864,59878],{"className":59863},[5308],[75,59865,59867],{"className":59866,"style":5942},[5312],[75,59868,59869,59872],{"style":52323},[75,59870],{"className":59871,"style":5321},[5320],[75,59873,59875],{"className":59874},[5325,5326,5327,5328],[75,59876,32181],{"className":59877,"style":35138},[5258,5259,5328],[75,59879,5336],{"className":59880},[5335],[75,59882,59884],{"className":59883},[5308],[75,59885,59887],{"className":59886,"style":5343},[5312],[75,59888],{}," increases symmetrically with ",[75,59891,59893,59919],{"className":59892},[5148],[75,59894,59896],{"className":59895},[5152],[5154,59897,59898],{"xmlns":5156},[5158,59899,59900,59916],{},[5161,59901,59902,59904,59906,59908,59914],{},[5168,59903,7452],{"stretchy":5170},[5164,59905,5894],{},[5168,59907,5199],{},[5179,59909,59910,59912],{},[5164,59911,5894],{},[5184,59913,5186],{},[5168,59915,7452],{"stretchy":5170},[5238,59917,59918],{"encoding":5240},"\\lvert f-f_0\\rvert",[75,59920,59922,59943],{"className":59921,"ariaHidden":5196},[5245],[75,59923,59925,59928,59931,59934,59937,59940],{"className":59924},[5249],[75,59926],{"className":59927,"style":5254},[5253],[75,59929,7452],{"className":59930},[5263],[75,59932,5894],{"className":59933,"style":6251},[5258,5259],[75,59935],{"className":59936,"style":6294},[5274],[75,59938,5199],{"className":59939},[5477],[75,59941],{"className":59942,"style":6294},[5274],[75,59944,59946,59949,59989],{"className":59945},[5249],[75,59947],{"className":59948,"style":5254},[5253],[75,59950,59952,59955],{"className":59951},[5258],[75,59953,5894],{"className":59954,"style":6251},[5258,5259],[75,59956,59958],{"className":59957},[5299],[75,59959,59961,59981],{"className":59960},[5303,5304],[75,59962,59964,59978],{"className":59963},[5308],[75,59965,59967],{"className":59966,"style":5313},[5312],[75,59968,59969,59972],{"style":6267},[75,59970],{"className":59971,"style":5321},[5320],[75,59973,59975],{"className":59974},[5325,5326,5327,5328],[75,59976,5186],{"className":59977},[5258,5328],[75,59979,5336],{"className":59980},[5335],[75,59982,59984],{"className":59983},[5308],[75,59985,59987],{"className":59986,"style":5343},[5312],[75,59988],{},[75,59990,7452],{"className":59991},[5270],". 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.",[12425,59994,59996],{"id":59995},"quantum-information-interpretation","Quantum Information Interpretation",[12,59998,59999,60000,60070],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to ",[75,60001,60003,60021],{"className":60002},[5148],[75,60004,60006],{"className":60005},[5152],[5154,60007,60008],{"xmlns":5156},[5158,60009,60010,60018],{},[5161,60011,60012],{},[5179,60013,60014,60016],{},[5164,60015,5232],{},[5164,60017,31430],{},[5238,60019,60020],{"encoding":5240},"\\sigma_x",[75,60022,60024],{"className":60023,"ariaHidden":5196},[5245],[75,60025,60027,60030],{"className":60026},[5249],[75,60028],{"className":60029,"style":12488},[5253],[75,60031,60033,60036],{"className":60032},[5258],[75,60034,5232],{"className":60035,"style":5416},[5258,5259],[75,60037,60039],{"className":60038},[5299],[75,60040,60042,60062],{"className":60041},[5303,5304],[75,60043,60045,60059],{"className":60044},[5308],[75,60046,60048],{"className":60047,"style":6264},[5312],[75,60049,60050,60053],{"style":12509},[75,60051],{"className":60052,"style":5321},[5320],[75,60054,60056],{"className":60055},[5325,5326,5327,5328],[75,60057,31430],{"className":60058},[5258,5259,5328],[75,60060,5336],{"className":60061},[5335],[75,60063,60065],{"className":60064},[5308],[75,60066,60068],{"className":60067,"style":5343},[5312],[75,60069],{},", so the microwave pulse 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nt":6582},[5238,60370,60371],{"encoding":5240},"\\theta=\\Omega\\tau.",[75,60373,60375,60394],{"className":60374,"ariaHidden":5196},[5245],[75,60376,60378,60382,60385,60388,60391],{"className":60377},[5249],[75,60379],{"className":60380,"style":60381},[5253],"height:0.6944em;",[75,60383,7494],{"className":60384,"style":7612},[5258,5259],[75,60386],{"className":60387,"style":5275},[5274],[75,60389,142],{"className":60390},[5279],[75,60392],{"className":60393,"style":5275},[5274],[75,60395,60397,60400,60403,60406],{"className":60396},[5249],[75,60398],{"className":60399,"style":12648},[5253],[75,60401,12635],{"className":60402},[5258],[75,60404,51024],{"className":60405,"style":51096},[5258,5259],[75,60407,6543],{"className":60408},[5258],[12,60410,60411,60412,60440],{},"A calibrated resonant ",[75,60413,60415,60428],{"className":60414},[5148],[75,60416,60418],{"className":60417},[5152],[5154,60419,60420],{"xmlns":5156},[5158,60421,60422,60426],{},[5161,60423,60424],{},[5164,60425,5889],{},[5238,60427,35323],{"encoding":5240},[75,60429,60431],{"className":60430,"ariaHidden":5196},[5245],[75,60432,60434,60437],{"className":60433},[5249],[75,60435],{"className":60436,"style":35333},[5253],[75,60438,5889],{"className":60439,"style":5416},[5258,5259],"-pulse 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corresponds to a quantum ",[75,60612,60614,60627],{"className":60613},[5148],[75,60615,60617],{"className":60616},[5152],[5154,60618,60619],{"xmlns":5156},[5158,60620,60621,60625],{},[5161,60622,60623],{},[5164,60624,5820],{},[5238,60626,5820],{"encoding":5240},[75,60628,60630],{"className":60629,"ariaHidden":5196},[5245],[75,60631,60633,60636],{"className":60632},[5249],[75,60634],{"className":60635,"style":12648},[5253],[75,60637,5820],{"className":60638,"style":5955},[5258,5259]," gate up to a global phase. A resonant ",[75,60641,60643,60661],{"className":60642},[5148],[75,60644,60646],{"className":60645},[5152],[5154,60647,60648],{"xmlns":5156},[5158,60649,60650,60658],{},[5161,60651,60652,60654,60656],{},[5164,60653,5889],{},[5164,60655,2810],{"mathvariant":6582},[5184,60657,270],{},[5238,60659,60660],{"encoding":5240},"\\pi\u002F2",[75,60662,60664],{"className":60663,"ariaHidden":5196},[5245],[75,60665,60667,60670,60673],{"className":60666},[5249],[75,60668],{"className":60669,"style":5254},[5253],[75,60671,5889],{"className":60672,"style":5416},[5258,5259],[75,60674,6608],{"className":60675},[5258],"-pulse prepares a balanced coherent superposition,",[75,60678,60680],{"className":60679},[31410],[75,60681,60683,60729],{"className":60682},[5148],[75,60684,60686],{"className":60685},[5152],[5154,60687,60688],{"xmlns":5156,"display":31419},[5158,60689,60690,60726],{},[5161,60691,60692,60694,60696,60698,60700,60724],{},[5168,60693,7452],{"stretchy":5170},[5164,60695,51207],{},[5168,60697,7457],{"stretchy":5170},[5168,60699,7460],{},[5201,60701,60702,60720],{},[5161,60703,60704,60706,60708,60710,60712,60714,60716,60718],{},[5168,60705,7452],{"stretchy":5170},[5164,60707,51207],{},[5168,60709,7457],{"stretchy":5170},[5168,60711,5199],{},[5164,60713,7491],{},[5168,60715,7452],{"stretchy":5170},[5164,60717,7486],{},[5168,60719,7457],{"stretchy":5170},[33250,60721,60722],{},[5184,60723,270],{},[5168,60725,7469],{"separator":5196},[5238,60727,60728],{"encoding":5240},"\\lvert g\\rangle\\rightarrow\\frac{\\lvert g\\rangle-i\\lvert e\\rangle}{\\sqrt{2}},",[75,60730,60732,60756],{"className":60731,"ariaHidden":5196},[5245],[75,60733,60735,60738,60741,60744,60747,60750,60753],{"className":60734},[5249],[75,60736],{"className":60737,"style":5254},[5253],[75,60739,7452],{"className":60740},[5263],[75,60742,51207],{"className":60743,"style":5416},[5258,5259],[75,60745,7457],{"className":60746},[5270],[75,60748],{"className":60749,"style":5275},[5274],[75,60751,7460],{"className":60752},[5279],[75,60754],{"className":60755,"style":5275},[5274],[75,60757,60759,60763,60899],{"className":60758},[5249],[75,60760],{"className":60761,"style":60762},[5253],"height:2.357em;vertical-align:-0.93em;",[75,60764,60766,60769,60896],{"className":60765},[5258],[75,60767],{"className":60768},[5263,5380],[75,60770,60772],{"className":60771},[5201],[75,60773,60775,60887],{"className":60774},[5303,5304],[75,60776,60778,60884],{"className":60777},[5308],[75,60779,60782,60838,60846],{"className":60780,"style":60781},[5312],"height:1.427em;",[75,60783,60785,60788],{"style":60784},"top:-2.2028em;",[75,60786],{"className":60787,"style":5400},[5320],[75,60789,60791],{"className":60790},[5258],[75,60792,60794],{"className":60793},[5258,14668],[75,60795,60797,60830],{"className":60796},[5303,5304],[75,60798,60800,60827],{"className":60799},[5308],[75,60801,60803,60815],{"className":60802,"style":34020},[5312],[75,60804,60806,60809],{"className":60805,"style":33435},[33434],[75,60807],{"className":60808,"style":5400},[5320],[75,60810,60812],{"className":60811,"style":33442},[5258],[75,60813,270],{"className":60814},[5258],[75,60816,60817,60820],{"style":34035},[75,60818],{"className":60819,"style":5400},[5320],[75,60821,60823],{"className":60822,"style":33456},[33455],[33458,60824,60825],{"xmlns":33460,"width":33461,"height":33462,"viewBox":33463,"preserveAspectRatio":33464},[33466,60826],{"d":33468},[75,60828,5336],{"className":60829},[5335],[75,60831,60833],{"className":60832},[5308],[75,60834,60836],{"className":60835,"style":34055},[5312],[75,60837],{},[75,60839,60840,60843],{"style":5447},[75,60841],{"className":60842,"style":5400},[5320],[75,60844],{"className":60845,"style":5455},[5454],[75,60847,60848,60851],{"style":34001},[75,60849],{"className":60850,"style":5400},[5320],[75,60852,60854,60857,60860,60863,60866,60869,60872,60875,60878,60881],{"className":60853},[5258],[75,60855,7452],{"className":60856},[5263],[75,60858,51207],{"className":60859,"style":5416},[5258,5259],[75,60861,7457],{"className":60862},[5270],[75,60864],{"className":60865,"style":6294},[5274],[75,60867,5199],{"className":60868},[5477],[75,60870],{"className":60871,"style":6294},[5274],[75,60873,7491],{"className":60874},[5258,5259],[75,60876,7452],{"className":60877},[5263],[75,60879,7486],{"className":60880},[5258,5259],[75,60882,7457],{"className":60883},[5270],[75,60885,5336],{"className":60886},[5335],[75,60888,60890],{"className":60889},[5308],[75,60891,60894],{"className":60892,"style":60893},[5312],"height:0.93em;",[75,60895],{},[75,60897],{"className":60898},[5270,5380],[75,60900,7469],{"className":60901},[7544],[12,60903,60904],{},"with a phase convention set by the microwave drive phase.",[12,60906,60907],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[75,60909,60911],{"className":60910},[31410],[75,60912,60914,60983],{"className":60913},[5148],[75,60915,60917],{"className":60916},[5152],[5154,60918,60919],{"xmlns":5156,"display":31419},[5158,60920,60921,60980],{},[5161,60922,60923,60933,60935,60937,60939,60941,60943,60945,60978],{},[5179,60924,60925,60927],{},[5164,60926,32181],{},[36328,60928,60929,60931],{"accent":5196},[5164,60930,33004],{},[5168,60932,36335],{},[5168,60934,197],{"stretchy":5170},[5164,60936,7494],{},[5168,60938,5175],{"stretchy":5170},[5168,60940,142],{},[5164,60942,243],{},[5168,60944,5191],{},[5161,60946,60947,60949,60951,60961,60967,60969,60976],{},[5168,60948,197],{"fence":5196},[5168,60950,5199],{},[5201,60952,60953,60959],{},[5161,60954,60955,60957],{},[5164,60956,7491],{},[5164,60958,7494],{},[5184,60960,270],{},[36328,60962,60963,60965],{"accent":5196},[5164,60964,33004],{},[5168,60966,36335],{},[5168,60968,32287],{},[36328,60970,60971,60973],{"accent":5196},[5164,60972,5232],{},[5168,60974,60975],{},"⃗",[5168,60977,5175],{"fence":5196},[5168,60979,7469],{"separator":5196},[5238,60981,60982],{"encoding":5240},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[75,60984,60986,61083],{"className":60985,"ariaHidden":5196},[5245],[75,60987,60989,60992,61065,61068,61071,61074,61077,61080],{"className":60988},[5249],[75,60990],{"className":60991,"style":5254},[5253],[75,60993,60995,60998],{"className":60994},[5258],[75,60996,32181],{"className":60997,"style":35138},[5258,5259],[75,60999,61001],{"className":61000},[5299],[75,61002,61004,61057],{"className":61003},[5303,5304],[75,61005,61007,61054],{"className":61006},[5308],[75,61008,61011],{"className":61009,"style":61010},[5312],"height:0.3361em;",[75,61012,61013,61016],{"style":36104},[75,61014],{"className":61015,"style":5321},[5320],[75,61017,61019],{"className":61018},[5325,5326,5327,5328],[75,61020,61022],{"className":61021},[5258,5328],[75,61023,61025],{"className":61024},[5258,36352,5328],[75,61026,61028],{"className":61027},[5303],[75,61029,61031],{"className":61030},[5308],[75,61032,61034,61043],{"className":61033,"style":60381},[5312],[75,61035,61037,61040],{"style":61036},"top:-2.7em;",[75,61038],{"className":61039,"style":5321},[5320],[75,61041,33004],{"className":61042},[5258,5259,5328],[75,61044,61045,61048],{"style":61036},[75,61046],{"className":61047,"style":5321},[5320],[75,61049,61051],{"className":61050,"style":36380},[36379],[75,61052,36335],{"className":61053},[5258,5328],[75,61055,5336],{"className":61056},[5335],[75,61058,61060],{"className":61059},[5308],[75,61061,61063],{"className":61062,"style":5343},[5312],[75,61064],{},[75,61066,197],{"className":61067},[5263],[75,61069,7494],{"className":61070,"style":7612},[5258,5259],[75,61072,5175],{"className":61073},[5270],[75,61075],{"className":61076,"style":5275},[5274],[75,61078,142],{"className":61079},[5279],[75,61081],{"className":61082,"style":5275},[5274],[75,61084,61086,61089,61092,61095,61263,61266],{"className":61085},[5249],[75,61087],{"className":61088,"style":31648},[5253],[75,61090,243],{"className":61091},[5353],[75,61093],{"className":61094,"style":5349},[5274],[75,61096,61098,61104,61107,61172,61203,61206,61209,61212,61257],{"className":61097},[5360],[75,61099,61101],{"className":61100,"style":5365},[5263,5364],[75,61102,197],{"className":61103},[5369,5327],[75,61105,5199],{"className":61106},[5258],[75,61108,61110,61113,61169],{"className":61109},[5258],[75,61111],{"className":61112},[5263,5380],[75,61114,61116],{"className":61115},[5201],[75,61117,61119,61161],{"className":61118},[5303,5304],[75,61120,61122,61158],{"className":61121},[5308],[75,61123,61125,61136,61144],{"className":61124,"style":60242},[5312],[75,61126,61127,61130],{"style":33981},[75,61128],{"className":61129,"style":5400},[5320],[75,61131,61133],{"className":61132},[5258],[75,61134,270],{"className":61135},[5258],[75,61137,61138,61141],{"style":5447},[75,61139],{"className":61140,"style":5400},[5320],[75,61142],{"className":61143,"style":5455},[5454],[75,61145,61146,61149],{"style":34001},[75,61147],{"className":61148,"style":5400},[5320],[75,61150,61152,61155],{"className":61151},[5258],[75,61153,7491],{"className":61154},[5258,5259],[75,61156,7494],{"className":61157,"style":7612},[5258,5259],[75,61159,5336],{"className":61160},[5335],[75,61162,61164],{"className":61163},[5308],[75,61165,61167],{"className":61166,"style":34067},[5312],[75,61168],{},[75,61170],{"className":61171},[5270,5380],[75,61173,61175],{"className":61174},[5258,36352],[75,61176,61178],{"className":61177},[5303],[75,61179,61181],{"className":61180},[5308],[75,61182,61184,61192],{"className":61183,"style":60381},[5312],[75,61185,61186,61189],{"style":33435},[75,61187],{"className":61188,"style":5400},[5320],[75,61190,33004],{"className":61191},[5258,5259],[75,61193,61194,61197],{"style":33435},[75,61195],{"className":61196,"style":5400},[5320],[75,61198,61200],{"className":61199,"style":36380},[36379],[7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0 471 714","xMinYMin",[33466,61255],{"d":61256},"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",[75,61258,61260],{"className":61259,"style":5365},[5270,5364],[75,61261,5175],{"className":61262},[5369,5327],[75,61264],{"className":61265,"style":5349},[5274],[75,61267,7469],{"className":61268},[7544],[12,61270,15807],{},[75,61272,61274],{"className":61273},[31410],[75,61275,61277,61339],{"className":61276},[5148],[75,61278,61280],{"className":61279},[5152],[5154,61281,61282],{"xmlns":5156,"display":31419},[5158,61283,61284,61336],{},[5161,61285,61286,61292,61294,61304,61306,61308,61310,61312,61314,61316,61318,61320,61322,61324,61326,61332,61334],{},[36328,61287,61288,61290],{"accent":5196},[5164,61289,33004],{},[5168,61291,36335],{},[5168,61293,142],{},[5201,61295,61296,61298],{},[5184,61297,7477],{},[5179,61299,61300,61302],{},[5164,61301,12635],{"mathvariant":6582},[5164,61303,32181],{},[5168,61305,197],{"stretchy":5170},[5164,61307,12635],{"mathvariant":6582},[5168,61309,7469],{"separator":5196},[5184,61311,5186],{},[5168,61313,7469],{"separator":5196},[5164,61315,12553],{"mathvariant":6582},[5168,61317,5175],{"stretchy":5170},[5168,61319,7469],{"separator":5196},[5274,61321],{"width":32297},[5164,61323,7494],{},[5168,61325,142],{},[5179,61327,61328,61330],{},[5164,61329,12635],{"mathvariant":6582},[5164,61331,32181],{},[5164,61333,51024],{},[5164,61335,6543],{"mathvariant":6582},[5238,61337,61338],{"encoding":5240},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[75,61340,61342,61388,61543],{"className":61341,"ariaHidden":5196},[5245],[75,61343,61345,61348,61379,61382,61385],{"className":61344},[5249],[75,61346],{"className":61347,"style":60381},[5253],[75,61349,61351],{"className":61350},[5258,36352],[75,61352,61354],{"className":61353},[5303],[75,61355,61357],{"className":61356},[5308],[75,61358,61360,61368],{"className":61359,"style":60381},[5312],[75,61361,61362,61365],{"style":33435},[75,61363],{"className":61364,"style":5400},[5320],[75,61366,33004],{"className":61367},[5258,5259],[75,61369,61370,61373],{"style":33435},[75,61371],{"className":61372,"style":5400},[5320],[75,61374,61376],{"className":61375,"style":36380},[36379],[75,61377,36335],{"className":61378},[5258],[75,61380],{"className":61381,"style":5275},[5274],[75,61383,142],{"className":61384},[5279],[75,61386],{"className":61387,"style":5275},[5274],[75,61389,61391,61395,61495,61498,61501,61504,61507,61510,61513,61516,61519,61522,61525,61528,61531,61534,61537,61540],{"className":61390},[5249],[75,61392],{"className":61393,"style":61394},[5253],"height:2.1574em;vertical-align:-0.836em;",[75,61396,61398,61401,61492],{"className":61397},[5258],[75,61399],{"className":61400},[5263,5380],[75,61402,61404],{"className":61403},[5201],[75,61405,61407,61483],{"className":61406},[5303,5304],[75,61408,61410,61480],{"className":61409},[5308],[75,61411,61413,61461,61469],{"className":61412,"style":57620},[5312],[75,61414,61415,61418],{"style":33981},[75,61416],{"className":61417,"style":5400},[5320],[75,61419,61421],{"className":61420},[5258],[75,61422,61424,61427],{"className":61423},[5258],[75,61425,12635],{"className":61426},[5258],[75,61428,61430],{"className":61429},[5299],[75,61431,61433,61453],{"className":61432},[5303,5304],[75,61434,61436,61450],{"className":61435},[5308],[75,61437,61439],{"className":61438,"style":5942},[5312],[75,61440,61441,61444],{"style":5316},[75,61442],{"className":61443,"style":5321},[5320],[75,61445,61447],{"className":61446},[5325,5326,5327,5328],[75,61448,32181],{"className":61449,"style":35138},[5258,5259,5328],[75,61451,5336],{"className":61452},[5335],[75,61454,61456],{"className":61455},[5308],[75,61457,61459],{"className":61458,"style":5343},[5312],[75,61460],{},[75,61462,61463,61466],{"style":5447},[75,61464],{"className":61465,"style":5400},[5320],[75,61467],{"className":61468,"style":5455},[5454],[75,61470,61471,61474],{"style":34001},[75,61472],{"className":61473,"style":5400},[5320],[75,61475,61477],{"className":61476},[5258],[75,61478,7477],{"className":61479},[5258],[75,61481,5336],{"className":61482},[5335],[75,61484,61486],{"className":61485},[5308],[75,61487,61490],{"className":61488,"style":61489},[5312],"height:0.836em;",[75,61491],{},[75,61493],{"className":61494},[5270,5380],[75,61496,197],{"className":61497},[5263],[75,61499,12635],{"className":61500},[5258],[75,61502,7469],{"className":61503},[7544],[75,61505],{"className":61506,"style":5349},[5274],[75,61508,5186],{"className":61509},[5258],[75,61511,7469],{"className":61512},[7544],[75,61514],{"className":61515,"style":5349},[5274],[75,61517,12553],{"className":61518},[5258],[75,61520,5175],{"className":61521},[5270],[75,61523,7469],{"className":61524},[7544],[75,61526],{"className":61527,"style":32516},[5274],[75,61529],{"className":61530,"style":5349},[5274],[75,61532,7494],{"className":61533,"style":7612},[5258,5259],[75,61535],{"className":61536,"style":5275},[5274],[75,61538,142],{"className":61539},[5279],[75,61541],{"className":61542,"style":5275},[5274],[75,61544,61546,61549,61589,61592],{"className":61545},[5249],[75,61547],{"className":61548,"style":12572},[5253],[75,61550,61552,61555],{"className":61551},[5258],[75,61553,12635],{"className":61554},[5258],[75,61556,61558],{"className":61557},[5299],[75,61559,61561,61581],{"className":61560},[5303,5304],[75,61562,61564,61578],{"className":61563},[5308],[75,61565,61567],{"className":61566,"style":5942},[5312],[75,61568,61569,61572],{"style":5316},[75,61570],{"className":61571,"style":5321},[5320],[75,61573,61575],{"className":61574},[5325,5326,5327,5328],[75,61576,32181],{"className":61577,"style":35138},[5258,5259,5328],[75,61579,5336],{"className":61580},[5335],[75,61582,61584],{"className":61583},[5308],[75,61585,61587],{"className":61586,"style":5343},[5312],[75,61588],{},[75,61590,51024],{"className":61591,"style":51096},[5258,5259],[75,61593,6543],{"className":61594},[5258],[12,61596,61597],{},"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.",[12425,61599,61601],{"id":61600},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[12,61603,61604],{},"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.",[12,61606,61607,61608,61646,61647,61675,61676,61711],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[75,61609,61611,61628],{"className":61610},[5148],[75,61612,61614],{"className":61613},[5152],[5154,61615,61616],{"xmlns":5156},[5158,61617,61618,61626],{},[5161,61619,61620,61622,61624],{},[5168,61621,7452],{"stretchy":5170},[5164,61623,51207],{},[5168,61625,7457],{"stretchy":5170},[5238,61627,51212],{"encoding":5240},[75,61629,61631],{"className":61630,"ariaHidden":5196},[5245],[75,61632,61634,61637,61640,61643],{"className":61633},[5249],[75,61635],{"className":61636,"style":5254},[5253],[75,61638,7452],{"className":61639},[5263],[75,61641,51207],{"className":61642,"style":5416},[5258,5259],[75,61644,7457],{"className":61645},[5270],", a properly timed ",[75,61648,61650,61663],{"className":61649},[5148],[75,61651,61653],{"className":61652},[5152],[5154,61654,61655],{"xmlns":5156},[5158,61656,61657,61661],{},[5161,61658,61659],{},[5164,61660,5889],{},[5238,61662,35323],{"encoding":5240},[75,61664,61666],{"className":61665,"ariaHidden":5196},[5245],[75,61667,61669,61672],{"className":61668},[5249],[75,61670],{"className":61671,"style":35333},[5253],[75,61673,5889],{"className":61674,"style":5416},[5258,5259],"-pulse moves the state to the excited-state pole. A ",[75,61677,61679,61696],{"className":61678},[5148],[75,61680,61682],{"className":61681},[5152],[5154,61683,61684],{"xmlns":5156},[5158,61685,61686,61694],{},[5161,61687,61688,61690,61692],{},[5164,61689,5889],{},[5164,61691,2810],{"mathvariant":6582},[5184,61693,270],{},[5238,61695,60660],{"encoding":5240},[75,61697,61699],{"className":61698,"ariaHidden":5196},[5245],[75,61700,61702,61705,61708],{"className":61701},[5249],[75,61703],{"className":61704,"style":5254},[5253],[75,61706,5889],{"className":61707,"style":5416},[5258,5259],[75,61709,6608],{"className":61710},[5258],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[12,61713,61714,61715,61743],{},"Detuning tilts the rotation axis toward the ",[75,61716,61718,61731],{"className":61717},[5148],[75,61719,61721],{"className":61720},[5152],[5154,61722,61723],{"xmlns":5156},[5158,61724,61725,61729],{},[5161,61726,61727],{},[5164,61728,31538],{},[5238,61730,31538],{"encoding":5240},[75,61732,61734],{"className":61733,"ariaHidden":5196},[5245],[75,61735,61737,61740],{"className":61736},[5249],[75,61738],{"className":61739,"style":35333},[5253],[75,61741,31538],{"className":61742,"style":32002},[5258,5259],"-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",[75,61745,61747],{"className":61746},[31410],[75,61748,61750,61780],{"className":61749},[5148],[75,61751,61753],{"className":61752},[5152],[5154,61754,61755],{"xmlns":5156,"display":31419},[5158,61756,61757,61777],{},[5161,61758,61759,61775],{},[5201,61760,61761,61767],{},[5218,61762,61763,61765],{},[5164,61764,12635],{"mathvariant":6582},[5184,61766,270],{},[39536,61768,61769,61771,61773],{},[5164,61770,12635],{"mathvariant":6582},[5164,61772,32181],{},[5184,61774,270],{},[5164,61776,6543],{"mathvariant":6582},[5238,61778,61779],{"encoding":5240},"\\frac{\\Omega^2}{\\Omega_R^2}.",[75,61781,61783],{"className":61782,"ariaHidden":5196},[5245],[75,61784,61786,61789,61925],{"className":61785},[5249],[75,61787],{"className":61788,"style":56136},[5253],[75,61790,61792,61795,61922],{"className":61791},[5258],[75,61793],{"className":61794},[5263,5380],[75,61796,61798],{"className":61797},[5201],[75,61799,61801,61914],{"className":61800},[5303,5304],[75,61802,61804,61911],{"className":61803},[5308],[75,61805,61807,61866,61874],{"className":61806,"style":56155},[5312],[75,61808,61809,61812],{"style":33981},[75,61810],{"className":61811,"style":5400},[5320],[75,61813,61815],{"className":61814},[5258],[75,61816,61818,61821],{"className":61817},[5258],[75,61819,12635],{"className":61820},[5258],[75,61822,61824],{"className":61823},[5299],[75,61825,61827,61858],{"className":61826},[5303,5304],[75,61828,61830,61855],{"className":61829},[5308],[75,61831,61833,61844],{"className":61832,"style":56182},[5312],[75,61834,61835,61838],{"style":56185},[75,61836],{"className":61837,"style":5321},[5320],[75,61839,61841],{"className":61840},[5325,5326,5327,5328],[75,61842,32181],{"className":61843,"style":35138},[5258,5259,5328],[75,61845,61846,61849],{"style":56197},[75,61847],{"className":61848,"style":5321},[5320],[75,61850,61852],{"className":61851},[5325,5326,5327,5328],[75,61853,270],{"className":61854},[5258,5328],[75,61856,5336],{"className":61857},[5335],[75,61859,61861],{"className":61860},[5308],[75,61862,61864],{"className":61863,"style":56216},[5312],[75,61865],{},[75,61867,61868,61871],{"style":5447},[75,61869],{"className":61870,"style":5400},[5320],[75,61872],{"className":61873,"style":5455},[5454],[75,61875,61876,61879],{"style":34001},[75,61877],{"className":61878,"style":5400},[5320],[75,61880,61882],{"className":61881},[5258],[75,61883,61885,61888],{"className":61884},[5258],[75,61886,12635],{"className":61887},[5258],[75,61889,61891],{"className":61890},[5299],[75,61892,61894],{"className":61893},[5303],[75,61895,61897],{"className":61896},[5308],[75,61898,61900],{"className":61899,"style":32216},[5312],[75,61901,61902,61905],{"style":7596},[75,61903],{"className":61904,"style":5321},[5320],[75,61906,61908],{"className":61907},[5325,5326,5327,5328],[75,61909,270],{"className":61910},[5258,5328],[75,61912,5336],{"className":61913},[5335],[75,61915,61917],{"className":61916},[5308],[75,61918,61920],{"className":61919,"style":56273},[5312],[75,61921],{},[75,61923],{"className":61924},[5270,5380],[75,61926,6543],{"className":61927},[5258],[12,61929,61930,61931,61970],{},"As ",[75,61932,61934,61952],{"className":61933},[5148],[75,61935,61937],{"className":61936},[5152],[5154,61938,61939],{"xmlns":5156},[5158,61940,61941,61949],{},[5161,61942,61943,61945,61947],{},[5168,61944,7452],{"stretchy":5170},[5164,61946,12553],{"mathvariant":6582},[5168,61948,7452],{"stretchy":5170},[5238,61950,61951],{"encoding":5240},"\\lvert\\Delta\\rvert",[75,61953,61955],{"className":61954,"ariaHidden":5196},[5245],[75,61956,61958,61961,61964,61967],{"className":61957},[5249],[75,61959],{"className":61960,"style":5254},[5253],[75,61962,7452],{"className":61963},[5263],[75,61965,12553],{"className":61966},[5258],[75,61968,7452],{"className":61969},[5270]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[12425,61972,61974],{"id":61973},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[12,61976,61977,61978,62006,62007,62105],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[75,61979,61981,61994],{"className":61980},[5148],[75,61982,61984],{"className":61983},[5152],[5154,61985,61986],{"xmlns":5156},[5158,61987,61988,61992],{},[5161,61989,61990],{},[5164,61991,5894],{},[5238,61993,5894],{"encoding":5240},[75,61995,61997],{"className":61996,"ariaHidden":5196},[5245],[75,61998,62000,62003],{"className":61999},[5249],[75,62001],{"className":62002,"style":51131},[5253],[75,62004,5894],{"className":62005,"style":6251},[5258,5259],", the trace ",[75,62008,62010,62038],{"className":62009},[5148],[75,62011,62013],{"className":62012},[5152],[5154,62014,62015],{"xmlns":5156},[5158,62016,62017,62035],{},[5161,62018,62019,62025,62027,62029,62031,62033],{},[5179,62020,62021,62023],{},[5164,62022,51013],{},[5164,62024,7486],{},[5168,62026,197],{"stretchy":5170},[5164,62028,51024],{},[5168,62030,7469],{"separator":5196},[5164,62032,5894],{},[5168,62034,5175],{"stretchy":5170},[5238,62036,62037],{"encoding":5240},"P_e(\\tau,f)",[75,62039,62041],{"className":62040,"ariaHidden":5196},[5245],[75,62042,62044,62047,62087,62090,62093,62096,62099,62102],{"className":62043},[5249],[75,62045],{"className":62046,"style":5254},[5253],[75,62048,62050,62053],{"className":62049},[5258],[75,62051,51013],{"className":62052,"style":9817},[5258,5259],[75,62054,62056],{"className":62055},[5299],[75,62057,62059,62079],{"className":62058},[5303,5304],[75,62060,62062,62076],{"className":62061},[5308],[75,62063,62065],{"className":62064,"style":6264},[5312],[75,62066,62067,62070],{"style":33755},[75,62068],{"className":62069,"style":5321},[5320],[75,62071,62073],{"className":62072},[5325,5326,5327,5328],[75,62074,7486],{"className":62075},[5258,5259,5328],[75,62077,5336],{"className":62078},[5335],[75,62080,62082],{"className":62081},[5308],[75,62083,62085],{"className":62084,"style":5343},[5312],[75,62086],{},[75,62088,197],{"className":62089},[5263],[75,62091,51024],{"className":62092,"style":51096},[5258,5259],[75,62094,7469],{"className":62095},[7544],[75,62097],{"className":62098,"style":5349},[5274],[75,62100,5894],{"className":62101,"style":6251},[5258,5259],[75,62103,5175],{"className":62104},[5270]," is treated as a time-domain signal. The spectral amplitude is",[75,62107,62109],{"className":62108},[31410],[75,62110,62112,62176],{"className":62111},[5148],[75,62113,62115],{"className":62114},[5152],[5154,62116,62117],{"xmlns":5156,"display":31419},[5158,62118,62119,62173],{},[5161,62120,62121,62123,62125,62127,62129,62131,62133,62135,62171],{},[5164,62122,33564],{},[5168,62124,197],{"stretchy":5170},[5164,62126,52287],{},[5168,62128,7469],{"separator":5196},[5164,62130,5894],{},[5168,62132,5175],{"stretchy":5170},[5168,62134,142],{},[5161,62136,62137,62139,62147,62169],{},[5168,62138,7452],{"fence":5196},[5179,62140,62141,62145],{},[5164,62142,62144],{"mathvariant":62143},"script","F",[5164,62146,51024],{},[5161,62148,62149,62151,62157,62159,62161,62163,62165,62167],{},[5168,62150,12219],{"fence":5196},[5179,62152,62153,62155],{},[5164,62154,51013],{},[5164,62156,7486],{},[5168,62158,197],{"stretchy":5170},[5164,62160,51024],{},[5168,62162,7469],{"separator":5196},[5164,62164,5894],{},[5168,62166,5175],{"stretchy":5170},[5168,62168,12224],{"fence":5196},[5168,62170,7452],{"fence":5196},[5164,62172,6543],{"mathvariant":6582},[5238,62174,62175],{"encoding":5240},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[75,62177,62179,62215],{"className":62178,"ariaHidden":5196},[5245],[75,62180,62182,62185,62188,62191,62194,62197,62200,62203,62206,62209,62212],{"className":62181},[5249],[75,62183],{"className":62184,"style":5254},[5253],[75,62186,33564],{"className":62187,"style":5929},[5258,5259],[75,62189,197],{"className":62190},[5263],[75,62192,52287],{"className":62193,"style":52308},[5258,5259],[75,62195,7469],{"className":62196},[7544],[75,62198],{"className":62199,"style":5349},[5274],[75,62201,5894],{"className":62202,"style":6251},[5258,5259],[75,62204,5175],{"className":62205},[5270],[75,62207],{"className":62208,"style":5275},[5274],[75,62210,142],{"className":62211},[5279],[75,62213],{"className":62214,"style":5275},[5274],[75,62216,62218,62221,62346,62349],{"className":62217},[5249],[75,62219],{"className":62220,"style":5254},[5253],[75,62222,62224,62227,62273,62276,62343],{"className":62223},[5360],[75,62225,7452],{"className":62226,"style":5365},[5263,5364],[75,62228,62230,62235],{"className":62229},[5258],[75,62231,62144],{"className":62232,"style":62234},[5258,62233],"mathcal","margin-right:0.0993em;",[75,62236,62238],{"className":62237},[5299],[75,62239,62241,62265],{"className":62240},[5303,5304],[75,62242,62244,62262],{"className":62243},[5308],[75,62245,62247],{"className":62246,"style":6264},[5312],[75,62248,62250,62253],{"style":62249},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[75,62251],{"className":62252,"style":5321},[5320],[75,62254,62256],{"className":62255},[5325,5326,5327,5328],[75,62257,62259],{"className":62258},[5258,5328],[75,62260,51024],{"className":62261,"style":51096},[5258,5259,5328],[75,62263,5336],{"className":62264},[5335],[75,62266,62268],{"className":62267},[5308],[75,62269,62271],{"className":62270,"style":5343},[5312],[75,62272],{},[75,62274],{"className":62275,"style":5349},[5274],[75,62277,62279,62282,62322,62325,62328,62331,62334,62337,62340],{"className":62278},[5360],[75,62280,12219],{"className":62281,"style":5365},[5263,5364],[75,62283,62285,62288],{"className":62284},[5258],[75,62286,51013],{"className":62287,"style":9817},[5258,5259],[75,62289,62291],{"className":62290},[5299],[75,62292,62294,62314],{"className":62293},[5303,5304],[75,62295,62297,62311],{"className":62296},[5308],[75,62298,62300],{"className":62299,"style":6264},[5312],[75,62301,62302,62305],{"style":33755},[75,62303],{"className":62304,"style":5321},[5320],[75,62306,62308],{"className":62307},[5325,5326,5327,5328],[75,62309,7486],{"className":62310},[5258,5259,5328],[75,62312,5336],{"className":62313},[5335],[75,62315,62317],{"className":62316},[5308],[75,62318,62320],{"className":62319,"style":5343},[5312],[75,62321],{},[75,62323,197],{"className":62324},[5263],[75,62326,51024],{"className":62327,"style":51096},[5258,5259],[75,62329,7469],{"className":62330},[7544],[75,62332],{"className":62333,"style":5349},[5274],[75,62335,5894],{"className":62336,"style":6251},[5258,5259],[75,62338,5175],{"className":62339},[5270],[75,62341,12224],{"className":62342,"style":5365},[5270,5364],[75,62344,7452],{"className":62345,"style":5365},[5270,5364],[75,62347],{"className":62348,"style":5349},[5274],[75,62350,6543],{"className":62351},[5258],[12,62353,62354],{},"This probability expression contains a squared sinusoid. Using the identity",[75,62356,62358],{"className":62357},[31410],[75,62359,62361,62437],{"className":62360},[5148],[75,62362,62364],{"className":62363},[5152],[5154,62365,62366],{"xmlns":5156,"display":31419},[5158,62367,62368,62434],{},[5161,62369,62370,62380,62400,62402,62408],{},[5218,62371,62372,62378],{},[5161,62373,62374,62376],{},[5164,62375,1179],{},[5168,62377,5191],{},[5184,62379,270],{},[5161,62381,62382,62384,62398],{},[5168,62383,197],{"fence":5196},[5201,62385,62386,62396],{},[5161,62387,62388,62394],{},[5179,62389,62390,62392],{},[5164,62391,12635],{"mathvariant":6582},[5164,62393,32181],{},[5164,62395,51024],{},[5184,62397,270],{},[5168,62399,5175],{"fence":5196},[5168,62401,142],{},[5201,62403,62404,62406],{},[5184,62405,7477],{},[5184,62407,270],{},[5161,62409,62410,62412,62414,62416,62418,62420,62422,62428,62430,62432],{},[5168,62411,32864],{"fence":5196},[5184,62413,7477],{},[5168,62415,5199],{},[5164,62417,2845],{},[5168,62419,5191],{},[5168,62421,197],{"stretchy":5170},[5179,62423,62424,62426],{},[5164,62425,12635],{"mathvariant":6582},[5164,62427,32181],{},[5164,62429,51024],{},[5168,62431,5175],{"stretchy":5170},[5168,62433,32873],{"fence":5196},[5238,62435,62436],{"encoding":5240},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[75,62438,62440,62604],{"className":62439,"ariaHidden":5196},[5245],[75,62441,62443,62446,62475,62478,62595,62598,62601],{"className":62442},[5249],[75,62444],{"className":62445,"style":31648},[5253],[75,62447,62449,62452],{"className":62448},[5353],[75,62450,1179],{"className":62451},[5353],[75,62453,62455],{"className":62454},[5299],[75,62456,62458],{"className":62457},[5303],[75,62459,62461],{"className":62460},[5308],[75,62462,62464],{"className":62463,"style":56300},[5312],[75,62465,62466,62469],{"style":56303},[75,62467],{"className":62468,"style":5321},[5320],[75,62470,62472],{"className":62471},[5325,5326,5327,5328],[75,62473,270],{"className":62474},[5258,5328],[75,62476],{"className":62477,"style":5349},[5274],[75,62479,62481,62487,62589],{"className":62480},[5360],[75,62482,62484],{"className":62483,"style":5365},[5263,5364],[75,62485,197],{"className":62486},[5369,5327],[75,62488,62490,62493,62586],{"className":62489},[5258],[75,62491],{"className":62492},[5263,5380],[75,62494,62496],{"className":62495},[5201],[75,62497,62499,62578],{"className":62498},[5303,5304],[75,62500,62502,62575],{"className":62501},[5308],[75,62503,62505,62516,62524],{"className":62504,"style":56343},[5312],[75,62506,62507,62510],{"style":33981},[75,62508],{"className":62509,"style":5400},[5320],[75,62511,62513],{"className":62512},[5258],[75,62514,270],{"className":62515},[5258],[75,62517,62518,62521],{"style":5447},[75,62519],{"className":62520,"style":5400},[5320],[75,62522],{"className":62523,"style":5455},[5454],[75,62525,62526,62529],{"style":34001},[75,62527],{"className":62528,"style":5400},[5320],[75,62530,62532,62572],{"className":62531},[5258],[75,62533,62535,62538],{"className":62534},[5258],[75,62536,12635],{"className":62537},[5258],[75,62539,62541],{"className":62540},[5299],[75,62542,62544,62564],{"className":62543},[5303,5304],[75,62545,62547,62561],{"className":62546},[5308],[75,62548,62550],{"className":62549,"style":5942},[5312],[75,62551,62552,62555],{"style":5316},[75,62553],{"className":62554,"style":5321},[5320],[75,62556,62558],{"className":62557},[5325,5326,5327,5328],[75,62559,32181],{"className":62560,"style":35138},[5258,5259,5328],[75,62562,5336],{"className":62563},[5335],[75,62565,62567],{"className":62566},[5308],[75,62568,62570],{"className":62569,"style":5343},[5312],[75,62571],{},[75,62573,51024],{"className":62574,"style":51096},[5258,5259],[75,62576,5336],{"className":62577},[5335],[75,62579,62581],{"className":62580},[5308],[75,62582,62584],{"className":62583,"style":34067},[5312],[75,62585],{},[75,62587],{"className":62588},[5270,5380],[75,62590,62592],{"className":62591,"style":5365},[5270,5364],[75,62593,5175],{"className":62594},[5369,5327],[75,62596],{"className":62597,"style":5275},[5274],[75,62599,142],{"className":62600},[5279],[75,62602],{"className":62603,"style":5275},[5274],[75,62605,62607,62610,62672,62675],{"className":62606},[5249],[75,62608],{"className":62609,"style":58107},[5253],[75,62611,62613,62616,62669],{"className":62612},[5258],[75,62614],{"className":62615},[5263,5380],[75,62617,62619],{"className":62618},[5201],[75,62620,62622,62661],{"className":62621},[5303,5304],[75,62623,62625,62658],{"className":62624},[5308],[75,62626,62628,62639,62647],{"className":62627,"style":57620},[5312],[75,62629,62630,62633],{"style":33981},[75,62631],{"className":62632,"style":5400},[5320],[75,62634,62636],{"className":62635},[5258],[75,62637,270],{"className":62638},[5258],[75,62640,62641,62644],{"style":5447},[75,62642],{"className":62643,"style":5400},[5320],[75,62645],{"className":62646,"style":5455},[5454],[75,62648,62649,62652],{"style":34001},[75,62650],{"className":62651,"styl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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.",[12425,62966,62968],{"id":62967},"computational-methodology","Computational Methodology",[12,62970,62971,62972,63000,63001,63029,63030,32154,63058,63086,63087,63221,63222,63441],{},"The notebook uses a deterministic closed-form workflow, so it does not require numerical time stepping of the Schrödinger equation. 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detuning 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evaluates the generalized Rabi angular frequency 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and calculates the analytic excited-state probability",[75,63443,63445],{"className":63444},[31410],[75,63446,63448,63526],{"className":63447},[5148],[75,63449,63451],{"className":63450},[5152],[5154,63452,63453],{"xmlns":5156,"display":31419},[5158,63454,63455,63523],{},[5161,63456,63457,63463,63465,63467,63469,63471,63473,63475,63491,63501,63521],{},[5179,63458,63459,63461],{},[5164,63460,51013],{},[5164,63462,7486],{},[5168,63464,197],{"stretchy":5170},[5164,63466,5894],{},[5168,63468,7469],{"separator":5196},[5164,63470,51024],{},[5168,63472,5175],{"stretchy":5170},[5168,63474,142],{},[5201,63476,63477,63483],{},[5218,63478,63479,63481],{},[5164,63480,12635],{"mathvariant":6582},[5184,63482,270],{},[39536,63484,63485,63487,63489],{},[5164,63486,12635],{"mathvariant":6582},[5164,63488,32181],{},[5184,63490,270],{},[5218,63492,63493,63499],{},[5161,63494,63495,63497],{},[5164,63496,1179],{},[5168,63498,5191],{},[5184,63500,270],{},[5161,63502,63503,63505,63519],{},[5168,63504,197],{"fence":5196},[5201,63506,63507,63517],{},[5161,63508,63509,63515],{},[5179,63510,63511,63513],{},[5164,63512,12635],{"mathvariant":6582},[5164,63514,32181],{},[5164,63516,51024],{},[5184,63518,270],{},[5168,63520,5175],{"fence":5196},[5164,63522,6543],{"mathvariant":6582},[5238,63524,63525],{"encoding":5240},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[75,63527,63529,63602],{"className":63528,"ariaHidden":5196},[5245],[75,63530,63532,63535,63575,63578,63581,63584,63587,63590,63593,63596,63599],{"className":63531},[5249],[75,63533],{"className":63534,"style":5254},[5253],[75,63536,63538,63541],{"className":63537},[5258],[75,63539,51013],{"className":63540,"style":9817},[5258,5259],[75,63542,63544],{"className":63543},[5299],[75,63545,63547,63567],{"className":63546},[5303,5304],[75,63548,63550,63564],{"className":63549},[5308],[75,63551,63553],{"className":63552,"style":6264},[5312],[75,63554,63555,63558],{"style":33755},[75,63556],{"className":63557,"style":5321},[5320],[75,63559,63561],{"className":63560},[5325,5326,5327,5328],[75,63562,7486],{"className":63563},[5258,5259,5328],[75,63565,5336],{"className":63566},[5335],[75,63568,63570],{"className":63569},[5308],[75,63571,63573],{"className":63572,"style":5343},[5312],[75,63574],{},[75,63576,197],{"className":63577},[5263],[75,63579,5894],{"className":63580,"style":6251},[5258,5259],[75,63582,7469],{"className":63583},[7544],[75,63585],{"className":63586,"style":5349},[5274],[75,63588,51024],{"className":63589,"style":51096},[5258,5259],[75,63591,5175],{"className":63592},[5270],[75,63594],{"className":63595,"style":5275},[5274],[75,63597,142],{"className":63598},[5279],[75,63600],{"className":63601,"style":5275},[5274],[75,63603,63605,63608,63744,63747,63776,63779,63896,63899],{"className":63604},[5249],[75,63606],{"className":63607,"style":56136},[5253],[75,63609,63611,63614,63741],{"className":63610},[5258],[75,63612],{"className":63613},[5263,5380],[75,63615,63617],{"className":63616},[5201],[75,63618,63620,63733],{"className":63619},[5303,5304],[75,63621,63623,63730],{"className":63622},[5308],[75,63624,63626,63685,63693],{"className":63625,"style":56155},[5312],[75,63627,63628,63631],{"style":33981},[75,63629],{"className":63630,"style":5400},[5320],[75,63632,63634],{"className":63633},[5258],[75,63635,63637,63640],{"className":63636},[5258],[75,63638,12635],{"className":63639},[5258],[75,63641,63643],{"className":63642},[5299],[75,63644,63646,63677],{"className":63645},[5303,5304],[75,63647,63649,63674],{"className":63648},[5308],[75,63650,63652,63663],{"className":63651,"style":56182},[5312],[75,63653,63654,63657],{"style":56185},[75,63655],{"className":63656,"style":5321},[5320],[75,63658,63660],{"className":63659},[5325,5326,5327,5328],[75,63661,32181],{"className":63662,"style":35138},[5258,5259,5328],[75,63664,63665,63668],{"style":56197},[75,63666],{"className":63667,"style":5321},[5320],[75,63669,63671],{"className":63670},[5325,5326,5327,5328],[75,63672,270],{"className":63673},[5258,5328],[75,63675,5336],{"className":63676},[5335],[75,63678,63680],{"className":63679},[5308],[75,63681,63683],{"className":63682,"style":56216},[5312],[75,63684],{},[75,63686,63687,63690],{"style":5447},[75,63688],{"className":63689,"style":5400},[5320],[75,63691],{"className":63692,"style":5455},[5454],[75,63694,63695,63698],{"style":34001},[75,63696],{"className":63697,"style":5400},[5320],[75,63699,63701],{"className":63700},[5258],[75,63702,63704,63707],{"className":63703},[5258],[75,63705,12635],{"className":63706},[5258],[75,63708,63710],{"className":63709},[5299],[75,63711,63713],{"className":63712},[5303],[75,63714,63716],{"className":63715},[5308],[75,63717,63719],{"className":63718,"style":32216},[5312],[75,63720,63721,63724],{"style":7596},[75,63722],{"className":63723,"style":5321},[5320],[75,63725,63727],{"className":63726},[5325,5326,5327,5328],[75,63728,270],{"className":63729},[5258,5328],[75,63731,5336],{"className":63732},[5335],[75,63734,63736],{"className":63735},[5308],[75,63737,63739],{"className":63738,"style":56273},[5312],[75,63740],{},[75,63742],{"className":63743},[5270,5380],[75,63745],{"className":63746,"style":5349},[5274],[75,63748,63750,63753],{"className":63749},[5353],[75,63751,1179],{"className":63752},[5353],[75,63754,63756],{"className":63755},[5299],[75,63757,63759],{"className":63758},[5303],[75,63760,63762],{"className":63761},[5308],[75,63763,63765],{"className":63764,"style":56300},[5312],[75,63766,63767,63770],{"style":56303},[75,63768],{"className":63769,"style":5321},[5320],[75,63771,63773],{"className":63772},[5325,5326,5327,5328],[75,63774,270],{"className":63775},[5258,5328],[75,63777],{"className":63778,"style":5349},[5274],[75,63780,63782,63788,63890],{"className":63781},[5360],[75,63783,63785],{"className":63784,"style":5365},[5263,5364],[75,63786,197],{"className":63787},[5369,5327],[75,63789,63791,63794,63887],{"className":63790},[5258],[75,63792],{"className":63793},[5263,5380],[75,63795,63797],{"className":63796},[5201],[75,63798,63800,63879],{"className":63799},[5303,5304],[75,63801,63803,63876],{"className":63802},[5308],[75,63804,63806,63817,63825],{"className":63805,"style":56343},[5312],[75,63807,63808,63811],{"style":33981},[75,63809],{"className":63810,"style":5400},[5320],[75,63812,63814],{"className":63813},[5258],[75,63815,270],{"className":63816},[5258],[75,63818,63819,63822],{"style":5447},[75,63820],{"className":63821,"style":5400},[5320],[75,63823],{"className":63824,"style":5455},[5454],[75,63826,63827,63830],{"style":34001},[75,63828],{"className":63829,"style":5400},[5320],[75,63831,63833,63873],{"className":63832},[5258],[75,63834,63836,63839],{"className":63835},[5258],[75,63837,12635],{"className":63838},[5258],[75,63840,63842],{"className":63841},[5299],[75,63843,63845,63865],{"className":63844},[5303,5304],[75,63846,63848,63862],{"className":63847},[5308],[75,63849,63851],{"className":63850,"style":5942},[5312],[75,63852,63853,63856],{"style":5316},[75,63854],{"className":63855,"style":5321},[5320],[75,63857,63859],{"className":63858},[5325,5326,5327,5328],[75,63860,32181],{"className":63861,"style":35138},[5258,5259,5328],[75,63863,5336],{"className":63864},[5335],[75,63866,63868],{"className":63867},[5308],[75,63869,63871],{"className":63870,"style":5343},[5312],[75,63872],{},[75,63874,51024],{"className":63875,"style":51096},[5258,5259],[75,63877,5336],{"className":63878},[5335],[75,63880,63882],{"className":63881},[5308],[75,63883,63885],{"className":63884,"style":34067},[5312],[75,63886],{},[75,63888],{"className":63889},[5270,5380],[75,63891,63893],{"className":63892,"style":5365},[5270,5364],[75,63894,5175],{"className":63895},[5369,5327],[75,63897],{"className":63898,"style":5349},[5274],[75,63900,6543],{"className":63901},[5258],[12,63903,63904,63905,64020],{},"The routine optionally applies the damping envelope ",[75,63906,63908,63942],{"className":63907},[5148],[75,63909,63911],{"className":63910},[5152],[5154,63912,63913],{"xmlns":5156},[5158,63914,63915,63939],{},[5161,63916,63917,63919,63921,63923,63925,63927,63929,63937],{},[5164,63918,243],{},[5168,63920,5191],{},[5168,63922,197],{"stretchy":5170},[5168,63924,5199],{},[5164,63926,51024],{},[5164,63928,2810],{"mathvariant":6582},[39536,63930,63931,63933,63935],{},[5164,63932,9676],{},[5184,63934,270],{},[5168,63936,52551],{},[5168,63938,5175],{"stretchy":5170},[5238,63940,63941],{"encoding":5240},"\\exp(-\\tau\u002FT_2^\\ast)",[75,63943,63945],{"className":63944,"ariaHidden":5196},[5245],[75,63946,63948,63951,63954,63957,63960,63963,63966,64017],{"className":63947},[5249],[75,63949],{"className":63950,"style":5254},[5253],[75,63952,243],{"className":63953},[5353],[75,63955,197],{"className":63956},[5263],[75,63958,5199],{"className":63959},[5258],[75,63961,51024],{"className":63962,"style":51096},[5258,5259],[75,63964,2810],{"className":63965},[5258],[75,63967,63969,63972],{"className":63968},[5258],[75,63970,9676],{"className":63971,"style":9817},[5258,5259],[75,63973,63975],{"className":63974},[5299],[75,63976,63978,64009],{"className":63977},[5303,5304],[75,63979,63981,64006],{"className":63980},[5308],[75,63982,63984,63995],{"className":63983,"style":52583},[5312],[75,63985,63986,63989],{"style":52586},[75,63987],{"className":63988,"style":5321},[5320],[75,63990,63992],{"className":63991},[5325,5326,5327,5328],[75,63993,270],{"className":63994},[5258,5328],[75,63996,63997,64000],{"style":7596},[75,63998],{"className":63999,"style":5321},[5320],[75,64001,64003],{"className":64002},[5325,5326,5327,5328],[75,64004,52551],{"className":64005},[5477,5328],[75,64007,5336],{"className":64008},[5335],[75,64010,64012],{"className":64011},[5308],[75,64013,64015],{"className":64014,"style":52616},[5312],[75,64016],{},[75,64018,5175],{"className":64019},[5270]," and clips the result to the physical interval",[75,64022,64024],{"className":64023},[31410],[75,64025,64027,64054],{"className":64026},[5148],[75,64028,64030],{"className":64029},[5152],[5154,64031,64032],{"xmlns":5156,"display":31419},[5158,64033,64034,64051],{},[5161,64035,64036,64038,64041,64047,64049],{},[5184,64037,5186],{},[5168,64039,64040],{},"≤",[5179,64042,64043,64045],{},[5164,64044,51013],{},[5164,64046,7486],{},[5168,64048,64040],{},[5184,64050,53656],{},[5238,64052,64053],{"encoding":5240},"0\\le P_e\\le 1.",[75,64055,64057,64076,64131],{"className":64056,"ariaHidden":5196},[5245],[75,64058,64060,64064,64067,64070,64073],{"className":64059},[5249],[75,64061],{"className":64062,"style":64063},[5253],"height:0.7804em;vertical-align:-0.136em;",[75,64065,5186],{"className":64066},[5258],[75,64068],{"className":64069,"style":5275},[5274],[75,64071,64040],{"className":64072},[5279],[75,64074],{"className":64075,"style":5275},[5274],[75,64077,64079,64082,64122,64125,64128],{"className":64078},[5249],[75,64080],{"className":64081,"style":12572},[5253],[75,64083,64085,64088],{"className":64084},[5258],[75,64086,51013],{"className":64087,"style":9817},[5258,5259],[75,64089,64091],{"className":64090},[5299],[75,64092,64094,64114],{"className":64093},[5303,5304],[75,64095,64097,64111],{"className":64096},[5308],[75,64098,64100],{"className":64099,"style":6264},[5312],[75,64101,64102,64105],{"style":33755},[75,64103],{"className":64104,"style":5321},[5320],[75,64106,64108],{"className":64107},[5325,5326,5327,5328],[75,64109,7486],{"className":64110},[5258,5259,5328],[75,64112,5336],{"className":64113},[5335],[75,64115,64117],{"className":64116},[5308],[75,64118,64120],{"className":64119,"style":5343},[5312],[75,64121],{},[75,64123],{"className":64124,"style":5275},[5274],[75,64126,64040],{"className":64127},[5279],[75,64129],{"className":64130,"style":5275},[5274],[75,64132,64134,64137],{"className":64133},[5249],[75,64135],{"className":64136,"style":32836},[5253],[75,64138,53656],{"className":64139},[5258],[12,64141,64142],{},"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.",[12425,64144,64146],{"id":64145},"graphical-interpretation","Graphical Interpretation",[12,64148,5792,64149,64247],{},[75,64150,64152,64180],{"className":64151},[5148],[75,64153,64155],{"className":64154},[5152],[5154,64156,64157],{"xmlns":5156},[5158,64158,64159,64177],{},[5161,64160,64161,64167,64169,64171,64173,64175],{},[5179,64162,64163,64165],{},[5164,64164,51013],{},[5164,64166,7486],{},[5168,64168,197],{"stretchy":5170},[5164,64170,5894],{},[5168,64172,7469],{"separator":5196},[5164,64174,51024],{},[5168,64176,5175],{"stretchy":5170},[5238,64178,64179],{"encoding":5240},"P_e(f,\\tau)",[75,64181,64183],{"className":64182,"ariaHidden":5196},[5245],[75,64184,64186,64189,64229,64232,64235,64238,64241,64244],{"className":64185},[5249],[75,64187],{"className":64188,"style":5254},[5253],[75,64190,64192,64195],{"className":64191},[5258],[75,64193,51013],{"className":64194,"style":9817},[5258,5259],[75,64196,64198],{"className":64197},[5299],[75,64199,64201,64221],{"className":64200},[5303,5304],[75,64202,64204,64218],{"className":64203},[5308],[75,64205,64207],{"className":64206,"style":6264},[5312],[75,64208,64209,64212],{"style":33755},[75,64210],{"className":64211,"style":5321},[5320],[75,64213,64215],{"className":64214},[5325,5326,5327,5328],[75,64216,7486],{"className":64217},[5258,5259,5328],[75,64219,5336],{"className":64220},[5335],[75,64222,64224],{"className":64223},[5308],[75,64225,64227],{"className":64226,"style":5343},[5312],[75,64228],{},[75,64230,197],{"className":64231},[5263],[75,64233,5894],{"className":64234,"style":6251},[5258,5259],[75,64236,7469],{"className":64237},[7544],[75,64239],{"className":64240,"style":5349},[5274],[75,64242,51024],{"className":64243,"style":51096},[5258,5259],[75,64245,5175],{"className":64246},[5270]," 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.",[12,64249,64250],{},"The resonant interaction region appears near",[75,64252,64254],{"className":64253},[31410],[75,64255,64257,64280],{"className":64256},[5148],[75,64258,64260],{"className":64259},[5152],[5154,64261,64262],{"xmlns":5156,"display":31419},[5158,64263,64264,64278],{},[5161,64265,64266,64268,64270,64276],{},[5164,64267,5894],{},[5168,64269,142],{},[5179,64271,64272,64274],{},[5164,64273,5894],{},[5184,64275,5186],{},[5164,64277,6543],{"mathvariant":6582},[5238,64279,53353],{"encoding":5240},[75,64281,64283,64301],{"className":64282,"ariaHidden":5196},[5245],[75,64284,64286,64289,64292,64295,64298],{"className":64285},[5249],[75,64287],{"className":64288,"style":51131},[5253],[75,64290,5894],{"className":64291,"style":6251},[5258,5259],[75,64293],{"className":64294,"style":5275},[5274],[75,64296,142],{"className":64297},[5279],[75,64299],{"className":64300,"style":5275},[5274],[75,64302,64304,64307,64347],{"className":64303},[5249],[75,64305],{"className":64306,"style":51131},[5253],[75,64308,64310,64313],{"className":64309},[5258],[75,64311,5894],{"className":64312,"style":6251},[5258,5259],[75,64314,64316],{"className":64315},[5299],[75,64317,64319,64339],{"className":64318},[5303,5304],[75,64320,64322,64336],{"className":64321},[5308],[75,64323,64325],{"className":64324,"style":5313},[5312],[75,64326,64327,64330],{"style":6267},[75,64328],{"className":64329,"style":5321},[5320],[75,64331,64333],{"className":64332},[5325,5326,5327,5328],[75,64334,5186],{"className":64335},[5258,5328],[75,64337,5336],{"className":64338},[5335],[75,64340,64342],{"className":64341},[5308],[75,64343,64345],{"className":64344,"style":5343},[5312],[75,64346],{},[75,64348,6543],{"className":64349},[5258],[12,64351,64352,64353,64432],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[75,64354,64356,64390],{"className":64355},[5148],[75,64357,64359],{"className":64358},[5152],[5154,64360,64361],{"xmlns":5156},[5158,64362,64363,64387],{},[5161,64364,64365,64367,64369,64371,64373,64375,64377,64379],{},[5164,64366,12635],{"mathvariant":6582},[5164,64368,2810],{"mathvariant":6582},[5184,64370,270],{},[5164,64372,5889],{},[5168,64374,142],{},[5184,64376,752],{},[32328,64378,32330],{},[5161,64380,64381,64383,64385],{},[5164,64382,57719],{"mathvariant":6582},[5164,64384,9119],{"mathvariant":6582},[5164,64386,31538],{"mathvariant":6582},[5238,64388,64389],{"encoding":5240},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[75,64391,64393,64414],{"className":64392,"ariaHidden":5196},[5245],[75,64394,64396,64399,64402,64405,64408,64411],{"className":64395},[5249],[75,64397],{"className":64398,"style":5254},[5253],[75,64400,52033],{"className":64401},[5258],[75,64403,5889],{"className":64404,"style":5416},[5258,5259],[75,64406],{"className":64407,"style":5275},[5274],[75,64409,142],{"className":64410},[5279],[75,64412],{"className":64413,"style":5275},[5274],[75,64415,64417,64420,64423,64426],{"className":64416},[5249],[75,64418],{"className":64419,"style":12648},[5253],[75,64421,752],{"className":64422},[5258],[75,64424,32330],{"className":64425},[5274,53204],[75,64427,64429],{"className":64428},[5258],[75,64430,15986],{"className":64431},[5258,32585],", the ideal resonant maxima appear near",[75,64434,64436],{"className":64435},[31410],[75,64437,64439,64512],{"className":64438},[5148],[75,64440,64442],{"className":64441},[5152],[5154,64443,64444],{"xmlns":5156,"display":31419},[5158,64445,64446,64509],{},[5161,64447,64448,64450,64452,64454,64456,64462,64464,64466,64469,64471,64477,64479,64481,64484,64486,64492,64494,64496,64499,64501,64507],{},[5164,64449,51024],{},[5168,64451,142],{},[5184,64453,57891],{},[32328,64455,32330],{},[5161,64457,64458,64460],{},[5164,64459,33004],{"mathvariant":6582},[5164,64461,57900],{"mathvariant":6582},[5168,64463,7469],{"separator":5196},[32328,64465,32330],{},[5184,64467,64468],{},"75",[32328,64470,32330],{},[5161,64472,64473,64475],{},[5164,64474,33004],{"mathvariant":6582},[5164,64476,57900],{"mathvariant":6582},[5168,64478,7469],{"separator":5196},[32328,64480,32330],{},[5184,64482,64483],{},"125",[32328,64485,32330],{},[5161,64487,64488,64490],{},[5164,64489,33004],{"mathvariant":6582},[5164,64491,57900],{"mathvariant":6582},[5168,64493,7469],{"separator":5196},[32328,64495,32330],{},[5184,64497,64498],{},"175",[32328,64500,32330],{},[5161,64502,64503,64505],{},[5164,64504,33004],{"mathvariant":6582},[5164,64506,57900],{"mathvariant":6582},[5168,64508,7469],{"separator":5196},[5238,64510,64511],{"encoding":5240},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[75,64513,64515,64533],{"className":64514,"ariaHidden":5196},[5245],[75,64516,64518,64521,64524,64527,64530],{"className":64517},[5249],[75,64519],{"className":64520,"style":35333},[5253],[75,64522,51024],{"className":64523,"style":51096},[5258,5259],[75,64525],{"className":64526,"style":5275},[5274],[75,64528,142],{"className":64529},[5279],[75,64531],{"className":64532,"style":5275},[5274],[75,64534,64536,64540,64543,64546,64552,64555,64558,64561,64564,64567,64573,64576,64579,64582,64585,64588,64594,64597,64600,64603,64606,64609,64615],{"className":64535},[5249],[75,64537],{"className":64538,"style":64539},[5253],"height:0.8389em;vertical-align:-0.1944em;",[75,64541,57891],{"className":64542},[5258],[75,64544,32330],{"className":64545},[5274,53204],[75,64547,64549],{"className":64548},[5258],[75,64550,57982],{"className":64551},[5258,32585],[75,64553,7469],{"className":64554},[7544],[75,64556,32330],{"className":64557},[5274],[75,64559],{"className":64560,"style":5349},[5274],[75,64562,64468],{"className":64563},[5258],[75,64565,32330],{"className":64566},[5274,53204],[75,64568,64570],{"className":64569},[5258],[75,64571,57982],{"className":64572},[5258,32585],[75,64574,7469],{"className":64575},[7544],[75,64577,32330],{"className":64578},[5274],[75,64580],{"className":64581,"style":5349},[5274],[75,64583,64483],{"className":64584},[5258],[75,64586,32330],{"className":64587},[5274,53204],[75,64589,64591],{"className":64590},[5258],[75,64592,57982],{"className":64593},[5258,32585],[75,64595,7469],{"className":64596},[7544],[75,64598,32330],{"className":64599},[5274],[75,64601],{"className":64602,"style":5349},[5274],[75,64604,64498],{"className":64605},[5258],[75,64607,32330],{"className":64608},[5274,53204],[75,64610,64612],{"className":64611},[5258],[75,64613,57982],{"className":64614},[5258,32585],[75,64616,7469],{"className":64617},[7544],[12,64619,64620],{},"before the optional phenomenological damping factor is applied.",[12,64622,64623,64624,64693,64694,64820],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[75,64625,64627,64644],{"className":64626},[5148],[75,64628,64630],{"className":64629},[5152],[5154,64631,64632],{"xmlns":5156},[5158,64633,64634,64642],{},[5161,64635,64636],{},[5179,64637,64638,64640],{},[5164,64639,12635],{"mathvariant":6582},[5164,64641,32181],{},[5238,64643,52214],{"encoding":5240},[75,64645,64647],{"className":64646,"ariaHidden":5196},[5245],[75,64648,64650,64653],{"className":64649},[5249],[75,64651],{"className":64652,"style":12572},[5253],[75,64654,64656,64659],{"className":64655},[5258],[75,64657,12635],{"className":64658},[5258],[75,64660,64662],{"className":64661},[5299],[75,64663,64665,64685],{"className":64664},[5303,5304],[75,64666,64668,64682],{"className":64667},[5308],[75,64669,64671],{"className":64670,"style":5942},[5312],[75,64672,64673,64676],{"style":5316},[75,64674],{"className":64675,"style":5321},[5320],[75,64677,64679],{"className":64678},[5325,5326,5327,5328],[75,64680,32181],{"className":64681,"style":35138},[5258,5259,5328],[75,64683,5336],{"className":64684},[5335],[75,64686,64688],{"className":64687},[5308],[75,64689,64691],{"className":64690,"style":5343},[5312],[75,64692],{},", which increases with detuning, and from the amplitude coefficient ",[75,64695,64697,64725],{"className":64696},[5148],[75,64698,64700],{"className":64699},[5152],[5154,64701,64702],{"xmlns":5156},[5158,64703,64704,64722],{},[5161,64705,64706,64712,64714],{},[5218,64707,64708,64710],{},[5164,64709,12635],{"mathvariant":6582},[5184,64711,270],{},[5164,64713,2810],{"mathvariant":6582},[39536,64715,64716,64718,64720],{},[5164,64717,12635],{"mathvariant":6582},[5164,64719,32181],{},[5184,64721,270],{},[5238,64723,64724],{"encoding":5240},"\\Omega^2\u002F\\Omega_R^2",[75,64726,64728],{"className":64727,"ariaHidden":5196},[5245],[75,64729,64731,64735,64764,64767],{"className":64730},[5249],[75,64732],{"className":64733,"style":64734},[5253],"height:1.0894em;vertical-align:-0.2753em;",[75,64736,64738,64741],{"className":64737},[5258],[75,64739,12635],{"className":64740},[5258],[75,64742,64744],{"className":64743},[5299],[75,64745,64747],{"className":64746},[5303],[75,64748,64750],{"className":64749},[5308],[75,64751,64753],{"className":64752,"style":32216},[5312],[75,64754,64755,64758],{"style":7596},[75,64756],{"className":64757,"style":5321},[5320],[75,64759,64761],{"className":64760},[5325,5326,5327,5328],[75,64762,270],{"className":64763},[5258,5328],[75,64765,2810],{"className":64766},[5258],[75,64768,64770,64773],{"className":64769},[5258],[75,64771,12635],{"className":64772},[5258],[75,64774,64776],{"className":64775},[5299],[75,64777,64779,64811],{"className":64778},[5303,5304],[75,64780,64782,64808],{"className":64781},[5308],[75,64783,64785,64797],{"className":64784,"style":32216},[5312],[75,64786,64788,64791],{"style":64787},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[75,64789],{"className":64790,"style":5321},[5320],[75,64792,64794],{"className":64793},[5325,5326,5327,5328],[75,64795,32181],{"className":64796,"style":35138},[5258,5259,5328],[75,64798,64799,64802],{"style":7596},[75,64800],{"className":64801,"style":5321},[5320],[75,64803,64805],{"className":64804},[5325,5326,5327,5328],[75,64806,270],{"className":64807},[5258,5328],[75,64809,5336],{"className":64810},[5335],[75,64812,64814],{"className":64813},[5308],[75,64815,64818],{"className":64816,"style":64817},[5312],"height:0.2753em;",[75,64819],{},", which decreases with detuning.",[12,64822,64823],{},"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.",[12,64825,64826,64827,64924],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[75,64828,64830,64857],{"className":64829},[5148],[75,64831,64833],{"className":64832},[5152],[5154,64834,64835],{"xmlns":5156},[5158,64836,64837,64855],{},[5161,64838,64839,64845,64847,64849,64851,64853],{},[5179,64840,64841,64843],{},[5164,64842,51013],{},[5164,64844,7486],{},[5168,64846,197],{"stretchy":5170},[5164,64848,51024],{},[5168,64850,7469],{"separator":5196},[5164,64852,5894],{},[5168,64854,5175],{"stretchy":5170},[5238,64856,62037],{"encoding":5240},[75,64858,64860],{"className":64859,"ariaHidden":5196},[5245],[75,64861,64863,64866,64906,64909,64912,64915,64918,64921],{"className":64862},[5249],[75,64864],{"className":64865,"style":5254},[5253],[75,64867,64869,64872],{"className":64868},[5258],[75,64870,51013],{"className":64871,"style":9817},[5258,5259],[75,64873,64875],{"className":64874},[5299],[75,64876,64878,64898],{"className":64877},[5303,5304],[75,64879,64881,64895],{"className":64880},[5308],[75,64882,64884],{"className":64883,"style":6264},[5312],[75,64885,64886,64889],{"style":33755},[75,64887],{"className":64888,"style":5321},[5320],[75,64890,64892],{"className":64891},[5325,5326,5327,5328],[75,64893,7486],{"className":64894},[5258,5259,5328],[75,64896,5336],{"className":64897},[5335],[75,64899,64901],{"className":64900},[5308],[75,64902,64904],{"className":64903,"style":5343},[5312],[75,64905],{},[75,64907,197],{"className":64908},[5263],[75,64910,51024],{"className":64911,"style":51096},[5258,5259],[75,64913,7469],{"className":64914},[7544],[75,64916],{"className":64917,"style":5349},[5274],[75,64919,5894],{"className":64920,"style":6251},[5258,5259],[75,64922,5175],{"className":64923},[5270],". An overlaid theoretical curve follows",[75,64926,64928],{"className":64927},[31410],[75,64929,64931,65003],{"className":64930},[5148],[75,64932,64934],{"className":64933},[5152],[5154,64935,64936],{"xmlns":5156,"display":31419},[5158,64937,64938,65000],{},[5161,64939,64940,64946,64948,64950,64952,64954,64998],{},[5179,64941,64942,64944],{},[5164,64943,52287],{},[5164,64945,32181],{},[5168,64947,197],{"stretchy":5170},[5164,64949,5894],{},[5168,64951,5175],{"stretchy":5170},[5168,64953,142],{},[33250,64955,64956],{},[5161,64957,64958,64978,64980,64982,64984,64986,64992],{},[5218,64959,64960,64976],{},[5161,64961,64962,64964,64974],{},[5168,64963,197],{"fence":5196},[5201,64965,64966,64968],{},[5164,64967,12635],{"mathvariant":6582},[5161,64969,64970,64972],{},[5184,64971,270],{},[5164,64973,5889],{},[5168,64975,5175],{"fence":5196},[5184,64977,270],{},[5168,64979,3251],{},[5168,64981,197],{"stretchy":5170},[5164,64983,5894],{},[5168,64985,5199],{},[5179,64987,64988,64990],{},[5164,64989,5894],{},[5184,64991,5186],{},[5218,64993,64994,64996],{},[5168,64995,5175],{"stretchy":5170},[5184,64997,270],{},[5164,64999,6543],{"mathvariant":6582},[5238,65001,65002],{"encoding":5240},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[75,65004,65006,65070],{"className":65005,"ariaHidden":5196},[5245],[75,65007,65009,65012,65052,65055,65058,65061,65064,65067],{"className":65008},[5249],[75,65010],{"className":65011,"style":5254},[5253],[75,65013,65015,65018],{"className":65014},[5258],[75,65016,52287],{"className":65017,"style":52308},[5258,5259],[75,65019,65021],{"className":65020},[5299],[75,65022,65024,65044],{"className":65023},[5303,5304],[75,65025,65027,65041],{"className":65026},[5308],[75,65028,65030],{"className":65029,"style":5942},[5312],[75,65031,65032,65035],{"style":52323},[75,65033],{"className":65034,"style":5321},[5320],[75,65036,65038],{"className":65037},[5325,5326,5327,5328],[75,65039,32181],{"className":65040,"style":35138},[5258,5259,5328],[75,65042,5336],{"className":65043},[5335],[75,65045,65047],{"className":65046},[5308],[75,65048,65050],{"className":65049,"style":5343},[5312],[75,65051],{},[75,65053,197],{"className":65054},[5263],[75,65056,5894],{"className":65057,"style":6251},[5258,5259],[75,65059,5175],{"className":65060},[5270],[75,65062],{"className":65063,"style":5275},[5274],[75,65065,142],{"className":65066},[5279],[75,65068],{"className":65069,"style":5275},[5274],[75,65071,65073,65076,65319],{"className":65072},[5249],[75,65074],{"className":65075,"style":59330},[5253],[75,65077,65079],{"className":65078},[5258,14668],[75,65080,65082,65311],{"className":65081},[5303,5304],[75,65083,65085,65308],{"className":65084},[5308],[75,65086,65088,65296],{"className":65087,"style":59343},[5312],[75,65089,65091,65094],{"className":65090,"style":59347},[33434],[75,65092],{"className":65093,"style":59351},[5320],[75,65095,65097,65203,65206,65209,65212,65215,65218,65221,65224,65227,65267],{"className":65096,"style":56575},[5258],[75,65098,65100,65180],{"className":65099},[5360],[75,65101,65103,65109,65174],{"className":65102},[5360],[75,65104,65106],{"className":65105,"style":5365},[5263,5364],[75,65107,197],{"className":65108},[5369,5327],[75,65110,65112,65115,65171],{"className":65111},[5258],[75,65113],{"className":65114},[5263,5380],[75,65116,65118],{"className":65117},[5201],[75,65119,65121,65163],{"className":65120},[5303,5304],[75,65122,65124,65160],{"className":65123},[5308],[75,65125,65127,65141,65149],{"className":65126,"style":56343},[5312],[75,65128,65129,65132],{"style":33981},[75,65130],{"className":65131,"style":5400},[5320],[75,65133,65135,65138],{"className":65134},[5258],[75,65136,270],{"className":65137},[5258],[75,65139,5889],{"className":65140,"style":5416},[5258,5259],[75,65142,65143,65146],{"style":5447},[75,65144],{"className":65145,"style":5400},[5320],[75,65147],{"className":65148,"style":5455},[5454],[75,65150,65151,65154],{"style":34001},[75,65152],{"className":65153,"style":5400},[5320],[75,65155,65157],{"className":65156},[5258],[75,65158,12635],{"className":65159},[5258],[75,65161,5336],{"className":65162},[5335],[75,65164,65166],{"className":65165},[5308],[75,65167,65169],{"className":65168,"style":34067},[5312],[75,65170],{},[75,65172],{"className":65173},[5270,5380],[75,65175,65177],{"className":65176,"style":5365},[5270,5364],[75,65178,5175],{"className":65179},[5369,5327],[75,65181,65183],{"className":65182},[5299],[75,65184,65186],{"className":65185},[5303],[75,65187,65189],{"className":65188},[5308],[75,65190,65192],{"className":65191,"style":59450},[5312],[75,65193,65194,65197],{"style":59453},[75,65195],{"className":65196,"style":5321},[5320],[75,65198,65200],{"className":65199},[5325,5326,5327,5328],[75,65201,270],{"className":65202},[5258,5328],[75,65204],{"className":65205,"style":6294},[5274],[75,65207,3251],{"className":65208},[5477],[75,65210],{"className":65211,"style":6294},[5274],[75,65213,197],{"className":65214},[5263],[75,65216,5894],{"className":65217,"style":6251},[5258,5259],[75,65219],{"className":65220,"style":6294},[5274],[75,65222,5199],{"className":65223},[5477],[75,65225],{"className":65226,"style":6294},[5274],[75,65228,65230,65233],{"className":65229},[5258],[75,65231,5894],{"className":65232,"style":6251},[5258,5259],[75,65234,65236],{"className":65235},[5299],[75,65237,65239,65259],{"className":65238},[5303,5304],[75,65240,65242,65256],{"className":65241},[5308],[75,65243,65245],{"className":65244,"style":5313},[5312],[75,65246,65247,65250],{"style":6267},[75,65248],{"className":65249,"style":5321},[5320],[75,65251,65253],{"className":65252},[5325,5326,5327,5328],[75,65254,5186],{"className":65255},[5258,5328],[75,65257,5336],{"className":65258},[5335],[75,65260,65262],{"className":65261},[5308],[75,65263,65265],{"className":65264,"style":5343},[5312],[75,65266],{},[75,65268,65270,65273],{"className":65269},[5270],[75,65271,5175],{"className":65272},[5270],[75,65274,65276],{"className":65275},[5299],[75,65277,65279],{"className":65278},[5303],[75,65280,65282],{"className":65281},[5308],[75,65283,65285],{"className":65284,"style":56594},[5312],[75,65286,65287,65290],{"style":56597},[75,65288],{"className":65289,"style":5321},[5320],[75,65291,65293],{"className":65292},[5325,5326,5327,5328],[75,65294,270],{"className":65295},[5258,5328],[75,65297,65298,65301],{"style":59558},[75,65299],{"className":65300,"style":59351},[5320],[75,65302,65304],{"className":65303,"style":59565},[33455],[33458,65305,65306],{"xmlns":33460,"width":33461,"height":59568,"viewBox":59569,"preserveAspectRatio":33464},[33466,65307],{"d":59572},[75,65309,5336],{"className":65310},[5335],[75,65312,65314],{"className":65313},[5308],[75,65315,65317],{"className":65316,"style":59582},[5312],[75,65318],{},[75,65320,6543],{"className":65321},[5258],[12425,65323,65325],{"id":65324},"primary-modeling-assumptions","Primary Modeling Assumptions",[12,65327,65328],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[7629,65330,65331,65341],{},[7632,65332,65333],{},[7635,65334,65335,65338],{},[7638,65336,65337],{},"Assumption",[7638,65339,65340],{},"Implication",[7645,65342,65343,65427,65435,65442,65450,65487,65577,65734,65782,65790,65798,65806],{},[7635,65344,65345,65348],{},[7650,65346,65347],{},"Two-level truncation",[7650,65349,65350,65351,32154,65389,6543],{},"The simulated model includes only ",[75,65352,65354,65371],{"className":65353},[5148],[75,65355,65357],{"className":65356},[5152],[5154,65358,65359],{"xmlns":5156},[5158,65360,65361,65369],{},[5161,65362,65363,65365,65367],{},[5168,65364,7452],{"stretchy":5170},[5164,65366,51207],{},[5168,65368,7457],{"stretchy":5170},[5238,65370,51212],{"encoding":5240},[75,65372,65374],{"className":65373,"ariaHidden":5196},[5245],[75,65375,65377,65380,65383,65386],{"className":65376},[5249],[75,65378],{"className":65379,"style":5254},[5253],[75,65381,7452],{"className":65382},[5263],[75,65384,51207],{"className":65385,"style":5416},[5258,5259],[75,65387,7457],{"className":65388},[5270],[75,65390,65392,65409],{"className":65391},[5148],[75,65393,65395],{"className":65394},[5152],[5154,65396,65397],{"xmlns":5156},[5158,65398,65399,65407],{},[5161,65400,65401,65403,65405],{},[5168,65402,7452],{"stretchy":5170},[5164,65404,7486],{},[5168,65406,7457],{"stretchy":5170},[5238,65408,51258],{"encoding":5240},[75,65410,65412],{"className":65411,"ariaHidden":5196},[5245],[75,65413,65415,65418,65421,65424],{"className":65414},[5249],[75,65416],{"className":65417,"style":5254},[5253],[75,65419,7452],{"className":65420},[5263],[75,65422,7486],{"className":65423},[5258,5259],[75,65425,7457],{"className":65426},[5270],[7635,65428,65429,65432],{},[7650,65430,65431],{},"Classical drive",[7650,65433,65434],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[7635,65436,65437,65439],{},[7650,65438,52629],{},[7650,65440,65441],{},"Fast counter-rotating terms are omitted.",[7635,65443,65444,65447],{},[7650,65445,65446],{},"Square pulse",[7650,65448,65449],{},"The drive is assumed to turn on and off instantaneously.",[7635,65451,65452,65455],{},[7650,65453,65454],{},"Uniform drive amplitude",[7650,65456,65457,65458,65486],{},"The parameter ",[75,65459,65461,65474],{"className":65460},[5148],[75,65462,65464],{"className":65463},[5152],[5154,65465,65466],{"xmlns":5156},[5158,65467,65468,65472],{},[5161,65469,65470],{},[5164,65471,12635],{"mathvariant":6582},[5238,65473,12638],{"encoding":5240},[75,65475,65477],{"className":65476,"ariaHidden":5196},[5245],[75,65478,65480,65483],{"className":65479},[5249],[75,65481],{"className":65482,"style":12648},[5253],[75,65484,12635],{"className":65485},[5258]," remains constant across the frequency sweep.",[7635,65488,65489,65492],{},[7650,65490,65491],{},"Optional damping envelope",[7650,65493,65457,65494,65576],{},[75,65495,65497,65516],{"className":65496},[5148],[75,65498,65500],{"className":65499},[5152],[5154,65501,65502],{"xmlns":5156},[5158,65503,65504,65514],{},[5161,65505,65506],{},[39536,65507,65508,65510,65512],{},[5164,65509,9676],{},[5184,65511,270],{},[5168,65513,52551],{},[5238,65515,52554],{"encoding":5240},[75,65517,65519],{"className":65518,"ariaHidden":5196},[5245],[75,65520,65522,65525],{"className":65521},[5249],[75,65523],{"className":65524,"style":52564},[5253],[75,65526,65528,65531],{"className":65527},[5258],[75,65529,9676],{"className":65530,"style":9817},[5258,5259],[75,65532,65534],{"className":65533},[5299],[75,65535,65537,65568],{"className":65536},[5303,5304],[75,65538,65540,65565],{"className":65539},[5308],[75,65541,65543,65554],{"className":65542,"style":52583},[5312],[75,65544,65545,65548],{"style":52586},[75,65546],{"className":65547,"style":5321},[5320],[75,65549,65551],{"className":65550},[5325,5326,5327,5328],[75,65552,270],{"className":65553},[5258,5328],[75,65555,65556,65559],{"style":7596},[75,65557],{"className":65558,"style":5321},[5320],[75,65560,65562],{"className":65561},[5325,5326,5327,5328],[75,65563,52551],{"className":65564},[5477,5328],[75,65566,5336],{"className":65567},[5335],[75,65569,65571],{"className":65570},[5308],[75,65572,65574],{"className":65573,"style":52616},[5312],[75,65575],{}," enters as a simple exponential contrast envelope.",[7635,65578,65579,65653],{},[7650,65580,65581,65582,65652],{},"No explicit ",[75,65583,65585,65603],{"className":65584},[5148],[75,65586,65588],{"className":65587},[5152],[5154,65589,65590],{"xmlns":5156},[5158,65591,65592,65600],{},[5161,65593,65594],{},[5179,65595,65596,65598],{},[5164,65597,9676],{},[5184,65599,7477],{},[5238,65601,65602],{"encoding":5240},"T_1",[75,65604,65606],{"className":65605,"ariaHidden":5196},[5245],[75,65607,65609,65612],{"className":65608},[5249],[75,65610],{"className":65611,"style":12572},[5253],[75,65613,65615,65618],{"className":65614},[5258],[75,65616,9676],{"className":65617,"style":9817},[5258,5259],[75,65619,65621],{"className":65620},[5299],[75,65622,65624,65644],{"className":65623},[5303,5304],[75,65625,65627,65641],{"className":65626},[5308],[75,65628,65630],{"className":65629,"style":5313},[5312],[75,65631,65632,65635],{"style":33755},[75,65633],{"className":65634,"style":5321},[5320],[75,65636,65638],{"className":65637},[5325,5326,5327,5328],[75,65639,7477],{"className":65640},[5258,5328],[75,65642,5336],{"className":65643},[5335],[75,65645,65647],{"className":65646},[5308],[75,65648,65650],{"className":65649,"style":5343},[5312],[75,65651],{}," relaxation",[7650,65654,65655,65656,65694,65695,65733],{},"Energy decay from ",[75,65657,65659,65676],{"className":65658},[5148],[75,65660,65662],{"className":65661},[5152],[5154,65663,65664],{"xmlns":5156},[5158,65665,65666,65674],{},[5161,65667,65668,65670,65672],{},[5168,65669,7452],{"stretchy":5170},[5164,65671,7486],{},[5168,65673,7457],{"stretchy":5170},[5238,65675,51258],{"encoding":5240},[75,65677,65679],{"className":65678,"ariaHidden":5196},[5245],[75,65680,65682,65685,65688,65691],{"className":65681},[5249],[75,65683],{"className":65684,"style":5254},[5253],[75,65686,7452],{"className":65687},[5263],[75,65689,7486],{"className":65690},[5258,5259],[75,65692,7457],{"className":65693},[5270]," to ",[75,65696,65698,65715],{"className":65697},[5148],[75,65699,65701],{"className":65700},[5152],[5154,65702,65703],{"xmlns":5156},[5158,65704,65705,65713],{},[5161,65706,65707,65709,65711],{},[5168,65708,7452],{"stretchy":5170},[5164,65710,51207],{},[5168,65712,7457],{"stretchy":5170},[5238,65714,51212],{"encoding":5240},[75,65716,65718],{"className":65717,"ariaHidden":5196},[5245],[75,65719,65721,65724,65727,65730],{"className":65720},[5249],[75,65722],{"className":65723,"style":5254},[5253],[75,65725,7452],{"className":65726},[5263],[75,65728,51207],{"className":65729,"style":5416},[5258,5259],[75,65731,7457],{"className":65732},[5270]," is omitted.",[7635,65735,65736,65739],{},[7650,65737,65738],{},"No higher-state leakage",[7650,65740,65741,65742,65781],{},"Higher levels, including a transmon ",[75,65743,65745,65763],{"className":65744},[5148],[75,65746,65748],{"className":65747},[5152],[5154,65749,65750],{"xmlns":5156},[5158,65751,65752,65760],{},[5161,65753,65754,65756,65758],{},[5168,65755,7452],{"stretchy":5170},[5164,65757,5894],{},[5168,65759,7457],{"stretchy":5170},[5238,65761,65762],{"encoding":5240},"\\lvert f\\rangle",[75,65764,65766],{"className":65765,"ariaHidden":5196},[5245],[75,65767,65769,65772,65775,65778],{"className":65768},[5249],[75,65770],{"className":65771,"style":5254},[5253],[75,65773,7452],{"className":65774},[5263],[75,65776,5894],{"className":65777,"style":6251},[5258,5259],[75,65779,7457],{"className":65780},[5270]," state, are excluded.",[7635,65783,65784,65787],{},[7650,65785,65786],{},"No AC Stark shift",[7650,65788,65789],{},"Drive-induced transition-frequency shifts are excluded.",[7635,65791,65792,65795],{},[7650,65793,65794],{},"No Bloch-Siegert shift",[7650,65796,65797],{},"Counter-rotating corrections are excluded.",[7635,65799,65800,65803],{},[7650,65801,65802],{},"No experimental readout model",[7650,65804,65805],{},"Assignment errors and finite signal-to-noise effects are omitted.",[7635,65807,65808,65811],{},[7650,65809,65810],{},"No pulse-envelope shaping",[7650,65812,65813],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[12,65815,65816],{},"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.",[12425,65818,65820],{"id":65819},"notebook-control-knobs","Notebook Control Knobs",[12,65822,65823],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[7629,65825,65826,65838],{},[7632,65827,65828],{},[7635,65829,65830,65833,65836],{},[7638,65831,65832],{},"Parameter",[7638,65834,65835],{"align":12437},"Default",[7638,65837,51184],{},[7645,65839,65840,65924,65977,65992,66007,66022,66036,66051,66065],{},[7635,65841,65842,65847,65851],{},[7650,65843,65844],{},[72,65845,65846],{},"F0_GHZ",[7650,65848,65849],{"align":12437},[72,65850,53117],{},[7650,65852,65853,65854,65923],{},"Qubit transition frequency ",[75,65855,65857,65874],{"className":65856},[5148],[75,65858,65860],{"className":65859},[5152],[5154,65861,65862],{"xmlns":5156},[5158,65863,65864,65872],{},[5161,65865,65866],{},[5179,65867,65868,65870],{},[5164,65869,5894],{},[5184,65871,5186],{},[5238,65873,51554],{"encoding":5240},[75,65875,65877],{"className":65876,"ariaHidden":5196},[5245],[75,65878,65880,65883],{"className":65879},[5249],[75,65881],{"className":65882,"style":51131},[5253],[75,65884,65886,65889],{"className":65885},[5258],[75,65887,5894],{"className":65888,"style":6251},[5258,5259],[75,65890,65892],{"className":65891},[5299],[75,65893,65895,65915],{"className":65894},[5303,5304],[75,65896,65898,65912],{"className":65897},[5308],[75,65899,65901],{"className":65900,"style":5313},[5312],[75,65902,65903,65906],{"style":6267},[75,65904],{"className":65905,"style":5321},[5320],[75,65907,65909],{"className":65908},[5325,5326,5327,5328],[75,65910,5186],{"className":65911},[5258,5328],[75,65913,5336],{"className":65914},[5335],[75,65916,65918],{"className":65917},[5308],[75,65919,65921],{"className":65920,"style":5343},[5312],[75,65922],{}," in GHz",[7635,65925,65926,65931,65936],{},[7650,65927,65928],{},[72,65929,65930],{},"OMEGA_ONRESONANCE_MHZ",[7650,65932,65933],{"align":12437},[72,65934,65935],{},"20.0",[7650,65937,65938,65939,65976],{},"On-resonance Rabi rate ",[75,65940,65942,65961],{"className":65941},[5148],[75,65943,65945],{"className":65944},[5152],[5154,65946,65947],{"xmlns":5156},[5158,65948,65949,65959],{},[5161,65950,65951,65953,65955,65957],{},[5164,65952,12635],{"mathvariant":6582},[5164,65954,2810],{"mathvariant":6582},[5184,65956,270],{},[5164,65958,5889],{},[5238,65960,52020],{"encoding":5240},[75,65962,65964],{"className":65963,"ariaHidden":5196},[5245],[75,65965,65967,65970,65973],{"className":65966},[5249],[75,65968],{"className":65969,"style":5254},[5253],[75,65971,52033],{"className":65972},[5258],[75,65974,5889],{"className":65975,"style":5416},[5258,5259]," in MHz",[7635,65978,65979,65984,65989],{},[7650,65980,65981],{},[72,65982,65983],{},"FREQ_MIN_GHZ",[7650,65985,65986],{"align":12437},[72,65987,65988],{},"4.90",[7650,65990,65991],{},"Minimum swept drive frequency in GHz",[7635,65993,65994,65999,66004],{},[7650,65995,65996],{},[72,65997,65998],{},"FREQ_MAX_GHZ",[7650,66000,66001],{"align":12437},[72,66002,66003],{},"5.10",[7650,66005,66006],{},"Maximum swept drive frequency in GHz",[7635,66008,66009,66014,66019],{},[7650,66010,66011],{},[72,66012,66013],{},"N_FREQ",[7650,66015,66016],{"align":12437},[72,66017,66018],{},"401",[7650,66020,66021],{},"Number of drive-frequency samples",[7635,66023,66024,66029,66033],{},[7650,66025,66026],{},[72,66027,66028],{},"DUR_MIN_NS",[7650,66030,66031],{"align":12437},[72,66032,14324],{},[7650,66034,66035],{},"Minimum pulse duration in ns",[7635,66037,66038,66043,66048],{},[7650,66039,66040],{},[72,66041,66042],{},"DUR_MAX_NS",[7650,66044,66045],{"align":12437},[72,66046,66047],{},"200.0",[7650,66049,66050],{},"Maximum pulse duration in ns",[7635,66052,66053,66058,66062],{},[7650,66054,66055],{},[72,66056,66057],{},"N_DUR",[7650,66059,66060],{"align":12437},[72,66061,66018],{},[7650,66063,66064],{},"Number of pulse-duration samples",[7635,66066,66067,66072,66076],{},[7650,66068,66069],{},[72,66070,66071],{},"T2STAR_NS",[7650,66073,66074],{"align":12437},[72,66075,14285],{},[7650,66077,66078,66079,66082],{},"Optional damping time in ns, with ",[72,66080,66081],{},"None"," disabling damping",[12,66084,66085],{},"Additional cross-section plotting cells expose localized slice controls.",[7629,66087,66088,66098],{},[7632,66089,66090],{},[7635,66091,66092,66094,66096],{},[7638,66093,65832],{},[7638,66095,65835],{"align":12437},[7638,66097,51184],{},[7645,66099,66100,66199],{},[7635,66101,66102,66107,66111],{},[7650,66103,66104],{},[72,66105,66106],{},"F_CROSS_GHZ",[7650,66108,66109],{"align":12437},[72,66110,53117],{},[7650,66112,66113,66114],{},"Drive-frequency slice used to plot ",[75,66115,66117,66141],{"className":66116},[5148],[75,66118,66120],{"className":66119},[5152],[5154,66121,66122],{"xmlns":5156},[5158,66123,66124,66138],{},[5161,66125,66126,66132,66134,66136],{},[5179,66127,66128,66130],{},[5164,66129,51013],{},[5164,66131,7486],{},[5168,66133,197],{"stretchy":5170},[5164,66135,51024],{},[5168,66137,5175],{"stretchy":5170},[5238,66139,66140],{"encoding":5240},"P_e(\\tau)",[75,66142,66144],{"className":66143,"ariaHidden":5196},[5245],[75,66145,66147,66150,66190,66193,66196],{"className":66146},[5249],[75,66148],{"className":66149,"style":5254},[5253],[75,66151,66153,66156],{"className":66152},[5258],[75,66154,51013],{"className":66155,"style":9817},[5258,5259],[75,66157,66159],{"className":66158},[5299],[75,66160,66162,66182],{"className":66161},[5303,5304],[75,66163,66165,66179],{"className":66164},[5308],[75,66166,66168],{"className":66167,"style":6264},[5312],[75,66169,66170,66173],{"style":33755},[75,66171],{"className":66172,"style":5321},[5320],[75,66174,66176],{"className":66175},[5325,5326,5327,5328],[75,66177,7486],{"className":66178},[5258,5259,5328],[75,66180,5336],{"className":66181},[5335],[75,66183,66185],{"className":66184},[5308],[75,66186,66188],{"className":66187,"style":5343},[5312],[75,66189],{},[75,66191,197],{"className":66192},[5263],[75,66194,51024],{"className":66195,"style":51096},[5258,5259],[75,66197,5175],{"className":66198},[5270],[7635,66200,66201,66206,66211],{},[7650,66202,66203],{},[72,66204,66205],{},"T_CROSS_NS",[7650,66207,66208],{"align":12437},[72,66209,66210],{},"100.0",[7650,66212,66213,66214],{},"Pulse-duration slice used to plot ",[75,66215,66217,66241],{"className":66216},[5148],[75,66218,66220],{"className":66219},[5152],[5154,66221,66222],{"xmlns":5156},[5158,66223,66224,66238],{},[5161,66225,66226,66232,66234,66236],{},[5179,66227,66228,66230],{},[5164,66229,51013],{},[5164,66231,7486],{},[5168,66233,197],{"stretchy":5170},[5164,66235,5894],{},[5168,66237,5175],{"stretchy":5170},[5238,66239,66240],{"encoding":5240},"P_e(f)",[75,66242,66244],{"className":66243,"ariaHidden":5196},[5245],[75,66245,66247,66250,66290,66293,66296],{"className":66246},[5249],[75,66248],{"className":66249,"style":5254},[5253],[75,66251,66253,66256],{"className":66252},[5258],[75,66254,51013],{"className":66255,"style":9817},[5258,5259],[75,66257,66259],{"className":66258},[5299],[75,66260,66262,66282],{"className":66261},[5303,5304],[75,66263,66265,66279],{"className":66264},[5308],[75,66266,66268],{"className":66267,"style":6264},[5312],[75,66269,66270,66273],{"style":33755},[75,66271],{"className":66272,"style":5321},[5320],[75,66274,66276],{"className":66275},[5325,5326,5327,5328],[75,66277,7486],{"className":66278},[5258,5259,5328],[75,66280,5336],{"className":66281},[5335],[75,66283,66285],{"className":66284},[5308],[75,66286,66288],{"className":66287,"style":5343},[5312],[75,66289],{},[75,66291,197],{"className":66292},[5263],[75,66294,5894],{"className":66295,"style":6251},[5258,5259],[75,66297,5175],{"className":66298},[5270],[12,66300,66301],{},"Separate FFT processing controls support spectral analysis.",[7629,66303,66304,66314],{},[7632,66305,66306],{},[7635,66307,66308,66310,66312],{},[7638,66309,65832],{},[7638,66311,65835],{"align":12437},[7638,66313,51184],{},[7645,66315,66316,66330,66345,66359,66374,66389],{},[7635,66317,66318,66323,66327],{},[7650,66319,66320],{},[72,66321,66322],{},"FFT_PAD",[7650,66324,66325],{"align":12437},[72,66326,877],{},[7650,66328,66329],{},"Zero-padding factor used in the Fourier transform",[7635,66331,66332,66337,66342],{},[7650,66333,66334],{},[72,66335,66336],{},"WINDOW",[7650,66338,66339],{"align":12437},[72,66340,66341],{},"\"hann\"",[7650,66343,66344],{},"Window function applied along the pulse-duration axis",[7635,66346,66347,66352,66356],{},[7650,66348,66349],{},[72,66350,66351],{},"DETREND_MEAN",[7650,66353,66354],{"align":12437},[72,66355,10504],{},[7650,66357,66358],{},"Flag that subtracts the mean before the FFT",[7635,66360,66361,66366,66371],{},[7650,66362,66363],{},[72,66364,66365],{},"AMP_MODE",[7650,66367,66368],{"align":12437},[72,66369,66370],{},"\"magnitude\"",[7650,66372,66373],{},"Display mode for magnitude, power, or dB-scaled amplitude",[7635,66375,66376,66381,66386],{},[7650,66377,66378],{},[72,66379,66380],{},"NORM_MODE",[7650,66382,66383],{"align":12437},[72,66384,66385],{},"\"global\"",[7650,66387,66388],{},"Normalization mode applied globally or per drive-frequency column",[7635,66390,66391,66396,66401],{},[7650,66392,66393],{},[72,66394,66395],{},"FREQ_UNITS",[7650,66397,66398],{"align":12437},[72,66399,66400],{},"\"MHz\"",[7650,66402,66403],{},"Frequency units used on the Fourier-frequency axis",[12,66405,66406],{},"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.",[65,66408,66410],{"className":67,"code":66409,"language":69,"meta":70,"style":70},"# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n",[72,66411,66412,66417,66424],{"__ignoreMap":70},[75,66413,66414],{"class":77,"line":78},[75,66415,66416],{"class":118},"# @title Install dependencies\n",[75,66418,66419,66421],{"class":77,"line":95},[75,66420,82],{"class":81},[75,66422,66423],{"class":85}," sys\n",[75,66425,66426,66429,66432,66434],{"class":77,"line":108},[75,66427,66428],{"class":141},"%",[75,66430,66431],{"class":85},"pip ",[75,66433,200],{"class":141},[75,66435,66436],{"class":85},"q install pyvista pythreejs trame\n",[65,66438,66440],{"className":67,"code":66439,"language":69,"meta":70,"style":70},"# @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",[72,66441,66442,66447,66453,66464,66474,66484,66496,66500,66507,66523,66533,66541,66550,66554,66567,66580,66584,66589,66605,66620,66636,66652,66668,66683,66699,66714,66733,66737],{"__ignoreMap":70},[75,66443,66444],{"class":77,"line":78},[75,66445,66446],{"class":118},"# @title Controls & imports\n",[75,66448,66449,66451],{"class":77,"line":95},[75,66450,82],{"class":81},[75,66452,14088],{"class":85},[75,66454,66455,66457,66459,66461],{"class":77,"line":108},[75,66456,2700],{"class":81},[75,66458,14076],{"class":85},[75,66460,82],{"class":81},[75,66462,66463],{"class":85}," Optional, Tuple\n",[75,66465,66466,66468,66470,66472],{"class":77,"line":115},[75,66467,82],{"class":81},[75,66469,86],{"class":85},[75,66471,89],{"class":81},[75,66473,92],{"class":85},[75,66475,66476,66478,66480,66482],{"class":77,"line":122},[75,66477,82],{"class":81},[75,66479,100],{"class":85},[75,66481,89],{"class":81},[75,66483,105],{"class":85},[75,66485,66486,66488,66491,66493],{"class":77,"line":128},[75,66487,82],{"class":81},[75,66489,66490],{"class":85}," pandas ",[75,66492,89],{"class":81},[75,66494,66495],{"class":85}," pd\n",[75,66497,66498],{"class":77,"line":149},[75,66499,112],{"emptyLinePlaceholder":111},[75,66501,66502,66505],{"class":77,"line":154},[75,66503,66504],{"class":81},"try",[75,66506,5800],{"class":85},[75,66508,66509,66512,66515,66517,66520],{"class":77,"line":160},[75,66510,66511],{"class":81},"    import",[75,66513,66514],{"class":85}," pyvista ",[75,66516,89],{"class":81},[75,66518,66519],{"class":85}," pv  ",[75,66521,66522],{"class":118},"# optional; only for 3D interactive\n",[75,66524,66525,66528,66530],{"class":77,"line":171},[75,66526,66527],{"class":145},"    PV_AVAILABLE",[75,66529,14153],{"class":141},[75,66531,66532],{"class":145}," True\n",[75,66534,66535,66538],{"class":77,"line":182},[75,66536,66537],{"class":81},"except",[75,66539,66540],{"class":85}," Exception:\n",[75,66542,66543,66545,66547],{"class":77,"line":222},[75,66544,66527],{"class":145},[75,66546,14153],{"class":141},[75,66548,66549],{"class":145}," False\n",[75,66551,66552],{"class":77,"line":227},[75,66553,112],{"emptyLinePlaceholder":111},[75,66555,66556,66558,66560,66562,66564],{"class":77,"line":233},[75,66557,131],{"class":85},[75,66559,135],{"class":134},[75,66561,138],{"class":85},[75,66563,142],{"class":141},[75,66565,66566],{"class":145}," 250\n",[75,66568,66569,66571,66574,66576,66578],{"class":77,"line":286},[75,66570,131],{"class":85},[75,66572,66573],{"class":134},"'axes.grid'",[75,66575,138],{"class":85},[75,66577,142],{"class":141},[75,66579,66549],{"class":145},[75,66581,66582],{"class":77,"line":291},[75,66583,112],{"emptyLinePlaceholder":111},[75,66585,66586],{"class":77,"line":297},[75,66587,66588],{"class":118},"# ---------- Control knobs ----------\n",[75,66590,66591,66593,66595,66597,66599,66602],{"class":77,"line":321},[75,66592,65846],{"class":145},[75,66594,14391],{"class":85},[75,66596,14150],{"class":141},[75,66598,14153],{"class":141},[75,66600,66601],{"class":145}," 5.000",[75,66603,66604],{"class":118},"        # Qubit transition freq f0 [GHz]\n",[75,66606,66607,66609,66611,66613,66615,66617],{"class":77,"line":336},[75,66608,65930],{"class":145},[75,66610,14391],{"class":85},[75,66612,14150],{"class":141},[75,66614,14153],{"class":141},[75,66616,18198],{"class":145},[75,66618,66619],{"class":118},"  # Ω\u002F2π on-resonance [MHz]\n",[75,66621,66622,66624,66626,66628,66630,66633],{"class":77,"line":351},[75,66623,65983],{"class":145},[75,66625,14391],{"class":85},[75,66627,14150],{"class":141},[75,66629,14153],{"class":141},[75,66631,66632],{"class":145}," 4.90",[75,66634,66635],{"class":118},"   # Sweep min f [GHz]\n",[75,66637,66638,66640,66642,66644,66646,66649],{"class":77,"line":366},[75,66639,65998],{"class":145},[75,66641,14391],{"class":85},[75,66643,14150],{"class":141},[75,66645,14153],{"class":141},[75,66647,66648],{"class":145}," 5.10",[75,66650,66651],{"class":118},"   # Sweep max f [GHz]\n",[75,66653,66654,66656,66658,66660,66662,66665],{"class":77,"line":381},[75,66655,66013],{"class":145},[75,66657,14391],{"class":85},[75,66659,14299],{"class":141},[75,66661,14153],{"class":141},[75,66663,66664],{"class":145}," 401",[75,66666,66667],{"class":118},"            # # of frequency points\n",[75,66669,66670,66672,66674,66676,66678,66680],{"class":77,"line":984},[75,66671,66028],{"class":145},[75,66673,14391],{"class":85},[75,66675,14150],{"class":141},[75,66677,14153],{"class":141},[75,66679,14744],{"class":145},[75,66681,66682],{"class":118},"      # Sweep min duration τ [ns]\n",[75,66684,66685,66687,66689,66691,66693,66696],{"class":77,"line":997},[75,66686,66042],{"class":145},[75,66688,14391],{"class":85},[75,66690,14150],{"class":141},[75,66692,14153],{"class":141},[75,66694,66695],{"class":145}," 200.0",[75,66697,66698],{"class":118},"    # Sweep max duration τ [ns]\n",[75,66700,66701,66703,66705,66707,66709,66711],{"class":77,"line":1325},[75,66702,66057],{"class":145},[75,66704,14391],{"class":85},[75,66706,14299],{"class":141},[75,66708,14153],{"class":141},[75,66710,66664],{"class":145},[75,66712,66713],{"class":118},"             # # of duration points\n",[75,66715,66716,66718,66721,66723,66725,66727,66730],{"class":77,"line":1338},[75,66717,66071],{"class":145},[75,66719,66720],{"class":85},": Optional[",[75,66722,14150],{"class":141},[75,66724,138],{"class":85},[75,66726,142],{"class":141},[75,66728,66729],{"class":145}," 500.0",[75,66731,66732],{"class":118},"  # None to disable exp(-τ\u002FT2*)\n",[75,66734,66735],{"class":77,"line":1343},[75,66736,112],{"emptyLinePlaceholder":111},[75,66738,66739,66741,66743,66746],{"class":77,"line":1356},[75,66740,45904],{"class":141},[75,66742,197],{"class":85},[75,66744,66745],{"class":134},"'Controls loaded.'",[75,66747,318],{"class":85},[65,66749,66752],{"className":66750,"code":66751,"language":34131,"meta":70},[41256],"Controls loaded.\n",[72,66753,66751],{"__ignoreMap":70},[65,66755,66757],{"className":67,"code":66756,"language":69,"meta":70,"style":70},"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",[72,66758,66759,66768,66775,66782,66793,66804,66822,66826,66831,66835,66840,66845,66850,66855,66860,66865,66869,66874,66879,66883,66897,66912,66926,66940,66962,66966,66992,67014,67043,67047,67078,67098,67117,67140,67144,67169,67197,67211,67215],{"__ignoreMap":70},[75,66760,66761,66763,66766],{"class":77,"line":78},[75,66762,14380],{"class":81},[75,66764,66765],{"class":193}," rabi_probability",[75,66767,43495],{"class":85},[75,66769,66770,66773],{"class":77,"line":95},[75,66771,66772],{"class":14388},"    f_ghz",[75,66774,43789],{"class":85},[75,66776,66777,66780],{"class":77,"line":108},[75,66778,66779],{"class":14388},"    tau_ns",[75,66781,43789],{"class":85},[75,66783,66784,66787,66789,66791],{"class":77,"line":115},[75,66785,66786],{"class":14388},"    f0_ghz",[75,66788,14391],{"class":85},[75,66790,14150],{"class":141},[75,66792,41629],{"class":85},[75,66794,66795,66798,66800,66802],{"class":77,"line":122},[75,66796,66797],{"class":14388},"    omega_onres_mhz",[75,66799,14391],{"class":85},[75,66801,14150],{"class":141},[75,66803,41629],{"class":85},[75,66805,66806,66809,66811,66813,66815,66817,66820],{"class":77,"line":128},[75,66807,66808],{"class":14388},"    t2star_ns",[75,66810,66720],{"class":85},[75,66812,14150],{"class":141},[75,66814,138],{"class":85},[75,66816,142],{"class":141},[75,66818,66819],{"class":145}," None",[75,66821,41629],{"class":85},[75,66823,66824],{"class":77,"line":149},[75,66825,42347],{"class":85},[75,66827,66828],{"class":77,"line":154},[75,66829,66830],{"class":134},"    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[75,66832,66833],{"class":77,"line":160},[75,66834,112],{"emptyLinePlaceholder":111},[75,66836,66837],{"class":77,"line":171},[75,66838,66839],{"class":134},"    Args:\n",[75,66841,66842],{"class":77,"line":182},[75,66843,66844],{"class":134},"        f_ghz: 1D array of drive frequencies [GHz].\n",[75,66846,66847],{"class":77,"line":222},[75,66848,66849],{"class":134},"        tau_ns: 1D array of pulse durations [ns].\n",[75,66851,66852],{"class":77,"line":227},[75,66853,66854],{"class":134},"        f0_ghz: Qubit frequency on resonance [GHz].\n",[75,66856,66857],{"class":77,"line":233},[75,66858,66859],{"class":134},"        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[75,66861,66862],{"class":77,"line":286},[75,66863,66864],{"class":134},"        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[75,66866,66867],{"class":77,"line":291},[75,66868,112],{"emptyLinePlaceholder":111},[75,66870,66871],{"class":77,"line":297},[75,66872,66873],{"class":134},"    Returns:\n",[75,66875,66876],{"class":77,"line":321},[75,66877,66878],{"class":134},"        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[75,66880,66881],{"class":77,"line":336},[75,66882,41832],{"class":134},[75,66884,66885,66888,66890,66892,66894],{"class":77,"line":351},[75,66886,66887],{"class":85},"    two_pi ",[75,66889,142],{"class":141},[75,66891,15263],{"class":145},[75,66893,273],{"class":141},[75,66895,66896],{"class":85}," math.pi\n",[75,66898,66899,66902,66904,66907,66909],{"class":77,"line":366},[75,66900,66901],{"class":85},"    f ",[75,66903,142],{"class":141},[75,66905,66906],{"class":85}," f_ghz ",[75,66908,500],{"class":141},[75,66910,66911],{"class":145}," 1e9\n",[75,66913,66914,66917,66919,66922,66924],{"class":77,"line":381},[75,66915,66916],{"class":85},"    f0 ",[75,66918,142],{"class":141},[75,66920,66921],{"class":85}," f0_ghz ",[75,66923,500],{"class":141},[75,66925,66911],{"class":145},[75,66927,66928,66931,66933,66936,66938],{"class":77,"line":984},[75,66929,66930],{"class":85},"    tau ",[75,66932,142],{"class":141},[75,66934,66935],{"class":85}," tau_ns ",[75,66937,500],{"class":141},[75,66939,18416],{"class":145},[75,66941,66942,66945,66947,66950,66952,66955,66957,66959],{"class":77,"line":997},[75,66943,66944],{"class":85},"    omega ",[75,66946,142],{"class":141},[75,66948,66949],{"class":85}," two_pi ",[75,66951,500],{"class":141},[75,66953,66954],{"class":85}," omega_onres_mhz ",[75,66956,500],{"class":141},[75,66958,18456],{"class":145},[75,66960,66961],{"class":118},"  # rad\u002Fs\n",[75,66963,66964],{"class":77,"line":1325},[75,66965,112],{"emptyLinePlaceholder":111},[75,66967,66968,66971,66974,66976,66978,66980,66983,66985,66987,66990],{"class":77,"line":1338},[75,66969,66970],{"class":85},"    F, ",[75,66972,66973],{"class":145},"TAU",[75,66975,14153],{"class":141},[75,66977,190],{"class":85},[75,66979,42911],{"class":193},[75,66981,66982],{"class":85},"(f, tau, ",[75,66984,42917],{"class":309},[75,66986,142],{"class":141},[75,66988,66989],{"class":134},"'xy'",[75,66991,318],{"class":85},[75,66993,66994,66997,66999,67001,67003,67006,67008,67011],{"class":77,"line":1343},[75,66995,66996],{"class":85},"    Delta ",[75,66998,142],{"class":141},[75,67000,66949],{"class":85},[75,67002,500],{"class":141},[75,67004,67005],{"class":85}," (F ",[75,67007,200],{"class":141},[75,67009,67010],{"class":85}," f0)               ",[75,67012,67013],{"class":118},"# rad\u002Fs\n",[75,67015,67016,67019,67021,67023,67025,67028,67030,67032,67034,67037,67039,67041],{"class":77,"line":1356},[75,67017,67018],{"class":85},"    Omega_R ",[75,67020,142],{"class":141},[75,67022,190],{"class":85},[75,67024,14668],{"class":193},[75,67026,67027],{"class":85},"(omega",[75,67029,258],{"class":141},[75,67031,270],{"class":145},[75,67033,11902],{"class":141},[75,67035,67036],{"class":85}," Delta",[75,67038,258],{"class":141},[75,67040,270],{"class":145},[75,67042,318],{"class":85},[75,67044,67045],{"class":77,"line":1887},[75,67046,112],{"emptyLinePlaceholder":111},[75,67048,67049,67052,67054,67057,67059,67062,67064,67067,67069,67072,67074,67076],{"class":77,"line":1900},[75,67050,67051],{"class":81},"    with",[75,67053,190],{"class":85},[75,67055,67056],{"class":193},"errstate",[75,67058,197],{"class":85},[75,67060,67061],{"class":309},"divide",[75,67063,142],{"class":141},[75,67065,67066],{"class":134},"'ignore'",[75,67068,206],{"class":85},[75,67070,67071],{"class":309},"invalid",[75,67073,142],{"class":141},[75,67075,67066],{"class":134},[75,67077,15605],{"class":85},[75,67079,67080,67083,67085,67088,67090,67093,67095],{"class":77,"line":1913},[75,67081,67082],{"class":85},"        amp ",[75,67084,142],{"class":141},[75,67086,67087],{"class":85}," (omega ",[75,67089,2810],{"class":141},[75,67091,67092],{"class":85}," Omega_R) ",[75,67094,258],{"class":141},[75,67096,67097],{"class":145}," 2\n",[75,67099,67100,67103,67105,67107,67109,67112,67114],{"class":77,"line":2657},[75,67101,67102],{"class":85},"        phase ",[75,67104,142],{"class":141},[75,67106,15214],{"class":145},[75,67108,273],{"class":141},[75,67110,67111],{"class":85}," Omega_R ",[75,67113,500],{"class":141},[75,67115,67116],{"class":145}," TAU\n",[75,67118,67119,67122,67124,67127,67129,67131,67133,67136,67138],{"class":77,"line":3177},[75,67120,67121],{"class":85},"        Pe ",[75,67123,142],{"class":141},[75,67125,67126],{"class":85}," amp ",[75,67128,500],{"class":141},[75,67130,190],{"class":85},[75,67132,1179],{"class":193},[75,67134,67135],{"class":85},"(phase) ",[75,67137,258],{"class":141},[75,67139,67097],{"class":145},[75,67141,67142],{"class":77,"line":3188},[75,67143,112],{"emptyLinePlaceholder":111},[75,67145,67146,67148,67151,67154,67157,67159,67161,67163,67165,67167],{"class":77,"line":3203},[75,67147,14433],{"class":81},[75,67149,67150],{"class":85}," t2star_ns ",[75,67152,67153],{"class":81},"is",[75,67155,67156],{"class":81}," not",[75,67158,66819],{"class":145},[75,67160,14747],{"class":81},[75,67162,67150],{"class":85},[75,67164,2907],{"class":141},[75,67166,14744],{"class":145},[75,67168,5800],{"class":85},[75,67170,67171,67174,67176,67178,67180,67182,67184,67186,67188,67191,67193,67195],{"class":77,"line":3218},[75,67172,67173],{"class":85},"        decay ",[75,67175,142],{"class":141},[75,67177,190],{"class":85},[75,67179,243],{"class":193},[75,67181,197],{"class":85},[75,67183,200],{"class":141},[75,67185,66973],{"class":145},[75,67187,264],{"class":141},[75,67189,67190],{"class":85}," (t2star_ns ",[75,67192,500],{"class":141},[75,67194,11899],{"class":145},[75,67196,283],{"class":85},[75,67198,67199,67201,67203,67206,67208],{"class":77,"line":3224},[75,67200,67121],{"class":85},[75,67202,142],{"class":141},[75,67204,67205],{"class":85}," Pe ",[75,67207,500],{"class":141},[75,67209,67210],{"class":85}," decay\n",[75,67212,67213],{"class":77,"line":3229},[75,67214,112],{"emptyLinePlaceholder":111},[75,67216,67217,67219,67221,67223,67226,67228,67230,67232],{"class":77,"line":3235},[75,67218,14467],{"class":81},[75,67220,190],{"class":85},[75,67222,16789],{"class":193},[75,67224,67225],{"class":85},"(Pe, ",[75,67227,14324],{"class":145},[75,67229,206],{"class":85},[75,67231,630],{"class":145},[75,67233,318],{"class":85},[65,67235,67237],{"className":67,"code":67236,"language":69,"meta":70,"style":70},"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",[72,67238,67239,67264,67289,67313],{"__ignoreMap":70},[75,67240,67241,67244,67246,67248,67250,67252,67254,67256,67258,67260,67262],{"class":77,"line":78},[75,67242,67243],{"class":85},"freq_ghz ",[75,67245,142],{"class":141},[75,67247,190],{"class":85},[75,67249,194],{"class":193},[75,67251,197],{"class":85},[75,67253,65983],{"class":145},[75,67255,206],{"class":85},[75,67257,65998],{"class":145},[75,67259,206],{"class":85},[75,67261,66013],{"class":145},[75,67263,318],{"class":85},[75,67265,67266,67269,67271,67273,67275,67277,67279,67281,67283,67285,67287],{"class":77,"line":95},[75,67267,67268],{"class":85},"dur_ns ",[75,67270,142],{"class":141},[75,67272,190],{"class":85},[75,67274,194],{"class":193},[75,67276,197],{"class":85},[75,67278,66028],{"class":145},[75,67280,206],{"class":85},[75,67282,66042],{"class":145},[75,67284,206],{"class":85},[75,67286,66057],{"class":145},[75,67288,318],{"class":85},[75,67290,67291,67294,67296,67298,67301,67303,67305,67307,67309,67311],{"class":77,"line":108},[75,67292,67293],{"class":85},"Pe ",[75,67295,142],{"class":141},[75,67297,66765],{"class":193},[75,67299,67300],{"class":85},"(freq_ghz, dur_ns, ",[75,67302,65846],{"class":145},[75,67304,206],{"class":85},[75,67306,65930],{"class":145},[75,67308,206],{"class":85},[75,67310,66071],{"class":145},[75,67312,318],{"class":85},[75,67314,67315,67317,67319,67322],{"class":77,"line":115},[75,67316,45904],{"class":141},[75,67318,197],{"class":85},[75,67320,67321],{"class":134},"'Grid shape (τ × f):'",[75,67323,67324],{"class":85},", Pe.shape)\n",[65,67326,67329],{"className":67327,"code":67328,"language":34131,"meta":70},[41256],"Grid shape (τ × f): (401, 401)\n",[72,67330,67328],{"__ignoreMap":70},[65,67332,67334],{"className":67,"code":67333,"language":69,"meta":70,"style":70},"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",[72,67335,67336,67351,67356,67370,67383,67396,67409,67422,67435,67448,67461,67474,67479,67498,67502],{"__ignoreMap":70},[75,67337,67338,67341,67343,67346,67349],{"class":77,"line":78},[75,67339,67340],{"class":85},"controls_table ",[75,67342,142],{"class":141},[75,67344,67345],{"class":85}," pd.",[75,67347,67348],{"class":193},"DataFrame",[75,67350,43495],{"class":85},[75,67352,67353],{"class":77,"line":95},[75,67354,67355],{"class":85},"    [\n",[75,67357,67358,67361,67364,67366,67368],{"class":77,"line":108},[75,67359,67360],{"class":85},"        [",[75,67362,67363],{"class":134},"'f0 (GHz)'",[75,67365,206],{"class":85},[75,67367,65846],{"class":145},[75,67369,18742],{"class":85},[75,67371,67372,67374,67377,67379,67381],{"class":77,"line":115},[75,67373,67360],{"class":85},[75,67375,67376],{"class":134},"'Ω\u002F2π on-res (MHz)'",[75,67378,206],{"class":85},[75,67380,65930],{"class":145},[75,67382,18742],{"class":85},[75,67384,67385,67387,67390,67392,67394],{"class":77,"line":122},[75,67386,67360],{"class":85},[75,67388,67389],{"class":134},"'f sweep min (GHz)'",[75,67391,206],{"class":85},[75,67393,65983],{"class":145},[75,67395,18742],{"class":85},[75,67397,67398,67400,67403,67405,67407],{"class":77,"line":128},[75,67399,67360],{"class":85},[75,67401,67402],{"class":134},"'f sweep max (GHz)'",[75,67404,206],{"class":85},[75,67406,65998],{"class":145},[75,67408,18742],{"class":85},[75,67410,67411,67413,67416,67418,67420],{"class":77,"line":149},[75,67412,67360],{"class":85},[75,67414,67415],{"class":134},"'N_f'",[75,67417,206],{"class":85},[75,67419,66013],{"class":145},[75,67421,18742],{"class":85},[75,67423,67424,67426,67429,67431,67433],{"class":77,"line":154},[75,67425,67360],{"class":85},[75,67427,67428],{"class":134},"'τ sweep min (ns)'",[75,67430,206],{"class":85},[75,67432,66028],{"class":145},[75,67434,18742],{"class":85},[75,67436,67437,67439,67442,67444,67446],{"class":77,"line":160},[75,67438,67360],{"class":85},[75,67440,67441],{"class":134},"'τ sweep max (ns)'",[75,67443,206],{"class":85},[75,67445,66042],{"class":145},[75,67447,18742],{"class":85},[75,67449,67450,67452,67455,67457,67459],{"class":77,"line":171},[75,67451,67360],{"class":85},[75,67453,67454],{"class":134},"'N_τ'",[75,67456,206],{"class":85},[75,67458,66057],{"class":145},[75,67460,18742],{"class":85},[75,67462,67463,67465,67468,67470,67472],{"class":77,"line":182},[75,67464,67360],{"class":85},[75,67466,67467],{"class":134},"'T2* (ns; None=off)'",[75,67469,206],{"class":85},[75,67471,66071],{"class":145},[75,67473,18742],{"class":85},[75,67475,67476],{"class":77,"line":222},[75,67477,67478],{"class":85},"    ],\n",[75,67480,67481,67484,67486,67488,67491,67493,67496],{"class":77,"line":227},[75,67482,67483],{"class":309},"    columns",[75,67485,142],{"class":141},[75,67487,32864],{"class":85},[75,67489,67490],{"class":134},"'Parameter'",[75,67492,206],{"class":85},[75,67494,67495],{"class":134},"'Value'",[75,67497,10451],{"class":85},[75,67499,67500],{"class":77,"line":233},[75,67501,318],{"class":85},[75,67503,67504],{"class":77,"line":286},[75,67505,67506],{"class":85},"controls_table\n",[65,67508,67511],{"className":67509,"code":67510,"language":34131,"meta":70},[41256],"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",[72,67512,67510],{"__ignoreMap":70},[65,67514,67516],{"className":67,"code":67515,"language":69,"meta":70,"style":70},"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",[72,67517,67518,67544,67569,67600,67613,67626,67649,67658],{"__ignoreMap":70},[75,67519,67520,67522,67524,67526,67528,67530,67532,67534,67536,67538,67540,67542],{"class":77,"line":78},[75,67521,11073],{"class":85},[75,67523,142],{"class":141},[75,67525,3093],{"class":85},[75,67527,3096],{"class":193},[75,67529,197],{"class":85},[75,67531,870],{"class":309},[75,67533,142],{"class":141},[75,67535,197],{"class":85},[75,67537,13806],{"class":145},[75,67539,206],{"class":85},[75,67541,203],{"class":145},[75,67543,283],{"class":85},[75,67545,67546,67549,67551,67553,67556,67559,67562,67564,67567],{"class":77,"line":95},[75,67547,67548],{"class":85},"c ",[75,67550,142],{"class":141},[75,67552,43490],{"class":85},[75,67554,67555],{"class":193},"pcolormesh",[75,67557,67558],{"class":85},"(freq_ghz, dur_ns, Pe, ",[75,67560,67561],{"class":309},"shading",[75,67563,142],{"class":141},[75,67565,67566],{"class":134},"'auto'",[75,67568,318],{"class":85},[75,67570,67571,67574,67576,67578,67580,67583,67585,67587,67589,67591,67593,67595,67598],{"class":77,"line":108},[75,67572,67573],{"class":85},"cb ",[75,67575,142],{"class":141},[75,67577,44104],{"class":85},[75,67579,16495],{"class":193},[75,67581,67582],{"class":85},"(c, ",[75,67584,15602],{"class":309},[75,67586,142],{"class":141},[75,67588,44116],{"class":85},[75,67590,573],{"class":309},[75,67592,142],{"class":141},[75,67594,15977],{"class":81},[75,67596,67597],{"class":15980},"'$P_e$'",[75,67599,318],{"class":85},[75,67601,67602,67604,67606,67608,67611],{"class":77,"line":115},[75,67603,11108],{"class":85},[75,67605,3193],{"class":193},[75,67607,197],{"class":85},[75,67609,67610],{"class":134},"'Drive frequency f (GHz)'",[75,67612,318],{"class":85},[75,67614,67615,67617,67619,67621,67624],{"class":77,"line":122},[75,67616,11108],{"class":85},[75,67618,3208],{"class":193},[75,67620,197],{"class":85},[75,67622,67623],{"class":134},"'Pulse duration τ (ns)'",[75,67625,318],{"class":85},[75,67627,67628,67630,67632,67634,67637,67640,67642,67644,67647],{"class":77,"line":128},[75,67629,11108],{"class":85},[75,67631,3182],{"class":193},[75,67633,197],{"class":85},[75,67635,67636],{"class":134},"'Rabi excited-state probability $P_e(f,",[75,67638,67639],{"class":141},"\\\\",[75,67641,7469],{"class":134},[75,67643,67639],{"class":141},[75,67645,67646],{"class":134},"tau)$'",[75,67648,318],{"class":85},[75,67650,67651,67654,67656],{"class":77,"line":149},[75,67652,67653],{"class":85},"fig.",[75,67655,3285],{"class":193},[75,67657,389],{"class":85},[75,67659,67660,67662,67664],{"class":77,"line":154},[75,67661,300],{"class":85},[75,67663,386],{"class":193},[75,67665,389],{"class":85},[391,67667],{"alt":67668,"src":67669},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[65,67671,67673],{"className":67,"code":67672,"language":69,"meta":70,"style":70},"# @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",[72,67674,67675,67680,67685,67689,67696,67703,67709,67713,67718,67723,67727,67736,67740,67749,67761,67773,67785,67797,67802,67806,67832,67847,67854,67861,67865,67869,67878,67888,67892,67897,67907,67911,67921,67940,67949,67968,67972,67981,67990,67999,68008,68017,68021,68031,68041,68051,68061,68071,68075,68085,68095,68105,68115,68125,68129,68139,68157,68161,68165,68187,68192,68211,68215,68219,68228,68244,68252,68257,68268,68273,68277,68286,68291,68309,68321,68326,68330,68339,68343,68347,68354,68364,68394,68405,68409,68417,68422,68427,68434,68441,68448,68455,68462,68469,68477,68482,68492,68496,68504,68512,68516,68520,68526,68532,68538,68544,68550,68554,68564,68568,68572,68576,68587,68591,68601,68611,68623,68635,68639,68645,68658,68662,68674,68689,68708,68715,68720,68724,68733,68738,68749,68753,68757,68761,68765,68787,68792,68810,68814,68818,68827,68838,68849,68860,68877,68882,68905,68927,68942,68946,68972,68976,68999,69020,69040,69044,69057,69071,69088,69092,69096,69110,69115,69120,69125,69129,69133,69162,69188,69213,69217,69230,69244,69248,69252,69266,69271,69276,69281,69286,69291,69295,69299,69312,69317,69328,69336,69346,69351,69356,69361,69368,69384,69393,69402,69411,69420,69443,69449,69454,69459,69464,69474,69483,69492,69497,69503,69518,69529,69544,69554,69562,69571,69576,69598,69603,69619,69636,69641,69654,69667,69672,69680,69685,69690,69700,69708,69716,69728,69740,69749,69755,69765,69775,69784,69793,69804,69816,69827,69832,69837,69842,69852,69859,69867,69878,69889,69898,69904,69914,69924,69933,69942,69951,69963,69975,69984,69989,69994,69999,70009,70016,70023,70034,70045,70054,70060,70070,70078,70085,70092,70101,70110,70119,70128,70139,70151,70163,70172,70177,70182,70187,70197,70204,70211,70223,70234,70243,70249,70265,70270,70279,70289,70298,70307,70313,70322,70334,70346,70358,70363,70368,70373,70391,70397,70406,70413,70420,70428,70439,70450,70461,70466,70475,70482,70489,70497,70518,70529,70540,70551,70562,70567,70572,70577,70591,70603,70609,70614,70621,70631,70636,70648,70659,70669,70680,70685,70690,70702,70713,70722,70733,70738,70743,70753,70759,70773,70778,70788,70793,70805,70810,70815,70837,70842,70865,70876,70915,70925,70935,70954,70972,70977,70982,70994,71004,71014,71019,71024,71032,71042,71057,71068,71080,71085,71092,71101,71115,71126,71138,71143,71148,71158,71166,71176,71197,71208,71220,71225,71232,71241,71260,71271,71283,71288,71297,71302,71312,71320,71329,71343,71354,71380,71385,71392,71401,71414,71425,71448,71453,71458,71472,71480,71489,71502,71524,71535,71540,71547,71556,71569,71590,71601,71606,71611,71621,71631,71639,71649,71658,71670,71682,71694,71699,71706,71715,71724,71735,71746,71757,71762,71769,71778,71787,71798,71809,71830,71835,71842,71851,71860,71871,71882,71903,71908,71913,71918,71928,71939,71951,71963,71975,71987,71992,71997,72006,72011,72021,72036,72051,72056,72068,72093,72118,72143,72167,72193,72200,72206,72231,72252,72274,72279,72286,72292,72314,72335,72356,72361,72368,72374,72403,72408,72415,72421,72448,72453,72460,72466,72489],{"__ignoreMap":70},[75,67676,67677],{"class":77,"line":78},[75,67678,67679],{"class":118},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[75,67681,67682],{"class":77,"line":95},[75,67683,67684],{"class":134},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[75,67686,67687],{"class":77,"line":108},[75,67688,112],{"emptyLinePlaceholder":111},[75,67690,67691,67693],{"class":77,"line":115},[75,67692,82],{"class":81},[75,67694,67695],{"class":85}," importlib.util\n",[75,67697,67698,67700],{"class":77,"line":122},[75,67699,82],{"class":81},[75,67701,67702],{"class":85}," subprocess\n",[75,67704,67705,67707],{"class":77,"line":128},[75,67706,82],{"class":81},[75,67708,66423],{"class":85},[75,67710,67711],{"class":77,"line":149},[75,67712,112],{"emptyLinePlaceholder":111},[75,67714,67715],{"class":77,"line":154},[75,67716,67717],{"class":118},"# -------------------------------------------------------------------------\n",[75,67719,67720],{"class":77,"line":160},[75,67721,67722],{"class":118},"# Control knobs\n",[75,67724,67725],{"class":77,"line":171},[75,67726,67717],{"class":118},[75,67728,67729,67732,67734],{"class":77,"line":182},[75,67730,67731],{"class":145},"INSTALL_BLOCH_DEPENDENCIES",[75,67733,14153],{"class":141},[75,67735,66532],{"class":145},[75,67737,67738],{"class":77,"line":222},[75,67739,112],{"emptyLinePlaceholder":111},[75,67741,67742,67745,67747],{"class":77,"line":227},[75,67743,67744],{"class":145},"BLOCH_REQUIRED_PACKAGES",[75,67746,14153],{"class":141},[75,67748,42099],{"class":85},[75,67750,67751,67754,67756,67759],{"class":77,"line":233},[75,67752,67753],{"class":134},"    \"numpy\"",[75,67755,14391],{"class":85},[75,67757,67758],{"class":134},"\"numpy\"",[75,67760,41629],{"class":85},[75,67762,67763,67766,67768,67771],{"class":77,"line":286},[75,67764,67765],{"class":134},"    \"matplotlib\"",[75,67767,14391],{"class":85},[75,67769,67770],{"class":134},"\"matplotlib\"",[75,67772,41629],{"class":85},[75,67774,67775,67778,67780,67783],{"class":77,"line":291},[75,67776,67777],{"class":134},"    \"scipy\"",[75,67779,14391],{"class":85},[75,67781,67782],{"class":134},"\"scipy\"",[75,67784,41629],{"class":85},[75,67786,67787,67790,67792,67795],{"class":77,"line":297},[75,67788,67789],{"class":134},"    \"qutip\"",[75,67791,14391],{"class":85},[75,67793,67794],{"class":134},"\"qutip\"",[75,67796,41629],{"class":85},[75,67798,67799],{"class":77,"line":321},[75,67800,67801],{"class":85},"}\n",[75,67803,67804],{"class":77,"line":336},[75,67805,112],{"emptyLinePlaceholder":111},[75,67807,67808,67811,67813,67816,67819,67821,67823,67826,67828,67830],{"class":77,"line":351},[75,67809,67810],{"class":145},"BLOCH_F0_GHZ",[75,67812,14153],{"class":141},[75,67814,67815],{"class":141}," globals",[75,67817,67818],{"class":85},"().",[75,67820,42235],{"class":193},[75,67822,197],{"class":85},[75,67824,67825],{"class":134},"\"F0_GHZ\"",[75,67827,206],{"class":85},[75,67829,53117],{"class":145},[75,67831,318],{"class":85},[75,67833,67834,67837,67839,67841,67843,67845],{"class":77,"line":366},[75,67835,67836],{"class":145},"BLOCH_OMEGA_ONRESONANCE_MHZ",[75,67838,14153],{"class":141},[75,67840,67815],{"class":141},[75,67842,67818],{"class":85},[75,67844,42235],{"class":193},[75,67846,43495],{"class":85},[75,67848,67849,67852],{"class":77,"line":381},[75,67850,67851],{"class":134},"    \"OMEGA_ONRESONANCE_MHZ\"",[75,67853,41629],{"class":85},[75,67855,67856,67859],{"class":77,"line":984},[75,67857,67858],{"class":145},"    20.0",[75,67860,41629],{"class":85},[75,67862,67863],{"class":77,"line":997},[75,67864,318],{"class":85},[75,67866,67867],{"class":77,"line":1325},[75,67868,112],{"emptyLinePlaceholder":111},[75,67870,67871,67874,67876],{"class":77,"line":1338},[75,67872,67873],{"class":145},"BLOCH_RESONANT_DETUNING_MHZ",[75,67875,14153],{"class":141},[75,67877,49997],{"class":145},[75,67879,67880,67883,67885],{"class":77,"line":1343},[75,67881,67882],{"class":145},"BLOCH_DETUNED_DETUNING_MHZ",[75,67884,14153],{"class":141},[75,67886,67887],{"class":145}," 40.0\n",[75,67889,67890],{"class":77,"line":1356},[75,67891,112],{"emptyLinePlaceholder":111},[75,67893,67894],{"class":77,"line":1887},[75,67895,67896],{"class":118},"# Use None to automatically plot one resonant pi pulse.\n",[75,67898,67899,67902,67904],{"class":77,"line":1900},[75,67900,67901],{"class":145},"BLOCH_DURATION_NS",[75,67903,14153],{"class":141},[75,67905,67906],{"class":145}," None\n",[75,67908,67909],{"class":77,"line":1913},[75,67910,112],{"emptyLinePlaceholder":111},[75,67912,67913,67916,67918],{"class":77,"line":2657},[75,67914,67915],{"class":145},"BLOCH_NUM_TIME_POINTS",[75,67917,14153],{"class":141},[75,67919,67920],{"class":145}," 300\n",[75,67922,67923,67926,67928,67930,67933,67935,67938],{"class":77,"line":3177},[75,67924,67925],{"class":145},"BLOCH_FIGSIZE",[75,67927,14153],{"class":141},[75,67929,267],{"class":85},[75,67931,67932],{"class":145},"8.8",[75,67934,206],{"class":85},[75,67936,67937],{"class":145},"7.8",[75,67939,318],{"class":85},[75,67941,67942,67945,67947],{"class":77,"line":3188},[75,67943,67944],{"class":145},"BLOCH_DPI",[75,67946,14153],{"class":141},[75,67948,66566],{"class":145},[75,67950,67951,67954,67956,67958,67960,67962,67964,67966],{"class":77,"line":3203},[75,67952,67953],{"class":145},"BLOCH_VIEW",[75,67955,14153],{"class":141},[75,67957,16404],{"class":85},[75,67959,200],{"class":141},[75,67961,44441],{"class":145},[75,67963,206],{"class":85},[75,67965,57891],{"class":145},[75,67967,10451],{"class":85},[75,67969,67970],{"class":77,"line":3218},[75,67971,112],{"emptyLinePlaceholder":111},[75,67973,67974,67977,67979],{"class":77,"line":3224},[75,67975,67976],{"class":145},"BLOCH_SHOW_CONTROL_AXIS_ARROWS",[75,67978,14153],{"class":141},[75,67980,66532],{"class":145},[75,67982,67983,67986,67988],{"class":77,"line":3229},[75,67984,67985],{"class":145},"BLOCH_SHOW_ARROW_TIP_LABELS",[75,67987,14153],{"class":141},[75,67989,66532],{"class":145},[75,67991,67992,67995,67997],{"class":77,"line":3235},[75,67993,67994],{"class":145},"BLOCH_SHOW_ENDPOINT_MARKERS",[75,67996,14153],{"class":141},[75,67998,66532],{"class":145},[75,68000,68001,68004,68006],{"class":77,"line":3263},[75,68002,68003],{"class":145},"BLOCH_SHOW_LEGEND",[75,68005,14153],{"class":141},[75,68007,66532],{"class":145},[75,68009,68010,68013,68015],{"class":77,"line":3275},[75,68011,68012],{"class":145},"BLOCH_PRINT_SUMMARY",[75,68014,14153],{"class":141},[75,68016,66532],{"class":145},[75,68018,68019],{"class":77,"line":3280},[75,68020,112],{"emptyLinePlaceholder":111},[75,68022,68023,68026,68028],{"class":77,"line":3290},[75,68024,68025],{"class":145},"COLOR_RESONANT_TRAJECTORY",[75,68027,14153],{"class":141},[75,68029,68030],{"class":134}," \"tab:blue\"\n",[75,68032,68033,68036,68038],{"class":77,"line":4431},[75,68034,68035],{"class":145},"COLOR_DETUNED_TRAJECTORY",[75,68037,14153],{"class":141},[75,68039,68040],{"class":134}," \"tab:orange\"\n",[75,68042,68043,68046,68048],{"class":77,"line":4454},[75,68044,68045],{"class":145},"COLOR_RESONANT_AXIS",[75,68047,14153],{"class":141},[75,68049,68050],{"class":134}," \"tab:green\"\n",[75,68052,68053,68056,68058],{"class":77,"line":4463},[75,68054,68055],{"class":145},"COLOR_DETUNED_AXIS",[75,68057,14153],{"class":141},[75,68059,68060],{"class":134}," \"tab:red\"\n",[75,68062,68063,68066,68068],{"class":77,"line":4468},[75,68064,68065],{"class":145},"COLOR_INITIAL_STATE",[75,68067,14153],{"class":141},[75,68069,68070],{"class":134}," \"black\"\n",[75,68072,68073],{"class":77,"line":4477},[75,68074,112],{"emptyLinePlaceholder":111},[75,68076,68077,68080,68082],{"class":77,"line":5105},[75,68078,68079],{"class":145},"TRAJECTORY_LINEWIDTH",[75,68081,14153],{"class":141},[75,68083,68084],{"class":145}," 2.4\n",[75,68086,68087,68090,68092],{"class":77,"line":5114},[75,68088,68089],{"class":145},"CONTROL_AXIS_LINEWIDTH",[75,68091,14153],{"class":141},[75,68093,68094],{"class":145}," 2.2\n",[75,68096,68097,68100,68102],{"class":77,"line":10795},[75,68098,68099],{"class":145},"ENDPOINT_MARKER_SIZE",[75,68101,14153],{"class":141},[75,68103,68104],{"class":145}," 45\n",[75,68106,68107,68110,68112],{"class":77,"line":14778},[75,68108,68109],{"class":145},"ARROW_LENGTH_RATIO",[75,68111,14153],{"class":141},[75,68113,68114],{"class":145}," 0.12\n",[75,68116,68117,68120,68122],{"class":77,"line":14783},[75,68118,68119],{"class":145},"ARROW_LABEL_SCALE",[75,68121,14153],{"class":141},[75,68123,68124],{"class":145}," 1.16\n",[75,68126,68127],{"class":77,"line":14819},[75,68128,112],{"emptyLinePlaceholder":111},[75,68130,68131,68134,68136],{"class":77,"line":14825},[75,68132,68133],{"class":145},"LEGEND_LOCATION",[75,68135,14153],{"class":141},[75,68137,68138],{"class":134}," \"upper left\"\n",[75,68140,68141,68144,68146,68148,68151,68153,68155],{"class":77,"line":14838},[75,68142,68143],{"class":145},"LEGEND_BBOX_TO_ANCHOR",[75,68145,14153],{"class":141},[75,68147,267],{"class":85},[75,68149,68150],{"class":145},"1.02",[75,68152,206],{"class":85},[75,68154,68150],{"class":145},[75,68156,318],{"class":85},[75,68158,68159],{"class":77,"line":14893},[75,68160,112],{"emptyLinePlaceholder":111},[75,68162,68163],{"class":77,"line":14951},[75,68164,112],{"emptyLinePlaceholder":111},[75,68166,68167,68169,68172,68174,68177,68179,68181,68183,68185],{"class":77,"line":14959},[75,68168,14380],{"class":81},[75,68170,68171],{"class":193}," package_is_available",[75,68173,197],{"class":85},[75,68175,68176],{"class":14388},"package_name",[75,68178,14391],{"class":85},[75,68180,41840],{"class":141},[75,68182,14419],{"class":85},[75,68184,14246],{"class":141},[75,68186,5800],{"class":85},[75,68188,68189],{"class":77,"line":14964},[75,68190,68191],{"class":134},"    \"\"\"Return True when a package can be imported.\"\"\"\n",[75,68193,68194,68196,68199,68202,68205,68207,68209],{"class":77,"line":14983},[75,68195,14467],{"class":81},[75,68197,68198],{"class":85}," importlib.util.",[75,68200,68201],{"class":193},"find_spec",[75,68203,68204],{"class":85},"(package_name) ",[75,68206,67153],{"class":81},[75,68208,67156],{"class":81},[75,68210,67906],{"class":145},[75,68212,68213],{"class":77,"line":14989},[75,68214,112],{"emptyLinePlaceholder":111},[75,68216,68217],{"class":77,"line":15012},[75,68218,112],{"emptyLinePlaceholder":111},[75,68220,68221,68223,68226],{"class":77,"line":15017},[75,68222,14380],{"class":81},[75,68224,68225],{"class":193}," install_missing_packages",[75,68227,43495],{"class":85},[75,68229,68230,68233,68236,68238,68240,68242],{"class":77,"line":15025},[75,68231,68232],{"class":14388},"    required_packages",[75,68234,68235],{"class":85},": dict[",[75,68237,41840],{"class":141},[75,68239,206],{"class":85},[75,68241,41840],{"class":141},[75,68243,18742],{"class":85},[75,68245,68246,68248,68250],{"class":77,"line":15038},[75,68247,14419],{"class":85},[75,68249,66081],{"class":145},[75,68251,5800],{"class":85},[75,68253,68254],{"class":77,"line":15051},[75,68255,68256],{"class":134},"    \"\"\"Install missing notebook packages using uv pip with a pip fallback.\"\"\"\n",[75,68258,68259,68261,68263,68266],{"class":77,"line":15058},[75,68260,14433],{"class":81},[75,68262,67156],{"class":81},[75,68264,68265],{"class":145}," INSTALL_BLOCH_DEPENDENCIES",[75,68267,5800],{"class":85},[75,68269,68270],{"class":77,"line":15069},[75,68271,68272],{"class":81},"        return\n",[75,68274,68275],{"class":77,"line":15080},[75,68276,112],{"emptyLinePlaceholder":111},[75,68278,68279,68282,68284],{"class":77,"line":15085},[75,68280,68281],{"class":85},"    missing_package_names ",[75,68283,142],{"class":141},[75,68285,10409],{"class":85},[75,68287,68288],{"class":77,"line":15090},[75,68289,68290],{"class":85},"        pip_name\n",[75,68292,68293,68296,68299,68301,68304,68307],{"class":77,"line":15096},[75,68294,68295],{"class":81},"        for",[75,68297,68298],{"class":85}," import_name, pip_name ",[75,68300,3153],{"class":81},[75,68302,68303],{"class":85}," 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   bloch_axis.",[75,71925,5058],{"class":193},[75,71927,43495],{"class":85},[75,71929,71931,71934,71936],{"class":77,"line":71930},458,[75,71932,71933],{"class":309},"        handles",[75,71935,142],{"class":141},[75,71937,71938],{"class":85},"legend_handles,\n",[75,71940,71942,71945,71947,71949],{"class":77,"line":71941},459,[75,71943,71944],{"class":309},"        loc",[75,71946,142],{"class":141},[75,71948,68133],{"class":145},[75,71950,41629],{"class":85},[75,71952,71954,71957,71959,71961],{"class":77,"line":71953},460,[75,71955,71956],{"class":309},"        bbox_to_anchor",[75,71958,142],{"class":141},[75,71960,68143],{"class":145},[75,71962,41629],{"class":85},[75,71964,71966,71969,71971,71973],{"class":77,"line":71965},461,[75,71967,71968],{"class":309},"        frameon",[75,71970,142],{"class":141},[75,71972,10504],{"class":145},[75,71974,41629],{"class":85},[75,71976,71978,71981,71983,71985],{"class":77,"line":71977},462,[75,71979,71980],{"class":309},"        borderaxespad",[75,71982,142],{"class":141},[75,71984,14324],{"class":145},[75,71986,41629],{"class":85},[75,71988,71990],{"class":77,"line":71989},463,[75,71991,15291],{"class":85},[75,71993,71995],{"class":77,"line":71994},464,[75,71996,112],{"emptyLinePlaceholder":111},[75,71998,72000,72002,72004],{"class":77,"line":71999},465,[75,72001,300],{"class":85},[75,72003,386],{"class":193},[75,72005,389],{"class":85},[75,72007,72009],{"class":77,"line":72008},466,[75,72010,112],{"emptyLinePlaceholder":111},[75,72012,72014,72016,72019],{"class":77,"line":72013},467,[75,72015,14737],{"class":81},[75,72017,72018],{"class":145}," BLOCH_PRINT_SUMMARY",[75,72020,5800],{"class":85},[75,72022,72024,72027,72029,72032,72034],{"class":77,"line":72023},468,[75,72025,72026],{"class":85},"    resonant_axis ",[75,72028,142],{"class":141},[75,72030,72031],{"class":85}," resonant_trajectory[",[75,72033,71340],{"class":134},[75,72035,10451],{"class":85},[75,72037,72039,72042,72044,72047,72049],{"class":77,"line":72038},469,[75,72040,72041],{"class":85},"    detuned_axis ",[75,72043,142],{"class":141},[75,72045,72046],{"class":85}," detuned_trajectory[",[75,72048,71340],{"class":134},[75,72050,10451],{"class":85},[75,72052,72054],{"class":77,"line":72053},470,[75,72055,112],{"emptyLinePlaceholder":111},[75,72057,72059,72061,72063,72066],{"class":77,"line":72058},471,[75,72060,44885],{"class":141},[75,72062,197],{"class":85},[75,72064,72065],{"class":134},"\"Bloch-sphere trajectory summary\"",[75,72067,318],{"class":85},[75,72069,72071,72073,72075,72077,72080,72083,72086,72088,72091],{"class":77,"line":72070},472,[75,72072,44885],{"class":141},[75,72074,197],{"class":85},[75,72076,5894],{"class":81},[75,72078,72079],{"class":134},"\"Natural transition frequency f0: ",[75,72081,72082],{"class":145},"{BLOCH_F0_GHZ",[75,72084,72085],{"class":81},":.6f",[75,72087,12224],{"class":145},[75,72089,72090],{"class":134}," GHz\"",[75,72092,318],{"class":85},[75,72094,72096,72098,72100,72102,72105,72107,72110,72112,72114,72116],{"class":77,"line":72095},473,[75,72097,44885],{"class":141},[75,72099,197],{"class":85},[75,72101,5894],{"class":81},[75,72103,72104],{"class":134},"\"Resonant drive frequency: ",[75,72106,12219],{"class":145},[75,72108,72109],{"class":85},"resonant_drive_ghz",[75,72111,72085],{"class":81},[75,72113,12224],{"class":145},[75,72115,72090],{"class":134},[75,72117,318],{"class":85},[75,72119,72121,72123,72125,72127,72130,72132,72135,72137,72139,72141],{"class":77,"line":72120},474,[75,72122,44885],{"class":141},[75,72124,197],{"class":85},[75,72126,5894],{"class":81},[75,72128,72129],{"class":134},"\"Detuned drive frequency: ",[75,72131,12219],{"class":145},[75,72133,72134],{"class":85},"detuned_drive_ghz",[75,72136,72085],{"class":81},[75,72138,12224],{"class":145},[75,72140,72090],{"class":134},[75,72142,318],{"class":85},[75,72144,72146,72148,72150,72152,72155,72158,72160,72162,72165],{"class":77,"line":72145},475,[75,72147,44885],{"class":141},[75,72149,197],{"class":85},[75,72151,5894],{"class":81},[75,72153,72154],{"class":134},"\"On-resonance Rabi rate: ",[75,72156,72157],{"class":145},"{BLOCH_OMEGA_ONRESONANCE_MHZ",[75,72159,44369],{"class":81},[75,72161,12224],{"class":145},[75,72163,72164],{"class":134}," MHz\"",[75,72166,318],{"class":85},[75,72168,72170,72172,72174,72176,72179,72181,72184,72186,72188,72191],{"class":77,"line":72169},476,[75,72171,44885],{"class":141},[75,72173,197],{"class":85},[75,72175,5894],{"class":81},[75,72177,72178],{"class":134},"\"Plotted duration: ",[75,72180,12219],{"class":145},[75,72182,72183],{"class":85},"bloch_duration_ns",[75,72185,44369],{"class":81},[75,72187,12224],{"class":145},[75,72189,72190],{"class":134}," ns\"",[75,72192,318],{"class":85},[75,72194,72196,72198],{"class":77,"line":72195},477,[75,72197,44885],{"class":141},[75,72199,43495],{"class":85},[75,72201,72203],{"class":77,"line":72202},478,[75,72204,72205],{"class":134},"        \"Resonant control axis n_res: \"\n",[75,72207,72209,72212,72215,72217,72220,72222,72224,72226,72228],{"class":77,"line":72208},479,[75,72210,72211],{"class":81},"        f",[75,72213,72214],{"class":134},"\"(",[75,72216,12219],{"class":145},[75,72218,72219],{"class":85},"resonant_axis[",[75,72221,5186],{"class":145},[75,72223,32873],{"class":85},[75,72225,72085],{"class":81},[75,72227,12224],{"class":145},[75,72229,72230],{"class":134},", \"\n",[75,72232,72234,72236,72238,72240,72242,72244,72246,72248,72250],{"class":77,"line":72233},480,[75,72235,72211],{"class":81},[75,72237,12216],{"class":134},[75,72239,12219],{"class":145},[75,72241,72219],{"class":85},[75,72243,7477],{"class":145},[75,72245,32873],{"class":85},[75,72247,72085],{"class":81},[75,72249,12224],{"class":145},[75,72251,72230],{"class":134},[75,72253,72255,72257,72259,72261,72263,72265,72267,72269,72271],{"class":77,"line":72254},481,[75,72256,72211],{"class":81},[75,72258,12216],{"class":134},[75,72260,12219],{"class":145},[75,72262,72219],{"class":85},[75,72264,270],{"class":145},[75,72266,32873],{"class":85},[75,72268,72085],{"class":81},[75,72270,12224],{"class":145},[75,72272,72273],{"class":134},")\"\n",[75,72275,72277],{"class":77,"line":72276},482,[75,72278,15291],{"class":85},[75,72280,72282,72284],{"class":77,"line":72281},483,[75,72283,44885],{"class":141},[75,72285,43495],{"class":85},[75,72287,72289],{"class":77,"line":72288},484,[75,72290,72291],{"class":134},"        \"Detuned control axis n_det: \"\n",[75,72293,72295,72297,72299,72301,72304,72306,72308,72310,72312],{"class":77,"line":72294},485,[75,72296,72211],{"class":81},[75,72298,72214],{"class":134},[75,72300,12219],{"class":145},[75,72302,72303],{"class":85},"detuned_axis[",[75,72305,5186],{"class":145},[75,72307,32873],{"class":85},[75,72309,72085],{"class":81},[75,72311,12224],{"class":145},[75,72313,72230],{"class":134},[75,72315,72317,72319,72321,72323,72325,72327,72329,72331,72333],{"class":77,"line":72316},486,[75,72318,72211],{"class":81},[75,72320,12216],{"class":134},[75,72322,12219],{"class":145},[75,72324,72303],{"class":85},[75,72326,7477],{"class":145},[75,72328,32873],{"class":85},[75,72330,72085],{"class":81},[75,72332,12224],{"class":145},[75,72334,72230],{"class":134},[75,72336,72338,72340,72342,72344,72346,72348,72350,72352,72354],{"class":77,"line":72337},487,[75,72339,72211],{"class":81},[75,72341,12216],{"class":134},[75,72343,12219],{"class":145},[75,72345,72303],{"class":85},[75,72347,270],{"class":145},[75,72349,32873],{"class":85},[75,72351,72085],{"class":81},[75,72353,12224],{"class":145},[75,72355,72273],{"class":134},[75,72357,72359],{"class":77,"line":72358},488,[75,72360,15291],{"class":85},[75,72362,72364,72366],{"class":77,"line":72363},489,[75,72365,44885],{"class":141},[75,72367,43495],{"class":85},[75,72369,72371],{"class":77,"line":72370},490,[75,72372,72373],{"class":134},"        \"Resonant final excited-state probability: \"\n",[75,72375,72377,72379,72381,72383,72385,72388,72391,72393,72395,72397,72399,72401],{"class":77,"line":72376},491,[75,72378,72211],{"class":81},[75,72380,12216],{"class":134},[75,72382,12219],{"class":145},[75,72384,71051],{"class":85},[75,72386,72387],{"class":134},"'excited_probability'",[75,72389,72390],{"class":85},"][",[75,72392,200],{"class":141},[75,72394,7477],{"class":145},[75,72396,32873],{"class":85},[75,72398,72085],{"class":81},[75,72400,12224],{"class":145},[75,72402,45544],{"class":134},[75,72404,72406],{"class":77,"line":72405},492,[75,72407,15291],{"class":85},[75,72409,72411,72413],{"class":77,"line":72410},493,[75,72412,44885],{"class":141},[75,72414,43495],{"class":85},[75,72416,72418],{"class":77,"line":72417},494,[75,72419,72420],{"class":134},"        \"Detuned final excited-state probability: \"\n",[75,72422,72424,72426,72428,72430,72432,72434,72436,72438,72440,72442,72444,72446],{"class":77,"line":72423},495,[75,72425,72211],{"class":81},[75,72427,12216],{"class":134},[75,72429,12219],{"class":145},[75,72431,71110],{"class":85},[75,72433,72387],{"class":134},[75,72435,72390],{"class":85},[75,72437,200],{"class":141},[75,72439,7477],{"class":145},[75,72441,32873],{"class":85},[75,72443,72085],{"class":81},[75,72445,12224],{"class":145},[75,72447,45544],{"class":134},[75,72449,72451],{"class":77,"line":72450},496,[75,72452,15291],{"class":85},[75,72454,72456,72458],{"class":77,"line":72455},497,[75,72457,44885],{"class":141},[75,72459,43495],{"class":85},[75,72461,72463],{"class":77,"line":72462},498,[75,72464,72465],{"class":134},"        \"Detuned generalized Rabi frequency: \"\n",[75,72467,72469,72471,72473,72475,72477,72480,72482,72484,72486],{"class":77,"line":72468},499,[75,72470,72211],{"class":81},[75,72472,12216],{"class":134},[75,72474,12219],{"class":145},[75,72476,71110],{"class":85},[75,72478,72479],{"class":134},"'generalized_rabi_mhz'",[75,72481,32873],{"class":85},[75,72483,72085],{"class":81},[75,72485,12224],{"class":145},[75,72487,72488],{"class":134}," MHz\"\n",[75,72490,72492],{"class":77,"line":72491},500,[75,72493,15291],{"class":85},[391,72495],{"alt":72496,"src":72497},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[65,72499,72502],{"className":72500,"code":72501,"language":34131,"meta":70},[41256],"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",[72,72503,72501],{"__ignoreMap":70},[65,72505,72507],{"className":67,"code":72506,"language":69,"meta":70,"style":70},"# --- 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",[72,72508,72509,72514,72526,72536,72546,72550,72555,72571,72588,72609,72613,72618,72623,72647,72673,72703,72721,72736,72740,72745,72768,72794,72822,72839,72853,72857,72862,72884,72888,72893,72928,72954,72958,72971,72976,72999,73019,73030,73049,73053,73057,73070,73098,73112,73125,73129,73134,73171,73186,73190,73198],{"__ignoreMap":70},[75,72510,72511],{"class":77,"line":78},[75,72512,72513],{"class":118},"# --- High-resolution 3D surface with colormap & legend  ---\n",[75,72515,72516,72518,72521,72523],{"class":77,"line":95},[75,72517,2700],{"class":81},[75,72519,72520],{"class":85}," mpl_toolkits.mplot3d ",[75,72522,82],{"class":81},[75,72524,72525],{"class":85}," Axes3D\n",[75,72527,72528,72530,72532,72534],{"class":77,"line":108},[75,72529,82],{"class":81},[75,72531,86],{"class":85},[75,72533,89],{"class":81},[75,72535,92],{"class":85},[75,72537,72538,72540,72542,72544],{"class":77,"line":115},[75,72539,82],{"class":81},[75,72541,100],{"class":85},[75,72543,89],{"class":81},[75,72545,105],{"class":85},[75,72547,72548],{"class":77,"line":122},[75,72549,112],{"emptyLinePlaceholder":111},[75,72551,72552],{"class":77,"line":128},[75,72553,72554],{"class":118},"# Controls for surface quality and appearance\n",[75,72556,72557,72560,72562,72564,72566,72568],{"class":77,"line":149},[75,72558,72559],{"class":145},"UPSAMPLE_FACTOR",[75,72561,14391],{"class":85},[75,72563,14299],{"class":141},[75,72565,14153],{"class":141},[75,72567,261],{"class":145},[75,72569,72570],{"class":118},"           # 2=~4× more faces; try 3 for even smoother\n",[75,72572,72573,72576,72578,72580,72582,72585],{"class":77,"line":154},[75,72574,72575],{"class":145},"SURFACE_CMAP",[75,72577,14391],{"class":85},[75,72579,41840],{"class":141},[75,72581,14153],{"class":141},[75,72583,72584],{"class":134}," 'inferno'",[75,72586,72587],{"class":118},"      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[75,72589,72590,72593,72595,72598,72600,72602,72604,72606],{"class":77,"line":160},[75,72591,72592],{"class":145},"Z_MIN",[75,72594,206],{"class":85},[75,72596,72597],{"class":145},"Z_MAX",[75,72599,14153],{"class":141},[75,72601,14744],{"class":145},[75,72603,206],{"class":85},[75,72605,630],{"class":145},[75,72607,72608],{"class":118},"            # clamp\u002Fnormalize expected Pe range\n",[75,72610,72611],{"class":77,"line":171},[75,72612,112],{"emptyLinePlaceholder":111},[75,72614,72615],{"class":77,"line":182},[75,72616,72617],{"class":118},"# --- Bilinear upsampling (SciPy-free) ---\n",[75,72619,72620],{"class":77,"line":222},[75,72621,72622],{"class":118},"# 1) Upsample frequency axis for each fixed τ\n",[75,72624,72625,72628,72630,72632,72634,72637,72639,72642,72644],{"class":77,"line":227},[75,72626,72627],{"class":85},"freq_hi ",[75,72629,142],{"class":141},[75,72631,190],{"class":85},[75,72633,194],{"class":193},[75,72635,72636],{"class":85},"(freq_ghz.",[75,72638,2804],{"class":193},[75,72640,72641],{"class":85},"(), freq_ghz.",[75,72643,2816],{"class":193},[75,72645,72646],{"class":85},"(),\n",[75,72648,72649,72652,72654,72656,72658,72661,72663,72665,72667,72669,72671],{"class":77,"line":233},[75,72650,72651],{"class":145},"                      UPSAMPLE_FACTOR",[75,72653,273],{"class":141},[75,72655,267],{"class":85},[75,72657,2930],{"class":141},[75,72659,72660],{"class":85},"(freq_ghz) ",[75,72662,200],{"class":141},[75,72664,1437],{"class":145},[75,72666,14860],{"class":85},[75,72668,3251],{"class":141},[75,72670,1437],{"class":145},[75,72672,318],{"class":85},[75,72674,72675,72678,72680,72682,72684,72686,72688,72691,72693,72696,72698,72701],{"class":77,"line":286},[75,72676,72677],{"class":85},"Pe_tau_f ",[75,72679,142],{"class":141},[75,72681,190],{"class":85},[75,72683,16295],{"class":193},[75,72685,16298],{"class":85},[75,72687,2930],{"class":141},[75,72689,72690],{"class":85},"(dur_ns), ",[75,72692,2930],{"class":141},[75,72694,72695],{"class":85},"(freq_hi)), ",[75,72697,12100],{"class":309},[75,72699,72700],{"class":141},"=float",[75,72702,318],{"class":85},[75,72704,72705,72707,72710,72712,72714,72716,72718],{"class":77,"line":291},[75,72706,3147],{"class":81},[75,72708,72709],{"class":85}," i ",[75,72711,3153],{"class":81},[75,72713,3245],{"class":141},[75,72715,197],{"class":85},[75,72717,2930],{"class":141},[75,72719,72720],{"class":85},"(dur_ns)):\n",[75,72722,72723,72726,72728,72730,72733],{"class":77,"line":297},[75,72724,72725],{"class":85},"    Pe_tau_f[i, :] ",[75,72727,142],{"class":141},[75,72729,190],{"class":85},[75,72731,72732],{"class":193},"interp",[75,72734,72735],{"class":85},"(freq_hi, freq_ghz, Pe[i, :])\n",[75,72737,72738],{"class":77,"line":321},[75,72739,112],{"emptyLinePlaceholder":111},[75,72741,72742],{"class":77,"line":336},[75,72743,72744],{"class":118},"# 2) Upsample τ axis for each fixed frequency\n",[75,72746,72747,72750,72752,72754,72756,72759,72761,72764,72766],{"class":77,"line":351},[75,72748,72749],{"class":85},"dur_hi ",[75,72751,142],{"class":141},[75,72753,190],{"class":85},[75,72755,194],{"class":193},[75,72757,72758],{"class":85},"(dur_ns.",[75,72760,2804],{"class":193},[75,72762,72763],{"class":85},"(), dur_ns.",[75,72765,2816],{"class":193},[75,72767,72646],{"class":85},[75,72769,72770,72773,72775,72777,72779,72782,72784,72786,72788,72790,72792],{"class":77,"line":366},[75,72771,72772],{"class":145},"                     UPSAMPLE_FACTOR",[75,72774,273],{"class":141},[75,72776,267],{"class":85},[75,72778,2930],{"class":141},[75,72780,72781],{"class":85},"(dur_ns) ",[75,72783,200],{"class":141},[75,72785,1437],{"class":145},[75,72787,14860],{"class":85},[75,72789,3251],{"class":141},[75,72791,1437],{"class":145},[75,72793,318],{"class":85},[75,72795,72796,72799,72801,72803,72805,72807,72809,72812,72814,72816,72818,72820],{"class":77,"line":381},[75,72797,72798],{"class":85},"Pe_hi ",[75,72800,142],{"class":141},[75,72802,190],{"class":85},[75,72804,16295],{"class":193},[75,72806,16298],{"class":85},[75,72808,2930],{"class":141},[75,72810,72811],{"class":85},"(dur_hi), ",[75,72813,2930],{"class":141},[75,72815,72695],{"class":85},[75,72817,12100],{"class":309},[75,72819,72700],{"class":141},[75,72821,318],{"class":85},[75,72823,72824,72826,72828,72830,72832,72834,72836],{"class":77,"line":984},[75,72825,3147],{"class":81},[75,72827,3240],{"class":85},[75,72829,3153],{"class":81},[75,72831,3245],{"class":141},[75,72833,197],{"class":85},[75,72835,2930],{"class":141},[75,72837,72838],{"class":85},"(freq_hi)):\n",[75,72840,72841,72844,72846,72848,72850],{"class":77,"line":997},[75,72842,72843],{"class":85},"    Pe_hi[:, j] ",[75,72845,142],{"class":141},[75,72847,190],{"class":85},[75,72849,72732],{"class":193},[75,72851,72852],{"class":85},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[75,72854,72855],{"class":77,"line":1325},[75,72856,112],{"emptyLinePlaceholder":111},[75,72858,72859],{"class":77,"line":1338},[75,72860,72861],{"class":118},"# Mesh for plotting\n",[75,72863,72864,72867,72869,72871,72873,72876,72878,72880,72882],{"class":77,"line":1343},[75,72865,72866],{"class":85},"F_hi, T_hi ",[75,72868,142],{"class":141},[75,72870,190],{"class":85},[75,72872,42911],{"class":193},[75,72874,72875],{"class":85},"(freq_hi, dur_hi, ",[75,72877,42917],{"class":309},[75,72879,142],{"class":141},[75,72881,66989],{"class":134},[75,72883,318],{"class":85},[75,72885,72886],{"class":77,"line":1356},[75,72887,112],{"emptyLinePlaceholder":111},[75,72889,72890],{"class":77,"line":1887},[75,72891,72892],{"class":118},"# --- Plot ---\n",[75,72894,72895,72898,72900,72902,72904,72906,72908,72910,72912,72914,72916,72918,72920,72922,72924,72926],{"class":77,"line":1900},[75,72896,72897],{"class":85},"fig ",[75,72899,142],{"class":141},[75,72901,3093],{"class":85},[75,72903,865],{"class":193},[75,72905,197],{"class":85},[75,72907,870],{"class":309},[75,72909,142],{"class":141},[75,72911,197],{"class":85},[75,72913,877],{"class":145},[75,72915,206],{"class":85},[75,72917,882],{"class":145},[75,72919,10496],{"class":85},[75,72921,15863],{"class":309},[75,72923,142],{"class":141},[75,72925,41574],{"class":145},[75,72927,318],{"class":85},[75,72929,72930,72933,72935,72937,72939,72941,72943,72945,72947,72949,72952],{"class":77,"line":1913},[75,72931,72932],{"class":85},"ax ",[75,72934,142],{"class":141},[75,72936,44104],{"class":85},[75,72938,47308],{"class":193},[75,72940,197],{"class":85},[75,72942,47313],{"class":145},[75,72944,206],{"class":85},[75,72946,47318],{"class":309},[75,72948,142],{"class":141},[75,72950,72951],{"class":134},"'3d'",[75,72953,318],{"class":85},[75,72955,72956],{"class":77,"line":2657},[75,72957,112],{"emptyLinePlaceholder":111},[75,72959,72960,72963,72965,72967,72969],{"class":77,"line":3177},[75,72961,72962],{"class":85},"surf ",[75,72964,142],{"class":141},[75,72966,43490],{"class":85},[75,72968,47470],{"class":193},[75,72970,43495],{"class":85},[75,72972,72973],{"class":77,"line":3188},[75,72974,72975],{"class":85},"    F_hi, T_hi, Pe_hi,\n",[75,72977,72978,72981,72983,72985,72987,72989,72991,72993,72996],{"class":77,"line":3203},[75,72979,72980],{"class":309},"    rstride",[75,72982,142],{"class":141},[75,72984,7477],{"class":145},[75,72986,206],{"class":85},[75,72988,47503],{"class":309},[75,72990,142],{"class":141},[75,72992,7477],{"class":145},[75,72994,72995],{"class":85},",             ",[75,72997,72998],{"class":118},"# use every row\u002Fcolumn (after upsampling)\n",[75,73000,73001,73004,73006,73008,73010,73013,73015,73017],{"class":77,"line":3218},[75,73002,73003],{"class":309},"    linewidth",[75,73005,142],{"class":141},[75,73007,5186],{"class":145},[75,73009,206],{"class":85},[75,73011,73012],{"class":309},"antialiased",[75,73014,142],{"class":141},[75,73016,10504],{"class":145},[75,73018,41629],{"class":85},[75,73020,73021,73024,73026,73028],{"class":77,"line":3224},[75,73022,73023],{"class":309},"    cmap",[75,73025,142],{"class":141},[75,73027,72575],{"class":145},[75,73029,41629],{"class":85},[75,73031,73032,73035,73037,73039,73041,73043,73045,73047],{"class":77,"line":3229},[75,73033,73034],{"class":309},"    vmin",[75,73036,142],{"class":141},[75,73038,72592],{"class":145},[75,73040,206],{"class":85},[75,73042,44081],{"class":309},[75,73044,142],{"class":141},[75,73046,72597],{"class":145},[75,73048,41629],{"class":85},[75,73050,73051],{"class":77,"line":3235},[75,73052,318],{"class":85},[75,73054,73055],{"class":77,"line":3263},[75,73056,112],{"emptyLinePlaceholder":111},[75,73058,73059,73061,73063,73065,73068],{"class":77,"line":3275},[75,73060,11108],{"class":85},[75,73062,3193],{"class":193},[75,73064,197],{"class":85},[75,73066,73067],{"class":134},"'f (GHz)'",[75,73069,318],{"class":85},[75,73071,73072,73074,73076,73078,73080,73082,73085,73088,73090,73092,73094,73096],{"class":77,"line":3280},[75,73073,11108],{"class":85},[75,73075,3208],{"class":193},[75,73077,197],{"class":85},[75,73079,15977],{"class":81},[75,73081,16007],{"class":15980},[75,73083,73084],{"class":141},"\\t",[75,73086,73087],{"class":15980},"au$ ",[75,73089,197],{"class":145},[75,73091,57982],{"class":15980},[75,73093,5175],{"class":145},[75,73095,15991],{"class":15980},[75,73097,318],{"class":85},[75,73099,73100,73102,73104,73106,73108,73110],{"class":77,"line":3290},[75,73101,11108],{"class":85},[75,73103,47643],{"class":193},[75,73105,197],{"class":85},[75,73107,15977],{"class":81},[75,73109,67597],{"class":15980},[75,73111,318],{"class":85},[75,73113,73114,73116,73118,73120,73123],{"class":77,"line":4431},[75,73115,11108],{"class":85},[75,73117,3182],{"class":193},[75,73119,197],{"class":85},[75,73121,73122],{"class":134},"'Rabi $P_e$ surface'",[75,73124,318],{"class":85},[75,73126,73127],{"class":77,"line":4454},[75,73128,112],{"emptyLinePlaceholder":111},[75,73130,73131],{"class":77,"line":4463},[75,73132,73133],{"class":118},"# Colorbar (legend for the colormap)\n",[75,73135,73136,73138,73140,73142,73144,73147,73149,73151,73153,73156,73158,73160,73162,73164,73166,73169],{"class":77,"line":4468},[75,73137,67573],{"class":85},[75,73139,142],{"class":141},[75,73141,44104],{"class":85},[75,73143,16495],{"class":193},[75,73145,73146],{"class":85},"(surf, ",[75,73148,15602],{"class":309},[75,73150,142],{"class":141},[75,73152,44116],{"class":85},[75,73154,73155],{"class":309},"shrink",[75,73157,142],{"class":141},[75,73159,11125],{"class":145},[75,73161,206],{"class":85},[75,73163,44129],{"class":309},[75,73165,142],{"class":141},[75,73167,73168],{"class":145},"0.08",[75,73170,318],{"class":85},[75,73172,73173,73176,73178,73180,73182,73184],{"class":77,"line":4477},[75,73174,73175],{"class":85},"cb.",[75,73177,44144],{"class":193},[75,73179,197],{"class":85},[75,73181,15977],{"class":81},[75,73183,67597],{"class":15980},[75,73185,318],{"class":85},[75,73187,73188],{"class":77,"line":5105},[75,73189,112],{"emptyLinePlaceholder":111},[75,73191,73192,73194,73196],{"class":77,"line":5114},[75,73193,67653],{"class":85},[75,73195,3285],{"class":193},[75,73197,389],{"class":85},[75,73199,73200,73202,73204],{"class":77,"line":10795},[75,73201,300],{"class":85},[75,73203,386],{"class":193},[75,73205,389],{"class":85},[391,73207],{"alt":73208,"src":73209},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[65,73211,73213],{"className":67,"code":73212,"language":69,"meta":70,"style":70},"# @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",[72,73214,73215,73220,73225,73240,73255,73259,73269,73279,73283,73319,73324,73349,73370,73417,73431,73454,73458,73463,73484,73494,73498,73525,73542,73568,73582,73587,73632,73649,73657,73665,73669,73674,73695,73705,73709,73735,73752,73764,73778,73824,73840,73848],{"__ignoreMap":70},[75,73216,73217],{"class":77,"line":78},[75,73218,73219],{"class":118},"# @title Cross-sections\n",[75,73221,73222],{"class":77,"line":95},[75,73223,73224],{"class":118},"# Slicing controls\n",[75,73226,73227,73229,73231,73233,73235,73237],{"class":77,"line":108},[75,73228,66106],{"class":145},[75,73230,14391],{"class":85},[75,73232,14150],{"class":141},[75,73234,14153],{"class":141},[75,73236,66601],{"class":145},[75,73238,73239],{"class":118},"  # drive frequency slice [GHz]\n",[75,73241,73242,73244,73246,73248,73250,73252],{"class":77,"line":115},[75,73243,66205],{"class":145},[75,73245,14391],{"class":85},[75,73247,14150],{"class":141},[75,73249,14153],{"class":141},[75,73251,18228],{"class":145},[75,73253,73254],{"class":118},"   # duration slice [ns]\n",[75,73256,73257],{"class":77,"line":122},[75,73258,112],{"emptyLinePlaceholder":111},[75,73260,73261,73263,73265,73267],{"class":77,"line":128},[75,73262,82],{"class":81},[75,73264,86],{"class":85},[75,73266,89],{"class":81},[75,73268,92],{"class":85},[75,73270,73271,73273,73275,73277],{"class":77,"line":149},[75,73272,82],{"class":81},[75,73274,100],{"class":85},[75,73276,89],{"class":81},[75,73278,105],{"class":85},[75,73280,73281],{"class":77,"line":154},[75,73282,112],{"emptyLinePlaceholder":111},[75,73284,73285,73287,73290,73292,73295,73297,73300,73302,73304,73306,73309,73311,73313,73315,73317],{"class":77,"line":160},[75,73286,14380],{"class":81},[75,73288,73289],{"class":193}," _nearest_index",[75,73291,197],{"class":85},[75,73293,73294],{"class":14388},"vec",[75,73296,19065],{"class":85},[75,73298,73299],{"class":14388},"value",[75,73301,14391],{"class":85},[75,73303,14150],{"class":141},[75,73305,206],{"class":85},[75,73307,73308],{"class":14388},"name",[75,73310,14391],{"class":85},[75,73312,41840],{"class":141},[75,73314,14419],{"class":85},[75,73316,14299],{"class":141},[75,73318,5800],{"class":85},[75,73320,73321],{"class":77,"line":171},[75,73322,73323],{"class":134},"    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[75,73325,73326,73329,73331,73333,73336,73338,73341,73343,73345,73347],{"class":77,"line":182},[75,73327,73328],{"class":85},"    vmin, vmax ",[75,73330,142],{"class":141},[75,73332,68874],{"class":141},[75,73334,73335],{"class":85},"(vec.",[75,73337,2804],{"class":193},[75,73339,73340],{"class":85},"()), ",[75,73342,14150],{"class":141},[75,73344,73335],{"class":85},[75,73346,2816],{"class":193},[75,73348,15965],{"class":85},[75,73350,73351,73353,73356,73358,73361,73363,73365,73367],{"class":77,"line":222},[75,73352,14433],{"class":81},[75,73354,73355],{"class":85}," value ",[75,73357,14442],{"class":141},[75,73359,73360],{"class":85}," vmin ",[75,73362,46510],{"class":81},[75,73364,73355],{"class":85},[75,73366,2907],{"class":141},[75,73368,73369],{"class":85}," vmax:\n",[75,73371,73372,73374,73376,73378,73381,73383,73385,73387,73389,73391,73393,73395,73398,73400,73402,73404,73406,73408,73410,73412,73415],{"class":77,"line":227},[75,73373,48354],{"class":141},[75,73375,197],{"class":85},[75,73377,5894],{"class":81},[75,73379,73380],{"class":134},"\"[Note] ",[75,73382,12219],{"class":145},[75,73384,73308],{"class":85},[75,73386,12224],{"class":145},[75,73388,142],{"class":134},[75,73390,12219],{"class":145},[75,73392,73299],{"class":85},[75,73394,12224],{"class":145},[75,73396,73397],{"class":134}," is outside sweep; clamping to [",[75,73399,12219],{"class":145},[75,73401,47199],{"class":85},[75,73403,12224],{"class":145},[75,73405,206],{"class":134},[75,73407,12219],{"class":145},[75,73409,44081],{"class":85},[75,73411,12224],{"class":145},[75,73413,73414],{"class":134},"].\"",[75,73416,318],{"class":85},[75,73418,73419,73422,73424,73426,73428],{"class":77,"line":233},[75,73420,73421],{"class":85},"        value ",[75,73423,142],{"class":141},[75,73425,190],{"class":85},[75,73427,16789],{"class":193},[75,73429,73430],{"class":85},"(value, vmin, vmax)\n",[75,73432,73433,73435,73437,73439,73442,73444,73446,73449,73451],{"class":77,"line":286},[75,73434,14467],{"class":81},[75,73436,10297],{"class":141},[75,73438,15804],{"class":85},[75,73440,73441],{"class":193},"argmin",[75,73443,15804],{"class":85},[75,73445,2145],{"class":193},[75,73447,73448],{"class":85},"(vec ",[75,73450,200],{"class":141},[75,73452,73453],{"class":85}," value)))\n",[75,73455,73456],{"class":77,"line":291},[75,73457,112],{"emptyLinePlaceholder":111},[75,73459,73460],{"class":77,"line":297},[75,73461,73462],{"class":118},"# --- Slice at f = F_CROSS_GHZ ---\n",[75,73464,73465,73468,73470,73472,73475,73477,73479,73482],{"class":77,"line":321},[75,73466,73467],{"class":85},"j ",[75,73469,142],{"class":141},[75,73471,73289],{"class":193},[75,73473,73474],{"class":85},"(freq_ghz, ",[75,73476,66106],{"class":145},[75,73478,206],{"class":85},[75,73480,73481],{"class":134},"\"F_CROSS_GHZ\"",[75,73483,318],{"class":85},[75,73485,73486,73489,73491],{"class":77,"line":336},[75,73487,73488],{"class":85},"Pe_vs_tau ",[75,73490,142],{"class":141},[75,73492,73493],{"class":85}," Pe[:, j]\n",[75,73495,73496],{"class":77,"line":351},[75,73497,112],{"emptyLinePlaceholder":111},[75,73499,73500,73502,73504,73506,73508,73510,73512,73514,73516,73518,73520,73523],{"class":77,"line":366},[75,73501,11073],{"class":85},[75,73503,142],{"class":141},[75,73505,3093],{"class":85},[75,73507,3096],{"class":193},[75,73509,197],{"class":85},[75,73511,870],{"class":309},[75,73513,142],{"class":141},[75,73515,197],{"class":85},[75,73517,882],{"class":145},[75,73519,206],{"class":85},[75,73521,73522],{"class":145},"3.5",[75,73524,283],{"class":85},[75,73526,73527,73529,73531,73534,73536,73538,73540],{"class":77,"line":381},[75,73528,11108],{"class":85},[75,73530,303],{"class":193},[75,73532,73533],{"class":85},"(dur_ns, Pe_vs_tau, ",[75,73535,310],{"class":309},[75,73537,142],{"class":141},[75,73539,315],{"class":134},[75,73541,318],{"class":85},[75,73543,73544,73546,73548,73550,73552,73554,73556,73558,73560,73562,73564,73566],{"class":77,"line":984},[75,73545,11108],{"class":85},[75,73547,3193],{"class":193},[75,73549,197],{"class":85},[75,73551,15977],{"class":81},[75,73553,16007],{"class":15980},[75,73555,73084],{"class":141},[75,73557,73087],{"class":15980},[75,73559,197],{"class":145},[75,73561,57982],{"class":15980},[75,73563,5175],{"class":145},[75,73565,15991],{"class":15980},[75,73567,318],{"class":85},[75,73569,73570,73572,73574,73576,73578,73580],{"class":77,"line":997},[75,73571,11108],{"class":85},[75,73573,3208],{"class":193},[75,73575,197],{"class":85},[75,73577,15977],{"class":81},[75,73579,67597],{"class":15980},[75,73581,318],{"class":85},[75,73583,73584],{"class":77,"line":1325},[75,73585,73586],{"class":118},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[75,73588,73589,73591,73593,73595,73597,73600,73602,73604,73607,73609,73612,73614,73616,73618,73620,73623,73625,73627,73630],{"class":77,"line":1338},[75,73590,11108],{"class":85},[75,73592,3182],{"class":193},[75,73594,197],{"class":85},[75,73596,15977],{"class":81},[75,73598,73599],{"class":15980},"'$P_e",[75,73601,197],{"class":145},[75,73603,73084],{"class":141},[75,73605,73606],{"class":15980},"au",[75,73608,5175],{"class":145},[75,73610,73611],{"class":15980},"$ at f = '",[75,73613,11902],{"class":141},[75,73615,44783],{"class":81},[75,73617,15991],{"class":134},[75,73619,12219],{"class":145},[75,73621,73622],{"class":85},"freq_ghz[j]",[75,73624,72085],{"class":81},[75,73626,12224],{"class":145},[75,73628,73629],{"class":134}," GHz'",[75,73631,318],{"class":85},[75,73633,73634,73636,73639,73641,73643,73645,73647],{"class":77,"line":1343},[75,73635,11108],{"class":85},[75,73637,73638],{"class":193},"set_ylim",[75,73640,197],{"class":85},[75,73642,14324],{"class":145},[75,73644,206],{"class":85},[75,73646,630],{"class":145},[75,73648,318],{"class":85},[75,73650,73651,73653,73655],{"class":77,"line":1356},[75,73652,67653],{"class":85},[75,73654,3285],{"class":193},[75,73656,389],{"class":85},[75,73658,73659,73661,73663],{"class":77,"line":1887},[75,73660,300],{"class":85},[75,73662,386],{"class":193},[75,73664,389],{"class":85},[75,73666,73667],{"class":77,"line":1900},[75,73668,112],{"emptyLinePlaceholder":111},[75,73670,73671],{"class":77,"line":1913},[75,73672,73673],{"class":118},"# --- Slice at τ = T_CROSS_NS ---\n",[75,73675,73676,73679,73681,73683,73686,73688,73690,73693],{"class":77,"line":2657},[75,73677,73678],{"class":85},"i ",[75,73680,142],{"class":141},[75,73682,73289],{"class":193},[75,73684,73685],{"class":85},"(dur_ns, ",[75,73687,66205],{"class":145},[75,73689,206],{"class":85},[75,73691,73692],{"class":134},"\"T_CROSS_NS\"",[75,73694,318],{"class":85},[75,73696,73697,73700,73702],{"class":77,"line":3177},[75,73698,73699],{"class":85},"Pe_vs_f ",[75,73701,142],{"class":141},[75,73703,73704],{"class":85}," Pe[i, :]\n",[75,73706,73707],{"class":77,"line":3188},[75,73708,112],{"emptyLinePlaceholder":111},[75,73710,73711,73713,73715,73717,73719,73721,73723,73725,73727,73729,73731,73733],{"class":77,"line":3203},[75,73712,11073],{"class":85},[75,73714,142],{"class":141},[75,73716,3093],{"class":85},[75,73718,3096],{"class":193},[75,73720,197],{"class":85},[75,73722,870],{"class":309},[75,73724,142],{"class":141},[75,73726,197],{"class":85},[75,73728,882],{"class":145},[75,73730,206],{"class":85},[75,73732,73522],{"class":145},[75,73734,283],{"class":85},[75,73736,73737,73739,73741,73744,73746,73748,73750],{"class":77,"line":3218},[75,73738,11108],{"class":85},[75,73740,303],{"class":193},[75,73742,73743],{"class":85},"(freq_ghz, Pe_vs_f, ",[75,73745,310],{"class":309},[75,73747,142],{"class":141},[75,73749,315],{"class":134},[75,73751,318],{"class":85},[75,73753,73754,73756,73758,73760,73762],{"class":77,"line":3224},[75,73755,11108],{"class":85},[75,73757,3193],{"class":193},[75,73759,197],{"class":85},[75,73761,73067],{"class":134},[75,73763,318],{"class":85},[75,73765,73766,73768,73770,73772,73774,73776],{"class":77,"line":3229},[75,73767,11108],{"class":85},[75,73769,3208],{"class":193},[75,73771,197],{"class":85},[75,73773,15977],{"class":81},[75,73775,67597],{"class":15980},[75,73777,318],{"class":85},[75,73779,73780,73782,73784,73786,73788,73790,73792,73794,73796,73799,73801,73804,73806,73808,73810,73812,73815,73817,73819,73822],{"class":77,"line":3235},[75,73781,11108],{"class":85},[75,73783,3182],{"class":193},[75,73785,197],{"class":85},[75,73787,15977],{"class":81},[75,73789,73599],{"class":15980},[75,73791,197],{"class":145},[75,73793,5894],{"class":15980},[75,73795,5175],{"class":145},[75,73797,73798],{"class":15980},"$ at $",[75,73800,73084],{"class":141},[75,73802,73803],{"class":15980},"au$ = '",[75,73805,11902],{"class":141},[75,73807,44783],{"class":81},[75,73809,15991],{"class":134},[75,73811,12219],{"class":145},[75,73813,73814],{"class":85},"dur_ns[i]",[75,73816,44369],{"class":81},[75,73818,12224],{"class":145},[75,73820,73821],{"class":134}," ns'",[75,73823,318],{"class":85},[75,73825,73826,73828,73830,73832,73834,73836,73838],{"class":77,"line":3263},[75,73827,11108],{"class":85},[75,73829,73638],{"class":193},[75,73831,197],{"class":85},[75,73833,14324],{"class":145},[75,73835,206],{"class":85},[75,73837,630],{"class":145},[75,73839,318],{"class":85},[75,73841,73842,73844,73846],{"class":77,"line":3275},[75,73843,67653],{"class":85},[75,73845,3285],{"class":193},[75,73847,389],{"class":85},[75,73849,73850,73852,73854],{"class":77,"line":3280},[75,73851,300],{"class":85},[75,73853,386],{"class":193},[75,73855,389],{"class":85},[391,73857],{"alt":73858,"src":73859},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[391,73861],{"alt":73862,"src":73863},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[65,73865,73867],{"className":67,"code":73866,"language":69,"meta":70,"style":70},"# @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",[72,73868,73869,73874,73884,73894,73898,73903,73919,73935,73951,73967,73983,73999,74003,74008,74034,74051,74063,74084,74088,74097,74104,74111,74126,74141,74156,74171,74186,74201,74206,74211,74215,74219,74224,74229,74234,74239,74244,74249,74254,74259,74263,74267,74272,74277,74281,74298,74315,74334,74338,74343,74375,74385,74427,74439,74443,74448,74457,74471,74478,74515,74529,74533,74538,74573,74599,74603,74608,74626,74644,74650,74663,74667,74672,74690,74730,74744,74750,74772,74776,74793,74822,74826,74831,74849,74863,74869,74882,74886,74893,74897,74902,74913,74918,74928,74938,74948,74958,74968,74978,74982],{"__ignoreMap":70},[75,73870,73871],{"class":77,"line":78},[75,73872,73873],{"class":118},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[75,73875,73876,73878,73880,73882],{"class":77,"line":95},[75,73877,82],{"class":81},[75,73879,86],{"class":85},[75,73881,89],{"class":81},[75,73883,92],{"class":85},[75,73885,73886,73888,73890,73892],{"class":77,"line":108},[75,73887,82],{"class":81},[75,73889,100],{"class":85},[75,73891,89],{"class":81},[75,73893,105],{"class":85},[75,73895,73896],{"class":77,"line":115},[75,73897,112],{"emptyLinePlaceholder":111},[75,73899,73900],{"class":77,"line":122},[75,73901,73902],{"class":118},"# -------- Control knobs (FFT & display) --------\n",[75,73904,73905,73907,73909,73911,73913,73916],{"class":77,"line":128},[75,73906,66322],{"class":145},[75,73908,14391],{"class":85},[75,73910,14299],{"class":141},[75,73912,14153],{"class":141},[75,73914,73915],{"class":145}," 8",[75,73917,73918],{"class":118},"                 # zero-padding factor (2, 4, 8, ...)\n",[75,73920,73921,73923,73925,73927,73929,73932],{"class":77,"line":149},[75,73922,66336],{"class":145},[75,73924,14391],{"class":85},[75,73926,41840],{"class":141},[75,73928,14153],{"class":141},[75,73930,73931],{"class":134}," \"hann\"",[75,73933,73934],{"class":118},"             # \"hann\", \"boxcar\"\n",[75,73936,73937,73939,73941,73943,73945,73948],{"class":77,"line":154},[75,73938,66351],{"class":145},[75,73940,14391],{"class":85},[75,73942,14246],{"class":141},[75,73944,14153],{"class":141},[75,73946,73947],{"class":145}," True",[75,73949,73950],{"class":118},"        # subtract mean along τ before FFT\n",[75,73952,73953,73955,73957,73959,73961,73964],{"class":77,"line":160},[75,73954,66365],{"class":145},[75,73956,14391],{"class":85},[75,73958,41840],{"class":141},[75,73960,14153],{"class":141},[75,73962,73963],{"class":134}," \"magnitude\"",[75,73965,73966],{"class":118},"      # \"magnitude\" | \"power\" | \"db\"\n",[75,73968,73969,73971,73973,73975,73977,73980],{"class":77,"line":171},[75,73970,66380],{"class":145},[75,73972,14391],{"class":85},[75,73974,41840],{"class":141},[75,73976,14153],{"class":141},[75,73978,73979],{"class":134}," \"global\"",[75,73981,73982],{"class":118},"        # \"global\" | \"per_f\" (column-wise)\n",[75,73984,73985,73987,73989,73991,73993,73996],{"class":77,"line":182},[75,73986,66395],{"class":145},[75,73988,14391],{"class":85},[75,73990,41840],{"class":141},[75,73992,14153],{"class":141},[75,73994,73995],{"class":134}," \"MHz\"",[75,73997,73998],{"class":118},"          # \"MHz\" or \"GHz\" for Fourier axis\n",[75,74000,74001],{"class":77,"line":222},[75,74002,112],{"emptyLinePlaceholder":111},[75,74004,74005],{"class":77,"line":227},[75,74006,74007],{"class":118},"# -------- Helpers --------\n",[75,74009,74010,74012,74015,74017,74019,74021,74023,74025,74028,74030,74032],{"class":77,"line":233},[75,74011,14380],{"class":81},[75,74013,74014],{"class":193}," _make_window",[75,74016,197],{"class":85},[75,74018,33004],{"class":14388},[75,74020,14391],{"class":85},[75,74022,14299],{"class":141},[75,74024,206],{"class":85},[75,74026,74027],{"class":14388},"kind",[75,74029,14391],{"class":85},[75,74031,41840],{"class":141},[75,74033,42347],{"class":85},[75,74035,74036,74038,74041,74043,74045,74047,74049],{"class":77,"line":286},[75,74037,14433],{"class":81},[75,74039,74040],{"class":85}," kind.",[75,74042,42241],{"class":193},[75,74044,2819],{"class":85},[75,74046,42294],{"class":141},[75,74048,73931],{"class":134},[75,74050,5800],{"class":85},[75,74052,74053,74055,74057,74060],{"class":77,"line":291},[75,74054,14450],{"class":81},[75,74056,190],{"class":85},[75,74058,74059],{"class":193},"hanning",[75,74061,74062],{"class":85},"(n)\n",[75,74064,74065,74067,74069,74072,74075,74077,74079,74081],{"class":77,"line":297},[75,74066,14467],{"class":81},[75,74068,190],{"class":85},[75,74070,74071],{"class":193},"ones",[75,74073,74074],{"class":85},"(n, ",[75,74076,12100],{"class":309},[75,74078,72700],{"class":141},[75,74080,216],{"class":85},[75,74082,74083],{"class":118},"# boxcar\n",[75,74085,74086],{"class":77,"line":321},[75,74087,112],{"emptyLinePlaceholder":111},[75,74089,74090,74092,74095],{"class":77,"line":336},[75,74091,14380],{"class":81},[75,74093,74094],{"class":193}," rfft_spectrogram",[75,74096,43495],{"class":85},[75,74098,74099,74102],{"class":77,"line":351},[75,74100,74101],{"class":14388},"    Pe",[75,74103,43789],{"class":85},[75,74105,74106,74109],{"class":77,"line":366},[75,74107,74108],{"class":14388},"    dur_ns",[75,74110,43789],{"class":85},[75,74112,74113,74116,74118,74120,74122,74124],{"class":77,"line":381},[75,74114,74115],{"class":14388},"    pad",[75,74117,14391],{"class":85},[75,74119,14299],{"class":141},[75,74121,14153],{"class":141},[75,74123,73915],{"class":145},[75,74125,41629],{"class":85},[75,74127,74128,74131,74133,74135,74137,74139],{"class":77,"line":984},[75,74129,74130],{"class":14388},"    window",[75,74132,14391],{"class":85},[75,74134,41840],{"class":141},[75,74136,14153],{"class":141},[75,74138,73931],{"class":134},[75,74140,41629],{"class":85},[75,74142,74143,74146,74148,74150,74152,74154],{"class":77,"line":997},[75,74144,74145],{"class":14388},"    detrend_mean",[75,74147,14391],{"class":85},[75,74149,14246],{"class":141},[75,74151,14153],{"class":141},[75,74153,73947],{"class":145},[75,74155,41629],{"class":85},[75,74157,74158,74161,74163,74165,74167,74169],{"class":77,"line":1325},[75,74159,74160],{"class":14388},"    amp_mode",[75,74162,14391],{"class":85},[75,74164,41840],{"class":141},[75,74166,14153],{"class":141},[75,74168,73963],{"class":134},[75,74170,41629],{"class":85},[75,74172,74173,74176,74178,74180,74182,74184],{"class":77,"line":1338},[75,74174,74175],{"class":14388},"    norm_mode",[75,74177,14391],{"class":85},[75,74179,41840],{"class":141},[75,74181,14153],{"class":141},[75,74183,73979],{"class":134},[75,74185,41629],{"class":85},[75,74187,74188,74191,74193,74195,74197,74199],{"class":77,"line":1343},[75,74189,74190],{"class":14388},"    freq_units",[75,74192,14391],{"class":85},[75,74194,41840],{"class":141},[75,74196,14153],{"class":141},[75,74198,73995],{"class":134},[75,74200,41629],{"class":85},[75,74202,74203],{"class":77,"line":1356},[75,74204,74205],{"class":85},") -> tuple[np.ndarray, np.ndarray]:\n",[75,74207,74208],{"class":77,"line":1887},[75,74209,74210],{"class":134},"    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[75,74212,74213],{"class":77,"line":1900},[75,74214,112],{"emptyLinePlaceholder":111},[75,74216,74217],{"class":77,"line":1913},[75,74218,66839],{"class":134},[75,74220,74221],{"class":77,"line":2657},[75,74222,74223],{"class":134},"        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[75,74225,74226],{"class":77,"line":3177},[75,74227,74228],{"class":134},"        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[75,74230,74231],{"class":77,"line":3188},[75,74232,74233],{"class":134},"        pad: zero-padding factor (≥1).\n",[75,74235,74236],{"class":77,"line":3203},[75,74237,74238],{"class":134},"        window: \"hann\" or \"boxcar\".\n",[75,74240,74241],{"class":77,"line":3218},[75,74242,74243],{"class":134},"        detrend_mean: subtract column-wise mean before FFT.\n",[75,74245,74246],{"class":77,"line":3224},[75,74247,74248],{"class":134},"        amp_mode: \"magnitude\", \"power\", or \"db\".\n",[75,74250,74251],{"class":77,"line":3229},[75,74252,74253],{"class":134},"        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[75,74255,74256],{"class":77,"line":3235},[75,74257,74258],{"class":134},"        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[75,74260,74261],{"class":77,"line":3263},[75,74262,112],{"emptyLinePlaceholder":111},[75,74264,74265],{"class":77,"line":3275},[75,74266,66873],{"class":134},[75,74268,74269],{"class":77,"line":3280},[75,74270,74271],{"class":134},"        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[75,74273,74274],{"class":77,"line":3290},[75,74275,74276],{"class":134},"        S:     2D nonnegative spectrogram (N_w, N_f).\n",[75,74278,74279],{"class":77,"line":4431},[75,74280,41832],{"class":134},[75,74282,74283,74286,74288,74290,74292,74294,74296],{"class":77,"line":4454},[75,74284,74285],{"class":85},"    Pe ",[75,74287,142],{"class":141},[75,74289,190],{"class":85},[75,74291,42229],{"class":193},[75,74293,67225],{"class":85},[75,74295,14150],{"class":141},[75,74297,318],{"class":85},[75,74299,74300,74303,74305,74307,74309,74311,74313],{"class":77,"line":4463},[75,74301,74302],{"class":85},"    dur_ns ",[75,74304,142],{"class":141},[75,74306,190],{"class":85},[75,74308,42229],{"class":193},[75,74310,73685],{"class":85},[75,74312,14150],{"class":141},[75,74314,318],{"class":85},[75,74316,74317,74319,74322,74324,74326,74328,74331],{"class":77,"line":4468},[75,74318,43899],{"class":81},[75,74320,74321],{"class":85}," Pe.shape[",[75,74323,5186],{"class":145},[75,74325,138],{"class":85},[75,74327,42294],{"class":141},[75,74329,74330],{"class":85}," dur_ns.size, ",[75,74332,74333],{"class":134},"\"Pe must be shaped (N_tau, N_f).\"\n",[75,74335,74336],{"class":77,"line":4477},[75,74337,112],{"emptyLinePlaceholder":111},[75,74339,74340],{"class":77,"line":5105},[75,74341,74342],{"class":118},"    # Uniform sampling assumption (true for np.linspace)\n",[75,74344,74345,74348,74350,74352,74355,74357,74359,74361,74364,74366,74368,74370,74372],{"class":77,"line":5114},[75,74346,74347],{"class":85},"    dt ",[75,74349,142],{"class":141},[75,74351,68874],{"class":141},[75,74353,74354],{"class":85},"(dur_ns[",[75,74356,7477],{"class":145},[75,74358,138],{"class":85},[75,74360,200],{"class":141},[75,74362,74363],{"class":85}," dur_ns[",[75,74365,5186],{"class":145},[75,74367,42286],{"class":85},[75,74369,500],{"class":141},[75,74371,11899],{"class":145},[75,74373,74374],{"class":118},"               # [s]\n",[75,74376,74377,74380,74382],{"class":77,"line":10795},[75,74378,74379],{"class":85},"    n_tau ",[75,74381,142],{"class":141},[75,74383,74384],{"class":85}," dur_ns.size\n",[75,74386,74387,74390,74392,74394,74396,74398,74401,74403,74405,74408,74410,74413,74416,74418,74420,74422,74424],{"class":77,"line":14778},[75,74388,74389],{"class":85},"    nfft ",[75,74391,142],{"class":141},[75,74393,10297],{"class":141},[75,74395,197],{"class":85},[75,74397,270],{"class":145},[75,74399,74400],{"class":141}," **",[75,74402,10297],{"class":141},[75,74404,15804],{"class":85},[75,74406,74407],{"class":193},"ceil",[75,74409,15804],{"class":85},[75,74411,74412],{"class":193},"log2",[75,74414,74415],{"class":85},"(n_tau))) ",[75,74417,500],{"class":141},[75,74419,14597],{"class":141},[75,74421,197],{"class":85},[75,74423,7477],{"class":145},[75,74425,74426],{"class":85},", pad))\n",[75,74428,74429,74432,74434,74436],{"class":77,"line":14783},[75,74430,74431],{"class":85},"    w ",[75,74433,142],{"class":141},[75,74435,74014],{"class":193},[75,74437,74438],{"class":85},"(n_tau, window)\n",[75,74440,74441],{"class":77,"line":14819},[75,74442,112],{"emptyLinePlaceholder":111},[75,74444,74445],{"class":77,"line":14825},[75,74446,74447],{"class":118},"    # Prepare output\n",[75,74449,74450,74453,74455],{"class":77,"line":14838},[75,74451,74452],{"class":85},"    S_list ",[75,74454,142],{"class":141},[75,74456,44753],{"class":85},[75,74458,74459,74461,74463,74466,74469],{"class":77,"line":14893},[75,74460,47073],{"class":85},[75,74462,142],{"class":141},[75,74464,74465],{"class":85}," Pe.",[75,74467,74468],{"class":193},"copy",[75,74470,389],{"class":85},[75,74472,74473,74475],{"class":77,"line":14951},[75,74474,14433],{"class":81},[75,74476,74477],{"class":85}," detrend_mean:\n",[75,74479,74480,74483,74485,74488,74490,74493,74496,74498,74500,74502,74504,74506,74509,74511,74513],{"class":77,"line":14959},[75,74481,74482],{"class":85},"        X ",[75,74484,142],{"class":141},[75,74486,74487],{"class":85}," X ",[75,74489,200],{"class":141},[75,74491,74492],{"class":85}," X.",[75,74494,74495],{"class":193},"mean",[75,74497,197],{"class":85},[75,74499,42331],{"class":309},[75,74501,142],{"class":141},[75,74503,5186],{"class":145},[75,74505,206],{"class":85},[75,74507,74508],{"class":309},"keepdims",[75,74510,142],{"class":141},[75,74512,10504],{"class":145},[75,74514,318],{"class":85},[75,74516,74517,74519,74522,74525,74527],{"class":77,"line":14964},[75,74518,47073],{"class":85},[75,74520,74521],{"class":141},"*=",[75,74523,74524],{"class":85}," w[:, ",[75,74526,66081],{"class":145},[75,74528,10451],{"class":85},[75,74530,74531],{"class":77,"line":14983},[75,74532,112],{"emptyLinePlaceholder":111},[75,74534,74535],{"class":77,"line":14989},[75,74536,74537],{"class":118},"    # Real FFT along τ (axis=0)\n",[75,74539,74540,74543,74545,74548,74551,74554,74556,74558,74561,74563,74565,74567,74570],{"class":77,"line":15012},[75,74541,74542],{"class":85},"    Xf ",[75,74544,142],{"class":141},[75,74546,74547],{"class":85}," np.fft.",[75,74549,74550],{"class":193},"rfft",[75,74552,74553],{"class":85},"(X, ",[75,74555,33004],{"class":309},[75,74557,142],{"class":141},[75,74559,74560],{"class":85},"nfft, ",[75,74562,42331],{"class":309},[75,74564,142],{"class":141},[75,74566,5186],{"class":145},[75,74568,74569],{"class":85},")                     ",[75,74571,74572],{"class":118},"# shape (N_w, N_f)\n",[75,74574,74575,74578,74580,74582,74585,74588,74591,74593,74596],{"class":77,"line":15017},[75,74576,74577],{"class":85},"    f_fft_hz ",[75,74579,142],{"class":141},[75,74581,74547],{"class":85},[75,74583,74584],{"class":193},"rfftfreq",[75,74586,74587],{"class":85},"(nfft, ",[75,74589,74590],{"class":309},"d",[75,74592,142],{"class":141},[75,74594,74595],{"class":85},"dt)                  ",[75,74597,74598],{"class":118},"# [Hz]\n",[75,74600,74601],{"class":77,"line":15025},[75,74602,112],{"emptyLinePlaceholder":111},[75,74604,74605],{"class":77,"line":15038},[75,74606,74607],{"class":118},"    # Amplitude selection\n",[75,74609,74610,74612,74615,74617,74619,74621,74624],{"class":77,"line":15051},[75,74611,14433],{"class":81},[75,74613,74614],{"class":85}," amp_mode.",[75,74616,42241],{"class":193},[75,74618,2819],{"class":85},[75,74620,42294],{"class":141},[75,74622,74623],{"class":134}," \"power\"",[75,74625,5800],{"class":85},[75,74627,74628,74631,74633,74635,74637,74640,74642],{"class":77,"line":15058},[75,74629,74630],{"class":85},"        S ",[75,74632,142],{"class":141},[75,74634,190],{"class":85},[75,74636,2145],{"class":193},[75,74638,74639],{"class":85},"(Xf) ",[75,74641,258],{"class":141},[75,74643,67097],{"class":145},[75,74645,74646,74648],{"class":77,"line":15069},[75,74647,69559],{"class":81},[75,74649,5800],{"class":85},[75,74651,74652,74654,74656,74658,74660],{"class":77,"line":15080},[75,74653,74630],{"class":85},[75,74655,142],{"class":141},[75,74657,190],{"class":85},[75,74659,2145],{"class":193},[75,74661,74662],{"class":85},"(Xf)\n",[75,74664,74665],{"class":77,"line":15085},[75,74666,112],{"emptyLinePlaceholder":111},[75,74668,74669],{"class":77,"line":15090},[75,74670,74671],{"class":118},"    # Normalization\n",[75,74673,74674,74676,74679,74681,74683,74685,74688],{"class":77,"line":15096},[75,74675,14433],{"class":81},[75,74677,74678],{"class":85}," norm_mode.",[75,74680,42241],{"class":193},[75,74682,2819],{"class":85},[75,74684,42294],{"class":141},[75,74686,74687],{"class":134}," \"per_f\"",[75,74689,5800],{"class":85},[75,74691,74692,74695,74697,74699,74702,74705,74707,74709,74711,74713,74715,74717,74719,74721,74723,74725,74728],{"class":77,"line":15101},[75,74693,74694],{"class":85},"        col_max ",[75,74696,142],{"class":141},[75,74698,190],{"class":85},[75,74700,74701],{"class":193},"maximum",[75,74703,74704],{"class":85},"(S.",[75,74706,2816],{"class":193},[75,74708,197],{"class":85},[75,74710,42331],{"class":309},[75,74712,142],{"class":141},[75,74714,5186],{"class":145},[75,74716,206],{"class":85},[75,74718,74508],{"class":309},[75,74720,142],{"class":141},[75,74722,10504],{"class":145},[75,74724,10496],{"class":85},[75,74726,74727],{"class":145},"1e-15",[75,74729,318],{"class":85},[75,74731,74732,74734,74736,74739,74741],{"class":77,"line":15151},[75,74733,74630],{"class":85},[75,74735,142],{"class":141},[75,74737,74738],{"class":85}," S ",[75,74740,2810],{"class":141},[75,74742,74743],{"class":85}," col_max\n",[75,74745,74746,74748],{"class":77,"line":15160},[75,74747,69559],{"class":81},[75,74749,5800],{"class":85},[75,74751,74752,74754,74756,74758,74760,74762,74764,74766,74768,74770],{"class":77,"line":15206},[75,74753,74630],{"class":85},[75,74755,142],{"class":141},[75,74757,74738],{"class":85},[75,74759,2810],{"class":141},[75,74761,14597],{"class":141},[75,74763,74704],{"class":85},[75,74765,2816],{"class":193},[75,74767,16413],{"class":85},[75,74769,74727],{"class":145},[75,74771,318],{"class":85},[75,74773,74774],{"class":77,"line":15235},[75,74775,112],{"emptyLinePlaceholder":111},[75,74777,74778,74780,74782,74784,74786,74788,74791],{"class":77,"line":15288},[75,74779,14433],{"class":81},[75,74781,74614],{"class":85},[75,74783,42241],{"class":193},[75,74785,2819],{"class":85},[75,74787,42294],{"class":141},[75,74789,74790],{"class":134}," \"db\"",[75,74792,5800],{"class":85},[75,74794,74795,74797,74799,74801,74803,74805,74808,74810,74812,74815,74817,74819],{"class":77,"line":15294},[75,74796,74630],{"class":85},[75,74798,142],{"class":141},[75,74800,18198],{"class":145},[75,74802,273],{"class":141},[75,74804,190],{"class":85},[75,74806,74807],{"class":193},"log10",[75,74809,15804],{"class":85},[75,74811,74701],{"class":193},[75,74813,74814],{"class":85},"(S, ",[75,74816,17740],{"class":145},[75,74818,48032],{"class":85},[75,74820,74821],{"class":118},"# dB, clipped floor\n",[75,74823,74824],{"class":77,"line":15311},[75,74825,112],{"emptyLinePlaceholder":111},[75,74827,74828],{"class":77,"line":15316},[75,74829,74830],{"class":118},"    # Units\n",[75,74832,74833,74835,74838,74840,74842,74844,74847],{"class":77,"line":15363},[75,74834,14433],{"class":81},[75,74836,74837],{"class":85}," freq_units.",[75,74839,44202],{"class":193},[75,74841,2819],{"class":85},[75,74843,42294],{"class":141},[75,74845,74846],{"class":134}," \"GHz\"",[75,74848,5800],{"class":85},[75,74850,74851,74854,74856,74859,74861],{"class":77,"line":15371},[75,74852,74853],{"class":85},"        f_fft ",[75,74855,142],{"class":141},[75,74857,74858],{"class":85}," f_fft_hz ",[75,74860,500],{"class":141},[75,74862,18416],{"class":145},[75,74864,74865,74867],{"class":77,"line":15419},[75,74866,69559],{"class":81},[75,74868,5800],{"class":85},[75,74870,74871,74873,74875,74877,74879],{"class":77,"line":15442},[75,74872,74853],{"class":85},[75,74874,142],{"class":141},[75,74876,74858],{"class":85},[75,74878,500],{"class":141},[75,74880,74881],{"class":145}," 1e-6\n",[75,74883,74884],{"class":77,"line":15482},[75,74885,112],{"emptyLinePlaceholder":111},[75,74887,74888,74890],{"class":77,"line":15520},[75,74889,14467],{"class":81},[75,74891,74892],{"class":85}," f_fft, S\n",[75,74894,74895],{"class":77,"line":15525},[75,74896,112],{"emptyLinePlaceholder":111},[75,74898,74899],{"class":77,"line":15540},[75,74900,74901],{"class":118},"# Compute spectrogram with current grid\n",[75,74903,74904,74907,74909,74911],{"class":77,"line":15545},[75,74905,74906],{"class":85},"f_fft_axis, S_fft ",[75,74908,142],{"class":141},[75,74910,74094],{"class":193},[75,74912,43495],{"class":85},[75,74914,74915],{"class":77,"line":15551},[75,74916,74917],{"class":85},"    Pe, dur_ns,\n",[75,74919,74920,74922,74924,74926],{"class":77,"line":15592},[75,74921,74115],{"class":309},[75,74923,142],{"class":141},[75,74925,66322],{"class":145},[75,74927,41629],{"class":85},[75,74929,74930,74932,74934,74936],{"class":77,"line":15608},[75,74931,74130],{"class":309},[75,74933,142],{"class":141},[75,74935,66336],{"class":145},[75,74937,41629],{"class":85},[75,74939,74940,74942,74944,74946],{"class":77,"line":15624},[75,74941,74145],{"class":309},[75,74943,142],{"class":141},[75,74945,66351],{"class":145},[75,74947,41629],{"class":85},[75,74949,74950,74952,74954,74956],{"class":77,"line":15652},[75,74951,74160],{"class":309},[75,74953,142],{"class":141},[75,74955,66365],{"class":145},[75,74957,41629],{"class":85},[75,74959,74960,74962,74964,74966],{"class":77,"line":15657},[75,74961,74175],{"class":309},[75,74963,142],{"class":141},[75,74965,66380],{"class":145},[75,74967,41629],{"class":85},[75,74969,74970,74972,74974,74976],{"class":77,"line":15662},[75,74971,74190],{"class":309},[75,74973,142],{"class":141},[75,74975,66395],{"class":145},[75,74977,41629],{"class":85},[75,74979,74980],{"class":77,"line":15668},[75,74981,318],{"class":85},[75,74983,74984,74986,74988,74991,74994,74997,75000,75002,75004,75007,75009,75011,75013,75015,75017],{"class":77,"line":15673},[75,74985,45904],{"class":141},[75,74987,197],{"class":85},[75,74989,74990],{"class":134},"\"FFT grid:\"",[75,74992,74993],{"class":85},", S_fft.shape, ",[75,74995,74996],{"class":134},"\"| Fourier freq range:\"",[75,74998,74999],{"class":85},", f_fft_axis[",[75,75001,5186],{"class":145},[75,75003,12097],{"class":85},[75,75005,75006],{"class":134},"\"→\"",[75,75008,74999],{"class":85},[75,75010,200],{"class":141},[75,75012,7477],{"class":145},[75,75014,12097],{"class":85},[75,75016,66395],{"class":145},[75,75018,318],{"class":85},[65,75020,75023],{"className":75021,"code":75022,"language":34131,"meta":70},[41256],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[72,75024,75022],{"__ignoreMap":70},[65,75026,75028],{"className":67,"code":75027,"language":69,"meta":70,"style":70},"# @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",[72,75029,75030,75035,75041,75054,75068,75095,75115,75147,75170,75189,75203,75209],{"__ignoreMap":70},[75,75031,75032],{"class":77,"line":78},[75,75033,75034],{"class":118},"# @title Theoretical Rabi ridge\n",[75,75036,75037,75039],{"class":77,"line":95},[75,75038,82],{"class":81},[75,75040,14088],{"class":85},[75,75042,75043,75046,75048,75050,75052],{"class":77,"line":108},[75,75044,75045],{"class":85},"two_pi ",[75,75047,142],{"class":141},[75,75049,15263],{"class":145},[75,75051,273],{"class":141},[75,75053,66896],{"class":85},[75,75055,75056,75059,75061,75064,75066],{"class":77,"line":115},[75,75057,75058],{"class":85},"f0 ",[75,75060,142],{"class":141},[75,75062,75063],{"class":145}," F0_GHZ",[75,75065,273],{"class":141},[75,75067,66911],{"class":145},[75,75069,75070,75073,75075,75077,75079,75082,75084,75087,75089,75092],{"class":77,"line":122},[75,75071,75072],{"class":85},"delta_hz ",[75,75074,142],{"class":141},[75,75076,66949],{"class":85},[75,75078,500],{"class":141},[75,75080,75081],{"class":85}," (freq_ghz",[75,75083,500],{"class":141},[75,75085,75086],{"class":145},"1e9",[75,75088,14186],{"class":141},[75,75090,75091],{"class":85}," f0)   ",[75,75093,75094],{"class":118},"# rad\u002Fs detuning\n",[75,75096,75097,75100,75102,75104,75106,75109,75111,75113],{"class":77,"line":128},[75,75098,75099],{"class":85},"omega_onres ",[75,75101,142],{"class":141},[75,75103,66949],{"class":85},[75,75105,500],{"class":141},[75,75107,75108],{"class":145}," OMEGA_ONRESONANCE_MHZ",[75,75110,273],{"class":141},[75,75112,18456],{"class":145},[75,75114,66961],{"class":118},[75,75116,75117,75120,75122,75124,75126,75129,75131,75133,75135,75138,75140,75142,75145],{"class":77,"line":149},[75,75118,75119],{"class":85},"Omega_R ",[75,75121,142],{"class":141},[75,75123,190],{"class":85},[75,75125,14668],{"class":193},[75,75127,75128],{"class":85},"(omega_onres",[75,75130,258],{"class":141},[75,75132,270],{"class":145},[75,75134,11902],{"class":141},[75,75136,75137],{"class":85}," delta_hz",[75,75139,258],{"class":141},[75,75141,270],{"class":145},[75,75143,75144],{"class":85},")     ",[75,75146,67013],{"class":118},[75,75148,75149,75152,75154,75156,75158,75160,75162,75164,75167],{"class":77,"line":154},[75,75150,75151],{"class":85},"ridge_hz ",[75,75153,142],{"class":141},[75,75155,67111],{"class":85},[75,75157,2810],{"class":141},[75,75159,267],{"class":85},[75,75161,14678],{"class":145},[75,75163,273],{"class":141},[75,75165,75166],{"class":85}," math.pi)                ",[75,75168,75169],{"class":118},"# Hz\n",[75,75171,75172,75174,75177,75179,75181,75183,75185,75187],{"class":77,"line":160},[75,75173,14737],{"class":81},[75,75175,75176],{"class":145}," FREQ_UNITS",[75,75178,6543],{"class":85},[75,75180,44202],{"class":193},[75,75182,2819],{"class":85},[75,75184,42294],{"class":141},[75,75186,74846],{"class":134},[75,75188,5800],{"class":85},[75,75190,75191,75194,75196,75199,75201],{"class":77,"line":171},[75,75192,75193],{"class":85},"    ridge_axis ",[75,75195,142],{"class":141},[75,75197,75198],{"class":85}," ridge_hz ",[75,75200,500],{"class":141},[75,75202,18416],{"class":145},[75,75204,75205,75207],{"class":77,"line":182},[75,75206,14773],{"class":81},[75,75208,5800],{"class":85},[75,75210,75211,75213,75215,75217,75219,75222],{"class":77,"line":222},[75,75212,75193],{"class":85},[75,75214,142],{"class":141},[75,75216,75198],{"class":85},[75,75218,500],{"class":141},[75,75220,75221],{"class":145}," 1e-6",[75,75223,75224],{"class":118},"  # MHz\n",[65,75226,75228],{"className":67,"code":75227,"language":69,"meta":70,"style":70},"# @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",[72,75229,75230,75235,75262,75293,75312,75354,75358,75401,75414,75435,75473,75490,75498],{"__ignoreMap":70},[75,75231,75232],{"class":77,"line":78},[75,75233,75234],{"class":118},"# @title 2D heatmap\n",[75,75236,75237,75239,75241,75243,75245,75247,75249,75251,75253,75255,75257,75260],{"class":77,"line":95},[75,75238,11073],{"class":85},[75,75240,142],{"class":141},[75,75242,3093],{"class":85},[75,75244,3096],{"class":193},[75,75246,197],{"class":85},[75,75248,870],{"class":309},[75,75250,142],{"class":141},[75,75252,197],{"class":85},[75,75254,17827],{"class":145},[75,75256,206],{"class":85},[75,75258,75259],{"class":145},"5.5",[75,75261,283],{"class":85},[75,75263,75264,75267,75269,75271,75273,75276,75278,75280,75282,75284,75286,75288,75291],{"class":77,"line":108},[75,75265,75266],{"class":85},"im ",[75,75268,142],{"class":141},[75,75270,43490],{"class":85},[75,75272,67555],{"class":193},[75,75274,75275],{"class":85},"(freq_ghz, f_fft_axis, S_fft, ",[75,75277,67561],{"class":309},[75,75279,142],{"class":141},[75,75281,16467],{"class":134},[75,75283,206],{"class":85},[75,75285,16480],{"class":309},[75,75287,142],{"class":141},[75,75289,75290],{"class":134},"\"inferno\"",[75,75292,318],{"class":85},[75,75294,75295,75297,75299,75301,75303,75305,75307,75309],{"class":77,"line":115},[75,75296,67573],{"class":85},[75,75298,142],{"class":141},[75,75300,44104],{"class":85},[75,75302,16495],{"class":193},[75,75304,44109],{"class":85},[75,75306,15602],{"class":309},[75,75308,142],{"class":141},[75,75310,75311],{"class":85},"ax)\n",[75,75313,75314,75316,75318,75320,75323,75325,75327,75330,75333,75336,75338,75340,75342,75344,75347,75349,75352],{"class":77,"line":122},[75,75315,73175],{"class":85},[75,75317,44144],{"class":193},[75,75319,197],{"class":85},[75,75321,75322],{"class":134},"\"FFT amplitude\"",[75,75324,11902],{"class":141},[75,75326,267],{"class":85},[75,75328,75329],{"class":134},"\" (dB)\"",[75,75331,75332],{"class":81}," if",[75,75334,75335],{"class":145}," AMP_MODE",[75,75337,6543],{"class":85},[75,75339,42241],{"class":193},[75,75341,44205],{"class":85},[75,75343,42294],{"class":141},[75,75345,75346],{"class":134},"\"db\"",[75,75348,16742],{"class":81},[75,75350,75351],{"class":134}," \" (arb.)\"",[75,75353,283],{"class":85},[75,75355,75356],{"class":77,"line":128},[75,75357,112],{"emptyLinePlaceholder":111},[75,75359,75360,75362,75364,75367,75369,75371,75374,75376,75378,75380,75382,75384,75386,75388,75390,75393,75396,75399],{"class":77,"line":149},[75,75361,11108],{"class":85},[75,75363,303],{"class":193},[75,75365,75366],{"class":85},"(freq_ghz, ridge_axis, ",[75,75368,15639],{"class":134},[75,75370,206],{"class":85},[75,75372,75373],{"class":309},"lw",[75,75375,142],{"class":141},[75,75377,15584],{"class":145},[75,75379,206],{"class":85},[75,75381,573],{"class":309},[75,75383,142],{"class":141},[75,75385,15977],{"class":81},[75,75387,70890],{"class":15980},[75,75389,16535],{"class":141},[75,75391,75392],{"class":15980},"mega_R\u002F2",[75,75394,75395],{"class":141},"\\p",[75,75397,75398],{"class":15980},"i$\"",[75,75400,318],{"class":85},[75,75402,75403,75405,75407,75409,75412],{"class":77,"line":154},[75,75404,11108],{"class":85},[75,75406,3193],{"class":193},[75,75408,197],{"class":85},[75,75410,75411],{"class":134},"\"Drive frequency f (GHz)\"",[75,75413,318],{"class":85},[75,75415,75416,75418,75420,75422,75424,75427,75430,75433],{"class":77,"line":160},[75,75417,11108],{"class":85},[75,75419,3208],{"class":193},[75,75421,197],{"class":85},[75,75423,5894],{"class":81},[75,75425,75426],{"class":134},"\"Fourier frequency (",[75,75428,75429],{"class":145},"{FREQ_UNITS}",[75,75431,75432],{"class":134},")\"",[75,75434,318],{"class":85},[75,75436,75437,75439,75441,75443,75446,75448,75450,75453,75456,75458,75461,75463,75466,75468,75471],{"class":77,"line":171},[75,75438,11108],{"class":85},[75,75440,3182],{"class":193},[75,75442,197],{"class":85},[75,75444,75445],{"class":134},"\"Fourier transform along pulse length: $|",[75,75447,67639],{"class":141},[75,75449,62233],{"class":134},[75,75451,75452],{"class":145},"{F}",[75,75454,75455],{"class":134},"_",[75,75457,67639],{"class":141},[75,75459,75460],{"class":134},"tau",[75,75462,67639],{"class":141},[75,75464,75465],{"class":134},"{P_e",[75,75467,67639],{"class":141},[75,75469,75470],{"class":134},"}|$\"",[75,75472,318],{"class":85},[75,75474,75475,75477,75479,75481,75483,75485,75488],{"class":77,"line":182},[75,75476,11108],{"class":85},[75,75478,5058],{"class":193},[75,75480,197],{"class":85},[75,75482,43124],{"class":309},[75,75484,142],{"class":141},[75,75486,75487],{"class":134},"\"upper right\"",[75,75489,318],{"class":85},[75,75491,75492,75494,75496],{"class":77,"line":222},[75,75493,67653],{"class":85},[75,75495,3285],{"class":193},[75,75497,389],{"class":85},[75,75499,75500,75502,75504],{"class":77,"line":227},[75,75501,300],{"class":85},[75,75503,386],{"class":193},[75,75505,389],{"class":85},[391,75507],{"alt":75508,"src":75509},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[65,75511,75513],{"className":67,"code":75512,"language":69,"meta":70,"style":70},"# --- 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",[72,75514,75515,75520,75530,75540,75550,75554,75559,75576,75593,75610,75627,75631,75636,75653,75669,75673,75678,75691,75695,75700,75738,75759,75763,75783,75796,75810,75814,75819,75824,75858,75887,75891,75896,75930,75940,75950,75954,75958,75984,75988,75993,76027,76051,76063,76073,76091,76109,76119,76135,76139,76143,76156,76175,76211,76224,76262,76266,76271,76295,76299,76307],{"__ignoreMap":70},[75,75516,75517],{"class":77,"line":78},[75,75518,75519],{"class":118},"# --- Adaptive high-res 3D surface ---\n",[75,75521,75522,75524,75526,75528],{"class":77,"line":95},[75,75523,2700],{"class":81},[75,75525,72520],{"class":85},[75,75527,82],{"class":81},[75,75529,72525],{"class":85},[75,75531,75532,75534,75536,75538],{"class":77,"line":108},[75,75533,82],{"class":81},[75,75535,86],{"class":85},[75,75537,89],{"class":81},[75,75539,92],{"class":85},[75,75541,75542,75544,75546,75548],{"class":77,"line":115},[75,75543,82],{"class":81},[75,75545,100],{"class":85},[75,75547,89],{"class":81},[75,75549,105],{"class":85},[75,75551,75552],{"class":77,"line":122},[75,75553,112],{"emptyLinePlaceholder":111},[75,75555,75556],{"class":77,"line":128},[75,75557,75558],{"class":118},"# ------ Control knobs ------\n",[75,75560,75561,75564,75566,75568,75570,75573],{"class":77,"line":149},[75,75562,75563],{"class":145},"UPSAMPLE_F_FREQ",[75,75565,14391],{"class":85},[75,75567,14299],{"class":141},[75,75569,14153],{"class":141},[75,75571,75572],{"class":145}," 3",[75,75574,75575],{"class":118},"       # try 2–3; upsample the drive-frequency axis only\n",[75,75577,75578,75581,75583,75585,75587,75590],{"class":77,"line":154},[75,75579,75580],{"class":145},"MAX_FACES",[75,75582,14391],{"class":85},[75,75584,14299],{"class":141},[75,75586,14153],{"class":141},[75,75588,75589],{"class":145}," 300_000",[75,75591,75592],{"class":118},"       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[75,75594,75595,75598,75600,75602,75604,75607],{"class":77,"line":160},[75,75596,75597],{"class":145},"MIN_ROWS",[75,75599,14391],{"class":85},[75,75601,14299],{"class":141},[75,75603,14153],{"class":141},[75,75605,75606],{"class":145}," 128",[75,75608,75609],{"class":118},"            # keep at least this many Fourier-frequency rows\n",[75,75611,75612,75615,75617,75619,75621,75624],{"class":77,"line":171},[75,75613,75614],{"class":145},"MAX_ROWS",[75,75616,14391],{"class":85},[75,75618,14299],{"class":141},[75,75620,14153],{"class":141},[75,75622,75623],{"class":145}," 1024",[75,75625,75626],{"class":118},"           # hard cap for rows to keep plotting fast\n",[75,75628,75629],{"class":77,"line":182},[75,75630,112],{"emptyLinePlaceholder":111},[75,75632,75633],{"class":77,"line":222},[75,75634,75635],{"class":118},"# Preserve global color limits so the legend stays consistent\n",[75,75637,75638,75641,75643,75645,75647,75650],{"class":77,"line":227},[75,75639,75640],{"class":85},"vmin ",[75,75642,142],{"class":141},[75,75644,68874],{"class":141},[75,75646,15804],{"class":85},[75,75648,75649],{"class":193},"nanmin",[75,75651,75652],{"class":85},"(S_fft))\n",[75,75654,75655,75658,75660,75662,75664,75667],{"class":77,"line":233},[75,75656,75657],{"class":85},"vmax ",[75,75659,142],{"class":141},[75,75661,68874],{"class":141},[75,75663,15804],{"class":85},[75,75665,75666],{"class":193},"nanmax",[75,75668,75652],{"class":85},[75,75670,75671],{"class":77,"line":286},[75,75672,112],{"emptyLinePlaceholder":111},[75,75674,75675],{"class":77,"line":291},[75,75676,75677],{"class":118},"# Shapes\n",[75,75679,75680,75683,75685,75688],{"class":77,"line":297},[75,75681,75682],{"class":85},"n_rows, n_cols ",[75,75684,142],{"class":141},[75,75686,75687],{"class":85}," S_fft.shape        ",[75,75689,75690],{"class":118},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[75,75692,75693],{"class":77,"line":321},[75,75694,112],{"emptyLinePlaceholder":111},[75,75696,75697],{"class":77,"line":336},[75,75698,75699],{"class":118},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[75,75701,75702,75705,75707,75709,75711,75713,75715,75718,75720,75722,75724,75726,75728,75730,75733,75735],{"class":77,"line":351},[75,75703,75704],{"class":85},"n_cols_up ",[75,75706,142],{"class":141},[75,75708,14762],{"class":141},[75,75710,197],{"class":85},[75,75712,75563],{"class":145},[75,75714,273],{"class":141},[75,75716,75717],{"class":85}," (n_cols ",[75,75719,200],{"class":141},[75,75721,1437],{"class":145},[75,75723,14860],{"class":85},[75,75725,3251],{"class":141},[75,75727,1437],{"class":145},[75,75729,206],{"class":85},[75,75731,75732],{"class":145},"1200",[75,75734,216],{"class":85},[75,75736,75737],{"class":118},"# sensible upper bound\n",[75,75739,75740,75742,75744,75746,75748,75750,75752,75754,75756],{"class":77,"line":366},[75,75741,72627],{"class":85},[75,75743,142],{"class":141},[75,75745,190],{"class":85},[75,75747,194],{"class":193},[75,75749,72636],{"class":85},[75,75751,2804],{"class":193},[75,75753,72641],{"class":85},[75,75755,2816],{"class":193},[75,75757,75758],{"class":85},"(), n_cols_up)\n",[75,75760,75761],{"class":77,"line":381},[75,75762,112],{"emptyLinePlaceholder":111},[75,75764,75765,75768,75770,75772,75774,75777,75779,75781],{"class":77,"line":984},[75,75766,75767],{"class":85},"S_up_f ",[75,75769,142],{"class":141},[75,75771,190],{"class":85},[75,75773,16295],{"class":193},[75,75775,75776],{"class":85},"((n_rows, n_cols_up), ",[75,75778,12100],{"class":309},[75,75780,72700],{"class":141},[75,75782,318],{"class":85},[75,75784,75785,75787,75789,75791,75793],{"class":77,"line":997},[75,75786,3147],{"class":81},[75,75788,72709],{"class":85},[75,75790,3153],{"class":81},[75,75792,3245],{"class":141},[75,75794,75795],{"class":85},"(n_rows):\n",[75,75797,75798,75801,75803,75805,75807],{"class":77,"line":1325},[75,75799,75800],{"class":85},"    S_up_f[i, :] ",[75,75802,142],{"class":141},[75,75804,190],{"class":85},[75,75806,72732],{"class":193},[75,75808,75809],{"class":85},"(freq_hi, freq_ghz, S_fft[i, :])\n",[75,75811,75812],{"class":77,"line":1338},[75,75813,112],{"emptyLinePlaceholder":111},[75,75815,75816],{"class":77,"line":1343},[75,75817,75818],{"class":118},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[75,75820,75821],{"class":77,"line":1356},[75,75822,75823],{"class":118},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[75,75825,75826,75829,75831,75833,75835,75837,75839,75841,75844,75846,75848,75850,75852,75854,75856],{"class":77,"line":1887},[75,75827,75828],{"class":85},"n_rows_cap ",[75,75830,142],{"class":141},[75,75832,10297],{"class":141},[75,75834,197],{"class":85},[75,75836,75580],{"class":145},[75,75838,264],{"class":141},[75,75840,14597],{"class":141},[75,75842,75843],{"class":85},"(n_cols_up ",[75,75845,200],{"class":141},[75,75847,1437],{"class":145},[75,75849,206],{"class":85},[75,75851,7477],{"class":145},[75,75853,14936],{"class":85},[75,75855,3251],{"class":141},[75,75857,179],{"class":145},[75,75859,75860,75863,75865,75867,75869,75871,75874,75876,75878,75880,75882,75884],{"class":77,"line":1900},[75,75861,75862],{"class":85},"n_rows_plot ",[75,75864,142],{"class":141},[75,75866,10297],{"class":141},[75,75868,15804],{"class":85},[75,75870,16789],{"class":193},[75,75872,75873],{"class":85},"(n_rows_cap, ",[75,75875,75597],{"class":145},[75,75877,206],{"class":85},[75,75879,2804],{"class":141},[75,75881,197],{"class":85},[75,75883,75614],{"class":145},[75,75885,75886],{"class":85},", n_rows)))\n",[75,75888,75889],{"class":77,"line":1913},[75,75890,112],{"emptyLinePlaceholder":111},[75,75892,75893],{"class":77,"line":2657},[75,75894,75895],{"class":118},"# Down-sample ν-axis evenly to n_rows_plot\n",[75,75897,75898,75901,75903,75905,75907,75909,75911,75914,75916,75918,75921,75924,75926,75928],{"class":77,"line":3177},[75,75899,75900],{"class":85},"row_idx ",[75,75902,142],{"class":141},[75,75904,190],{"class":85},[75,75906,194],{"class":193},[75,75908,197],{"class":85},[75,75910,5186],{"class":145},[75,75912,75913],{"class":85},", n_rows ",[75,75915,200],{"class":141},[75,75917,1437],{"class":145},[75,75919,75920],{"class":85},", n_rows_plot).",[75,75922,75923],{"class":193},"astype",[75,75925,197],{"class":85},[75,75927,14299],{"class":141},[75,75929,318],{"class":85},[75,75931,75932,75935,75937],{"class":77,"line":3188},[75,75933,75934],{"class":85},"f_fft_plot ",[75,75936,142],{"class":141},[75,75938,75939],{"class":85}," f_fft_axis[row_idx]\n",[75,75941,75942,75945,75947],{"class":77,"line":3203},[75,75943,75944],{"class":85},"S_final ",[75,75946,142],{"class":141},[75,75948,75949],{"class":85}," S_up_f[row_idx, :]\n",[75,75951,75952],{"class":77,"line":3218},[75,75953,112],{"emptyLinePlaceholder":111},[75,75955,75956],{"class":77,"line":3224},[75,75957,72861],{"class":118},[75,75959,75960,75963,75966,75968,75970,75972,75975,75977,75979,75982],{"class":77,"line":3229},[75,75961,75962],{"class":85},"F2_hi, ",[75,75964,75965],{"class":145},"FFOUR_hi",[75,75967,14153],{"class":141},[75,75969,190],{"class":85},[75,75971,42911],{"class":193},[75,75973,75974],{"class":85},"(freq_hi, f_fft_plot, ",[75,75976,42917],{"class":309},[75,75978,142],{"class":141},[75,75980,75981],{"class":134},"\"xy\"",[75,75983,318],{"class":85},[75,75985,75986],{"class":77,"line":3235},[75,75987,112],{"emptyLinePlaceholder":111},[75,75989,75990],{"class":77,"line":3263},[75,75991,75992],{"class":118},"# 3) Plot (keeps 'inferno')\n",[75,75994,75995,75997,75999,76001,76003,76005,76007,76009,76011,76013,76015,76017,76019,76021,76023,76025],{"class":77,"line":3275},[75,75996,72897],{"class":85},[75,75998,142],{"class":141},[75,76000,3093],{"class":85},[75,76002,865],{"class":193},[75,76004,197],{"class":85},[75,76006,870],{"class":309},[75,76008,142],{"class":141},[75,76010,197],{"class":85},[75,76012,877],{"class":145},[75,76014,206],{"class":85},[75,76016,882],{"class":145},[75,76018,10496],{"class":85},[75,76020,15863],{"class":309},[75,76022,142],{"class":141},[75,76024,41574],{"class":145},[75,76026,318],{"class":85},[75,76028,76029,76031,76033,76035,76037,76039,76041,76043,76045,76047,76049],{"class":77,"line":3280},[75,76030,72932],{"class":85},[75,76032,142],{"class":141},[75,76034,44104],{"class":85},[75,76036,47308],{"class":193},[75,76038,197],{"class":85},[75,76040,47313],{"class":145},[75,76042,206],{"class":85},[75,76044,47318],{"class":309},[75,76046,142],{"class":141},[75,76048,47323],{"class":134},[75,76050,318],{"class":85},[75,76052,76053,76055,76057,76059,76061],{"class":77,"line":3290},[75,76054,72962],{"class":85},[75,76056,142],{"class":141},[75,76058,43490],{"class":85},[75,76060,47470],{"class":193},[75,76062,43495],{"class":85},[75,76064,76065,76068,76070],{"class":77,"line":4431},[75,76066,76067],{"class":85},"    F2_hi, ",[75,76069,75965],{"class":145},[75,76071,76072],{"class":85},", S_final,\n",[75,76074,76075,76077,76079,76081,76083,76085,76087,76089],{"class":77,"line":4454},[75,76076,72980],{"class":309},[75,76078,142],{"class":141},[75,76080,7477],{"class":145},[75,76082,206],{"class":85},[75,76084,47503],{"class":309},[75,76086,142],{"class":141},[75,76088,7477],{"class":145},[75,76090,41629],{"class":85},[75,76092,76093,76095,76097,76099,76101,76103,76105,76107],{"class":77,"line":4463},[75,76094,73003],{"class":309},[75,76096,142],{"class":141},[75,76098,5186],{"class":145},[75,76100,206],{"class":85},[75,76102,73012],{"class":309},[75,76104,142],{"class":141},[75,76106,10504],{"class":145},[75,76108,41629],{"class":85},[75,76110,76111,76113,76115,76117],{"class":77,"line":4468},[75,76112,73023],{"class":309},[75,76114,142],{"class":141},[75,76116,75290],{"class":134},[75,76118,41629],{"class":85},[75,76120,76121,76123,76125,76128,76130,76132],{"class":77,"line":4477},[75,76122,73034],{"class":309},[75,76124,142],{"class":141},[75,76126,76127],{"class":85},"vmin, ",[75,76129,44081],{"class":309},[75,76131,142],{"class":141},[75,76133,76134],{"class":85},"vmax\n",[75,76136,76137],{"class":77,"line":5105},[75,76138,318],{"class":85},[75,76140,76141],{"class":77,"line":5114},[75,76142,112],{"emptyLinePlaceholder":111},[75,76144,76145,76147,76149,76151,76154],{"class":77,"line":10795},[75,76146,11108],{"class":85},[75,76148,3193],{"class":193},[75,76150,197],{"class":85},[75,76152,76153],{"class":134},"\"f (GHz)\"",[75,76155,318],{"class":85},[75,76157,76158,76160,76162,76164,76166,76169,76171,76173],{"class":77,"line":14778},[75,76159,11108],{"class":85},[75,76161,3208],{"class":193},[75,76163,197],{"class":85},[75,76165,5894],{"class":81},[75,76167,76168],{"class":134},"\"Fourier freq (",[75,76170,75429],{"class":145},[75,76172,75432],{"class":134},[75,76174,318],{"class":85},[75,76176,76177,76179,76181,76183,76185,76187,76189,76191,76193,76195,76197,76199,76201,76203,76205,76207,76209],{"class":77,"line":14783},[75,76178,11108],{"class":85},[75,76180,47643],{"class":193},[75,76182,197],{"class":85},[75,76184,75322],{"class":134},[75,76186,11902],{"class":141},[75,76188,267],{"class":85},[75,76190,75329],{"class":134},[75,76192,75332],{"class":81},[75,76194,75335],{"class":145},[75,76196,6543],{"class":85},[75,76198,42241],{"class":193},[75,76200,2819],{"class":85},[75,76202,42294],{"class":141},[75,76204,74790],{"class":134},[75,76206,16742],{"class":81},[75,76208,75351],{"class":134},[75,76210,283],{"class":85},[75,76212,76213,76215,76217,76219,76222],{"class":77,"line":14819},[75,76214,11108],{"class":85},[75,76216,3182],{"class":193},[75,76218,197],{"class":85},[75,76220,76221],{"class":134},"\"Rabi FFT surface (adaptive resolution)\"",[75,76223,318],{"class":85},[75,76225,76226,76228,76230,76232,76234,76236,76238,76240,76242,76244,76246,76248,76250,76252,76254,76256,76258,76260],{"class":77,"line":14825},[75,76227,67653],{"class":85},[75,76229,16495],{"class":193},[75,76231,73146],{"class":85},[75,76233,15602],{"class":309},[75,76235,142],{"class":141},[75,76237,44116],{"class":85},[75,76239,73155],{"class":309},[75,76241,142],{"class":141},[75,76243,11125],{"class":145},[75,76245,206],{"class":85},[75,76247,44129],{"class":309},[75,76249,142],{"class":141},[75,76251,73168],{"class":145},[75,76253,206],{"class":85},[75,76255,573],{"class":309},[75,76257,142],{"class":141},[75,76259,3213],{"class":134},[75,76261,318],{"class":85},[75,76263,76264],{"class":77,"line":14838},[75,76265,112],{"emptyLinePlaceholder":111},[75,76267,76268],{"class":77,"line":14893},[75,76269,76270],{"class":118},"# A clearer default view\n",[75,76272,76273,76275,76277,76279,76281,76283,76285,76287,76289,76291,76293],{"class":77,"line":14951},[75,76274,11108],{"class":85},[75,76276,47332],{"class":193},[75,76278,197],{"class":85},[75,76280,47337],{"class":309},[75,76282,142],{"class":141},[75,76284,57891],{"class":145},[75,76286,206],{"class":85},[75,76288,47345],{"class":309},[75,76290,48477],{"class":141},[75,76292,44441],{"class":145},[75,76294,318],{"class":85},[75,76296,76297],{"class":77,"line":14959},[75,76298,112],{"emptyLinePlaceholder":111},[75,76300,76301,76303,76305],{"class":77,"line":14964},[75,76302,67653],{"class":85},[75,76304,3285],{"class":193},[75,76306,389],{"class":85},[75,76308,76309,76311,76313],{"class":77,"line":14983},[75,76310,300],{"class":85},[75,76312,386],{"class":193},[75,76314,389],{"class":85},[391,76316],{"alt":76317,"src":76318},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[65,76320,76322],{"className":67,"code":76321,"language":69,"meta":70,"style":70},"# @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",[72,76323,76324,76329,76352,76384,76388,76393,76417,76445,76462,76506,76524,76560,76587,76595,76608,76612,76617,76643,76669,76686,76698,76734,76763],{"__ignoreMap":70},[75,76325,76326],{"class":77,"line":78},[75,76327,76328],{"class":118},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[75,76330,76331,76334,76336,76338,76340,76342,76344,76346,76349],{"class":77,"line":95},[75,76332,76333],{"class":145},"F_STAR_GHZ",[75,76335,14391],{"class":85},[75,76337,14150],{"class":141},[75,76339,14153],{"class":141},[75,76341,68874],{"class":141},[75,76343,197],{"class":85},[75,76345,65846],{"class":145},[75,76347,76348],{"class":85},")   ",[75,76350,76351],{"class":118},"# slice near resonance\n",[75,76353,76354,76357,76359,76361,76364,76366,76368,76370,76373,76375,76378,76381],{"class":77,"line":108},[75,76355,76356],{"class":145},"NU_STAR",[75,76358,14153],{"class":141},[75,76360,68874],{"class":141},[75,76362,76363],{"class":85},"(ridge_axis[np.",[75,76365,73441],{"class":193},[75,76367,15804],{"class":85},[75,76369,2145],{"class":193},[75,76371,76372],{"class":85},"(freq_ghz ",[75,76374,200],{"class":141},[75,76376,76377],{"class":145}," F_STAR_GHZ",[75,76379,76380],{"class":85},"))])  ",[75,76382,76383],{"class":118},"# near Ω_R\u002F2π\n",[75,76385,76386],{"class":77,"line":115},[75,76387,112],{"emptyLinePlaceholder":111},[75,76389,76390],{"class":77,"line":122},[75,76391,76392],{"class":118},"# Slice at f = F_STAR_GHZ\n",[75,76394,76395,76397,76399,76401,76403,76405,76407,76409,76411,76413,76415],{"class":77,"line":128},[75,76396,73467],{"class":85},[75,76398,142],{"class":141},[75,76400,10297],{"class":141},[75,76402,15804],{"class":85},[75,76404,73441],{"class":193},[75,76406,15804],{"class":85},[75,76408,2145],{"class":193},[75,76410,76372],{"class":85},[75,76412,200],{"class":141},[75,76414,76377],{"class":145},[75,76416,48568],{"class":85},[75,76418,76419,76421,76423,76425,76427,76429,76431,76433,76435,76438,76440,76443],{"class":77,"line":149},[75,76420,11073],{"class":85},[75,76422,142],{"class":141},[75,76424,3093],{"class":85},[75,76426,3096],{"class":193},[75,76428,197],{"class":85},[75,76430,870],{"class":309},[75,76432,142],{"class":141},[75,76434,197],{"class":85},[75,76436,76437],{"class":145},"6.0",[75,76439,206],{"class":85},[75,76441,76442],{"class":145},"3.8",[75,76444,283],{"class":85},[75,76446,76447,76449,76451,76454,76456,76458,76460],{"class":77,"line":154},[75,76448,11108],{"class":85},[75,76450,303],{"class":193},[75,76452,76453],{"class":85},"(f_fft_axis, S_fft[:, j], ",[75,76455,310],{"class":309},[75,76457,142],{"class":141},[75,76459,315],{"class":134},[75,76461,318],{"class":85},[75,76463,76464,76466,76468,76471,76474,76476,76478,76480,76482,76484,76486,76488,76490,76492,76494,76496,76498,76500,76502,76504],{"class":77,"line":160},[75,76465,11108],{"class":85},[75,76467,11213],{"class":193},[75,76469,76470],{"class":85},"(ridge_axis[j], ",[75,76472,76473],{"class":309},"ls",[75,76475,142],{"class":141},[75,76477,15639],{"class":134},[75,76479,206],{"class":85},[75,76481,75373],{"class":309},[75,76483,142],{"class":141},[75,76485,630],{"class":145},[75,76487,206],{"class":85},[75,76489,573],{"class":309},[75,76491,142],{"class":141},[75,76493,15977],{"class":81},[75,76495,70890],{"class":15980},[75,76497,16535],{"class":141},[75,76499,75392],{"class":15980},[75,76501,75395],{"class":141},[75,76503,75398],{"class":15980},[75,76505,318],{"class":85},[75,76507,76508,76510,76512,76514,76516,76518,76520,76522],{"class":77,"line":171},[75,76509,11108],{"class":85},[75,76511,3193],{"class":193},[75,76513,197],{"class":85},[75,76515,5894],{"class":81},[75,76517,75426],{"class":134},[75,76519,75429],{"class":145},[75,76521,75432],{"class":134},[75,76523,318],{"class":85},[75,76525,76526,76528,76530,76532,76534,76536,76538,76540,76542,76544,76546,76548,76550,76552,76554,76556,76558],{"class":77,"line":182},[75,76527,11108],{"class":85},[75,76529,3208],{"class":193},[75,76531,197],{"class":85},[75,76533,75322],{"class":134},[75,76535,11902],{"class":141},[75,76537,267],{"class":85},[75,76539,75329],{"class":134},[75,76541,75332],{"class":81},[75,76543,75335],{"class":145},[75,76545,6543],{"class":85},[75,76547,42241],{"class":193},[75,76549,44205],{"class":85},[75,76551,42294],{"class":141},[75,76553,75346],{"class":134},[75,76555,16742],{"class":81},[75,76557,75351],{"class":134},[75,76559,283],{"class":85},[75,76561,76562,76564,76566,76568,76571,76574,76576,76578,76580,76582,76585],{"class":77,"line":222},[75,76563,11108],{"class":85},[75,76565,3182],{"class":193},[75,76567,197],{"class":85},[75,76569,76570],{"class":81},"fr",[75,76572,76573],{"class":134},"\"Slice at $f=",[75,76575,12219],{"class":145},[75,76577,73622],{"class":85},[75,76579,72085],{"class":81},[75,76581,12224],{"class":145},[75,76583,76584],{"class":134},"$ GHz\"",[75,76586,318],{"class":85},[75,76588,76589,76591,76593],{"class":77,"line":227},[75,76590,11108],{"class":85},[75,76592,5058],{"class":193},[75,76594,389],{"class":85},[75,76596,76597,76599,76601,76604,76606],{"class":77,"line":233},[75,76598,67653],{"class":85},[75,76600,3285],{"class":193},[75,76602,76603],{"class":85},"(); plt.",[75,76605,386],{"class":193},[75,76607,389],{"class":85},[75,76609,76610],{"class":77,"line":286},[75,76611,112],{"emptyLinePlaceholder":111},[75,76613,76614],{"class":77,"line":291},[75,76615,76616],{"class":118},"# Slice at ν = NU_STAR\n",[75,76618,76619,76621,76623,76625,76627,76629,76631,76633,76636,76638,76641],{"class":77,"line":297},[75,76620,73678],{"class":85},[75,76622,142],{"class":141},[75,76624,10297],{"class":141},[75,76626,15804],{"class":85},[75,76628,73441],{"class":193},[75,76630,15804],{"class":85},[75,76632,2145],{"class":193},[75,76634,76635],{"class":85},"(f_fft_axis ",[75,76637,200],{"class":141},[75,76639,76640],{"class":145}," NU_STAR",[75,76642,48568],{"class":85},[75,76644,76645,76647,76649,76651,76653,76655,76657,76659,76661,76663,76665,76667],{"class":77,"line":321},[75,76646,11073],{"class":85},[75,76648,142],{"class":141},[75,76650,3093],{"class":85},[75,76652,3096],{"class":193},[75,76654,197],{"class":85},[75,76656,870],{"class":309},[75,76658,142],{"class":141},[75,76660,197],{"class":85},[75,76662,76437],{"class":145},[75,76664,206],{"class":85},[75,76666,76442],{"class":145},[75,76668,283],{"class":85},[75,76670,76671,76673,76675,76678,76680,76682,76684],{"class":77,"line":336},[75,76672,11108],{"class":85},[75,76674,303],{"class":193},[75,76676,76677],{"class":85},"(freq_ghz, S_fft[i, :], ",[75,76679,310],{"class":309},[75,76681,142],{"class":141},[75,76683,315],{"class":134},[75,76685,318],{"class":85},[75,76687,76688,76690,76692,76694,76696],{"class":77,"line":351},[75,76689,11108],{"class":85},[75,76691,3193],{"class":193},[75,76693,197],{"class":85},[75,76695,75411],{"class":134},[75,76697,318],{"class":85},[75,76699,76700,76702,76704,76706,76708,76710,76712,76714,76716,76718,76720,76722,76724,76726,76728,76730,76732],{"class":77,"line":366},[75,76701,11108],{"class":85},[75,76703,3208],{"class":193},[75,76705,197],{"class":85},[75,76707,75322],{"class":134},[75,76709,11902],{"class":141},[75,76711,267],{"class":85},[75,76713,75329],{"class":134},[75,76715,75332],{"class":81},[75,76717,75335],{"class":145},[75,76719,6543],{"class":85},[75,76721,42241],{"class":193},[75,76723,44205],{"class":85},[75,76725,42294],{"class":141},[75,76727,75346],{"class":134},[75,76729,16742],{"class":81},[75,76731,75351],{"class":134},[75,76733,283],{"class":85},[75,76735,76736,76738,76740,76742,76744,76747,76749,76752,76754,76756,76759,76761],{"class":77,"line":381},[75,76737,11108],{"class":85},[75,76739,3182],{"class":193},[75,76741,197],{"class":85},[75,76743,76570],{"class":81},[75,76745,76746],{"class":134},"\"Slice at Fourier freq ≈ ",[75,76748,12219],{"class":145},[75,76750,76751],{"class":85},"f_fft_axis[i]",[75,76753,44369],{"class":81},[75,76755,12224],{"class":145},[75,76757,76758],{"class":145}," {FREQ_UNITS}",[75,76760,12216],{"class":134},[75,76762,318],{"class":85},[75,76764,76765,76767,76769,76771,76773],{"class":77,"line":984},[75,76766,67653],{"class":85},[75,76768,3285],{"class":193},[75,76770,76603],{"class":85},[75,76772,386],{"class":193},[75,76774,389],{"class":85},[391,76776],{"alt":76777,"src":76778},"Output 8 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-08.webp",[391,76780],{"alt":76781,"src":76782},"Output 9 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-09.webp",[12425,76784,76786],{"id":76785},"run-it-yourself","Run it yourself",[12,76788,76789,76790,76792,76793,6543],{},"Get ",[19,76791,22],{"href":50990}," or ",[19,76794,76796],{"href":76795},"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",[12364,76798,76799],{},"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: 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.sVyAn{--shiki-default:#E06C75}",{"title":70,"searchDepth":95,"depth":95,"links":76801},[76802,76803,76804,76805,76806,76807,76808,76809,76810,76811,76812,76813,76814,76815,76816],{"id":51168,"depth":95,"text":51169},{"id":52640,"depth":95,"text":52641},{"id":53429,"depth":95,"text":53430},{"id":54769,"depth":95,"text":54770},{"id":55898,"depth":95,"text":55899},{"id":58379,"depth":95,"text":58380},{"id":58896,"depth":95,"text":58897},{"id":59995,"depth":95,"text":59996},{"id":61600,"depth":95,"text":61601},{"id":61973,"depth":95,"text":61974},{"id":62967,"depth":95,"text":62968},{"id":64145,"depth":95,"text":64146},{"id":65324,"depth":95,"text":65325},{"id":65819,"depth":95,"text":65820},{"id":76785,"depth":95,"text":76786},[12374,12375,12376,76818],"Rabi oscillations",[76820],{"username":7,"name":12380,"role":12381,"bio":12382,"links":76821},[76822,76823],{"label":12385,"href":12386},{"label":12388,"href":50990},{"username":7,"name":12380,"role":12390},"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":50980,"description":76825},"blog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[12406,12407],"IAzB0ZaRHqWwL_egvokIhbtpcSM2VZ_XYnWDf-q-QyM",{"id":76837,"title":76838,"authors":76839,"body":76840,"breadcrumb":81408,"builders":81410,"byline":81415,"challenge":12391,"courseAuthor":12391,"courseLead":12391,"dek":12391,"description":81416,"draft":12393,"extension":12394,"eyebrow":12395,"featured":12393,"finish":12391,"fork":12391,"hero":12391,"heroAlt":12391,"heroCta":12391,"heroImage":12391,"homepageFeatured":12393,"kind":12396,"lessonCount":12391,"meta":81417,"navigation":111,"newsItems":12391,"next":12391,"ogImage":81418,"order":12391,"outcomes":12391,"path":81419,"publishDate":76829,"readingTime":81420,"related":81421,"relatedProjects":12391,"seo":81422,"stem":81423,"tags":81424,"track":12391,"trackName":12391,"__hash__":81425},"blog\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications.md","A Sampling-Budget & Bit-String Analogy for Qubit Classifications",[7],{"type":9,"value":76841,"toc":81404},[76842,76849,81245,81251,81254,81257,81263,81269,81271,81273,81392,81394,81401],[12,76843,76844],{},[15,76845,17,76846,12423],{},[19,76847,22],{"href":76848},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[65,76850,76852],{"className":67,"code":76851,"language":69,"meta":70,"style":70},"# 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",[72,76853,76854,76859,76864,76869,76874,76879,76884,76889,76894,76899,76904,76908,76915,76921,76931,76942,76946,76956,76966,76976,76987,76999,77003,77015,77019,77024,77028,77074,77078,77083,77088,77092,77105,77119,77134,77157,77193,77258,77291,77295,77299,77346,77350,77355,77360,77364,77388,77408,77421,77427,77456,77478,77482,77486,77491,77495,77511,77520,77528,77538,77545,77549,77553,77565,77572,77579,77586,77591,77596,77602,77609,77616,77621,77625,77631,77638,77645,77650,77654,77660,77667,77674,77679,77683,77689,77696,77703,77708,77712,77716,77720,77725,77729,77741,77753,77765,77792,77800,77812,77822,77827,77834,77844,77848,77852,77856,77880,77884,77888,77893,77897,77929,77933,77938,77942,77975,77979,77983,78014,78018,78023,78027,78038,78054,78071,78104,78116,78120,78129,78137,78145,78153,78161,78169,78177,78185,78199,78212,78225,78229,78240,78278,78285,78289,78293,78332,78336,78341,78345,78362,78366,78370,78375,78379,78394,78403,78407,78418,78432,78454,78464,78489,78493,78507,78511,78516,78538,78558,78581,78595,78613,78630,78634,78638,78643,78647,78652,78669,78706,78737,78741,78751,78769,78773,78778,78803,78827,78831,78842,78846,78851,78855,78860,78875,78897,78914,78935,78939,78944,78949,78954,78981,79007,79031,79036,79040,79068,79121,79157,79170,79192,79200,79210,79238,79246,79250,79255,79277,79286,79306,79330,79334,79356,79381,79436,79440,79479,79492,79513,79527,79544,79552,79562,79587,79595,79599,79610,79614,79619,79623,79639,79651,79670,79690,79713,79735,79754,79771,79776,79780,79784,79789,79793,79805,79810,79814,79845,79849,79930,79934,79938,79949,79954,79959,79964,79969,79973,79981,79988,79999,80006,80013,80017,80028,80033,80038,80043,80048,80052,80066,80071,80076,80081,80086,80091,80095,80099,80111,80115,80119,80130,80135,80140,80145,80150,80154,80159,80164,80168,80188,80200,80210,80220,80240,80300,80331,80335,80350,80366,80416,80420,80435,80446,80457,80469,80474,80489,80504,80509,80513,80518,80536,80541,80545,80550,80558,80570,80575,80599,80603,80618,80645,80649,80654,80665,80692,80696,80701,80714,80722,80733,80741,80745,80763,80785,80797,80801,80806,80811,80827,80835,80849,80857,80883,80887,80904,80925,80936,80948,80959,80971,80975,80982,80989,80993,80997,81012,81026,81037,81047,81056,81060,81064,81074,81101,81112,81116,81121,81144,81155,81170,81192,81214],{"__ignoreMap":70},[75,76855,76856],{"class":77,"line":78},[75,76857,76858],{"class":118},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[75,76860,76861],{"class":77,"line":95},[75,76862,76863],{"class":118},"# generates tables and figures, so you can iterate further.\n",[75,76865,76866],{"class":77,"line":108},[75,76867,76868],{"class":118},"#\n",[75,76870,76871],{"class":77,"line":115},[75,76872,76873],{"class":118},"# Contents created in the current directory:\n",[75,76875,76876],{"class":77,"line":122},[75,76877,76878],{"class":118},"#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n",[75,76880,76881],{"class":77,"line":128},[75,76882,76883],{"class":118},"#  - summary.csv                                 (summary table)\n",[75,76885,76886],{"class":77,"line":149},[75,76887,76888],{"class":118},"#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n",[75,76890,76891],{"class":77,"line":154},[75,76892,76893],{"class":118},"#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n",[75,76895,76896],{"class":77,"line":160},[75,76897,76898],{"class":118},"#  - fig_required_shots.png                      (required shots for ±ε)\n",[75,76900,76901],{"class":77,"line":171},[75,76902,76903],{"class":118},"#  - simulation_config.json                      (run parameters)\n",[75,76905,76906],{"class":77,"line":182},[75,76907,112],{"emptyLinePlaceholder":111},[75,76909,76910,76912],{"class":77,"line":222},[75,76911,82],{"class":81},[75,76913,76914],{"class":85}," json\n",[75,76916,76917,76919],{"class":77,"line":227},[75,76918,82],{"class":81},[75,76920,14088],{"class":85},[75,76922,76923,76925,76927,76929],{"class":77,"line":233},[75,76924,2700],{"class":81},[75,76926,14064],{"class":85},[75,76928,82],{"class":81},[75,76930,14069],{"class":85},[75,76932,76933,76935,76937,76939],{"class":77,"line":286},[75,76934,2700],{"class":81},[75,76936,14076],{"class":85},[75,76938,82],{"class":81},[75,76940,76941],{"class":85}," List, Dict, Tuple\n",[75,76943,76944],{"class":77,"line":291},[75,76945,112],{"emptyLinePlaceholder":111},[75,76947,76948,76950,76952,76954],{"class":77,"line":297},[75,76949,82],{"class":81},[75,76951,86],{"class":85},[75,76953,89],{"class":81},[75,76955,92],{"class":85},[75,76957,76958,76960,76962,76964],{"class":77,"line":321},[75,76959,82],{"class":81},[75,76961,66490],{"class":85},[75,76963,89],{"class":81},[75,76965,66495],{"class":85},[75,76967,76968,76970,76972,76974],{"class":77,"line":336},[75,76969,82],{"class":81},[75,76971,100],{"class":85},[75,76973,89],{"class":81},[75,76975,105],{"class":85},[75,76977,76978,76980,76982,76984],{"class":77,"line":351},[75,76979,2700],{"class":81},[75,76981,41727],{"class":85},[75,76983,82],{"class":81},[75,76985,76986],{"class":85}," display, Markdown\n",[75,76988,76989,76991,76994,76996],{"class":77,"line":366},[75,76990,82],{"class":81},[75,76992,76993],{"class":85}," nbformat ",[75,76995,89],{"class":81},[75,76997,76998],{"class":85}," nbf\n",[75,77000,77001],{"class":77,"line":381},[75,77002,112],{"emptyLinePlaceholder":111},[75,77004,77005,77007,77009,77011,77013],{"class":77,"line":984},[75,77006,131],{"class":85},[75,77008,135],{"class":134},[75,77010,138],{"class":85},[75,77012,142],{"class":141},[75,77014,146],{"class":145},[75,77016,77017],{"class":77,"line":997},[75,77018,112],{"emptyLinePlaceholder":111},[75,77020,77021],{"class":77,"line":1325},[75,77022,77023],{"class":118},"# ---------- Helper math ----------\n",[75,77025,77026],{"class":77,"line":1338},[75,77027,112],{"emptyLinePlaceholder":111},[75,77029,77030,77032,77035,77037,77039,77041,77043,77045,77047,77049,77051,77053,77055,77057,77059,77061,77064,77066,77068,77070,77072],{"class":77,"line":1343},[75,77031,14380],{"class":81},[75,77033,77034],{"class":193}," wilson_ci",[75,77036,197],{"class":85},[75,77038,44273],{"class":14388},[75,77040,14391],{"class":85},[75,77042,14299],{"class":141},[75,77044,206],{"class":85},[75,77046,33004],{"class":14388},[75,77048,14391],{"class":85},[75,77050,14299],{"class":141},[75,77052,206],{"class":85},[75,77054,31538],{"class":14388},[75,77056,14391],{"class":85},[75,77058,14150],{"class":141},[75,77060,14153],{"class":141},[75,77062,77063],{"class":145}," 1.96",[75,77065,14807],{"class":85},[75,77067,14150],{"class":141},[75,77069,206],{"class":85},[75,77071,14150],{"class":141},[75,77073,14816],{"class":85},[75,77075,77076],{"class":77,"line":1356},[75,77077,41832],{"class":134},[75,77079,77080],{"class":77,"line":1887},[75,77081,77082],{"class":134},"    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[75,77084,77085],{"class":77,"line":1900},[75,77086,77087],{"class":134},"    Returns (low, high). Works for edge cases k=0 or k=n.\n",[75,77089,77090],{"class":77,"line":1913},[75,77091,41832],{"class":134},[75,77093,77094,77096,77099,77101,77103],{"class":77,"line":2657},[75,77095,14433],{"class":81},[75,77097,77098],{"class":85}," n ",[75,77100,42294],{"class":141},[75,77102,10251],{"class":145},[75,77104,5800],{"class":85},[75,77106,77107,77109,77111,77113,77115,77117],{"class":77,"line":3177},[75,77108,14450],{"class":81},[75,77110,267],{"class":85},[75,77112,14324],{"class":145},[75,77114,206],{"class":85},[75,77116,630],{"class":145},[75,77118,318],{"class":85},[75,77120,77121,77124,77126,77129,77131],{"class":77,"line":3188},[75,77122,77123],{"class":85},"    p_hat ",[75,77125,142],{"class":141},[75,77127,77128],{"class":85}," k ",[75,77130,2810],{"class":141},[75,77132,77133],{"class":85}," n\n",[75,77135,77136,77138,77140,77142,77144,77147,77149,77151,77153,77155],{"class":77,"line":3203},[75,77137,17323],{"class":85},[75,77139,142],{"class":141},[75,77141,14846],{"class":145},[75,77143,11902],{"class":141},[75,77145,77146],{"class":85}," (z ",[75,77148,258],{"class":141},[75,77150,261],{"class":145},[75,77152,14860],{"class":85},[75,77154,2810],{"class":141},[75,77156,77133],{"class":85},[75,77158,77159,77162,77164,77167,77169,77171,77173,77175,77177,77179,77181,77183,77185,77188,77190],{"class":77,"line":3218},[75,77160,77161],{"class":85},"    center ",[75,77163,142],{"class":141},[75,77165,77166],{"class":85}," (p_hat ",[75,77168,3251],{"class":141},[75,77170,77146],{"class":85},[75,77172,258],{"class":141},[75,77174,261],{"class":145},[75,77176,14860],{"class":85},[75,77178,2810],{"class":141},[75,77180,267],{"class":85},[75,77182,270],{"class":145},[75,77184,273],{"class":141},[75,77186,77187],{"class":85}," n)) ",[75,77189,2810],{"class":141},[75,77191,77192],{"class":85}," denom\n",[75,77194,77195,77198,77200,77202,77204,77206,77208,77211,77213,77215,77217,77219,77222,77224,77227,77229,77231,77233,77235,77237,77239,77241,77243,77245,77247,77249,77251,77254,77256],{"class":77,"line":3224},[75,77196,77197],{"class":85},"    span ",[75,77199,142],{"class":141},[75,77201,77146],{"class":85},[75,77203,500],{"class":141},[75,77205,14665],{"class":85},[75,77207,14668],{"class":193},[75,77209,77210],{"class":85},"((p_hat ",[75,77212,500],{"class":141},[75,77214,267],{"class":85},[75,77216,7477],{"class":145},[75,77218,14186],{"class":141},[75,77220,77221],{"class":85}," p_hat) ",[75,77223,2810],{"class":141},[75,77225,77226],{"class":85}," n) ",[75,77228,3251],{"class":141},[75,77230,77146],{"class":85},[75,77232,258],{"class":141},[75,77234,261],{"class":145},[75,77236,14860],{"class":85},[75,77238,2810],{"class":141},[75,77240,267],{"class":85},[75,77242,13162],{"class":145},[75,77244,273],{"class":141},[75,77246,77098],{"class":85},[75,77248,258],{"class":141},[75,77250,261],{"class":145},[75,77252,77253],{"class":85},"))) ",[75,77255,2810],{"class":141},[75,77257,77192],{"class":85},[75,77259,77260,77262,77264,77266,77268,77270,77273,77275,77278,77280,77282,77284,77286,77288],{"class":77,"line":3229},[75,77261,14467],{"class":81},[75,77263,267],{"class":85},[75,77265,2816],{"class":141},[75,77267,197],{"class":85},[75,77269,14324],{"class":145},[75,77271,77272],{"class":85},", center ",[75,77274,200],{"class":141},[75,77276,77277],{"class":85}," span), ",[75,77279,2804],{"class":141},[75,77281,197],{"class":85},[75,77283,630],{"class":145},[75,77285,77272],{"class":85},[75,77287,3251],{"class":141},[75,77289,77290],{"class":85}," span))\n",[75,77292,77293],{"class":77,"line":3235},[75,77294,112],{"emptyLinePlaceholder":111},[75,77296,77297],{"class":77,"line":3263},[75,77298,112],{"emptyLinePlaceholder":111},[75,77300,77301,77303,77306,77308,77310,77312,77314,77316,77319,77321,77323,77325,77328,77330,77332,77334,77336,77338,77340,77342,77344],{"class":77,"line":3275},[75,77302,14380],{"class":81},[75,77304,77305],{"class":193}," shots_for_epsilon",[75,77307,197],{"class":85},[75,77309,12],{"class":14388},[75,77311,14391],{"class":85},[75,77313,14150],{"class":141},[75,77315,206],{"class":85},[75,77317,77318],{"class":14388},"epsilon",[75,77320,14391],{"class":85},[75,77322,14150],{"class":141},[75,77324,14153],{"class":141},[75,77326,77327],{"class":145}," 0.01",[75,77329,206],{"class":85},[75,77331,31538],{"class":14388},[75,77333,14391],{"class":85},[75,77335,14150],{"class":141},[75,77337,14153],{"class":141},[75,77339,77063],{"class":145},[75,77341,14419],{"class":85},[75,77343,14299],{"class":141},[75,77345,5800],{"class":85},[75,77347,77348],{"class":77,"line":3280},[75,77349,41832],{"class":134},[75,77351,77352],{"class":77,"line":3290},[75,77353,77354],{"class":134},"    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[75,77356,77357],{"class":77,"line":4431},[75,77358,77359],{"class":134},"    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[75,77361,77362],{"class":77,"line":4454},[75,77363,41832],{"class":134},[75,77365,77366,77369,77371,77373,77375,77377,77380,77382,77384,77386],{"class":77,"line":4463},[75,77367,77368],{"class":85},"    p ",[75,77370,142],{"class":141},[75,77372,14762],{"class":141},[75,77374,197],{"class":85},[75,77376,2816],{"class":141},[75,77378,77379],{"class":85},"(p, ",[75,77381,14324],{"class":145},[75,77383,10496],{"class":85},[75,77385,630],{"class":145},[75,77387,318],{"class":85},[75,77389,77390,77393,77395,77397,77399,77401,77403,77405],{"class":77,"line":4468},[75,77391,77392],{"class":85},"    var ",[75,77394,142],{"class":141},[75,77396,44854],{"class":85},[75,77398,500],{"class":141},[75,77400,267],{"class":85},[75,77402,630],{"class":145},[75,77404,14186],{"class":141},[75,77406,77407],{"class":85}," p)\n",[75,77409,77410,77412,77415,77417,77419],{"class":77,"line":4477},[75,77411,14433],{"class":81},[75,77413,77414],{"class":85}," var ",[75,77416,42294],{"class":141},[75,77418,14744],{"class":145},[75,77420,5800],{"class":85},[75,77422,77423,77425],{"class":77,"line":5105},[75,77424,14450],{"class":81},[75,77426,179],{"class":145},[75,77428,77429,77431,77433,77435,77437,77439,77441,77443,77445,77447,77450,77452,77454],{"class":77,"line":5114},[75,77430,46886],{"class":85},[75,77432,142],{"class":141},[75,77434,77146],{"class":85},[75,77436,258],{"class":141},[75,77438,261],{"class":145},[75,77440,14860],{"class":85},[75,77442,500],{"class":141},[75,77444,77414],{"class":85},[75,77446,2810],{"class":141},[75,77448,77449],{"class":85}," (epsilon ",[75,77451,258],{"class":141},[75,77453,261],{"class":145},[75,77455,318],{"class":85},[75,77457,77458,77460,77462,77464,77466,77468,77470,77473,77475],{"class":77,"line":10795},[75,77459,14467],{"class":81},[75,77461,14597],{"class":141},[75,77463,197],{"class":85},[75,77465,7477],{"class":145},[75,77467,206],{"class":85},[75,77469,14299],{"class":141},[75,77471,77472],{"class":85},"(math.",[75,77474,74407],{"class":193},[75,77476,77477],{"class":85},"(n)))\n",[75,77479,77480],{"class":77,"line":14778},[75,77481,112],{"emptyLinePlaceholder":111},[75,77483,77484],{"class":77,"line":14783},[75,77485,112],{"emptyLinePlaceholder":111},[75,77487,77488],{"class":77,"line":14819},[75,77489,77490],{"class":118},"# ---------- Domain model ----------\n",[75,77492,77493],{"class":77,"line":14825},[75,77494,112],{"emptyLinePlaceholder":111},[75,77496,77497,77500,77502,77505,77507,77509],{"class":77,"line":14838},[75,77498,77499],{"class":193},"@dataclass",[75,77501,197],{"class":85},[75,77503,77504],{"class":309},"frozen",[75,77506,142],{"class":141},[75,77508,10504],{"class":145},[75,77510,318],{"class":85},[75,77512,77513,77515,77518],{"class":77,"line":14893},[75,77514,14136],{"class":81},[75,77516,77517],{"class":14139}," QubitModel",[75,77519,5800],{"class":85},[75,77521,77522,77525],{"class":77,"line":14951},[75,77523,77524],{"class":85},"    name: ",[75,77526,77527],{"class":141},"str\n",[75,77529,77530,77533,77535],{"class":77,"line":14959},[75,77531,77532],{"class":85},"    satisfaction_prob: ",[75,77534,14150],{"class":141},[75,77536,77537],{"class":118},"  # Probability a single shot meets the analytical criterion (bit = 1)\n",[75,77539,77540,77543],{"class":77,"line":14964},[75,77541,77542],{"class":85},"    description: ",[75,77544,77527],{"class":141},[75,77546,77547],{"class":77,"line":14983},[75,77548,112],{"emptyLinePlaceholder":111},[75,77550,77551],{"class":77,"line":14989},[75,77552,112],{"emptyLinePlaceholder":111},[75,77554,77555,77558,77561,77563],{"class":77,"line":15012},[75,77556,77557],{"class":145},"DEFAULT_MODELS",[75,77559,77560],{"class":85},": List[QubitModel] ",[75,77562,142],{"class":141},[75,77564,10409],{"class":85},[75,77566,77567,77570],{"class":77,"line":15017},[75,77568,77569],{"class":193},"    QubitModel",[75,77571,43495],{"class":85},[75,77573,77574,77577],{"class":77,"line":15025},[75,77575,77576],{"class":134},"        \"Toy working qubit\"",[75,77578,41629],{"class":85},[75,77580,77581,77584],{"class":77,"line":15038},[75,77582,77583],{"class":145},"        0.50",[75,77585,41629],{"class":85},[75,77587,77588],{"class":77,"line":15051},[75,77589,77590],{"class":134},"        \"Simple demonstrator; about half shots meet the criterion.\"\n",[75,77592,77593],{"class":77,"line":15058},[75,77594,77595],{"class":85},"    ),\n",[75,77597,77598,77600],{"class":77,"line":15069},[75,77599,77569],{"class":193},[75,77601,43495],{"class":85},[75,77603,77604,77607],{"class":77,"line":15080},[75,77605,77606],{"class":134},"        \"NISQ qubit\"",[75,77608,41629],{"class":85},[75,77610,77611,77614],{"class":77,"line":15085},[75,77612,77613],{"class":145},"        0.60",[75,77615,41629],{"class":85},[75,77617,77618],{"class":77,"line":15090},[75,77619,77620],{"class":134},"        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[75,77622,77623],{"class":77,"line":15096},[75,77624,77595],{"class":85},[75,77626,77627,77629],{"class":77,"line":15101},[75,77628,77569],{"class":193},[75,77630,43495],{"class":85},[75,77632,77633,77636],{"class":77,"line":15151},[75,77634,77635],{"class":134},"        \"Fault-tolerant (logical)\"",[75,77637,41629],{"class":85},[75,77639,77640,77643],{"class":77,"line":15160},[75,77641,77642],{"class":145},"        0.90",[75,77644,41629],{"class":85},[75,77646,77647],{"class":77,"line":15206},[75,77648,77649],{"class":134},"        \"Error-corrected logical qubit; high single-shot success probability.\"\n",[75,77651,77652],{"class":77,"line":15235},[75,77653,77595],{"class":85},[75,77655,77656,77658],{"class":77,"line":15288},[75,77657,77569],{"class":193},[75,77659,43495],{"class":85},[75,77661,77662,77665],{"class":77,"line":15294},[75,77663,77664],{"class":134},"        \"Topologically protected\"",[75,77666,41629],{"class":85},[75,77668,77669,77672],{"class":77,"line":15311},[75,77670,77671],{"class":145},"        0.98",[75,77673,41629],{"class":85},[75,77675,77676],{"class":77,"line":15316},[75,77677,77678],{"class":134},"        \"Intrinsic protection; very high single-shot success probability.\"\n",[75,77680,77681],{"class":77,"line":15363},[75,77682,77595],{"class":85},[75,77684,77685,77687],{"class":77,"line":15371},[75,77686,77569],{"class":193},[75,77688,43495],{"class":85},[75,77690,77691,77694],{"class":77,"line":15419},[75,77692,77693],{"class":134},"        \"Ideal qubit\"",[75,77695,41629],{"class":85},[75,77697,77698,77701],{"class":77,"line":15442},[75,77699,77700],{"class":145},"        1.00",[75,77702,41629],{"class":85},[75,77704,77705],{"class":77,"line":15482},[75,77706,77707],{"class":134},"        \"Theoretical perfect qubit; always successful in a single shot.\"\n",[75,77709,77710],{"class":77,"line":15520},[75,77711,77595],{"class":85},[75,77713,77714],{"class":77,"line":15525},[75,77715,10451],{"class":85},[75,77717,77718],{"class":77,"line":15540},[75,77719,112],{"emptyLinePlaceholder":111},[75,77721,77722],{"class":77,"line":15545},[75,77723,77724],{"class":118},"# ---------- Simulation configuration ----------\n",[75,77726,77727],{"class":77,"line":15551},[75,77728,112],{"emptyLinePlaceholder":111},[75,77730,77731,77734,77737,77739],{"class":77,"line":15592},[75,77732,77733],{"class":145},"CONFIG",[75,77735,77736],{"class":85},": Dict ",[75,77738,142],{"class":141},[75,77740,42099],{"class":85},[75,77742,77743,77746,77748,77751],{"class":77,"line":15608},[75,77744,77745],{"class":134},"    \"random_seed\"",[75,77747,14391],{"class":85},[75,77749,77750],{"class":145},"123456",[75,77752,41629],{"class":85},[75,77754,77755,77758,77760,77763],{"class":77,"line":15624},[75,77756,77757],{"class":134},"    \"shots_per_model\"",[75,77759,14391],{"class":85},[75,77761,77762],{"class":145},"10000",[75,77764,41629],{"class":85},[75,77766,77767,77770,77773,77776,77778,77781,77783,77786,77789],{"class":77,"line":15652},[75,77768,77769],{"class":134},"    \"epsilons\"",[75,77771,77772],{"class":85},": [",[75,77774,77775],{"class":145},"0.02",[75,77777,206],{"class":85},[75,77779,77780],{"class":145},"0.01",[75,77782,206],{"class":85},[75,77784,77785],{"class":145},"0.005",[75,77787,77788],{"class":85},"],  ",[75,77790,77791],{"class":118},"# ±2%, ±1%, ±0.5% targets\n",[75,77793,77794,77797],{"class":77,"line":15657},[75,77795,77796],{"class":134},"    \"cost_model\"",[75,77798,77799],{"class":85},": {\n",[75,77801,77802,77805,77807,77810],{"class":77,"line":15662},[75,77803,77804],{"class":134},"        \"default_cost_per_shot_usd\"",[75,77806,14391],{"class":85},[75,77808,77809],{"class":145},"0.00005",[75,77811,41629],{"class":85},[75,77813,77814,77817,77819],{"class":77,"line":15668},[75,77815,77816],{"class":134},"        \"fixed_overhead_usd\"",[75,77818,14391],{"class":85},[75,77820,77821],{"class":145},"0.10\n",[75,77823,77824],{"class":77,"line":15673},[75,77825,77826],{"class":85},"    },\n",[75,77828,77829,77832],{"class":77,"line":15698},[75,77830,77831],{"class":134},"    \"timing\"",[75,77833,77799],{"class":85},[75,77835,77836,77839,77841],{"class":77,"line":15723},[75,77837,77838],{"class":134},"        \"time_per_shot_seconds\"",[75,77840,14391],{"class":85},[75,77842,77843],{"class":145},"2e-6\n",[75,77845,77846],{"class":77,"line":15728},[75,77847,42218],{"class":85},[75,77849,77850],{"class":77,"line":15762},[75,77851,67801],{"class":85},[75,77853,77854],{"class":77,"line":15792},[75,77855,112],{"emptyLinePlaceholder":111},[75,77857,77858,77861,77863,77866,77869,77871,77873,77875,77878],{"class":77,"line":15829},[75,77859,77860],{"class":85},"rng ",[75,77862,142],{"class":141},[75,77864,77865],{"class":85}," np.random.",[75,77867,77868],{"class":193},"default_rng",[75,77870,197],{"class":85},[75,77872,77733],{"class":145},[75,77874,32864],{"class":85},[75,77876,77877],{"class":134},"\"random_seed\"",[75,77879,15759],{"class":85},[75,77881,77882],{"class":77,"line":15834},[75,77883,112],{"emptyLinePlaceholder":111},[75,77885,77886],{"class":77,"line":15840},[75,77887,112],{"emptyLinePlaceholder":111},[75,77889,77890],{"class":77,"line":15873},[75,77891,77892],{"class":118},"# ---------- Core simulation ----------\n",[75,77894,77895],{"class":77,"line":15931},[75,77896,112],{"emptyLinePlaceholder":111},[75,77898,77899,77901,77904,77906,77908,77910,77912,77914,77917,77919,77921,77923,77926],{"class":77,"line":15953},[75,77900,14380],{"class":81},[75,77902,77903],{"class":193}," simulate_bitstring",[75,77905,197],{"class":85},[75,77907,12],{"class":14388},[75,77909,14391],{"class":85},[75,77911,14150],{"class":141},[75,77913,206],{"class":85},[75,77915,77916],{"class":14388},"shots",[75,77918,14391],{"class":85},[75,77920,14299],{"class":141},[75,77922,206],{"class":85},[75,77924,77925],{"class":14388},"rng_",[75,77927,77928],{"class":85},": np.random.Generator) -> np.ndarray:\n",[75,77930,77931],{"class":77,"line":15968},[75,77932,41832],{"class":134},[75,77934,77935],{"class":77,"line":15996},[75,77936,77937],{"class":134},"    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[75,77939,77940],{"class":77,"line":16026},[75,77941,41832],{"class":134},[75,77943,77944,77946,77949,77952,77954,77956,77959,77962,77964,77967,77969,77971,77973],{"class":77,"line":16047},[75,77945,14467],{"class":81},[75,77947,77948],{"class":85}," rng_.",[75,77950,77951],{"class":193},"binomial",[75,77953,197],{"class":85},[75,77955,7477],{"class":145},[75,77957,77958],{"class":85},", p, ",[75,77960,77961],{"class":309},"size",[75,77963,142],{"class":141},[75,77965,77966],{"class":85},"shots).",[75,77968,75923],{"class":193},[75,77970,197],{"class":85},[75,77972,14299],{"class":141},[75,77974,318],{"class":85},[75,77976,77977],{"class":77,"line":16056},[75,77978,112],{"emptyLinePlaceholder":111},[75,77980,77981],{"class":77,"line":16065},[75,77982,112],{"emptyLinePlaceholder":111},[75,77984,77985,77987,77990,77992,77995,77998,78001,78003,78006,78009,78011],{"class":77,"line":16070},[75,77986,14380],{"class":81},[75,77988,77989],{"class":193}," summarize_model",[75,77991,197],{"class":85},[75,77993,77994],{"class":14388},"model",[75,77996,77997],{"class":85},": QubitModel, ",[75,77999,78000],{"class":14388},"bits",[75,78002,19065],{"class":85},[75,78004,78005],{"class":14388},"epsilons",[75,78007,78008],{"class":85},": List[",[75,78010,14150],{"class":141},[75,78012,78013],{"class":85},"]) -> Dict:\n",[75,78015,78016],{"class":77,"line":16076},[75,78017,41832],{"class":134},[75,78019,78020],{"class":77,"line":16107},[75,78021,78022],{"class":134},"    Calculate summary statistics for a given model and its simulated bitstring.\n",[75,78024,78025],{"class":77,"line":16153},[75,78026,41832],{"class":134},[75,78028,78029,78031,78033,78035],{"class":77,"line":16174},[75,78030,46886],{"class":85},[75,78032,142],{"class":141},[75,78034,10297],{"class":141},[75,78036,78037],{"class":85},"(bits.size)\n",[75,78039,78040,78043,78045,78047,78050,78052],{"class":77,"line":16185},[75,78041,78042],{"class":85},"    ones ",[75,78044,142],{"class":141},[75,78046,10297],{"class":141},[75,78048,78049],{"class":85},"(bits.",[75,78051,43089],{"class":193},[75,78053,15965],{"class":85},[75,78055,78056,78059,78061,78063,78066,78068],{"class":77,"line":16208},[75,78057,78058],{"class":85},"    zeros ",[75,78060,142],{"class":141},[75,78062,10297],{"class":141},[75,78064,78065],{"class":85},"(n ",[75,78067,200],{"class":141},[75,78069,78070],{"class":85}," ones)\n",[75,78072,78073,78076,78078,78081,78083,78085,78087,78089,78091,78093,78095,78097,78099,78102],{"class":77,"line":16235},[75,78074,78075],{"class":85},"    frac ",[75,78077,142],{"class":141},[75,78079,78080],{"class":85}," ones ",[75,78082,2810],{"class":141},[75,78084,77098],{"class":85},[75,78086,14737],{"class":81},[75,78088,77098],{"class":85},[75,78090,2907],{"class":141},[75,78092,10251],{"class":145},[75,78094,16742],{"class":81},[75,78096,68874],{"class":141},[75,78098,197],{"class":85},[75,78100,78101],{"class":134},"\"nan\"",[75,78103,318],{"class":85},[75,78105,78106,78109,78111,78113],{"class":77,"line":16256},[75,78107,78108],{"class":85},"    lo, hi ",[75,78110,142],{"class":141},[75,78112,77034],{"class":193},[75,78114,78115],{"class":85},"(ones, n)\n",[75,78117,78118],{"class":77,"line":16265},[75,78119,112],{"emptyLinePlaceholder":111},[75,78121,78122,78125,78127],{"class":77,"line":16274},[75,78123,78124],{"class":85},"    row ",[75,78126,142],{"class":141},[75,78128,42099],{"class":85},[75,78130,78131,78134],{"class":77,"line":16279},[75,78132,78133],{"class":134},"        \"Model\"",[75,78135,78136],{"class":85},": model.name,\n",[75,78138,78139,78142],{"class":77,"line":16285},[75,78140,78141],{"class":134},"        \"Description\"",[75,78143,78144],{"class":85},": model.description,\n",[75,78146,78147,78150],{"class":77,"line":16311},[75,78148,78149],{"class":134},"        \"Configured p (target)\"",[75,78151,78152],{"class":85},": model.satisfaction_prob,\n",[75,78154,78155,78158],{"class":77,"line":16329},[75,78156,78157],{"class":134},"        \"Shots\"",[75,78159,78160],{"class":85},": n,\n",[75,78162,78163,78166],{"class":77,"line":16347},[75,78164,78165],{"class":134},"        \"Ones (successes)\"",[75,78167,78168],{"class":85},": ones,\n",[75,78170,78171,78174],{"class":77,"line":16360},[75,78172,78173],{"class":134},"        \"Zeros (failures)\"",[75,78175,78176],{"class":85},": zeros,\n",[75,78178,78179,78182],{"class":77,"line":16365},[75,78180,78181],{"class":134},"        \"Ones fraction (observed)\"",[75,78183,78184],{"class":85},": frac,\n",[75,78186,78187,78190,78193,78196],{"class":77,"line":16396},[75,78188,78189],{"class":134},"        \"Wilson CI 95",[75,78191,78192],{"class":145},"% lo",[75,78194,78195],{"class":134},"w\"",[75,78197,78198],{"class":85},": lo,\n",[75,78200,78201,78203,78206,78209],{"class":77,"line":16441},[75,78202,78189],{"class":134},[75,78204,78205],{"class":145},"% hi",[75,78207,78208],{"class":134},"gh\"",[75,78210,78211],{"class":85},": hi,\n",[75,78213,78214,78217,78220,78222],{"class":77,"line":16490},[75,78215,78216],{"class":134},"        \"CI width (95%)\"",[75,78218,78219],{"class":85},": hi ",[75,78221,200],{"class":141},[75,78223,78224],{"class":85}," lo,\n",[75,78226,78227],{"class":77,"line":16522},[75,78228,42218],{"class":85},[75,78230,78231,78233,78235,78237],{"class":77,"line":16551},[75,78232,15611],{"class":81},[75,78234,14865],{"class":85},[75,78236,3153],{"class":81},[75,78238,78239],{"class":85}," epsilons:\n",[75,78241,78242,78245,78247,78250,78252,78254,78257,78259,78262,78264,78266,78269,78271,78273,78275],{"class":77,"line":16574},[75,78243,78244],{"class":85},"        row[",[75,78246,5894],{"class":81},[75,78248,78249],{"class":134},"\"Required shots for ±",[75,78251,12219],{"class":145},[75,78253,14299],{"class":141},[75,78255,78256],{"class":85},"(eps",[75,78258,500],{"class":141},[75,78260,78261],{"class":145},"100",[75,78263,5175],{"class":85},[75,78265,12224],{"class":145},[75,78267,78268],{"class":134},"%\"",[75,78270,138],{"class":85},[75,78272,142],{"class":141},[75,78274,77305],{"class":193},[75,78276,78277],{"class":85},"(model.satisfaction_prob, eps)\n",[75,78279,78280,78282],{"class":77,"line":16618},[75,78281,14467],{"class":81},[75,78283,78284],{"class":85}," row\n",[75,78286,78287],{"class":77,"line":16627},[75,78288,112],{"emptyLinePlaceholder":111},[75,78290,78291],{"class":77,"line":16632},[75,78292,112],{"emptyLinePlaceholder":111},[75,78294,78295,78297,78300,78302,78304,78306,78308,78310,78313,78315,78317,78319,78322,78324,78326,78328,78330],{"class":77,"line":16638},[75,78296,14380],{"class":81},[75,78298,78299],{"class":193}," apply_costs",[75,78301,197],{"class":85},[75,78303,77916],{"class":14388},[75,78305,14391],{"class":85},[75,78307,14299],{"class":141},[75,78309,206],{"class":85},[75,78311,78312],{"class":14388},"cost_per_shot",[75,78314,14391],{"class":85},[75,78316,14150],{"class":141},[75,78318,206],{"class":85},[75,78320,78321],{"class":14388},"fixed_overhead",[75,78323,14391],{"class":85},[75,78325,14150],{"class":141},[75,78327,14419],{"class":85},[75,78329,14150],{"class":141},[75,78331,5800],{"class":85},[75,78333,78334],{"class":77,"line":16654},[75,78335,41832],{"class":134},[75,78337,78338],{"class":77,"line":16671},[75,78339,78340],{"class":134},"    Calculate illustrative cost based on shots and cost model.\n",[75,78342,78343],{"class":77,"line":16684},[75,78344,41832],{"class":134},[75,78346,78347,78349,78352,78354,78357,78359],{"class":77,"line":16701},[75,78348,14467],{"class":81},[75,78350,78351],{"class":85}," fixed_overhead ",[75,78353,3251],{"class":141},[75,78355,78356],{"class":85}," shots ",[75,78358,500],{"class":141},[75,78360,78361],{"class":85}," cost_per_shot\n",[75,78363,78364],{"class":77,"line":16713},[75,78365,112],{"emptyLinePlaceholder":111},[75,78367,78368],{"class":77,"line":16748},[75,78369,112],{"emptyLinePlaceholder":111},[75,78371,78372],{"class":77,"line":16753},[75,78373,78374],{"class":118},"# ---------- Run the simulation ----------\n",[75,78376,78377],{"class":77,"line":16779},[75,78378,112],{"emptyLinePlaceholder":111},[75,78380,78381,78384,78386,78389,78391],{"class":77,"line":16800},[75,78382,78383],{"class":85},"all_bitstrings: Dict[",[75,78385,41840],{"class":141},[75,78387,78388],{"class":85},", np.ndarray] ",[75,78390,142],{"class":141},[75,78392,78393],{"class":85}," {}\n",[75,78395,78396,78399,78401],{"class":77,"line":16805},[75,78397,78398],{"class":85},"summary_rows: List[Dict] ",[75,78400,142],{"class":141},[75,78402,44753],{"class":85},[75,78404,78405],{"class":77,"line":16836},[75,78406,112],{"emptyLinePlaceholder":111},[75,78408,78409,78411,78413,78416],{"class":77,"line":16869},[75,78410,45904],{"class":141},[75,78412,197],{"class":85},[75,78414,78415],{"class":134},"\"Running simulation...\"",[75,78417,318],{"class":85},[75,78419,78420,78422,78425,78427,78430],{"class":77,"line":16906},[75,78421,3147],{"class":81},[75,78423,78424],{"class":85}," m ",[75,78426,3153],{"class":81},[75,78428,78429],{"class":145}," DEFAULT_MODELS",[75,78431,5800],{"class":85},[75,78433,78434,78437,78439,78441,78444,78446,78448,78451],{"class":77,"line":16933},[75,78435,78436],{"class":85},"    bits ",[75,78438,142],{"class":141},[75,78440,77903],{"class":193},[75,78442,78443],{"class":85},"(m.satisfaction_prob, ",[75,78445,77733],{"class":145},[75,78447,32864],{"class":85},[75,78449,78450],{"class":134},"\"shots_per_model\"",[75,78452,78453],{"class":85},"], rng)\n",[75,78455,78456,78459,78461],{"class":77,"line":16956},[75,78457,78458],{"class":85},"    all_bitstrings[m.name] ",[75,78460,142],{"class":141},[75,78462,78463],{"class":85}," bits\n",[75,78465,78466,78469,78471,78473,78476,78479,78481,78483,78486],{"class":77,"line":16996},[75,78467,78468],{"class":85},"    summary_rows.",[75,78470,44804],{"class":193},[75,78472,197],{"class":85},[75,78474,78475],{"class":193},"summarize_model",[75,78477,78478],{"class":85},"(m, bits, ",[75,78480,77733],{"class":145},[75,78482,32864],{"class":85},[75,78484,78485],{"class":134},"\"epsilons\"",[75,78487,78488],{"class":85},"]))\n",[75,78490,78491],{"class":77,"line":48243},[75,78492,112],{"emptyLinePlaceholder":111},[75,78494,78495,78498,78500,78502,78504],{"class":77,"line":48248},[75,78496,78497],{"class":85},"summary_df ",[75,78499,142],{"class":141},[75,78501,67345],{"class":85},[75,78503,67348],{"class":193},[75,78505,78506],{"class":85},"(summary_rows)\n",[75,78508,78509],{"class":77,"line":48263},[75,78510,112],{"emptyLinePlaceholder":111},[75,78512,78513],{"class":77,"line":48296},[75,78514,78515],{"class":118},"# Add cost estimates\n",[75,78517,78518,78521,78523,78526,78528,78531,78533,78536],{"class":77,"line":48302},[75,78519,78520],{"class":85},"cps ",[75,78522,142],{"class":141},[75,78524,78525],{"class":145}," CONFIG",[75,78527,32864],{"class":85},[75,78529,78530],{"class":134},"\"cost_model\"",[75,78532,72390],{"class":85},[75,78534,78535],{"class":134},"\"default_cost_per_shot_usd\"",[75,78537,10451],{"class":85},[75,78539,78540,78543,78545,78547,78549,78551,78553,78556],{"class":77,"line":48314},[75,78541,78542],{"class":85},"over ",[75,78544,142],{"class":141},[75,78546,78525],{"class":145},[75,78548,32864],{"class":85},[75,78550,78530],{"class":134},[75,78552,72390],{"class":85},[75,78554,78555],{"class":134},"\"fixed_overhead_usd\"",[75,78557,10451],{"class":85},[75,78559,78560,78562,78565,78567,78569,78572,78575,78577,78579],{"class":77,"line":48332},[75,78561,14737],{"class":81},[75,78563,78564],{"class":85}," cps ",[75,78566,2907],{"class":141},[75,78568,10251],{"class":145},[75,78570,78571],{"class":81}," or",[75,78573,78574],{"class":85}," over ",[75,78576,2907],{"class":141},[75,78578,10251],{"class":145},[75,78580,5800],{"class":85},[75,78582,78583,78586,78589,78591,78593],{"class":77,"line":48351},[75,78584,78585],{"class":85},"    summary_df[",[75,78587,78588],{"class":134},"\"Illustrative cost (USD)\"",[75,78590,138],{"class":85},[75,78592,142],{"class":141},[75,78594,10409],{"class":85},[75,78596,78597,78600,78602,78604,78607,78610],{"class":77,"line":48374},[75,78598,78599],{"class":193},"        apply_costs",[75,78601,197],{"class":85},[75,78603,14299],{"class":141},[75,78605,78606],{"class":85},"(row[",[75,78608,78609],{"class":134},"\"Shots\"",[75,78611,78612],{"class":85},"]), cps, over)\n",[75,78614,78615,78617,78620,78622,78625,78628],{"class":77,"line":48383},[75,78616,68295],{"class":81},[75,78618,78619],{"class":85}," _, row ",[75,78621,3153],{"class":81},[75,78623,78624],{"class":85}," summary_df.",[75,78626,78627],{"class":193},"iterrows",[75,78629,389],{"class":85},[75,78631,78632],{"class":77,"line":48388},[75,78633,68325],{"class":85},[75,78635,78636],{"class":77,"line":48393},[75,78637,112],{"emptyLinePlaceholder":111},[75,78639,78640],{"class":77,"line":48399},[75,78641,78642],{"class":118},"# ---------- Save artifacts ----------\n",[75,78644,78645],{"class":77,"line":48414},[75,78646,112],{"emptyLinePlaceholder":111},[75,78648,78649],{"class":77,"line":48425},[75,78650,78651],{"class":118},"# Save raw bitstrings and summary CSV\n",[75,78653,78654,78656,78659,78661,78664,78666],{"class":77,"line":48462},[75,78655,3147],{"class":81},[75,78657,78658],{"class":85}," name, arr ",[75,78660,3153],{"class":81},[75,78662,78663],{"class":85}," all_bitstrings.",[75,78665,68306],{"class":193},[75,78667,78668],{"class":85},"():\n",[75,78670,78671,78674,78676,78678,78681,78683,78686,78689,78691,78694,78696,78699,78701,78703],{"class":77,"line":48484},[75,78672,78673],{"class":85},"    out_path ",[75,78675,142],{"class":141},[75,78677,44783],{"class":81},[75,78679,78680],{"class":134},"\"bitstrings_",[75,78682,12219],{"class":145},[75,78684,78685],{"class":85},"name.",[75,78687,78688],{"class":193},"replace",[75,78690,197],{"class":85},[75,78692,78693],{"class":134},"' '",[75,78695,206],{"class":85},[75,78697,78698],{"class":134},"'_'",[75,78700,5175],{"class":85},[75,78702,12224],{"class":145},[75,78704,78705],{"class":134},".csv\"\n",[75,78707,78708,78711,78713,78716,78719,78722,78725,78728,78731,78733,78735],{"class":77,"line":48495},[75,78709,78710],{"class":85},"    pd.",[75,78712,67348],{"class":193},[75,78714,78715],{"class":85},"({",[75,78717,78718],{"class":134},"\"bit\"",[75,78720,78721],{"class":85},": arr}).",[75,78723,78724],{"class":193},"to_csv",[75,78726,78727],{"class":85},"(out_path, ",[75,78729,78730],{"class":309},"index",[75,78732,142],{"class":141},[75,78734,376],{"class":145},[75,78736,318],{"class":85},[75,78738,78739],{"class":77,"line":48512},[75,78740,112],{"emptyLinePlaceholder":111},[75,78742,78743,78746,78748],{"class":77,"line":48524},[75,78744,78745],{"class":85},"summary_csv_path ",[75,78747,142],{"class":141},[75,78749,78750],{"class":134}," \"summary.csv\"\n",[75,78752,78753,78756,78758,78761,78763,78765,78767],{"class":77,"line":48529},[75,78754,78755],{"class":85},"summary_df.",[75,78757,78724],{"class":193},[75,78759,78760],{"class":85},"(summary_csv_path, ",[75,78762,78730],{"class":309},[75,78764,142],{"class":141},[75,78766,376],{"class":145},[75,78768,318],{"class":85},[75,78770,78771],{"class":77,"line":48534},[75,78772,112],{"emptyLinePlaceholder":111},[75,78774,78775],{"class":77,"line":69314},[75,78776,78777],{"class":118},"# Save config JSON\n",[75,78779,78780,78783,78786,78788,78791,78793,78796,78798,78800],{"class":77,"line":69319},[75,78781,78782],{"class":81},"with",[75,78784,78785],{"class":141}," open",[75,78787,197],{"class":85},[75,78789,78790],{"class":134},"\"simulation_config.json\"",[75,78792,206],{"class":85},[75,78794,78795],{"class":134},"\"w\"",[75,78797,14860],{"class":85},[75,78799,89],{"class":81},[75,78801,78802],{"class":85}," f:\n",[75,78804,78805,78808,78811,78813,78815,78818,78821,78823,78825],{"class":77,"line":69330},[75,78806,78807],{"class":85},"    json.",[75,78809,78810],{"class":193},"dump",[75,78812,197],{"class":85},[75,78814,77733],{"class":145},[75,78816,78817],{"class":85},", f, ",[75,78819,78820],{"class":309},"indent",[75,78822,142],{"class":141},[75,78824,270],{"class":145},[75,78826,318],{"class":85},[75,78828,78829],{"class":77,"line":69338},[75,78830,112],{"emptyLinePlaceholder":111},[75,78832,78833,78835,78837,78840],{"class":77,"line":69348},[75,78834,45904],{"class":141},[75,78836,197],{"class":85},[75,78838,78839],{"class":134},"\"Saved data to CSV and JSON files.\"",[75,78841,318],{"class":85},[75,78843,78844],{"class":77,"line":69353},[75,78845,112],{"emptyLinePlaceholder":111},[75,78847,78848],{"class":77,"line":69358},[75,78849,78850],{"class":118},"# ---------- Visualizations ----------\n",[75,78852,78853],{"class":77,"line":69363},[75,78854,112],{"emptyLinePlaceholder":111},[75,78856,78857],{"class":77,"line":69370},[75,78858,78859],{"class":118},"# Figure 1: Observed ones fraction with Wilson CI\n",[75,78861,78862,78865,78868,78870,78873],{"class":77,"line":69386},[75,78863,78864],{"class":85},"plt.style.",[75,78866,78867],{"class":193},"use",[75,78869,197],{"class":85},[75,78871,78872],{"class":134},"'seaborn-v0_8-whitegrid'",[75,78874,318],{"class":85},[75,78876,78877,78879,78881,78883,78885,78887,78889,78891,78893,78895],{"class":77,"line":69395},[75,78878,300],{"class":85},[75,78880,865],{"class":193},[75,78882,197],{"class":85},[75,78884,870],{"class":309},[75,78886,142],{"class":141},[75,78888,197],{"class":85},[75,78890,70331],{"class":145},[75,78892,206],{"class":85},[75,78894,882],{"class":145},[75,78896,283],{"class":85},[75,78898,78899,78901,78903,78905,78907,78909,78911],{"class":77,"line":69404},[75,78900,17925],{"class":85},[75,78902,142],{"class":141},[75,78904,190],{"class":85},[75,78906,10675],{"class":193},[75,78908,197],{"class":85},[75,78910,2930],{"class":141},[75,78912,78913],{"class":85},"(summary_df))\n",[75,78915,78916,78919,78921,78924,78927,78930,78933],{"class":77,"line":69413},[75,78917,78918],{"class":85},"y ",[75,78920,142],{"class":141},[75,78922,78923],{"class":85}," summary_df[",[75,78925,78926],{"class":134},"\"Ones fraction (observed)\"",[75,78928,78929],{"class":85},"].",[75,78931,78932],{"class":193},"to_numpy",[75,78934,389],{"class":85},[75,78936,78937],{"class":77,"line":69422},[75,78938,112],{"emptyLinePlaceholder":111},[75,78940,78941],{"class":77,"line":69445},[75,78942,78943],{"class":118},"# --- FIX APPLIED HERE ---\n",[75,78945,78946],{"class":77,"line":69451},[75,78947,78948],{"class":118},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[75,78950,78951],{"class":77,"line":69456},[75,78952,78953],{"class":118},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[75,78955,78956,78959,78961,78964,78966,78968,78971,78973,78975,78977,78979],{"class":77,"line":69461},[75,78957,78958],{"class":85},"lower_error ",[75,78960,142],{"class":141},[75,78962,78963],{"class":85}," y ",[75,78965,200],{"class":141},[75,78967,78923],{"class":85},[75,78969,78970],{"class":134},"\"Wilson CI 95",[75,78972,78192],{"class":145},[75,78974,78195],{"class":134},[75,78976,78929],{"class":85},[75,78978,78932],{"class":193},[75,78980,389],{"class":85},[75,78982,78983,78986,78988,78990,78992,78994,78996,78998,79000,79002,79004],{"class":77,"line":69466},[75,78984,78985],{"class":85},"upper_error ",[75,78987,142],{"class":141},[75,78989,78923],{"class":85},[75,78991,78970],{"class":134},[75,78993,78205],{"class":145},[75,78995,78208],{"class":134},[75,78997,78929],{"class":85},[75,78999,78932],{"class":193},[75,79001,2819],{"class":85},[75,79003,200],{"class":141},[75,79005,79006],{"class":85}," y\n",[75,79008,79009,79012,79014,79016,79018,79021,79023,79026,79028],{"class":77,"line":69476},[75,79010,79011],{"class":85},"yerr ",[75,79013,142],{"class":141},[75,79015,190],{"class":85},[75,79017,69263],{"class":193},[75,79019,79020],{"class":85},"([np.",[75,79022,2145],{"class":193},[75,79024,79025],{"class":85},"(lower_error), np.",[75,79027,2145],{"class":193},[75,79029,79030],{"class":85},"(upper_error)])\n",[75,79032,79033],{"class":77,"line":69485},[75,79034,79035],{"class":118},"# --- END OF FIX ---\n",[75,79037,79038],{"class":77,"line":69494},[75,79039,112],{"emptyLinePlaceholder":111},[75,79041,79042,79044,79046,79049,79051,79053,79056,79058,79061,79063,79066],{"class":77,"line":69499},[75,79043,300],{"class":85},[75,79045,17945],{"class":193},[75,79047,79048],{"class":85},"(x, y, ",[75,79050,310],{"class":309},[75,79052,142],{"class":141},[75,79054,79055],{"class":134},"'skyblue'",[75,79057,206],{"class":85},[75,79059,79060],{"class":309},"edgecolor",[75,79062,142],{"class":141},[75,79064,79065],{"class":134},"'black'",[75,79067,318],{"class":85},[75,79069,79070,79072,79075,79077,79080,79082,79085,79088,79090,79092,79094,79097,79099,79101,79103,79105,79107,79110,79112,79115,79117,79119],{"class":77,"line":69505},[75,79071,300],{"class":85},[75,79073,79074],{"class":193},"errorbar",[75,79076,79048],{"class":85},[75,79078,79079],{"class":309},"yerr",[75,79081,142],{"class":141},[75,79083,79084],{"class":85},"yerr, ",[75,79086,79087],{"class":309},"fmt",[75,79089,142],{"class":141},[75,79091,43580],{"class":134},[75,79093,206],{"class":85},[75,79095,79096],{"class":309},"capsize",[75,79098,142],{"class":141},[75,79100,203],{"class":145},[75,79102,206],{"class":85},[75,79104,310],{"class":309},[75,79106,142],{"class":141},[75,79108,79109],{"class":134},"\"black\"",[75,79111,206],{"class":85},[75,79113,79114],{"class":309},"elinewidth",[75,79116,142],{"class":141},[75,79118,15584],{"class":145},[75,79120,318],{"class":85},[75,79122,79123,79125,79127,79130,79133,79135,79138,79140,79143,79145,79147,79149,79151,79153,79155],{"class":77,"line":69520},[75,79124,300],{"class":85},[75,79126,17955],{"class":193},[75,79128,79129],{"class":85},"(x, summary_df[",[75,79131,79132],{"class":134},"\"Model\"",[75,79134,78929],{"class":85},[75,79136,79137],{"class":193},"tolist",[75,79139,16413],{"class":85},[75,79141,79142],{"class":309},"rotation",[75,79144,142],{"class":141},[75,79146,57891],{"class":145},[75,79148,206],{"class":85},[75,79150,43197],{"class":309},[75,79152,142],{"class":141},[75,79154,43251],{"class":134},[75,79156,318],{"class":85},[75,79158,79159,79161,79163,79165,79168],{"class":77,"line":69531},[75,79160,300],{"class":85},[75,79162,356],{"class":193},[75,79164,197],{"class":85},[75,79166,79167],{"class":134},"\"Observed Ones Fraction\"",[75,79169,318],{"class":85},[75,79171,79172,79174,79176,79178,79181,79183,79185,79187,79190],{"class":77,"line":69546},[75,79173,300],{"class":85},[75,79175,326],{"class":193},[75,79177,197],{"class":85},[75,79179,79180],{"class":134},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[75,79182,206],{"class":85},[75,79184,43519],{"class":309},[75,79186,142],{"class":141},[75,79188,79189],{"class":145},"14",[75,79191,318],{"class":85},[75,79193,79194,79196,79198],{"class":77,"line":69556},[75,79195,300],{"class":85},[75,79197,3285],{"class":193},[75,79199,389],{"class":85},[75,79201,79202,79205,79207],{"class":77,"line":69564},[75,79203,79204],{"class":85},"fig1_path ",[75,79206,142],{"class":141},[75,79208,79209],{"class":134}," \"fig_ones_fraction.png\"\n",[75,79211,79212,79214,79217,79220,79222,79224,79226,79228,79231,79233,79236],{"class":77,"line":69573},[75,79213,300],{"class":85},[75,79215,79216],{"class":193},"savefig",[75,79218,79219],{"class":85},"(fig1_path, ",[75,79221,15863],{"class":309},[75,79223,142],{"class":141},[75,79225,41574],{"class":145},[75,79227,206],{"class":85},[75,79229,79230],{"class":309},"bbox_inches",[75,79232,142],{"class":141},[75,79234,79235],{"class":134},"'tight'",[75,79237,318],{"class":85},[75,79239,79240,79242,79244],{"class":77,"line":69578},[75,79241,300],{"class":85},[75,79243,386],{"class":193},[75,79245,389],{"class":85},[75,79247,79248],{"class":77,"line":69600},[75,79249,112],{"emptyLinePlaceholder":111},[75,79251,79252],{"class":77,"line":69605},[75,79253,79254],{"class":118},"# Figure 2: Required shots at several ±epsilon targets\n",[75,79256,79257,79259,79261,79263,79265,79267,79269,79271,79273,79275],{"class":77,"line":69621},[75,79258,300],{"class":85},[75,79260,865],{"class":193},[75,79262,197],{"class":85},[75,79264,870],{"class":309},[75,79266,142],{"class":141},[75,79268,197],{"class":85},[75,79270,70331],{"class":145},[75,79272,206],{"class":85},[75,79274,882],{"class":145},[75,79276,283],{"class":85},[75,79278,79279,79282,79284],{"class":77,"line":69638},[75,79280,79281],{"class":85},"width ",[75,79283,142],{"class":141},[75,79285,14646],{"class":145},[75,79287,79288,79290,79292,79294,79296,79298,79300,79302,79304],{"class":77,"line":69643},[75,79289,17925],{"class":85},[75,79291,142],{"class":141},[75,79293,190],{"class":85},[75,79295,10675],{"class":193},[75,79297,197],{"class":85},[75,79299,2930],{"class":141},[75,79301,197],{"class":85},[75,79303,77557],{"class":145},[75,79305,283],{"class":85},[75,79307,79308,79311,79313,79315,79318,79320,79323,79325,79328],{"class":77,"line":69656},[75,79309,79310],{"class":85},"colors ",[75,79312,142],{"class":141},[75,79314,16404],{"class":85},[75,79316,79317],{"class":134},"'#ff9999'",[75,79319,7469],{"class":85},[75,79321,79322],{"class":134},"'#66b3ff'",[75,79324,7469],{"class":85},[75,79326,79327],{"class":134},"'#99ff99'",[75,79329,10451],{"class":85},[75,79331,79332],{"class":77,"line":69669},[75,79333,112],{"emptyLinePlaceholder":111},[75,79335,79336,79338,79341,79343,79345,79347,79349,79351,79353],{"class":77,"line":69674},[75,79337,3147],{"class":81},[75,79339,79340],{"class":85}," i, eps ",[75,79342,3153],{"class":81},[75,79344,3156],{"class":141},[75,79346,197],{"class":85},[75,79348,77733],{"class":145},[75,79350,32864],{"class":85},[75,79352,78485],{"class":134},[75,79354,79355],{"class":85},"]):\n",[75,79357,79358,79361,79363,79365,79368,79371,79373,79375,79377,79379],{"class":77,"line":69682},[75,79359,79360],{"class":85},"    req ",[75,79362,142],{"class":141},[75,79364,16404],{"class":85},[75,79366,79367],{"class":193},"shots_for_epsilon",[75,79369,79370],{"class":85},"(m.satisfaction_prob, eps) ",[75,79372,3147],{"class":81},[75,79374,78424],{"class":85},[75,79376,3153],{"class":81},[75,79378,78429],{"class":145},[75,79380,10451],{"class":85},[75,79382,79383,79385,79387,79390,79392,79394,79396,79399,79401,79403,79405,79408,79410,79412,79415,79417,79419,79421,79423,79425,79428,79430,79432,79434],{"class":77,"line":69687},[75,79384,44537],{"class":85},[75,79386,17945],{"class":193},[75,79388,79389],{"class":85},"(x ",[75,79391,3251],{"class":141},[75,79393,72709],{"class":85},[75,79395,500],{"class":141},[75,79397,79398],{"class":85}," width, req, width, ",[75,79400,573],{"class":309},[75,79402,142],{"class":141},[75,79404,5894],{"class":81},[75,79406,79407],{"class":134},"\"±",[75,79409,12219],{"class":145},[75,79411,14407],{"class":85},[75,79413,79414],{"class":81},":.1%",[75,79416,12224],{"class":145},[75,79418,12216],{"class":134},[75,79420,206],{"class":85},[75,79422,310],{"class":309},[75,79424,142],{"class":141},[75,79426,79427],{"class":85},"colors[i], ",[75,79429,79060],{"class":309},[75,79431,142],{"class":141},[75,79433,79065],{"class":134},[75,79435,318],{"class":85},[75,79437,79438],{"class":77,"line":69692},[75,79439,112],{"emptyLinePlaceholder":111},[75,79441,79442,79444,79446,79448,79450,79453,79455,79457,79459,79461,79463,79465,79467,79469,79471,79473,79475,79477],{"class":77,"line":69702},[75,79443,300],{"class":85},[75,79445,17955],{"class":193},[75,79447,79389],{"class":85},[75,79449,3251],{"class":141},[75,79451,79452],{"class":85}," width, [m.name ",[75,79454,3147],{"class":81},[75,79456,78424],{"class":85},[75,79458,3153],{"class":81},[75,79460,78429],{"class":145},[75,79462,12097],{"class":85},[75,79464,79142],{"class":309},[75,79466,142],{"class":141},[75,79468,57891],{"class":145},[75,79470,206],{"class":85},[75,79472,43197],{"class":309},[75,79474,142],{"class":141},[75,79476,43251],{"class":134},[75,79478,318],{"class":85},[75,79480,79481,79483,79485,79487,79490],{"class":77,"line":69710},[75,79482,300],{"class":85},[75,79484,356],{"class":193},[75,79486,197],{"class":85},[75,79488,79489],{"class":134},"\"Required Shots (Log Scale)\"",[75,79491,318],{"class":85},[75,79493,79494,79496,79498,79500,79503,79505,79507,79509,79511],{"class":77,"line":69718},[75,79495,300],{"class":85},[75,79497,326],{"class":193},[75,79499,197],{"class":85},[75,79501,79502],{"class":134},"\"Approximate Shots to Estimate Success Rate within ±ε\"",[75,79504,206],{"class":85},[75,79506,43519],{"class":309},[75,79508,142],{"class":141},[75,79510,79189],{"class":145},[75,79512,318],{"class":85},[75,79514,79515,79517,79520,79522,79525],{"class":77,"line":69730},[75,79516,300],{"class":85},[75,79518,79519],{"class":193},"yscale",[75,79521,197],{"class":85},[75,79523,79524],{"class":134},"'log'",[75,79526,318],{"class":85},[75,79528,79529,79531,79533,79535,79537,79539,79542],{"class":77,"line":69742},[75,79530,300],{"class":85},[75,79532,5058],{"class":193},[75,79534,197],{"class":85},[75,79536,326],{"class":309},[75,79538,142],{"class":141},[75,79540,79541],{"class":134},"\"Error Tolerance\"",[75,79543,318],{"class":85},[75,79545,79546,79548,79550],{"class":77,"line":69751},[75,79547,300],{"class":85},[75,79549,3285],{"class":193},[75,79551,389],{"class":85},[75,79553,79554,79557,79559],{"class":77,"line":69757},[75,79555,79556],{"class":85},"fig2_path ",[75,79558,142],{"class":141},[75,79560,79561],{"class":134}," \"fig_required_shots.png\"\n",[75,79563,79564,79566,79568,79571,79573,79575,79577,79579,79581,79583,79585],{"class":77,"line":69767},[75,79565,300],{"class":85},[75,79567,79216],{"class":193},[75,79569,79570],{"class":85},"(fig2_path, ",[75,79572,15863],{"class":309},[75,79574,142],{"class":141},[75,79576,41574],{"class":145},[75,79578,206],{"class":85},[75,79580,79230],{"class":309},[75,79582,142],{"class":141},[75,79584,79235],{"class":134},[75,79586,318],{"class":85},[75,79588,79589,79591,79593],{"class":77,"line":69777},[75,79590,300],{"class":85},[75,79592,386],{"class":193},[75,79594,389],{"class":85},[75,79596,79597],{"class":77,"line":69786},[75,79598,112],{"emptyLinePlaceholder":111},[75,79600,79601,79603,79605,79608],{"class":77,"line":69795},[75,79602,45904],{"class":141},[75,79604,197],{"class":85},[75,79606,79607],{"class":134},"\"Generated and saved plots.\"",[75,79609,318],{"class":85},[75,79611,79612],{"class":77,"line":69806},[75,79613,112],{"emptyLinePlaceholder":111},[75,79615,79616],{"class":77,"line":69818},[75,79617,79618],{"class":118},"# ---------- Display interactive table ----------\n",[75,79620,79621],{"class":77,"line":69829},[75,79622,112],{"emptyLinePlaceholder":111},[75,79624,79625,79627,79629,79632,79634,79637],{"class":77,"line":69834},[75,79626,45886],{"class":193},[75,79628,197],{"class":85},[75,79630,79631],{"class":193},"Markdown",[75,79633,197],{"class":85},[75,79635,79636],{"class":134},"\"##  Qubit Sampling Budget - Summary\"",[75,79638,283],{"class":85},[75,79640,79641,79643,79646,79649],{"class":77,"line":69839},[75,79642,45886],{"class":193},[75,79644,79645],{"class":85},"(summary_df.style.",[75,79647,79648],{"class":193},"format",[75,79650,41564],{"class":85},[75,79652,79653,79656,79658,79660,79662,79664,79666,79668],{"class":77,"line":69844},[75,79654,79655],{"class":134},"    \"Configured p (target)\"",[75,79657,14391],{"class":85},[75,79659,12216],{"class":134},[75,79661,12219],{"class":145},[75,79663,45507],{"class":81},[75,79665,12224],{"class":145},[75,79667,12216],{"class":134},[75,79669,41629],{"class":85},[75,79671,79672,79675,79677,79679,79681,79684,79686,79688],{"class":77,"line":69854},[75,79673,79674],{"class":134},"    \"Ones fraction (observed)\"",[75,79676,14391],{"class":85},[75,79678,12216],{"class":134},[75,79680,12219],{"class":145},[75,79682,79683],{"class":81},":.4f",[75,79685,12224],{"class":145},[75,79687,12216],{"class":134},[75,79689,41629],{"class":85},[75,79691,79692,79695,79697,79699,79701,79703,79705,79707,79709,79711],{"class":77,"line":69861},[75,79693,79694],{"class":134},"    \"Wilson CI 95",[75,79696,78192],{"class":145},[75,79698,78195],{"class":134},[75,79700,14391],{"class":85},[75,79702,12216],{"class":134},[75,79704,12219],{"class":145},[75,79706,79683],{"class":81},[75,79708,12224],{"class":145},[75,79710,12216],{"class":134},[75,79712,41629],{"class":85},[75,79714,79715,79717,79719,79721,79723,79725,79727,79729,79731,79733],{"class":77,"line":69869},[75,79716,79694],{"class":134},[75,79718,78205],{"class":145},[75,79720,78208],{"class":134},[75,79722,14391],{"class":85},[75,79724,12216],{"class":134},[75,79726,12219],{"class":145},[75,79728,79683],{"class":81},[75,79730,12224],{"class":145},[75,79732,12216],{"class":134},[75,79734,41629],{"class":85},[75,79736,79737,79740,79742,79744,79746,79748,79750,79752],{"class":77,"line":69880},[75,79738,79739],{"class":134},"    \"CI width (95%)\"",[75,79741,14391],{"class":85},[75,79743,12216],{"class":134},[75,79745,12219],{"class":145},[75,79747,79683],{"class":81},[75,79749,12224],{"class":145},[75,79751,12216],{"class":134},[75,79753,41629],{"class":85},[75,79755,79756,79759,79761,79763,79765,79767,79769],{"class":77,"line":69891},[75,79757,79758],{"class":134},"    \"Illustrative cost (USD)\"",[75,79760,14391],{"class":85},[75,79762,70890],{"class":134},[75,79764,12219],{"class":145},[75,79766,45507],{"class":81},[75,79768,12224],{"class":145},[75,79770,45544],{"class":134},[75,79772,79773],{"class":77,"line":69900},[75,79774,79775],{"class":85},"}))\n",[75,79777,79778],{"class":77,"line":69906},[75,79779,112],{"emptyLinePlaceholder":111},[75,79781,79782],{"class":77,"line":69916},[75,79783,112],{"emptyLinePlaceholder":111},[75,79785,79786],{"class":77,"line":69926},[75,79787,79788],{"class":118},"# ---------- Build a ready-to-run Colab notebook ----------\n",[75,79790,79791],{"class":77,"line":69935},[75,79792,112],{"emptyLinePlaceholder":111},[75,79794,79795,79798,79800,79803],{"class":77,"line":69944},[75,79796,79797],{"class":85},"md_intro ",[75,79799,142],{"class":141},[75,79801,79802],{"class":81}," r",[75,79804,41449],{"class":15980},[75,79806,79807],{"class":77,"line":69953},[75,79808,79809],{"class":118},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[75,79811,79812],{"class":77,"line":69965},[75,79813,112],{"emptyLinePlaceholder":111},[75,79815,79816,79818,79821,79823,79826,79828,79830,79832,79835,79837,79840,79842],{"class":77,"line":69977},[75,79817,258],{"class":145},[75,79819,79820],{"class":15980},"High school level ",[75,79822,197],{"class":145},[75,79824,79825],{"class":15980},"concise",[75,79827,5175],{"class":145},[75,79829,41083],{"class":15980},[75,79831,258],{"class":145},[75,79833,79834],{"class":15980}," We treat each measurement ",[75,79836,197],{"class":145},[75,79838,79839],{"class":15980},"“single shot”",[75,79841,5175],{"class":145},[75,79843,79844],{"class":15980}," 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",[75,79846,79847],{"class":77,"line":69986},[75,79848,112],{"emptyLinePlaceholder":111},[75,79850,79851,79853,79856,79858,79860,79862,79864,79866,79869,79871,79874,79876,79879,79881,79884,79886,79889,79891,79894,79896,79899,79902,79905,79908,79910,79912,79915,79917,79919,79922,79925,79927],{"class":77,"line":69991},[75,79852,258],{"class":145},[75,79854,79855],{"class":15980},"Graduate level ",[75,79857,197],{"class":145},[75,79859,79825],{"class":15980},[75,79861,5175],{"class":145},[75,79863,41083],{"class":15980},[75,79865,258],{"class":145},[75,79867,79868],{"class":15980}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[75,79870,197],{"class":145},[75,79872,79873],{"class":15980},"noisy intermediate-scale quantum",[75,79875,5175],{"class":145},[75,79877,79878],{"class":15980},", fault-tolerant ",[75,79880,197],{"class":145},[75,79882,79883],{"class":15980},"logical",[75,79885,5175],{"class":145},[75,79887,79888],{"class":15980},", topologically protected, and ideal. We report Wilson score intervals ",[75,79890,197],{"class":145},[75,79892,79893],{"class":15980},"95%",[75,79895,5175],{"class":145},[75,79897,79898],{"class":15980}," for observed proportions and estimate the required shot budget $n ",[75,79900,79901],{"class":141},"\\a",[75,79903,79904],{"class":15980},"pprox z^2",[75,79906,79907],{"class":141},"\\,",[75,79909,12],{"class":15980},[75,79911,197],{"class":145},[75,79913,79914],{"class":15980},"1-p",[75,79916,5175],{"class":145},[75,79918,2810],{"class":15980},[75,79920,79921],{"class":141},"\\e",[75,79923,79924],{"class":15980},"psilon^2$ to bound absolute error by ±$",[75,79926,79921],{"class":141},[75,79928,79929],{"class":15980},"psilon$ at ~95% confidence.\n",[75,79931,79932],{"class":77,"line":69996},[75,79933,41449],{"class":15980},[75,79935,79936],{"class":77,"line":70001},[75,79937,112],{"emptyLinePlaceholder":111},[75,79939,79940,79943,79945,79947],{"class":77,"line":70011},[75,79941,79942],{"class":85},"code_config ",[75,79944,142],{"class":141},[75,79946,79802],{"class":81},[75,79948,41449],{"class":15980},[75,79950,79951],{"class":77,"line":70018},[75,79952,79953],{"class":118},"# @title Configuration\n",[75,79955,79956],{"class":77,"line":70025},[75,79957,79958],{"class":15980},"from dataclasses import dataclass\n",[75,79960,79961],{"class":77,"line":70036},[75,79962,79963],{"class":15980},"from typing import List, Dict\n",[75,79965,79966],{"class":77,"line":70047},[75,79967,79968],{"class":15980},"import numpy as np\n",[75,79970,79971],{"class":77,"line":70056},[75,79972,112],{"emptyLinePlaceholder":111},[75,79974,79975,79978],{"class":77,"line":70062},[75,79976,79977],{"class":15980},"RANDOM_SEED = 579345  ",[75,79979,79980],{"class":118},"# @param {type:\"number\"}\n",[75,79982,79983,79986],{"class":77,"line":70072},[75,79984,79985],{"class":15980},"SHOTS_PER_MODEL = 10000  ",[75,79987,79980],{"class":118},[75,79989,79990,79993,79996],{"class":77,"line":70080},[75,79991,79992],{"class":15980},"EPSILONS = ",[75,79994,79995],{"class":145},"[0.02, 0.01, 0.005]",[75,79997,79998],{"class":118},"  # @param\n",[75,80000,80001,80004],{"class":77,"line":70087},[75,80002,80003],{"class":15980},"COST_PER_SHOT_USD = 0.00005  ",[75,80005,79980],{"class":118},[75,80007,80008,80011],{"class":77,"line":70094},[75,80009,80010],{"class":15980},"FIXED_OVERHEAD_USD = 0.10    ",[75,80012,79980],{"class":118},[75,80014,80015],{"class":77,"line":70103},[75,80016,112],{"emptyLinePlaceholder":111},[75,80018,80019,80021,80023,80026],{"class":77,"line":70112},[75,80020,77499],{"class":15980},[75,80022,197],{"class":145},[75,80024,80025],{"class":15980},"frozen=True",[75,80027,318],{"class":145},[75,80029,80030],{"class":77,"line":70121},[75,80031,80032],{"class":15980},"class QubitModel:\n",[75,80034,80035],{"class":77,"line":70130},[75,80036,80037],{"class":15980},"    name: str\n",[75,80039,80040],{"class":77,"line":70141},[75,80041,80042],{"class":15980},"    satisfaction_prob: float\n",[75,80044,80045],{"class":77,"line":70153},[75,80046,80047],{"class":15980},"    description: str\n",[75,80049,80050],{"class":77,"line":70165},[75,80051,112],{"emptyLinePlaceholder":111},[75,80053,80054,80057,80060,80063],{"class":77,"line":70174},[75,80055,80056],{"class":15980},"MODELS: List",[75,80058,80059],{"class":145},"[QubitModel]",[75,80061,80062],{"class":15980}," = ",[75,80064,80065],{"class":145},"[\n",[75,80067,80068],{"class":77,"line":70179},[75,80069,80070],{"class":145},"    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[75,80072,80073],{"class":77,"line":70184},[75,80074,80075],{"class":145},"    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[75,80077,80078],{"class":77,"line":70189},[75,80079,80080],{"class":145},"    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[75,80082,80083],{"class":77,"line":70199},[75,80084,80085],{"class":145},"    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[75,80087,80088],{"class":77,"line":70206},[75,80089,80090],{"class":145},"    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[75,80092,80093],{"class":77,"line":70213},[75,80094,10451],{"class":145},[75,80096,80097],{"class":77,"line":70225},[75,80098,112],{"emptyLinePlaceholder":111},[75,80100,80101,80104,80106,80109],{"class":77,"line":70236},[75,80102,80103],{"class":15980},"rng = np.random.default_rng",[75,80105,197],{"class":145},[75,80107,80108],{"class":15980},"RANDOM_SEED",[75,80110,318],{"class":145},[75,80112,80113],{"class":77,"line":70245},[75,80114,41449],{"class":15980},[75,80116,80117],{"class":77,"line":70251},[75,80118,112],{"emptyLinePlaceholder":111},[75,80120,80121,80124,80126,80128],{"class":77,"line":70267},[75,80122,80123],{"class":85},"code_lib_and_run ",[75,80125,142],{"class":141},[75,80127,79802],{"class":81},[75,80129,41449],{"class":15980},[75,80131,80132],{"class":77,"line":70272},[75,80133,80134],{"class":118},"# @title Run Simulation & Display Results\n",[75,80136,80137],{"class":77,"line":70281},[75,80138,80139],{"class":15980},"import math\n",[75,80141,80142],{"class":77,"line":70291},[75,80143,80144],{"class":15980},"from typing import Tuple, Dict\n",[75,80146,80147],{"class":77,"line":70300},[75,80148,80149],{"class":15980},"import pandas as pd\n",[75,80151,80152],{"class":77,"line":70309},[75,80153,79968],{"class":15980},[75,80155,80156],{"class":77,"line":70315},[75,80157,80158],{"class":15980},"import matplotlib.pyplot as plt\n",[75,80160,80161],{"class":77,"line":70324},[75,80162,80163],{"class":15980},"from IPython.display import display, Markdown\n",[75,80165,80166],{"class":77,"line":70336},[75,80167,112],{"emptyLinePlaceholder":111},[75,80169,80170,80173,80175,80178,80180,80183,80186],{"class":77,"line":70348},[75,80171,80172],{"class":15980},"def wilson_ci",[75,80174,197],{"class":145},[75,80176,80177],{"class":15980},"k: int, n: int, z: float = 1.96",[75,80179,5175],{"class":145},[75,80181,80182],{"class":15980}," -> Tuple",[75,80184,80185],{"class":145},"[float, float]",[75,80187,5800],{"class":15980},[75,80189,80190,80193,80195,80198],{"class":77,"line":70360},[75,80191,80192],{"class":15980},"    if n == 0: return ",[75,80194,197],{"class":145},[75,80196,80197],{"class":15980},"0.0, 1.0",[75,80199,318],{"class":145},[75,80201,80202,80205,80207],{"class":77,"line":70365},[75,80203,80204],{"class":15980},"    p_hat, z2_n = k \u002F n, z",[75,80206,258],{"class":145},[75,80208,80209],{"class":15980},"2 \u002F n\n",[75,80211,80212,80215,80217],{"class":77,"line":70370},[75,80213,80214],{"class":15980},"    denom = 1.0 ",[75,80216,3251],{"class":145},[75,80218,80219],{"class":15980}," z2_n\n",[75,80221,80222,80225,80227,80230,80232,80235,80237],{"class":77,"line":70375},[75,80223,80224],{"class":15980},"    center = ",[75,80226,197],{"class":145},[75,80228,80229],{"class":15980},"p_hat ",[75,80231,3251],{"class":145},[75,80233,80234],{"class":15980}," z2_n \u002F 2",[75,80236,5175],{"class":145},[75,80238,80239],{"class":15980}," \u002F denom\n",[75,80241,80242,80245,80247,80250,80252,80255,80257,80259,80261,80263,80266,80268,80271,80273,80276,80278,80281,80283,80286,80288,80291,80293,80295,80298],{"class":77,"line":70393},[75,80243,80244],{"class":15980},"    span = ",[75,80246,197],{"class":145},[75,80248,80249],{"class":15980},"z ",[75,80251,500],{"class":145},[75,80253,80254],{"class":15980}," math.sqrt",[75,80256,197],{"class":145},[75,80258,80229],{"class":15980},[75,80260,500],{"class":145},[75,80262,267],{"class":145},[75,80264,80265],{"class":15980},"1 - p_hat",[75,80267,5175],{"class":145},[75,80269,80270],{"class":15980}," \u002F n ",[75,80272,3251],{"class":145},[75,80274,80275],{"class":15980}," z",[75,80277,258],{"class":145},[75,80279,80280],{"class":15980},"2 \u002F ",[75,80282,197],{"class":145},[75,80284,80285],{"class":15980},"4 ",[75,80287,500],{"class":145},[75,80289,80290],{"class":15980}," n",[75,80292,258],{"class":145},[75,80294,270],{"class":15980},[75,80296,80297],{"class":145},")))",[75,80299,80239],{"class":15980},[75,80301,80302,80305,80307,80309,80311,80314,80316,80319,80321,80324,80326,80329],{"class":77,"line":70399},[75,80303,80304],{"class":15980},"    return ",[75,80306,197],{"class":145},[75,80308,2816],{"class":15980},[75,80310,197],{"class":145},[75,80312,80313],{"class":15980},"0.0, center - span",[75,80315,5175],{"class":145},[75,80317,80318],{"class":15980},", min",[75,80320,197],{"class":145},[75,80322,80323],{"class":15980},"1.0, center ",[75,80325,3251],{"class":145},[75,80327,80328],{"class":15980}," span",[75,80330,283],{"class":145},[75,80332,80333],{"class":77,"line":70408},[75,80334,112],{"emptyLinePlaceholder":111},[75,80336,80337,80340,80342,80345,80347],{"class":77,"line":70415},[75,80338,80339],{"class":15980},"def shots_for_epsilon",[75,80341,197],{"class":145},[75,80343,80344],{"class":15980},"p: float, epsilon: float = 0.01, z: float = 1.96",[75,80346,5175],{"class":145},[75,80348,80349],{"class":15980}," -> int:\n",[75,80351,80352,80355,80357,80359,80361,80363],{"class":77,"line":70422},[75,80353,80354],{"class":15980},"    if p ",[75,80356,500],{"class":145},[75,80358,267],{"class":145},[75,80360,79914],{"class":15980},[75,80362,5175],{"class":145},[75,80364,80365],{"class":15980}," == 0: return 1\n",[75,80367,80368,80371,80373,80376,80378,80381,80383,80385,80387,80390,80392,80394,80396,80398,80400,80402,80405,80407,80409,80411,80413],{"class":77,"line":70430},[75,80369,80370],{"class":15980},"    return max",[75,80372,197],{"class":145},[75,80374,80375],{"class":15980},"1, int",[75,80377,197],{"class":145},[75,80379,80380],{"class":15980},"math.ceil",[75,80382,16298],{"class":145},[75,80384,31538],{"class":15980},[75,80386,258],{"class":145},[75,80388,80389],{"class":15980},"2 ",[75,80391,500],{"class":145},[75,80393,44854],{"class":15980},[75,80395,500],{"class":145},[75,80397,267],{"class":145},[75,80399,79914],{"class":15980},[75,80401,17487],{"class":145},[75,80403,80404],{"class":15980}," \u002F ",[75,80406,197],{"class":145},[75,80408,77318],{"class":15980},[75,80410,258],{"class":145},[75,80412,270],{"class":15980},[75,80414,80415],{"class":145},"))))\n",[75,80417,80418],{"class":77,"line":70441},[75,80419,112],{"emptyLinePlaceholder":111},[75,80421,80422,80425,80427,80430,80432],{"class":77,"line":70452},[75,80423,80424],{"class":15980},"def summarize_model",[75,80426,197],{"class":145},[75,80428,80429],{"class":15980},"model, bits: np.ndarray, epsilons",[75,80431,5175],{"class":145},[75,80433,80434],{"class":15980}," -> Dict:\n",[75,80436,80437,80440,80442,80444],{"class":77,"line":70463},[75,80438,80439],{"class":15980},"    n = len",[75,80441,197],{"class":145},[75,80443,78000],{"class":15980},[75,80445,318],{"class":145},[75,80447,80448,80451,80453,80455],{"class":77,"line":70468},[75,80449,80450],{"class":15980},"    ones = sum",[75,80452,197],{"class":145},[75,80454,78000],{"class":15980},[75,80456,318],{"class":145},[75,80458,80459,80462,80464,80467],{"class":77,"line":70477},[75,80460,80461],{"class":15980},"    lo, hi = wilson_ci",[75,80463,197],{"class":145},[75,80465,80466],{"class":15980},"ones, n",[75,80468,318],{"class":145},[75,80470,80471],{"class":77,"line":70484},[75,80472,80473],{"class":15980},"    row = {\n",[75,80475,80476,80479,80481,80484,80486],{"class":77,"line":70491},[75,80477,80478],{"class":15980},"        \"Model\": model.name, \"p ",[75,80480,197],{"class":145},[75,80482,80483],{"class":15980},"target",[75,80485,5175],{"class":145},[75,80487,80488],{"class":15980},"\": model.satisfaction_prob, \"Shots\": n,\n",[75,80490,80491,80494,80496,80499,80501],{"class":77,"line":70499},[75,80492,80493],{"class":15980},"        \"Ones\": ones, \"p ",[75,80495,197],{"class":145},[75,80497,80498],{"class":15980},"observed",[75,80500,5175],{"class":145},[75,80502,80503],{"class":15980},"\": ones \u002F n if n > 0 else 0,\n",[75,80505,80506],{"class":77,"line":70520},[75,80507,80508],{"class":15980},"        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[75,80510,80511],{"class":77,"line":70531},[75,80512,42218],{"class":15980},[75,80514,80515],{"class":77,"line":70542},[75,80516,80517],{"class":15980},"    for eps in epsilons:\n",[75,80519,80520,80523,80526,80529,80531,80534],{"class":77,"line":70553},[75,80521,80522],{"class":15980},"        row",[75,80524,80525],{"class":145},"[f\"Shots for ±{eps:.1%}\"]",[75,80527,80528],{"class":15980}," = shots_for_epsilon",[75,80530,197],{"class":145},[75,80532,80533],{"class":15980},"model.satisfaction_prob, eps",[75,80535,318],{"class":145},[75,80537,80538],{"class":77,"line":70564},[75,80539,80540],{"class":15980},"    return row\n",[75,80542,80543],{"class":77,"line":70569},[75,80544,112],{"emptyLinePlaceholder":111},[75,80546,80547],{"class":77,"line":70574},[75,80548,80549],{"class":118},"# --- Run Simulation ---\n",[75,80551,80552,80555],{"class":77,"line":70579},[75,80553,80554],{"class":15980},"summary_rows = ",[75,80556,80557],{"class":145},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[75,80559,80560,80563,80565,80568],{"class":77,"line":70593},[75,80561,80562],{"class":15980},"summary_df = pd.DataFrame",[75,80564,197],{"class":145},[75,80566,80567],{"class":15980},"summary_rows",[75,80569,318],{"class":145},[75,80571,80572],{"class":77,"line":70605},[75,80573,80574],{"class":15980},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[75,80576,80577,80580,80583,80586,80588,80591,80594,80596],{"class":77,"line":70611},[75,80578,80579],{"class":15980},"    summary_df",[75,80581,80582],{"class":145},"[\"Cost (USD)\"]",[75,80584,80585],{"class":15980}," = FIXED_OVERHEAD_USD ",[75,80587,3251],{"class":145},[75,80589,80590],{"class":15980}," summary_df",[75,80592,80593],{"class":145},"[\"Shots\"]",[75,80595,273],{"class":145},[75,80597,80598],{"class":15980}," COST_PER_SHOT_USD\n",[75,80600,80601],{"class":77,"line":70616},[75,80602,112],{"emptyLinePlaceholder":111},[75,80604,80605,80607,80609,80611,80613,80615],{"class":77,"line":70623},[75,80606,45886],{"class":15980},[75,80608,197],{"class":145},[75,80610,79631],{"class":15980},[75,80612,197],{"class":145},[75,80614,12216],{"class":15980},[75,80616,80617],{"class":118},"### Simulation Summary\"))\n",[75,80619,80620,80622,80624,80627,80629,80632,80634,80637,80639,80642],{"class":77,"line":70633},[75,80621,45886],{"class":15980},[75,80623,197],{"class":145},[75,80625,80626],{"class":15980},"summary_df.style.format",[75,80628,197],{"class":145},[75,80630,80631],{"class":15980},"precision=4",[75,80633,5175],{"class":145},[75,80635,80636],{"class":15980},".background_gradient",[75,80638,197],{"class":145},[75,80640,80641],{"class":15980},"cmap='viridis', subset=",[75,80643,80644],{"class":145},"['p (observed)', 'CI width']))\n",[75,80646,80647],{"class":77,"line":70638},[75,80648,112],{"emptyLinePlaceholder":111},[75,80650,80651],{"class":77,"line":70650},[75,80652,80653],{"class":118},"# --- Plotting ---\n",[75,80655,80656,80659,80661,80663],{"class":77,"line":70661},[75,80657,80658],{"class":15980},"plt.style.use",[75,80660,197],{"class":145},[75,80662,78872],{"class":15980},[75,80664,318],{"class":145},[75,80666,80667,80670,80672,80675,80677,80680,80682,80685,80687,80690],{"class":77,"line":70671},[75,80668,80669],{"class":15980},"fig, ",[75,80671,197],{"class":145},[75,80673,80674],{"class":15980},"ax1, ax2",[75,80676,5175],{"class":145},[75,80678,80679],{"class":15980}," = plt.subplots",[75,80681,197],{"class":145},[75,80683,80684],{"class":15980},"1, 2, figsize=",[75,80686,197],{"class":145},[75,80688,80689],{"class":15980},"16, 6",[75,80691,283],{"class":145},[75,80693,80694],{"class":77,"line":70682},[75,80695,112],{"emptyLinePlaceholder":111},[75,80697,80698],{"class":77,"line":70687},[75,80699,80700],{"class":118},"# Plot 1\n",[75,80702,80703,80706,80709,80712],{"class":77,"line":70692},[75,80704,80705],{"class":15980},"y = summary_df",[75,80707,80708],{"class":145},"[\"p (observed)\"]",[75,80710,80711],{"class":15980},".to_numpy",[75,80713,389],{"class":145},[75,80715,80716,80719],{"class":77,"line":70704},[75,80717,80718],{"class":15980},"lower_error = y - summary_df",[75,80720,80721],{"class":145},"[\"CI 95% low\"]\n",[75,80723,80724,80727,80730],{"class":77,"line":70715},[75,80725,80726],{"class":15980},"upper_error = summary_df",[75,80728,80729],{"class":145},"[\"CI 95% high\"]",[75,80731,80732],{"class":15980}," - y\n",[75,80734,80735,80738],{"class":77,"line":70724},[75,80736,80737],{"class":15980},"yerr = np.vstack",[75,80739,80740],{"class":145},"([np.abs(lower_error), np.abs(upper_error)])\n",[75,80742,80743],{"class":77,"line":70735},[75,80744,112],{"emptyLinePlaceholder":111},[75,80746,80747,80750,80752,80755,80758,80761],{"class":77,"line":70740},[75,80748,80749],{"class":15980},"ax1.bar",[75,80751,197],{"class":145},[75,80753,80754],{"class":15980},"summary_df",[75,80756,80757],{"class":145},"[\"Model\"]",[75,80759,80760],{"class":15980},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[75,80762,318],{"class":145},[75,80764,80765,80768,80770,80773,80775,80778,80780,80783],{"class":77,"line":70745},[75,80766,80767],{"class":15980},"ax1.set_title",[75,80769,197],{"class":145},[75,80771,80772],{"class":15980},"\"Observed Success Rate ",[75,80774,197],{"class":145},[75,80776,80777],{"class":15980},"95% Wilson CI",[75,80779,5175],{"class":145},[75,80781,80782],{"class":15980},"\", fontsize=14",[75,80784,318],{"class":145},[75,80786,80787,80790,80792,80795],{"class":77,"line":70755},[75,80788,80789],{"class":15980},"ax1.tick_params",[75,80791,197],{"class":145},[75,80793,80794],{"class":15980},"axis='x', rotation=30, labelsize=10",[75,80796,318],{"class":145},[75,80798,80799],{"class":77,"line":70761},[75,80800,112],{"emptyLinePlaceholder":111},[75,80802,80803],{"class":77,"line":70775},[75,80804,80805],{"class":118},"# Plot 2\n",[75,80807,80808],{"class":77,"line":70780},[75,80809,80810],{"class":15980},"width = 0.2\n",[75,80812,80813,80816,80818,80820,80822,80825],{"class":77,"line":70790},[75,80814,80815],{"class":15980},"x = np.arange",[75,80817,197],{"class":145},[75,80819,2930],{"class":15980},[75,80821,197],{"class":145},[75,80823,80824],{"class":15980},"MODELS",[75,80826,283],{"class":145},[75,80828,80829,80832],{"class":77,"line":70795},[75,80830,80831],{"class":15980},"colors = ",[75,80833,80834],{"class":145},"['#ff9999','#66b3ff','#99ff99']\n",[75,80836,80837,80840,80842,80845,80847],{"class":77,"line":70807},[75,80838,80839],{"class":15980},"for i, eps in enumerate",[75,80841,197],{"class":145},[75,80843,80844],{"class":15980},"EPSILONS",[75,80846,5175],{"class":145},[75,80848,5800],{"class":15980},[75,80850,80851,80854],{"class":77,"line":70812},[75,80852,80853],{"class":15980},"    req = summary_df",[75,80855,80856],{"class":145},"[f\"Shots for ±{eps:.1%}\"]\n",[75,80858,80859,80862,80864,80866,80868,80870,80872,80875,80878,80881],{"class":77,"line":70817},[75,80860,80861],{"class":15980},"    ax2.bar",[75,80863,197],{"class":145},[75,80865,17925],{"class":15980},[75,80867,3251],{"class":145},[75,80869,72709],{"class":15980},[75,80871,500],{"class":145},[75,80873,80874],{"class":15980}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[75,80876,80877],{"class":145},"[i]",[75,80879,80880],{"class":15980},", edgecolor='black'",[75,80882,318],{"class":145},[75,80884,80885],{"class":77,"line":70839},[75,80886,112],{"emptyLinePlaceholder":111},[75,80888,80889,80892,80894,80896,80898,80901],{"class":77,"line":70844},[75,80890,80891],{"class":15980},"ax2.set_xticks",[75,80893,197],{"class":145},[75,80895,17925],{"class":15980},[75,80897,3251],{"class":145},[75,80899,80900],{"class":15980}," width, summary_df",[75,80902,80903],{"class":145},"[\"Model\"])\n",[75,80905,80906,80909,80911,80914,80916,80919,80921,80923],{"class":77,"line":70867},[75,80907,80908],{"class":15980},"ax2.set_ylabel",[75,80910,197],{"class":145},[75,80912,80913],{"class":15980},"\"Required Shots ",[75,80915,197],{"class":145},[75,80917,80918],{"class":15980},"Log Scale",[75,80920,5175],{"class":145},[75,80922,12216],{"class":15980},[75,80924,318],{"class":145},[75,80926,80927,80930,80932,80934],{"class":77,"line":70878},[75,80928,80929],{"class":15980},"ax2.set_yscale",[75,80931,197],{"class":145},[75,80933,79524],{"class":15980},[75,80935,318],{"class":145},[75,80937,80938,80941,80943,80946],{"class":77,"line":70917},[75,80939,80940],{"class":15980},"ax2.set_title",[75,80942,197],{"class":145},[75,80944,80945],{"class":15980},"\"Shots Needed for ±ε Tolerance\", fontsize=14",[75,80947,318],{"class":145},[75,80949,80950,80953,80955,80957],{"class":77,"line":70927},[75,80951,80952],{"class":15980},"ax2.tick_params",[75,80954,197],{"class":145},[75,80956,80794],{"class":15980},[75,80958,318],{"class":145},[75,80960,80961,80964,80966,80969],{"class":77,"line":70937},[75,80962,80963],{"class":15980},"ax2.legend",[75,80965,197],{"class":145},[75,80967,80968],{"class":15980},"title=\"Tolerance\"",[75,80970,318],{"class":145},[75,80972,80973],{"class":77,"line":70956},[75,80974,112],{"emptyLinePlaceholder":111},[75,80976,80977,80980],{"class":77,"line":70974},[75,80978,80979],{"class":15980},"plt.tight_layout",[75,80981,389],{"class":145},[75,80983,80984,80987],{"class":77,"line":70979},[75,80985,80986],{"class":15980},"plt.show",[75,80988,389],{"class":145},[75,80990,80991],{"class":77,"line":70984},[75,80992,41449],{"class":15980},[75,80994,80995],{"class":77,"line":70996},[75,80996,112],{"emptyLinePlaceholder":111},[75,80998,80999,81002,81004,81007,81010],{"class":77,"line":71006},[75,81000,81001],{"class":85},"nb 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nbf.v4.",[75,81032,81033],{"class":193},"new_markdown_cell",[75,81035,81036],{"class":85},"(md_intro),\n",[75,81038,81039,81041,81044],{"class":77,"line":71026},[75,81040,81030],{"class":85},[75,81042,81043],{"class":193},"new_code_cell",[75,81045,81046],{"class":85},"(code_config),\n",[75,81048,81049,81051,81053],{"class":77,"line":71034},[75,81050,81030],{"class":85},[75,81052,81043],{"class":193},[75,81054,81055],{"class":85},"(code_lib_and_run),\n",[75,81057,81058],{"class":77,"line":71044},[75,81059,10451],{"class":85},[75,81061,81062],{"class":77,"line":71059},[75,81063,112],{"emptyLinePlaceholder":111},[75,81065,81066,81069,81071],{"class":77,"line":71070},[75,81067,81068],{"class":85},"nb_path ",[75,81070,142],{"class":141},[75,81072,81073],{"class":134}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[75,81075,81076,81078,81080,81083,81085,81087,81090,81092,81095,81097,81099],{"class":77,"line":71082},[75,81077,78782],{"class":81},[75,81079,78785],{"class":141},[75,81081,81082],{"class":85},"(nb_path, ",[75,81084,78795],{"class":134},[75,81086,206],{"class":85},[75,81088,81089],{"class":309},"encoding",[75,81091,142],{"class":141},[75,81093,81094],{"class":134},"\"utf-8\"",[75,81096,14860],{"class":85},[75,81098,89],{"class":81},[75,81100,78802],{"class":85},[75,81102,81103,81106,81109],{"class":77,"line":71087},[75,81104,81105],{"class":85},"    nbf.",[75,81107,81108],{"class":193},"write",[75,81110,81111],{"class":85},"(nb, f)\n",[75,81113,81114],{"class":77,"line":71094},[75,81115,112],{"emptyLinePlaceholder":111},[75,81117,81118],{"class":77,"line":71103},[75,81119,81120],{"class":118},"# ---------- Final confirmation ----------\n",[75,81122,81123,81125,81127,81129,81131,81133,81135,81138,81140,81142],{"class":77,"line":71117},[75,81124,45904],{"class":141},[75,81126,197],{"class":85},[75,81128,12216],{"class":134},[75,81130,44377],{"class":141},[75,81132,12216],{"class":134},[75,81134,11902],{"class":141},[75,81136,81137],{"class":134}," \"=\"",[75,81139,500],{"class":141},[75,81141,58284],{"class":145},[75,81143,318],{"class":85},[75,81145,81146,81148,81150,81153],{"class":77,"line":71128},[75,81147,45904],{"class":141},[75,81149,197],{"class":85},[75,81151,81152],{"class":134},"\"SCRIPT COMPLETE\"",[75,81154,318],{"class":85},[75,81156,81157,81159,81161,81164,81166,81168],{"class":77,"line":71140},[75,81158,45904],{"class":141},[75,81160,197],{"class":85},[75,81162,81163],{"class":134},"\"=\"",[75,81165,500],{"class":141},[75,81167,58284],{"class":145},[75,81169,318],{"class":85},[75,81171,81172,81174,81176,81178,81181,81183,81186,81188,81190],{"class":77,"line":71145},[75,81173,45904],{"class":141},[75,81175,197],{"class":85},[75,81177,5894],{"class":81},[75,81179,81180],{"class":134},"\"Google Colab notebook saved to: ",[75,81182,12219],{"class":145},[75,81184,81185],{"class":85},"nb_path",[75,81187,12224],{"class":145},[75,81189,12216],{"class":134},[75,81191,318],{"class":85},[75,81193,81194,81196,81198,81200,81203,81205,81208,81210,81212],{"class":77,"line":71150},[75,81195,45904],{"class":141},[75,81197,197],{"class":85},[75,81199,5894],{"class":81},[75,81201,81202],{"class":134},"\"Summary data saved to: ",[75,81204,12219],{"class":145},[75,81206,81207],{"class":85},"summary_csv_path",[75,81209,12224],{"class":145},[75,81211,12216],{"class":134},[75,81213,318],{"class":85},[75,81215,81216,81218,81220,81222,81225,81227,81230,81232,81234,81236,81239,81241,81243],{"class":77,"line":71160},[75,81217,45904],{"class":141},[75,81219,197],{"class":85},[75,81221,5894],{"class":81},[75,81223,81224],{"class":134},"\"Plots saved to: ",[75,81226,12219],{"class":145},[75,81228,81229],{"class":85},"fig1_path",[75,81231,12224],{"class":145},[75,81233,206],{"class":134},[75,81235,12219],{"class":145},[75,81237,81238],{"class":85},"fig2_path",[75,81240,12224],{"class":145},[75,81242,12216],{"class":134},[75,81244,318],{"class":85},[65,81246,81249],{"className":81247,"code":81248,"language":34131,"meta":70},[41256],"Running simulation...\nSaved data to CSV and JSON files.\n",[72,81250,81248],{"__ignoreMap":70},[391,81252],{"alt":67668,"src":81253},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[391,81255],{"alt":72496,"src":81256},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[65,81258,81261],{"className":81259,"code":81260,"language":34131,"meta":70},[41256],"Generated and saved plots.\n",[72,81262,81260],{"__ignoreMap":70},[65,81264,81267],{"className":81265,"code":81266,"language":34131,"meta":70},[41256],"==================================================\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",[72,81268,81266],{"__ignoreMap":70},[7621,81270],{},[12425,81272,13965],{"id":13964},[81274,81275,81276,81283,81290,81297,81304,81311,81318,81325,81332,81339,81346,81353,81360,81371,81378,81385],"ol",{},[31,81277,81278,81279,81282],{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[15,81280,81281],{},"The American Statistician",", 52(2), pp. 119-126.",[31,81284,81285,81286,81289],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[15,81287,81288],{},"Statistical Science",", 16(2), pp. 101-133.",[31,81291,81292,81293,81296],{},"Cochran, W.G. (1977) ",[15,81294,81295],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[31,81298,81299,81300,81303],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[15,81301,81302],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[31,81305,81306,81307,81310],{},"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', ",[15,81308,81309],{},"Nature",", 585, pp. 357–362.",[31,81312,81313,81314,81317],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[15,81315,81316],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[31,81319,81320,81321,81324],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[15,81322,81323],{},"Annals of Physics",", 303(1), pp. 2-30.",[31,81326,81327,81328,81331],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[15,81329,81330],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[31,81333,81334,81335,81338],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[15,81336,81337],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[31,81340,81341,81342,81345],{},"Newman, M. (2013) ",[15,81343,81344],{},"Computational Physics",". CreateSpace Independent Publishing Platform.",[31,81347,81348,81349,81352],{},"Nielsen, M.A. and Chuang, I.L. (2010) ",[15,81350,81351],{},"Quantum Computation and Quantum Information: 10th Anniversary Edition",". Cambridge: Cambridge University Press.",[31,81354,81355,81356,81359],{},"Preskill, J. (2018) 'Quantum Computing in the NISQ era and beyond', ",[15,81357,81358],{},"Quantum",", 2, p. 79.",[31,81361,81362,81363,81366,81367,6543],{},"The pandas development team (2020) ",[15,81364,81365],{},"pandas-dev\u002Fpandas: Pandas",". Zenodo. Available at: ",[19,81368,81370],{"href":81369},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",[31,81372,81373,81374,81377],{},"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', ",[15,81375,81376],{},"Nature Methods",", 17, pp. 261–272.",[31,81379,81380,81381,81384],{},"Wilde, M.M. (2017) ",[15,81382,81383],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[31,81386,81387,81388,81391],{},"Wilson, E.B. (1927) 'Probable inference, the law of succession, and statistical inference', ",[15,81389,81390],{},"Journal of the American Statistical Association",", 22(158), pp. 209-212.",[12425,81393,76786],{"id":76785},[12,81395,76789,81396,76792,81398,6543],{},[19,81397,22],{"href":76848},[19,81399,76796],{"href":81400},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[12364,81402,81403],{},"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":70,"searchDepth":95,"depth":95,"links":81405},[81406,81407],{"id":13964,"depth":95,"text":13965},{"id":76785,"depth":95,"text":76786},[12374,12375,12376,81409],"Sampling budget",[81411],{"username":7,"name":12380,"role":12381,"bio":12382,"links":81412},[81413,81414],{"label":12385,"href":12386},{"label":12388,"href":76848},{"username":7,"name":12380,"role":12390},"A Sampling-Budget & Bit-String Analogy for Qubit Classifications. 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