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bands away from poles suggest high entangling-rate with low static ",[61,1263,1265,1280],{"className":1264},[64],[61,1266,1268],{"className":1267},[68],[70,1269,1270],{"xmlns":72},[74,1271,1272,1278],{},[77,1273,1274,1276],{},[83,1275,90],{},[83,1277,90],{},[95,1279,520],{"encoding":97},[61,1281,1283],{"className":1282,"ariaHidden":103},[102],[61,1284,1286,1289,1292],{"className":1285},[107],[61,1287],{"className":1288,"style":296},[111],[61,1290,90],{"className":1291,"style":161},[116,120],[61,1293,90],{"className":1294,"style":161},[116,120]," (good for echoed-CR).",[36,1297,1298,1301,1304],{},[55,1299,1300],{"align":41},"6",[55,1302,1303],{},"Echoed-CR coefficients (two-segment echo)",[55,1305,1306,1307,1335,1336,1370,1371,1399,1400,1435,1436,1469],{},"Echo preserves odd-in-",[61,1308,1310,1323],{"className":1309},[64],[61,1311,1313],{"className":1312},[68],[70,1314,1315],{"xmlns":72},[74,1316,1317,1321],{},[77,1318,1319],{},[83,1320,283],{"mathvariant":197},[95,1322,286],{"encoding":97},[61,1324,1326],{"className":1325,"ariaHidden":103},[102],[61,1327,1329,1332],{"className":1328},[107],[61,1330],{"className":1331,"style":296},[111],[61,1333,283],{"className":1334},[116]," ",[61,1337,1339,1355],{"className":1338},[64],[61,1340,1342],{"className":1341},[68],[70,1343,1344],{"xmlns":72},[74,1345,1346,1352],{},[77,1347,1348,1350],{},[83,1349,90],{},[83,1351,93],{},[95,1353,1354],{"encoding":97},"ZX",[61,1356,1358],{"className":1357,"ariaHidden":103},[102],[61,1359,1361,1364,1367],{"className":1360},[107],[61,1362],{"className":1363,"style":296},[111],[61,1365,90],{"className":1366,"style":161},[116,120],[61,1368,93],{"className":1369,"style":165},[116,120]," and suppresses even-in-",[61,1372,1374,1387],{"className":1373},[64],[61,1375,1377],{"className":1376},[68],[70,1378,1379],{"xmlns":72},[74,1380,1381,1385],{},[77,1382,1383],{},[83,1384,283],{"mathvariant":197},[95,1386,286],{"encoding":97},[61,1388,1390],{"className":1389,"ariaHidden":103},[102],[61,1391,1393,1396],{"className":1392},[107],[61,1394],{"className":1395,"style":296},[111],[61,1397,283],{"className":1398},[116]," terms (",[61,1401,1403,1420],{"className":1402},[64],[61,1404,1406],{"className":1405},[68],[70,1407,1408],{"xmlns":72},[74,1409,1410,1417],{},[77,1411,1412,1415],{},[83,1413,1414],{},"I",[83,1416,90],{},[95,1418,1419],{"encoding":97},"IZ",[61,1421,1423],{"className":1422,"ariaHidden":103},[102],[61,1424,1426,1429,1432],{"className":1425},[107],[61,1427],{"className":1428,"style":296},[111],[61,1430,1414],{"className":1431,"style":165},[116,120],[61,1433,90],{"className":1434,"style":161},[116,120],", ",[61,1437,1439,1454],{"className":1438},[64],[61,1440,1442],{"className":1441},[68],[70,1443,1444],{"xmlns":72},[74,1445,1446,1452],{},[77,1447,1448,1450],{},[83,1449,90],{},[83,1451,90],{},[95,1453,520],{"encoding":97},[61,1455,1457],{"className":1456,"ariaHidden":103},[102],[61,1458,1460,1463,1466],{"className":1459},[107],[61,1461],{"className":1462,"style":296},[111],[61,1464,90],{"className":1465,"style":161},[116,120],[61,1467,90],{"className":1468,"style":161},[116,120],") to first order.",[36,1471,1472,1475,1559],{},[55,1473,1474],{"align":41},"7",[55,1476,1477,1478,180,1520,300],{},"Conditional target Rabi (control ",[61,1479,1481,1502],{"className":1480},[64],[61,1482,1484],{"className":1483},[68],[70,1485,1486],{"xmlns":72},[74,1487,1488,1499],{},[77,1489,1490,1492,1496],{},[586,1491,589],{"stretchy":588},[1493,1494,1495],"mn",{},"0",[586,1497,1498],{"stretchy":588},"⟩",[95,1500,1501],{"encoding":97},"\\lvert0\\rangle",[61,1503,1505],{"className":1504,"ariaHidden":103},[102],[61,1506,1508,1511,1514,1517],{"className":1507},[107],[61,1509],{"className":1510,"style":627},[111],[61,1512,589],{"className":1513},[631],[61,1515,1495],{"className":1516},[116],[61,1518,1498],{"className":1519},[730],[61,1521,1523,1541],{"className":1522},[64],[61,1524,1526],{"className":1525},[68],[70,1527,1528],{"xmlns":72},[74,1529,1530,1538],{},[77,1531,1532,1534,1536],{},[586,1533,589],{"stretchy":588},[1493,1535,57],{},[586,1537,1498],{"stretchy":588},[95,1539,1540],{"encoding":97},"\\lvert1\\rangle",[61,1542,1544],{"className":1543,"ariaHidden":103},[102],[61,1545,1547,1550,1553,1556],{"className":1546},[107],[61,1548],{"className":1549,"style":627},[111],[61,1551,589],{"className":1552},[631],[61,1554,57],{"className":1555},[116],[61,1557,1498],{"className":1558},[730],[55,1560,1561,1562,1595,1596,1630],{},"Target oscillation frequency\u002Fphase depends on control state due to the ",[61,1563,1565,1580],{"className":1564},[64],[61,1566,1568],{"className":1567},[68],[70,1569,1570],{"xmlns":72},[74,1571,1572,1578],{},[77,1573,1574,1576],{},[83,1575,90],{},[83,1577,93],{},[95,1579,1354],{"encoding":97},[61,1581,1583],{"className":1582,"ariaHidden":103},[102],[61,1584,1586,1589,1592],{"className":1585},[107],[61,1587],{"className":1588,"style":296},[111],[61,1590,90],{"className":1591,"style":161},[116,120],[61,1593,93],{"className":1594,"style":165},[116,120]," term (and any residual ",[61,1597,1599,1615],{"className":1598},[64],[61,1600,1602],{"className":1601},[68],[70,1603,1604],{"xmlns":72},[74,1605,1606,1612],{},[77,1607,1608,1610],{},[83,1609,1414],{},[83,1611,93],{},[95,1613,1614],{"encoding":97},"IX",[61,1616,1618],{"className":1617,"ariaHidden":103},[102],[61,1619,1621,1624,1627],{"className":1620},[107],[61,1622],{"className":1623,"style":296},[111],[61,1625,1414],{"className":1626,"style":165},[116,120],[61,1628,93],{"className":1629,"style":165},[116,120],").",[1632,1633],"hr",{},[1635,1636,1638],"h3",{"id":1637},"references","References",[1640,1641,1642,1690,1705],"ul",{},[1643,1644,1645,1646,1649,1650,1683,1684,1687],"li",{},"The Quantum Aviary — ",[15,1647,1648],{},"How the Cross-Resonance Gate Works"," (overview of ",[61,1651,1653,1668],{"className":1652},[64],[61,1654,1656],{"className":1655},[68],[70,1657,1658],{"xmlns":72},[74,1659,1660,1666],{},[77,1661,1662,1664],{},[83,1663,90],{},[83,1665,93],{},[95,1667,1354],{"encoding":97},[61,1669,1671],{"className":1670,"ariaHidden":103},[102],[61,1672,1674,1677,1680],{"className":1673},[107],[61,1675],{"className":1676,"style":296},[111],[61,1678,90],{"className":1679,"style":161},[116,120],[61,1681,93],{"className":1682,"style":165},[116,120],", echo, and intuition)",[1685,1686],"br",{},[19,1688,1689],{"href":1689},"https:\u002F\u002Fthequantumaviary.blogspot.com\u002F2021\u002F07\u002Fhow-cross-resonance-gate-works.html",[1643,1691,1692,1693,1435,1696,1699,1700,1702],{},"Malekakhlagh, Magesan & McKay — ",[15,1694,1695],{},"First-principles analysis of cross-resonance gate operation",[814,1697,1698],{},"Phys. Rev. A"," 102, 042605 (2020)",[1685,1701],{},[19,1703,1704],{"href":1704},"https:\u002F\u002Flink.aps.org\u002Fdoi\u002F10.1103\u002FPhysRevA.102.042605",[1643,1706,1707,1708,1435,1711,1713,1714,1716],{},"Magesan & Gambetta — ",[15,1709,1710],{},"Effective Hamiltonian models of the cross-resonance gate",[814,1712,1698],{}," 101, 052308 (2020)",[1685,1715],{},[19,1717,1718],{"href":1718},"https:\u002F\u002Flink.aps.org\u002Fdoi\u002F10.1103\u002FPhysRevA.101.052308",[1720,1721,1726],"pre",{"className":1722,"code":1723,"language":1724,"meta":1725,"style":1725},"language-python shiki shiki-themes one-dark-pro","#@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","python","",[1727,1728,1729,1737,1744,1761,1774,1782,1796,1809,1814,1820,1826,1831,1838,1851,1871,1887,1906,1923,1939,1955,1972,1977,1983,2017,2031,2059,2072,2077,2090,2095,2100,2106,2111,2160,2167,2185,2205,2219,2224,2256,2262,2278,2302,2324,2340,2377,2394,2402,2441,2473,2518,2539,2544,2580,2586,2599,2657,2715,2723,2728,2747,2753,2776,2781,2789,2802,2815,2822,2833,2844,2849,2854,2860,2865,2915,2924,2970,2999,3052,3058,3075,3080,3127,3135,3183,3206,3246,3284,3289,3304,3309,3315,3356,3372,3388,3420,3425,3430,3436,3441,3467,3492,3497,3532,3562,3600,3605,3611,3649,3713,3736,3751,3780,3811,3833,3843,3853,3858,3864,3895,3941,3962,3973,3996,4023,4044,4053,4062,4067,4073,4099,4118,4136,4149,4154,4185,4232,4281,4313,4342,4365,4409,4418,4423,4429,4445,4462,4475,4492,4504,4539,4544,4570,4591,4596,4627,4660,4697,4724,4747,4787],"code",{"__ignoreMap":1725},[61,1730,1733],{"class":1731,"line":1732},"line",1,[61,1734,1736],{"class":1735},"sV9Aq","#@title Cross-Resonance (CR) gate — slices & heatmaps (PRA 2020 formulas)\n",[61,1738,1740],{"class":1731,"line":1739},2,[61,1741,1743],{"emptyLinePlaceholder":1742},true,"\n",[61,1745,1747,1751,1755,1758],{"class":1731,"line":1746},3,[61,1748,1750],{"class":1749},"seHd6","from",[61,1752,1754],{"class":1753},"sn6KH"," dataclasses ",[61,1756,1757],{"class":1749},"import",[61,1759,1760],{"class":1753}," dataclass\n",[61,1762,1764,1766,1769,1771],{"class":1731,"line":1763},4,[61,1765,1750],{"class":1749},[61,1767,1768],{"class":1753}," typing ",[61,1770,1757],{"class":1749},[61,1772,1773],{"class":1753}," Tuple\n",[61,1775,1777,1779],{"class":1731,"line":1776},5,[61,1778,1757],{"class":1749},[61,1780,1781],{"class":1753}," math\n",[61,1783,1785,1787,1790,1793],{"class":1731,"line":1784},6,[61,1786,1757],{"class":1749},[61,1788,1789],{"class":1753}," numpy ",[61,1791,1792],{"class":1749},"as",[61,1794,1795],{"class":1753}," np\n",[61,1797,1799,1801,1804,1806],{"class":1731,"line":1798},7,[61,1800,1757],{"class":1749},[61,1802,1803],{"class":1753}," matplotlib.pyplot ",[61,1805,1792],{"class":1749},[61,1807,1808],{"class":1753}," plt\n",[61,1810,1812],{"class":1731,"line":1811},8,[61,1813,1743],{"emptyLinePlaceholder":1742},[61,1815,1817],{"class":1731,"line":1816},9,[61,1818,1819],{"class":1735},"# -----------------------------\n",[61,1821,1823],{"class":1731,"line":1822},10,[61,1824,1825],{"class":1735},"# Control knobs (MHz)\n",[61,1827,1829],{"class":1731,"line":1828},11,[61,1830,1819],{"class":1735},[61,1832,1834],{"class":1731,"line":1833},12,[61,1835,1837],{"class":1836},"sVbv2","@dataclass\n",[61,1839,1841,1844,1848],{"class":1731,"line":1840},13,[61,1842,1843],{"class":1749},"class",[61,1845,1847],{"class":1846},"sU0A5"," CRKnobs",[61,1849,1850],{"class":1753},":\n",[61,1852,1854,1857,1861,1864,1868],{"class":1731,"line":1853},14,[61,1855,1856],{"class":1753},"    f_c: ",[61,1858,1860],{"class":1859},"sjrmR","float",[61,1862,1863],{"class":1859}," =",[61,1865,1867],{"class":1866},"sVC51"," 5114.0",[61,1869,1870],{"class":1735},"        # control frequency\n",[61,1872,1874,1877,1879,1881,1884],{"class":1731,"line":1873},15,[61,1875,1876],{"class":1753},"    f_t: ",[61,1878,1860],{"class":1859},[61,1880,1863],{"class":1859},[61,1882,1883],{"class":1866}," 4914.0",[61,1885,1886],{"class":1735},"        # target frequency\n",[61,1888,1890,1893,1895,1897,1900,1903],{"class":1731,"line":1889},16,[61,1891,1892],{"class":1753},"    alpha_c: ",[61,1894,1860],{"class":1859},[61,1896,1863],{"class":1859},[61,1898,1899],{"class":1859}," -",[61,1901,1902],{"class":1866},"330.0",[61,1904,1905],{"class":1735},"    # control anharmonicity (negative)\n",[61,1907,1909,1912,1914,1916,1918,1920],{"class":1731,"line":1908},17,[61,1910,1911],{"class":1753},"    alpha_t: ",[61,1913,1860],{"class":1859},[61,1915,1863],{"class":1859},[61,1917,1899],{"class":1859},[61,1919,1902],{"class":1866},[61,1921,1922],{"class":1735},"    # target anharmonicity\n",[61,1924,1926,1929,1931,1933,1936],{"class":1731,"line":1925},18,[61,1927,1928],{"class":1753},"    J: ",[61,1930,1860],{"class":1859},[61,1932,1863],{"class":1859},[61,1934,1935],{"class":1866}," 3.8",[61,1937,1938],{"class":1735},"             # exchange coupling\n",[61,1940,1942,1945,1947,1949,1952],{"class":1731,"line":1941},19,[61,1943,1944],{"class":1753},"    Omega0: ",[61,1946,1860],{"class":1859},[61,1948,1863],{"class":1859},[61,1950,1951],{"class":1866}," 30.0",[61,1953,1954],{"class":1735},"       # default drive for Δ-slices\n",[61,1956,1958,1961,1964,1966,1969],{"class":1731,"line":1957},20,[61,1959,1960],{"class":1753},"    use_energy_basis: ",[61,1962,1963],{"class":1859},"bool",[61,1965,1863],{"class":1859},[61,1967,1968],{"class":1866}," False",[61,1970,1971],{"class":1735},"  # toggle ν-corrections (~10–15%)\n",[61,1973,1975],{"class":1731,"line":1974},21,[61,1976,1743],{"emptyLinePlaceholder":1742},[61,1978,1980],{"class":1731,"line":1979},22,[61,1981,1982],{"class":1735},"    # Sweep grids\n",[61,1984,1986,1989,1991,1993,1995,1998,2001,2004,2007,2010,2012,2014],{"class":1731,"line":1985},23,[61,1987,1988],{"class":1753},"    delta_span: Tuple[",[61,1990,1860],{"class":1859},[61,1992,1435],{"class":1753},[61,1994,1860],{"class":1859},[61,1996,1997],{"class":1753},"] ",[61,1999,2000],{"class":1859},"=",[61,2002,2003],{"class":1753}," (",[61,2005,2006],{"class":1859},"-",[61,2008,2009],{"class":1866},"500.0",[61,2011,1435],{"class":1753},[61,2013,2009],{"class":1866},[61,2015,2016],{"class":1753},")\n",[61,2018,2020,2023,2026,2028],{"class":1731,"line":2019},24,[61,2021,2022],{"class":1753},"    delta_points: ",[61,2024,2025],{"class":1859},"int",[61,2027,1863],{"class":1859},[61,2029,2030],{"class":1866}," 1801\n",[61,2032,2034,2037,2039,2041,2043,2045,2047,2049,2052,2054,2057],{"class":1731,"line":2033},25,[61,2035,2036],{"class":1753},"    omega_span: Tuple[",[61,2038,1860],{"class":1859},[61,2040,1435],{"class":1753},[61,2042,1860],{"class":1859},[61,2044,1997],{"class":1753},[61,2046,2000],{"class":1859},[61,2048,2003],{"class":1753},[61,2050,2051],{"class":1866},"0.0",[61,2053,1435],{"class":1753},[61,2055,2056],{"class":1866},"60.0",[61,2058,2016],{"class":1753},[61,2060,2062,2065,2067,2069],{"class":1731,"line":2061},26,[61,2063,2064],{"class":1753},"    omega_points: ",[61,2066,2025],{"class":1859},[61,2068,1863],{"class":1859},[61,2070,2071],{"class":1866}," 361\n",[61,2073,2075],{"class":1731,"line":2074},27,[61,2076,1743],{"emptyLinePlaceholder":1742},[61,2078,2080,2083,2085,2087],{"class":1731,"line":2079},28,[61,2081,2082],{"class":1753},"K ",[61,2084,2000],{"class":1859},[61,2086,1847],{"class":1836},[61,2088,2089],{"class":1753},"()\n",[61,2091,2093],{"class":1731,"line":2092},29,[61,2094,1743],{"emptyLinePlaceholder":1742},[61,2096,2098],{"class":1731,"line":2097},30,[61,2099,1819],{"class":1735},[61,2101,2103],{"class":1731,"line":2102},31,[61,2104,2105],{"class":1735},"# Helpers & ν-coefficients\n",[61,2107,2109],{"class":1731,"line":2108},32,[61,2110,1819],{"class":1735},[61,2112,2114,2117,2120,2122,2125,2128,2130,2132,2135,2137,2139,2141,2144,2146,2148,2150,2153,2156,2158],{"class":1731,"line":2113},33,[61,2115,2116],{"class":1749},"def",[61,2118,2119],{"class":1836}," _safe_div",[61,2121,849],{"class":1753},[61,2123,19],{"class":2124},"sb9H8",[61,2126,2127],{"class":1753},": ",[61,2129,1860],{"class":1859},[61,2131,1435],{"class":1753},[61,2133,2134],{"class":2124},"b",[61,2136,2127],{"class":1753},[61,2138,1860],{"class":1859},[61,2140,1435],{"class":1753},[61,2142,2143],{"class":2124},"eps",[61,2145,2127],{"class":1753},[61,2147,1860],{"class":1859},[61,2149,1863],{"class":1859},[61,2151,2152],{"class":1866}," 1e-12",[61,2154,2155],{"class":1753},") -> ",[61,2157,1860],{"class":1859},[61,2159,1850],{"class":1753},[61,2161,2163],{"class":1731,"line":2162},34,[61,2164,2166],{"class":2165},"subq3","    \"\"\"Return a\u002Fb with a small guard to avoid blow-ups at poles.\"\"\"\n",[61,2168,2170,2173,2176,2179,2182],{"class":1731,"line":2169},35,[61,2171,2172],{"class":1749},"    if",[61,2174,2175],{"class":1859}," abs",[61,2177,2178],{"class":1753},"(b) ",[61,2180,2181],{"class":1859},"\u003C",[61,2183,2184],{"class":1753}," eps:\n",[61,2186,2188,2191,2194,2197,2199,2202],{"class":1731,"line":2187},36,[61,2189,2190],{"class":1749},"        return",[61,2192,2193],{"class":1753}," np.",[61,2195,2196],{"class":1836},"sign",[61,2198,2178],{"class":1753},[61,2200,2201],{"class":1859},"*",[61,2203,2204],{"class":1866}," 1e12\n",[61,2206,2208,2211,2214,2216],{"class":1731,"line":2207},37,[61,2209,2210],{"class":1749},"    return",[61,2212,2213],{"class":1753}," a ",[61,2215,602],{"class":1859},[61,2217,2218],{"class":1753}," b\n",[61,2220,2222],{"class":1731,"line":2221},38,[61,2223,1743],{"emptyLinePlaceholder":1742},[61,2225,2227,2229,2232,2234,2237,2239,2241,2243,2246,2248,2250,2252,2254],{"class":1731,"line":2226},39,[61,2228,2116],{"class":1749},[61,2230,2231],{"class":1836}," _epsilon_from_alpha_over_omega",[61,2233,849],{"class":1753},[61,2235,2236],{"class":2124},"alpha_mhz",[61,2238,2127],{"class":1753},[61,2240,1860],{"class":1859},[61,2242,1435],{"class":1753},[61,2244,2245],{"class":2124},"omega_mhz",[61,2247,2127],{"class":1753},[61,2249,1860],{"class":1859},[61,2251,2155],{"class":1753},[61,2253,1860],{"class":1859},[61,2255,1850],{"class":1753},[61,2257,2259],{"class":1731,"line":2258},40,[61,2260,2261],{"class":2165},"    \"\"\"Solve ε from (9 − 4r) ε² + 16(1 − r) ε + 64 r = 0, with r = α\u002Fω.\"\"\"\n",[61,2263,2265,2268,2270,2273,2275],{"class":1731,"line":2264},41,[61,2266,2267],{"class":1753},"    r ",[61,2269,2000],{"class":1859},[61,2271,2272],{"class":1753}," alpha_mhz ",[61,2274,602],{"class":1859},[61,2276,2277],{"class":1753}," omega_mhz\n",[61,2279,2281,2284,2286,2288,2291,2293,2296,2299],{"class":1731,"line":2280},42,[61,2282,2283],{"class":1753},"    A ",[61,2285,2000],{"class":1859},[61,2287,2003],{"class":1753},[61,2289,2290],{"class":1866},"9.0",[61,2292,1899],{"class":1859},[61,2294,2295],{"class":1866}," 4.0",[61,2297,2298],{"class":1859}," *",[61,2300,2301],{"class":1753}," r)\n",[61,2303,2305,2308,2310,2313,2315,2317,2320,2322],{"class":1731,"line":2304},43,[61,2306,2307],{"class":1753},"    B ",[61,2309,2000],{"class":1859},[61,2311,2312],{"class":1866}," 16.0",[61,2314,2298],{"class":1859},[61,2316,2003],{"class":1753},[61,2318,2319],{"class":1866},"1.0",[61,2321,1899],{"class":1859},[61,2323,2301],{"class":1753},[61,2325,2327,2330,2332,2335,2337],{"class":1731,"line":2326},44,[61,2328,2329],{"class":1753},"    C ",[61,2331,2000],{"class":1859},[61,2333,2334],{"class":1866}," 64.0",[61,2336,2298],{"class":1859},[61,2338,2339],{"class":1753}," r\n",[61,2341,2343,2346,2348,2351,2354,2356,2359,2361,2363,2365,2368,2370,2373,2375],{"class":1731,"line":2342},45,[61,2344,2345],{"class":1753},"    disc ",[61,2347,2000],{"class":1859},[61,2349,2350],{"class":1859}," max",[61,2352,2353],{"class":1753},"(B ",[61,2355,2201],{"class":1859},[61,2357,2358],{"class":1753}," B ",[61,2360,2006],{"class":1859},[61,2362,2295],{"class":1866},[61,2364,2298],{"class":1859},[61,2366,2367],{"class":1753}," A ",[61,2369,2201],{"class":1859},[61,2371,2372],{"class":1753}," C, ",[61,2374,2051],{"class":1866},[61,2376,2016],{"class":1753},[61,2378,2380,2382,2384,2387,2389,2392],{"class":1731,"line":2379},46,[61,2381,2172],{"class":1749},[61,2383,2175],{"class":1859},[61,2385,2386],{"class":1753},"(A) ",[61,2388,2181],{"class":1859},[61,2390,2391],{"class":1866}," 1e-15",[61,2393,1850],{"class":1753},[61,2395,2397,2399],{"class":1731,"line":2396},47,[61,2398,2190],{"class":1749},[61,2400,2401],{"class":1866}," 0.2\n",[61,2403,2405,2408,2410,2412,2414,2417,2420,2423,2426,2429,2431,2433,2436,2438],{"class":1731,"line":2404},48,[61,2406,2407],{"class":1753},"    e1 ",[61,2409,2000],{"class":1859},[61,2411,2003],{"class":1753},[61,2413,2006],{"class":1859},[61,2415,2416],{"class":1753},"B ",[61,2418,2419],{"class":1859},"+",[61,2421,2422],{"class":1753}," math.",[61,2424,2425],{"class":1836},"sqrt",[61,2427,2428],{"class":1753},"(disc)) ",[61,2430,602],{"class":1859},[61,2432,2003],{"class":1753},[61,2434,2435],{"class":1866},"2.0",[61,2437,2298],{"class":1859},[61,2439,2440],{"class":1753}," A)\n",[61,2442,2444,2447,2449,2451,2453,2455,2457,2459,2461,2463,2465,2467,2469,2471],{"class":1731,"line":2443},49,[61,2445,2446],{"class":1753},"    e2 ",[61,2448,2000],{"class":1859},[61,2450,2003],{"class":1753},[61,2452,2006],{"class":1859},[61,2454,2416],{"class":1753},[61,2456,2006],{"class":1859},[61,2458,2422],{"class":1753},[61,2460,2425],{"class":1836},[61,2462,2428],{"class":1753},[61,2464,602],{"class":1859},[61,2466,2003],{"class":1753},[61,2468,2435],{"class":1866},[61,2470,2298],{"class":1859},[61,2472,2440],{"class":1753},[61,2474,2476,2479,2481,2484,2487,2490,2493,2496,2499,2501,2504,2507,2510,2512,2515],{"class":1731,"line":2475},50,[61,2477,2478],{"class":1753},"    roots ",[61,2480,2000],{"class":1859},[61,2482,2483],{"class":1753}," [e ",[61,2485,2486],{"class":1749},"for",[61,2488,2489],{"class":1753}," e ",[61,2491,2492],{"class":1749},"in",[61,2494,2495],{"class":1753}," (e1, e2) ",[61,2497,2498],{"class":1749},"if",[61,2500,2489],{"class":1753},[61,2502,2503],{"class":1859},">",[61,2505,2506],{"class":1866}," 0.0",[61,2508,2509],{"class":1749}," and",[61,2511,2193],{"class":1753},[61,2513,2514],{"class":1836},"isfinite",[61,2516,2517],{"class":1753},"(e)]\n",[61,2519,2521,2523,2526,2529,2531,2534,2537],{"class":1731,"line":2520},51,[61,2522,2210],{"class":1749},[61,2524,2525],{"class":1859}," min",[61,2527,2528],{"class":1753},"(roots) ",[61,2530,2498],{"class":1749},[61,2532,2533],{"class":1753}," roots ",[61,2535,2536],{"class":1749},"else",[61,2538,2401],{"class":1866},[61,2540,2542],{"class":1731,"line":2541},52,[61,2543,1743],{"emptyLinePlaceholder":1742},[61,2545,2547,2549,2552,2554,2556,2558,2560,2562,2564,2566,2568,2571,2573,2575,2577],{"class":1731,"line":2546},53,[61,2548,2116],{"class":1749},[61,2550,2551],{"class":1836}," _nu_energy_basis",[61,2553,849],{"class":1753},[61,2555,2236],{"class":2124},[61,2557,2127],{"class":1753},[61,2559,1860],{"class":1859},[61,2561,1435],{"class":1753},[61,2563,2245],{"class":2124},[61,2565,2127],{"class":1753},[61,2567,1860],{"class":1859},[61,2569,2570],{"class":1753},") -> Tuple[",[61,2572,1860],{"class":1859},[61,2574,1435],{"class":1753},[61,2576,1860],{"class":1859},[61,2578,2579],{"class":1753},"]:\n",[61,2581,2583],{"class":1731,"line":2582},54,[61,2584,2585],{"class":2165},"    \"\"\"Energy-basis ν_01, ν_12 up to O(ε²).\"\"\"\n",[61,2587,2589,2592,2594,2596],{"class":1731,"line":2588},55,[61,2590,2591],{"class":1753},"    eps ",[61,2593,2000],{"class":1859},[61,2595,2231],{"class":1836},[61,2597,2598],{"class":1753},"(alpha_mhz, omega_mhz)\n",[61,2600,2602,2605,2607,2610,2612,2614,2616,2619,2622,2625,2627,2630,2632,2634,2637,2639,2642,2644,2646,2649,2652,2655],{"class":1731,"line":2601},56,[61,2603,2604],{"class":1753},"    nu01 ",[61,2606,2000],{"class":1859},[61,2608,2609],{"class":1866}," 1.0",[61,2611,1899],{"class":1859},[61,2613,2003],{"class":1753},[61,2615,2319],{"class":1866},[61,2617,2618],{"class":1859}," \u002F",[61,2620,2621],{"class":1866}," 8.0",[61,2623,2624],{"class":1753},") ",[61,2626,2201],{"class":1859},[61,2628,2629],{"class":1753}," eps ",[61,2631,2006],{"class":1859},[61,2633,2003],{"class":1753},[61,2635,2636],{"class":1866},"11.0",[61,2638,2618],{"class":1859},[61,2640,2641],{"class":1866}," 256.0",[61,2643,2624],{"class":1753},[61,2645,2201],{"class":1859},[61,2647,2648],{"class":1753}," (eps ",[61,2650,2651],{"class":1859},"**",[61,2653,2654],{"class":1866}," 2",[61,2656,2016],{"class":1753},[61,2658,2660,2663,2665,2667,2669,2671,2674,2676,2678,2680,2682,2685,2687,2690,2692,2694,2696,2698,2700,2703,2705,2707,2709,2711,2713],{"class":1731,"line":2659},57,[61,2661,2662],{"class":1753},"    nu12 ",[61,2664,2000],{"class":1859},[61,2666,2003],{"class":1753},[61,2668,2319],{"class":1866},[61,2670,1899],{"class":1859},[61,2672,2673],{"class":1866}," 0.25",[61,2675,2298],{"class":1859},[61,2677,2629],{"class":1753},[61,2679,2006],{"class":1859},[61,2681,2003],{"class":1753},[61,2683,2684],{"class":1866},"73.0",[61,2686,2618],{"class":1859},[61,2688,2689],{"class":1866}," 512.0",[61,2691,2624],{"class":1753},[61,2693,2201],{"class":1859},[61,2695,2648],{"class":1753},[61,2697,2651],{"class":1859},[61,2699,2654],{"class":1866},[61,2701,2702],{"class":1753},")) ",[61,2704,602],{"class":1859},[61,2706,2422],{"class":1753},[61,2708,2425],{"class":1836},[61,2710,849],{"class":1753},[61,2712,2435],{"class":1866},[61,2714,2016],{"class":1753},[61,2716,2718,2720],{"class":1731,"line":2717},58,[61,2719,2210],{"class":1749},[61,2721,2722],{"class":1753}," nu01, nu12\n",[61,2724,2726],{"class":1731,"line":2725},59,[61,2727,1743],{"emptyLinePlaceholder":1742},[61,2729,2731,2733,2736,2739,2741,2743,2745],{"class":1731,"line":2730},60,[61,2732,2116],{"class":1749},[61,2734,2735],{"class":1836}," _nu_kerr",[61,2737,2738],{"class":1753},"() -> Tuple[",[61,2740,1860],{"class":1859},[61,2742,1435],{"class":1753},[61,2744,1860],{"class":1859},[61,2746,2579],{"class":1753},[61,2748,2750],{"class":1731,"line":2749},61,[61,2751,2752],{"class":2165},"    \"\"\"Kerr-limit charge matrix elements.\"\"\"\n",[61,2754,2756,2758,2760,2762,2764,2766,2768,2770,2772,2774],{"class":1731,"line":2755},62,[61,2757,2210],{"class":1749},[61,2759,2609],{"class":1866},[61,2761,1435],{"class":1753},[61,2763,2319],{"class":1866},[61,2765,2618],{"class":1859},[61,2767,2422],{"class":1753},[61,2769,2425],{"class":1836},[61,2771,849],{"class":1753},[61,2773,2435],{"class":1866},[61,2775,2016],{"class":1753},[61,2777,2779],{"class":1731,"line":2778},63,[61,2780,1743],{"emptyLinePlaceholder":1742},[61,2782,2784,2786],{"class":1731,"line":2783},64,[61,2785,2498],{"class":1749},[61,2787,2788],{"class":1753}," K.use_energy_basis:\n",[61,2790,2792,2795,2797,2799],{"class":1731,"line":2791},65,[61,2793,2794],{"class":1753},"    nu_c01, nu_c12 ",[61,2796,2000],{"class":1859},[61,2798,2551],{"class":1836},[61,2800,2801],{"class":1753},"(K.alpha_c, K.f_c)\n",[61,2803,2805,2808,2810,2812],{"class":1731,"line":2804},66,[61,2806,2807],{"class":1753},"    nu_t01, nu_t12 ",[61,2809,2000],{"class":1859},[61,2811,2551],{"class":1836},[61,2813,2814],{"class":1753},"(K.alpha_t, K.f_t)\n",[61,2816,2818,2820],{"class":1731,"line":2817},67,[61,2819,2536],{"class":1749},[61,2821,1850],{"class":1753},[61,2823,2825,2827,2829,2831],{"class":1731,"line":2824},68,[61,2826,2794],{"class":1753},[61,2828,2000],{"class":1859},[61,2830,2735],{"class":1836},[61,2832,2089],{"class":1753},[61,2834,2836,2838,2840,2842],{"class":1731,"line":2835},69,[61,2837,2807],{"class":1753},[61,2839,2000],{"class":1859},[61,2841,2735],{"class":1836},[61,2843,2089],{"class":1753},[61,2845,2847],{"class":1731,"line":2846},70,[61,2848,1743],{"emptyLinePlaceholder":1742},[61,2850,2852],{"class":1731,"line":2851},71,[61,2853,1819],{"class":1735},[61,2855,2857],{"class":1731,"line":2856},72,[61,2858,2859],{"class":1735},"# Leading-order rates (MHz)\n",[61,2861,2863],{"class":1731,"line":2862},73,[61,2864,1819],{"class":1735},[61,2866,2868,2870,2873,2875,2878,2880,2882,2884,2887,2889,2891,2893,2896,2898,2900,2902,2905,2907,2909,2911,2913],{"class":1731,"line":2867},74,[61,2869,2116],{"class":1749},[61,2871,2872],{"class":1836}," omega_zx_mhz",[61,2874,849],{"class":1753},[61,2876,2877],{"class":2124},"delta_ct",[61,2879,2127],{"class":1753},[61,2881,1860],{"class":1859},[61,2883,1435],{"class":1753},[61,2885,2886],{"class":2124},"J",[61,2888,2127],{"class":1753},[61,2890,1860],{"class":1859},[61,2892,1435],{"class":1753},[61,2894,2895],{"class":2124},"Omega",[61,2897,2127],{"class":1753},[61,2899,1860],{"class":1859},[61,2901,1435],{"class":1753},[61,2903,2904],{"class":2124},"alpha_c",[61,2906,2127],{"class":1753},[61,2908,1860],{"class":1859},[61,2910,2155],{"class":1753},[61,2912,1860],{"class":1859},[61,2914,1850],{"class":1753},[61,2916,2918,2921],{"class":1731,"line":2917},75,[61,2919,2920],{"class":1749},"    r",[61,2922,2923],{"class":2165},"\"\"\"Entangling rate ω_ZX ≈ JΩ[(Δ+α_c)^{-1} − Δ^{-1}] with small ν-correction.\"\"\"\n",[61,2925,2927,2930,2932,2935,2937,2940,2942,2944,2947,2949,2951,2954,2956,2959,2961,2963,2965,2967],{"class":1731,"line":2926},76,[61,2928,2929],{"class":1753},"    kerr ",[61,2931,2000],{"class":1859},[61,2933,2934],{"class":1753}," J ",[61,2936,2201],{"class":1859},[61,2938,2939],{"class":1753}," Omega ",[61,2941,2201],{"class":1859},[61,2943,2003],{"class":1753},[61,2945,2946],{"class":1836},"_safe_div",[61,2948,849],{"class":1753},[61,2950,2319],{"class":1866},[61,2952,2953],{"class":1753},", delta_ct ",[61,2955,2419],{"class":1859},[61,2957,2958],{"class":1753}," alpha_c) ",[61,2960,2006],{"class":1859},[61,2962,2119],{"class":1836},[61,2964,849],{"class":1753},[61,2966,2319],{"class":1866},[61,2968,2969],{"class":1753},", delta_ct))\n",[61,2971,2973,2976,2978,2981,2983,2985,2987,2989,2991,2994,2996],{"class":1731,"line":2972},77,[61,2974,2975],{"class":1753},"    eb ",[61,2977,2000],{"class":1859},[61,2979,2980],{"class":1866}," 0.5",[61,2982,2298],{"class":1859},[61,2984,2934],{"class":1753},[61,2986,2201],{"class":1859},[61,2988,2939],{"class":1753},[61,2990,2201],{"class":1859},[61,2992,2993],{"class":1753}," nu_t01 ",[61,2995,2201],{"class":1859},[61,2997,2998],{"class":1753}," (\n",[61,3000,3002,3005,3007,3009,3011,3013,3015,3017,3019,3021,3023,3025,3027,3030,3032,3035,3037,3039,3041,3043,3045,3047,3049],{"class":1731,"line":3001},78,[61,3003,3004],{"class":1753},"        (nu_c12 ",[61,3006,2651],{"class":1859},[61,3008,2654],{"class":1866},[61,3010,2624],{"class":1753},[61,3012,2201],{"class":1859},[61,3014,2119],{"class":1836},[61,3016,849],{"class":1753},[61,3018,2319],{"class":1866},[61,3020,2953],{"class":1753},[61,3022,2419],{"class":1859},[61,3024,2958],{"class":1753},[61,3026,2006],{"class":1859},[61,3028,3029],{"class":1866}," 2.0",[61,3031,2298],{"class":1859},[61,3033,3034],{"class":1753}," (nu_c01 ",[61,3036,2651],{"class":1859},[61,3038,2654],{"class":1866},[61,3040,2624],{"class":1753},[61,3042,2201],{"class":1859},[61,3044,2119],{"class":1836},[61,3046,849],{"class":1753},[61,3048,2319],{"class":1866},[61,3050,3051],{"class":1753},", delta_ct)\n",[61,3053,3055],{"class":1731,"line":3054},79,[61,3056,3057],{"class":1753},"    )\n",[61,3059,3061,3063,3065,3067,3070,3072],{"class":1731,"line":3060},80,[61,3062,2210],{"class":1749},[61,3064,2980],{"class":1866},[61,3066,2298],{"class":1859},[61,3068,3069],{"class":1753}," (kerr ",[61,3071,2419],{"class":1859},[61,3073,3074],{"class":1753}," eb)\n",[61,3076,3078],{"class":1731,"line":3077},81,[61,3079,1743],{"emptyLinePlaceholder":1742},[61,3081,3083,3085,3088,3090,3092,3094,3096,3098,3100,3102,3104,3106,3108,3110,3112,3114,3117,3119,3121,3123,3125],{"class":1731,"line":3082},82,[61,3084,2116],{"class":1749},[61,3086,3087],{"class":1836}," omega_zz_mhz",[61,3089,849],{"class":1753},[61,3091,2877],{"class":2124},[61,3093,2127],{"class":1753},[61,3095,1860],{"class":1859},[61,3097,1435],{"class":1753},[61,3099,2886],{"class":2124},[61,3101,2127],{"class":1753},[61,3103,1860],{"class":1859},[61,3105,1435],{"class":1753},[61,3107,2904],{"class":2124},[61,3109,2127],{"class":1753},[61,3111,1860],{"class":1859},[61,3113,1435],{"class":1753},[61,3115,3116],{"class":2124},"alpha_t",[61,3118,2127],{"class":1753},[61,3120,1860],{"class":1859},[61,3122,2155],{"class":1753},[61,3124,1860],{"class":1859},[61,3126,1850],{"class":1753},[61,3128,3130,3132],{"class":1731,"line":3129},83,[61,3131,2920],{"class":1749},[61,3133,3134],{"class":2165},"\"\"\"Static ZZ ≈ J²[(Δ−α_t)^{-1} − (Δ+α_c)^{-1}] with small ν-correction.\"\"\"\n",[61,3136,3138,3140,3142,3145,3147,3149,3151,3153,3155,3157,3159,3161,3163,3165,3168,3170,3172,3174,3176,3178,3180],{"class":1731,"line":3137},84,[61,3139,2929],{"class":1753},[61,3141,2000],{"class":1859},[61,3143,3144],{"class":1753}," (J ",[61,3146,2651],{"class":1859},[61,3148,2654],{"class":1866},[61,3150,2624],{"class":1753},[61,3152,2201],{"class":1859},[61,3154,2003],{"class":1753},[61,3156,2946],{"class":1836},[61,3158,849],{"class":1753},[61,3160,2319],{"class":1866},[61,3162,2953],{"class":1753},[61,3164,2006],{"class":1859},[61,3166,3167],{"class":1753}," alpha_t) ",[61,3169,2006],{"class":1859},[61,3171,2119],{"class":1836},[61,3173,849],{"class":1753},[61,3175,2319],{"class":1866},[61,3177,2953],{"class":1753},[61,3179,2419],{"class":1859},[61,3181,3182],{"class":1753}," alpha_c))\n",[61,3184,3186,3188,3190,3192,3194,3196,3198,3200,3202,3204],{"class":1731,"line":3185},85,[61,3187,2975],{"class":1753},[61,3189,2000],{"class":1859},[61,3191,2980],{"class":1866},[61,3193,2298],{"class":1859},[61,3195,3144],{"class":1753},[61,3197,2651],{"class":1859},[61,3199,2654],{"class":1866},[61,3201,2624],{"class":1753},[61,3203,2201],{"class":1859},[61,3205,2998],{"class":1753},[61,3207,3209,3212,3214,3216,3218,3220,3223,3225,3227,3229,3231,3233,3235,3237,3239,3241,3243],{"class":1731,"line":3208},86,[61,3210,3211],{"class":1753},"        (nu_c01 ",[61,3213,2651],{"class":1859},[61,3215,2654],{"class":1866},[61,3217,2624],{"class":1753},[61,3219,2201],{"class":1859},[61,3221,3222],{"class":1753}," (nu_t12 ",[61,3224,2651],{"class":1859},[61,3226,2654],{"class":1866},[61,3228,2624],{"class":1753},[61,3230,2201],{"class":1859},[61,3232,2119],{"class":1836},[61,3234,849],{"class":1753},[61,3236,2319],{"class":1866},[61,3238,2953],{"class":1753},[61,3240,2006],{"class":1859},[61,3242,3167],{"class":1753},[61,3244,3245],{"class":1859},"-\n",[61,3247,3249,3252,3254,3256,3258,3260,3263,3265,3267,3269,3271,3273,3275,3277,3279,3281],{"class":1731,"line":3248},87,[61,3250,3251],{"class":1753},"        (nu_t01 ",[61,3253,2651],{"class":1859},[61,3255,2654],{"class":1866},[61,3257,2624],{"class":1753},[61,3259,2201],{"class":1859},[61,3261,3262],{"class":1753}," (nu_c12 ",[61,3264,2651],{"class":1859},[61,3266,2654],{"class":1866},[61,3268,2624],{"class":1753},[61,3270,2201],{"class":1859},[61,3272,2119],{"class":1836},[61,3274,849],{"class":1753},[61,3276,2319],{"class":1866},[61,3278,2953],{"class":1753},[61,3280,2419],{"class":1859},[61,3282,3283],{"class":1753}," alpha_c)\n",[61,3285,3287],{"class":1731,"line":3286},88,[61,3288,3057],{"class":1753},[61,3290,3292,3294,3296,3298,3300,3302],{"class":1731,"line":3291},89,[61,3293,2210],{"class":1749},[61,3295,2980],{"class":1866},[61,3297,2298],{"class":1859},[61,3299,3069],{"class":1753},[61,3301,2419],{"class":1859},[61,3303,3074],{"class":1753},[61,3305,3307],{"class":1731,"line":3306},90,[61,3308,1743],{"emptyLinePlaceholder":1742},[61,3310,3312],{"class":1731,"line":3311},91,[61,3313,3314],{"class":1735},"# Region boundaries (poles) for I–V:\n",[61,3316,3318,3321,3323,3326,3328,3330,3332,3335,3337,3339,3341,3343,3346,3348,3351,3353],{"class":1731,"line":3317},92,[61,3319,3320],{"class":1866},"REGION_LINES",[61,3322,1863],{"class":1859},[61,3324,3325],{"class":1753}," (K.alpha_t, ",[61,3327,2051],{"class":1866},[61,3329,1435],{"class":1753},[61,3331,2006],{"class":1859},[61,3333,3334],{"class":1753},"K.alpha_c ",[61,3336,602],{"class":1859},[61,3338,3029],{"class":1866},[61,3340,1435],{"class":1753},[61,3342,2006],{"class":1859},[61,3344,3345],{"class":1753},"K.alpha_c, ",[61,3347,2006],{"class":1859},[61,3349,3350],{"class":1866},"1.5",[61,3352,2298],{"class":1859},[61,3354,3355],{"class":1753}," K.alpha_c)\n",[61,3357,3359,3361,3364,3366,3369],{"class":1731,"line":3358},93,[61,3360,2116],{"class":1749},[61,3362,3363],{"class":1836}," _annotate_regions",[61,3365,849],{"class":1753},[61,3367,3368],{"class":2124},"ax",[61,3370,3371],{"class":1753},"):\n",[61,3373,3375,3378,3381,3383,3386],{"class":1731,"line":3374},94,[61,3376,3377],{"class":1749},"    for",[61,3379,3380],{"class":1753}," x ",[61,3382,2492],{"class":1749},[61,3384,3385],{"class":1866}," REGION_LINES",[61,3387,1850],{"class":1753},[61,3389,3391,3394,3397,3400,3404,3406,3409,3411,3414,3416,3418],{"class":1731,"line":3390},95,[61,3392,3393],{"class":1753},"        ax.",[61,3395,3396],{"class":1836},"axvline",[61,3398,3399],{"class":1753},"(x, ",[61,3401,3403],{"class":3402},"s_ZVi","linestyle",[61,3405,2000],{"class":1859},[61,3407,3408],{"class":2165},"\"--\"",[61,3410,1435],{"class":1753},[61,3412,3413],{"class":3402},"linewidth",[61,3415,2000],{"class":1859},[61,3417,2319],{"class":1866},[61,3419,2016],{"class":1753},[61,3421,3423],{"class":1731,"line":3422},96,[61,3424,1743],{"emptyLinePlaceholder":1742},[61,3426,3428],{"class":1731,"line":3427},97,[61,3429,1819],{"class":1735},[61,3431,3433],{"class":1731,"line":3432},98,[61,3434,3435],{"class":1735},"# Slices\n",[61,3437,3439],{"class":1731,"line":3438},99,[61,3440,1819],{"class":1735},[61,3442,3444,3447,3449,3451,3454,3457,3459,3462,3464],{"class":1731,"line":3443},100,[61,3445,3446],{"class":1866},"DELTA",[61,3448,1863],{"class":1859},[61,3450,2193],{"class":1753},[61,3452,3453],{"class":1836},"linspace",[61,3455,3456],{"class":1753},"(K.delta_span[",[61,3458,1495],{"class":1866},[61,3460,3461],{"class":1753},"], K.delta_span[",[61,3463,57],{"class":1866},[61,3465,3466],{"class":1753},"], K.delta_points)\n",[61,3468,3470,3473,3475,3477,3479,3482,3484,3487,3489],{"class":1731,"line":3469},101,[61,3471,3472],{"class":1866},"OMEGA",[61,3474,1863],{"class":1859},[61,3476,2193],{"class":1753},[61,3478,3453],{"class":1836},[61,3480,3481],{"class":1753},"(K.omega_span[",[61,3483,1495],{"class":1866},[61,3485,3486],{"class":1753},"], K.omega_span[",[61,3488,57],{"class":1866},[61,3490,3491],{"class":1753},"], K.omega_points)\n",[61,3493,3495],{"class":1731,"line":3494},102,[61,3496,1743],{"emptyLinePlaceholder":1742},[61,3498,3500,3503,3505,3507,3510,3513,3516,3519,3521,3524,3526,3529],{"class":1731,"line":3499},103,[61,3501,3502],{"class":1753},"wzx_slice ",[61,3504,2000],{"class":1859},[61,3506,2193],{"class":1753},[61,3508,3509],{"class":1836},"array",[61,3511,3512],{"class":1753},"([",[61,3514,3515],{"class":1836},"omega_zx_mhz",[61,3517,3518],{"class":1753},"(d, K.J, K.Omega0, K.alpha_c) ",[61,3520,2486],{"class":1749},[61,3522,3523],{"class":1753}," d ",[61,3525,2492],{"class":1749},[61,3527,3528],{"class":1866}," DELTA",[61,3530,3531],{"class":1753},"])\n",[61,3533,3535,3538,3540,3542,3544,3546,3549,3552,3554,3556,3558,3560],{"class":1731,"line":3534},104,[61,3536,3537],{"class":1753},"wzz_slice ",[61,3539,2000],{"class":1859},[61,3541,2193],{"class":1753},[61,3543,3509],{"class":1836},[61,3545,3512],{"class":1753},[61,3547,3548],{"class":1836},"omega_zz_mhz",[61,3550,3551],{"class":1753},"(d, K.J, K.alpha_c, K.alpha_t) ",[61,3553,2486],{"class":1749},[61,3555,3523],{"class":1753},[61,3557,2492],{"class":1749},[61,3559,3528],{"class":1866},[61,3561,3531],{"class":1753},[61,3563,3565,3568,3570,3572,3575,3578,3581,3583,3585,3588,3590,3592,3595,3597],{"class":1731,"line":3564},105,[61,3566,3567],{"class":1753},"ratio_slice ",[61,3569,2000],{"class":1859},[61,3571,2193],{"class":1753},[61,3573,3574],{"class":1836},"abs",[61,3576,3577],{"class":1753},"(np.",[61,3579,3580],{"class":1836},"where",[61,3582,3577],{"class":1753},[61,3584,3574],{"class":1836},[61,3586,3587],{"class":1753},"(wzz_slice) ",[61,3589,2503],{"class":1859},[61,3591,2152],{"class":1866},[61,3593,3594],{"class":1753},", wzx_slice ",[61,3596,602],{"class":1859},[61,3598,3599],{"class":1753}," wzz_slice, np.nan))\n",[61,3601,3603],{"class":1731,"line":3602},106,[61,3604,1743],{"emptyLinePlaceholder":1742},[61,3606,3608],{"class":1731,"line":3607},107,[61,3609,3610],{"class":1735},"# Plot 1: ω_ZX vs Δ_ct\n",[61,3612,3614,3617,3620,3622,3625,3627,3629,3632,3634,3636,3639,3642,3644,3647],{"class":1731,"line":3613},108,[61,3615,3616],{"class":1753},"plt.",[61,3618,3619],{"class":1836},"figure",[61,3621,849],{"class":1753},[61,3623,3624],{"class":3402},"figsize",[61,3626,2000],{"class":1859},[61,3628,849],{"class":1753},[61,3630,3631],{"class":1866},"8",[61,3633,1435],{"class":1753},[61,3635,1026],{"class":1866},[61,3637,3638],{"class":1753},"), ",[61,3640,3641],{"class":3402},"dpi",[61,3643,2000],{"class":1859},[61,3645,3646],{"class":1866},"150",[61,3648,2016],{"class":1753},[61,3650,3652,3654,3657,3659,3661,3664,3667,3669,3672,3675,3678,3680,3683,3686,3689,3692,3695,3698,3701,3703,3706,3708,3711],{"class":1731,"line":3651},109,[61,3653,3616],{"class":1753},[61,3655,3656],{"class":1836},"plot",[61,3658,849],{"class":1753},[61,3660,3446],{"class":1866},[61,3662,3663],{"class":1753},", wzx_slice, ",[61,3665,3666],{"class":3402},"label",[61,3668,2000],{"class":1859},[61,3670,3671],{"class":1749},"rf",[61,3673,3674],{"class":2165},"'$\\omega_",[61,3676,3677],{"class":1859},"{{",[61,3679,1354],{"class":2165},[61,3681,3682],{"class":1859},"}}",[61,3684,3685],{"class":2165},"$ (Ω=",[61,3687,3688],{"class":1866},"{",[61,3690,3691],{"class":1753},"K.Omega0",[61,3693,3694],{"class":1749},":.1f",[61,3696,3697],{"class":1866},"}",[61,3699,3700],{"class":2165}," MHz)'",[61,3702,1435],{"class":1753},[61,3704,3705],{"class":3402},"color",[61,3707,2000],{"class":1859},[61,3709,3710],{"class":2165},"'blue'",[61,3712,2016],{"class":1753},[61,3714,3716,3718,3721,3723,3725,3727,3729,3731,3734],{"class":1731,"line":3715},110,[61,3717,3616],{"class":1753},[61,3719,3720],{"class":1836},"axhline",[61,3722,849],{"class":1753},[61,3724,2051],{"class":1866},[61,3726,1435],{"class":1753},[61,3728,3403],{"class":3402},[61,3730,2000],{"class":1859},[61,3732,3733],{"class":2165},"\":\"",[61,3735,2016],{"class":1753},[61,3737,3739,3742,3745,3748],{"class":1731,"line":3738},111,[61,3740,3741],{"class":1836},"_annotate_regions",[61,3743,3744],{"class":1753},"(plt.",[61,3746,3747],{"class":1836},"gca",[61,3749,3750],{"class":1753},"())\n",[61,3752,3754,3756,3759,3761,3764,3768,3770,3773,3775,3778],{"class":1731,"line":3753},112,[61,3755,3616],{"class":1753},[61,3757,3758],{"class":1836},"xlabel",[61,3760,849],{"class":1753},[61,3762,3763],{"class":1749},"r",[61,3765,3767],{"class":3766},"sVyAn","'$\\Delta_{ct}$ ",[61,3769,849],{"class":1866},[61,3771,3772],{"class":3766},"MHz",[61,3774,300],{"class":1866},[61,3776,3777],{"class":3766},"'",[61,3779,2016],{"class":1753},[61,3781,3783,3785,3788,3790,3792,3795,3798,3801,3803,3805,3807,3809],{"class":1731,"line":3782},113,[61,3784,3616],{"class":1753},[61,3786,3787],{"class":1836},"ylabel",[61,3789,849],{"class":1753},[61,3791,3763],{"class":1749},[61,3793,3794],{"class":3766},"'$",[61,3796,3797],{"class":1859},"\\o",[61,3799,3800],{"class":3766},"mega_{ZX}$ ",[61,3802,849],{"class":1866},[61,3804,3772],{"class":3766},[61,3806,300],{"class":1866},[61,3808,3777],{"class":3766},[61,3810,2016],{"class":1753},[61,3812,3814,3816,3819,3821,3823,3826,3828,3831],{"class":1731,"line":3813},114,[61,3815,3616],{"class":1753},[61,3817,3818],{"class":1836},"title",[61,3820,849],{"class":1753},[61,3822,3763],{"class":1749},[61,3824,3825],{"class":3766},"'Leading-order $",[61,3827,3797],{"class":1859},[61,3829,3830],{"class":3766},"mega_{ZX}$ vs detuning'",[61,3832,2016],{"class":1753},[61,3834,3836,3838,3841],{"class":1731,"line":3835},115,[61,3837,3616],{"class":1753},[61,3839,3840],{"class":1836},"legend",[61,3842,2089],{"class":1753},[61,3844,3846,3848,3851],{"class":1731,"line":3845},116,[61,3847,3616],{"class":1753},[61,3849,3850],{"class":1836},"show",[61,3852,2089],{"class":1753},[61,3854,3856],{"class":1731,"line":3855},117,[61,3857,1743],{"emptyLinePlaceholder":1742},[61,3859,3861],{"class":1731,"line":3860},118,[61,3862,3863],{"class":1735},"# Plot 2: ω_ZZ vs Δ_ct\n",[61,3865,3867,3869,3871,3873,3875,3877,3879,3881,3883,3885,3887,3889,3891,3893],{"class":1731,"line":3866},119,[61,3868,3616],{"class":1753},[61,3870,3619],{"class":1836},[61,3872,849],{"class":1753},[61,3874,3624],{"class":3402},[61,3876,2000],{"class":1859},[61,3878,849],{"class":1753},[61,3880,3631],{"class":1866},[61,3882,1435],{"class":1753},[61,3884,1026],{"class":1866},[61,3886,3638],{"class":1753},[61,3888,3641],{"class":3402},[61,3890,2000],{"class":1859},[61,3892,3646],{"class":1866},[61,3894,2016],{"class":1753},[61,3896,3898,3900,3902,3904,3906,3909,3911,3913,3915,3917,3919,3922,3924,3927,3929,3931,3933,3935,3937,3939],{"class":1731,"line":3897},120,[61,3899,3616],{"class":1753},[61,3901,3656],{"class":1836},[61,3903,849],{"class":1753},[61,3905,3446],{"class":1866},[61,3907,3908],{"class":1753},", wzz_slice, ",[61,3910,3666],{"class":3402},[61,3912,2000],{"class":1859},[61,3914,3763],{"class":1749},[61,3916,3794],{"class":3766},[61,3918,3797],{"class":1859},[61,3920,3921],{"class":3766},"mega_{ZZ}$ ",[61,3923,849],{"class":1866},[61,3925,3926],{"class":3766},"static; lowest order",[61,3928,300],{"class":1866},[61,3930,3777],{"class":3766},[61,3932,1435],{"class":1753},[61,3934,3705],{"class":3402},[61,3936,2000],{"class":1859},[61,3938,3710],{"class":2165},[61,3940,2016],{"class":1753},[61,3942,3944,3946,3948,3950,3952,3954,3956,3958,3960],{"class":1731,"line":3943},121,[61,3945,3616],{"class":1753},[61,3947,3720],{"class":1836},[61,3949,849],{"class":1753},[61,3951,2051],{"class":1866},[61,3953,1435],{"class":1753},[61,3955,3403],{"class":3402},[61,3957,2000],{"class":1859},[61,3959,3733],{"class":2165},[61,3961,2016],{"class":1753},[61,3963,3965,3967,3969,3971],{"class":1731,"line":3964},122,[61,3966,3741],{"class":1836},[61,3968,3744],{"class":1753},[61,3970,3747],{"class":1836},[61,3972,3750],{"class":1753},[61,3974,3976,3978,3980,3982,3984,3986,3988,3990,3992,3994],{"class":1731,"line":3975},123,[61,3977,3616],{"class":1753},[61,3979,3758],{"class":1836},[61,3981,849],{"class":1753},[61,3983,3763],{"class":1749},[61,3985,3767],{"class":3766},[61,3987,849],{"class":1866},[61,3989,3772],{"class":3766},[61,3991,300],{"class":1866},[61,3993,3777],{"class":3766},[61,3995,2016],{"class":1753},[61,3997,3999,4001,4003,4005,4007,4009,4011,4013,4015,4017,4019,4021],{"class":1731,"line":3998},124,[61,4000,3616],{"class":1753},[61,4002,3787],{"class":1836},[61,4004,849],{"class":1753},[61,4006,3763],{"class":1749},[61,4008,3794],{"class":3766},[61,4010,3797],{"class":1859},[61,4012,3921],{"class":3766},[61,4014,849],{"class":1866},[61,4016,3772],{"class":3766},[61,4018,300],{"class":1866},[61,4020,3777],{"class":3766},[61,4022,2016],{"class":1753},[61,4024,4026,4028,4030,4032,4034,4037,4039,4042],{"class":1731,"line":4025},125,[61,4027,3616],{"class":1753},[61,4029,3818],{"class":1836},[61,4031,849],{"class":1753},[61,4033,3763],{"class":1749},[61,4035,4036],{"class":3766},"'Lowest-order $",[61,4038,3797],{"class":1859},[61,4040,4041],{"class":3766},"mega_{ZZ}$ vs detuning'",[61,4043,2016],{"class":1753},[61,4045,4047,4049,4051],{"class":1731,"line":4046},126,[61,4048,3616],{"class":1753},[61,4050,3840],{"class":1836},[61,4052,2089],{"class":1753},[61,4054,4056,4058,4060],{"class":1731,"line":4055},127,[61,4057,3616],{"class":1753},[61,4059,3850],{"class":1836},[61,4061,2089],{"class":1753},[61,4063,4065],{"class":1731,"line":4064},128,[61,4066,1743],{"emptyLinePlaceholder":1742},[61,4068,4070],{"class":1731,"line":4069},129,[61,4071,4072],{"class":1735},"# Heatmap 1: ω_ZX(Δ, Ω)\n",[61,4074,4076,4079,4081,4083,4086,4089,4091,4094,4096],{"class":1731,"line":4075},130,[61,4077,4078],{"class":1753},"wzx_map ",[61,4080,2000],{"class":1859},[61,4082,2193],{"class":1753},[61,4084,4085],{"class":1836},"empty",[61,4087,4088],{"class":1753},"((",[61,4090,3446],{"class":1866},[61,4092,4093],{"class":1753},".size, ",[61,4095,3472],{"class":1866},[61,4097,4098],{"class":1753},".size))\n",[61,4100,4102,4104,4107,4109,4112,4114,4116],{"class":1731,"line":4101},131,[61,4103,2486],{"class":1749},[61,4105,4106],{"class":1753}," i, d ",[61,4108,2492],{"class":1749},[61,4110,4111],{"class":1859}," enumerate",[61,4113,849],{"class":1753},[61,4115,3446],{"class":1866},[61,4117,3371],{"class":1753},[61,4119,4121,4123,4126,4128,4130,4132,4134],{"class":1731,"line":4120},132,[61,4122,3377],{"class":1749},[61,4124,4125],{"class":1753}," j, om ",[61,4127,2492],{"class":1749},[61,4129,4111],{"class":1859},[61,4131,849],{"class":1753},[61,4133,3472],{"class":1866},[61,4135,3371],{"class":1753},[61,4137,4139,4142,4144,4146],{"class":1731,"line":4138},133,[61,4140,4141],{"class":1753},"        wzx_map[i, j] ",[61,4143,2000],{"class":1859},[61,4145,2872],{"class":1836},[61,4147,4148],{"class":1753},"(d, K.J, om, K.alpha_c)\n",[61,4150,4152],{"class":1731,"line":4151},134,[61,4153,1743],{"emptyLinePlaceholder":1742},[61,4155,4157,4159,4161,4163,4165,4167,4169,4171,4173,4175,4177,4179,4181,4183],{"class":1731,"line":4156},135,[61,4158,3616],{"class":1753},[61,4160,3619],{"class":1836},[61,4162,849],{"class":1753},[61,4164,3624],{"class":3402},[61,4166,2000],{"class":1859},[61,4168,849],{"class":1753},[61,4170,3631],{"class":1866},[61,4172,1435],{"class":1753},[61,4174,1300],{"class":1866},[61,4176,3638],{"class":1753},[61,4178,3641],{"class":3402},[61,4180,2000],{"class":1859},[61,4182,3646],{"class":1866},[61,4184,2016],{"class":1753},[61,4186,4188,4191,4193,4196,4198,4200,4203,4206,4208,4210,4213,4215,4217,4219,4221,4223,4225,4227,4229],{"class":1731,"line":4187},136,[61,4189,4190],{"class":1753},"extent ",[61,4192,2000],{"class":1859},[61,4194,4195],{"class":1753}," [",[61,4197,3472],{"class":1866},[61,4199,332],{"class":1753},[61,4201,4202],{"class":1836},"min",[61,4204,4205],{"class":1753},"(), ",[61,4207,3472],{"class":1866},[61,4209,332],{"class":1753},[61,4211,4212],{"class":1836},"max",[61,4214,4205],{"class":1753},[61,4216,3446],{"class":1866},[61,4218,332],{"class":1753},[61,4220,4202],{"class":1836},[61,4222,4205],{"class":1753},[61,4224,3446],{"class":1866},[61,4226,332],{"class":1753},[61,4228,4212],{"class":1836},[61,4230,4231],{"class":1753},"()]\n",[61,4233,4235,4237,4240,4243,4246,4248,4251,4253,4256,4258,4261,4263,4266,4268,4271,4274,4276,4279],{"class":1731,"line":4234},137,[61,4236,3616],{"class":1753},[61,4238,4239],{"class":1836},"imshow",[61,4241,4242],{"class":1753},"(wzx_map, ",[61,4244,4245],{"class":3402},"origin",[61,4247,2000],{"class":1859},[61,4249,4250],{"class":2165},"\"lower\"",[61,4252,1435],{"class":1753},[61,4254,4255],{"class":3402},"aspect",[61,4257,2000],{"class":1859},[61,4259,4260],{"class":2165},"\"auto\"",[61,4262,1435],{"class":1753},[61,4264,4265],{"class":3402},"extent",[61,4267,2000],{"class":1859},[61,4269,4270],{"class":1753},"extent, ",[61,4272,4273],{"class":3402},"cmap",[61,4275,2000],{"class":1859},[61,4277,4278],{"class":2165},"\"magma\"",[61,4280,2016],{"class":1753},[61,4282,4284,4286,4289,4291,4293,4295,4297,4299,4301,4303,4305,4307,4309,4311],{"class":1731,"line":4283},138,[61,4285,3616],{"class":1753},[61,4287,4288],{"class":1836},"colorbar",[61,4290,849],{"class":1753},[61,4292,3666],{"class":3402},[61,4294,2000],{"class":1859},[61,4296,3763],{"class":1749},[61,4298,3794],{"class":3766},[61,4300,3797],{"class":1859},[61,4302,3800],{"class":3766},[61,4304,849],{"class":1866},[61,4306,3772],{"class":3766},[61,4308,300],{"class":1866},[61,4310,3777],{"class":3766},[61,4312,2016],{"class":1753},[61,4314,4316,4318,4320,4322,4324,4326,4329,4332,4334,4336,4338,4340],{"class":1731,"line":4315},139,[61,4317,3616],{"class":1753},[61,4319,3758],{"class":1836},[61,4321,849],{"class":1753},[61,4323,3763],{"class":1749},[61,4325,3794],{"class":3766},[61,4327,4328],{"class":1859},"\\O",[61,4330,4331],{"class":3766},"mega$ ",[61,4333,849],{"class":1866},[61,4335,3772],{"class":3766},[61,4337,300],{"class":1866},[61,4339,3777],{"class":3766},[61,4341,2016],{"class":1753},[61,4343,4345,4347,4349,4351,4353,4355,4357,4359,4361,4363],{"class":1731,"line":4344},140,[61,4346,3616],{"class":1753},[61,4348,3787],{"class":1836},[61,4350,849],{"class":1753},[61,4352,3763],{"class":1749},[61,4354,3767],{"class":3766},[61,4356,849],{"class":1866},[61,4358,3772],{"class":3766},[61,4360,300],{"class":1866},[61,4362,3777],{"class":3766},[61,4364,2016],{"class":1753},[61,4366,4368,4370,4372,4374,4376,4378,4380,4383,4385,4388,4390,4393,4395,4398,4400,4403,4405,4407],{"class":1731,"line":4367},141,[61,4369,3616],{"class":1753},[61,4371,3818],{"class":1836},[61,4373,849],{"class":1753},[61,4375,3763],{"class":1749},[61,4377,3794],{"class":3766},[61,4379,3797],{"class":1859},[61,4381,4382],{"class":3766},"mega_{ZX}",[61,4384,849],{"class":1866},[61,4386,4387],{"class":3766},"\\Delta_{ct}, ",[61,4389,4328],{"class":1859},[61,4391,4392],{"class":3766},"mega",[61,4394,300],{"class":1866},[61,4396,4397],{"class":3766},"$ ",[61,4399,849],{"class":1866},[61,4401,4402],{"class":3766},"leading-order",[61,4404,300],{"class":1866},[61,4406,3777],{"class":3766},[61,4408,2016],{"class":1753},[61,4410,4412,4414,4416],{"class":1731,"line":4411},142,[61,4413,3616],{"class":1753},[61,4415,3850],{"class":1836},[61,4417,2089],{"class":1753},[61,4419,4421],{"class":1731,"line":4420},143,[61,4422,1743],{"emptyLinePlaceholder":1742},[61,4424,4426],{"class":1731,"line":4425},144,[61,4427,4428],{"class":1735},"# Heatmap 2: |ω_ZX\u002Fω_ZZ|(Δ, Ω) (clipped for visibility)\n",[61,4430,4432,4435,4437,4439,4442],{"class":1731,"line":4431},145,[61,4433,4434],{"class":1753},"ratio_map ",[61,4436,2000],{"class":1859},[61,4438,2193],{"class":1753},[61,4440,4441],{"class":1836},"empty_like",[61,4443,4444],{"class":1753},"(wzx_map)\n",[61,4446,4448,4450,4452,4454,4456,4458,4460],{"class":1731,"line":4447},146,[61,4449,2486],{"class":1749},[61,4451,4106],{"class":1753},[61,4453,2492],{"class":1749},[61,4455,4111],{"class":1859},[61,4457,849],{"class":1753},[61,4459,3446],{"class":1866},[61,4461,3371],{"class":1753},[61,4463,4465,4468,4470,4472],{"class":1731,"line":4464},147,[61,4466,4467],{"class":1753},"    zz ",[61,4469,2000],{"class":1859},[61,4471,3087],{"class":1836},[61,4473,4474],{"class":1753},"(d, K.J, K.alpha_c, K.alpha_t)\n",[61,4476,4478,4480,4482,4484,4486,4488,4490],{"class":1731,"line":4477},148,[61,4479,3377],{"class":1749},[61,4481,4125],{"class":1753},[61,4483,2492],{"class":1749},[61,4485,4111],{"class":1859},[61,4487,849],{"class":1753},[61,4489,3472],{"class":1866},[61,4491,3371],{"class":1753},[61,4493,4495,4498,4500,4502],{"class":1731,"line":4494},149,[61,4496,4497],{"class":1753},"        zx ",[61,4499,2000],{"class":1859},[61,4501,2872],{"class":1836},[61,4503,4148],{"class":1753},[61,4505,4507,4510,4512,4514,4517,4519,4522,4524,4526,4529,4531,4533,4536],{"class":1731,"line":4506},150,[61,4508,4509],{"class":1753},"        ratio_map[i, j] ",[61,4511,2000],{"class":1859},[61,4513,2175],{"class":1859},[61,4515,4516],{"class":1753},"(zx ",[61,4518,602],{"class":1859},[61,4520,4521],{"class":1753}," zz) ",[61,4523,2498],{"class":1749},[61,4525,2175],{"class":1859},[61,4527,4528],{"class":1753},"(zz) ",[61,4530,2503],{"class":1859},[61,4532,2152],{"class":1866},[61,4534,4535],{"class":1749}," else",[61,4537,4538],{"class":1753}," np.nan\n",[61,4540,4542],{"class":1731,"line":4541},151,[61,4543,1743],{"emptyLinePlaceholder":1742},[61,4545,4547,4550,4552,4554,4557,4560,4562,4565,4568],{"class":1731,"line":4546},152,[61,4548,4549],{"class":1753},"clip_max ",[61,4551,2000],{"class":1859},[61,4553,2193],{"class":1753},[61,4555,4556],{"class":1836},"nanpercentile",[61,4558,4559],{"class":1753},"(ratio_map[np.",[61,4561,2514],{"class":1836},[61,4563,4564],{"class":1753},"(ratio_map)], ",[61,4566,4567],{"class":1866},"99.5",[61,4569,2016],{"class":1753},[61,4571,4573,4576,4578,4580,4583,4586,4588],{"class":1731,"line":4572},153,[61,4574,4575],{"class":1753},"ratio_map_vis ",[61,4577,2000],{"class":1859},[61,4579,2193],{"class":1753},[61,4581,4582],{"class":1836},"clip",[61,4584,4585],{"class":1753},"(ratio_map, ",[61,4587,2051],{"class":1866},[61,4589,4590],{"class":1753},", clip_max)\n",[61,4592,4594],{"class":1731,"line":4593},154,[61,4595,1743],{"emptyLinePlaceholder":1742},[61,4597,4599,4601,4603,4605,4607,4609,4611,4613,4615,4617,4619,4621,4623,4625],{"class":1731,"line":4598},155,[61,4600,3616],{"class":1753},[61,4602,3619],{"class":1836},[61,4604,849],{"class":1753},[61,4606,3624],{"class":3402},[61,4608,2000],{"class":1859},[61,4610,849],{"class":1753},[61,4612,3631],{"class":1866},[61,4614,1435],{"class":1753},[61,4616,1300],{"class":1866},[61,4618,3638],{"class":1753},[61,4620,3641],{"class":3402},[61,4622,2000],{"class":1859},[61,4624,3646],{"class":1866},[61,4626,2016],{"class":1753},[61,4628,4630,4632,4634,4637,4639,4641,4643,4645,4647,4649,4651,4653,4655,4657],{"class":1731,"line":4629},156,[61,4631,3616],{"class":1753},[61,4633,4239],{"class":1836},[61,4635,4636],{"class":1753},"(ratio_map_vis, ",[61,4638,4245],{"class":3402},[61,4640,2000],{"class":1859},[61,4642,4250],{"class":2165},[61,4644,1435],{"class":1753},[61,4646,4255],{"class":3402},[61,4648,2000],{"class":1859},[61,4650,4260],{"class":2165},[61,4652,1435],{"class":1753},[61,4654,4265],{"class":3402},[61,4656,2000],{"class":1859},[61,4658,4659],{"class":1753},"extent)\n",[61,4661,4663,4665,4667,4669,4671,4673,4675,4677,4680,4682,4685,4687,4690,4692,4695],{"class":1731,"line":4662},157,[61,4664,3616],{"class":1753},[61,4666,4288],{"class":1836},[61,4668,849],{"class":1753},[61,4670,3666],{"class":3402},[61,4672,2000],{"class":1859},[61,4674,3763],{"class":1749},[61,4676,3794],{"class":3766},[61,4678,4679],{"class":1753},"|",[61,4681,3797],{"class":1859},[61,4683,4684],{"class":3766},"mega_{ZX}\u002F",[61,4686,3797],{"class":1859},[61,4688,4689],{"class":3766},"mega_{ZZ}",[61,4691,4679],{"class":1753},[61,4693,4694],{"class":3766},"$'",[61,4696,2016],{"class":1753},[61,4698,4700,4702,4704,4706,4708,4710,4712,4714,4716,4718,4720,4722],{"class":1731,"line":4699},158,[61,4701,3616],{"class":1753},[61,4703,3758],{"class":1836},[61,4705,849],{"class":1753},[61,4707,3763],{"class":1749},[61,4709,3794],{"class":3766},[61,4711,4328],{"class":1859},[61,4713,4331],{"class":3766},[61,4715,849],{"class":1866},[61,4717,3772],{"class":3766},[61,4719,300],{"class":1866},[61,4721,3777],{"class":3766},[61,4723,2016],{"class":1753},[61,4725,4727,4729,4731,4733,4735,4737,4739,4741,4743,4745],{"class":1731,"line":4726},159,[61,4728,3616],{"class":1753},[61,4730,3787],{"class":1836},[61,4732,849],{"class":1753},[61,4734,3763],{"class":1749},[61,4736,3767],{"class":3766},[61,4738,849],{"class":1866},[61,4740,3772],{"class":3766},[61,4742,300],{"class":1866},[61,4744,3777],{"class":3766},[61,4746,2016],{"class":1753},[61,4748,4750,4752,4754,4756,4758,4761,4763,4765,4767,4769,4771,4773,4776,4778,4781,4783,4785],{"class":1731,"line":4749},160,[61,4751,3616],{"class":1753},[61,4753,3818],{"class":1836},[61,4755,849],{"class":1753},[61,4757,3763],{"class":1749},[61,4759,4760],{"class":3766},"'Heuristic $",[61,4762,4679],{"class":1753},[61,4764,3797],{"class":1859},[61,4766,4684],{"class":3766},[61,4768,3797],{"class":1859},[61,4770,4689],{"class":3766},[61,4772,4679],{"class":1753},[61,4774,4775],{"class":3766},"$ map ",[61,4777,849],{"class":1866},[61,4779,4780],{"class":3766},"clipped at 99.5th pct",[61,4782,300],{"class":1866},[61,4784,3777],{"class":3766},[61,4786,2016],{"class":1753},[61,4788,4790,4792,4794],{"class":1731,"line":4789},161,[61,4791,3616],{"class":1753},[61,4793,3850],{"class":1836},[61,4795,2089],{"class":1753},[4797,4798],"blog-figure",{"alt":4799,"src":4800},"Output 1 of the notebook","\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-01.webp",[4797,4802],{"alt":4803,"src":4804},"Output 2 of the notebook","\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-02.webp",[4797,4806],{"alt":4807,"src":4808},"Output 3 of the notebook","\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-03.webp",[4797,4810],{"alt":4811,"src":4812},"Output 4 of the notebook","\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-04.webp",[1720,4814,4816],{"className":1722,"code":4815,"language":1724,"meta":1725,"style":1725},"#@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",[1727,4817,4818,4823,4827,4833,4843,4853,4857,4911,4916,4935,4965,5005,5040,5083,5121,5143,5211,5257,5335,5406,5413,5417,5422,5436,5448,5462,5477,5481,5505,5547,5551,5563,5585,5595,5599,5611,5615,5647,5702,5726,5746,5756,5766,5779,5792],{"__ignoreMap":1725},[61,4819,4820],{"class":1731,"line":1732},[61,4821,4822],{"class":1735},"#@title Echoed-CR demonstration (two-segment echo, Eqs. 28a–d in PRA 2020)\n",[61,4824,4825],{"class":1731,"line":1739},[61,4826,1743],{"emptyLinePlaceholder":1742},[61,4828,4829,4831],{"class":1731,"line":1746},[61,4830,1757],{"class":1749},[61,4832,1781],{"class":1753},[61,4834,4835,4837,4839,4841],{"class":1731,"line":1763},[61,4836,1757],{"class":1749},[61,4838,1789],{"class":1753},[61,4840,1792],{"class":1749},[61,4842,1795],{"class":1753},[61,4844,4845,4847,4849,4851],{"class":1731,"line":1776},[61,4846,1757],{"class":1749},[61,4848,1803],{"class":1753},[61,4850,1792],{"class":1749},[61,4852,1808],{"class":1753},[61,4854,4855],{"class":1731,"line":1784},[61,4856,1743],{"emptyLinePlaceholder":1742},[61,4858,4859,4861,4864,4866,4869,4871,4873,4875,4878,4880,4882,4884,4887,4889,4891,4893,4896,4898,4900,4902,4905,4907,4909],{"class":1731,"line":1798},[61,4860,2116],{"class":1749},[61,4862,4863],{"class":1836}," omega_echo_coeffs",[61,4865,849],{"class":1753},[61,4867,4868],{"class":2124},"zx_mhz",[61,4870,2127],{"class":1753},[61,4872,1860],{"class":1859},[61,4874,1435],{"class":1753},[61,4876,4877],{"class":2124},"ix_mhz",[61,4879,2127],{"class":1753},[61,4881,1860],{"class":1859},[61,4883,1435],{"class":1753},[61,4885,4886],{"class":2124},"iz_mhz",[61,4888,2127],{"class":1753},[61,4890,1860],{"class":1859},[61,4892,1435],{"class":1753},[61,4894,4895],{"class":2124},"zz_mhz",[61,4897,2127],{"class":1753},[61,4899,1860],{"class":1859},[61,4901,1435],{"class":1753},[61,4903,4904],{"class":2124},"tau_ns",[61,4906,2127],{"class":1753},[61,4908,1860],{"class":1859},[61,4910,3371],{"class":1753},[61,4912,4913],{"class":1731,"line":1811},[61,4914,4915],{"class":2165},"    \"\"\"Return (u_ii, u_iy, u_iz, u_zx) after a +Ω \u002F Xπ \u002F −Ω echo; MHz→rad\u002Fns inside.\"\"\"\n",[61,4917,4918,4921,4923,4925,4927,4930,4932],{"class":1731,"line":1816},[61,4919,4920],{"class":1753},"    conv ",[61,4922,2000],{"class":1859},[61,4924,3029],{"class":1866},[61,4926,2298],{"class":1859},[61,4928,4929],{"class":1753}," math.pi ",[61,4931,602],{"class":1859},[61,4933,4934],{"class":1866}," 1000.0\n",[61,4936,4937,4940,4942,4945,4947,4950,4952,4955,4957,4960,4962],{"class":1731,"line":1822},[61,4938,4939],{"class":1753},"    zx, ix, iz, zz ",[61,4941,2000],{"class":1859},[61,4943,4944],{"class":1753}," zx_mhz",[61,4946,2201],{"class":1859},[61,4948,4949],{"class":1753},"conv, ix_mhz",[61,4951,2201],{"class":1859},[61,4953,4954],{"class":1753},"conv, iz_mhz",[61,4956,2201],{"class":1859},[61,4958,4959],{"class":1753},"conv, zz_mhz",[61,4961,2201],{"class":1859},[61,4963,4964],{"class":1753},"conv\n",[61,4966,4967,4970,4972,4974,4976,4979,4981,4984,4986,4988,4991,4994,4996,4999,5001,5003],{"class":1731,"line":1828},[61,4968,4969],{"class":1753},"    w_plus  ",[61,4971,2000],{"class":1859},[61,4973,2422],{"class":1753},[61,4975,2425],{"class":1836},[61,4977,4978],{"class":1753},"((zx ",[61,4980,2419],{"class":1859},[61,4982,4983],{"class":1753}," ix)",[61,4985,2651],{"class":1859},[61,4987,337],{"class":1866},[61,4989,4990],{"class":1859}," +",[61,4992,4993],{"class":1753}," (iz ",[61,4995,2419],{"class":1859},[61,4997,4998],{"class":1753}," zz)",[61,5000,2651],{"class":1859},[61,5002,337],{"class":1866},[61,5004,2016],{"class":1753},[61,5006,5007,5010,5012,5014,5016,5018,5020,5022,5024,5026,5028,5030,5032,5034,5036,5038],{"class":1731,"line":1833},[61,5008,5009],{"class":1753},"    w_minus ",[61,5011,2000],{"class":1859},[61,5013,2422],{"class":1753},[61,5015,2425],{"class":1836},[61,5017,4978],{"class":1753},[61,5019,2006],{"class":1859},[61,5021,4983],{"class":1753},[61,5023,2651],{"class":1859},[61,5025,337],{"class":1866},[61,5027,4990],{"class":1859},[61,5029,4993],{"class":1753},[61,5031,2006],{"class":1859},[61,5033,4998],{"class":1753},[61,5035,2651],{"class":1859},[61,5037,337],{"class":1866},[61,5039,2016],{"class":1753},[61,5041,5042,5045,5047,5049,5052,5054,5057,5059,5062,5064,5067,5070,5072,5074,5076,5078,5080],{"class":1731,"line":1840},[61,5043,5044],{"class":1753},"    c_plus, s_plus ",[61,5046,2000],{"class":1859},[61,5048,2422],{"class":1753},[61,5050,5051],{"class":1836},"cos",[61,5053,849],{"class":1753},[61,5055,5056],{"class":1866},"0.5",[61,5058,2201],{"class":1859},[61,5060,5061],{"class":1753},"w_plus",[61,5063,2201],{"class":1859},[61,5065,5066],{"class":1753},"tau_ns), math.",[61,5068,5069],{"class":1836},"sin",[61,5071,849],{"class":1753},[61,5073,5056],{"class":1866},[61,5075,2201],{"class":1859},[61,5077,5061],{"class":1753},[61,5079,2201],{"class":1859},[61,5081,5082],{"class":1753},"tau_ns)\n",[61,5084,5085,5088,5090,5092,5094,5096,5098,5100,5103,5105,5107,5109,5111,5113,5115,5117,5119],{"class":1731,"line":1853},[61,5086,5087],{"class":1753},"    c_minus, s_minus ",[61,5089,2000],{"class":1859},[61,5091,2422],{"class":1753},[61,5093,5051],{"class":1836},[61,5095,849],{"class":1753},[61,5097,5056],{"class":1866},[61,5099,2201],{"class":1859},[61,5101,5102],{"class":1753},"w_minus",[61,5104,2201],{"class":1859},[61,5106,5066],{"class":1753},[61,5108,5069],{"class":1836},[61,5110,849],{"class":1753},[61,5112,5056],{"class":1866},[61,5114,2201],{"class":1859},[61,5116,5102],{"class":1753},[61,5118,2201],{"class":1859},[61,5120,5082],{"class":1753},[61,5122,5123,5126,5128,5131,5133,5136,5138,5141],{"class":1731,"line":1873},[61,5124,5125],{"class":1753},"    denom ",[61,5127,2000],{"class":1859},[61,5129,5130],{"class":1753}," (w_plus",[61,5132,2201],{"class":1859},[61,5134,5135],{"class":1753},"w_minus ",[61,5137,2419],{"class":1859},[61,5139,5140],{"class":1866}," 1e-30",[61,5142,2016],{"class":1753},[61,5144,5145,5148,5150,5153,5155,5158,5160,5163,5165,5167,5169,5172,5174,5176,5178,5181,5183,5185,5187,5190,5192,5194,5196,5198,5201,5203,5206,5208],{"class":1731,"line":1889},[61,5146,5147],{"class":1753},"    u_ii ",[61,5149,2000],{"class":1859},[61,5151,5152],{"class":1753}," c_plus",[61,5154,2201],{"class":1859},[61,5156,5157],{"class":1753},"c_minus ",[61,5159,2419],{"class":1859},[61,5161,5162],{"class":1753}," ((ix",[61,5164,2651],{"class":1859},[61,5166,337],{"class":1866},[61,5168,1899],{"class":1859},[61,5170,5171],{"class":1753}," iz",[61,5173,2651],{"class":1859},[61,5175,337],{"class":1866},[61,5177,1899],{"class":1859},[61,5179,5180],{"class":1753}," zx",[61,5182,2651],{"class":1859},[61,5184,337],{"class":1866},[61,5186,4990],{"class":1859},[61,5188,5189],{"class":1753}," zz",[61,5191,2651],{"class":1859},[61,5193,337],{"class":1866},[61,5195,300],{"class":1753},[61,5197,602],{"class":1859},[61,5199,5200],{"class":1753},"denom)",[61,5202,2201],{"class":1859},[61,5204,5205],{"class":1753},"s_plus",[61,5207,2201],{"class":1859},[61,5209,5210],{"class":1753},"s_minus\n",[61,5212,5213,5216,5218,5220,5223,5225,5228,5230,5233,5235,5238,5240,5243,5245,5248,5250,5253,5255],{"class":1731,"line":1908},[61,5214,5215],{"class":1753},"    u_iy ",[61,5217,2000],{"class":1859},[61,5219,2654],{"class":1866},[61,5221,5222],{"class":1749},"j",[61,5224,2201],{"class":1859},[61,5226,5227],{"class":1753},"(zx",[61,5229,2201],{"class":1859},[61,5231,5232],{"class":1753},"zz ",[61,5234,2006],{"class":1859},[61,5236,5237],{"class":1753}," ix",[61,5239,2201],{"class":1859},[61,5241,5242],{"class":1753},"iz)",[61,5244,602],{"class":1859},[61,5246,5247],{"class":1753},"denom ",[61,5249,2201],{"class":1859},[61,5251,5252],{"class":1753}," s_plus",[61,5254,2201],{"class":1859},[61,5256,5210],{"class":1753},[61,5258,5259,5262,5264,5267,5269,5271,5274,5276,5279,5281,5284,5286,5288,5291,5293,5296,5298,5301,5303,5305,5307,5309,5311,5313,5315,5317,5320,5322,5324,5326,5328,5330,5332],{"class":1731,"line":1925},[61,5260,5261],{"class":1753},"    u_iz ",[61,5263,2000],{"class":1859},[61,5265,5266],{"class":1866}," 1",[61,5268,5222],{"class":1749},[61,5270,2201],{"class":1859},[61,5272,5273],{"class":1753},"((zz ",[61,5275,2006],{"class":1859},[61,5277,5278],{"class":1753}," iz)",[61,5280,602],{"class":1859},[61,5282,5283],{"class":1753},"(w_minus ",[61,5285,2419],{"class":1859},[61,5287,5140],{"class":1866},[61,5289,5290],{"class":1753},"))",[61,5292,2201],{"class":1859},[61,5294,5295],{"class":1753},"c_plus",[61,5297,2201],{"class":1859},[61,5299,5300],{"class":1753},"s_minus ",[61,5302,2006],{"class":1859},[61,5304,5266],{"class":1866},[61,5306,5222],{"class":1749},[61,5308,2201],{"class":1859},[61,5310,5273],{"class":1753},[61,5312,2419],{"class":1859},[61,5314,5278],{"class":1753},[61,5316,602],{"class":1859},[61,5318,5319],{"class":1753},"(w_plus ",[61,5321,2419],{"class":1859},[61,5323,5140],{"class":1866},[61,5325,5290],{"class":1753},[61,5327,2201],{"class":1859},[61,5329,5205],{"class":1753},[61,5331,2201],{"class":1859},[61,5333,5334],{"class":1753},"c_minus\n",[61,5336,5337,5340,5342,5344,5346,5348,5351,5353,5356,5358,5360,5362,5364,5366,5368,5370,5372,5374,5376,5378,5380,5382,5384,5386,5388,5390,5392,5394,5396,5398,5400,5402,5404],{"class":1731,"line":1941},[61,5338,5339],{"class":1753},"    u_zx ",[61,5341,2000],{"class":1859},[61,5343,5266],{"class":1866},[61,5345,5222],{"class":1749},[61,5347,2201],{"class":1859},[61,5349,5350],{"class":1753},"((ix ",[61,5352,2006],{"class":1859},[61,5354,5355],{"class":1753}," zx)",[61,5357,602],{"class":1859},[61,5359,5283],{"class":1753},[61,5361,2419],{"class":1859},[61,5363,5140],{"class":1866},[61,5365,5290],{"class":1753},[61,5367,2201],{"class":1859},[61,5369,5295],{"class":1753},[61,5371,2201],{"class":1859},[61,5373,5300],{"class":1753},[61,5375,2006],{"class":1859},[61,5377,5266],{"class":1866},[61,5379,5222],{"class":1749},[61,5381,2201],{"class":1859},[61,5383,5350],{"class":1753},[61,5385,2419],{"class":1859},[61,5387,5355],{"class":1753},[61,5389,602],{"class":1859},[61,5391,5319],{"class":1753},[61,5393,2419],{"class":1859},[61,5395,5140],{"class":1866},[61,5397,5290],{"class":1753},[61,5399,2201],{"class":1859},[61,5401,5205],{"class":1753},[61,5403,2201],{"class":1859},[61,5405,5334],{"class":1753},[61,5407,5408,5410],{"class":1731,"line":1957},[61,5409,2210],{"class":1749},[61,5411,5412],{"class":1753}," u_ii, u_iy, u_iz, u_zx\n",[61,5414,5415],{"class":1731,"line":1974},[61,5416,1743],{"emptyLinePlaceholder":1742},[61,5418,5419],{"class":1731,"line":1979},[61,5420,5421],{"class":1735},"# Representative point near Region III speed-up:\n",[61,5423,5424,5427,5429,5431,5433],{"class":1731,"line":1985},[61,5425,5426],{"class":1753},"f_c, f_t ",[61,5428,2000],{"class":1859},[61,5430,1867],{"class":1866},[61,5432,1435],{"class":1753},[61,5434,5435],{"class":1866},"4914.0\n",[61,5437,5438,5441,5443,5445],{"class":1731,"line":2019},[61,5439,5440],{"class":1753},"alpha_c ",[61,5442,2000],{"class":1859},[61,5444,1899],{"class":1859},[61,5446,5447],{"class":1866},"330.0\n",[61,5449,5450,5453,5455,5457,5459],{"class":1731,"line":2033},[61,5451,5452],{"class":1753},"J, Omega ",[61,5454,2000],{"class":1859},[61,5456,1935],{"class":1866},[61,5458,1435],{"class":1753},[61,5460,5461],{"class":1866},"30.0\n",[61,5463,5464,5467,5469,5472,5474],{"class":1731,"line":2061},[61,5465,5466],{"class":1753},"delta0 ",[61,5468,2000],{"class":1859},[61,5470,5471],{"class":1753}," f_c ",[61,5473,2006],{"class":1859},[61,5475,5476],{"class":1753}," f_t\n",[61,5478,5479],{"class":1731,"line":2074},[61,5480,1743],{"emptyLinePlaceholder":1742},[61,5482,5483,5485,5487,5489,5491,5493,5495,5497,5499,5501,5503],{"class":1731,"line":2079},[61,5484,2116],{"class":1749},[61,5486,2872],{"class":1836},[61,5488,849],{"class":1753},[61,5490,2877],{"class":2124},[61,5492,1435],{"class":1753},[61,5494,2886],{"class":2124},[61,5496,1435],{"class":1753},[61,5498,2895],{"class":2124},[61,5500,1435],{"class":1753},[61,5502,2904],{"class":2124},[61,5504,3371],{"class":1753},[61,5506,5507,5509,5512,5514,5516,5518,5520,5522,5524,5527,5529,5531,5533,5535,5538,5541,5544],{"class":1731,"line":2092},[61,5508,2210],{"class":1749},[61,5510,5511],{"class":1753}," J",[61,5513,2201],{"class":1859},[61,5515,2895],{"class":1753},[61,5517,2201],{"class":1859},[61,5519,849],{"class":1753},[61,5521,2319],{"class":1866},[61,5523,602],{"class":1859},[61,5525,5526],{"class":1753},"(delta_ct ",[61,5528,2419],{"class":1859},[61,5530,2958],{"class":1753},[61,5532,2006],{"class":1859},[61,5534,2609],{"class":1866},[61,5536,5537],{"class":1859},"\u002Fmax",[61,5539,5540],{"class":1753},"(delta_ct, ",[61,5542,5543],{"class":1866},"1e-12",[61,5545,5546],{"class":1753},"))\n",[61,5548,5549],{"class":1731,"line":2097},[61,5550,1743],{"emptyLinePlaceholder":1742},[61,5552,5553,5556,5558,5560],{"class":1731,"line":2102},[61,5554,5555],{"class":1753},"w_zx ",[61,5557,2000],{"class":1859},[61,5559,2872],{"class":1836},[61,5561,5562],{"class":1753},"(delta0, J, Omega, alpha_c)\n",[61,5564,5565,5568,5570,5572,5574,5577,5579,5582],{"class":1731,"line":2108},[61,5566,5567],{"class":1753},"w_ix, w_iz, w_zz ",[61,5569,2000],{"class":1859},[61,5571,2609],{"class":1866},[61,5573,1435],{"class":1753},[61,5575,5576],{"class":1866},"10.0",[61,5578,1435],{"class":1753},[61,5580,5581],{"class":1866},"0.1",[61,5583,5584],{"class":1735},"   # illustrative single-qubit terms \u002F static ZZ\n",[61,5586,5587,5590,5592],{"class":1731,"line":2113},[61,5588,5589],{"class":1753},"tau_ns ",[61,5591,2000],{"class":1859},[61,5593,5594],{"class":1866}," 100.0\n",[61,5596,5597],{"class":1731,"line":2162},[61,5598,1743],{"emptyLinePlaceholder":1742},[61,5600,5601,5604,5606,5608],{"class":1731,"line":2169},[61,5602,5603],{"class":1753},"u_ii, u_iy, u_iz, u_zx ",[61,5605,2000],{"class":1859},[61,5607,4863],{"class":1836},[61,5609,5610],{"class":1753},"(w_zx, w_ix, w_iz, w_zz, tau_ns)\n",[61,5612,5613],{"class":1731,"line":2187},[61,5614,1743],{"emptyLinePlaceholder":1742},[61,5616,5617,5619,5621,5623,5625,5627,5629,5632,5634,5637,5639,5641,5643,5645],{"class":1731,"line":2207},[61,5618,3616],{"class":1753},[61,5620,3619],{"class":1836},[61,5622,849],{"class":1753},[61,5624,3624],{"class":3402},[61,5626,2000],{"class":1859},[61,5628,849],{"class":1753},[61,5630,5631],{"class":1866},"7.5",[61,5633,1435],{"class":1753},[61,5635,5636],{"class":1866},"4.8",[61,5638,3638],{"class":1753},[61,5640,3641],{"class":3402},[61,5642,2000],{"class":1859},[61,5644,3646],{"class":1866},[61,5646,2016],{"class":1753},[61,5648,5649,5652,5654,5656,5658,5660,5662,5665,5667,5669,5671,5673,5675,5677,5680,5682,5684,5686,5688,5690,5692,5695,5697,5699],{"class":1731,"line":2221},[61,5650,5651],{"class":1753},"labels ",[61,5653,2000],{"class":1859},[61,5655,4195],{"class":1753},[61,5657,3763],{"class":1749},[61,5659,3794],{"class":3766},[61,5661,4679],{"class":1753},[61,5663,5664],{"class":3766},"u_{ZX}",[61,5666,4679],{"class":1753},[61,5668,4694],{"class":3766},[61,5670,1435],{"class":1753},[61,5672,3763],{"class":1749},[61,5674,3794],{"class":3766},[61,5676,4679],{"class":1753},[61,5678,5679],{"class":3766},"u_{IY}",[61,5681,4679],{"class":1753},[61,5683,4694],{"class":3766},[61,5685,1435],{"class":1753},[61,5687,3763],{"class":1749},[61,5689,3794],{"class":3766},[61,5691,4679],{"class":1753},[61,5693,5694],{"class":3766},"u_{IZ}",[61,5696,4679],{"class":1753},[61,5698,4694],{"class":3766},[61,5700,5701],{"class":1753},"]\n",[61,5703,5704,5707,5709,5711,5713,5716,5718,5721,5723],{"class":1731,"line":2226},[61,5705,5706],{"class":1753},"vals ",[61,5708,2000],{"class":1859},[61,5710,4195],{"class":1753},[61,5712,3574],{"class":1859},[61,5714,5715],{"class":1753},"(u_zx), ",[61,5717,3574],{"class":1859},[61,5719,5720],{"class":1753},"(u_iy), ",[61,5722,3574],{"class":1859},[61,5724,5725],{"class":1753},"(u_iz)]\n",[61,5727,5728,5731,5733,5735,5738,5740,5743],{"class":1731,"line":2258},[61,5729,5730],{"class":1753},"x ",[61,5732,2000],{"class":1859},[61,5734,2193],{"class":1753},[61,5736,5737],{"class":1836},"arange",[61,5739,849],{"class":1753},[61,5741,5742],{"class":1859},"len",[61,5744,5745],{"class":1753},"(labels))\n",[61,5747,5748,5750,5753],{"class":1731,"line":2264},[61,5749,3616],{"class":1753},[61,5751,5752],{"class":1836},"bar",[61,5754,5755],{"class":1753},"(x, vals)\n",[61,5757,5758,5760,5763],{"class":1731,"line":2280},[61,5759,3616],{"class":1753},[61,5761,5762],{"class":1836},"xticks",[61,5764,5765],{"class":1753},"(x, labels)\n",[61,5767,5768,5770,5772,5774,5777],{"class":1731,"line":2304},[61,5769,3616],{"class":1753},[61,5771,3787],{"class":1836},[61,5773,849],{"class":1753},[61,5775,5776],{"class":2165},"'Coefficient magnitude (unitless)'",[61,5778,2016],{"class":1753},[61,5780,5781,5783,5785,5787,5790],{"class":1731,"line":2326},[61,5782,3616],{"class":1753},[61,5784,3818],{"class":1836},[61,5786,849],{"class":1753},[61,5788,5789],{"class":2165},"'Echoed-CR: effective coefficients after two-segment echo'",[61,5791,2016],{"class":1753},[61,5793,5794,5796,5798],{"class":1731,"line":2342},[61,5795,3616],{"class":1753},[61,5797,3850],{"class":1836},[61,5799,2089],{"class":1753},[4797,5801],{"alt":5802,"src":5803},"Output 5 of the notebook","\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-05.webp",[1720,5805,5807],{"className":1722,"code":5806,"language":1724,"meta":1725,"style":1725},"#@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",[1727,5808,5809,5814,5819,5823,5833,5839,5849,5859,5863,5867,5872,5876,5880,5889,5902,5915,5930,5945,5958,5972,5976,5981,5995,6010,6025,6040,6044,6049,6061,6073,6077,6087,6091,6095,6100,6104,6144,6179,6183,6205,6225,6229,6251,6267,6271,6293,6309,6313,6318,6323,6367,6403,6407,6451,6493,6497,6501,6506,6510,6528,6551,6565,6586,6600,6604,6633,6637,6652,6667,6683,6706,6718,6730,6742,6765,6785,6805,6825,6843,6847,6851,6856,6860,6886,6902,6918,6934,6956,6970,7011,7018,7022,7027,7063,7098,7133,7168,7172,7176,7181,7185,7197,7215,7219,7224,7236,7248,7252,7257,7307,7353,7357,7361,7366,7370,7401,7427,7454,7467,7480,7493,7501],{"__ignoreMap":1725},[61,5810,5811],{"class":1731,"line":1732},[61,5812,5813],{"class":1735},"#@title Conditional target Rabi under CR effective Hamiltonian\n",[61,5815,5816],{"class":1731,"line":1739},[61,5817,5818],{"class":1735},"# we diagonalize once and propagate.\n",[61,5820,5821],{"class":1731,"line":1746},[61,5822,1743],{"emptyLinePlaceholder":1742},[61,5824,5825,5827,5829,5831],{"class":1731,"line":1763},[61,5826,1750],{"class":1749},[61,5828,1754],{"class":1753},[61,5830,1757],{"class":1749},[61,5832,1760],{"class":1753},[61,5834,5835,5837],{"class":1731,"line":1776},[61,5836,1757],{"class":1749},[61,5838,1781],{"class":1753},[61,5840,5841,5843,5845,5847],{"class":1731,"line":1784},[61,5842,1757],{"class":1749},[61,5844,1789],{"class":1753},[61,5846,1792],{"class":1749},[61,5848,1795],{"class":1753},[61,5850,5851,5853,5855,5857],{"class":1731,"line":1798},[61,5852,1757],{"class":1749},[61,5854,1803],{"class":1753},[61,5856,1792],{"class":1749},[61,5858,1808],{"class":1753},[61,5860,5861],{"class":1731,"line":1811},[61,5862,1743],{"emptyLinePlaceholder":1742},[61,5864,5865],{"class":1731,"line":1816},[61,5866,1819],{"class":1735},[61,5868,5869],{"class":1731,"line":1822},[61,5870,5871],{"class":1735},"# Control knobs (MHz unless noted)\n",[61,5873,5874],{"class":1731,"line":1828},[61,5875,1819],{"class":1735},[61,5877,5878],{"class":1731,"line":1833},[61,5879,1837],{"class":1836},[61,5881,5882,5884,5887],{"class":1731,"line":1840},[61,5883,1843],{"class":1749},[61,5885,5886],{"class":1846}," RabiKnobs",[61,5888,1850],{"class":1753},[61,5890,5891,5893,5895,5897,5899],{"class":1731,"line":1853},[61,5892,1856],{"class":1753},[61,5894,1860],{"class":1859},[61,5896,1863],{"class":1859},[61,5898,1867],{"class":1866},[61,5900,5901],{"class":1735},"          # control frequency\n",[61,5903,5904,5906,5908,5910,5912],{"class":1731,"line":1873},[61,5905,1876],{"class":1753},[61,5907,1860],{"class":1859},[61,5909,1863],{"class":1859},[61,5911,1883],{"class":1866},[61,5913,5914],{"class":1735},"          # target frequency\n",[61,5916,5917,5919,5921,5923,5925,5927],{"class":1731,"line":1889},[61,5918,1892],{"class":1753},[61,5920,1860],{"class":1859},[61,5922,1863],{"class":1859},[61,5924,1899],{"class":1859},[61,5926,1902],{"class":1866},[61,5928,5929],{"class":1735},"      # control anharmonicity\n",[61,5931,5932,5934,5936,5938,5940,5942],{"class":1731,"line":1908},[61,5933,1911],{"class":1753},[61,5935,1860],{"class":1859},[61,5937,1863],{"class":1859},[61,5939,1899],{"class":1859},[61,5941,1902],{"class":1866},[61,5943,5944],{"class":1735},"      # target anharmonicity\n",[61,5946,5947,5949,5951,5953,5955],{"class":1731,"line":1925},[61,5948,1928],{"class":1753},[61,5950,1860],{"class":1859},[61,5952,1863],{"class":1859},[61,5954,1935],{"class":1866},[61,5956,5957],{"class":1735},"               # exchange coupling\n",[61,5959,5960,5963,5965,5967,5969],{"class":1731,"line":1941},[61,5961,5962],{"class":1753},"    Omega: ",[61,5964,1860],{"class":1859},[61,5966,1863],{"class":1859},[61,5968,1951],{"class":1866},[61,5970,5971],{"class":1735},"          # CR drive amplitude on control\n",[61,5973,5974],{"class":1731,"line":1957},[61,5975,1743],{"emptyLinePlaceholder":1742},[61,5977,5978],{"class":1731,"line":1974},[61,5979,5980],{"class":1735},"    # Residuals (typical orders, tweak for your device)\n",[61,5982,5983,5986,5988,5990,5992],{"class":1731,"line":1979},[61,5984,5985],{"class":1753},"    wIX_mhz: ",[61,5987,1860],{"class":1859},[61,5989,1863],{"class":1859},[61,5991,2609],{"class":1866},[61,5993,5994],{"class":1735},"         # IX (drive crosstalk on target)\n",[61,5996,5997,6000,6002,6004,6007],{"class":1731,"line":1985},[61,5998,5999],{"class":1753},"    wIZ_khz: ",[61,6001,1860],{"class":1859},[61,6003,1863],{"class":1859},[61,6005,6006],{"class":1866}," 20.0",[61,6008,6009],{"class":1735},"        # IZ (AC Stark on target)\n",[61,6011,6012,6015,6017,6019,6022],{"class":1731,"line":2019},[61,6013,6014],{"class":1753},"    wZI_mhz: ",[61,6016,1860],{"class":1859},[61,6018,1863],{"class":1859},[61,6020,6021],{"class":1866}," 10.0",[61,6023,6024],{"class":1735},"        # ZI (AC Stark on control)\n",[61,6026,6027,6030,6032,6034,6037],{"class":1731,"line":2033},[61,6028,6029],{"class":1753},"    wZZ_khz: ",[61,6031,1860],{"class":1859},[61,6033,1863],{"class":1859},[61,6035,6036],{"class":1866}," 100.0",[61,6038,6039],{"class":1735},"       # ZZ (static coupling residual)\n",[61,6041,6042],{"class":1731,"line":2061},[61,6043,1743],{"emptyLinePlaceholder":1742},[61,6045,6046],{"class":1731,"line":2074},[61,6047,6048],{"class":1735},"    # Time axis\n",[61,6050,6051,6054,6056,6058],{"class":1731,"line":2079},[61,6052,6053],{"class":1753},"    t_end_ns: ",[61,6055,1860],{"class":1859},[61,6057,1863],{"class":1859},[61,6059,6060],{"class":1866}," 400.0\n",[61,6062,6063,6066,6068,6070],{"class":1731,"line":2092},[61,6064,6065],{"class":1753},"    n_steps: ",[61,6067,2025],{"class":1859},[61,6069,1863],{"class":1859},[61,6071,6072],{"class":1866}," 2001\n",[61,6074,6075],{"class":1731,"line":2097},[61,6076,1743],{"emptyLinePlaceholder":1742},[61,6078,6079,6081,6083,6085],{"class":1731,"line":2102},[61,6080,2082],{"class":1753},[61,6082,2000],{"class":1859},[61,6084,5886],{"class":1836},[61,6086,2089],{"class":1753},[61,6088,6089],{"class":1731,"line":2108},[61,6090,1743],{"emptyLinePlaceholder":1742},[61,6092,6093],{"class":1731,"line":2113},[61,6094,1819],{"class":1735},[61,6096,6097],{"class":1731,"line":2162},[61,6098,6099],{"class":1735},"# Helpers and literature-rate formulas\n",[61,6101,6102],{"class":1731,"line":2169},[61,6103,1819],{"class":1735},[61,6105,6106,6108,6110,6112,6114,6116,6118,6120,6122,6124,6126,6128,6130,6132,6134,6136,6138,6140,6142],{"class":1731,"line":2187},[61,6107,2116],{"class":1749},[61,6109,2119],{"class":1836},[61,6111,849],{"class":1753},[61,6113,19],{"class":2124},[61,6115,2127],{"class":1753},[61,6117,1860],{"class":1859},[61,6119,1435],{"class":1753},[61,6121,2134],{"class":2124},[61,6123,2127],{"class":1753},[61,6125,1860],{"class":1859},[61,6127,1435],{"class":1753},[61,6129,2143],{"class":2124},[61,6131,2127],{"class":1753},[61,6133,1860],{"class":1859},[61,6135,1863],{"class":1859},[61,6137,2152],{"class":1866},[61,6139,2155],{"class":1753},[61,6141,1860],{"class":1859},[61,6143,1850],{"class":1753},[61,6145,6146,6148,6150,6152,6155,6157,6159,6161,6164,6166,6168,6170,6172,6174,6176],{"class":1731,"line":2207},[61,6147,2210],{"class":1749},[61,6149,2213],{"class":1753},[61,6151,602],{"class":1859},[61,6153,6154],{"class":1753}," (b ",[61,6156,2498],{"class":1749},[61,6158,2175],{"class":1859},[61,6160,2178],{"class":1753},[61,6162,6163],{"class":1859},">=",[61,6165,2629],{"class":1753},[61,6167,2536],{"class":1749},[61,6169,2193],{"class":1753},[61,6171,2196],{"class":1836},[61,6173,2178],{"class":1753},[61,6175,2201],{"class":1859},[61,6177,6178],{"class":1753}," eps)\n",[61,6180,6181],{"class":1731,"line":2221},[61,6182,1743],{"emptyLinePlaceholder":1742},[61,6184,6185,6187,6190,6192,6195,6197,6199,6201,6203],{"class":1731,"line":2226},[61,6186,2116],{"class":1749},[61,6188,6189],{"class":1836}," hz_to_rad_per_ns",[61,6191,849],{"class":1753},[61,6193,6194],{"class":2124},"hz",[61,6196,2127],{"class":1753},[61,6198,1860],{"class":1859},[61,6200,2155],{"class":1753},[61,6202,1860],{"class":1859},[61,6204,1850],{"class":1753},[61,6206,6207,6209,6211,6213,6215,6217,6220,6222],{"class":1731,"line":2258},[61,6208,2210],{"class":1749},[61,6210,3029],{"class":1866},[61,6212,2298],{"class":1859},[61,6214,4929],{"class":1753},[61,6216,2201],{"class":1859},[61,6218,6219],{"class":1753}," hz ",[61,6221,2201],{"class":1859},[61,6223,6224],{"class":1866}," 1e-9\n",[61,6226,6227],{"class":1731,"line":2264},[61,6228,1743],{"emptyLinePlaceholder":1742},[61,6230,6231,6233,6236,6238,6241,6243,6245,6247,6249],{"class":1731,"line":2280},[61,6232,2116],{"class":1749},[61,6234,6235],{"class":1836}," mhz_to_rad_per_ns",[61,6237,849],{"class":1753},[61,6239,6240],{"class":2124},"mhz",[61,6242,2127],{"class":1753},[61,6244,1860],{"class":1859},[61,6246,2155],{"class":1753},[61,6248,1860],{"class":1859},[61,6250,1850],{"class":1753},[61,6252,6253,6255,6257,6260,6262,6265],{"class":1731,"line":2304},[61,6254,2210],{"class":1749},[61,6256,6189],{"class":1836},[61,6258,6259],{"class":1753},"(mhz ",[61,6261,2201],{"class":1859},[61,6263,6264],{"class":1866}," 1e6",[61,6266,2016],{"class":1753},[61,6268,6269],{"class":1731,"line":2326},[61,6270,1743],{"emptyLinePlaceholder":1742},[61,6272,6273,6275,6278,6280,6283,6285,6287,6289,6291],{"class":1731,"line":2342},[61,6274,2116],{"class":1749},[61,6276,6277],{"class":1836}," khz_to_rad_per_ns",[61,6279,849],{"class":1753},[61,6281,6282],{"class":2124},"khz",[61,6284,2127],{"class":1753},[61,6286,1860],{"class":1859},[61,6288,2155],{"class":1753},[61,6290,1860],{"class":1859},[61,6292,1850],{"class":1753},[61,6294,6295,6297,6299,6302,6304,6307],{"class":1731,"line":2379},[61,6296,2210],{"class":1749},[61,6298,6189],{"class":1836},[61,6300,6301],{"class":1753},"(khz ",[61,6303,2201],{"class":1859},[61,6305,6306],{"class":1866}," 1e3",[61,6308,2016],{"class":1753},[61,6310,6311],{"class":1731,"line":2396},[61,6312,1743],{"emptyLinePlaceholder":1742},[61,6314,6315],{"class":1731,"line":2404},[61,6316,6317],{"class":1735},"# Leading-order CR analytics (weak drive, effective model)\n",[61,6319,6320],{"class":1731,"line":2443},[61,6321,6322],{"class":1735},"# 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",[61,6324,6325,6327,6329,6331,6333,6335,6337,6339,6341,6343,6345,6347,6349,6351,6353,6355,6357,6359,6361,6363,6365],{"class":1731,"line":2475},[61,6326,2116],{"class":1749},[61,6328,2872],{"class":1836},[61,6330,849],{"class":1753},[61,6332,2877],{"class":2124},[61,6334,2127],{"class":1753},[61,6336,1860],{"class":1859},[61,6338,1435],{"class":1753},[61,6340,2886],{"class":2124},[61,6342,2127],{"class":1753},[61,6344,1860],{"class":1859},[61,6346,1435],{"class":1753},[61,6348,2895],{"class":2124},[61,6350,2127],{"class":1753},[61,6352,1860],{"class":1859},[61,6354,1435],{"class":1753},[61,6356,2904],{"class":2124},[61,6358,2127],{"class":1753},[61,6360,1860],{"class":1859},[61,6362,2155],{"class":1753},[61,6364,1860],{"class":1859},[61,6366,1850],{"class":1753},[61,6368,6369,6371,6373,6375,6377,6379,6381,6383,6385,6387,6389,6391,6393,6395,6397,6399,6401],{"class":1731,"line":2520},[61,6370,2210],{"class":1749},[61,6372,2934],{"class":1753},[61,6374,2201],{"class":1859},[61,6376,2939],{"class":1753},[61,6378,2201],{"class":1859},[61,6380,2003],{"class":1753},[61,6382,2946],{"class":1836},[61,6384,849],{"class":1753},[61,6386,2319],{"class":1866},[61,6388,2953],{"class":1753},[61,6390,2419],{"class":1859},[61,6392,2958],{"class":1753},[61,6394,2006],{"class":1859},[61,6396,2119],{"class":1836},[61,6398,849],{"class":1753},[61,6400,2319],{"class":1866},[61,6402,2969],{"class":1753},[61,6404,6405],{"class":1731,"line":2541},[61,6406,1743],{"emptyLinePlaceholder":1742},[61,6408,6409,6411,6413,6415,6417,6419,6421,6423,6425,6427,6429,6431,6433,6435,6437,6439,6441,6443,6445,6447,6449],{"class":1731,"line":2546},[61,6410,2116],{"class":1749},[61,6412,3087],{"class":1836},[61,6414,849],{"class":1753},[61,6416,2877],{"class":2124},[61,6418,2127],{"class":1753},[61,6420,1860],{"class":1859},[61,6422,1435],{"class":1753},[61,6424,2886],{"class":2124},[61,6426,2127],{"class":1753},[61,6428,1860],{"class":1859},[61,6430,1435],{"class":1753},[61,6432,2904],{"class":2124},[61,6434,2127],{"class":1753},[61,6436,1860],{"class":1859},[61,6438,1435],{"class":1753},[61,6440,3116],{"class":2124},[61,6442,2127],{"class":1753},[61,6444,1860],{"class":1859},[61,6446,2155],{"class":1753},[61,6448,1860],{"class":1859},[61,6450,1850],{"class":1753},[61,6452,6453,6455,6457,6459,6461,6463,6465,6467,6469,6471,6473,6475,6477,6479,6481,6483,6485,6487,6489,6491],{"class":1731,"line":2582},[61,6454,2210],{"class":1749},[61,6456,3144],{"class":1753},[61,6458,2651],{"class":1859},[61,6460,2654],{"class":1866},[61,6462,2624],{"class":1753},[61,6464,2201],{"class":1859},[61,6466,2003],{"class":1753},[61,6468,2946],{"class":1836},[61,6470,849],{"class":1753},[61,6472,2319],{"class":1866},[61,6474,2953],{"class":1753},[61,6476,2006],{"class":1859},[61,6478,3167],{"class":1753},[61,6480,2006],{"class":1859},[61,6482,2119],{"class":1836},[61,6484,849],{"class":1753},[61,6486,2319],{"class":1866},[61,6488,2953],{"class":1753},[61,6490,2419],{"class":1859},[61,6492,3182],{"class":1753},[61,6494,6495],{"class":1731,"line":2588},[61,6496,1743],{"emptyLinePlaceholder":1742},[61,6498,6499],{"class":1731,"line":2601},[61,6500,1819],{"class":1735},[61,6502,6503],{"class":1731,"line":2659},[61,6504,6505],{"class":1735},"# Build H_eff (4×4) in rad\u002Fns\n",[61,6507,6508],{"class":1731,"line":2717},[61,6509,1819],{"class":1735},[61,6511,6512,6515,6517,6519,6522,6524,6526],{"class":1731,"line":2725},[61,6513,6514],{"class":1753},"I2 ",[61,6516,2000],{"class":1859},[61,6518,2193],{"class":1753},[61,6520,6521],{"class":1836},"eye",[61,6523,849],{"class":1753},[61,6525,337],{"class":1866},[61,6527,2016],{"class":1753},[61,6529,6530,6533,6535,6537,6539,6542,6544,6546,6548],{"class":1731,"line":2730},[61,6531,6532],{"class":1753},"X ",[61,6534,2000],{"class":1859},[61,6536,2193],{"class":1753},[61,6538,3509],{"class":1836},[61,6540,6541],{"class":1753},"([[",[61,6543,2051],{"class":1866},[61,6545,1435],{"class":1753},[61,6547,2319],{"class":1866},[61,6549,6550],{"class":1753},"],\n",[61,6552,6553,6556,6558,6560,6562],{"class":1731,"line":2749},[61,6554,6555],{"class":1753},"              [",[61,6557,2319],{"class":1866},[61,6559,1435],{"class":1753},[61,6561,2051],{"class":1866},[61,6563,6564],{"class":1753},"]])\n",[61,6566,6567,6570,6572,6574,6576,6578,6580,6582,6584],{"class":1731,"line":2755},[61,6568,6569],{"class":1753},"Z ",[61,6571,2000],{"class":1859},[61,6573,2193],{"class":1753},[61,6575,3509],{"class":1836},[61,6577,6541],{"class":1753},[61,6579,2319],{"class":1866},[61,6581,1435],{"class":1753},[61,6583,2051],{"class":1866},[61,6585,6550],{"class":1753},[61,6587,6588,6590,6592,6594,6596,6598],{"class":1731,"line":2778},[61,6589,6555],{"class":1753},[61,6591,2051],{"class":1866},[61,6593,1435],{"class":1753},[61,6595,2006],{"class":1859},[61,6597,2319],{"class":1866},[61,6599,6564],{"class":1753},[61,6601,6602],{"class":1731,"line":2783},[61,6603,1743],{"emptyLinePlaceholder":1742},[61,6605,6606,6608,6611,6613,6615,6617,6619,6622,6625,6627,6630],{"class":1731,"line":2791},[61,6607,2116],{"class":1749},[61,6609,6610],{"class":1836}," kron2",[61,6612,849],{"class":1753},[61,6614,19],{"class":2124},[61,6616,1435],{"class":1753},[61,6618,2134],{"class":2124},[61,6620,6621],{"class":1753},"): ",[61,6623,6624],{"class":1749},"return",[61,6626,2193],{"class":1753},[61,6628,6629],{"class":1836},"kron",[61,6631,6632],{"class":1753},"(a, b)\n",[61,6634,6635],{"class":1731,"line":2804},[61,6636,1743],{"emptyLinePlaceholder":1742},[61,6638,6639,6641,6644,6646,6649],{"class":1731,"line":2817},[61,6640,2116],{"class":1749},[61,6642,6643],{"class":1836}," build_heff_radns",[61,6645,849],{"class":1753},[61,6647,6648],{"class":2124},"K",[61,6650,6651],{"class":1753},": RabiKnobs) -> np.ndarray:\n",[61,6653,6654,6657,6659,6662,6664],{"class":1731,"line":2824},[61,6655,6656],{"class":1753},"    Δ ",[61,6658,2000],{"class":1859},[61,6660,6661],{"class":1753}," K.f_c ",[61,6663,2006],{"class":1859},[61,6665,6666],{"class":1753}," K.f_t\n",[61,6668,6669,6672,6674,6676,6678,6680],{"class":1731,"line":2835},[61,6670,6671],{"class":1753},"    w_zx ",[61,6673,2000],{"class":1859},[61,6675,6235],{"class":1836},[61,6677,849],{"class":1753},[61,6679,3515],{"class":1836},[61,6681,6682],{"class":1753},"(Δ, K.J, K.Omega, K.alpha_c))\n",[61,6684,6685,6688,6690,6692,6694,6696,6699,6701,6703],{"class":1731,"line":2846},[61,6686,6687],{"class":1753},"    w_zz ",[61,6689,2000],{"class":1859},[61,6691,6235],{"class":1836},[61,6693,849],{"class":1753},[61,6695,3548],{"class":1836},[61,6697,6698],{"class":1753},"(Δ, K.J, K.alpha_c, K.alpha_t)) ",[61,6700,2419],{"class":1859},[61,6702,6277],{"class":1836},[61,6704,6705],{"class":1753},"(K.wZZ_khz)\n",[61,6707,6708,6711,6713,6715],{"class":1731,"line":2851},[61,6709,6710],{"class":1753},"    w_ix ",[61,6712,2000],{"class":1859},[61,6714,6235],{"class":1836},[61,6716,6717],{"class":1753},"(K.wIX_mhz)\n",[61,6719,6720,6723,6725,6727],{"class":1731,"line":2856},[61,6721,6722],{"class":1753},"    w_iz ",[61,6724,2000],{"class":1859},[61,6726,6277],{"class":1836},[61,6728,6729],{"class":1753},"(K.wIZ_khz)\n",[61,6731,6732,6735,6737,6739],{"class":1731,"line":2862},[61,6733,6734],{"class":1753},"    w_zi ",[61,6736,2000],{"class":1859},[61,6738,6235],{"class":1836},[61,6740,6741],{"class":1753},"(K.wZI_mhz)\n",[61,6743,6744,6746,6749,6751,6753,6755,6757,6759,6762],{"class":1731,"line":2867},[61,6745,2210],{"class":1749},[61,6747,6748],{"class":1753}," ((w_ix",[61,6750,602],{"class":1859},[61,6752,2435],{"class":1866},[61,6754,2624],{"class":1753},[61,6756,2201],{"class":1859},[61,6758,6610],{"class":1836},[61,6760,6761],{"class":1753},"(I2, X) ",[61,6763,6764],{"class":1859},"+\n",[61,6766,6767,6770,6772,6774,6776,6778,6780,6783],{"class":1731,"line":2917},[61,6768,6769],{"class":1753},"            (w_iz",[61,6771,602],{"class":1859},[61,6773,2435],{"class":1866},[61,6775,2624],{"class":1753},[61,6777,2201],{"class":1859},[61,6779,6610],{"class":1836},[61,6781,6782],{"class":1753},"(I2, Z) ",[61,6784,6764],{"class":1859},[61,6786,6787,6790,6792,6794,6796,6798,6800,6803],{"class":1731,"line":2926},[61,6788,6789],{"class":1753},"            (w_zi",[61,6791,602],{"class":1859},[61,6793,2435],{"class":1866},[61,6795,2624],{"class":1753},[61,6797,2201],{"class":1859},[61,6799,6610],{"class":1836},[61,6801,6802],{"class":1753},"(Z,  I2) ",[61,6804,6764],{"class":1859},[61,6806,6807,6810,6812,6814,6816,6818,6820,6823],{"class":1731,"line":2972},[61,6808,6809],{"class":1753},"            (w_zx",[61,6811,602],{"class":1859},[61,6813,2435],{"class":1866},[61,6815,2624],{"class":1753},[61,6817,2201],{"class":1859},[61,6819,6610],{"class":1836},[61,6821,6822],{"class":1753},"(Z,  X) ",[61,6824,6764],{"class":1859},[61,6826,6827,6830,6832,6834,6836,6838,6840],{"class":1731,"line":3001},[61,6828,6829],{"class":1753},"            (w_zz",[61,6831,602],{"class":1859},[61,6833,2435],{"class":1866},[61,6835,2624],{"class":1753},[61,6837,2201],{"class":1859},[61,6839,6610],{"class":1836},[61,6841,6842],{"class":1753},"(Z,  Z))\n",[61,6844,6845],{"class":1731,"line":3054},[61,6846,1743],{"emptyLinePlaceholder":1742},[61,6848,6849],{"class":1731,"line":3060},[61,6850,1819],{"class":1735},[61,6852,6853],{"class":1731,"line":3077},[61,6854,6855],{"class":1735},"# Time evolution via diagonalization\n",[61,6857,6858],{"class":1731,"line":3082},[61,6859,1819],{"class":1735},[61,6861,6862,6864,6867,6869,6872,6875,6878,6880,6883],{"class":1731,"line":3129},[61,6863,2116],{"class":1749},[61,6865,6866],{"class":1836}," evolve_state_constH",[61,6868,849],{"class":1753},[61,6870,6871],{"class":2124},"H",[61,6873,6874],{"class":1753},": np.ndarray, ",[61,6876,6877],{"class":2124},"psi0",[61,6879,6874],{"class":1753},[61,6881,6882],{"class":2124},"t_ns",[61,6884,6885],{"class":1753},": np.ndarray) -> np.ndarray:\n",[61,6887,6888,6891,6893,6896,6899],{"class":1731,"line":3137},[61,6889,6890],{"class":1753},"    E, V ",[61,6892,2000],{"class":1859},[61,6894,6895],{"class":1753}," np.linalg.",[61,6897,6898],{"class":1836},"eigh",[61,6900,6901],{"class":1753},"(H)\n",[61,6903,6904,6907,6909,6912,6915],{"class":1731,"line":3185},[61,6905,6906],{"class":1753},"    Vinv ",[61,6908,2000],{"class":1859},[61,6910,6911],{"class":1753}," V.",[61,6913,6914],{"class":1836},"conj",[61,6916,6917],{"class":1753},"().T\n",[61,6919,6920,6923,6925,6928,6931],{"class":1731,"line":3208},[61,6921,6922],{"class":1753},"    amps0 ",[61,6924,2000],{"class":1859},[61,6926,6927],{"class":1753}," Vinv ",[61,6929,6930],{"class":1859},"@",[61,6932,6933],{"class":1753}," psi0\n",[61,6935,6936,6939,6941,6943,6945,6948,6951,6954],{"class":1731,"line":3248},[61,6937,6938],{"class":1753},"    out ",[61,6940,2000],{"class":1859},[61,6942,2193],{"class":1753},[61,6944,4085],{"class":1836},[61,6946,6947],{"class":1753},"((psi0.size, t_ns.size), ",[61,6949,6950],{"class":3402},"dtype",[61,6952,6953],{"class":1859},"=complex",[61,6955,2016],{"class":1753},[61,6957,6958,6960,6963,6965,6967],{"class":1731,"line":3286},[61,6959,3377],{"class":1749},[61,6961,6962],{"class":1753}," k, t ",[61,6964,2492],{"class":1749},[61,6966,4111],{"class":1859},[61,6968,6969],{"class":1753},"(t_ns):\n",[61,6971,6972,6975,6977,6980,6982,6985,6988,6990,6992,6994,6996,6998,7001,7003,7006,7008],{"class":1731,"line":3291},[61,6973,6974],{"class":1753},"        out[:, k] ",[61,6976,2000],{"class":1859},[61,6978,6979],{"class":1753}," V ",[61,6981,6930],{"class":1859},[61,6983,6984],{"class":1753}," (np.",[61,6986,6987],{"class":1836},"exp",[61,6989,849],{"class":1753},[61,6991,2006],{"class":1859},[61,6993,57],{"class":1866},[61,6995,5222],{"class":1749},[61,6997,2298],{"class":1859},[61,6999,7000],{"class":1753}," E ",[61,7002,2201],{"class":1859},[61,7004,7005],{"class":1753}," t) ",[61,7007,2201],{"class":1859},[61,7009,7010],{"class":1753}," amps0)\n",[61,7012,7013,7015],{"class":1731,"line":3306},[61,7014,2210],{"class":1749},[61,7016,7017],{"class":1753}," out\n",[61,7019,7020],{"class":1731,"line":3311},[61,7021,1743],{"emptyLinePlaceholder":1742},[61,7023,7024],{"class":1731,"line":3317},[61,7025,7026],{"class":1735},"# Basis |00>, |01>, |10>, |11>\n",[61,7028,7029,7032,7034,7036,7038,7040,7042,7044,7046,7048,7050,7052,7054,7057,7059,7061],{"class":1731,"line":3358},[61,7030,7031],{"class":1753},"e00 ",[61,7033,2000],{"class":1859},[61,7035,2193],{"class":1753},[61,7037,3509],{"class":1836},[61,7039,3512],{"class":1753},[61,7041,57],{"class":1866},[61,7043,862],{"class":1753},[61,7045,1495],{"class":1866},[61,7047,862],{"class":1753},[61,7049,1495],{"class":1866},[61,7051,862],{"class":1753},[61,7053,1495],{"class":1866},[61,7055,7056],{"class":1753},"], ",[61,7058,6950],{"class":3402},[61,7060,6953],{"class":1859},[61,7062,2016],{"class":1753},[61,7064,7065,7068,7070,7072,7074,7076,7078,7080,7082,7084,7086,7088,7090,7092,7094,7096],{"class":1731,"line":3374},[61,7066,7067],{"class":1753},"e01 ",[61,7069,2000],{"class":1859},[61,7071,2193],{"class":1753},[61,7073,3509],{"class":1836},[61,7075,3512],{"class":1753},[61,7077,1495],{"class":1866},[61,7079,862],{"class":1753},[61,7081,57],{"class":1866},[61,7083,862],{"class":1753},[61,7085,1495],{"class":1866},[61,7087,862],{"class":1753},[61,7089,1495],{"class":1866},[61,7091,7056],{"class":1753},[61,7093,6950],{"class":3402},[61,7095,6953],{"class":1859},[61,7097,2016],{"class":1753},[61,7099,7100,7103,7105,7107,7109,7111,7113,7115,7117,7119,7121,7123,7125,7127,7129,7131],{"class":1731,"line":3390},[61,7101,7102],{"class":1753},"e10 ",[61,7104,2000],{"class":1859},[61,7106,2193],{"class":1753},[61,7108,3509],{"class":1836},[61,7110,3512],{"class":1753},[61,7112,1495],{"class":1866},[61,7114,862],{"class":1753},[61,7116,1495],{"class":1866},[61,7118,862],{"class":1753},[61,7120,57],{"class":1866},[61,7122,862],{"class":1753},[61,7124,1495],{"class":1866},[61,7126,7056],{"class":1753},[61,7128,6950],{"class":3402},[61,7130,6953],{"class":1859},[61,7132,2016],{"class":1753},[61,7134,7135,7138,7140,7142,7144,7146,7148,7150,7152,7154,7156,7158,7160,7162,7164,7166],{"class":1731,"line":3422},[61,7136,7137],{"class":1753},"e11 ",[61,7139,2000],{"class":1859},[61,7141,2193],{"class":1753},[61,7143,3509],{"class":1836},[61,7145,3512],{"class":1753},[61,7147,1495],{"class":1866},[61,7149,862],{"class":1753},[61,7151,1495],{"class":1866},[61,7153,862],{"class":1753},[61,7155,1495],{"class":1866},[61,7157,862],{"class":1753},[61,7159,57],{"class":1866},[61,7161,7056],{"class":1753},[61,7163,6950],{"class":3402},[61,7165,6953],{"class":1859},[61,7167,2016],{"class":1753},[61,7169,7170],{"class":1731,"line":3427},[61,7171,1743],{"emptyLinePlaceholder":1742},[61,7173,7174],{"class":1731,"line":3432},[61,7175,1819],{"class":1735},[61,7177,7178],{"class":1731,"line":3438},[61,7179,7180],{"class":1735},"# Conditional Rabi simulation\n",[61,7182,7183],{"class":1731,"line":3443},[61,7184,1819],{"class":1735},[61,7186,7187,7190,7192,7194],{"class":1731,"line":3469},[61,7188,7189],{"class":1753},"H ",[61,7191,2000],{"class":1859},[61,7193,6643],{"class":1836},[61,7195,7196],{"class":1753},"(K)\n",[61,7198,7199,7202,7204,7206,7208,7210,7212],{"class":1731,"line":3494},[61,7200,7201],{"class":1753},"t_ns ",[61,7203,2000],{"class":1859},[61,7205,2193],{"class":1753},[61,7207,3453],{"class":1836},[61,7209,849],{"class":1753},[61,7211,2051],{"class":1866},[61,7213,7214],{"class":1753},", K.t_end_ns, K.n_steps)\n",[61,7216,7217],{"class":1731,"line":3499},[61,7218,1743],{"emptyLinePlaceholder":1742},[61,7220,7221],{"class":1731,"line":3534},[61,7222,7223],{"class":1735},"# Prepare control in |0> (|00>) vs |1> (|10>), target initially |0>\n",[61,7225,7226,7229,7231,7233],{"class":1731,"line":3564},[61,7227,7228],{"class":1753},"ψ_c0 ",[61,7230,2000],{"class":1859},[61,7232,6866],{"class":1836},[61,7234,7235],{"class":1753},"(H, e00, t_ns)\n",[61,7237,7238,7241,7243,7245],{"class":1731,"line":3602},[61,7239,7240],{"class":1753},"ψ_c1 ",[61,7242,2000],{"class":1859},[61,7244,6866],{"class":1836},[61,7246,7247],{"class":1753},"(H, e10, t_ns)\n",[61,7249,7250],{"class":1731,"line":3607},[61,7251,1743],{"emptyLinePlaceholder":1742},[61,7253,7254],{"class":1731,"line":3613},[61,7255,7256],{"class":1735},"# Target |1> probability = P(|01>) + P(|11>)\n",[61,7258,7259,7262,7264,7266,7268,7271,7273,7276,7278,7281,7283,7285,7287,7289,7291,7294,7296,7298,7300,7302,7304],{"class":1731,"line":3651},[61,7260,7261],{"class":1753},"p_t1_c0 ",[61,7263,2000],{"class":1859},[61,7265,2193],{"class":1753},[61,7267,3574],{"class":1836},[61,7269,7270],{"class":1753},"(e01.",[61,7272,6914],{"class":1836},[61,7274,7275],{"class":1753},"() ",[61,7277,6930],{"class":1859},[61,7279,7280],{"class":1753}," ψ_c0)",[61,7282,2651],{"class":1859},[61,7284,337],{"class":1866},[61,7286,4990],{"class":1859},[61,7288,2193],{"class":1753},[61,7290,3574],{"class":1836},[61,7292,7293],{"class":1753},"(e11.",[61,7295,6914],{"class":1836},[61,7297,7275],{"class":1753},[61,7299,6930],{"class":1859},[61,7301,7280],{"class":1753},[61,7303,2651],{"class":1859},[61,7305,7306],{"class":1866},"2\n",[61,7308,7309,7312,7314,7316,7318,7320,7322,7324,7326,7329,7331,7333,7335,7337,7339,7341,7343,7345,7347,7349,7351],{"class":1731,"line":3715},[61,7310,7311],{"class":1753},"p_t1_c1 ",[61,7313,2000],{"class":1859},[61,7315,2193],{"class":1753},[61,7317,3574],{"class":1836},[61,7319,7270],{"class":1753},[61,7321,6914],{"class":1836},[61,7323,7275],{"class":1753},[61,7325,6930],{"class":1859},[61,7327,7328],{"class":1753}," ψ_c1)",[61,7330,2651],{"class":1859},[61,7332,337],{"class":1866},[61,7334,4990],{"class":1859},[61,7336,2193],{"class":1753},[61,7338,3574],{"class":1836},[61,7340,7293],{"class":1753},[61,7342,6914],{"class":1836},[61,7344,7275],{"class":1753},[61,7346,6930],{"class":1859},[61,7348,7328],{"class":1753},[61,7350,2651],{"class":1859},[61,7352,7306],{"class":1866},[61,7354,7355],{"class":1731,"line":3738},[61,7356,1743],{"emptyLinePlaceholder":1742},[61,7358,7359],{"class":1731,"line":3753},[61,7360,1819],{"class":1735},[61,7362,7363],{"class":1731,"line":3782},[61,7364,7365],{"class":1735},"# Plot\n",[61,7367,7368],{"class":1731,"line":3813},[61,7369,1819],{"class":1735},[61,7371,7372,7374,7376,7378,7380,7382,7384,7387,7389,7391,7393,7395,7397,7399],{"class":1731,"line":3835},[61,7373,3616],{"class":1753},[61,7375,3619],{"class":1836},[61,7377,849],{"class":1753},[61,7379,3624],{"class":3402},[61,7381,2000],{"class":1859},[61,7383,849],{"class":1753},[61,7385,7386],{"class":1866},"8.2",[61,7388,1435],{"class":1753},[61,7390,5636],{"class":1866},[61,7392,3638],{"class":1753},[61,7394,3641],{"class":3402},[61,7396,2000],{"class":1859},[61,7398,3646],{"class":1866},[61,7400,2016],{"class":1753},[61,7402,7403,7405,7407,7410,7412,7414,7417,7419,7421,7423,7425],{"class":1731,"line":3845},[61,7404,3616],{"class":1753},[61,7406,3656],{"class":1836},[61,7408,7409],{"class":1753},"(t_ns, p_t1_c0, ",[61,7411,3666],{"class":3402},[61,7413,2000],{"class":1859},[61,7415,7416],{"class":2165},"'Control |0⟩, init |00⟩'",[61,7418,1435],{"class":1753},[61,7420,3705],{"class":3402},[61,7422,2000],{"class":1859},[61,7424,3710],{"class":2165},[61,7426,2016],{"class":1753},[61,7428,7429,7431,7433,7436,7438,7440,7443,7445,7447,7449,7452],{"class":1731,"line":3855},[61,7430,3616],{"class":1753},[61,7432,3656],{"class":1836},[61,7434,7435],{"class":1753},"(t_ns, p_t1_c1, ",[61,7437,3666],{"class":3402},[61,7439,2000],{"class":1859},[61,7441,7442],{"class":2165},"'Control |1⟩, init |10⟩'",[61,7444,1435],{"class":1753},[61,7446,3705],{"class":3402},[61,7448,2000],{"class":1859},[61,7450,7451],{"class":2165},"'orange'",[61,7453,2016],{"class":1753},[61,7455,7456,7458,7460,7462,7465],{"class":1731,"line":3860},[61,7457,3616],{"class":1753},[61,7459,3758],{"class":1836},[61,7461,849],{"class":1753},[61,7463,7464],{"class":2165},"'Time (ns)'",[61,7466,2016],{"class":1753},[61,7468,7469,7471,7473,7475,7478],{"class":1731,"line":3866},[61,7470,3616],{"class":1753},[61,7472,3787],{"class":1836},[61,7474,849],{"class":1753},[61,7476,7477],{"class":2165},"'Target |1⟩ probability'",[61,7479,2016],{"class":1753},[61,7481,7482,7484,7486,7488,7491],{"class":1731,"line":3897},[61,7483,3616],{"class":1753},[61,7485,3818],{"class":1836},[61,7487,849],{"class":1753},[61,7489,7490],{"class":2165},"'Conditional target Rabi under CR effective Hamiltonian'",[61,7492,2016],{"class":1753},[61,7494,7495,7497,7499],{"class":1731,"line":3943},[61,7496,3616],{"class":1753},[61,7498,3840],{"class":1836},[61,7500,2089],{"class":1753},[61,7502,7503,7505,7507],{"class":1731,"line":3964},[61,7504,3616],{"class":1753},[61,7506,3850],{"class":1836},[61,7508,2089],{"class":1753},[4797,7510],{"alt":7511,"src":7512},"Output 6 of the notebook","\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-06.webp",[7514,7515,7516],"style",{},"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":1725,"searchDepth":1739,"depth":1739,"links":7518},[7519],{"id":27,"depth":1739,"text":28,"children":7520},[7521],{"id":1637,"depth":1746,"text":1638},[7523,7524,7525,7526],"Qollab","Explore","Expert notes","Cross-resonance gate",[7528],{"username":7,"name":7529,"role":7530,"bio":7531,"links":7532},"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.",[7533,7536],{"label":7534,"href":7535},"GitHub ↗","https:\u002F\u002Fgithub.com\u002FOJB-Quantum",{"label":7537,"href":21},"Original notebook ↗",{"username":7,"name":7529,"role":7539},"Quantum Hardware Engineer, University of Minnesota",null,"How the ZX entangling rate and static ZZ vary with detuning and drive amplitude, plotted from the PRA 2020 cross-resonance formulas.",false,"md","Deep dive · Hardware","article",{},"\u002F_content\u002Fimages\u002Fcross-resonance-gate-visualization\u002Foutput-01.png","\u002Fblog\u002Fexpert-notes\u002Fcross-resonance-gate-visualization","2026-08-06","9 min 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Shapes and Envelopes, Visualized",[7],{"type":9,"value":14100,"toc":26236},[14101,14109,14112,14123,14146,14401,14404,14671,14674,15002,15005,15355,15358,15596,15599,15906,15909,16261,16265,16647,16651,16654,17253,17257,17809,17813,18435,18439,19069,19073,19076,19083,19089,19479,19689,19695,20250,20450,20456,21047,21279,21285,21511,21513,21518,24091,24681,24685,24689,25144,25148,25521,25525,26229,26233],[12,14102,14103],{},[15,14104,17,14105,14108],{},[19,14106,22],{"href":14107},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FPulse_Shapes_and_Envelopes_Visualization.ipynb",". The text, code, and figures below are Onri's, as published.",[12,14110,14111],{},"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.",[1640,14113,14114,14117,14120],{},[1643,14115,14116],{},"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.",[1643,14118,14119],{},"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.",[1643,14121,14122],{},"Optimal control techniques, such as GRAPE or CRAB, can generate tailored pulse shapes that might include sinusoidal modulations.",[14124,14125,14126],"blockquote",{},[1640,14127,14128,14134,14140],{},[1643,14129,14130,14133],{},[814,14131,14132],{},"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.",[1643,14135,14136,14139],{},[814,14137,14138],{},"Chopped Random Basis (CRAB)",": A method for quantum optimal control that uses a randomly truncated basis to optimize control pulses.",[1643,14141,14142,14145],{},[814,14143,14144],{},"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.",[1720,14147,14149],{"className":1722,"code":14148,"language":1724,"meta":1725,"style":1725},"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",[1727,14150,14151,14161,14171,14175,14180,14185,14200,14204,14209,14219,14229,14261,14265,14270,14314,14318,14323,14340,14353,14366,14379,14393],{"__ignoreMap":1725},[61,14152,14153,14155,14157,14159],{"class":1731,"line":1732},[61,14154,1757],{"class":1749},[61,14156,1789],{"class":1753},[61,14158,1792],{"class":1749},[61,14160,1795],{"class":1753},[61,14162,14163,14165,14167,14169],{"class":1731,"line":1739},[61,14164,1757],{"class":1749},[61,14166,1803],{"class":1753},[61,14168,1792],{"class":1749},[61,14170,1808],{"class":1753},[61,14172,14173],{"class":1731,"line":1746},[61,14174,1743],{"emptyLinePlaceholder":1742},[61,14176,14177],{"class":1731,"line":1763},[61,14178,14179],{"class":1735},"# Onri prefers using rcParams to increase the quality of all plots to be higher than the default settings.\n",[61,14181,14182],{"class":1731,"line":1776},[61,14183,14184],{"class":1735},"# This only needs to be done once at beginning of every Jupyter Notebook.\n",[61,14186,14187,14190,14193,14195,14197],{"class":1731,"line":1784},[61,14188,14189],{"class":1753},"plt.rcParams[",[61,14191,14192],{"class":2165},"'figure.dpi'",[61,14194,1997],{"class":1753},[61,14196,2000],{"class":1859},[61,14198,14199],{"class":1866}," 200\n",[61,14201,14202],{"class":1731,"line":1798},[61,14203,1743],{"emptyLinePlaceholder":1742},[61,14205,14206],{"class":1731,"line":1811},[61,14207,14208],{"class":1735},"# Define the parameters for the Gaussian modulation function\n",[61,14210,14211,14214,14216],{"class":1731,"line":1816},[61,14212,14213],{"class":1753},"mean ",[61,14215,2000],{"class":1859},[61,14217,14218],{"class":1866}," 0\n",[61,14220,14221,14224,14226],{"class":1731,"line":1822},[61,14222,14223],{"class":1753},"std_dev ",[61,14225,2000],{"class":1859},[61,14227,14228],{"class":1866}," 1\n",[61,14230,14231,14234,14236,14238,14240,14242,14244,14246,14248,14250,14252,14255,14258],{"class":1731,"line":1828},[61,14232,14233],{"class":1753},"time ",[61,14235,2000],{"class":1859},[61,14237,2193],{"class":1753},[61,14239,3453],{"class":1836},[61,14241,849],{"class":1753},[61,14243,2006],{"class":1859},[61,14245,1026],{"class":1866},[61,14247,1435],{"class":1753},[61,14249,1026],{"class":1866},[61,14251,1435],{"class":1753},[61,14253,14254],{"class":1866},"1000",[61,14256,14257],{"class":1753},")  ",[61,14259,14260],{"class":1735},"# Generate time values from -5 to 5\n",[61,14262,14263],{"class":1731,"line":1833},[61,14264,1743],{"emptyLinePlaceholder":1742},[61,14266,14267],{"class":1731,"line":1840},[61,14268,14269],{"class":1735},"# Calculate the Gaussian modulation function\n",[61,14271,14272,14275,14277,14279,14281,14283,14285,14288,14290,14293,14295,14297,14299,14301,14303,14305,14308,14310,14312],{"class":1731,"line":1853},[61,14273,14274],{"class":1753},"gaussian_modulation ",[61,14276,2000],{"class":1859},[61,14278,2193],{"class":1753},[61,14280,6987],{"class":1836},[61,14282,849],{"class":1753},[61,14284,2006],{"class":1859},[61,14286,14287],{"class":1753},"(time ",[61,14289,2006],{"class":1859},[61,14291,14292],{"class":1753}," mean) ",[61,14294,2651],{"class":1859},[61,14296,2654],{"class":1866},[61,14298,2618],{"class":1859},[61,14300,2003],{"class":1753},[61,14302,337],{"class":1866},[61,14304,2298],{"class":1859},[61,14306,14307],{"class":1753}," std_dev ",[61,14309,2651],{"class":1859},[61,14311,2654],{"class":1866},[61,14313,5546],{"class":1753},[61,14315,14316],{"class":1731,"line":1873},[61,14317,1743],{"emptyLinePlaceholder":1742},[61,14319,14320],{"class":1731,"line":1889},[61,14321,14322],{"class":1735},"# Plot the Gaussian modulation function with blue color\n",[61,14324,14325,14327,14329,14332,14334,14336,14338],{"class":1731,"line":1908},[61,14326,3616],{"class":1753},[61,14328,3656],{"class":1836},[61,14330,14331],{"class":1753},"(time, gaussian_modulation, ",[61,14333,3705],{"class":3402},[61,14335,2000],{"class":1859},[61,14337,3710],{"class":2165},[61,14339,2016],{"class":1753},[61,14341,14342,14344,14346,14348,14351],{"class":1731,"line":1925},[61,14343,3616],{"class":1753},[61,14345,3818],{"class":1836},[61,14347,849],{"class":1753},[61,14349,14350],{"class":2165},"'Gaussian Modulation Function'",[61,14352,2016],{"class":1753},[61,14354,14355,14357,14359,14361,14364],{"class":1731,"line":1941},[61,14356,3616],{"class":1753},[61,14358,3758],{"class":1836},[61,14360,849],{"class":1753},[61,14362,14363],{"class":2165},"'Time'",[61,14365,2016],{"class":1753},[61,14367,14368,14370,14372,14374,14377],{"class":1731,"line":1957},[61,14369,3616],{"class":1753},[61,14371,3787],{"class":1836},[61,14373,849],{"class":1753},[61,14375,14376],{"class":2165},"'Amplitude'",[61,14378,2016],{"class":1753},[61,14380,14381,14383,14386,14388,14391],{"class":1731,"line":1974},[61,14382,3616],{"class":1753},[61,14384,14385],{"class":1836},"grid",[61,14387,849],{"class":1753},[61,14389,14390],{"class":1866},"False",[61,14392,2016],{"class":1753},[61,14394,14395,14397,14399],{"class":1731,"line":1979},[61,14396,3616],{"class":1753},[61,14398,3850],{"class":1836},[61,14400,2089],{"class":1753},[4797,14402],{"alt":4799,"src":14403},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-01.webp",[1720,14405,14407],{"className":1722,"code":14406,"language":1724,"meta":1725,"style":1725},"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",[1727,14408,14409,14419,14429,14433,14437,14446,14454,14462,14490,14494,14498,14543,14547,14552,14587,14599,14611,14623,14646,14651,14663],{"__ignoreMap":1725},[61,14410,14411,14413,14415,14417],{"class":1731,"line":1732},[61,14412,1757],{"class":1749},[61,14414,1789],{"class":1753},[61,14416,1792],{"class":1749},[61,14418,1795],{"class":1753},[61,14420,14421,14423,14425,14427],{"class":1731,"line":1739},[61,14422,1757],{"class":1749},[61,14424,1803],{"class":1753},[61,14426,1792],{"class":1749},[61,14428,1808],{"class":1753},[61,14430,14431],{"class":1731,"line":1746},[61,14432,1743],{"emptyLinePlaceholder":1742},[61,14434,14435],{"class":1731,"line":1763},[61,14436,14208],{"class":1735},[61,14438,14439,14442,14444],{"class":1731,"line":1776},[61,14440,14441],{"class":1753},"amplitude ",[61,14443,2000],{"class":1859},[61,14445,14228],{"class":1866},[61,14447,14448,14450,14452],{"class":1731,"line":1784},[61,14449,14213],{"class":1753},[61,14451,2000],{"class":1859},[61,14453,14218],{"class":1866},[61,14455,14456,14458,14460],{"class":1731,"line":1798},[61,14457,14223],{"class":1753},[61,14459,2000],{"class":1859},[61,14461,14228],{"class":1866},[61,14463,14464,14466,14468,14470,14472,14474,14476,14478,14480,14482,14484,14486,14488],{"class":1731,"line":1811},[61,14465,14233],{"class":1753},[61,14467,2000],{"class":1859},[61,14469,2193],{"class":1753},[61,14471,3453],{"class":1836},[61,14473,849],{"class":1753},[61,14475,2006],{"class":1859},[61,14477,1026],{"class":1866},[61,14479,1435],{"class":1753},[61,14481,1026],{"class":1866},[61,14483,1435],{"class":1753},[61,14485,14254],{"class":1866},[61,14487,14257],{"class":1753},[61,14489,14260],{"class":1735},[61,14491,14492],{"class":1731,"line":1816},[61,14493,1743],{"emptyLinePlaceholder":1742},[61,14495,14496],{"class":1731,"line":1822},[61,14497,14269],{"class":1735},[61,14499,14500,14502,14504,14507,14509,14511,14513,14515,14517,14519,14521,14523,14525,14527,14529,14531,14533,14535,14537,14539,14541],{"class":1731,"line":1828},[61,14501,14274],{"class":1753},[61,14503,2000],{"class":1859},[61,14505,14506],{"class":1753}," amplitude ",[61,14508,2201],{"class":1859},[61,14510,2193],{"class":1753},[61,14512,6987],{"class":1836},[61,14514,849],{"class":1753},[61,14516,2006],{"class":1859},[61,14518,14287],{"class":1753},[61,14520,2006],{"class":1859},[61,14522,14292],{"class":1753},[61,14524,2651],{"class":1859},[61,14526,2654],{"class":1866},[61,14528,2618],{"class":1859},[61,14530,2003],{"class":1753},[61,14532,337],{"class":1866},[61,14534,2298],{"class":1859},[61,14536,14307],{"class":1753},[61,14538,2651],{"class":1859},[61,14540,2654],{"class":1866},[61,14542,5546],{"class":1753},[61,14544,14545],{"class":1731,"line":1833},[61,14546,1743],{"emptyLinePlaceholder":1742},[61,14548,14549],{"class":1731,"line":1840},[61,14550,14551],{"class":1735},"# Plot the Gaussian modulation function\n",[61,14553,14554,14556,14558,14560,14562,14564,14567,14569,14571,14573,14576,14578,14580,14582,14585],{"class":1731,"line":1853},[61,14555,3616],{"class":1753},[61,14557,3656],{"class":1836},[61,14559,14331],{"class":1753},[61,14561,3705],{"class":3402},[61,14563,2000],{"class":1859},[61,14565,14566],{"class":2165},"'red'",[61,14568,1435],{"class":1753},[61,14570,3403],{"class":3402},[61,14572,2000],{"class":1859},[61,14574,14575],{"class":2165},"'dashed'",[61,14577,1435],{"class":1753},[61,14579,3666],{"class":3402},[61,14581,2000],{"class":1859},[61,14583,14584],{"class":2165},"'Gaussian Modulation'",[61,14586,2016],{"class":1753},[61,14588,14589,14591,14593,14595,14597],{"class":1731,"line":1873},[61,14590,3616],{"class":1753},[61,14592,3818],{"class":1836},[61,14594,849],{"class":1753},[61,14596,14350],{"class":2165},[61,14598,2016],{"class":1753},[61,14600,14601,14603,14605,14607,14609],{"class":1731,"line":1889},[61,14602,3616],{"class":1753},[61,14604,3758],{"class":1836},[61,14606,849],{"class":1753},[61,14608,14363],{"class":2165},[61,14610,2016],{"class":1753},[61,14612,14613,14615,14617,14619,14621],{"class":1731,"line":1908},[61,14614,3616],{"class":1753},[61,14616,3787],{"class":1836},[61,14618,849],{"class":1753},[61,14620,14376],{"class":2165},[61,14622,2016],{"class":1753},[61,14624,14625,14627,14630,14632,14634,14636,14638,14641,14643],{"class":1731,"line":1925},[61,14626,3616],{"class":1753},[61,14628,14629],{"class":1836},"ylim",[61,14631,849],{"class":1753},[61,14633,2006],{"class":1859},[61,14635,2319],{"class":1866},[61,14637,1435],{"class":1753},[61,14639,14640],{"class":1866},"1.1",[61,14642,14257],{"class":1753},[61,14644,14645],{"class":1735},"# Force the y-axis to display the range from -1.0 to 1.0\n",[61,14647,14648],{"class":1731,"line":1941},[61,14649,14650],{"class":1735},"# plt.legend()\n",[61,14652,14653,14655,14657,14659,14661],{"class":1731,"line":1957},[61,14654,3616],{"class":1753},[61,14656,14385],{"class":1836},[61,14658,849],{"class":1753},[61,14660,14390],{"class":1866},[61,14662,2016],{"class":1753},[61,14664,14665,14667,14669],{"class":1731,"line":1974},[61,14666,3616],{"class":1753},[61,14668,3850],{"class":1836},[61,14670,2089],{"class":1753},[4797,14672],{"alt":4803,"src":14673},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-02.webp",[1720,14675,14677],{"className":1722,"code":14676,"language":1724,"meta":1725,"style":1725},"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",[1727,14678,14679,14689,14699,14703,14708,14716,14724,14732,14762,14766,14771,14801,14805,14810,14854,14858,14863,14885,14924,14937,14949,14961,14982,14994],{"__ignoreMap":1725},[61,14680,14681,14683,14685,14687],{"class":1731,"line":1732},[61,14682,1757],{"class":1749},[61,14684,1789],{"class":1753},[61,14686,1792],{"class":1749},[61,14688,1795],{"class":1753},[61,14690,14691,14693,14695,14697],{"class":1731,"line":1739},[61,14692,1757],{"class":1749},[61,14694,1803],{"class":1753},[61,14696,1792],{"class":1749},[61,14698,1808],{"class":1753},[61,14700,14701],{"class":1731,"line":1746},[61,14702,1743],{"emptyLinePlaceholder":1742},[61,14704,14705],{"class":1731,"line":1763},[61,14706,14707],{"class":1735},"# Define the parameters for the Gaussian function\n",[61,14709,14710,14712,14714],{"class":1731,"line":1776},[61,14711,14441],{"class":1753},[61,14713,2000],{"class":1859},[61,14715,14228],{"class":1866},[61,14717,14718,14720,14722],{"class":1731,"line":1784},[61,14719,14213],{"class":1753},[61,14721,2000],{"class":1859},[61,14723,14218],{"class":1866},[61,14725,14726,14728,14730],{"class":1731,"line":1798},[61,14727,14223],{"class":1753},[61,14729,2000],{"class":1859},[61,14731,14228],{"class":1866},[61,14733,14734,14736,14738,14740,14742,14744,14746,14748,14750,14752,14754,14757,14759],{"class":1731,"line":1811},[61,14735,14233],{"class":1753},[61,14737,2000],{"class":1859},[61,14739,2193],{"class":1753},[61,14741,3453],{"class":1836},[61,14743,849],{"class":1753},[61,14745,2006],{"class":1859},[61,14747,1026],{"class":1866},[61,14749,1435],{"class":1753},[61,14751,1026],{"class":1866},[61,14753,1435],{"class":1753},[61,14755,14756],{"class":1866},"20",[61,14758,14257],{"class":1753},[61,14760,14761],{"class":1735},"# Reduce points for lollipop plot clarity\n",[61,14763,14764],{"class":1731,"line":1816},[61,14765,1743],{"emptyLinePlaceholder":1742},[61,14767,14768],{"class":1731,"line":1822},[61,14769,14770],{"class":1735},"# Increase the number of points for the lollipop plot\n",[61,14772,14773,14775,14777,14779,14781,14783,14785,14787,14789,14791,14793,14796,14798],{"class":1731,"line":1828},[61,14774,14233],{"class":1753},[61,14776,2000],{"class":1859},[61,14778,2193],{"class":1753},[61,14780,3453],{"class":1836},[61,14782,849],{"class":1753},[61,14784,2006],{"class":1859},[61,14786,1026],{"class":1866},[61,14788,1435],{"class":1753},[61,14790,1026],{"class":1866},[61,14792,1435],{"class":1753},[61,14794,14795],{"class":1866},"40",[61,14797,14257],{"class":1753},[61,14799,14800],{"class":1735},"# Doubling the number of points\n",[61,14802,14803],{"class":1731,"line":1833},[61,14804,1743],{"emptyLinePlaceholder":1742},[61,14806,14807],{"class":1731,"line":1840},[61,14808,14809],{"class":1735},"# Calculate the Gaussian function\n",[61,14811,14812,14814,14816,14818,14820,14822,14824,14826,14828,14830,14832,14834,14836,14838,14840,14842,14844,14846,14848,14850,14852],{"class":1731,"line":1853},[61,14813,14274],{"class":1753},[61,14815,2000],{"class":1859},[61,14817,14506],{"class":1753},[61,14819,2201],{"class":1859},[61,14821,2193],{"class":1753},[61,14823,6987],{"class":1836},[61,14825,849],{"class":1753},[61,14827,2006],{"class":1859},[61,14829,14287],{"class":1753},[61,14831,2006],{"class":1859},[61,14833,14292],{"class":1753},[61,14835,2651],{"class":1859},[61,14837,2654],{"class":1866},[61,14839,2618],{"class":1859},[61,14841,2003],{"class":1753},[61,14843,337],{"class":1866},[61,14845,2298],{"class":1859},[61,14847,14307],{"class":1753},[61,14849,2651],{"class":1859},[61,14851,2654],{"class":1866},[61,14853,5546],{"class":1753},[61,14855,14856],{"class":1731,"line":1873},[61,14857,1743],{"emptyLinePlaceholder":1742},[61,14859,14860],{"class":1731,"line":1889},[61,14861,14862],{"class":1735},"# Create the updated lollipop plot with blue color\n",[61,14864,14865,14867,14869,14871,14873,14875,14877,14879,14881,14883],{"class":1731,"line":1908},[61,14866,3616],{"class":1753},[61,14868,3619],{"class":1836},[61,14870,849],{"class":1753},[61,14872,3624],{"class":3402},[61,14874,2000],{"class":1859},[61,14876,849],{"class":1753},[61,14878,3631],{"class":1866},[61,14880,1435],{"class":1753},[61,14882,1300],{"class":1866},[61,14884,5546],{"class":1753},[61,14886,14887,14889,14892,14894,14897,14899,14902,14904,14907,14909,14912,14914,14917,14919,14922],{"class":1731,"line":1925},[61,14888,3616],{"class":1753},[61,14890,14891],{"class":1836},"stem",[61,14893,14331],{"class":1753},[61,14895,14896],{"class":3402},"linefmt",[61,14898,2000],{"class":1859},[61,14900,14901],{"class":2165},"'b-'",[61,14903,1435],{"class":1753},[61,14905,14906],{"class":3402},"markerfmt",[61,14908,2000],{"class":1859},[61,14910,14911],{"class":2165},"'bo'",[61,14913,1435],{"class":1753},[61,14915,14916],{"class":3402},"basefmt",[61,14918,2000],{"class":1859},[61,14920,14921],{"class":2165},"'k-'",[61,14923,2016],{"class":1753},[61,14925,14926,14928,14930,14932,14935],{"class":1731,"line":1941},[61,14927,3616],{"class":1753},[61,14929,3818],{"class":1836},[61,14931,849],{"class":1753},[61,14933,14934],{"class":2165},"'Lollipop Plot of Gaussian Function'",[61,14936,2016],{"class":1753},[61,14938,14939,14941,14943,14945,14947],{"class":1731,"line":1957},[61,14940,3616],{"class":1753},[61,14942,3758],{"class":1836},[61,14944,849],{"class":1753},[61,14946,14363],{"class":2165},[61,14948,2016],{"class":1753},[61,14950,14951,14953,14955,14957,14959],{"class":1731,"line":1974},[61,14952,3616],{"class":1753},[61,14954,3787],{"class":1836},[61,14956,849],{"class":1753},[61,14958,14376],{"class":2165},[61,14960,2016],{"class":1753},[61,14962,14963,14965,14967,14969,14971,14973,14975,14977,14979],{"class":1731,"line":1979},[61,14964,3616],{"class":1753},[61,14966,14629],{"class":1836},[61,14968,849],{"class":1753},[61,14970,2006],{"class":1859},[61,14972,2319],{"class":1866},[61,14974,1435],{"class":1753},[61,14976,14640],{"class":1866},[61,14978,14257],{"class":1753},[61,14980,14981],{"class":1735},"# Force the y-axis to display the range from -1.0 to 1.1\n",[61,14983,14984,14986,14988,14990,14992],{"class":1731,"line":1985},[61,14985,3616],{"class":1753},[61,14987,14385],{"class":1836},[61,14989,849],{"class":1753},[61,14991,14390],{"class":1866},[61,14993,2016],{"class":1753},[61,14995,14996,14998,15000],{"class":1731,"line":2019},[61,14997,3616],{"class":1753},[61,14999,3850],{"class":1836},[61,15001,2089],{"class":1753},[4797,15003],{"alt":4807,"src":15004},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-03.webp",[1720,15006,15008],{"className":1722,"code":15007,"language":1724,"meta":1725,"style":1725},"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",[1727,15009,15010,15020,15030,15034,15039,15047,15055,15063,15076,15104,15108,15112,15152,15156,15161,15195,15199,15204,15219,15223,15228,15261,15294,15307,15319,15331,15335,15347],{"__ignoreMap":1725},[61,15011,15012,15014,15016,15018],{"class":1731,"line":1732},[61,15013,1757],{"class":1749},[61,15015,1789],{"class":1753},[61,15017,1792],{"class":1749},[61,15019,1795],{"class":1753},[61,15021,15022,15024,15026,15028],{"class":1731,"line":1739},[61,15023,1757],{"class":1749},[61,15025,1803],{"class":1753},[61,15027,1792],{"class":1749},[61,15029,1808],{"class":1753},[61,15031,15032],{"class":1731,"line":1746},[61,15033,1743],{"emptyLinePlaceholder":1742},[61,15035,15036],{"class":1731,"line":1763},[61,15037,15038],{"class":1735},"# Define the parameters for the Gaussian-modulated sine wave\n",[61,15040,15041,15043,15045],{"class":1731,"line":1776},[61,15042,14441],{"class":1753},[61,15044,2000],{"class":1859},[61,15046,14228],{"class":1866},[61,15048,15049,15051,15053],{"class":1731,"line":1784},[61,15050,14213],{"class":1753},[61,15052,2000],{"class":1859},[61,15054,14218],{"class":1866},[61,15056,15057,15059,15061],{"class":1731,"line":1798},[61,15058,14223],{"class":1753},[61,15060,2000],{"class":1859},[61,15062,14228],{"class":1866},[61,15064,15065,15068,15070,15073],{"class":1731,"line":1811},[61,15066,15067],{"class":1753},"frequency ",[61,15069,2000],{"class":1859},[61,15071,15072],{"class":1866}," 5",[61,15074,15075],{"class":1735},"  # Frequency of the sine wave\n",[61,15077,15078,15080,15082,15084,15086,15088,15090,15092,15094,15096,15098,15100,15102],{"class":1731,"line":1816},[61,15079,14233],{"class":1753},[61,15081,2000],{"class":1859},[61,15083,2193],{"class":1753},[61,15085,3453],{"class":1836},[61,15087,849],{"class":1753},[61,15089,2006],{"class":1859},[61,15091,1026],{"class":1866},[61,15093,1435],{"class":1753},[61,15095,1026],{"class":1866},[61,15097,1435],{"class":1753},[61,15099,14254],{"class":1866},[61,15101,14257],{"class":1753},[61,15103,14260],{"class":1735},[61,15105,15106],{"class":1731,"line":1822},[61,15107,1743],{"emptyLinePlaceholder":1742},[61,15109,15110],{"class":1731,"line":1828},[61,15111,14809],{"class":1735},[61,15113,15114,15116,15118,15120,15122,15124,15126,15128,15130,15132,15134,15136,15138,15140,15142,15144,15146,15148,15150],{"class":1731,"line":1833},[61,15115,14274],{"class":1753},[61,15117,2000],{"class":1859},[61,15119,2193],{"class":1753},[61,15121,6987],{"class":1836},[61,15123,849],{"class":1753},[61,15125,2006],{"class":1859},[61,15127,14287],{"class":1753},[61,15129,2006],{"class":1859},[61,15131,14292],{"class":1753},[61,15133,2651],{"class":1859},[61,15135,2654],{"class":1866},[61,15137,2618],{"class":1859},[61,15139,2003],{"class":1753},[61,15141,337],{"class":1866},[61,15143,2298],{"class":1859},[61,15145,14307],{"class":1753},[61,15147,2651],{"class":1859},[61,15149,2654],{"class":1866},[61,15151,5546],{"class":1753},[61,15153,15154],{"class":1731,"line":1840},[61,15155,1743],{"emptyLinePlaceholder":1742},[61,15157,15158],{"class":1731,"line":1853},[61,15159,15160],{"class":1735},"# Calculate the sine wave\n",[61,15162,15163,15166,15168,15170,15172,15174,15176,15178,15180,15182,15185,15187,15190,15192],{"class":1731,"line":1873},[61,15164,15165],{"class":1753},"sine_wave ",[61,15167,2000],{"class":1859},[61,15169,14506],{"class":1753},[61,15171,2201],{"class":1859},[61,15173,2193],{"class":1753},[61,15175,5069],{"class":1836},[61,15177,849],{"class":1753},[61,15179,337],{"class":1866},[61,15181,2298],{"class":1859},[61,15183,15184],{"class":1753}," np.pi ",[61,15186,2201],{"class":1859},[61,15188,15189],{"class":1753}," frequency ",[61,15191,2201],{"class":1859},[61,15193,15194],{"class":1753}," time)\n",[61,15196,15197],{"class":1731,"line":1889},[61,15198,1743],{"emptyLinePlaceholder":1742},[61,15200,15201],{"class":1731,"line":1908},[61,15202,15203],{"class":1735},"# Modulate the sine wave with the Gaussian function\n",[61,15205,15206,15209,15211,15214,15216],{"class":1731,"line":1925},[61,15207,15208],{"class":1753},"gaussian_modulated_sine_wave ",[61,15210,2000],{"class":1859},[61,15212,15213],{"class":1753}," gaussian_modulation ",[61,15215,2201],{"class":1859},[61,15217,15218],{"class":1753}," sine_wave\n",[61,15220,15221],{"class":1731,"line":1941},[61,15222,1743],{"emptyLinePlaceholder":1742},[61,15224,15225],{"class":1731,"line":1957},[61,15226,15227],{"class":1735},"# Plot only the function of the Gaussian-modulated sine wave\n",[61,15229,15230,15232,15234,15236,15238,15240,15242,15244,15246,15248,15250,15252,15254,15256,15259],{"class":1731,"line":1974},[61,15231,3616],{"class":1753},[61,15233,3656],{"class":1836},[61,15235,14331],{"class":1753},[61,15237,3705],{"class":3402},[61,15239,2000],{"class":1859},[61,15241,14566],{"class":2165},[61,15243,1435],{"class":1753},[61,15245,3403],{"class":3402},[61,15247,2000],{"class":1859},[61,15249,14575],{"class":2165},[61,15251,1435],{"class":1753},[61,15253,3666],{"class":3402},[61,15255,2000],{"class":1859},[61,15257,15258],{"class":2165},"'Modulation'",[61,15260,2016],{"class":1753},[61,15262,15263,15265,15267,15270,15272,15275,15277,15279,15281,15283,15285,15287,15289,15291],{"class":1731,"line":1979},[61,15264,3616],{"class":1753},[61,15266,3656],{"class":1836},[61,15268,15269],{"class":1753},"(time, ",[61,15271,2006],{"class":1859},[61,15273,15274],{"class":1753},"gaussian_modulation, ",[61,15276,3705],{"class":3402},[61,15278,2000],{"class":1859},[61,15280,14566],{"class":2165},[61,15282,1435],{"class":1753},[61,15284,3403],{"class":3402},[61,15286,2000],{"class":1859},[61,15288,14575],{"class":2165},[61,15290,14257],{"class":1753},[61,15292,15293],{"class":1735},"# Add the negative function for completeness\n",[61,15295,15296,15298,15300,15302,15305],{"class":1731,"line":1985},[61,15297,3616],{"class":1753},[61,15299,3818],{"class":1836},[61,15301,849],{"class":1753},[61,15303,15304],{"class":2165},"'Upper & Lower Gaussian Modulation Function'",[61,15306,2016],{"class":1753},[61,15308,15309,15311,15313,15315,15317],{"class":1731,"line":2019},[61,15310,3616],{"class":1753},[61,15312,3758],{"class":1836},[61,15314,849],{"class":1753},[61,15316,14363],{"class":2165},[61,15318,2016],{"class":1753},[61,15320,15321,15323,15325,15327,15329],{"class":1731,"line":2033},[61,15322,3616],{"class":1753},[61,15324,3787],{"class":1836},[61,15326,849],{"class":1753},[61,15328,14376],{"class":2165},[61,15330,2016],{"class":1753},[61,15332,15333],{"class":1731,"line":2061},[61,15334,14650],{"class":1735},[61,15336,15337,15339,15341,15343,15345],{"class":1731,"line":2074},[61,15338,3616],{"class":1753},[61,15340,14385],{"class":1836},[61,15342,849],{"class":1753},[61,15344,14390],{"class":1866},[61,15346,2016],{"class":1753},[61,15348,15349,15351,15353],{"class":1731,"line":2079},[61,15350,3616],{"class":1753},[61,15352,3850],{"class":1836},[61,15354,2089],{"class":1753},[4797,15356],{"alt":4811,"src":15357},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-04.webp",[1720,15359,15361],{"className":1722,"code":15360,"language":1724,"meta":1725,"style":1725},"'''\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",[1727,15362,15363,15368,15373,15378,15383,15387,15391,15401,15411,15415,15420,15431,15442,15471,15475,15480,15510,15514,15519,15536,15549,15561,15573,15588],{"__ignoreMap":1725},[61,15364,15365],{"class":1731,"line":1732},[61,15366,15367],{"class":2165},"'''\n",[61,15369,15370],{"class":1731,"line":1739},[61,15371,15372],{"class":2165},"This plot represents a sine wave at a chosen frequency, commonly used to match the resonance frequency\n",[61,15374,15375],{"class":1731,"line":1746},[61,15376,15377],{"class":2165},"of a quantum device on a chip. In quantum computing, such signals serve as carrier waves for qubit\n",[61,15379,15380],{"class":1731,"line":1763},[61,15381,15382],{"class":2165},"control pulses before modulation is applied.\n",[61,15384,15385],{"class":1731,"line":1776},[61,15386,15367],{"class":2165},[61,15388,15389],{"class":1731,"line":1784},[61,15390,1743],{"emptyLinePlaceholder":1742},[61,15392,15393,15395,15397,15399],{"class":1731,"line":1798},[61,15394,1757],{"class":1749},[61,15396,1789],{"class":1753},[61,15398,1792],{"class":1749},[61,15400,1795],{"class":1753},[61,15402,15403,15405,15407,15409],{"class":1731,"line":1811},[61,15404,1757],{"class":1749},[61,15406,1803],{"class":1753},[61,15408,1792],{"class":1749},[61,15410,1808],{"class":1753},[61,15412,15413],{"class":1731,"line":1816},[61,15414,1743],{"emptyLinePlaceholder":1742},[61,15416,15417],{"class":1731,"line":1822},[61,15418,15419],{"class":1735},"# Define sine wave parameters\n",[61,15421,15422,15424,15426,15428],{"class":1731,"line":1828},[61,15423,14441],{"class":1753},[61,15425,2000],{"class":1859},[61,15427,5266],{"class":1866},[61,15429,15430],{"class":1735},"  # Peak amplitude of the wave\n",[61,15432,15433,15435,15437,15439],{"class":1731,"line":1833},[61,15434,15067],{"class":1753},[61,15436,2000],{"class":1859},[61,15438,15072],{"class":1866},[61,15440,15441],{"class":1735},"  # Frequency of the sine wave (in arbitrary units, could be GHz for quantum systems)\n",[61,15443,15444,15446,15448,15450,15452,15454,15456,15458,15460,15462,15464,15466,15468],{"class":1731,"line":1840},[61,15445,14233],{"class":1753},[61,15447,2000],{"class":1859},[61,15449,2193],{"class":1753},[61,15451,3453],{"class":1836},[61,15453,849],{"class":1753},[61,15455,2006],{"class":1859},[61,15457,1026],{"class":1866},[61,15459,1435],{"class":1753},[61,15461,1026],{"class":1866},[61,15463,1435],{"class":1753},[61,15465,14254],{"class":1866},[61,15467,14257],{"class":1753},[61,15469,15470],{"class":1735},"# Generate 1000 evenly spaced time values from -5 to 5\n",[61,15472,15473],{"class":1731,"line":1853},[61,15474,1743],{"emptyLinePlaceholder":1742},[61,15476,15477],{"class":1731,"line":1873},[61,15478,15479],{"class":1735},"# Compute the sine wave, representing an unmodulated carrier signal\n",[61,15481,15482,15484,15486,15488,15490,15492,15494,15496,15498,15500,15502,15504,15506,15508],{"class":1731,"line":1889},[61,15483,15165],{"class":1753},[61,15485,2000],{"class":1859},[61,15487,14506],{"class":1753},[61,15489,2201],{"class":1859},[61,15491,2193],{"class":1753},[61,15493,5069],{"class":1836},[61,15495,849],{"class":1753},[61,15497,337],{"class":1866},[61,15499,2298],{"class":1859},[61,15501,15184],{"class":1753},[61,15503,2201],{"class":1859},[61,15505,15189],{"class":1753},[61,15507,2201],{"class":1859},[61,15509,15194],{"class":1753},[61,15511,15512],{"class":1731,"line":1908},[61,15513,1743],{"emptyLinePlaceholder":1742},[61,15515,15516],{"class":1731,"line":1925},[61,15517,15518],{"class":1735},"# Plot the sine wave with blue color\n",[61,15520,15521,15523,15525,15528,15530,15532,15534],{"class":1731,"line":1941},[61,15522,3616],{"class":1753},[61,15524,3656],{"class":1836},[61,15526,15527],{"class":1753},"(time, sine_wave, ",[61,15529,3705],{"class":3402},[61,15531,2000],{"class":1859},[61,15533,3710],{"class":2165},[61,15535,2016],{"class":1753},[61,15537,15538,15540,15542,15544,15547],{"class":1731,"line":1957},[61,15539,3616],{"class":1753},[61,15541,3818],{"class":1836},[61,15543,849],{"class":1753},[61,15545,15546],{"class":2165},"'Unmodulated Sine Wave (Carrier Signal)'",[61,15548,2016],{"class":1753},[61,15550,15551,15553,15555,15557,15559],{"class":1731,"line":1974},[61,15552,3616],{"class":1753},[61,15554,3758],{"class":1836},[61,15556,849],{"class":1753},[61,15558,14363],{"class":2165},[61,15560,2016],{"class":1753},[61,15562,15563,15565,15567,15569,15571],{"class":1731,"line":1979},[61,15564,3616],{"class":1753},[61,15566,3787],{"class":1836},[61,15568,849],{"class":1753},[61,15570,14376],{"class":2165},[61,15572,2016],{"class":1753},[61,15574,15575,15577,15579,15581,15583,15585],{"class":1731,"line":1985},[61,15576,3616],{"class":1753},[61,15578,14385],{"class":1836},[61,15580,849],{"class":1753},[61,15582,14390],{"class":1866},[61,15584,14257],{"class":1753},[61,15586,15587],{"class":1735},"# No grid for a cleaner look\n",[61,15589,15590,15592,15594],{"class":1731,"line":2019},[61,15591,3616],{"class":1753},[61,15593,3850],{"class":1836},[61,15595,2089],{"class":1753},[4797,15597],{"alt":5802,"src":15598},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-05.webp",[1720,15600,15602],{"className":1722,"code":15601,"language":1724,"meta":1725,"style":1725},"'''\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",[1727,15603,15604,15608,15613,15618,15623,15627,15637,15647,15651,15655,15663,15671,15679,15689,15717,15721,15725,15765,15769,15773,15803,15807,15811,15823,15827,15832,15849,15862,15874,15886,15898],{"__ignoreMap":1725},[61,15605,15606],{"class":1731,"line":1732},[61,15607,15367],{"class":2165},[61,15609,15610],{"class":1731,"line":1739},[61,15611,15612],{"class":2165},"When a modulating pulse shape is applied to an unmodulated sine wave tone, then the result will be a\n",[61,15614,15615],{"class":1731,"line":1746},[61,15616,15617],{"class":2165},"modulated sine wave output as shown in the plot below. The example shown is modulation by a\n",[61,15619,15620],{"class":1731,"line":1763},[61,15621,15622],{"class":2165},"Gaussian wave.\n",[61,15624,15625],{"class":1731,"line":1776},[61,15626,15367],{"class":2165},[61,15628,15629,15631,15633,15635],{"class":1731,"line":1784},[61,15630,1757],{"class":1749},[61,15632,1789],{"class":1753},[61,15634,1792],{"class":1749},[61,15636,1795],{"class":1753},[61,15638,15639,15641,15643,15645],{"class":1731,"line":1798},[61,15640,1757],{"class":1749},[61,15642,1803],{"class":1753},[61,15644,1792],{"class":1749},[61,15646,1808],{"class":1753},[61,15648,15649],{"class":1731,"line":1811},[61,15650,1743],{"emptyLinePlaceholder":1742},[61,15652,15653],{"class":1731,"line":1816},[61,15654,15038],{"class":1735},[61,15656,15657,15659,15661],{"class":1731,"line":1822},[61,15658,14441],{"class":1753},[61,15660,2000],{"class":1859},[61,15662,14228],{"class":1866},[61,15664,15665,15667,15669],{"class":1731,"line":1828},[61,15666,14213],{"class":1753},[61,15668,2000],{"class":1859},[61,15670,14218],{"class":1866},[61,15672,15673,15675,15677],{"class":1731,"line":1833},[61,15674,14223],{"class":1753},[61,15676,2000],{"class":1859},[61,15678,14228],{"class":1866},[61,15680,15681,15683,15685,15687],{"class":1731,"line":1840},[61,15682,15067],{"class":1753},[61,15684,2000],{"class":1859},[61,15686,15072],{"class":1866},[61,15688,15075],{"class":1735},[61,15690,15691,15693,15695,15697,15699,15701,15703,15705,15707,15709,15711,15713,15715],{"class":1731,"line":1853},[61,15692,14233],{"class":1753},[61,15694,2000],{"class":1859},[61,15696,2193],{"class":1753},[61,15698,3453],{"class":1836},[61,15700,849],{"class":1753},[61,15702,2006],{"class":1859},[61,15704,1026],{"class":1866},[61,15706,1435],{"class":1753},[61,15708,1026],{"class":1866},[61,15710,1435],{"class":1753},[61,15712,14254],{"class":1866},[61,15714,14257],{"class":1753},[61,15716,14260],{"class":1735},[61,15718,15719],{"class":1731,"line":1873},[61,15720,1743],{"emptyLinePlaceholder":1742},[61,15722,15723],{"class":1731,"line":1889},[61,15724,14809],{"class":1735},[61,15726,15727,15729,15731,15733,15735,15737,15739,15741,15743,15745,15747,15749,15751,15753,15755,15757,15759,15761,15763],{"class":1731,"line":1908},[61,15728,14274],{"class":1753},[61,15730,2000],{"class":1859},[61,15732,2193],{"class":1753},[61,15734,6987],{"class":1836},[61,15736,849],{"class":1753},[61,15738,2006],{"class":1859},[61,15740,14287],{"class":1753},[61,15742,2006],{"class":1859},[61,15744,14292],{"class":1753},[61,15746,2651],{"class":1859},[61,15748,2654],{"class":1866},[61,15750,2618],{"class":1859},[61,15752,2003],{"class":1753},[61,15754,337],{"class":1866},[61,15756,2298],{"class":1859},[61,15758,14307],{"class":1753},[61,15760,2651],{"class":1859},[61,15762,2654],{"class":1866},[61,15764,5546],{"class":1753},[61,15766,15767],{"class":1731,"line":1925},[61,15768,1743],{"emptyLinePlaceholder":1742},[61,15770,15771],{"class":1731,"line":1941},[61,15772,15160],{"class":1735},[61,15774,15775,15777,15779,15781,15783,15785,15787,15789,15791,15793,15795,15797,15799,15801],{"class":1731,"line":1957},[61,15776,15165],{"class":1753},[61,15778,2000],{"class":1859},[61,15780,14506],{"class":1753},[61,15782,2201],{"class":1859},[61,15784,2193],{"class":1753},[61,15786,5069],{"class":1836},[61,15788,849],{"class":1753},[61,15790,337],{"class":1866},[61,15792,2298],{"class":1859},[61,15794,15184],{"class":1753},[61,15796,2201],{"class":1859},[61,15798,15189],{"class":1753},[61,15800,2201],{"class":1859},[61,15802,15194],{"class":1753},[61,15804,15805],{"class":1731,"line":1974},[61,15806,1743],{"emptyLinePlaceholder":1742},[61,15808,15809],{"class":1731,"line":1979},[61,15810,15203],{"class":1735},[61,15812,15813,15815,15817,15819,15821],{"class":1731,"line":1985},[61,15814,15208],{"class":1753},[61,15816,2000],{"class":1859},[61,15818,15213],{"class":1753},[61,15820,2201],{"class":1859},[61,15822,15218],{"class":1753},[61,15824,15825],{"class":1731,"line":2019},[61,15826,1743],{"emptyLinePlaceholder":1742},[61,15828,15829],{"class":1731,"line":2033},[61,15830,15831],{"class":1735},"# Plot the Gaussian-modulated sine wave with blue color\n",[61,15833,15834,15836,15838,15841,15843,15845,15847],{"class":1731,"line":2061},[61,15835,3616],{"class":1753},[61,15837,3656],{"class":1836},[61,15839,15840],{"class":1753},"(time, gaussian_modulated_sine_wave, ",[61,15842,3705],{"class":3402},[61,15844,2000],{"class":1859},[61,15846,3710],{"class":2165},[61,15848,2016],{"class":1753},[61,15850,15851,15853,15855,15857,15860],{"class":1731,"line":2074},[61,15852,3616],{"class":1753},[61,15854,3818],{"class":1836},[61,15856,849],{"class":1753},[61,15858,15859],{"class":2165},"'Gaussian-Modulated Sine Wave Pulse'",[61,15861,2016],{"class":1753},[61,15863,15864,15866,15868,15870,15872],{"class":1731,"line":2079},[61,15865,3616],{"class":1753},[61,15867,3758],{"class":1836},[61,15869,849],{"class":1753},[61,15871,14363],{"class":2165},[61,15873,2016],{"class":1753},[61,15875,15876,15878,15880,15882,15884],{"class":1731,"line":2092},[61,15877,3616],{"class":1753},[61,15879,3787],{"class":1836},[61,15881,849],{"class":1753},[61,15883,14376],{"class":2165},[61,15885,2016],{"class":1753},[61,15887,15888,15890,15892,15894,15896],{"class":1731,"line":2097},[61,15889,3616],{"class":1753},[61,15891,14385],{"class":1836},[61,15893,849],{"class":1753},[61,15895,14390],{"class":1866},[61,15897,2016],{"class":1753},[61,15899,15900,15902,15904],{"class":1731,"line":2102},[61,15901,3616],{"class":1753},[61,15903,3850],{"class":1836},[61,15905,2089],{"class":1753},[4797,15907],{"alt":7511,"src":15908},"\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-06.webp",[1720,15910,15912],{"className":1722,"code":15911,"language":1724,"meta":1725,"style":1725},"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",[1727,15913,15914,15924,15934,15938,15942,15950,15958,15966,15976,16004,16008,16012,16052,16056,16060,16090,16094,16098,16110,16114,16119,16133,16137,16142,16166,16200,16213,16225,16237,16241,16253],{"__ignoreMap":1725},[61,15915,15916,15918,15920,15922],{"class":1731,"line":1732},[61,15917,1757],{"class":1749},[61,15919,1789],{"class":1753},[61,15921,1792],{"class":1749},[61,15923,1795],{"class":1753},[61,15925,15926,15928,15930,15932],{"class":1731,"line":1739},[61,15927,1757],{"class":1749},[61,15929,1803],{"class":1753},[61,15931,1792],{"class":1749},[61,15933,1808],{"class":1753},[61,15935,15936],{"class":1731,"line":1746},[61,15937,1743],{"emptyLinePlaceholder":1742},[61,15939,15940],{"class":1731,"line":1763},[61,15941,15038],{"class":1735},[61,15943,15944,15946,15948],{"class":1731,"line":1776},[61,15945,14441],{"class":1753},[61,15947,2000],{"class":1859},[61,15949,14228],{"class":1866},[61,15951,15952,15954,15956],{"class":1731,"line":1784},[61,15953,14213],{"class":1753},[61,15955,2000],{"class":1859},[61,15957,14218],{"class":1866},[61,15959,15960,15962,15964],{"class":1731,"line":1798},[61,15961,14223],{"class":1753},[61,15963,2000],{"class":1859},[61,15965,14228],{"class":1866},[61,15967,15968,15970,15972,15974],{"class":1731,"line":1811},[61,15969,15067],{"class":1753},[61,15971,2000],{"class":1859},[61,15973,15072],{"class":1866},[61,15975,15075],{"class":1735},[61,15977,15978,15980,15982,15984,15986,15988,15990,15992,15994,15996,15998,16000,16002],{"class":1731,"line":1816},[61,15979,14233],{"class":1753},[61,15981,2000],{"class":1859},[61,15983,2193],{"class":1753},[61,15985,3453],{"class":1836},[61,15987,849],{"class":1753},[61,15989,2006],{"class":1859},[61,15991,1026],{"class":1866},[61,15993,1435],{"class":1753},[61,15995,1026],{"class":1866},[61,15997,1435],{"class":1753},[61,15999,14254],{"class":1866},[61,16001,14257],{"class":1753},[61,16003,14260],{"class":1735},[61,16005,16006],{"class":1731,"line":1822},[61,16007,1743],{"emptyLinePlaceholder":1742},[61,16009,16010],{"class":1731,"line":1828},[61,16011,14809],{"class":1735},[61,16013,16014,16016,16018,16020,16022,16024,16026,16028,16030,16032,16034,16036,16038,16040,16042,16044,16046,16048,16050],{"class":1731,"line":1833},[61,16015,14274],{"class":1753},[61,16017,2000],{"class":1859},[61,16019,2193],{"class":1753},[61,16021,6987],{"class":1836},[61,16023,849],{"class":1753},[61,16025,2006],{"class":1859},[61,16027,14287],{"class":1753},[61,16029,2006],{"class":1859},[61,16031,14292],{"class":1753},[61,16033,2651],{"class":1859},[61,16035,2654],{"class":1866},[61,16037,2618],{"class":1859},[61,16039,2003],{"class":1753},[61,16041,337],{"class":1866},[61,16043,2298],{"class":1859},[61,16045,14307],{"class":1753},[61,16047,2651],{"class":1859},[61,16049,2654],{"class":1866},[61,16051,5546],{"class":1753},[61,16053,16054],{"class":1731,"line":1840},[61,16055,1743],{"emptyLinePlaceholder":1742},[61,16057,16058],{"class":1731,"line":1853},[61,16059,15160],{"class":1735},[61,16061,16062,16064,16066,16068,16070,16072,16074,16076,16078,16080,16082,16084,16086,16088],{"class":1731,"line":1873},[61,16063,15165],{"class":1753},[61,16065,2000],{"class":1859},[61,16067,14506],{"class":1753},[61,16069,2201],{"class":1859},[61,16071,2193],{"class":1753},[61,16073,5069],{"class":1836},[61,16075,849],{"class":1753},[61,16077,337],{"class":1866},[61,16079,2298],{"class":1859},[61,16081,15184],{"class":1753},[61,16083,2201],{"class":1859},[61,16085,15189],{"class":1753},[61,16087,2201],{"class":1859},[61,16089,15194],{"class":1753},[61,16091,16092],{"class":1731,"line":1889},[61,16093,1743],{"emptyLinePlaceholder":1742},[61,16095,16096],{"class":1731,"line":1908},[61,16097,15203],{"class":1735},[61,16099,16100,16102,16104,16106,16108],{"class":1731,"line":1925},[61,16101,15208],{"class":1753},[61,16103,2000],{"class":1859},[61,16105,15213],{"class":1753},[61,16107,2201],{"class":1859},[61,16109,15218],{"class":1753},[61,16111,16112],{"class":1731,"line":1941},[61,16113,1743],{"emptyLinePlaceholder":1742},[61,16115,16116],{"class":1731,"line":1957},[61,16117,16118],{"class":1735},"# Calculate the upper envelope of the modulated sine wave\n",[61,16120,16121,16124,16126,16128,16130],{"class":1731,"line":1974},[61,16122,16123],{"class":1753},"upper_modulation ",[61,16125,2000],{"class":1859},[61,16127,2193],{"class":1753},[61,16129,3574],{"class":1836},[61,16131,16132],{"class":1753},"(gaussian_modulation)\n",[61,16134,16135],{"class":1731,"line":1979},[61,16136,1743],{"emptyLinePlaceholder":1742},[61,16138,16139],{"class":1731,"line":1985},[61,16140,16141],{"class":1735},"# Plot the Gaussian-modulated sine wave and its upper envelope\n",[61,16143,16144,16146,16148,16150,16152,16154,16156,16158,16160,16162,16164],{"class":1731,"line":2019},[61,16145,3616],{"class":1753},[61,16147,3656],{"class":1836},[61,16149,15840],{"class":1753},[61,16151,3705],{"class":3402},[61,16153,2000],{"class":1859},[61,16155,3710],{"class":2165},[61,16157,1435],{"class":1753},[61,16159,3666],{"class":3402},[61,16161,2000],{"class":1859},[61,16163,15859],{"class":2165},[61,16165,2016],{"class":1753},[61,16167,16168,16170,16172,16175,16177,16179,16181,16183,16185,16187,16189,16191,16193,16195,16198],{"class":1731,"line":2033},[61,16169,3616],{"class":1753},[61,16171,3656],{"class":1836},[61,16173,16174],{"class":1753},"(time, upper_modulation, ",[61,16176,3705],{"class":3402},[61,16178,2000],{"class":1859},[61,16180,14566],{"class":2165},[61,16182,1435],{"class":1753},[61,16184,3403],{"class":3402},[61,16186,2000],{"class":1859},[61,16188,14575],{"class":2165},[61,16190,1435],{"class":1753},[61,16192,3666],{"class":3402},[61,16194,2000],{"class":1859},[61,16196,16197],{"class":2165},"'Upper Modulation Function'",[61,16199,2016],{"class":1753},[61,16201,16202,16204,16206,16208,16211],{"class":1731,"line":2061},[61,16203,3616],{"class":1753},[61,16205,3818],{"class":1836},[61,16207,849],{"class":1753},[61,16209,16210],{"class":2165},"'Gaussian-Modulated Sine Wave with Upper Modulation Function'",[61,16212,2016],{"class":1753},[61,16214,16215,16217,16219,16221,16223],{"class":1731,"line":2074},[61,16216,3616],{"class":1753},[61,16218,3758],{"class":1836},[61,16220,849],{"class":1753},[61,16222,14363],{"class":2165},[61,16224,2016],{"class":1753},[61,16226,16227,16229,16231,16233,16235],{"class":1731,"line":2079},[61,16228,3616],{"class":1753},[61,16230,3787],{"class":1836},[61,16232,849],{"class":1753},[61,16234,14376],{"class":2165},[61,16236,2016],{"class":1753},[61,16238,16239],{"class":1731,"line":2092},[61,16240,14650],{"class":1735},[61,16242,16243,16245,16247,16249,16251],{"class":1731,"line":2097},[61,16244,3616],{"class":1753},[61,16246,14385],{"class":1836},[61,16248,849],{"class":1753},[61,16250,14390],{"class":1866},[61,16252,2016],{"class":1753},[61,16254,16255,16257,16259],{"class":1731,"line":2102},[61,16256,3616],{"class":1753},[61,16258,3850],{"class":1836},[61,16260,2089],{"class":1753},[4797,16262],{"alt":16263,"src":16264},"Output 7 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-07.webp",[1720,16266,16268],{"className":1722,"code":16267,"language":1724,"meta":1725,"style":1725},"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",[1727,16269,16270,16280,16290,16294,16298,16306,16314,16322,16332,16360,16364,16368,16408,16412,16416,16446,16450,16454,16466,16470,16475,16488,16492,16497,16521,16554,16586,16599,16611,16623,16627,16639],{"__ignoreMap":1725},[61,16271,16272,16274,16276,16278],{"class":1731,"line":1732},[61,16273,1757],{"class":1749},[61,16275,1789],{"class":1753},[61,16277,1792],{"class":1749},[61,16279,1795],{"class":1753},[61,16281,16282,16284,16286,16288],{"class":1731,"line":1739},[61,16283,1757],{"class":1749},[61,16285,1803],{"class":1753},[61,16287,1792],{"class":1749},[61,16289,1808],{"class":1753},[61,16291,16292],{"class":1731,"line":1746},[61,16293,1743],{"emptyLinePlaceholder":1742},[61,16295,16296],{"class":1731,"line":1763},[61,16297,15038],{"class":1735},[61,16299,16300,16302,16304],{"class":1731,"line":1776},[61,16301,14441],{"class":1753},[61,16303,2000],{"class":1859},[61,16305,14228],{"class":1866},[61,16307,16308,16310,16312],{"class":1731,"line":1784},[61,16309,14213],{"class":1753},[61,16311,2000],{"class":1859},[61,16313,14218],{"class":1866},[61,16315,16316,16318,16320],{"class":1731,"line":1798},[61,16317,14223],{"class":1753},[61,16319,2000],{"class":1859},[61,16321,14228],{"class":1866},[61,16323,16324,16326,16328,16330],{"class":1731,"line":1811},[61,16325,15067],{"class":1753},[61,16327,2000],{"class":1859},[61,16329,15072],{"class":1866},[61,16331,15075],{"class":1735},[61,16333,16334,16336,16338,16340,16342,16344,16346,16348,16350,16352,16354,16356,16358],{"class":1731,"line":1816},[61,16335,14233],{"class":1753},[61,16337,2000],{"class":1859},[61,16339,2193],{"class":1753},[61,16341,3453],{"class":1836},[61,16343,849],{"class":1753},[61,16345,2006],{"class":1859},[61,16347,1026],{"class":1866},[61,16349,1435],{"class":1753},[61,16351,1026],{"class":1866},[61,16353,1435],{"class":1753},[61,16355,14254],{"class":1866},[61,16357,14257],{"class":1753},[61,16359,14260],{"class":1735},[61,16361,16362],{"class":1731,"line":1822},[61,16363,1743],{"emptyLinePlaceholder":1742},[61,16365,16366],{"class":1731,"line":1828},[61,16367,14809],{"class":1735},[61,16369,16370,16372,16374,16376,16378,16380,16382,16384,16386,16388,16390,16392,16394,16396,16398,16400,16402,16404,16406],{"class":1731,"line":1833},[61,16371,14274],{"class":1753},[61,16373,2000],{"class":1859},[61,16375,2193],{"class":1753},[61,16377,6987],{"class":1836},[61,16379,849],{"class":1753},[61,16381,2006],{"class":1859},[61,16383,14287],{"class":1753},[61,16385,2006],{"class":1859},[61,16387,14292],{"class":1753},[61,16389,2651],{"class":1859},[61,16391,2654],{"class":1866},[61,16393,2618],{"class":1859},[61,16395,2003],{"class":1753},[61,16397,337],{"class":1866},[61,16399,2298],{"class":1859},[61,16401,14307],{"class":1753},[61,16403,2651],{"class":1859},[61,16405,2654],{"class":1866},[61,16407,5546],{"class":1753},[61,16409,16410],{"class":1731,"line":1840},[61,16411,1743],{"emptyLinePlaceholder":1742},[61,16413,16414],{"class":1731,"line":1853},[61,16415,15160],{"class":1735},[61,16417,16418,16420,16422,16424,16426,16428,16430,16432,16434,16436,16438,16440,16442,16444],{"class":1731,"line":1873},[61,16419,15165],{"class":1753},[61,16421,2000],{"class":1859},[61,16423,14506],{"class":1753},[61,16425,2201],{"class":1859},[61,16427,2193],{"class":1753},[61,16429,5069],{"class":1836},[61,16431,849],{"class":1753},[61,16433,337],{"class":1866},[61,16435,2298],{"class":1859},[61,16437,15184],{"class":1753},[61,16439,2201],{"class":1859},[61,16441,15189],{"class":1753},[61,16443,2201],{"class":1859},[61,16445,15194],{"class":1753},[61,16447,16448],{"class":1731,"line":1889},[61,16449,1743],{"emptyLinePlaceholder":1742},[61,16451,16452],{"class":1731,"line":1908},[61,16453,15203],{"class":1735},[61,16455,16456,16458,16460,16462,16464],{"class":1731,"line":1925},[61,16457,15208],{"class":1753},[61,16459,2000],{"class":1859},[61,16461,15213],{"class":1753},[61,16463,2201],{"class":1859},[61,16465,15218],{"class":1753},[61,16467,16468],{"class":1731,"line":1941},[61,16469,1743],{"emptyLinePlaceholder":1742},[61,16471,16472],{"class":1731,"line":1957},[61,16473,16474],{"class":1735},"# Calculate the function of the modulated sine wave\n",[61,16476,16477,16480,16482,16484,16486],{"class":1731,"line":1974},[61,16478,16479],{"class":1753},"modulation ",[61,16481,2000],{"class":1859},[61,16483,2193],{"class":1753},[61,16485,3574],{"class":1836},[61,16487,16132],{"class":1753},[61,16489,16490],{"class":1731,"line":1979},[61,16491,1743],{"emptyLinePlaceholder":1742},[61,16493,16494],{"class":1731,"line":1985},[61,16495,16496],{"class":1735},"# Plot the Gaussian-modulated sine wave and its modulation function\n",[61,16498,16499,16501,16503,16505,16507,16509,16511,16513,16515,16517,16519],{"class":1731,"line":2019},[61,16500,3616],{"class":1753},[61,16502,3656],{"class":1836},[61,16504,15840],{"class":1753},[61,16506,3705],{"class":3402},[61,16508,2000],{"class":1859},[61,16510,3710],{"class":2165},[61,16512,1435],{"class":1753},[61,16514,3666],{"class":3402},[61,16516,2000],{"class":1859},[61,16518,15859],{"class":2165},[61,16520,2016],{"class":1753},[61,16522,16523,16525,16527,16530,16532,16534,16536,16538,16540,16542,16544,16546,16548,16550,16552],{"class":1731,"line":2033},[61,16524,3616],{"class":1753},[61,16526,3656],{"class":1836},[61,16528,16529],{"class":1753},"(time, modulation, ",[61,16531,3705],{"class":3402},[61,16533,2000],{"class":1859},[61,16535,14566],{"class":2165},[61,16537,1435],{"class":1753},[61,16539,3403],{"class":3402},[61,16541,2000],{"class":1859},[61,16543,14575],{"class":2165},[61,16545,1435],{"class":1753},[61,16547,3666],{"class":3402},[61,16549,2000],{"class":1859},[61,16551,15258],{"class":2165},[61,16553,2016],{"class":1753},[61,16555,16556,16558,16560,16562,16564,16567,16569,16571,16573,16575,16577,16579,16581,16583],{"class":1731,"line":2061},[61,16557,3616],{"class":1753},[61,16559,3656],{"class":1836},[61,16561,15269],{"class":1753},[61,16563,2006],{"class":1859},[61,16565,16566],{"class":1753},"modulation, ",[61,16568,3705],{"class":3402},[61,16570,2000],{"class":1859},[61,16572,14566],{"class":2165},[61,16574,1435],{"class":1753},[61,16576,3403],{"class":3402},[61,16578,2000],{"class":1859},[61,16580,14575],{"class":2165},[61,16582,14257],{"class":1753},[61,16584,16585],{"class":1735},"# Add the negative modulation function for completeness\n",[61,16587,16588,16590,16592,16594,16597],{"class":1731,"line":2074},[61,16589,3616],{"class":1753},[61,16591,3818],{"class":1836},[61,16593,849],{"class":1753},[61,16595,16596],{"class":2165},"'Gaussian-Modulated Sine Wave with Modulation Function'",[61,16598,2016],{"class":1753},[61,16600,16601,16603,16605,16607,16609],{"class":1731,"line":2079},[61,16602,3616],{"class":1753},[61,16604,3758],{"class":1836},[61,16606,849],{"class":1753},[61,16608,14363],{"class":2165},[61,16610,2016],{"class":1753},[61,16612,16613,16615,16617,16619,16621],{"class":1731,"line":2092},[61,16614,3616],{"class":1753},[61,16616,3787],{"class":1836},[61,16618,849],{"class":1753},[61,16620,14376],{"class":2165},[61,16622,2016],{"class":1753},[61,16624,16625],{"class":1731,"line":2097},[61,16626,14650],{"class":1735},[61,16628,16629,16631,16633,16635,16637],{"class":1731,"line":2102},[61,16630,3616],{"class":1753},[61,16632,14385],{"class":1836},[61,16634,849],{"class":1753},[61,16636,14390],{"class":1866},[61,16638,2016],{"class":1753},[61,16640,16641,16643,16645],{"class":1731,"line":2108},[61,16642,3616],{"class":1753},[61,16644,3850],{"class":1836},[61,16646,2089],{"class":1753},[4797,16648],{"alt":16649,"src":16650},"Output 8 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-08.webp",[12,16652,16653],{},"Below are some common pulse envelope shapes used to modulate a sine wave tone.",[1720,16655,16657],{"className":1722,"code":16656,"language":1724,"meta":1725,"style":1725},"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",[1727,16658,16659,16669,16679,16691,16695,16700,16727,16731,16736,16759,16810,16855,16870,16890,16907,16933,16951,16955,16960,16976,16992,17006,17016,17026,17036,17047,17051,17056,17095,17110,17114,17128,17143,17153,17167,17181,17186,17190,17195,17221,17232,17236,17245],{"__ignoreMap":1725},[61,16660,16661,16663,16665,16667],{"class":1731,"line":1732},[61,16662,1757],{"class":1749},[61,16664,1789],{"class":1753},[61,16666,1792],{"class":1749},[61,16668,1795],{"class":1753},[61,16670,16671,16673,16675,16677],{"class":1731,"line":1739},[61,16672,1757],{"class":1749},[61,16674,1803],{"class":1753},[61,16676,1792],{"class":1749},[61,16678,1808],{"class":1753},[61,16680,16681,16683,16686,16688],{"class":1731,"line":1746},[61,16682,1750],{"class":1749},[61,16684,16685],{"class":1753}," scipy.signal.windows ",[61,16687,1757],{"class":1749},[61,16689,16690],{"class":1753}," blackmanharris, tukey\n",[61,16692,16693],{"class":1731,"line":1763},[61,16694,1743],{"emptyLinePlaceholder":1742},[61,16696,16697],{"class":1731,"line":1776},[61,16698,16699],{"class":1735},"# Define time vector\n",[61,16701,16702,16705,16707,16709,16711,16713,16715,16717,16719,16721,16723,16725],{"class":1731,"line":1784},[61,16703,16704],{"class":1753},"t ",[61,16706,2000],{"class":1859},[61,16708,2193],{"class":1753},[61,16710,3453],{"class":1836},[61,16712,849],{"class":1753},[61,16714,2006],{"class":1859},[61,16716,337],{"class":1866},[61,16718,1435],{"class":1753},[61,16720,337],{"class":1866},[61,16722,1435],{"class":1753},[61,16724,14254],{"class":1866},[61,16726,2016],{"class":1753},[61,16728,16729],{"class":1731,"line":1798},[61,16730,1743],{"emptyLinePlaceholder":1742},[61,16732,16733],{"class":1731,"line":1811},[61,16734,16735],{"class":1735},"# Define different pulse envelopes\n",[61,16737,16738,16741,16743,16745,16747,16749,16751,16753,16755,16757],{"class":1731,"line":1816},[61,16739,16740],{"class":1753},"gaussian ",[61,16742,2000],{"class":1859},[61,16744,2193],{"class":1753},[61,16746,6987],{"class":1836},[61,16748,849],{"class":1753},[61,16750,2006],{"class":1859},[61,16752,206],{"class":1753},[61,16754,2651],{"class":1859},[61,16756,337],{"class":1866},[61,16758,2016],{"class":1753},[61,16760,16761,16764,16766,16768,16770,16773,16775,16778,16780,16783,16785,16788,16790,16793,16795,16797,16799,16801,16803,16806,16808],{"class":1731,"line":1822},[61,16762,16763],{"class":1753},"sin2 ",[61,16765,2000],{"class":1859},[61,16767,2193],{"class":1753},[61,16769,5069],{"class":1836},[61,16771,16772],{"class":1753},"(np.pi ",[61,16774,2201],{"class":1859},[61,16776,16777],{"class":1753}," (t ",[61,16779,2006],{"class":1859},[61,16781,16782],{"class":1753}," t.",[61,16784,4202],{"class":1836},[61,16786,16787],{"class":1753},"()) ",[61,16789,602],{"class":1859},[61,16791,16792],{"class":1753}," (t.",[61,16794,4212],{"class":1836},[61,16796,7275],{"class":1753},[61,16798,2006],{"class":1859},[61,16800,16782],{"class":1753},[61,16802,4202],{"class":1836},[61,16804,16805],{"class":1753},"()))",[61,16807,2651],{"class":1859},[61,16809,7306],{"class":1866},[61,16811,16812,16815,16817,16819,16821,16823,16825,16827,16829,16831,16833,16835,16837,16839,16841,16843,16845,16847,16849,16851,16853],{"class":1731,"line":1828},[61,16813,16814],{"class":1753},"cos2 ",[61,16816,2000],{"class":1859},[61,16818,2193],{"class":1753},[61,16820,5051],{"class":1836},[61,16822,16772],{"class":1753},[61,16824,2201],{"class":1859},[61,16826,16777],{"class":1753},[61,16828,2006],{"class":1859},[61,16830,16782],{"class":1753},[61,16832,4202],{"class":1836},[61,16834,16787],{"class":1753},[61,16836,602],{"class":1859},[61,16838,16792],{"class":1753},[61,16840,4212],{"class":1836},[61,16842,7275],{"class":1753},[61,16844,2006],{"class":1859},[61,16846,16782],{"class":1753},[61,16848,4202],{"class":1836},[61,16850,16805],{"class":1753},[61,16852,2651],{"class":1859},[61,16854,7306],{"class":1866},[61,16856,16857,16860,16862,16864,16867],{"class":1731,"line":1833},[61,16858,16859],{"class":1753},"rectangular ",[61,16861,2000],{"class":1859},[61,16863,2193],{"class":1753},[61,16865,16866],{"class":1836},"ones_like",[61,16868,16869],{"class":1753},"(t)\n",[61,16871,16872,16875,16877,16880,16882,16884,16886,16888],{"class":1731,"line":1840},[61,16873,16874],{"class":1753},"rectangular[np.",[61,16876,3574],{"class":1836},[61,16878,16879],{"class":1753},"(t) ",[61,16881,2503],{"class":1859},[61,16883,5266],{"class":1866},[61,16885,1997],{"class":1753},[61,16887,2000],{"class":1859},[61,16889,14218],{"class":1866},[61,16891,16892,16895,16897,16900,16902,16904],{"class":1731,"line":1853},[61,16893,16894],{"class":1753},"blackman_harris ",[61,16896,2000],{"class":1859},[61,16898,16899],{"class":1836}," blackmanharris",[61,16901,849],{"class":1753},[61,16903,5742],{"class":1859},[61,16905,16906],{"class":1753},"(t))\n",[61,16908,16909,16912,16914,16917,16919,16921,16924,16927,16929,16931],{"class":1731,"line":1873},[61,16910,16911],{"class":1753},"tukey_envelope ",[61,16913,2000],{"class":1859},[61,16915,16916],{"class":1836}," tukey",[61,16918,849],{"class":1753},[61,16920,5742],{"class":1859},[61,16922,16923],{"class":1753},"(t), ",[61,16925,16926],{"class":3402},"alpha",[61,16928,2000],{"class":1859},[61,16930,5056],{"class":1866},[61,16932,2016],{"class":1753},[61,16934,16935,16938,16940,16942,16944,16946,16949],{"class":1731,"line":1889},[61,16936,16937],{"class":1753},"sech ",[61,16939,2000],{"class":1859},[61,16941,5266],{"class":1866},[61,16943,2618],{"class":1859},[61,16945,2193],{"class":1753},[61,16947,16948],{"class":1836},"cosh",[61,16950,16869],{"class":1753},[61,16952,16953],{"class":1731,"line":1908},[61,16954,1743],{"emptyLinePlaceholder":1742},[61,16956,16957],{"class":1731,"line":1925},[61,16958,16959],{"class":1735},"# List of envelopes and titles\n",[61,16961,16962,16965,16967,16970,16973],{"class":1731,"line":1941},[61,16963,16964],{"class":1753},"envelopes ",[61,16966,2000],{"class":1859},[61,16968,16969],{"class":1753}," [(gaussian, ",[61,16971,16972],{"class":2165},"\"Gaussian Modulation Function\"",[61,16974,16975],{"class":1753},"),\n",[61,16977,16978,16981,16984,16987,16990],{"class":1731,"line":1957},[61,16979,16980],{"class":1753},"             (sin2, ",[61,16982,16983],{"class":2165},"\"$Sin^",[61,16985,16986],{"class":1866},"{2}",[61,16988,16989],{"class":2165},"$ Modulation Function\"",[61,16991,16975],{"class":1753},[61,16993,16994,16997,17000,17002,17004],{"class":1731,"line":1974},[61,16995,16996],{"class":1753},"             (cos2, ",[61,16998,16999],{"class":2165},"\"$Cos^",[61,17001,16986],{"class":1866},[61,17003,16989],{"class":2165},[61,17005,16975],{"class":1753},[61,17007,17008,17011,17014],{"class":1731,"line":1979},[61,17009,17010],{"class":1753},"             (rectangular, ",[61,17012,17013],{"class":2165},"\"Rectangular Modulation Function\"",[61,17015,16975],{"class":1753},[61,17017,17018,17021,17024],{"class":1731,"line":1985},[61,17019,17020],{"class":1753},"             (blackman_harris, ",[61,17022,17023],{"class":2165},"\"Blackman-Harris Modulation Function\"",[61,17025,16975],{"class":1753},[61,17027,17028,17031,17034],{"class":1731,"line":2019},[61,17029,17030],{"class":1753},"             (tukey_envelope, ",[61,17032,17033],{"class":2165},"\"Tukey Modulation Function\"",[61,17035,16975],{"class":1753},[61,17037,17038,17041,17044],{"class":1731,"line":2033},[61,17039,17040],{"class":1753},"             (sech, ",[61,17042,17043],{"class":2165},"\"Sech Modulation Function\"",[61,17045,17046],{"class":1753},")]\n",[61,17048,17049],{"class":1731,"line":2061},[61,17050,1743],{"emptyLinePlaceholder":1742},[61,17052,17053],{"class":1731,"line":2074},[61,17054,17055],{"class":1735},"# Create subplots\n",[61,17057,17058,17061,17063,17066,17069,17071,17073,17075,17077,17079,17081,17083,17085,17088,17090,17093],{"class":1731,"line":2079},[61,17059,17060],{"class":1753},"fig, axes ",[61,17062,2000],{"class":1859},[61,17064,17065],{"class":1753}," plt.",[61,17067,17068],{"class":1836},"subplots",[61,17070,849],{"class":1753},[61,17072,570],{"class":1866},[61,17074,1435],{"class":1753},[61,17076,570],{"class":1866},[61,17078,1435],{"class":1753},[61,17080,3624],{"class":3402},[61,17082,2000],{"class":1859},[61,17084,849],{"class":1753},[61,17086,17087],{"class":1866},"12",[61,17089,1435],{"class":1753},[61,17091,17092],{"class":1866},"9",[61,17094,5546],{"class":1753},[61,17096,17097,17100,17102,17105,17108],{"class":1731,"line":2092},[61,17098,17099],{"class":1753},"axes ",[61,17101,2000],{"class":1859},[61,17103,17104],{"class":1753}," axes.",[61,17106,17107],{"class":1836},"flatten",[61,17109,2089],{"class":1753},[61,17111,17112],{"class":1731,"line":2097},[61,17113,1743],{"emptyLinePlaceholder":1742},[61,17115,17116,17118,17121,17123,17125],{"class":1731,"line":2102},[61,17117,2486],{"class":1749},[61,17119,17120],{"class":1753}," i, (envelope, title) ",[61,17122,2492],{"class":1749},[61,17124,4111],{"class":1859},[61,17126,17127],{"class":1753},"(envelopes):\n",[61,17129,17130,17133,17135,17138,17141],{"class":1731,"line":2108},[61,17131,17132],{"class":1753},"    axes[i].",[61,17134,3656],{"class":1836},[61,17136,17137],{"class":1753},"(t, envelope, ",[61,17139,17140],{"class":2165},"'b'",[61,17142,2016],{"class":1753},[61,17144,17145,17147,17150],{"class":1731,"line":2113},[61,17146,17132],{"class":1753},[61,17148,17149],{"class":1836},"set_title",[61,17151,17152],{"class":1753},"(title)\n",[61,17154,17155,17157,17160,17162,17165],{"class":1731,"line":2162},[61,17156,17132],{"class":1753},[61,17158,17159],{"class":1836},"set_xlabel",[61,17161,849],{"class":1753},[61,17163,17164],{"class":2165},"\"Time\"",[61,17166,2016],{"class":1753},[61,17168,17169,17171,17174,17176,17179],{"class":1731,"line":2169},[61,17170,17132],{"class":1753},[61,17172,17173],{"class":1836},"set_ylabel",[61,17175,849],{"class":1753},[61,17177,17178],{"class":2165},"\"Amplitude\"",[61,17180,2016],{"class":1753},[61,17182,17183],{"class":1731,"line":2187},[61,17184,17185],{"class":1735},"    # axes[i].legend([title])\n",[61,17187,17188],{"class":1731,"line":2207},[61,17189,1743],{"emptyLinePlaceholder":1742},[61,17191,17192],{"class":1731,"line":2221},[61,17193,17194],{"class":1735},"# Hide unused subplots\n",[61,17196,17197,17199,17202,17204,17207,17210,17212,17214,17216,17218],{"class":1731,"line":2226},[61,17198,2486],{"class":1749},[61,17200,17201],{"class":1753}," j ",[61,17203,2492],{"class":1749},[61,17205,17206],{"class":1859}," range",[61,17208,17209],{"class":1753},"(i ",[61,17211,2419],{"class":1859},[61,17213,5266],{"class":1866},[61,17215,1435],{"class":1753},[61,17217,5742],{"class":1859},[61,17219,17220],{"class":1753},"(axes)):\n",[61,17222,17223,17226,17229],{"class":1731,"line":2258},[61,17224,17225],{"class":1753},"    fig.",[61,17227,17228],{"class":1836},"delaxes",[61,17230,17231],{"class":1753},"(axes[j])\n",[61,17233,17234],{"class":1731,"line":2264},[61,17235,1743],{"emptyLinePlaceholder":1742},[61,17237,17238,17240,17243],{"class":1731,"line":2280},[61,17239,3616],{"class":1753},[61,17241,17242],{"class":1836},"tight_layout",[61,17244,2089],{"class":1753},[61,17246,17247,17249,17251],{"class":1731,"line":2304},[61,17248,3616],{"class":1753},[61,17250,3850],{"class":1836},[61,17252,2089],{"class":1753},[4797,17254],{"alt":17255,"src":17256},"Output 9 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-09.webp",[1720,17258,17260],{"className":1722,"code":17259,"language":1724,"meta":1725,"style":1725},"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",[1727,17261,17262,17272,17282,17292,17296,17301,17327,17331,17336,17358,17403,17448,17461,17480,17495,17518,17535,17539,17544,17558,17568,17578,17587,17596,17605,17614,17618,17622,17656,17668,17672,17686,17719,17727,17739,17751,17755,17759,17781,17789,17793,17801],{"__ignoreMap":1725},[61,17263,17264,17266,17268,17270],{"class":1731,"line":1732},[61,17265,1757],{"class":1749},[61,17267,1789],{"class":1753},[61,17269,1792],{"class":1749},[61,17271,1795],{"class":1753},[61,17273,17274,17276,17278,17280],{"class":1731,"line":1739},[61,17275,1757],{"class":1749},[61,17277,1803],{"class":1753},[61,17279,1792],{"class":1749},[61,17281,1808],{"class":1753},[61,17283,17284,17286,17288,17290],{"class":1731,"line":1746},[61,17285,1750],{"class":1749},[61,17287,16685],{"class":1753},[61,17289,1757],{"class":1749},[61,17291,16690],{"class":1753},[61,17293,17294],{"class":1731,"line":1763},[61,17295,1743],{"emptyLinePlaceholder":1742},[61,17297,17298],{"class":1731,"line":1776},[61,17299,17300],{"class":1735},"# Define time vector with reduced points for lollipop plot clarity\n",[61,17302,17303,17305,17307,17309,17311,17313,17315,17317,17319,17321,17323,17325],{"class":1731,"line":1784},[61,17304,16704],{"class":1753},[61,17306,2000],{"class":1859},[61,17308,2193],{"class":1753},[61,17310,3453],{"class":1836},[61,17312,849],{"class":1753},[61,17314,2006],{"class":1859},[61,17316,337],{"class":1866},[61,17318,1435],{"class":1753},[61,17320,337],{"class":1866},[61,17322,1435],{"class":1753},[61,17324,14795],{"class":1866},[61,17326,2016],{"class":1753},[61,17328,17329],{"class":1731,"line":1798},[61,17330,1743],{"emptyLinePlaceholder":1742},[61,17332,17333],{"class":1731,"line":1811},[61,17334,17335],{"class":1735},"# Define different modulation functions\n",[61,17337,17338,17340,17342,17344,17346,17348,17350,17352,17354,17356],{"class":1731,"line":1816},[61,17339,14274],{"class":1753},[61,17341,2000],{"class":1859},[61,17343,2193],{"class":1753},[61,17345,6987],{"class":1836},[61,17347,849],{"class":1753},[61,17349,2006],{"class":1859},[61,17351,206],{"class":1753},[61,17353,2651],{"class":1859},[61,17355,337],{"class":1866},[61,17357,2016],{"class":1753},[61,17359,17360,17363,17365,17367,17369,17371,17373,17375,17377,17379,17381,17383,17385,17387,17389,17391,17393,17395,17397,17399,17401],{"class":1731,"line":1822},[61,17361,17362],{"class":1753},"sin2_modulation ",[61,17364,2000],{"class":1859},[61,17366,2193],{"class":1753},[61,17368,5069],{"class":1836},[61,17370,16772],{"class":1753},[61,17372,2201],{"class":1859},[61,17374,16777],{"class":1753},[61,17376,2006],{"class":1859},[61,17378,16782],{"class":1753},[61,17380,4202],{"class":1836},[61,17382,16787],{"class":1753},[61,17384,602],{"class":1859},[61,17386,16792],{"class":1753},[61,17388,4212],{"class":1836},[61,17390,7275],{"class":1753},[61,17392,2006],{"class":1859},[61,17394,16782],{"class":1753},[61,17396,4202],{"class":1836},[61,17398,16805],{"class":1753},[61,17400,2651],{"class":1859},[61,17402,7306],{"class":1866},[61,17404,17405,17408,17410,17412,17414,17416,17418,17420,17422,17424,17426,17428,17430,17432,17434,17436,17438,17440,17442,17444,17446],{"class":1731,"line":1828},[61,17406,17407],{"class":1753},"cos2_modulation ",[61,17409,2000],{"class":1859},[61,17411,2193],{"class":1753},[61,17413,5051],{"class":1836},[61,17415,16772],{"class":1753},[61,17417,2201],{"class":1859},[61,17419,16777],{"class":1753},[61,17421,2006],{"class":1859},[61,17423,16782],{"class":1753},[61,17425,4202],{"class":1836},[61,17427,16787],{"class":1753},[61,17429,602],{"class":1859},[61,17431,16792],{"class":1753},[61,17433,4212],{"class":1836},[61,17435,7275],{"class":1753},[61,17437,2006],{"class":1859},[61,17439,16782],{"class":1753},[61,17441,4202],{"class":1836},[61,17443,16805],{"class":1753},[61,17445,2651],{"class":1859},[61,17447,7306],{"class":1866},[61,17449,17450,17453,17455,17457,17459],{"class":1731,"line":1833},[61,17451,17452],{"class":1753},"rectangular_modulation ",[61,17454,2000],{"class":1859},[61,17456,2193],{"class":1753},[61,17458,16866],{"class":1836},[61,17460,16869],{"class":1753},[61,17462,17463,17466,17468,17470,17472,17474,17476,17478],{"class":1731,"line":1840},[61,17464,17465],{"class":1753},"rectangular_modulation[np.",[61,17467,3574],{"class":1836},[61,17469,16879],{"class":1753},[61,17471,2503],{"class":1859},[61,17473,5266],{"class":1866},[61,17475,1997],{"class":1753},[61,17477,2000],{"class":1859},[61,17479,14218],{"class":1866},[61,17481,17482,17485,17487,17489,17491,17493],{"class":1731,"line":1853},[61,17483,17484],{"class":1753},"blackman_harris_modulation ",[61,17486,2000],{"class":1859},[61,17488,16899],{"class":1836},[61,17490,849],{"class":1753},[61,17492,5742],{"class":1859},[61,17494,16906],{"class":1753},[61,17496,17497,17500,17502,17504,17506,17508,17510,17512,17514,17516],{"class":1731,"line":1873},[61,17498,17499],{"class":1753},"tukey_modulation ",[61,17501,2000],{"class":1859},[61,17503,16916],{"class":1836},[61,17505,849],{"class":1753},[61,17507,5742],{"class":1859},[61,17509,16923],{"class":1753},[61,17511,16926],{"class":3402},[61,17513,2000],{"class":1859},[61,17515,5056],{"class":1866},[61,17517,2016],{"class":1753},[61,17519,17520,17523,17525,17527,17529,17531,17533],{"class":1731,"line":1889},[61,17521,17522],{"class":1753},"sech_modulation ",[61,17524,2000],{"class":1859},[61,17526,5266],{"class":1866},[61,17528,2618],{"class":1859},[61,17530,2193],{"class":1753},[61,17532,16948],{"class":1836},[61,17534,16869],{"class":1753},[61,17536,17537],{"class":1731,"line":1908},[61,17538,1743],{"emptyLinePlaceholder":1742},[61,17540,17541],{"class":1731,"line":1925},[61,17542,17543],{"class":1735},"# List of modulation functions and titles\n",[61,17545,17546,17549,17551,17554,17556],{"class":1731,"line":1941},[61,17547,17548],{"class":1753},"modulations ",[61,17550,2000],{"class":1859},[61,17552,17553],{"class":1753}," [(gaussian_modulation, ",[61,17555,16972],{"class":2165},[61,17557,16975],{"class":1753},[61,17559,17560,17563,17566],{"class":1731,"line":1957},[61,17561,17562],{"class":1753},"               (sin2_modulation, ",[61,17564,17565],{"class":2165},"\"Sin² Modulation Function\"",[61,17567,16975],{"class":1753},[61,17569,17570,17573,17576],{"class":1731,"line":1974},[61,17571,17572],{"class":1753},"               (cos2_modulation, ",[61,17574,17575],{"class":2165},"\"Cos² Modulation Function\"",[61,17577,16975],{"class":1753},[61,17579,17580,17583,17585],{"class":1731,"line":1979},[61,17581,17582],{"class":1753},"               (rectangular_modulation, ",[61,17584,17013],{"class":2165},[61,17586,16975],{"class":1753},[61,17588,17589,17592,17594],{"class":1731,"line":1985},[61,17590,17591],{"class":1753},"               (blackman_harris_modulation, ",[61,17593,17023],{"class":2165},[61,17595,16975],{"class":1753},[61,17597,17598,17601,17603],{"class":1731,"line":2019},[61,17599,17600],{"class":1753},"               (tukey_modulation, ",[61,17602,17033],{"class":2165},[61,17604,16975],{"class":1753},[61,17606,17607,17610,17612],{"class":1731,"line":2033},[61,17608,17609],{"class":1753},"               (sech_modulation, ",[61,17611,17043],{"class":2165},[61,17613,17046],{"class":1753},[61,17615,17616],{"class":1731,"line":2061},[61,17617,1743],{"emptyLinePlaceholder":1742},[61,17619,17620],{"class":1731,"line":2074},[61,17621,17055],{"class":1735},[61,17623,17624,17626,17628,17630,17632,17634,17636,17638,17640,17642,17644,17646,17648,17650,17652,17654],{"class":1731,"line":2079},[61,17625,17060],{"class":1753},[61,17627,2000],{"class":1859},[61,17629,17065],{"class":1753},[61,17631,17068],{"class":1836},[61,17633,849],{"class":1753},[61,17635,570],{"class":1866},[61,17637,1435],{"class":1753},[61,17639,570],{"class":1866},[61,17641,1435],{"class":1753},[61,17643,3624],{"class":3402},[61,17645,2000],{"class":1859},[61,17647,849],{"class":1753},[61,17649,17087],{"class":1866},[61,17651,1435],{"class":1753},[61,17653,17092],{"class":1866},[61,17655,5546],{"class":1753},[61,17657,17658,17660,17662,17664,17666],{"class":1731,"line":2092},[61,17659,17099],{"class":1753},[61,17661,2000],{"class":1859},[61,17663,17104],{"class":1753},[61,17665,17107],{"class":1836},[61,17667,2089],{"class":1753},[61,17669,17670],{"class":1731,"line":2097},[61,17671,1743],{"emptyLinePlaceholder":1742},[61,17673,17674,17676,17679,17681,17683],{"class":1731,"line":2102},[61,17675,2486],{"class":1749},[61,17677,17678],{"class":1753}," i, (modulation, title) ",[61,17680,2492],{"class":1749},[61,17682,4111],{"class":1859},[61,17684,17685],{"class":1753},"(modulations):\n",[61,17687,17688,17690,17692,17695,17697,17699,17701,17703,17705,17707,17709,17711,17713,17715,17717],{"class":1731,"line":2108},[61,17689,17132],{"class":1753},[61,17691,14891],{"class":1836},[61,17693,17694],{"class":1753},"(t, modulation, ",[61,17696,14896],{"class":3402},[61,17698,2000],{"class":1859},[61,17700,14901],{"class":2165},[61,17702,1435],{"class":1753},[61,17704,14906],{"class":3402},[61,17706,2000],{"class":1859},[61,17708,14911],{"class":2165},[61,17710,1435],{"class":1753},[61,17712,14916],{"class":3402},[61,17714,2000],{"class":1859},[61,17716,14921],{"class":2165},[61,17718,2016],{"class":1753},[61,17720,17721,17723,17725],{"class":1731,"line":2113},[61,17722,17132],{"class":1753},[61,17724,17149],{"class":1836},[61,17726,17152],{"class":1753},[61,17728,17729,17731,17733,17735,17737],{"class":1731,"line":2162},[61,17730,17132],{"class":1753},[61,17732,17159],{"class":1836},[61,17734,849],{"class":1753},[61,17736,17164],{"class":2165},[61,17738,2016],{"class":1753},[61,17740,17741,17743,17745,17747,17749],{"class":1731,"line":2169},[61,17742,17132],{"class":1753},[61,17744,17173],{"class":1836},[61,17746,849],{"class":1753},[61,17748,17178],{"class":2165},[61,17750,2016],{"class":1753},[61,17752,17753],{"class":1731,"line":2187},[61,17754,1743],{"emptyLinePlaceholder":1742},[61,17756,17757],{"class":1731,"line":2207},[61,17758,17194],{"class":1735},[61,17760,17761,17763,17765,17767,17769,17771,17773,17775,17777,17779],{"class":1731,"line":2221},[61,17762,2486],{"class":1749},[61,17764,17201],{"class":1753},[61,17766,2492],{"class":1749},[61,17768,17206],{"class":1859},[61,17770,17209],{"class":1753},[61,17772,2419],{"class":1859},[61,17774,5266],{"class":1866},[61,17776,1435],{"class":1753},[61,17778,5742],{"class":1859},[61,17780,17220],{"class":1753},[61,17782,17783,17785,17787],{"class":1731,"line":2226},[61,17784,17225],{"class":1753},[61,17786,17228],{"class":1836},[61,17788,17231],{"class":1753},[61,17790,17791],{"class":1731,"line":2258},[61,17792,1743],{"emptyLinePlaceholder":1742},[61,17794,17795,17797,17799],{"class":1731,"line":2264},[61,17796,3616],{"class":1753},[61,17798,17242],{"class":1836},[61,17800,2089],{"class":1753},[61,17802,17803,17805,17807],{"class":1731,"line":2280},[61,17804,3616],{"class":1753},[61,17806,3850],{"class":1836},[61,17808,2089],{"class":1753},[4797,17810],{"alt":17811,"src":17812},"Output 10 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-10.webp",[1720,17814,17816],{"className":1722,"code":17815,"language":1724,"meta":1725,"style":1725},"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",[1727,17817,17818,17828,17838,17848,17852,17856,17882,17886,17891,17915,17963,18008,18022,18042,18058,18082,18099,18103,18107,18115,18125,18155,18159,18164,18179,18193,18206,18216,18226,18236,18246,18250,18254,18288,18300,18304,18318,18332,18345,18353,18365,18377,18381,18385,18407,18415,18419,18427],{"__ignoreMap":1725},[61,17819,17820,17822,17824,17826],{"class":1731,"line":1732},[61,17821,1757],{"class":1749},[61,17823,1789],{"class":1753},[61,17825,1792],{"class":1749},[61,17827,1795],{"class":1753},[61,17829,17830,17832,17834,17836],{"class":1731,"line":1739},[61,17831,1757],{"class":1749},[61,17833,1803],{"class":1753},[61,17835,1792],{"class":1749},[61,17837,1808],{"class":1753},[61,17839,17840,17842,17844,17846],{"class":1731,"line":1746},[61,17841,1750],{"class":1749},[61,17843,16685],{"class":1753},[61,17845,1757],{"class":1749},[61,17847,16690],{"class":1753},[61,17849,17850],{"class":1731,"line":1763},[61,17851,1743],{"emptyLinePlaceholder":1742},[61,17853,17854],{"class":1731,"line":1776},[61,17855,16699],{"class":1735},[61,17857,17858,17860,17862,17864,17866,17868,17870,17872,17874,17876,17878,17880],{"class":1731,"line":1784},[61,17859,14233],{"class":1753},[61,17861,2000],{"class":1859},[61,17863,2193],{"class":1753},[61,17865,3453],{"class":1836},[61,17867,849],{"class":1753},[61,17869,2006],{"class":1859},[61,17871,1026],{"class":1866},[61,17873,1435],{"class":1753},[61,17875,1026],{"class":1866},[61,17877,1435],{"class":1753},[61,17879,14254],{"class":1866},[61,17881,2016],{"class":1753},[61,17883,17884],{"class":1731,"line":1798},[61,17885,1743],{"emptyLinePlaceholder":1742},[61,17887,17888],{"class":1731,"line":1811},[61,17889,17890],{"class":1735},"# Define different modulated sine waves\n",[61,17892,17893,17896,17898,17900,17902,17904,17906,17909,17911,17913],{"class":1731,"line":1816},[61,17894,17895],{"class":1753},"gaussian_modulated_sine ",[61,17897,2000],{"class":1859},[61,17899,2193],{"class":1753},[61,17901,6987],{"class":1836},[61,17903,849],{"class":1753},[61,17905,2006],{"class":1859},[61,17907,17908],{"class":1753},"time",[61,17910,2651],{"class":1859},[61,17912,337],{"class":1866},[61,17914,2016],{"class":1753},[61,17916,17917,17920,17922,17924,17926,17928,17930,17933,17935,17938,17940,17942,17944,17947,17949,17951,17953,17955,17957,17959,17961],{"class":1731,"line":1822},[61,17918,17919],{"class":1753},"sin2_modulated_sine ",[61,17921,2000],{"class":1859},[61,17923,2193],{"class":1753},[61,17925,5069],{"class":1836},[61,17927,16772],{"class":1753},[61,17929,2201],{"class":1859},[61,17931,17932],{"class":1753}," (time ",[61,17934,2006],{"class":1859},[61,17936,17937],{"class":1753}," time.",[61,17939,4202],{"class":1836},[61,17941,16787],{"class":1753},[61,17943,602],{"class":1859},[61,17945,17946],{"class":1753}," (time.",[61,17948,4212],{"class":1836},[61,17950,7275],{"class":1753},[61,17952,2006],{"class":1859},[61,17954,17937],{"class":1753},[61,17956,4202],{"class":1836},[61,17958,16805],{"class":1753},[61,17960,2651],{"class":1859},[61,17962,7306],{"class":1866},[61,17964,17965,17968,17970,17972,17974,17976,17978,17980,17982,17984,17986,17988,17990,17992,17994,17996,17998,18000,18002,18004,18006],{"class":1731,"line":1828},[61,17966,17967],{"class":1753},"cos2_modulated_sine ",[61,17969,2000],{"class":1859},[61,17971,2193],{"class":1753},[61,17973,5051],{"class":1836},[61,17975,16772],{"class":1753},[61,17977,2201],{"class":1859},[61,17979,17932],{"class":1753},[61,17981,2006],{"class":1859},[61,17983,17937],{"class":1753},[61,17985,4202],{"class":1836},[61,17987,16787],{"class":1753},[61,17989,602],{"class":1859},[61,17991,17946],{"class":1753},[61,17993,4212],{"class":1836},[61,17995,7275],{"class":1753},[61,17997,2006],{"class":1859},[61,17999,17937],{"class":1753},[61,18001,4202],{"class":1836},[61,18003,16805],{"class":1753},[61,18005,2651],{"class":1859},[61,18007,7306],{"class":1866},[61,18009,18010,18013,18015,18017,18019],{"class":1731,"line":1833},[61,18011,18012],{"class":1753},"rectangular_modulated_sine ",[61,18014,2000],{"class":1859},[61,18016,2193],{"class":1753},[61,18018,16866],{"class":1836},[61,18020,18021],{"class":1753},"(time)\n",[61,18023,18024,18027,18029,18032,18034,18036,18038,18040],{"class":1731,"line":1840},[61,18025,18026],{"class":1753},"rectangular_modulated_sine[np.",[61,18028,3574],{"class":1836},[61,18030,18031],{"class":1753},"(time) ",[61,18033,2503],{"class":1859},[61,18035,2654],{"class":1866},[61,18037,1997],{"class":1753},[61,18039,2000],{"class":1859},[61,18041,14218],{"class":1866},[61,18043,18044,18047,18049,18051,18053,18055],{"class":1731,"line":1853},[61,18045,18046],{"class":1753},"blackman_harris_modulated_sine ",[61,18048,2000],{"class":1859},[61,18050,16899],{"class":1836},[61,18052,849],{"class":1753},[61,18054,5742],{"class":1859},[61,18056,18057],{"class":1753},"(time))\n",[61,18059,18060,18063,18065,18067,18069,18071,18074,18076,18078,18080],{"class":1731,"line":1873},[61,18061,18062],{"class":1753},"tukey_modulated_sine ",[61,18064,2000],{"class":1859},[61,18066,16916],{"class":1836},[61,18068,849],{"class":1753},[61,18070,5742],{"class":1859},[61,18072,18073],{"class":1753},"(time), ",[61,18075,16926],{"class":3402},[61,18077,2000],{"class":1859},[61,18079,5056],{"class":1866},[61,18081,2016],{"class":1753},[61,18083,18084,18087,18089,18091,18093,18095,18097],{"class":1731,"line":1889},[61,18085,18086],{"class":1753},"sech_modulated_sine ",[61,18088,2000],{"class":1859},[61,18090,5266],{"class":1866},[61,18092,2618],{"class":1859},[61,18094,2193],{"class":1753},[61,18096,16948],{"class":1836},[61,18098,18021],{"class":1753},[61,18100,18101],{"class":1731,"line":1908},[61,18102,1743],{"emptyLinePlaceholder":1742},[61,18104,18105],{"class":1731,"line":1925},[61,18106,15419],{"class":1735},[61,18108,18109,18111,18113],{"class":1731,"line":1941},[61,18110,14441],{"class":1753},[61,18112,2000],{"class":1859},[61,18114,14228],{"class":1866},[61,18116,18117,18119,18121,18123],{"class":1731,"line":1957},[61,18118,15067],{"class":1753},[61,18120,2000],{"class":1859},[61,18122,15072],{"class":1866},[61,18124,15075],{"class":1735},[61,18126,18127,18129,18131,18133,18135,18137,18139,18141,18143,18145,18147,18149,18151,18153],{"class":1731,"line":1974},[61,18128,15165],{"class":1753},[61,18130,2000],{"class":1859},[61,18132,14506],{"class":1753},[61,18134,2201],{"class":1859},[61,18136,2193],{"class":1753},[61,18138,5069],{"class":1836},[61,18140,849],{"class":1753},[61,18142,337],{"class":1866},[61,18144,2298],{"class":1859},[61,18146,15184],{"class":1753},[61,18148,2201],{"class":1859},[61,18150,15189],{"class":1753},[61,18152,2201],{"class":1859},[61,18154,15194],{"class":1753},[61,18156,18157],{"class":1731,"line":1979},[61,18158,1743],{"emptyLinePlaceholder":1742},[61,18160,18161],{"class":1731,"line":1985},[61,18162,18163],{"class":1735},"# List of modulated sine waves and titles\n",[61,18165,18166,18169,18171,18174,18177],{"class":1731,"line":2019},[61,18167,18168],{"class":1753},"modulated_sines ",[61,18170,2000],{"class":1859},[61,18172,18173],{"class":1753}," [(gaussian_modulated_sine, ",[61,18175,18176],{"class":2165},"\"Gaussian Modulated Sine Wave\"",[61,18178,16975],{"class":1753},[61,18180,18181,18184,18186,18188,18191],{"class":1731,"line":2033},[61,18182,18183],{"class":1753},"                    (sin2_modulated_sine, ",[61,18185,16983],{"class":2165},[61,18187,16986],{"class":1866},[61,18189,18190],{"class":2165},"$ Modulated Sine Wave\"",[61,18192,16975],{"class":1753},[61,18194,18195,18198,18200,18202,18204],{"class":1731,"line":2061},[61,18196,18197],{"class":1753},"                    (cos2_modulated_sine, ",[61,18199,16999],{"class":2165},[61,18201,16986],{"class":1866},[61,18203,18190],{"class":2165},[61,18205,16975],{"class":1753},[61,18207,18208,18211,18214],{"class":1731,"line":2074},[61,18209,18210],{"class":1753},"                    (rectangular_modulated_sine, ",[61,18212,18213],{"class":2165},"\"Rectangular Modulated Sine Wave\"",[61,18215,16975],{"class":1753},[61,18217,18218,18221,18224],{"class":1731,"line":2079},[61,18219,18220],{"class":1753},"                    (blackman_harris_modulated_sine, ",[61,18222,18223],{"class":2165},"\"Blackman-Harris Modulated Sine Wave\"",[61,18225,16975],{"class":1753},[61,18227,18228,18231,18234],{"class":1731,"line":2092},[61,18229,18230],{"class":1753},"                    (tukey_modulated_sine, ",[61,18232,18233],{"class":2165},"\"Tukey Modulated Sine Wave\"",[61,18235,16975],{"class":1753},[61,18237,18238,18241,18244],{"class":1731,"line":2097},[61,18239,18240],{"class":1753},"                    (sech_modulated_sine, ",[61,18242,18243],{"class":2165},"\"Sech Modulated Sine Wave\"",[61,18245,17046],{"class":1753},[61,18247,18248],{"class":1731,"line":2102},[61,18249,1743],{"emptyLinePlaceholder":1742},[61,18251,18252],{"class":1731,"line":2108},[61,18253,17055],{"class":1735},[61,18255,18256,18258,18260,18262,18264,18266,18268,18270,18272,18274,18276,18278,18280,18282,18284,18286],{"class":1731,"line":2113},[61,18257,17060],{"class":1753},[61,18259,2000],{"class":1859},[61,18261,17065],{"class":1753},[61,18263,17068],{"class":1836},[61,18265,849],{"class":1753},[61,18267,570],{"class":1866},[61,18269,1435],{"class":1753},[61,18271,570],{"class":1866},[61,18273,1435],{"class":1753},[61,18275,3624],{"class":3402},[61,18277,2000],{"class":1859},[61,18279,849],{"class":1753},[61,18281,17087],{"class":1866},[61,18283,1435],{"class":1753},[61,18285,17092],{"class":1866},[61,18287,5546],{"class":1753},[61,18289,18290,18292,18294,18296,18298],{"class":1731,"line":2162},[61,18291,17099],{"class":1753},[61,18293,2000],{"class":1859},[61,18295,17104],{"class":1753},[61,18297,17107],{"class":1836},[61,18299,2089],{"class":1753},[61,18301,18302],{"class":1731,"line":2169},[61,18303,1743],{"emptyLinePlaceholder":1742},[61,18305,18306,18308,18311,18313,18315],{"class":1731,"line":2187},[61,18307,2486],{"class":1749},[61,18309,18310],{"class":1753}," i, (modulated_sine, title) ",[61,18312,2492],{"class":1749},[61,18314,4111],{"class":1859},[61,18316,18317],{"class":1753},"(modulated_sines):\n",[61,18319,18320,18323,18325,18328,18330],{"class":1731,"line":2207},[61,18321,18322],{"class":1753},"    modulated_wave ",[61,18324,2000],{"class":1859},[61,18326,18327],{"class":1753}," modulated_sine ",[61,18329,2201],{"class":1859},[61,18331,15218],{"class":1753},[61,18333,18334,18336,18338,18341,18343],{"class":1731,"line":2221},[61,18335,17132],{"class":1753},[61,18337,3656],{"class":1836},[61,18339,18340],{"class":1753},"(time, modulated_wave, ",[61,18342,17140],{"class":2165},[61,18344,2016],{"class":1753},[61,18346,18347,18349,18351],{"class":1731,"line":2226},[61,18348,17132],{"class":1753},[61,18350,17149],{"class":1836},[61,18352,17152],{"class":1753},[61,18354,18355,18357,18359,18361,18363],{"class":1731,"line":2258},[61,18356,17132],{"class":1753},[61,18358,17159],{"class":1836},[61,18360,849],{"class":1753},[61,18362,17164],{"class":2165},[61,18364,2016],{"class":1753},[61,18366,18367,18369,18371,18373,18375],{"class":1731,"line":2264},[61,18368,17132],{"class":1753},[61,18370,17173],{"class":1836},[61,18372,849],{"class":1753},[61,18374,17178],{"class":2165},[61,18376,2016],{"class":1753},[61,18378,18379],{"class":1731,"line":2280},[61,18380,1743],{"emptyLinePlaceholder":1742},[61,18382,18383],{"class":1731,"line":2304},[61,18384,17194],{"class":1735},[61,18386,18387,18389,18391,18393,18395,18397,18399,18401,18403,18405],{"class":1731,"line":2326},[61,18388,2486],{"class":1749},[61,18390,17201],{"class":1753},[61,18392,2492],{"class":1749},[61,18394,17206],{"class":1859},[61,18396,17209],{"class":1753},[61,18398,2419],{"class":1859},[61,18400,5266],{"class":1866},[61,18402,1435],{"class":1753},[61,18404,5742],{"class":1859},[61,18406,17220],{"class":1753},[61,18408,18409,18411,18413],{"class":1731,"line":2342},[61,18410,17225],{"class":1753},[61,18412,17228],{"class":1836},[61,18414,17231],{"class":1753},[61,18416,18417],{"class":1731,"line":2379},[61,18418,1743],{"emptyLinePlaceholder":1742},[61,18420,18421,18423,18425],{"class":1731,"line":2396},[61,18422,3616],{"class":1753},[61,18424,17242],{"class":1836},[61,18426,2089],{"class":1753},[61,18428,18429,18431,18433],{"class":1731,"line":2404},[61,18430,3616],{"class":1753},[61,18432,3850],{"class":1836},[61,18434,2089],{"class":1753},[4797,18436],{"alt":18437,"src":18438},"Output 11 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-11.webp",[1720,18440,18442],{"className":1722,"code":18441,"language":1724,"meta":1725,"style":1725},"import numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.signal.windows import blackmanharris, tukey\n\n# Define time vector\ntime = np.linspace(-5, 5, 1000)\n\n# Define different modulated sine waves\ngaussian_modulated_sine = np.exp(-time**2)\nsin2_modulated_sine = np.sin(np.pi * (time - time.min()) \u002F (time.max() - time.min()))**2\ncos2_modulated_sine = np.cos(np.pi * (time - time.min()) \u002F (time.max() - time.min()))**2\nrectangular_modulated_sine = np.ones_like(time)\nrectangular_modulated_sine[np.abs(time) > 2] = 0\nblackman_harris_modulated_sine = blackmanharris(len(time))\ntukey_modulated_sine = tukey(len(time), alpha=0.5)\nsech_modulated_sine = 1 \u002F np.cosh(time)\n\n# Define sine wave parameters\namplitude = 1\nfrequency = 5  # Frequency of the sine wave\nsine_wave = amplitude * np.sin(2 * np.pi * frequency * time)\n\n# List of modulated sine waves and titles\nmodulated_sines = [(gaussian_modulated_sine, \"Gaussian Modulated Sine Wave\"),\n                   (sin2_modulated_sine, \"$Sin^{2}$ Modulated Sine Wave\"),\n                   (cos2_modulated_sine, \"$Cos^{2}$ Modulated Sine Wave\"),\n                   (rectangular_modulated_sine, \"Rectangular Modulated Sine Wave\"),\n                   (blackman_harris_modulated_sine, \"Blackman-Harris Modulated Sine Wave\"),\n                   (tukey_modulated_sine, \"Tukey Modulated Sine Wave\"),\n                   (sech_modulated_sine, \"Sech Modulated Sine Wave\")]\n\n# Create subplots\nfig, axes = plt.subplots(3, 3, figsize=(12, 9))\naxes = axes.flatten()\n\nfor i, (modulated_sine, title) in enumerate(modulated_sines):\n    modulated_wave = modulated_sine * sine_wave\n    axes[i].plot(time, modulated_wave, 'b')\n    axes[i].plot(time, modulated_sine, 'r', linestyle='--', label='Upper Modulation Function')\n    axes[i].set_title(title)\n    axes[i].set_xlabel(\"Time\")\n    axes[i].set_ylabel(\"Amplitude\")\n    axes[i].legend()\n\n# Hide unused subplots\nfor j in range(i + 1, len(axes)):\n    fig.delaxes(axes[j])\n\nplt.tight_layout()\nplt.show()\n",[1727,18443,18444,18454,18464,18474,18478,18482,18508,18512,18516,18538,18582,18626,18638,18656,18670,18692,18708,18712,18716,18724,18734,18764,18768,18772,18784,18797,18810,18819,18828,18837,18846,18850,18854,18888,18900,18904,18916,18928,18940,18971,18979,18991,19003,19011,19015,19019,19041,19049,19053,19061],{"__ignoreMap":1725},[61,18445,18446,18448,18450,18452],{"class":1731,"line":1732},[61,18447,1757],{"class":1749},[61,18449,1789],{"class":1753},[61,18451,1792],{"class":1749},[61,18453,1795],{"class":1753},[61,18455,18456,18458,18460,18462],{"class":1731,"line":1739},[61,18457,1757],{"class":1749},[61,18459,1803],{"class":1753},[61,18461,1792],{"class":1749},[61,18463,1808],{"class":1753},[61,18465,18466,18468,18470,18472],{"class":1731,"line":1746},[61,18467,1750],{"class":1749},[61,18469,16685],{"class":1753},[61,18471,1757],{"class":1749},[61,18473,16690],{"class":1753},[61,18475,18476],{"class":1731,"line":1763},[61,18477,1743],{"emptyLinePlaceholder":1742},[61,18479,18480],{"class":1731,"line":1776},[61,18481,16699],{"class":1735},[61,18483,18484,18486,18488,18490,18492,18494,18496,18498,18500,18502,18504,18506],{"class":1731,"line":1784},[61,18485,14233],{"class":1753},[61,18487,2000],{"class":1859},[61,18489,2193],{"class":1753},[61,18491,3453],{"class":1836},[61,18493,849],{"class":1753},[61,18495,2006],{"class":1859},[61,18497,1026],{"class":1866},[61,18499,1435],{"class":1753},[61,18501,1026],{"class":1866},[61,18503,1435],{"class":1753},[61,18505,14254],{"class":1866},[61,18507,2016],{"class":1753},[61,18509,18510],{"class":1731,"line":1798},[61,18511,1743],{"emptyLinePlaceholder":1742},[61,18513,18514],{"class":1731,"line":1811},[61,18515,17890],{"class":1735},[61,18517,18518,18520,18522,18524,18526,18528,18530,18532,18534,18536],{"class":1731,"line":1816},[61,18519,17895],{"class":1753},[61,18521,2000],{"class":1859},[61,18523,2193],{"class":1753},[61,18525,6987],{"class":1836},[61,18527,849],{"class":1753},[61,18529,2006],{"class":1859},[61,18531,17908],{"class":1753},[61,18533,2651],{"class":1859},[61,18535,337],{"class":1866},[61,18537,2016],{"class":1753},[61,18539,18540,18542,18544,18546,18548,18550,18552,18554,18556,18558,18560,18562,18564,18566,18568,18570,18572,18574,18576,18578,18580],{"class":1731,"line":1822},[61,18541,17919],{"class":1753},[61,18543,2000],{"class":1859},[61,18545,2193],{"class":1753},[61,18547,5069],{"class":1836},[61,18549,16772],{"class":1753},[61,18551,2201],{"class":1859},[61,18553,17932],{"class":1753},[61,18555,2006],{"class":1859},[61,18557,17937],{"class":1753},[61,18559,4202],{"class":1836},[61,18561,16787],{"class":1753},[61,18563,602],{"class":1859},[61,18565,17946],{"class":1753},[61,18567,4212],{"class":1836},[61,18569,7275],{"class":1753},[61,18571,2006],{"class":1859},[61,18573,17937],{"class":1753},[61,18575,4202],{"class":1836},[61,18577,16805],{"class":1753},[61,18579,2651],{"class":1859},[61,18581,7306],{"class":1866},[61,18583,18584,18586,18588,18590,18592,18594,18596,18598,18600,18602,18604,18606,18608,18610,18612,18614,18616,18618,18620,18622,18624],{"class":1731,"line":1828},[61,18585,17967],{"class":1753},[61,18587,2000],{"class":1859},[61,18589,2193],{"class":1753},[61,18591,5051],{"class":1836},[61,18593,16772],{"class":1753},[61,18595,2201],{"class":1859},[61,18597,17932],{"class":1753},[61,18599,2006],{"class":1859},[61,18601,17937],{"class":1753},[61,18603,4202],{"class":1836},[61,18605,16787],{"class":1753},[61,18607,602],{"class":1859},[61,18609,17946],{"class":1753},[61,18611,4212],{"class":1836},[61,18613,7275],{"class":1753},[61,18615,2006],{"class":1859},[61,18617,17937],{"class":1753},[61,18619,4202],{"class":1836},[61,18621,16805],{"class":1753},[61,18623,2651],{"class":1859},[61,18625,7306],{"class":1866},[61,18627,18628,18630,18632,18634,18636],{"class":1731,"line":1833},[61,18629,18012],{"class":1753},[61,18631,2000],{"class":1859},[61,18633,2193],{"class":1753},[61,18635,16866],{"class":1836},[61,18637,18021],{"class":1753},[61,18639,18640,18642,18644,18646,18648,18650,18652,18654],{"class":1731,"line":1840},[61,18641,18026],{"class":1753},[61,18643,3574],{"class":1836},[61,18645,18031],{"class":1753},[61,18647,2503],{"class":1859},[61,18649,2654],{"class":1866},[61,18651,1997],{"class":1753},[61,18653,2000],{"class":1859},[61,18655,14218],{"class":1866},[61,18657,18658,18660,18662,18664,18666,18668],{"class":1731,"line":1853},[61,18659,18046],{"class":1753},[61,18661,2000],{"class":1859},[61,18663,16899],{"class":1836},[61,18665,849],{"class":1753},[61,18667,5742],{"class":1859},[61,18669,18057],{"class":1753},[61,18671,18672,18674,18676,18678,18680,18682,18684,18686,18688,18690],{"class":1731,"line":1873},[61,18673,18062],{"class":1753},[61,18675,2000],{"class":1859},[61,18677,16916],{"class":1836},[61,18679,849],{"class":1753},[61,18681,5742],{"class":1859},[61,18683,18073],{"class":1753},[61,18685,16926],{"class":3402},[61,18687,2000],{"class":1859},[61,186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                  (sin2_modulated_sine, ",[61,18790,16983],{"class":2165},[61,18792,16986],{"class":1866},[61,18794,18190],{"class":2165},[61,18796,16975],{"class":1753},[61,18798,18799,18802,18804,18806,18808],{"class":1731,"line":2061},[61,18800,18801],{"class":1753},"                   (cos2_modulated_sine, ",[61,18803,16999],{"class":2165},[61,18805,16986],{"class":1866},[61,18807,18190],{"class":2165},[61,18809,16975],{"class":1753},[61,18811,18812,18815,18817],{"class":1731,"line":2074},[61,18813,18814],{"class":1753},"                   (rectangular_modulated_sine, ",[61,18816,18213],{"class":2165},[61,18818,16975],{"class":1753},[61,18820,18821,18824,18826],{"class":1731,"line":2079},[61,18822,18823],{"class":1753},"                   (blackman_harris_modulated_sine, ",[61,18825,18223],{"class":2165},[61,18827,16975],{"class":1753},[61,18829,18830,18833,18835],{"class":1731,"line":2092},[61,18831,18832],{"class":1753},"                   (tukey_modulated_sine, 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12 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-12.webp",[12,19074,19075],{},"Below are some applied examples in the context of common quantum gates.",[12,19077,19078,19079,19082],{},"Relevant equations for the ",[814,19080,19081],{},"modulated Gaussian pulses"," used in the plots for some common quantum gates:",[1635,19084,19086],{"id":19085},"_1-gaussian-envelope",[814,19087,19088],{},"1. Gaussian Envelope",[12,19090,19091,19092,19478],{},"The Gaussian envelope is used to smoothly shape the pulse to minimize spectral leakage:\n",[61,19093,19095,19173],{"className":19094},[64],[61,19096,19098],{"className":19097},[68],[70,19099,19100],{"xmlns":72},[74,19101,19102,19170],{},[77,19103,19104,19107,19109,19111,19113,19115,19121,19123,19126],{},[83,19105,19106],{},"A",[586,19108,849],{"stretchy":588},[83,19110,206],{},[586,19112,300],{"stretchy":588},[586,19114,2000],{},[80,19116,19117,19119],{},[83,19118,19106],{},[1493,19120,1495],{},[83,19122,6987],{},[586,19124,19125],{},"⁡",[77,19127,19128,19130,19133,19168],{},[586,19129,849],{"fence":103},[586,19131,19132],{},"−",[19134,19135,19136,19157],"mfrac",{},[77,19137,19138,19140,19142,19144,19150],{},[586,19139,849],{"stretchy":588},[83,19141,206],{},[586,19143,19132],{},[80,19145,19146,19148],{},[83,19147,206],{},[1493,19149,1495],{},[19151,19152,19153,19155],"msup",{},[586,19154,300],{"stretchy":588},[1493,19156,337],{},[77,19158,19159,19161],{},[1493,19160,337],{},[19151,19162,19163,19166],{},[83,19164,19165],{},"σ",[1493,19167,337],{},[586,19169,300],{"fence":103},[95,19171,19172],{"encoding":97},"A(t) = A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right)",[61,19174,19176,19205],{"className":19175,"ariaHidden":103},[102],[61,19177,19179,19182,19185,19188,19191,19194,19198,19202],{"className":19178},[107],[61,19180],{"className":19181,"style":627},[111],[61,19183,19106],{"className":19184},[116,120],[61,19186,849],{"className":19187},[631],[61,19189,206],{"className":19190},[116,120],[61,19192,300],{"className":19193},[730],[61,19195],{"className":19196,"style":19197},[981],"margin-right:0.2778em;",[61,19199,2000],{"className":19200},[19201],"mrel",[61,19203],{"className":19204,"style":19197},[981],[61,19206,19208,19212,19253,19256,19260,19263],{"className":19207},[107],[61,19209],{"className":19210,"style":19211},[111],"height:1.8em;vertical-align:-0.65em;",[61,19213,19215,19218],{"className":19214},[116],[61,19216,19106],{"className":19217},[116,120],[61,19219,19221],{"className":19220},[125],[61,19222,19224,19245],{"className":19223},[129,130],[61,19225,19227,19242],{"className":19226},[134],[61,19228,19231],{"className":19229,"style":19230},[138],"height:0.3011em;",[61,19232,19233,19236],{"style":241},[61,19234],{"className":19235,"style":147},[146],[61,19237,19239],{"className":19238},[151,152,153,154],[61,19240,1495],{"className":19241},[116,154],[61,19243,170],{"className":19244},[169],[61,19246,19248],{"className":19247},[134],[61,19249,19251],{"className":19250,"style":177},[138],[61,19252],{},[61,19254],{"className":19255,"style":982},[981],[61,19257,6987],{"className":19258},[19259],"mop",[61,19261],{"className":19262,"style":982},[981],[61,19264,19267,19277,19280,19472],{"className":19265},[19266],"minner",[61,19268,19272],{"className":19269,"style":19271},[631,19270],"delimcenter","top:0em;",[61,19273,849],{"className":19274},[19275,19276],"delimsizing","size2",[61,19278,19132],{"className":19279},[116],[61,19281,19283,19287,19469],{"className":19282},[116],[61,19284],{"className":19285},[631,19286],"nulldelimiter",[61,19288,19290],{"className":19289},[19134],[61,19291,19293,19460],{"className":19292},[129,130],[61,19294,19296,19457],{"className":19295},[134],[61,19297,19300,19350,19361],{"className":19298,"style":19299},[138],"height:1.1089em;",[61,19301,19303,19307],{"style":19302},"top:-2.655em;",[61,19304],{"className":19305,"style":19306},[146],"height:3em;",[61,19308,19310],{"className":19309},[151,152,153,154],[61,19311,19313,19316],{"className":19312},[116,154],[61,19314,337],{"className":19315},[116,154],[61,19317,19319,19322],{"className":19318},[116,154],[61,19320,19165],{"className":19321,"style":121},[116,120,154],[61,19323,19325],{"className":19324},[125],[61,19326,19328],{"className":19327},[129],[61,19329,19331],{"className":19330},[134],[61,19332,19335],{"className":19333,"style":19334},[138],"height:0.7463em;",[61,19336,19338,19342],{"style":19337},"top:-2.786em;margin-right:0.0714em;",[61,19339],{"className":19340,"style":19341},[146],"height:2.5em;",[61,19343,19347],{"className":19344},[151,19345,19346,154],"reset-size3","size1",[61,19348,337],{"className":19349},[116,154],[61,19351,19353,19356],{"style":19352},"top:-3.23em;",[61,19354],{"className":19355,"style":19306},[146],[61,19357],{"className":19358,"style":19360},[19359],"frac-line","border-bottom-width:0.04em;",[61,19362,19364,19367],{"style":19363},"top:-3.485em;",[61,19365],{"className":19366,"style":19306},[146],[61,19368,19370],{"className":19369},[151,152,153,154],[61,19371,19373,19376,19379,19383,19426],{"className":19372},[116,154],[61,19374,849],{"className":19375},[631,154],[61,19377,206],{"className":19378},[116,120,154],[61,19380,19132],{"className":19381},[19382,154],"mbin",[61,19384,19386,19389],{"className":19385},[116,154],[61,19387,206],{"className":19388},[116,120,154],[61,19390,19392],{"className":19391},[125],[61,19393,19395,19417],{"className":19394},[129,130],[61,19396,19398,19414],{"className":19397},[134],[61,19399,19402],{"className":19400,"style":19401},[138],"height:0.3173em;",[61,19403,19405,19408],{"style":19404},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[61,19406],{"className":19407,"style":19341},[146],[61,19409,19411],{"className":19410},[151,19345,19346,154],[61,19412,1495],{"className":19413},[116,154],[61,19415,170],{"className":19416},[169],[61,19418,19420],{"className":19419},[134],[61,19421,19424],{"className":19422,"style":19423},[138],"height:0.143em;",[61,19425],{},[61,19427,19429,19432],{"className":19428},[730,154],[61,19430,300],{"className":19431},[730,154],[61,19433,19435],{"className":19434},[125],[61,19436,19438],{"className":19437},[129],[61,19439,19441],{"className":19440},[134],[61,19442,19445],{"className":19443,"style":19444},[138],"height:0.8913em;",[61,19446,19448,19451],{"style":19447},"top:-2.931em;margin-right:0.0714em;",[61,19449],{"className":19450,"style":19341},[146],[61,19452,19454],{"className":19453},[151,19345,19346,154],[61,19455,337],{"className":19456},[116,154],[61,19458,170],{"className":19459},[169],[61,19461,19463],{"className":19462},[134],[61,19464,19467],{"className":19465,"style":19466},[138],"height:0.345em;",[61,19468],{},[61,19470],{"className":19471},[730,19286],[61,19473,19475],{"className":19474,"style":19271},[730,19270],[61,19476,300],{"className":19477},[19275,19276],"\nwhere:",[1640,19480,19481,19564,19647],{},[1643,19482,19483,19563],{},[61,19484,19486,19508],{"className":19485},[64],[61,19487,19489],{"className":19488},[68],[70,19490,19491],{"xmlns":72},[74,19492,19493,19505],{},[77,19494,19495,19497,19503],{},[586,19496,849],{"stretchy":588},[80,19498,19499,19501],{},[83,19500,19106],{},[1493,19502,1495],{},[586,19504,300],{"stretchy":588},[95,19506,19507],{"encoding":97},"( A_0 )",[61,19509,19511],{"className":19510,"ariaHidden":103},[102],[61,19512,19514,19517,19520,19560],{"className":19513},[107],[61,19515],{"className":19516,"style":627},[111],[61,19518,849],{"className":19519},[631],[61,19521,19523,19526],{"className":19522},[116],[61,19524,19106],{"className":19525},[116,120],[61,19527,19529],{"className":19528},[125],[61,19530,19532,19552],{"className":19531},[129,130],[61,19533,19535,19549],{"className":19534},[134],[61,19536,19538],{"className":19537,"style":19230},[138],[61,19539,19540,19543],{"style":241},[61,19541],{"className":19542,"style":147},[146],[61,19544,19546],{"className":19545},[151,152,153,154],[61,19547,1495],{"className":19548},[116,154],[61,19550,170],{"className":19551},[169],[61,19553,19555],{"className":19554},[134],[61,19556,19558],{"className":19557,"style":177},[138],[61,19559],{},[61,19561,300],{"className":19562},[730]," is the peak amplitude,",[1643,19565,19566,19646],{},[61,19567,19569,19591],{"className":19568},[64],[61,19570,19572],{"className":19571},[68],[70,19573,19574],{"xmlns":72},[74,19575,19576,19588],{},[77,19577,19578,19580,19586],{},[586,19579,849],{"stretchy":588},[80,19581,19582,19584],{},[83,19583,206],{},[1493,19585,1495],{},[586,19587,300],{"stretchy":588},[95,19589,19590],{"encoding":97},"( t_0 )",[61,19592,19594],{"className":19593,"ariaHidden":103},[102],[61,19595,19597,19600,19603,19643],{"className":19596},[107],[61,19598],{"className":19599,"style":627},[111],[61,19601,849],{"className":19602},[631],[61,19604,19606,19609],{"className":19605},[116],[61,19607,206],{"className":19608},[116,120],[61,19610,19612],{"className":19611},[125],[61,19613,19615,19635],{"className":19614},[129,130],[61,19616,19618,19632],{"className":19617},[134],[61,19619,19621],{"className":19620,"style":19230},[138],[61,19622,19623,19626],{"style":241},[61,19624],{"className":19625,"style":147},[146],[61,19627,19629],{"className":19628},[151,152,153,154],[61,19630,1495],{"className":19631},[116,154],[61,19633,170],{"className":19634},[169],[61,19636,19638],{"className":19637},[134],[61,19639,19641],{"className":19640,"style":177},[138],[61,19642],{},[61,19644,300],{"className":19645},[730]," is the pulse center,",[1643,19648,19649,19688],{},[61,19650,19652,19670],{"className":19651},[64],[61,19653,19655],{"className":19654},[68],[70,19656,19657],{"xmlns":72},[74,19658,19659,19667],{},[77,19660,19661,19663,19665],{},[586,19662,849],{"stretchy":588},[83,19664,19165],{},[586,19666,300],{"stretchy":588},[95,19668,19669],{"encoding":97},"( \\sigma )",[61,19671,19673],{"className":19672,"ariaHidden":103},[102],[61,19674,19676,19679,19682,19685],{"className":19675},[107],[61,19677],{"className":19678,"style":627},[111],[61,19680,849],{"className":19681},[631],[61,19683,19165],{"className":19684,"style":121},[116,120],[61,19686,300],{"className":19687},[730]," determines the width of the pulse.",[1635,19690,19692],{"id":19691},"_2-modulated-pulse-for-pauli-x-gate",[814,19693,19694],{},"2. 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modulation",[61,19706,19708,19816],{"className":19707},[64],[61,19709,19711],{"className":19710},[68],[70,19712,19713],{"xmlns":72},[74,19714,19715,19813],{},[77,19716,19717,19724,19726,19728,19730,19732,19738,19740,19742,19782,19784,19786,19788,19790,19793,19800,19802,19804,19811],{},[80,19718,19719,19722],{},[83,19720,19721],{},"E",[83,19723,93],{},[586,19725,849],{"stretchy":588},[83,19727,206],{},[586,19729,300],{"stretchy":588},[586,19731,2000],{},[80,19733,19734,19736],{},[83,19735,19106],{},[1493,19737,1495],{},[83,19739,6987],{},[586,19741,19125],{},[77,19743,19744,19746,19748,19780],{},[586,19745,849],{"fence":103},[586,19747,19132],{},[19134,19749,19750,19770],{},[77,19751,19752,19754,19756,19758,19764],{},[586,19753,849],{"stretchy":588},[83,19755,206],{},[586,19757,19132],{},[80,19759,19760,19762],{},[83,19761,206],{},[1493,19763,1495],{},[19151,19765,19766,19768],{},[586,19767,300],{"stretchy":588},[1493,19769,337],{},[77,19771,19772,19774],{},[1493,19773,337],{},[19151,19775,19776,19778],{},[83,19777,19165],{},[1493,19779,337],{},[586,19781,300],{"fence":103},[83,19783,5051],{},[586,19785,19125],{},[586,19787,849],{"stretchy":588},[1493,19789,337],{},[83,19791,19792],{},"π",[80,19794,19795,19798],{},[83,19796,19797],{},"f",[83,19799,203],{},[83,19801,206],{},[586,19803,2419],{},[80,19805,19806,19809],{},[83,19807,19808],{},"ϕ",[83,19810,93],{},[586,19812,300],{"stretchy":588},[95,19814,19815],{"encoding":97},"E_X(t) 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f_c )",[61,20280,20282],{"className":20281,"ariaHidden":103},[102],[61,20283,20285,20288,20291,20331],{"className":20284},[107],[61,20286],{"className":20287,"style":627},[111],[61,20289,849],{"className":20290},[631],[61,20292,20294,20297],{"className":20293},[116],[61,20295,19797],{"className":20296,"style":20151},[116,120],[61,20298,20300],{"className":20299},[125],[61,20301,20303,20323],{"className":20302},[129,130],[61,20304,20306,20320],{"className":20305},[134],[61,20307,20309],{"className":20308,"style":20164},[138],[61,20310,20311,20314],{"style":20167},[61,20312],{"className":20313,"style":147},[146],[61,20315,20317],{"className":20316},[151,152,153,154],[61,20318,203],{"className":20319},[116,120,154],[61,20321,170],{"className":20322},[169],[61,20324,20326],{"className":20325},[134],[61,20327,20329],{"className":20328,"style":177},[138],[61,20330],{},[61,20332,300],{"className":20333},[730]," is the carrier frequency (typically GHz range),",[1643,20336,20337,20439,20440,332],{},[61,20338,20340,20366],{"className":20339},[64],[61,20341,20343],{"className":20342},[68],[70,20344,20345],{"xmlns":72},[74,20346,20347,20363],{},[77,20348,20349,20351,20357,20359,20361],{},[586,20350,849],{"stretchy":588},[80,20352,20353,20355],{},[83,20354,19808],{},[83,20356,93],{},[586,20358,2000],{},[1493,20360,1495],{},[586,20362,300],{"stretchy":588},[95,20364,20365],{"encoding":97},"( \\phi_X = 0 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for an ",[814,20441,20442],{},"X gate",[1643,20444,20445,20446,20449],{},"The modulation is centered at the qubit frequency with ",[814,20447,20448],{},"zero phase shift",", leading to a cosine-like oscillation.",[1635,20451,20453],{"id":20452},"_3-modulated-pulse-for-pauli-y-gate",[814,20454,20455],{},"3. 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= A_0 \\exp\\left(-\\frac{(t - t_0)^2}{2\\sigma^2}\\right) \\cos(2\\pi f_c t + \\phi_Y)",[61,20620,20622,20686,20998],{"className":20621,"ariaHidden":103},[102],[61,20623,20625,20628,20668,20671,20674,20677,20680,20683],{"className":20624},[107],[61,20626],{"className":20627,"style":627},[111],[61,20629,20631,20634],{"className":20630},[116],[61,20632,19721],{"className":20633,"style":19831},[116,120],[61,20635,20637],{"className":20636},[125],[61,20638,20640,20660],{"className":20639},[129,130],[61,20641,20643,20657],{"className":20642},[134],[61,20644,20646],{"className":20645,"style":139},[138],[61,20647,20648,20651],{"style":19846},[61,20649],{"className":20650,"style":147},[146],[61,20652,20654],{"className":20653},[151,152,153,154],[61,20655,20529],{"className":20656,"style":20194},[116,120,154],[61,20658,170],{"className":20659},[169],[61,20661,20663],{"className":20662},[134],[61,20664,20666],{"className":20665,"style":177},[138],[61,20667],{},[61,20669,849],{"className":20670},[631],[61,20672,206],{"className":20673},[116,120],[61,20675,300],{"className":20676},[730],[61,20678],{"className":20679,"style":19197},[981],[61,20681,2000],{"className":20682},[19201],[61,20684],{"className":20685,"style":19197},[981],[61,20687,20689,20692,20732,20735,20738,20741,20931,20934,20937,20940,20943,20946,20986,20989,20992,20995],{"className":20688},[107],[61,20690],{"className":20691,"style":19211},[111],[61,20693,20695,20698],{"className":20694},[116],[61,20696,19106],{"className":20697},[116,120],[61,20699,20701],{"className":20700},[125],[61,20702,20704,20724],{"className":20703},[129,130],[61,20705,20707,20721],{"className":20706},[134],[61,20708,20710],{"className":20709,"style":19230},[138],[61,20711,20712,20715],{"style":241},[61,20713],{"className":20714,"style":147},[146],[61,20716,20718],{"className":20717},[151,152,153,154],[61,20719,1495],{"className":20720},[116,154],[61,20722,170],{"className":20723},[169],[61,20725,20727],{"className":20726},[134],[61,20728,20730],{"className":20729,"style":177},[138],[61,20731],{},[61,20733],{"className":20734,"style":982},[981],[61,20736,6987],{"className":20737},[19259],[61,20739],{"className":20740,"style":982},[981],[61,20742,20744,20750,20753,20925],{"className":20743},[19266],[61,20745,20747],{"className":20746,"style":19271},[631,19270],[61,20748,849],{"className":20749},[19275,19276],[61,20751,19132],{"className":20752},[116],[61,20754,20756,20759,20922],{"className":20755},[116],[61,20757],{"className":20758},[631,19286],[61,20760,20762],{"className":20761},[19134],[61,20763,20765,20914],{"className":20764},[129,130],[61,20766,20768,20911],{"className":20767},[134],[61,20769,20771,20814,20822],{"className":20770,"style":19299},[138],[61,20772,20773,20776],{"style":19302},[61,20774],{"className":20775,"style":19306},[146],[61,20777,20779],{"className":20778},[151,152,153,154],[61,20780,20782,20785],{"className":20781},[116,154],[61,20783,337],{"className":20784},[116,154],[61,20786,20788,20791],{"className":20787},[116,154],[61,20789,19165],{"className":20790,"style":121},[116,120,154],[61,20792,20794],{"className":20793},[125],[61,20795,20797],{"className":20796},[129],[61,20798,20800],{"className":20799},[134],[61,20801,20803],{"className":20802,"style":19334},[138],[61,20804,20805,20808],{"style":19337},[61,20806],{"className":20807,"style":19341},[146],[61,20809,20811],{"className":20810},[151,19345,19346,154],[61,20812,337],{"className":20813},[116,154],[61,20815,20816,20819],{"style":19352},[61,20817],{"className":20818,"style":19306},[146],[61,20820],{"className":20821,"style":19360},[19359],[61,20823,20824,20827],{"style":19363},[61,20825],{"className":20826,"style":19306},[146],[61,20828,20830],{"className":20829},[151,152,153,154],[61,20831,20833,20836,20839,20842,20882],{"className":20832},[116,154],[61,20834,849],{"className":20835},[631,154],[61,20837,206],{"className":20838},[116,120,154],[61,20840,19132],{"className":20841},[19382,154],[61,20843,20845,20848],{"className":20844},[116,154],[61,20846,206],{"className":20847},[116,120,154],[61,20849,20851],{"className":20850},[125],[61,20852,20854,20874],{"className":20853},[129,130],[61,20855,20857,20871],{"className":20856},[134],[61,20858,20860],{"className":20859,"style":19401},[138],[61,20861,20862,20865],{"style":19404},[61,20863],{"className":20864,"style":19341},[146],[61,20866,20868],{"className":20867},[151,19345,19346,154],[61,20869,1495],{"className":20870},[116,154],[61,20872,170],{"className":20873},[169],[61,20875,20877],{"className":20876},[134],[61,20878,20880],{"className":20879,"style":19423},[138],[61,20881],{},[61,20883,20885,20888],{"className":20884},[730,154],[61,20886,300],{"className":20887},[730,154],[61,20889,20891],{"className":20890},[125],[61,20892,20894],{"className":20893},[129],[61,20895,20897],{"className":20896},[134],[61,20898,20900],{"className":20899,"style":19444},[138],[61,20901,20902,20905],{"style":19447},[61,20903],{"className":20904,"style":19341},[146],[61,20906,20908],{"className":20907},[151,19345,19346,154],[61,20909,337],{"className":20910},[116,154],[61,20912,170],{"className":20913},[169],[61,20915,20917],{"className":20916},[134],[61,20918,20920],{"className":20919,"style":19466},[138],[61,20921],{},[61,20923],{"className":20924},[730,19286],[61,20926,20928],{"className":20927,"style":19271},[730,19270],[61,20929,300],{"className":20930},[19275,19276],[61,20932],{"className":20933,"style":982},[981],[61,20935,5051],{"className":20936},[19259],[61,20938,849],{"className":20939},[631],[61,20941,337],{"className":20942},[116],[61,20944,19792],{"className":20945,"style":121},[116,120],[61,20947,20949,20952],{"className":20948},[116],[61,20950,19797],{"className":20951,"style":20151},[116,120],[61,20953,20955],{"className":20954},[125],[61,20956,20958,20978],{"className":20957},[129,130],[61,20959,20961,20975],{"className":20960},[134],[61,20962,20964],{"className":20963,"style":20164},[138],[61,20965,20966,20969],{"style":20167},[61,20967],{"className":20968,"style":147},[146],[61,20970,20972],{"className":20971},[151,152,153,154],[61,20973,203],{"className":20974},[116,120,154],[61,20976,170],{"className":20977},[169],[61,20979,20981],{"className":20980},[134],[61,20982,20984],{"className":20983,"style":177},[138],[61,20985],{},[61,20987,206],{"className":20988},[116,120],[61,20990],{"className":20991,"style":20194},[981],[61,20993,2419],{"className":20994},[19382],[61,20996],{"className":20997,"style":20194},[981],[61,20999,21001,21004,21044],{"className":21000},[107],[61,21002],{"className":21003,"style":627},[111],[61,21005,21007,21010],{"className":21006},[116],[61,21008,19808],{"className":21009},[116,120],[61,21011,21013],{"className":21012},[125],[61,21014,21016,21036],{"className":21015},[129,130],[61,21017,21019,21033],{"className":21018},[134],[61,21020,21022],{"className":21021,"style":139},[138],[61,21023,21024,21027],{"style":241},[61,21025],{"className":21026,"style":147},[146],[61,21028,21030],{"className":21029},[151,152,153,154],[61,21031,20529],{"className":21032,"style":20194},[116,120,154],[61,21034,170],{"className":21035},[169],[61,21037,21039],{"className":21038},[134],[61,21040,21042],{"className":21041,"style":177},[138],[61,21043],{},[61,21045,300],{"className":21046},[730],[1640,21048,21049,21226],{},[1643,21050,21051,21225],{},[61,21052,21054,21084],{"className":21053},[64],[61,21055,21057],{"className":21056},[68],[70,21058,21059],{"xmlns":72},[74,21060,21061,21081],{},[77,21062,21063,21065,21071,21073,21079],{},[586,21064,849],{"stretchy":588},[80,21066,21067,21069],{},[83,21068,19808],{},[83,21070,20529],{},[586,21072,2000],{},[19134,21074,21075,21077],{},[83,21076,19792],{},[1493,21078,337],{},[586,21080,300],{"stretchy":588},[95,21082,21083],{"encoding":97},"( \\phi_Y = \\frac{\\pi}{2} )",[61,21085,21087,21145],{"className":21086,"ariaHidden":103},[102],[61,21088,21090,21093,21096,21136,21139,21142],{"className":21089},[107],[61,21091],{"className":21092,"style":627},[111],[61,21094,849],{"className":21095},[631],[61,21097,21099,21102],{"className":21098},[116],[61,21100,19808],{"className":21101},[116,120],[61,21103,21105],{"className":21104},[125],[61,21106,21108,21128],{"className":21107},[129,130],[61,21109,21111,21125],{"className":21110},[134],[61,21112,21114],{"className":21113,"style":139},[138],[61,21115,21116,21119],{"style":241},[61,21117],{"className":21118,"style":147},[146],[61,21120,21122],{"className":21121},[151,152,153,154],[61,21123,20529],{"className":21124,"style":20194},[116,120,154],[61,21126,170],{"className":21127},[169],[61,21129,21131],{"className":21130},[134],[61,21132,21134],{"className":21133,"style":177},[138],[61,21135],{},[61,21137],{"className":21138,"style":19197},[981],[61,21140,2000],{"className":21141},[19201],[61,21143],{"className":21144,"style":19197},[981],[61,21146,21148,21152,21222],{"className":21147},[107],[61,21149],{"className":21150,"style":21151},[111],"height:1.095em;vertical-align:-0.345em;",[61,21153,21155,21158,21219],{"className":21154},[116],[61,21156],{"className":21157},[631,19286],[61,21159,21161],{"className":21160},[19134],[61,21162,21164,21211],{"className":21163},[129,130],[61,21165,21167,21208],{"className":21166},[134],[61,21168,21171,21185,21193],{"className":21169,"style":21170},[138],"height:0.6954em;",[61,21172,21173,21176],{"style":19302},[61,21174],{"className":21175,"style":19306},[146],[61,21177,21179],{"className":21178},[151,152,153,154],[61,21180,21182],{"className":21181},[116,154],[61,21183,337],{"className":21184},[116,154],[61,21186,21187,21190],{"style":19352},[61,21188],{"className":21189,"style":19306},[146],[61,21191],{"className":21192,"style":19360},[19359],[61,21194,21196,21199],{"style":21195},"top:-3.394em;",[61,21197],{"className":21198,"style":19306},[146],[61,21200,21202],{"className":21201},[151,152,153,154],[61,21203,21205],{"className":21204},[116,154],[61,21206,19792],{"className":21207,"style":121},[116,120,154],[61,21209,170],{"className":21210},[169],[61,21212,21214],{"className":21213},[134],[61,21215,21217],{"className":21216,"style":19466},[138],[61,21218],{},[61,21220],{"className":21221},[730,19286],[61,21223,300],{"className":21224},[730]," shifts the phase by 90° (producing a sine-like waveform).",[1643,21227,21228,21229,21278],{},"The modulation has a ",[814,21230,21231,21277],{},[61,21232,21234,21256],{"className":21233},[64],[61,21235,21237],{"className":21236},[68],[70,21238,21239],{"xmlns":72},[74,21240,21241,21253],{},[77,21242,21243,21245,21247,21249,21251],{},[586,21244,849],{"stretchy":588},[83,21246,19792],{},[83,21248,602],{"mathvariant":197},[1493,21250,337],{},[586,21252,300],{"stretchy":588},[95,21254,21255],{"encoding":97},"( \\pi\u002F2 )",[61,21257,21259],{"className":21258,"ariaHidden":103},[102],[61,21260,21262,21265,21268,21271,21274],{"className":21261},[107],[61,21263],{"className":21264,"style":627},[111],[61,21266,849],{"className":21267},[631],[61,21269,19792],{"className":21270,"style":121},[116,120],[61,21272,20506],{"className":21273},[116],[61,21275,300],{"className":21276},[730]," phase shift",", which shifts the waveform by a quarter cycle, making it a sine-like oscillation.",[1635,21280,21282],{"id":21281},"_4-why-no-pulse-for-pauli-z-gate",[814,21283,21284],{},"4. Why No Pulse for Pauli-Z Gate?",[12,21286,21287,21288,21330,21331,1850,21334,21510],{},"For ",[814,21289,21290,21291],{},"Pauli-Z ",[61,21292,21294,21312],{"className":21293},[64],[61,21295,21297],{"className":21296},[68],[70,21298,21299],{"xmlns":72},[74,21300,21301,21309],{},[77,21302,21303,21305,21307],{},[586,21304,849],{"stretchy":588},[83,21306,90],{},[586,21308,300],{"stretchy":588},[95,21310,21311],{"encoding":97},"( Z )",[61,21313,21315],{"className":21314,"ariaHidden":103},[102],[61,21316,21318,21321,21324,21327],{"className":21317},[107],[61,21319],{"className":21320,"style":627},[111],[61,21322,849],{"className":21323},[631],[61,21325,90],{"className":21326,"style":161},[116,120],[61,21328,300],{"className":21329},[730],", no physical pulse is needed because it is implemented as a ",[814,21332,21333],{},"virtual phase shift",[61,21335,21337,21396],{"className":21336},[64],[61,21338,21340],{"className":21339},[68],[70,21341,21342],{"xmlns":72},[74,21343,21344,21393],{},[77,21345,21346,21348,21350,21352,21355,21357,21359,21361,21363,21366,21368,21370,21372,21374,21387,21389,21391],{},[83,21347,589],{"mathvariant":197},[1493,21349,1495],{},[586,21351,1498],{"stretchy":588},[586,21353,21354],{},"→",[83,21356,589],{"mathvariant":197},[1493,21358,1495],{},[586,21360,1498],{"stretchy":588},[586,21362,862],{"separator":103},[981,21364],{"width":21365},"1em",[83,21367,589],{"mathvariant":197},[1493,21369,57],{},[586,21371,1498],{"stretchy":588},[586,21373,21354],{},[19151,21375,21376,21379],{},[83,21377,21378],{},"e",[77,21380,21381,21384],{},[83,21382,21383],{},"i",[83,21385,21386],{},"θ",[83,21388,589],{"mathvariant":197},[1493,21390,57],{},[586,21392,1498],{"stretchy":588},[95,21394,21395],{"encoding":97},"|0\\rangle \\rightarrow |0\\rangle, \\quad |1\\rangle \\rightarrow e^{i\\theta}|1\\rangle",[61,21397,21399,21421,21459],{"className":21398,"ariaHidden":103},[102],[61,21400,21402,21405,21409,21412,21415,21418],{"className":21401},[107],[61,21403],{"className":21404,"style":627},[111],[61,21406,21408],{"className":21407},[116],"∣0",[61,21410,1498],{"className":21411},[730],[61,21413],{"className":21414,"style":19197},[981],[61,21416,21354],{"className":21417},[19201],[61,21419],{"className":21420,"style":19197},[981],[61,21422,21424,21427,21430,21433,21436,21440,21443,21447,21450,21453,21456],{"className":21423},[107],[61,21425],{"className":21426,"style":627},[111],[61,21428,21408],{"className":21429},[116],[61,21431,1498],{"className":21432},[730],[61,21434,862],{"className":21435},[977],[61,21437],{"className":21438,"style":21439},[981],"margin-right:1em;",[61,21441],{"className":21442,"style":982},[981],[61,21444,21446],{"className":21445},[116],"∣1",[61,21448,1498],{"className":21449},[730],[61,21451],{"className":21452,"style":19197},[981],[61,21454,21354],{"className":21455},[19201],[61,21457],{"className":21458,"style":19197},[981],[61,21460,21462,21466,21504,21507],{"className":21461},[107],[61,21463],{"className":21464,"style":21465},[111],"height:1.0991em;vertical-align:-0.25em;",[61,21467,21469,21472],{"className":21468},[116],[61,21470,21378],{"className":21471},[116,120],[61,21473,21475],{"className":21474},[125],[61,21476,21478],{"className":21477},[129],[61,21479,21481],{"className":21480},[134],[61,21482,21485],{"className":21483,"style":21484},[138],"height:0.8491em;",[61,21486,21488,21491],{"style":21487},"top:-3.063em;margin-right:0.05em;",[61,21489],{"className":21490,"style":147},[146],[61,21492,21494],{"className":21493},[151,152,153,154],[61,21495,21497,21500],{"className":21496},[116,154],[61,21498,21383],{"className":21499},[116,120,154],[61,21501,21386],{"className":21502,"style":21503},[116,120,154],"margin-right:0.0278em;",[61,21505,21446],{"className":21506},[116],[61,21508,1498],{"className":21509},[730],"\nThis shift is typically done in software rather than by applying a physical pulse.",[1632,21512],{},[12,21514,21515,21516,19082],{},"Relevant equations in table form, for the ",[814,21517,19081],{},[30,21519,21520,21530],{},[33,21521,21522],{},[36,21523,21524,21527],{},[39,21525,21526],{},"Gate",[39,21528,21529],{},"Equation",[50,21531,21532,21890,22433,22976,23521],{},[36,21533,21534,21539],{},[55,21535,21536],{},[814,21537,21538],{},"Gaussian 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Target Qubit 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61,23936,23938],{"className":23937},[129],[61,23939,23941],{"className":23940},[134],[61,23942,23944],{"className":23943,"style":19444},[138],[61,23945,23946,23949],{"style":19447},[61,23947],{"className":23948,"style":19341},[146],[61,23950,23952],{"className":23951},[151,19345,19346,154],[61,23953,337],{"className":23954},[116,154],[61,23956,170],{"className":23957},[169],[61,23959,23961],{"className":23960},[134],[61,23962,23964],{"className":23963,"style":19466},[138],[61,23965],{},[61,23967],{"className":23968},[730,19286],[61,23970,23972],{"className":23971,"style":19271},[730,19270],[61,23973,300],{"className":23974},[19275,19276],[61,23976],{"className":23977,"style":982},[981],[61,23979,5051],{"className":23980},[19259],[61,23982,849],{"className":23983},[631],[61,23985,337],{"className":23986},[116],[61,23988,19792],{"className":23989,"style":121},[116,120],[61,23991,23993,23996],{"className":23992},[116],[61,23994,19797],{"className":23995,"style":20151},[116,120],[61,23997,23999],{"className":23998},[125],[61,24000,24002,24022],{"className":24001},[129,130],[61,24003,24005,24019],{"className":24004},[134],[61,24006,24008],{"className":24007,"style":20164},[138],[61,24009,24010,24013],{"style":20167},[61,24011],{"className":24012,"style":147},[146],[61,24014,24016],{"className":24015},[151,152,153,154],[61,24017,203],{"className":24018},[116,120,154],[61,24020,170],{"className":24021},[169],[61,24023,24025],{"className":24024},[134],[61,24026,24028],{"className":24027,"style":177},[138],[61,24029],{},[61,24031,206],{"className":24032},[116,120],[61,24034],{"className":24035,"style":20194},[981],[61,24037,2419],{"className":24038},[19382],[61,24040],{"className":24041,"style":20194},[981],[61,24043,24045,24048,24088],{"className":24044},[107],[61,24046],{"className":24047,"style":627},[111],[61,24049,24051,24054],{"className":24050},[116],[61,24052,19808],{"className":24053},[116,120],[61,24055,24057],{"className":24056},[125],[61,24058,24060,24080],{"className":24059},[129,130],[61,24061,24063,24077],{"className":24062},[134],[61,24064,24066],{"className":24065,"style":139},[138],[61,24067,24068,24071],{"style":241},[61,24069],{"className":24070,"style":147},[146],[61,24072,24074],{"className":24073},[151,152,153,154],[61,24075,93],{"className":24076,"style":165},[116,120,154],[61,24078,170],{"className":24079},[169],[61,24081,24083],{"className":24082},[134],[61,24084,24086],{"className":24085,"style":177},[138],[61,24087],{},[61,24089,300],{"className":24090},[730],[1720,24092,24094],{"className":1722,"code":24093,"language":1724,"meta":1725,"style":1725},"import numpy as np\nimport matplotlib.pyplot as plt\n\n# Define parameters\nt0 = 0      # Pulse center\nsigma = 5   # Standard deviation of Gaussian modulation function (ns)\nA0 = 1      # Peak amplitude\n\n# Increase the number of points by 1.5x for denser lollipop visualization\nnum_points = int(40 * 1.5)\nt = np.linspace(-20, 20, num_points)  # Adjusted time axis\n\n# Compute Gaussian modulation function\ngaussian_modulation = A0 * np.exp(-(t - t0)**2 \u002F (2 * sigma**2))\n\n# Store modulation functions and titles in an array\nmodulations = [\n    (gaussian_modulation, \"Gaussian Modulation Function for Pauli-X Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for Pauli-Y Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for Hadamard Gate\"),\n    (gaussian_modulation, \"Gaussian Modulation Function for CNOT Gate Target Qubit\")\n]\n\n# Create a figure with constrained layout\nfig, axs = plt.subplots(2, 2, figsize=(12, 8), constrained_layout=True)\naxs = axs.flatten()\n\n# Lollipop plot setup\nfor ax, (modulation, title) in zip(axs, modulations):\n    ax.stem(t, modulation, linefmt='b-', markerfmt='bo', basefmt='k-')\n    ax.set_title(title)\n    ax.set_xlabel(\"Time (ns)\")\n    ax.set_ylabel(\"Amplitude\")\n\n# Memory Addressing with Lollipop Representation\nfig_combined, ax_combined = plt.subplots(figsize=(12, 6), constrained_layout=True)\n\n# Memory addresses as an index range\nmemory_addresses = np.arange(len(t))\n\n# Lollipop-style plot for memory addressing\nax_combined.stem(memory_addresses, gaussian_modulation, linefmt='b-', markerfmt='bo', basefmt='k-')\n\n# Labels and title\nax_combined.set_title(\"Memory Addressing with Lollipop Representation for Quantum Gate Control\")\nax_combined.set_xlabel(\"Memory Address Index\")\nax_combined.set_ylabel(\"Amplitude\")\nax_combined.grid(True)\n\n# Display plots\nplt.show()\n",[1727,24095,24096,24106,24116,24120,24125,24138,24150,24162,24166,24171,24192,24218,24222,24227,24275,24279,24284,24293,24303,24312,24321,24330,24334,24338,24343,24388,24402,24406,24411,24426,24459,24467,24480,24492,24496,24501,24536,24540,24545,24562,24566,24571,24605,24609,24614,24627,24640,24652,24664,24668,24673],{"__ignoreMap":1725},[61,24097,24098,24100,24102,24104],{"class":1731,"line":1732},[61,24099,1757],{"class":1749},[61,24101,1789],{"class":1753},[61,24103,1792],{"class":1749},[61,24105,1795],{"class":1753},[61,24107,24108,24110,24112,24114],{"class":1731,"line":1739},[61,24109,1757],{"class":1749},[61,24111,1803],{"class":1753},[61,24113,1792],{"class":1749},[61,24115,1808],{"class":1753},[61,24117,24118],{"class":1731,"line":1746},[61,24119,1743],{"emptyLinePlaceholder":1742},[61,24121,24122],{"class":1731,"line":1763},[61,24123,24124],{"class":1735},"# Define parameters\n",[61,24126,24127,24130,24132,24135],{"class":1731,"line":1776},[61,24128,24129],{"class":1753},"t0 ",[61,24131,2000],{"class":1859},[61,24133,24134],{"class":1866}," 0",[61,24136,24137],{"class":1735},"      # Pulse center\n",[61,24139,24140,24143,24145,24147],{"class":1731,"line":1784},[61,24141,24142],{"class":1753},"sigma ",[61,24144,2000],{"class":1859},[61,24146,15072],{"class":1866},[61,24148,24149],{"class":1735},"   # Standard deviation of Gaussian modulation function (ns)\n",[61,24151,24152,24155,24157,24159],{"class":1731,"line":1798},[61,24153,24154],{"class":1753},"A0 ",[61,24156,2000],{"class":1859},[61,24158,5266],{"class":1866},[61,24160,24161],{"class":1735},"      # Peak amplitude\n",[61,24163,24164],{"class":1731,"line":1811},[61,24165,1743],{"emptyLinePlaceholder":1742},[61,24167,24168],{"class":1731,"line":1816},[61,24169,24170],{"class":1735},"# Increase the number of points by 1.5x for denser lollipop visualization\n",[61,24172,24173,24176,24178,24181,24183,24185,24187,24190],{"class":1731,"line":1822},[61,24174,24175],{"class":1753},"num_points ",[61,24177,2000],{"class":1859},[61,24179,24180],{"class":1859}," int",[61,24182,849],{"class":1753},[61,24184,14795],{"class":1866},[61,24186,2298],{"class":1859},[61,24188,24189],{"class":1866}," 1.5",[61,24191,2016],{"class":1753},[61,24193,24194,24196,24198,24200,24202,24204,24206,24208,24210,24212,24215],{"class":1731,"line":1828},[61,24195,16704],{"class":1753},[61,24197,2000],{"class":1859},[61,24199,2193],{"class":1753},[61,24201,3453],{"class":1836},[61,24203,849],{"class":1753},[61,24205,2006],{"class":1859},[61,24207,14756],{"class":1866},[61,24209,1435],{"class":1753},[61,24211,14756],{"class":1866},[61,24213,24214],{"class":1753},", num_points)  ",[61,24216,24217],{"class":1735},"# Adjusted time axis\n",[61,24219,24220],{"class":1731,"line":1833},[61,24221,1743],{"emptyLinePlaceholder":1742},[61,24223,24224],{"class":1731,"line":1840},[61,24225,24226],{"class":1735},"# Compute Gaussian modulation function\n",[61,24228,24229,24231,24233,24236,24238,24240,24242,24244,24246,24249,24251,24254,24256,24258,24260,24262,24264,24266,24269,24271,24273],{"class":1731,"line":1853},[61,24230,14274],{"class":1753},[61,24232,2000],{"class":1859},[61,24234,24235],{"class":1753}," A0 ",[61,24237,2201],{"class":1859},[61,24239,2193],{"class":1753},[61,24241,6987],{"class":1836},[61,24243,849],{"class":1753},[61,24245,2006],{"class":1859},[61,24247,24248],{"class":1753},"(t ",[61,24250,2006],{"class":1859},[61,24252,24253],{"class":1753}," t0)",[61,24255,2651],{"class":1859},[61,24257,337],{"class":1866},[61,24259,2618],{"class":1859},[61,24261,2003],{"class":1753},[61,24263,337],{"class":1866},[61,24265,2298],{"class":1859},[61,24267,24268],{"class":1753}," sigma",[61,24270,2651],{"class":1859},[61,24272,337],{"class":1866},[61,24274,5546],{"class":1753},[61,24276,24277],{"class":1731,"line":1873},[61,24278,1743],{"emptyLinePlaceholder":1742},[61,24280,24281],{"class":1731,"line":1889},[61,24282,24283],{"class":1735},"# Store modulation functions and titles in an array\n",[61,24285,24286,24288,24290],{"class":1731,"line":1908},[61,24287,17548],{"class":1753},[61,24289,2000],{"class":1859},[61,24291,24292],{"class":1753}," [\n",[61,24294,24295,24298,24301],{"class":1731,"line":1925},[61,24296,24297],{"class":1753},"    (gaussian_modulation, ",[61,24299,24300],{"class":2165},"\"Gaussian Modulation Function for Pauli-X Gate\"",[61,24302,16975],{"class":1753},[61,24304,24305,24307,24310],{"class":1731,"line":1941},[61,24306,24297],{"class":1753},[61,24308,24309],{"class":2165},"\"Gaussian Modulation Function for Pauli-Y Gate\"",[61,24311,16975],{"class":1753},[61,24313,24314,24316,24319],{"class":1731,"line":1957},[61,24315,24297],{"class":1753},[61,24317,24318],{"class":2165},"\"Gaussian Modulation Function for Hadamard Gate\"",[61,24320,16975],{"class":1753},[61,24322,24323,24325,24328],{"class":1731,"line":1974},[61,24324,24297],{"class":1753},[61,24326,24327],{"class":2165},"\"Gaussian Modulation Function for CNOT Gate Target Qubit\"",[61,24329,2016],{"class":1753},[61,24331,24332],{"class":1731,"line":1979},[61,24333,5701],{"class":1753},[61,24335,24336],{"class":1731,"line":1985},[61,24337,1743],{"emptyLinePlaceholder":1742},[61,24339,24340],{"class":1731,"line":2019},[61,24341,24342],{"class":1735},"# Create a figure with constrained layout\n",[61,24344,24345,24348,24350,24352,24354,24356,24358,24360,24362,24364,24366,24368,24370,24372,24374,24376,24378,24381,24383,24386],{"class":1731,"line":2033},[61,24346,24347],{"class":1753},"fig, axs ",[61,24349,2000],{"class":1859},[61,24351,17065],{"class":1753},[61,24353,17068],{"class":1836},[61,24355,849],{"class":1753},[61,24357,337],{"class":1866},[61,24359,1435],{"class":1753},[61,24361,337],{"class":1866},[61,24363,1435],{"class":1753},[61,24365,3624],{"class":3402},[61,24367,2000],{"class":1859},[61,24369,849],{"class":1753},[61,24371,17087],{"class":1866},[61,24373,1435],{"class":1753},[61,24375,3631],{"class":1866},[61,24377,3638],{"class":1753},[61,24379,24380],{"class":3402},"constrained_layout",[61,24382,2000],{"class":1859},[61,24384,24385],{"class":1866},"True",[61,24387,2016],{"class":1753},[61,24389,24390,24393,24395,24398,24400],{"class":1731,"line":2061},[61,24391,24392],{"class":1753},"axs ",[61,24394,2000],{"class":1859},[61,24396,24397],{"class":1753}," axs.",[61,24399,17107],{"class":1836},[61,24401,2089],{"class":1753},[61,24403,24404],{"class":1731,"line":2074},[61,24405,1743],{"emptyLinePlaceholder":1742},[61,24407,24408],{"class":1731,"line":2079},[61,24409,24410],{"class":1735},"# Lollipop plot setup\n",[61,24412,24413,24415,24418,24420,24423],{"class":1731,"line":2092},[61,24414,2486],{"class":1749},[61,24416,24417],{"class":1753}," ax, (modulation, title) ",[61,24419,2492],{"class":1749},[61,24421,24422],{"class":1859}," zip",[61,24424,24425],{"class":1753},"(axs, modulations):\n",[61,24427,24428,24431,24433,24435,24437,24439,24441,24443,24445,24447,24449,24451,24453,24455,24457],{"class":1731,"line":2097},[61,24429,24430],{"class":1753},"    ax.",[61,24432,14891],{"class":1836},[61,24434,17694],{"class":1753},[61,24436,14896],{"class":3402},[61,24438,2000],{"class":1859},[61,24440,14901],{"class":2165},[61,24442,1435],{"class":1753},[61,24444,14906],{"class":3402},[61,24446,2000],{"class":1859},[61,24448,14911],{"class":2165},[61,24450,1435],{"class":1753},[61,24452,14916],{"class":3402},[61,24454,2000],{"class":1859},[61,24456,14921],{"class":2165},[61,24458,2016],{"class":1753},[61,24460,24461,24463,24465],{"class":1731,"line":2102},[61,24462,24430],{"class":1753},[61,24464,17149],{"class":1836},[61,24466,17152],{"class":1753},[61,24468,24469,24471,24473,24475,24478],{"class":1731,"line":2108},[61,24470,24430],{"class":1753},[61,24472,17159],{"class":1836},[61,24474,849],{"class":1753},[61,24476,24477],{"class":2165},"\"Time (ns)\"",[61,24479,2016],{"class":1753},[61,24481,24482,24484,24486,24488,24490],{"class":1731,"line":2113},[61,24483,24430],{"class":1753},[61,24485,17173],{"class":1836},[61,24487,849],{"class":1753},[61,24489,17178],{"class":2165},[61,24491,2016],{"class":1753},[61,24493,24494],{"class":1731,"line":2162},[61,24495,1743],{"emptyLinePlaceholder":1742},[61,24497,24498],{"class":1731,"line":2169},[61,24499,24500],{"class":1735},"# Memory Addressing with Lollipop Representation\n",[61,24502,24503,24506,24508,24510,24512,24514,24516,24518,24520,24522,24524,24526,24528,24530,24532,24534],{"class":1731,"line":2187},[61,24504,24505],{"class":1753},"fig_combined, ax_combined ",[61,24507,2000],{"class":1859},[61,24509,17065],{"class":1753},[61,24511,17068],{"class":1836},[61,24513,849],{"class":1753},[61,24515,3624],{"class":3402},[61,24517,2000],{"class":1859},[61,24519,849],{"class":1753},[61,24521,17087],{"class":1866},[61,24523,1435],{"class":1753},[61,24525,1300],{"class":1866},[61,24527,3638],{"class":1753},[61,24529,24380],{"class":3402},[61,24531,2000],{"class":1859},[61,24533,24385],{"class":1866},[61,24535,2016],{"class":1753},[61,24537,24538],{"class":1731,"line":2207},[61,24539,1743],{"emptyLinePlaceholder":1742},[61,24541,24542],{"class":1731,"line":2221},[61,24543,24544],{"class":1735},"# Memory addresses as an index range\n",[61,24546,24547,24550,24552,24554,24556,24558,24560],{"class":1731,"line":2226},[61,24548,24549],{"class":1753},"memory_addresses ",[61,24551,2000],{"class":1859},[61,24553,2193],{"class":1753},[61,24555,5737],{"class":1836},[61,24557,849],{"class":1753},[61,24559,5742],{"class":1859},[61,24561,16906],{"class":1753},[61,24563,24564],{"class":1731,"line":2258},[61,24565,1743],{"emptyLinePlaceholder":1742},[61,24567,24568],{"class":1731,"line":2264},[61,24569,24570],{"class":1735},"# Lollipop-style plot for memory addressing\n",[61,24572,24573,24576,24578,24581,24583,24585,24587,24589,24591,24593,24595,24597,24599,24601,24603],{"class":1731,"line":2280},[61,24574,24575],{"class":1753},"ax_combined.",[61,24577,14891],{"class":1836},[61,24579,24580],{"class":1753},"(memory_addresses, gaussian_modulation, ",[61,24582,14896],{"class":3402},[61,24584,2000],{"class":1859},[61,24586,14901],{"class":2165},[61,24588,1435],{"class":1753},[61,24590,14906],{"class":3402},[61,24592,2000],{"class":1859},[61,24594,14911],{"class":2165},[61,24596,1435],{"class":1753},[61,24598,14916],{"class":3402},[61,24600,2000],{"class":1859},[61,24602,14921],{"class":2165},[61,24604,2016],{"class":1753},[61,24606,24607],{"class":1731,"line":2304},[61,24608,1743],{"emptyLinePlaceholder":1742},[61,24610,24611],{"class":1731,"line":2326},[61,24612,24613],{"class":1735},"# Labels and title\n",[61,24615,24616,24618,24620,24622,24625],{"class":1731,"line":2342},[61,24617,24575],{"class":1753},[61,24619,17149],{"class":1836},[61,24621,849],{"class":1753},[61,24623,24624],{"class":2165},"\"Memory Addressing with Lollipop Representation for Quantum Gate Control\"",[61,24626,2016],{"class":1753},[61,24628,24629,24631,24633,24635,24638],{"class":1731,"line":2379},[61,24630,24575],{"class":1753},[61,24632,17159],{"class":1836},[61,24634,849],{"class":1753},[61,24636,24637],{"class":2165},"\"Memory Address Index\"",[61,24639,2016],{"class":1753},[61,24641,24642,24644,24646,24648,24650],{"class":1731,"line":2396},[61,24643,24575],{"class":1753},[61,24645,17173],{"class":1836},[61,24647,849],{"class":1753},[61,24649,17178],{"class":2165},[61,24651,2016],{"class":1753},[61,24653,24654,24656,24658,24660,24662],{"class":1731,"line":2404},[61,24655,24575],{"class":1753},[61,24657,14385],{"class":1836},[61,24659,849],{"class":1753},[61,24661,24385],{"class":1866},[61,24663,2016],{"class":1753},[61,24665,24666],{"class":1731,"line":2443},[61,24667,1743],{"emptyLinePlaceholder":1742},[61,24669,24670],{"class":1731,"line":2475},[61,24671,24672],{"class":1735},"# Display plots\n",[61,24674,24675,24677,24679],{"class":1731,"line":2520},[61,24676,3616],{"class":1753},[61,24678,3850],{"class":1836},[61,24680,2089],{"class":1753},[4797,24682],{"alt":24683,"src":24684},"Output 13 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-13.webp",[4797,24686],{"alt":24687,"src":24688},"Output 14 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-14.webp",[1720,24690,24692],{"className":1722,"code":24691,"language":1724,"meta":1725,"style":1725},"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",[1727,24693,24694,24704,24714,24718,24722,24732,24742,24752,24764,24768,24773,24791,24815,24819,24823,24867,24871,24876,24908,24939,24943,24948,24983,25016,25029,25042,25055,25084,25112,25124,25132,25136],{"__ignoreMap":1725},[61,24695,24696,24698,24700,24702],{"class":1731,"line":1732},[61,24697,1757],{"class":1749},[61,24699,1789],{"class":1753},[61,24701,1792],{"class":1749},[61,24703,1795],{"class":1753},[61,24705,24706,24708,24710,24712],{"class":1731,"line":1739},[61,24707,1757],{"class":1749},[61,24709,1803],{"class":1753},[61,24711,1792],{"class":1749},[61,24713,1808],{"class":1753},[61,24715,24716],{"class":1731,"line":1746},[61,24717,1743],{"emptyLinePlaceholder":1742},[61,24719,24720],{"class":1731,"line":1763},[61,24721,24124],{"class":1735},[61,24723,24724,24726,24728,24730],{"class":1731,"line":1776},[61,24725,24129],{"class":1753},[61,24727,2000],{"class":1859},[61,24729,24134],{"class":1866},[61,24731,24137],{"class":1735},[61,24733,24734,24736,24738,24740],{"class":1731,"line":1784},[61,24735,24142],{"class":1753},[61,24737,2000],{"class":1859},[61,24739,15072],{"class":1866},[61,24741,24149],{"class":1735},[61,24743,24744,24746,24748,24750],{"class":1731,"line":1798},[61,24745,24154],{"class":1753},[61,24747,2000],{"class":1859},[61,24749,5266],{"class":1866},[61,24751,24161],{"class":1735},[61,24753,24754,24756,24758,24761],{"class":1731,"line":1811},[61,24755,15067],{"class":1753},[61,24757,2000],{"class":1859},[61,24759,24760],{"class":1866}," 0.2",[61,24762,24763],{"class":1735},"  # Frequency of the carrier wave (normalized)\n",[61,24765,24766],{"class":1731,"line":1816},[61,24767,1743],{"emptyLinePlaceholder":1742},[61,24769,24770],{"class":1731,"line":1822},[61,24771,24772],{"class":1735},"# Increase the number of points by 1.5x for denser visualization\n",[61,24774,24775,24777,24779,24781,24783,24785,24787,24789],{"class":1731,"line":1828},[61,24776,24175],{"class":1753},[61,24778,2000],{"class":1859},[61,24780,24180],{"class":1859},[61,24782,849],{"class":1753},[61,24784,14795],{"class":1866},[61,24786,2298],{"class":1859},[61,24788,24189],{"class":1866},[61,24790,2016],{"class":1753},[61,24792,24793,24795,24797,24799,24801,24803,24805,24807,24809,24811,24813],{"class":1731,"line":1833},[61,24794,16704],{"class":1753},[61,24796,2000],{"class":1859},[61,24798,2193],{"class":1753},[61,24800,3453],{"class":1836},[61,24802,849],{"class":1753},[61,24804,2006],{"class":1859},[61,24806,14756],{"class":1866},[61,24808,1435],{"class":1753},[61,24810,14756],{"class":1866},[61,24812,24214],{"class":1753},[61,24814,24217],{"class":1735},[61,24816,24817],{"class":1731,"line":1840},[61,24818,1743],{"emptyLinePlaceholder":1742},[61,24820,24821],{"class":1731,"line":1853},[61,24822,24226],{"class":1735},[61,24824,24825,24827,24829,24831,24833,24835,24837,24839,24841,24843,24845,24847,24849,24851,24853,24855,24857,24859,24861,24863,24865],{"class":1731,"line":1873},[61,24826,14274],{"class":1753},[61,24828,2000],{"class":1859},[61,24830,24235],{"class":1753},[61,24832,2201],{"class":1859},[61,24834,2193],{"class":1753},[61,24836,6987],{"class":1836},[61,24838,849],{"class":1753},[61,24840,2006],{"class":1859},[61,24842,24248],{"class":1753},[61,24844,2006],{"class":1859},[61,24846,24253],{"class":1753},[61,24848,2651],{"class":1859},[61,24850,337],{"class":1866},[61,24852,2618],{"class":1859},[61,24854,2003],{"class":1753},[61,24856,337],{"class":1866},[61,24858,2298],{"class":1859},[61,24860,24268],{"class":1753},[61,24862,2651],{"class":1859},[61,24864,337],{"class":1866},[61,24866,5546],{"class":1753},[61,24868,24869],{"class":1731,"line":1889},[61,24870,1743],{"emptyLinePlaceholder":1742},[61,24872,24873],{"class":1731,"line":1908},[61,24874,24875],{"class":1735},"# Generate In-phase (I) and Quadrature (Q) components\n",[61,24877,24878,24881,24883,24885,24887,24889,24891,24893,24895,24897,24899,24901,24903,24905],{"class":1731,"line":1925},[61,24879,24880],{"class":1753},"I ",[61,24882,2000],{"class":1859},[61,24884,15213],{"class":1753},[61,24886,2201],{"class":1859},[61,24888,2193],{"class":1753},[61,24890,5051],{"class":1836},[61,24892,849],{"class":1753},[61,24894,337],{"class":1866},[61,24896,2298],{"class":1859},[61,24898,15184],{"class":1753},[61,24900,2201],{"class":1859},[61,24902,15189],{"class":1753},[61,24904,2201],{"class":1859},[61,24906,24907],{"class":1753}," t)\n",[61,24909,24910,24913,24915,24917,24919,24921,24923,24925,24927,24929,24931,24933,24935,24937],{"class":1731,"line":1941},[61,24911,24912],{"class":1753},"Q ",[61,24914,2000],{"class":1859},[61,24916,15213],{"class":1753},[61,24918,2201],{"class":1859},[61,24920,2193],{"class":1753},[61,24922,5069],{"class":1836},[61,24924,849],{"class":1753},[61,24926,337],{"class":1866},[61,24928,2298],{"class":1859},[61,24930,15184],{"class":1753},[61,24932,2201],{"class":1859},[61,24934,15189],{"class":1753},[61,24936,2201],{"class":1859},[61,24938,24907],{"class":1753},[61,24940,24941],{"class":1731,"line":1957},[61,24942,1743],{"emptyLinePlaceholder":1742},[61,24944,24945],{"class":1731,"line":1974},[61,24946,24947],{"class":1735},"# Plot IQ diagram\n",[61,24949,24950,24953,24955,24957,24959,24961,24963,24965,24967,24969,24971,24973,24975,24977,24979,24981],{"class":1731,"line":1979},[61,24951,24952],{"class":1753},"fig, ax ",[61,24954,2000],{"class":1859},[61,24956,17065],{"class":1753},[61,24958,17068],{"class":1836},[61,24960,849],{"class":1753},[61,24962,3624],{"class":3402},[61,24964,2000],{"class":1859},[61,24966,849],{"class":1753},[61,24968,1300],{"class":1866},[61,24970,1435],{"class":1753},[61,24972,1300],{"class":1866},[61,24974,3638],{"class":1753},[61,24976,24380],{"class":3402},[61,24978,2000],{"class":1859},[61,24980,24385],{"class":1866},[61,24982,2016],{"class":1753},[61,24984,24985,24988,24990,24993,24996,24998,25000,25002,25005,25007,25009,25011,25014],{"class":1731,"line":1985},[61,24986,24987],{"class":1753},"ax.",[61,24989,3656],{"class":1836},[61,24991,24992],{"class":1753},"(I, Q, ",[61,24994,24995],{"class":2165},"'bo-'",[61,24997,1435],{"class":1753},[61,24999,16926],{"class":3402},[61,25001,2000],{"class":1859},[61,25003,25004],{"class":1866},"0.7",[61,25006,1435],{"class":1753},[61,25008,3666],{"class":3402},[61,25010,2000],{"class":1859},[61,25012,25013],{"class":2165},"\"IQ Trajectory\"",[61,25015,2016],{"class":1753},[61,25017,25018,25020,25022,25024,25027],{"class":1731,"line":2019},[61,25019,24987],{"class":1753},[61,25021,17159],{"class":1836},[61,25023,849],{"class":1753},[61,25025,25026],{"class":2165},"\"In-phase (I)\"",[61,25028,2016],{"class":1753},[61,25030,25031,25033,25035,25037,25040],{"class":1731,"line":2033},[61,25032,24987],{"class":1753},[61,25034,17173],{"class":1836},[61,25036,849],{"class":1753},[61,25038,25039],{"class":2165},"\"Quadrature (Q)\"",[61,25041,2016],{"class":1753},[61,25043,25044,25046,25048,25050,25053],{"class":1731,"line":2061},[61,25045,24987],{"class":1753},[61,25047,17149],{"class":1836},[61,25049,849],{"class":1753},[61,25051,25052],{"class":2165},"\"IQ Diagram for Gaussian Modulated Pulse\"",[61,25054,2016],{"class":1753},[61,25056,25057,25059,25061,25063,25065,25067,25069,25071,25074,25076,25078,25080,25082],{"class":1731,"line":2074},[61,25058,24987],{"class":1753},[61,25060,3720],{"class":1836},[61,25062,849],{"class":1753},[61,25064,1495],{"class":1866},[61,25066,1435],{"class":1753},[61,25068,3705],{"class":3402},[61,25070,2000],{"class":1859},[61,25072,25073],{"class":2165},"'k'",[61,25075,1435],{"class":1753},[61,25077,3413],{"class":3402},[61,25079,2000],{"class":1859},[61,25081,5056],{"class":1866},[61,25083,2016],{"class":1753},[61,25085,25086,25088,25090,25092,25094,25096,25098,25100,25102,25104,25106,25108,25110],{"class":1731,"line":2079},[61,25087,24987],{"class":1753},[61,25089,3396],{"class":1836},[61,25091,849],{"class":1753},[61,25093,1495],{"class":1866},[61,25095,1435],{"class":1753},[61,25097,3705],{"class":3402},[61,25099,2000],{"class":1859},[61,25101,25073],{"class":2165},[61,25103,1435],{"class":1753},[61,25105,3413],{"class":3402},[61,25107,2000],{"class":1859},[61,25109,5056],{"class":1866},[61,25111,2016],{"class":1753},[61,25113,25114,25116,25118,25120,25122],{"class":1731,"line":2092},[61,25115,24987],{"class":1753},[61,25117,14385],{"class":1836},[61,25119,849],{"class":1753},[61,25121,24385],{"class":1866},[61,25123,2016],{"class":1753},[61,25125,25126,25128,25130],{"class":1731,"line":2097},[61,25127,24987],{"class":1753},[61,25129,3840],{"class":1836},[61,25131,2089],{"class":1753},[61,25133,25134],{"class":1731,"line":2102},[61,25135,1743],{"emptyLinePlaceholder":1742},[61,25137,25138,25140,25142],{"class":1731,"line":2108},[61,25139,3616],{"class":1753},[61,25141,3850],{"class":1836},[61,25143,2089],{"class":1753},[4797,25145],{"alt":25146,"src":25147},"Output 15 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-15.webp",[1720,25149,25151],{"className":1722,"code":25150,"language":1724,"meta":1725,"style":1725},"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",[1727,25152,25153,25163,25173,25177,25181,25191,25201,25211,25215,25219,25237,25261,25265,25269,25313,25317,25321,25329,25337,25345,25353,25361,25365,25369,25374,25416,25428,25432,25444,25477,25485,25497,25509,25513],{"__ignoreMap":1725},[61,25154,25155,25157,25159,25161],{"class":1731,"line":1732},[61,25156,1757],{"class":1749},[61,25158,1789],{"class":1753},[61,25160,1792],{"class":1749},[61,25162,1795],{"class":1753},[61,25164,25165,25167,25169,25171],{"class":1731,"line":1739},[61,25166,1757],{"class":1749},[61,25168,1803],{"class":1753},[61,25170,1792],{"class":1749},[61,25172,1808],{"class":1753},[61,25174,25175],{"class":1731,"line":1746},[61,25176,1743],{"emptyLinePlaceholder":1742},[61,25178,25179],{"class":1731,"line":1763},[61,25180,24124],{"class":1735},[61,25182,25183,25185,25187,25189],{"class":1731,"line":1776},[61,25184,24129],{"class":1753},[61,25186,2000],{"class":1859},[61,25188,24134],{"class":1866},[61,25190,24137],{"class":1735},[61,25192,25193,25195,25197,25199],{"class":1731,"line":1784},[61,25194,24142],{"class":1753},[61,25196,2000],{"class":1859},[61,25198,15072],{"class":1866},[61,25200,24149],{"class":1735},[61,25202,25203,25205,25207,25209],{"class":1731,"line":1798},[61,25204,24154],{"class":1753},[61,25206,2000],{"class":1859},[61,25208,5266],{"class":1866},[61,25210,24161],{"class":1735},[61,25212,25213],{"class":1731,"line":1811},[61,25214,1743],{"emptyLinePlaceholder":1742},[61,25216,25217],{"class":1731,"line":1816},[61,25218,24170],{"class":1735},[61,25220,25221,25223,25225,25227,25229,25231,25233,25235],{"class":1731,"line":1822},[61,25222,24175],{"class":1753},[61,25224,2000],{"class":1859},[61,25226,24180],{"class":1859},[61,25228,849],{"class":1753},[61,25230,14795],{"class":1866},[61,25232,2298],{"class":1859},[61,25234,24189],{"class":1866},[61,25236,2016],{"class":1753},[61,25238,25239,25241,25243,25245,25247,25249,25251,25253,25255,25257,25259],{"class":1731,"line":1828},[61,25240,16704],{"class":1753},[61,25242,2000],{"class":1859},[61,25244,2193],{"class":1753},[61,25246,3453],{"class":1836},[61,25248,849],{"class":1753},[61,25250,2006],{"class":1859},[61,25252,14756],{"class":1866},[61,25254,1435],{"class":1753},[61,25256,14756],{"class":1866},[61,25258,24214],{"class":1753},[61,25260,24217],{"class":1735},[61,25262,25263],{"class":1731,"line":1833},[61,25264,1743],{"emptyLinePlaceholder":1742},[61,25266,25267],{"class":1731,"line":1840},[61,25268,24226],{"class":1735},[61,25270,25271,25273,25275,25277,25279,25281,25283,25285,25287,25289,25291,25293,25295,25297,25299,25301,25303,25305,25307,25309,25311],{"class":1731,"line":1853},[61,25272,14274],{"class":1753},[61,25274,2000],{"class":1859},[61,25276,24235],{"class":1753},[61,25278,2201],{"class":1859},[61,25280,2193],{"class":1753},[61,25282,6987],{"class":1836},[61,25284,849],{"class":1753},[61,25286,2006],{"class":1859},[61,25288,24248],{"class":1753},[61,25290,2006],{"class":1859},[61,25292,24253],{"class":1753},[61,25294,2651],{"class":1859},[61,25296,337],{"class":1866},[61,25298,2618],{"class":1859},[61,25300,2003],{"class":1753},[61,25302,337],{"class":1866},[61,25304,2298],{"class":1859},[61,25306,24268],{"class":1753},[61,25308,2651],{"class":1859},[61,25310,337],{"class":1866},[61,25312,5546],{"class":1753},[61,25314,25315],{"class":1731,"line":1873},[61,25316,1743],{"emptyLinePlaceholder":1742},[61,25318,25319],{"class":1731,"line":1889},[61,25320,24283],{"class":1735},[61,25322,25323,25325,25327],{"class":1731,"line":1908},[61,25324,17548],{"class":1753},[61,25326,2000],{"class":1859},[61,25328,24292],{"class":1753},[61,25330,25331,25333,25335],{"class":1731,"line":1925},[61,25332,24297],{"class":1753},[61,25334,24300],{"class":2165},[61,25336,16975],{"class":1753},[61,25338,25339,25341,25343],{"class":1731,"line":1941},[61,25340,24297],{"class":1753},[61,25342,24309],{"class":2165},[61,25344,16975],{"class":1753},[61,25346,25347,25349,25351],{"class":1731,"line":1957},[61,25348,24297],{"class":1753},[61,25350,24318],{"class":2165},[61,25352,16975],{"class":1753},[61,25354,25355,25357,25359],{"class":1731,"line":1974},[61,25356,24297],{"class":1753},[61,25358,24327],{"class":2165},[61,25360,2016],{"class":1753},[61,25362,25363],{"class":1731,"line":1979},[61,25364,5701],{"class":1753},[61,25366,25367],{"class":1731,"line":1985},[61,25368,1743],{"emptyLinePlaceholder":1742},[61,25370,25371],{"class":1731,"line":2019},[61,25372,25373],{"class":1735},"# Create lollipop plots\n",[61,25375,25376,25378,25380,25382,25384,25386,25388,25390,25392,25394,25396,25398,25400,25402,25404,25406,25408,25410,25412,25414],{"class":1731,"line":2033},[61,25377,24347],{"class":1753},[61,25379,2000],{"class":1859},[61,25381,17065],{"class":1753},[61,25383,17068],{"class":1836},[61,25385,849],{"class":1753},[61,25387,337],{"class":1866},[61,25389,1435],{"class":1753},[61,25391,337],{"class":1866},[61,25393,1435],{"class":1753},[61,25395,3624],{"class":3402},[61,25397,2000],{"class":1859},[61,25399,849],{"class":1753},[61,25401,17087],{"class":1866},[61,25403,1435],{"class":1753},[61,25405,3631],{"class":1866},[61,25407,3638],{"class":1753},[61,25409,24380],{"class":3402},[61,25411,2000],{"class":1859},[61,25413,24385],{"class":1866},[61,25415,2016],{"class":1753},[61,25417,25418,25420,25422,25424,25426],{"class":1731,"line":2061},[61,25419,24392],{"class":1753},[61,25421,2000],{"class":1859},[61,25423,24397],{"class":1753},[61,25425,17107],{"class":1836},[61,25427,2089],{"class":1753},[61,25429,25430],{"class":1731,"line":2074},[61,25431,1743],{"emptyLinePlaceholder":1742},[61,25433,25434,25436,25438,25440,25442],{"class":1731,"line":2079},[61,25435,2486],{"class":1749},[61,25437,24417],{"class":1753},[61,25439,2492],{"class":1749},[61,25441,24422],{"class":1859},[61,25443,24425],{"class":1753},[61,25445,25446,25448,25450,25453,25455,25457,25459,25461,25463,25465,25467,25469,25471,25473,25475],{"class":1731,"line":2092},[61,25447,24430],{"class":1753},[61,25449,14891],{"class":1836},[61,25451,25452],{"class":1753},"(t, gaussian_modulation, ",[61,25454,14896],{"class":3402},[61,25456,2000],{"class":1859},[61,25458,14901],{"class":2165},[61,25460,1435],{"class":1753},[61,25462,14906],{"class":3402},[61,25464,2000],{"class":1859},[61,25466,14911],{"class":2165},[61,25468,1435],{"class":1753},[61,25470,14916],{"class":3402},[61,25472,2000],{"class":1859},[61,25474,14921],{"class":2165},[61,25476,2016],{"class":1753},[61,25478,25479,25481,25483],{"class":1731,"line":2097},[61,25480,24430],{"class":1753},[61,25482,17149],{"class":1836},[61,25484,17152],{"class":1753},[61,25486,25487,25489,25491,25493,25495],{"class":1731,"line":2102},[61,25488,24430],{"class":1753},[61,25490,17159],{"class":1836},[61,25492,849],{"class":1753},[61,25494,24477],{"class":2165},[61,25496,2016],{"class":1753},[61,25498,25499,25501,25503,25505,25507],{"class":1731,"line":2108},[61,25500,24430],{"class":1753},[61,25502,17173],{"class":1836},[61,25504,849],{"class":1753},[61,25506,17178],{"class":2165},[61,25508,2016],{"class":1753},[61,25510,25511],{"class":1731,"line":2113},[61,25512,1743],{"emptyLinePlaceholder":1742},[61,25514,25515,25517,25519],{"class":1731,"line":2162},[61,25516,3616],{"class":1753},[61,25518,3850],{"class":1836},[61,25520,2089],{"class":1753},[4797,25522],{"alt":25523,"src":25524},"Output 16 of the notebook","\u002F_content\u002Fimages\u002Fpulse-shapes-and-envelopes\u002Foutput-16.webp",[1720,25526,25528],{"className":1722,"code":25527,"language":1724,"meta":1725,"style":1725},"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",[1727,25529,25530,25540,25550,25554,25558,25568,25579,25589,25602,25614,25630,25646,25675,25679,25683,25728,25732,25737,25781,25821,25861,25904,25908,25913,25927,25937,25947,25957,25967,25978,25982,25987,26029,26041,26045,26059,26103,26134,26160,26183,26195,26207,26212,26217,26221],{"__ignoreMap":1725},[61,25531,25532,25534,25536,25538],{"class":1731,"line":1732},[61,25533,1757],{"class":1749},[61,25535,1789],{"class":1753},[61,25537,1792],{"class":1749},[61,25539,1795],{"class":1753},[61,25541,25542,25544,25546,25548],{"class":1731,"line":1739},[61,25543,1757],{"class":1749},[61,25545,1803],{"class":1753},[61,25547,1792],{"class":1749},[61,25549,1808],{"class":1753},[61,25551,25552],{"class":1731,"line":1746},[61,25553,1743],{"emptyLinePlaceholder":1742},[61,25555,25556],{"class":1731,"line":1763},[61,25557,24124],{"class":1735},[61,25559,25560,25562,25564,25566],{"class":1731,"line":1776},[61,25561,24129],{"class":1753},[61,25563,2000],{"class":1859},[61,25565,24134],{"class":1866},[61,25567,24137],{"class":1735},[61,25569,25570,25572,25574,25576],{"class":1731,"line":1784},[61,25571,24142],{"class":1753},[61,25573,2000],{"class":1859},[61,25575,15072],{"class":1866},[61,25577,25578],{"class":1735},"   # Standard deviation of Gaussian modulation (ns)\n",[61,25580,25581,25583,25585,25587],{"class":1731,"line":1798},[61,25582,24154],{"class":1753},[61,25584,2000],{"class":1859},[61,25586,5266],{"class":1866},[61,25588,24161],{"class":1735},[61,25590,25591,25594,25596,25599],{"class":1731,"line":1811},[61,25592,25593],{"class":1753},"fc ",[61,25595,2000],{"class":1859},[61,25597,25598],{"class":1866}," 5e9",[61,25600,25601],{"class":1735},"    # Carrier frequency (5 GHz, typical for superconducting qubits)\n",[61,25603,25604,25607,25609,25611],{"class":1731,"line":1816},[61,25605,25606],{"class":1753},"phi_X ",[61,25608,2000],{"class":1859},[61,25610,24134],{"class":1866},[61,25612,25613],{"class":1735},"           # X gate (Pauli-X) - $\\pi$ rotation around X-axis\n",[61,25615,25616,25619,25621,25623,25625,25627],{"class":1731,"line":1822},[61,25617,25618],{"class":1753},"phi_Y ",[61,25620,2000],{"class":1859},[61,25622,15184],{"class":1753},[61,25624,602],{"class":1859},[61,25626,2654],{"class":1866},[61,25628,25629],{"class":1735},"   # Y gate (Pauli-Y) - $\\pi\u002F2$ rotation around Y-axis\n",[61,25631,25632,25635,25637,25639,25641,25643],{"class":1731,"line":1828},[61,25633,25634],{"class":1753},"phi_H ",[61,25636,2000],{"class":1859},[61,25638,15184],{"class":1753},[61,25640,602],{"class":1859},[61,25642,2654],{"class":1866},[61,25644,25645],{"class":1735},"   # Hadamard gate (H) - $\\pi\u002F2$ rotation around Y-axis\n",[61,25647,25648,25650,25652,25654,25656,25658,25660,25662,25664,25666,25668,25670,25672],{"class":1731,"line":1833},[61,25649,16704],{"class":1753},[61,25651,2000],{"class":1859},[61,25653,2193],{"class":1753},[61,25655,3453],{"class":1836},[61,25657,849],{"class":1753},[61,25659,2006],{"class":1859},[61,25661,14756],{"class":1866},[61,25663,1435],{"class":1753},[61,25665,14756],{"class":1866},[61,25667,1435],{"class":1753},[61,25669,14254],{"class":1866},[61,25671,14257],{"class":1753},[61,25673,25674],{"class":1735},"# Time axis in nanoseconds\n",[61,25676,25677],{"class":1731,"line":1840},[61,25678,1743],{"emptyLinePlaceholder":1742},[61,25680,25681],{"class":1731,"line":1853},[61,25682,24226],{"class":1735},[61,25684,25685,25688,25690,25692,25694,25696,25698,25700,25702,25704,25706,25708,25710,25712,25714,25716,25718,25720,25722,25724,25726],{"class":1731,"line":1873},[61,25686,25687],{"class":1753},"gaussian_envelope ",[61,25689,2000],{"class":1859},[61,25691,24235],{"class":1753},[61,25693,2201],{"class":1859},[61,25695,2193],{"class":1753},[61,25697,6987],{"class":1836},[61,25699,849],{"class":1753},[61,25701,2006],{"class":1859},[61,25703,24248],{"class":1753},[61,25705,2006],{"class":1859},[61,25707,24253],{"class":1753},[61,25709,2651],{"class":1859},[61,25711,337],{"class":1866},[61,25713,2618],{"class":1859},[61,25715,2003],{"class":1753},[61,25717,337],{"class":1866},[61,25719,2298],{"class":1859},[61,25721,24268],{"class":1753},[61,25723,2651],{"class":1859},[61,25725,337],{"class":1866},[61,25727,5546],{"class":1753},[61,25729,25730],{"class":1731,"line":1889},[61,25731,1743],{"emptyLinePlaceholder":1742},[61,25733,25734],{"class":1731,"line":1908},[61,25735,25736],{"class":1735},"# Compute modulated pulses for X, Y, H, and CNOT gates\n",[61,25738,25739,25742,25744,25747,25749,25751,25753,25755,25757,25759,25761,25763,25766,25768,25771,25773,25776,25778],{"class":1731,"line":1925},[61,25740,25741],{"class":1753},"modulated_pulse_X ",[61,25743,2000],{"class":1859},[61,25745,25746],{"class":1753}," gaussian_envelope ",[61,25748,2201],{"class":1859},[61,25750,2193],{"class":1753},[61,25752,5051],{"class":1836},[61,25754,849],{"class":1753},[61,25756,337],{"class":1866},[61,25758,2298],{"class":1859},[61,25760,15184],{"class":1753},[61,25762,2201],{"class":1859},[61,25764,25765],{"class":1753}," fc ",[61,25767,2201],{"class":1859},[61,25769,25770],{"class":1753}," t ",[61,25772,2201],{"class":1859},[61,25774,25775],{"class":1866}," 1e-9",[61,25777,4990],{"class":1859},[61,25779,25780],{"class":1753}," phi_X)\n",[61,25782,25783,25786,25788,25790,25792,25794,25796,25798,25800,25802,25804,25806,25808,25810,25812,25814,25816,25818],{"class":1731,"line":1941},[61,25784,25785],{"class":1753},"modulated_pulse_Y ",[61,25787,2000],{"class":1859},[61,25789,25746],{"class":1753},[61,25791,2201],{"class":1859},[61,25793,2193],{"class":1753},[61,25795,5051],{"class":1836},[61,25797,849],{"class":1753},[61,25799,337],{"class":1866},[61,25801,2298],{"class":1859},[61,25803,15184],{"class":1753},[61,25805,2201],{"class":1859},[61,25807,25765],{"class":1753},[61,25809,2201],{"class":1859},[61,25811,25770],{"class":1753},[61,25813,2201],{"class":1859},[61,25815,25775],{"class":1866},[61,25817,4990],{"class":1859},[61,25819,25820],{"class":1753}," phi_Y)\n",[61,25822,25823,25826,25828,25830,25832,25834,25836,25838,25840,25842,25844,25846,25848,25850,25852,25854,25856,25858],{"class":1731,"line":1957},[61,25824,25825],{"class":1753},"modulated_pulse_H ",[61,25827,2000],{"class":1859},[61,25829,25746],{"class":1753},[61,25831,2201],{"class":1859},[61,25833,2193],{"class":1753},[61,25835,5051],{"class":1836},[61,25837,849],{"class":1753},[61,25839,337],{"class":1866},[61,25841,2298],{"class":1859},[61,25843,15184],{"class":1753},[61,25845,2201],{"class":1859},[61,25847,25765],{"class":1753},[61,25849,2201],{"class":1859},[61,25851,25770],{"class":1753},[61,25853,2201],{"class":1859},[61,25855,25775],{"class":1866},[61,25857,4990],{"class":1859},[61,25859,25860],{"class":1753}," phi_H)\n",[61,25862,25863,25866,25868,25870,25872,25874,25876,25878,25880,25882,25884,25886,25888,25890,25892,25894,25896,25898,25901],{"class":1731,"line":1974},[61,25864,25865],{"class":1753},"modulated_pulse_CNOT ",[61,25867,2000],{"class":1859},[61,25869,25746],{"class":1753},[61,25871,2201],{"class":1859},[61,25873,2193],{"class":1753},[61,25875,5051],{"class":1836},[61,25877,849],{"class":1753},[61,25879,337],{"class":1866},[61,25881,2298],{"class":1859},[61,25883,15184],{"class":1753},[61,25885,2201],{"class":1859},[61,25887,25765],{"class":1753},[61,25889,2201],{"class":1859},[61,25891,25770],{"class":1753},[61,25893,2201],{"class":1859},[61,25895,25775],{"class":1866},[61,25897,4990],{"class":1859},[61,25899,25900],{"class":1753}," phi_X)  ",[61,25902,25903],{"class":1735},"# Similar to X gate\n",[61,25905,25906],{"class":1731,"line":1979},[61,25907,1743],{"emptyLinePlaceholder":1742},[61,25909,25910],{"class":1731,"line":1985},[61,25911,25912],{"class":1735},"# Store pulses and titles in an array\n",[61,25914,25915,25918,25920,25922,25924],{"class":1731,"line":2019},[61,25916,25917],{"class":1753},"pulses ",[61,25919,2000],{"class":1859},[61,25921,2193],{"class":1753},[61,25923,3509],{"class":1836},[61,25925,25926],{"class":1753},"([\n",[61,25928,25929,25932,25935],{"class":1731,"line":2033},[61,25930,25931],{"class":1753},"    (modulated_pulse_X, ",[61,25933,25934],{"class":2165},"\"Pauli-X Gate\"",[61,25936,16975],{"class":1753},[61,25938,25939,25942,25945],{"class":1731,"line":2061},[61,25940,25941],{"class":1753},"    (modulated_pulse_Y, ",[61,25943,25944],{"class":2165},"\"Pauli-Y Gate\"",[61,25946,16975],{"class":1753},[61,25948,25949,25952,25955],{"class":1731,"line":2074},[61,25950,25951],{"class":1753},"    (modulated_pulse_H, ",[61,25953,25954],{"class":2165},"\"Hadamard Gate\"",[61,25956,16975],{"class":1753},[61,25958,25959,25962,25965],{"class":1731,"line":2079},[61,25960,25961],{"class":1753},"    (modulated_pulse_CNOT, ",[61,25963,25964],{"class":2165},"\"CNOT Gate Target Qubit\"",[61,25966,2016],{"class":1753},[61,25968,25969,25971,25973,25976],{"class":1731,"line":2092},[61,25970,7056],{"class":1753},[61,25972,6950],{"class":3402},[61,25974,25975],{"class":1859},"=object",[61,25977,2016],{"class":1753},[61,25979,25980],{"class":1731,"line":2097},[61,25981,1743],{"emptyLinePlaceholder":1742},[61,25983,25984],{"class":1731,"line":2102},[61,25985,25986],{"class":1735},"# Create plots with blue pulse lines\n",[61,25988,25989,25991,25993,25995,25997,25999,26001,26003,26005,26007,26009,26011,26013,26015,26017,26019,26021,26023,26025,26027],{"class":1731,"line":2108},[61,25990,24347],{"class":1753},[61,25992,2000],{"class":1859},[61,25994,17065],{"class":1753},[61,25996,17068],{"class":1836},[61,25998,849],{"class":1753},[61,26000,337],{"class":1866},[61,26002,1435],{"class":1753},[61,26004,337],{"class":1866},[61,26006,1435],{"class":1753},[61,26008,3624],{"class":3402},[61,26010,2000],{"class":1859},[61,26012,849],{"class":1753},[61,26014,17087],{"class":1866},[61,26016,1435],{"class":1753},[61,26018,3631],{"class":1866},[61,26020,3638],{"class":1753},[61,26022,24380],{"class":3402},[61,26024,2000],{"class":1859},[61,26026,24385],{"class":1866},[61,26028,2016],{"class":1753},[61,26030,26031,26033,26035,26037,26039],{"class":1731,"line":2113},[61,26032,24392],{"class":1753},[61,26034,2000],{"class":1859},[61,26036,24397],{"class":1753},[61,26038,17107],{"class":1836},[61,26040,2089],{"class":1753},[61,26042,26043],{"class":1731,"line":2162},[61,26044,1743],{"emptyLinePlaceholder":1742},[61,26046,26047,26049,26052,26054,26056],{"class":1731,"line":2169},[61,26048,2486],{"class":1749},[61,26050,26051],{"class":1753}," ax, (pulse, title) ",[61,26053,2492],{"class":1749},[61,26055,24422],{"class":1859},[61,26057,26058],{"class":1753},"(axs, pulses):\n",[61,26060,26061,26063,26065,26068,26070,26072,26074,26076,26078,26080,26082,26084,26086,26089,26091,26093,26095,26098,26100],{"class":1731,"line":2187},[61,26062,24430],{"class":1753},[61,26064,3656],{"class":1836},[61,26066,26067],{"class":1753},"(t, pulse, ",[61,26069,17140],{"class":2165},[61,26071,1435],{"class":1753},[61,26073,3413],{"class":3402},[61,26075,2000],{"class":1859},[61,26077,5056],{"class":1866},[61,26079,1435],{"class":1753},[61,26081,3666],{"class":3402},[61,26083,2000],{"class":1859},[61,26085,19797],{"class":1749},[61,26087,26088],{"class":2165},"\"",[61,26090,3688],{"class":1866},[61,26092,3818],{"class":1753},[61,26094,3697],{"class":1866},[61,26096,26097],{"class":2165}," Modulated Pulse\"",[61,26099,14257],{"class":1753},[61,26101,26102],{"class":1735},"# Blue line with reduced thickness\n",[61,26104,26105,26107,26109,26112,26115,26117,26119,26121,26123,26125,26127,26129,26132],{"class":1731,"line":2207},[61,26106,24430],{"class":1753},[61,26108,3656],{"class":1836},[61,26110,26111],{"class":1753},"(t, gaussian_envelope, ",[61,26113,26114],{"class":2165},"'r--'",[61,26116,1435],{"class":1753},[61,26118,3413],{"class":3402},[61,26120,2000],{"class":1859},[61,26122,2319],{"class":1866},[61,26124,1435],{"class":1753},[61,26126,3666],{"class":3402},[61,26128,2000],{"class":1859},[61,26130,26131],{"class":2165},"\"Gaussian Envelope\"",[61,26133,2016],{"class":1753},[61,26135,26136,26138,26140,26143,26145,26148,26150,26152,26154,26156,26158],{"class":1731,"line":2221},[61,26137,24430],{"class":1753},[61,26139,3656],{"class":1836},[61,26141,26142],{"class":1753},"(t, ",[61,26144,2006],{"class":1859},[61,26146,26147],{"class":1753},"gaussian_envelope, ",[61,26149,26114],{"class":2165},[61,26151,1435],{"class":1753},[61,26153,3413],{"class":3402},[61,26155,2000],{"class":1859},[61,26157,2319],{"class":1866},[61,26159,2016],{"class":1753},[61,26161,26162,26164,26166,26168,26170,26173,26175,26177,26179,26181],{"class":1731,"line":2226},[61,26163,24430],{"class":1753},[61,26165,17149],{"class":1836},[61,26167,849],{"class":1753},[61,26169,19797],{"class":1749},[61,26171,26172],{"class":2165},"\"Modulated Pulse for 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operator"," and ",[814,27100,27101],{},"Bloch vector",[61,27103,27105,27131],{"className":27104},[64],[61,27106,27108],{"className":27107},[68],[70,27109,27110],{"xmlns":72},[74,27111,27112,27128],{},[77,27113,27114,27117,27120],{},[83,27115,3763],{"mathvariant":27116},"bold",[586,27118,27119],{},"∈",[19151,27121,27122,27126],{},[83,27123,27125],{"mathvariant":27124},"double-struck","R",[1493,27127,570],{},[95,27129,27130],{"encoding":97},"\\mathbf r\\in\\mathbb R^3",[61,27132,27134,27154],{"className":27133,"ariaHidden":103},[102],[61,27135,27137,27141,27145,27148,27151],{"className":27136},[107],[61,27138],{"className":27139,"style":27140},[111],"height:0.5782em;vertical-align:-0.0391em;",[61,27142,3763],{"className":27143},[116,27144],"mathbf",[61,27146],{"className":27147,"style":19197},[981],[61,27149,27119],{"className":27150},[19201],[61,27152],{"className":27153,"style":19197},[981],[61,27155,27157,27161],{"className":27156},[107],[61,27158],{"className":27159,"style":27160},[111],"height:0.8141em;",[61,27162,27164,27168],{"className":27163},[116],[61,27165,27125],{"className":27166},[116,27167],"mathbb",[61,27169,27171],{"className":27170},[125],[61,27172,27174],{"className":27173},[129],[61,27175,27177],{"className":27176},[134],[61,27178,27180],{"className":27179,"style":27160},[138],[61,27181,27182,27185],{"style":21487},[61,27183],{"className":27184,"style":147},[146],[61,27186,27188],{"className":27187},[151,152,153,154],[61,27189,570],{"className":27190},[116,154],"):",[61,27193,27195],{"className":27194},[26354],[61,27196,27198,27296],{"className":27197},[64],[61,27199,27201],{"className":27200},[68],[70,27202,27203],{"xmlns":72,"display":26363},[74,27204,27205,27293],{},[77,27206,27207,27210,27212,27220,27223,27225,27227,27229,27232,27235,27237,27239,27242,27248,27250,27253,27255,27257,27259,27261,27267,27269,27271,27275,27277,27279,27291],{},[83,27208,27209],{},"ρ",[586,27211,2000],{},[26393,27213,27214],{"scriptlevel":1495,"displaystyle":588},[19134,27215,27216,27218],{},[1493,27217,57],{},[1493,27219,337],{},[586,27221,849],{"fence":588,"stretchy":103,"minsize":27222,"maxsize":27222},"1.2em",[83,27224,1414],{},[586,27226,2419],{},[83,27228,3763],{"mathvariant":27116},[586,27230,27231],{},"⋅",[83,27233,19165],{"mathvariant":27234},"bold-italic",[586,27236,300],{"fence":588,"stretchy":103,"minsize":27222,"maxsize":27222},[586,27238,862],{"separator":103},[981,27240],{"width":27241},"2em",[80,27243,27244,27246],{},[83,27245,3763],{},[83,27247,5222],{},[586,27249,2000],{},[83,27251,27252],{"mathvariant":197},"Tr",[586,27254,19125],{},[586,27256,849],{"stretchy":588},[83,27258,27209],{},[586,27260,862],{"separator":103},[80,27262,27263,27265],{},[83,27264,19165],{},[83,27266,5222],{},[586,27268,300],{"stretchy":588},[586,27270,862],{"separator":103},[27272,27273,27274],"mtext",{}," ",[83,27276,5222],{},[586,27278,27119],{},[77,27280,27281,27283,27285,27287,27289],{},[83,27282,26374],{},[586,27284,862],{"separator":103},[83,27286,26429],{},[586,27288,862],{"separator":103},[83,27290,26482],{},[83,27292,332],{"mathvariant":197},[95,27294,27295],{"encoding":97},"\\rho 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j\\in{x,y,z}.",[61,27297,27299,27318,27412,27431,27516,27606],{"className":27298,"ariaHidden":103},[102],[61,27300,27302,27306,27309,27312,27315],{"className":27301},[107],[61,27303],{"className":27304,"style":27305},[111],"height:0.625em;vertical-align:-0.1944em;",[61,27307,27209],{"className":27308},[116,120],[61,27310],{"className":27311,"style":19197},[981],[61,27313,2000],{"className":27314},[19201],[61,27316],{"className":27317,"style":19197},[981],[61,27319,27321,27325,27394,27400,27403,27406,27409],{"className":27320},[107],[61,27322],{"className":27323,"style":27324},[111],"height:1.2em;vertical-align:-0.35em;",[61,27326,27328,27331,27391],{"className":27327},[116],[61,27329],{"className":27330},[631,19286],[61,27332,27334],{"className":27333},[19134],[61,27335,27337,27383],{"className":27336},[129,130],[61,27338,27340,27380],{"className":27339},[134],[61,27341,27344,27358,27366],{"className":27342,"style":27343},[138],"height:0.8451em;",[61,27345,27346,27349],{"style":19302},[61,27347],{"className":27348,"style":19306},[146],[61,27350,27352],{"className":27351},[151,152,153,154],[61,27353,27355],{"className":27354},[116,154],[61,27356,337],{"className":27357},[116,154],[61,27359,27360,27363],{"style":19352},[61,27361],{"className":27362,"style":19306},[146],[61,27364],{"className":27365,"style":19360},[19359],[61,27367,27368,27371],{"style":21195},[61,27369],{"className":27370,"style":19306},[146],[61,27372,27374],{"className":27373},[151,152,153,154],[61,27375,27377],{"className":27376},[116,154],[61,27378,57],{"className":27379},[116,154],[61,27381,170],{"className":27382},[169],[61,27384,27386],{"className":27385},[134],[61,27387,27389],{"className":27388,"style":19466},[138],[61,27390],{},[61,27392],{"className":27393},[730,19286],[61,27395,27397],{"className":27396},[116],[61,27398,849],{"className":27399},[19275,19346],[61,27401,1414],{"className":27402,"style":165},[116,120],[61,27404],{"className":27405,"style":20194},[981],[61,27407,2419],{"className":27408},[19382],[61,27410],{"className":27411,"style":20194},[981],[61,27413,27415,27419,27422,27425,27428],{"className":27414},[107],[61,27416],{"className":27417,"style":27418},[111],"height:0.4445em;",[61,27420,3763],{"className":27421},[116,27144],[61,27423],{"className":27424,"style":20194},[981],[61,27426,27231],{"className":27427},[19382],[61,27429],{"className":27430,"style":20194},[981],[61,27432,27434,27437,27448,27454,27457,27461,27464,27507,27510,27513],{"className":27433},[107],[61,27435],{"className":27436,"style":27324},[111],[61,27438,27440],{"className":27439},[116],[61,27441,27443],{"className":27442},[116],[61,27444,19165],{"className":27445,"style":27447},[116,27446],"boldsymbol","margin-right:0.037em;",[61,27449,27451],{"className":27450},[116],[61,27452,300],{"className":27453},[19275,19346],[61,27455,862],{"className":27456},[977],[61,27458],{"className":27459,"style":27460},[981],"margin-right:2em;",[61,27462],{"className":27463,"style":982},[981],[61,27465,27467,27470],{"className":27466},[116],[61,27468,3763],{"className":27469,"style":21503},[116,120],[61,27471,27473],{"className":27472},[125],[61,27474,27476,27499],{"className":27475},[129,130],[61,27477,27479,27496],{"className":27478},[134],[61,27480,27483],{"className":27481,"style":27482},[138],"height:0.3117em;",[61,27484,27486,27489],{"style":27485},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[61,27487],{"className":27488,"style":147},[146],[61,27490,27492],{"className":27491},[151,152,153,154],[61,27493,5222],{"className":27494,"style":27495},[116,120,154],"margin-right:0.0572em;",[61,27497,170],{"className":27498},[169],[61,27500,27502],{"className":27501},[134],[61,27503,27505],{"className":27504,"style":26767},[138],[61,27506],{},[61,27508],{"className":27509,"style":19197},[981],[61,27511,2000],{"className":27512},[19201],[61,27514],{"className":27515,"style":19197},[981],[61,27517,27519,27523,27530,27533,27536,27539,27542,27582,27585,27588,27591,27594,27597,27600,27603],{"className":27518},[107],[61,27520],{"className":27521,"style":27522},[111],"height:1.0361em;vertical-align:-0.2861em;",[61,27524,27526],{"className":27525},[19259],[61,27527,27252],{"className":27528},[116,27529],"mathrm",[61,27531,849],{"className":27532},[631],[61,27534,27209],{"className":27535},[116,120],[61,27537,862],{"className":27538},[977],[61,27540],{"className":27541,"style":982},[981],[61,27543,27545,27548],{"className":27544},[116],[61,27546,19165],{"className":27547,"style":121},[116,120],[61,27549,27551],{"className":27550},[125],[61,27552,27554,27574],{"className":27553},[129,130],[61,27555,27557,27571],{"className":27556},[134],[61,27558,27560],{"className":27559,"style":27482},[138],[61,27561,27562,27565],{"style":142},[61,27563],{"className":27564,"style":147},[146],[61,27566,27568],{"className":27567},[151,152,153,154],[61,27569,5222],{"className":27570,"style":27495},[116,120,154],[61,27572,170],{"className":27573},[169],[61,27575,27577],{"className":2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is the 3D rotation associated with (U) (see §3–§4).",[1632,30246],{},[25,30248,30250,30251,30281],{"id":30249},"_3-pauli-gates-as-πpiπ-rotations","3. 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(",[61,30286,30288,30301],{"className":30287},[64],[61,30289,30291],{"className":30290},[68],[70,30292,30293],{"xmlns":72},[74,30294,30295,30299],{},[77,30296,30297],{},[83,30298,19792],{},[95,30300,30267],{"encoding":97},[61,30302,30304],{"className":30303,"ariaHidden":103},[102],[61,30305,30307,30310],{"className":30306},[107],[61,30308],{"className":30309,"style":30277},[111],[61,30311,19792],{"className":30312,"style":121},[116,120],")-rotations about Cartesian axes:",[61,30315,30317],{"className":30316},[26354],[61,30318,30320,30426],{"className":30319},[64],[61,30321,30323],{"className":30322},[68],[70,30324,30325],{"xmlns":72,"display":26363},[74,30326,30327,30423],{},[77,30328,30329,30331,30333,30357,30359,30361,30363,30365,30389,30391,30393,30395,30397,30421],{},[83,30330,93],{},[586,30332,2000],{},[19151,30334,30335,30337],{},[83,30336,21378],{},[77,30338,30339,30341,30343,30349,30351],{},[586,30340,19132],{},[83,30342,21383],{},[19134,30344,30345,30347],{},[83,30346,19792],{},[1493,30348,337],{},[586,30350,862],{"separator":103},[80,30352,30353,30355],{},[83,30354,19165],{},[83,30356,26374],{},[586,30358,862],{"separator":103},[981,30360],{"width":27241},[83,30362,20529],{},[586,30364,2000],{},[19151,30366,30367,30369],{},[83,30368,21378],{},[77,30370,30371,30373,30375,30381,30383],{},[586,30372,19132],{},[83,30374,21383],{},[19134,30376,30377,30379],{},[83,30378,19792],{},[1493,30380,337],{},[586,30382,862],{"separator":103},[80,30384,30385,30387],{},[83,30386,19165],{},[83,30388,26429],{},[586,30390,862],{"separator":103},[981,30392],{"width":27241},[83,30394,90],{},[586,30396,2000],{},[19151,30398,30399,30401],{},[83,30400,21378],{},[77,30402,30403,30405,30407,30413,30415],{},[586,30404,19132],{},[83,30406,21383],{},[19134,30408,30409,30411],{},[83,30410,19792],{},[1493,30412,337],{},[586,30414,862],{"separator":103},[80,30416,30417,30419],{},[83,30418,19165],{},[83,30420,26482],{},[83,30422,332],{"mathvariant":197},[95,30424,30425],{"encoding":97},"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}.",[61,30427,30429,30447,30632,30806],{"className":30428,"ariaHidden":103},[102],[61,30430,30432,30435,30438,30441,30444],{"className":30431},[107],[61,30433],{"className":30434,"style":296},[111],[61,30436,93],{"className":30437,"style":165},[116,120],[61,30439],{"className":30440,"style":19197},[981],[61,30442,2000],{"className":30443},[19201],[61,30445],{"className":30446,"style":19197},[981],[61,30448,30450,30454,30611,30614,30617,30620,30623,30626,30629],{"className":30449},[107],[61,30451],{"className":30452,"style":30453},[111],"height:1.0915em;vertical-align:-0.1944em;",[61,30455,30457,30460],{"className":30456},[116],[61,30458,21378],{"className":30459},[116,120],[61,30461,30463],{"className":30462},[125],[61,30464,30466],{"className":30465},[129],[61,30467,30469],{"className":30468},[134],[61,30470,30473],{"className":30471,"style":30472},[138],"height:0.8971em;",[61,30474,30476,30479],{"style":30475},"top:-3.413em;margin-right:0.05em;",[61,30477],{"className":30478,"style":19306},[146],[61,30480,30482],{"className":30481},[151,152,153,154],[61,30483,30485,30488,30491,30566,30569],{"className":30484},[116,154],[61,30486,19132],{"className":30487},[116,154],[61,30489,21383],{"className":30490},[116,120,154],[61,30492,30494,30498,30563],{"className":30493},[116,154],[61,30495],{"className":30496},[631,19286,151,19345,30497],"size6",[61,30499,30501],{"className":30500},[19134],[61,30502,30504,30554],{"className":30503},[129,130],[61,30505,30507,30551],{"className":30506},[134],[61,30508,30511,30526,30536],{"className":30509,"style":30510},[138],"height:0.6915em;",[61,30512,30514,30517],{"style":30513},"top:-2.656em;",[61,30515],{"className":30516,"style":19306},[146],[61,30518,30520],{"className":30519},[151,19345,19346,154],[61,30521,30523],{"className":30522},[116,154],[61,30524,337],{"className":30525},[116,154],[61,30527,30529,30532],{"style":30528},"top:-3.2255em;",[61,30530],{"className":30531,"style":19306},[146],[61,30533],{"className":30534,"style":30535},[19359,154],"border-bottom-width:0.049em;",[61,30537,30539,30542],{"style":30538},"top:-3.384em;",[61,30540],{"className":30541,"style":19306},[146],[61,30543,30545],{"className":30544},[151,19345,19346,154],[61,30546,30548],{"className":30547},[116,154],[61,30549,19792],{"className":30550,"style":121},[116,120,154],[61,30552,170],{"className":30553},[169],[61,30555,30557],{"className":30556},[134],[61,30558,30561],{"className":30559,"style":30560},[138],"height:0.344em;",[61,30562],{},[61,30564],{"className":30565},[730,19286,151,19345,30497],[61,30567,862],{"className":30568},[977,154],[61,30570,30572,30575],{"className":30571},[116,154],[61,30573,19165],{"className":30574,"style":121},[116,120,154],[61,30576,30578],{"className":30577},[125],[61,30579,30581,30603],{"className":30580},[129,130],[61,30582,30584,30600],{"className":30583},[134],[61,30585,30588],{"className":30586,"style":30587},[138],"height:0.1645em;",[61,30589,30591,30594],{"style":30590},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[61,30592],{"className":30593,"style":19341},[146],[61,30595,30597],{"className":30596},[151,19345,19346,154],[61,30598,26374],{"className":30599},[116,120,154],[61,30601,170],{"className":30602},[169],[61,30604,30606],{"className":30605},[134],[61,30607,30609],{"className":30608,"style":19423},[138],[61,30610],{},[61,30612,862],{"className":30613},[977],[61,30615],{"className":30616,"style":27460},[981],[61,30618],{"className":30619,"style":982},[981],[61,30621,20529],{"className":30622,"style":20194},[116,120],[61,30624],{"className":30625,"style":19197},[981],[61,30627,2000],{"className":30628},[19201],[61,30630],{"className":30631,"style":19197},[981],[61,30633,30635,30638,30785,30788,30791,30794,30797,30800,30803],{"className":30634},[107],[61,30636],{"className":30637,"style":30453},[111],[61,30639,30641,30644],{"className":30640},[116],[61,30642,21378],{"className":30643},[116,120],[61,30645,30647],{"className":30646},[125],[61,30648,30650],{"className":30649},[129],[61,30651,30653],{"className":30652},[134],[61,30654,30656],{"className":30655,"style":30472},[138],[61,30657,30658,30661],{"style":30475},[61,30659],{"className":30660,"style":19306},[146],[61,30662,30664],{"className":30663},[151,152,153,154],[61,30665,30667,30670,30673,30741,30744],{"className":30666},[116,154],[61,30668,19132],{"className":30669},[116,154],[61,30671,21383],{"className":30672},[116,120,154],[61,30674,30676,30679,30738],{"className":30675},[116,154],[61,30677],{"className":30678},[631,19286,151,19345,30497],[61,30680,30682],{"className":30681},[19134],[61,30683,30685,30730],{"className":30684},[129,130],[61,30686,30688,30727],{"className":30687},[134],[61,30689,30691,30705,30713],{"className":30690,"style":30510},[138],[61,30692,30693,30696],{"style":30513},[61,30694],{"className":30695,"style":19306},[146],[61,30697,30699],{"className":30698},[151,19345,19346,154],[61,30700,30702],{"className":30701},[116,154],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act on the Bloch vector by the corresponding rotation matrices (",[61,30965,30967,31019],{"className":30966},[64],[61,30968,30970],{"className":30969},[68],[70,30971,30972],{"xmlns":72},[74,30973,30974,31016],{},[77,30975,30976,30982,30984,30986,30988,30990,30996,30998,31000,31002,31004,31010,31012,31014],{},[80,30977,30978,30980],{},[83,30979,27125],{},[83,30981,26374],{},[586,30983,849],{"stretchy":588},[83,30985,19792],{},[586,30987,300],{"stretchy":588},[586,30989,862],{"separator":103},[80,30991,30992,30994],{},[83,30993,27125],{},[83,30995,26429],{},[586,30997,849],{"stretchy":588},[83,30999,19792],{},[586,31001,300],{"stretchy":588},[586,31003,862],{"separator":103},[80,31005,31006,31008],{},[83,31007,27125],{},[83,31009,26482],{},[586,31011,849],{"stretchy":588},[83,31013,19792],{},[586,31015,300],{"stretchy":588},[95,31017,31018],{"encoding":97},"R_x(\\pi),R_y(\\pi),R_z(\\pi)",[61,31020,31022],{"className":31021,"ariaHidden":103},[102],[61,31023,31025,31028,31069,31072,31075,31078,31081,31084,31124,31127,31130,31133,31136,31139,31179,31182,31185],{"className":31024},[107],[61,31026],{"className":31027,"style":27522},[111],[61,31029,31031,31034],{"className":31030},[116],[61,31032,27125],{"className":31033,"style":30082},[116,120],[61,31035,31037],{"className":31036},[125],[61,31038,31040,31061],{"className":31039},[129,130],[61,31041,31043,31058],{"className":31042},[134],[61,31044,31046],{"className":31045,"style":20164},[138],[61,31047,31049,31052],{"style":31048},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[61,31050],{"className":31051,"style":147},[146],[61,31053,31055],{"className":31054},[151,152,153,154],[61,31056,26374],{"className":31057},[116,120,154],[61,31059,170],{"className":31060},[169],[61,31062,31064],{"className":31063},[134],[61,31065,31067],{"className":31066,"style":177},[138],[61,31068],{},[61,31070,849],{"className":31071},[631],[61,31073,19792],{"className":31074,"style":121},[116,120],[61,31076,300],{"className":31077},[730],[61,31079,862],{"className":31080},[977],[61,31082],{"className":31083,"style":982},[981],[61,31085,31087,31090],{"className":31086},[116],[61,31088,27125],{"className":31089,"style":30082},[116,120],[61,31091,31093],{"className":31092},[125],[61,31094,31096,31116],{"className":31095},[129,130],[61,31097,31099,31113],{"className":31098},[134],[61,31100,31102],{"className":31101,"style":20164},[138],[61,31103,31104,31107],{"style":31048},[61,31105],{"className":31106,"style":147},[146],[61,31108,31110],{"className":31109},[151,152,153,154],[61,31111,26429],{"className":31112,"style":121},[116,120,154],[61,31114,170],{"className":31115},[169],[61,31117,31119],{"className":31118},[134],[61,31120,31122],{"className":31121,"style":26767},[138],[61,31123],{},[61,31125,849],{"className":31126},[631],[61,31128,19792],{"className":31129,"style":121},[116,120],[61,31131,300],{"className":31132},[730],[61,31134,862],{"className":31135},[977],[61,31137],{"className":31138,"style":982},[981],[61,31140,31142,31145],{"className":31141},[116],[61,31143,27125],{"className":31144,"style":30082},[116,120],[61,31146,31148],{"className":31147},[125],[61,31149,31151,31171],{"className":31150},[129,130],[61,31152,31154,31168],{"className":31153},[134],[61,31155,31157],{"className":31156,"style":20164},[138],[61,31158,31159,31162],{"style":31048},[61,31160],{"className":31161,"style":147},[146],[61,31163,31165],{"className":31164},[151,152,153,154],[61,31166,26482],{"className":31167,"style":26946},[116,120,154],[61,31169,170],{"className":31170},[169],[61,31172,31174],{"className":31173},[134],[61,31175,31177],{"className":31176,"style":177},[138],[61,31178],{},[61,31180,849],{"className":31181},[631],[61,31183,19792],{"className":31184,"style":121},[116,120],[61,31186,300],{"className":31187},[730],"), so ",[814,31190,29894],{}," implies the Wigner pattern simply rotates rigidly on (",[61,31193,31195,31212],{"className":31194},[64],[61,31196,31198],{"className":31197},[68],[70,31199,31200],{"xmlns":72},[74,31201,31202,31210],{},[77,31203,31204],{},[19151,31205,31206,31208],{},[83,31207,28508],{},[1493,31209,337],{},[95,31211,28513],{"encoding":97},[61,31213,31215],{"className":31214,"ariaHidden":103},[102],[61,31216,31218,31221],{"className":31217},[107],[61,31219],{"className":31220,"style":27160},[111],[61,31222,31224,31227],{"className":31223},[116],[61,31225,28508],{"className":31226,"style":19831},[116,120],[61,31228,31230],{"className":31229},[125],[61,31231,31233],{"className":31232},[129],[61,31234,31236],{"className":31235},[134],[61,31237,31239],{"className":31238,"style":27160},[138],[61,31240,31241,31244],{"style":21487},[61,31242],{"className":31243,"style":147},[146],[61,31245,31247],{"className":31246},[151,152,153,154],[61,31248,337],{"className":31249},[116,154],[1632,31251],{},[25,31253,31255],{"id":31254},"_4-general-su2-rotation-and-the-so3-map","4. 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center",[26387,32376,32377,32383,32397,32417,32423,32447,32457],{},[26390,32378,32379],{},[26393,32380,32381],{"scriptlevel":1495,"displaystyle":588},[1493,32382,1495],{},[26390,32384,32385],{},[26393,32386,32387],{"scriptlevel":1495,"displaystyle":588},[77,32388,32389,32391],{},[586,32390,19132],{},[80,32392,32393,32395],{},[83,32394,31276],{},[83,32396,26482],{},[26390,32398,32399],{},[26393,32400,32401],{"scriptlevel":1495,"displaystyle":588},[77,32402,32403,32409,32411],{},[80,32404,32405,32407],{},[83,32406,31276],{},[83,32408,26429],{},[27272,32410,27274],{},[80,32412,32413,32415],{},[83,32414,31276],{},[83,32416,26482],{},[26390,32418,32419],{},[26393,32420,32421],{"scriptlevel":1495,"displaystyle":588},[1493,32422,1495],{},[26390,32424,32425],{},[26393,32426,32427],{"scriptlevel":1495,"displaystyle":588},[77,32428,32429,32431,32437,32439,32441],{},[586,32430,19132],{},[80,32432,32433,32435],{},[83,32434,31276],{},[83,32436,26374],{},[27272,32438,27274],{},[586,32440,19132],{},[80,32442,32443,32445],{},[83,32444,31276],{},[83,32446,26429],{},[26390,32448,32449],{},[26393,32450,32451],{"scriptlevel":1495,"displaystyle":588},[80,32452,32453,32455],{},[83,32454,31276],{},[83,32456,26374],{},[26390,32458,32459],{},[26393,32460,32461],{"scriptlevel":1495,"displaystyle":588},[1493,32462,1495],{},[586,32464,300],{"fence":103},[83,32466,332],{"mathvariant":197},[95,32468,32469],{"encoding":97},"K(\\hat{\\mathbf 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0\n\\end{pmatrix}.",[61,32471,32473,32528],{"className":32472,"ariaHidden":103},[102],[61,32474,32476,32479,32482,32485,32516,32519,32522,32525],{"className":32475},[107],[61,32477],{"className":32478,"style":627},[111],[61,32480,6648],{"className":32481,"style":161},[116,120],[61,32483,849],{"className":32484},[631],[61,32486,32488],{"className":32487},[116,31296],[61,32489,32491],{"className":32490},[129],[61,32492,32494],{"className":32493},[134],[61,32495,32497,32505],{"className":32496,"style":31292},[138],[61,32498,32499,32502],{"style":28379},[61,32500],{"className":32501,"style":19306},[146],[61,32503,31276],{"className":32504},[116,27144],[61,32506,32507,32510],{"style":31316},[61,32508],{"className":32509,"style":19306},[146],[61,32511,32513],{"className":32512,"style":31324},[31323],[61,32514,31279],{"className":32515},[116],[61,32517,300],{"className":32518},[730],[61,32520],{"className":32521,"style":19197},[981],[61,32523,2000],{"className":32524},[19201],[61,32526],{"className":32527,"style":19197},[981],[61,32529,32531,32534,33081,33084],{"className":32530},[107],[61,32532],{"className":32533,"style":27324},[111],[61,32535,32537,32543,33075],{"className":32536},[19266],[61,32538,32540],{"className":32539,"style":19271},[631,19270],[61,32541,849],{"className":32542},[19275,19346],[61,32544,32546],{"className":32545},[116],[61,32547,32549,32586,32589,32592,32666,32669,32672,32786,32789,32792,32826,32829,32832,32958,32961,32964,33035,33038,33041],{"className":32548},[26382],[61,32550,32552],{"className":32551},[26611],[61,32553,32555,32577],{"className":32554},[129,130],[61,32556,32558,32574],{"className":32557},[134],[61,32559,32562],{"className":32560,"style":32561},[138],"height:0.85em;",[61,32563,32565,32568],{"style":32564},"top:-3.01em;",[61,32566],{"className":32567,"style":19306},[146],[61,32569,32571],{"className":32570},[116],[61,32572,1495],{"className":32573},[116],[61,32575,170],{"className":32576},[169],[61,32578,32580],{"className":32579},[134],[61,32581,32584],{"className":32582,"style":32583},[138],"height:0.35em;",[61,32585],{},[61,32587],{"className":32588,"style":26662},[26661],[61,32590],{"className":32591,"style":26662},[26661],[61,32593,32595],{"className":32594},[26611],[61,32596,32598,32658],{"className":32597},[129,130],[61,32599,32601,32655],{"className":32600},[134],[61,32602,32604],{"className":32603,"style":32561},[138],[61,32605,32606,32609],{"style":32564},[61,32607],{"className":32608,"style":19306},[146],[61,32610,32612,32615],{"className":32611},[116],[61,32613,19132],{"className":32614},[116],[61,32616,32618,32621],{"className":32617},[116],[61,32619,31276],{"className":32620},[116,120],[61,32622,32624],{"className":32623},[125],[61,32625,32627,32647],{"className":32626},[129,130],[61,32628,32630,32644],{"className":32629},[134],[61,32631,32633],{"className":32632,"style":20164},[138],[61,32634,32635,32638],{"style":241},[61,32636],{"className":32637,"style":147},[146],[61,32639,32641],{"className":32640},[151,152,153,154],[61,32642,26482],{"className":32643,"style":26946},[116,120,154],[61,32645,170],{"className":32646},[169],[61,32648,32650],{"className":32649},[134],[61,32651,32653],{"className":32652,"style":177},[138],[61,32654],{},[61,32656,170],{"className":32657},[169],[61,32659,32661],{"className":32660},[134],[61,32662,32664],{"className":32663,"style":32583},[138],[61,32665],{},[61,32667],{"className":32668,"style":26662},[26661],[61,32670],{"className":32671,"style":26662},[26661],[61,32673,32675],{"className":32674},[26611],[61,32676,32678,32778],{"className":32677},[129,130],[61,32679,32681,32775],{"className":32680},[134],[61,32682,32684],{"className":32683,"style":32561},[138],[61,32685,32686,32689],{"style":32564},[61,32687],{"className":32688,"style":19306},[146],[61,32690,32692,32732,32735],{"className":32691},[116],[61,32693,32695,32698],{"className":32694},[116],[61,32696,31276],{"className":32697},[116,120],[61,32699,32701],{"className":32700},[125],[61,32702,32704,32724],{"className":32703},[129,130],[61,32705,32707,32721],{"className":32706},[134],[61,32708,32710],{"className":32709,"style":20164},[138],[61,32711,32712,32715],{"style":241},[61,32713],{"className":32714,"style":147},[146],[61,32716,32718],{"className":32717},[151,152,153,154],[61,32719,26429],{"className":32720,"style":121},[116,120,154],[61,32722,170],{"className":32723},[169],[61,32725,32727],{"className":32726},[134],[61,32728,32730],{"className":32729,"style":26767},[138],[61,32731],{},[61,32733,27274],{"className":32734},[981],[61,32736,32738,32741],{"className":32737},[116],[61,32739,31276],{"className":32740},[116,120],[61,32742,32744],{"className":32743},[125],[61,32745,32747,32767],{"className":32746},[129,130],[61,32748,32750,32764],{"className":32749},[134],[61,32751,32753],{"className":32752,"style":20164},[138],[61,32754,32755,32758],{"style":241},[61,32756],{"className":32757,"style":147},[146],[61,32759,32761],{"className":32760},[151,152,153,154],[61,327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line sums reproducing measurement probabilities in mutually unbiased bases.",[1632,35108],{},[25,35110,35112],{"id":35111},"_7-color-scale-purestate-bounds","7. 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constants are useful for fixing a consistent color scale across frames.",[1632,35856],{},[25,35858,35860],{"id":35859},"_8-acronymnotation-glossary","8. Acronym\u002FNotation Glossary",[1640,35862,35863,35869,35875,35881,35887,36005],{},[1643,35864,35865,35868],{},[814,35866,35867],{},"SW"," — Stratonovich–Weyl (kernel\u002Fcorrespondence).",[1643,35870,35871,35874],{},[814,35872,35873],{},"SU(2)"," — Special Unitary group of degree two (spin‑(1\u002F2) rotations).",[1643,35876,35877,35880],{},[814,35878,35879],{},"SO(3)"," — Special Orthogonal group in 3D (real rotation matrices).",[1643,35882,35883,35886],{},[814,35884,35885],{},"WF"," — Wigner function.",[1643,35888,35889,35930,35931,35972,35973,36004],{},[814,35890,849,35891,300],{},[61,35892,35894,35912],{"className":35893},[64],[61,35895,35897],{"className":35896},[68],[70,35898,35899],{"xmlns":72},[74,35900,35901,35909],{},[77,35902,35903,35905,35907],{},[586,35904,589],{"stretchy":588},[83,35906,27662],{},[586,35908,1498],{"stretchy":588},[95,35910,35911],{"encoding":97},"\\lvert\\psi\\rangle",[61,35913,35915],{"className":35914,"ariaHidden":103},[102],[61,35916,35918,35921,35924,35927],{"className":35917},[107],[61,35919],{"className":35920,"style":627},[111],[61,35922,589],{"className":35923},[631],[61,35925,27662],{"className":35926,"style":121},[116,120],[61,35928,1498],{"className":35929},[730]," — State vector (ket); ",[814,35932,849,35933,300],{},[61,35934,35936,35954],{"className":35935},[64],[61,35937,35939],{"className":35938},[68],[70,35940,35941],{"xmlns":72},[74,35942,35943,35951],{},[77,35944,35945,35947,35949],{},[586,35946,27667],{"stretchy":588},[83,35948,27662],{},[586,35950,589],{"stretchy":588},[95,35952,35953],{"encoding":97},"\\langle\\psi\\rvert",[61,35955,35957],{"className":35956,"ariaHidden":103},[102],[61,35958,35960,35963,35966,35969],{"className":35959},[107],[61,35961],{"className":35962,"style":627},[111],[61,35964,27667],{"className":35965},[631],[61,35967,27662],{"className":35968,"style":121},[116,120],[61,35970,589],{"className":35971},[730]," — bra; ",[814,35974,849,35975,300],{},[61,35976,35978,35992],{"className":35977},[64],[61,35979,35981],{"className":35980},[68],[70,35982,35983],{"xmlns":72},[74,35984,35985,35989],{},[77,35986,35987],{},[83,35988,27209],{},[95,35990,35991],{"encoding":97},"\\rho",[61,35993,35995],{"className":35994,"ariaHidden":103},[102],[61,35996,35998,36001],{"className":35997},[107],[61,35999],{"className":36000,"style":27305},[111],[61,36002,27209],{"className":36003},[116,120]," — density operator.",[1643,36006,36007,36010,36011,36130,36131,1630],{},[814,36008,36009],{},"Adjoint map"," — Conjugation action (",[61,36012,36014,36046],{"className":36013},[64],[61,36015,36017],{"className":36016},[68],[70,36018,36019],{"xmlns":72},[74,36020,36021,36043],{},[77,36022,36023,36025,36028,36030,36033,36035,36037],{},[83,36024,29828],{},[586,36026,36027],{},":",[83,36029,27209],{},[586,36031,36032],{},"↦",[83,36034,29828],{},[83,36036,27209],{},[19151,36038,36039,36041],{},[83,36040,29828],{},[586,36042,29838],{"lspace":29837,"rspace":29837},[95,36044,36045],{"encoding":97},"U: \\rho\\mapsto U\\rho U^{\\dagger}",[61,36047,36049,36067,36085],{"className":36048,"ariaHidden":103},[102],[61,36050,36052,36055,36058,36061,36064],{"className":36051},[107],[61,36053],{"className":36054,"style":296},[111],[61,36056,29828],{"className":36057,"style":23693},[116,120],[61,36059],{"className":36060,"style":19197},[981],[61,36062,36027],{"className":36063},[19201],[61,36065],{"className":36066,"style":19197},[981],[61,36068,36070,36073,36076,36079,36082],{"className":36069},[107],[61,36071],{"className":36072,"style":31898},[111],[61,36074,27209],{"className":36075},[116,120],[61,36077],{"className":36078,"style":19197},[981],[61,36080,36032],{"className":36081},[19201],[61,36083],{"className":36084,"style":19197},[981],[61,36086,36088,36092,36095,36098],{"className":36087},[107],[61,36089],{"className":36090,"style":36091},[111],"height:1.0435em;vertical-align:-0.1944em;",[61,36093,29828],{"className":36094,"style":23693},[116,120],[61,36096,27209],{"className":36097},[116,120],[61,36099,36101,36104],{"className":36100},[116],[61,36102,29828],{"className":36103,"style":23693},[116,120],[61,36105,36107],{"className":36106},[125],[61,36108,36110],{"className":36109},[129],[61,36111,36113],{"className":36112},[134],[61,36114,36116],{"className":36115,"style":21484},[138],[61,36117,36118,36121],{"style":21487},[61,36119],{"className":36120,"style":147},[146],[61,36122,36124],{"className":36123},[151,152,153,154],[61,36125,36127],{"className":36126},[116,154],[61,36128,29838],{"className":36129},[116,154],") inducing (",[61,36132,36134,36154],{"className":36133},[64],[61,36135,36137],{"className":36136},[68],[70,36138,36139],{"xmlns":72},[74,36140,36141,36151],{},[77,36142,36143,36145,36147,36149],{},[83,36144,3763],{"mathvariant":27116},[586,36146,36032],{},[83,36148,27125],{},[83,36150,3763],{"mathvariant":27116},[95,36152,36153],{"encoding":97},"\\mathbf r\\mapsto R\\mathbf r",[61,36155,36157,36176],{"className":36156,"ariaHidden":103},[102],[61,36158,36160,36164,36167,36170,36173],{"className":36159},[107],[61,36161],{"className":36162,"style":36163},[111],"height:0.522em;vertical-align:-0.011em;",[61,36165,3763],{"className":36166},[116,27144],[61,36168],{"className":36169,"style":19197},[981],[61,36171,36032],{"className":36172},[19201],[61,36174],{"className":36175,"style":19197},[981],[61,36177,36179,36182,36185],{"className":36178},[107],[61,36180],{"className":36181,"style":296},[111],[61,36183,27125],{"className":36184,"style":30082},[116,120],[61,36186,3763],{"className":36187},[116,27144],[1632,36189],{},[25,36191,36193],{"id":36192},"_9-mindmap","9. Mind‑Map",[12,36195,36196],{},"Wigner for Qubits",[1720,36198,36202],{"className":36199,"code":36201,"language":29075},[36200],"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",[1727,36203,36201],{"__ignoreMap":1725},[1632,36205],{},[25,36207,36209],{"id":36208},"_10-minimal-substitutions-if-variations-are-needed","10. Minimal Substitutions (if variations are needed)",[1640,36211,36212,36274],{},[1643,36213,36214,36215,36244,36245,36273],{},"If a different SW kernel normalization is chosen, the prefactors in (",[61,36216,36218,36232],{"className":36217},[64],[61,36219,36221],{"className":36220},[68],[70,36222,36223],{"xmlns":72},[74,36224,36225,36229],{},[77,36226,36227],{},[83,36228,198],{"mathvariant":197},[95,36230,36231],{"encoding":97},"\\Delta",[61,36233,36235],{"className":36234,"ariaHidden":103},[102],[61,36236,36238,36241],{"className":36237},[107],[61,36239],{"className":36240,"style":296},[111],[61,36242,198],{"className":36243},[116],") and hence (",[61,36246,36248,36261],{"className":36247},[64],[61,36249,36251],{"className":36250},[68],[70,36252,36253],{"xmlns":72},[74,36254,36255,36259],{},[77,36256,36257],{},[83,36258,28581],{},[95,36260,28581],{"encoding":97},[61,36262,36264],{"className":36263,"ariaHidden":103},[102],[61,36265,36267,36270],{"className":36266},[107],[61,36268],{"className":36269,"style":296},[111],[61,36271,28581],{"className":36272,"style":23700},[116,120],") adjust accordingly, while covariance and linearity remain.",[1643,36275,36276,36277,36338,36339,36378],{},"Mixed states use the same formulas with (",[61,36278,36280,36302],{"className":36279},[64],[61,36281,36283],{"className":36282},[68],[70,36284,36285],{"xmlns":72},[74,36286,36287,36299],{},[77,36288,36289,36291,36293,36295,36297],{},[586,36290,27735],{"stretchy":588},[83,36292,3763],{"mathvariant":27116},[586,36294,27735],{"stretchy":588},[586,36296,2181],{},[1493,36298,57],{},[95,36300,36301],{"encoding":97},"\\lVert\\mathbf r\\rVert\u003C1",[61,36303,36305,36329],{"className":36304,"ariaHidden":103},[102],[61,36306,36308,36311,36314,36317,36320,36323,36326],{"className":36307},[107],[61,36309],{"className":36310,"style":627},[111],[61,36312,27735],{"className":36313},[631],[61,36315,3763],{"className":36316},[116,27144],[61,36318,27735],{"className":36319},[730],[61,36321],{"className":36322,"style":19197},[981],[61,36324,2181],{"className":36325},[19201],[61,36327],{"className":36328,"style":19197},[981],[61,36330,36332,36335],{"className":36331},[107],[61,36333],{"className":36334,"style":27780},[111],[61,36336,57],{"className":36337},[116],") (the extrema shrink linearly with (",[61,36340,36342,36360],{"className":36341},[64],[61,36343,36345],{"className":36344},[68],[70,36346,36347],{"xmlns":72},[74,36348,36349,36357],{},[77,36350,36351,36353,36355],{},[586,36352,27735],{"stretchy":588},[83,36354,3763],{"mathvariant":27116},[586,36356,27735],{"stretchy":588},[95,36358,36359],{"encoding":97},"\\lVert\\mathbf r\\rVert",[61,36361,36363],{"className":36362,"ariaHidden":103},[102],[61,36364,36366,36369,36372,36375],{"className":36365},[107],[61,36367],{"className":36368,"style":627},[111],[61,36370,27735],{"className":36371},[631],[61,36373,3763],{"className":36374},[116,27144],[61,36376,27735],{"className":36377},[730],")).",[1720,36380,36382],{"className":1722,"code":36381,"language":1724,"meta":1725,"style":1725},"# @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",[1727,36383,36384,36389,36394,36399,36403,36408,36413,36418,36422,36427,36432,36436,36440,36453,36457,36467,36479,36489,36493,36498,36509,36525,36540,36562,36574],{"__ignoreMap":1725},[61,36385,36386],{"class":1731,"line":1732},[61,36387,36388],{"class":1735},"# @title Cell 1 — Imports & global plotting config (DPI=200)\n",[61,36390,36391],{"class":1731,"line":1739},[61,36392,36393],{"class":2165},"\"\"\"\n",[61,36395,36396],{"class":1731,"line":1746},[61,36397,36398],{"class":2165},"Spin (SU(2)) Wigner-function animations for single-qubit gates.\n",[61,36400,36401],{"class":1731,"line":1763},[61,36402,1743],{"emptyLinePlaceholder":1742},[61,36404,36405],{"class":1731,"line":1776},[61,36406,36407],{"class":2165},"This notebook renders the qubit Wigner function\n",[61,36409,36410],{"class":1731,"line":1784},[61,36411,36412],{"class":2165},"    W(θ, φ) = 1\u002F2 + (√3\u002F2) * r · n(θ, φ)\n",[61,36414,36415],{"class":1731,"line":1798},[61,36416,36417],{"class":2165},"and animates its rigid rotation under Pauli and custom axis–angle gates.\n",[61,36419,36420],{"class":1731,"line":1811},[61,36421,1743],{"emptyLinePlaceholder":1742},[61,36423,36424],{"class":1731,"line":1816},[61,36425,36426],{"class":2165},"All titles and dynamic text are drawn *inside* the plotted region so they\n",[61,36428,36429],{"class":1731,"line":1822},[61,36430,36431],{"class":2165},"never clip in the saved GIFs. Use Cells 3 and 4 to generate animations.\n",[61,36433,36434],{"class":1731,"line":1828},[61,36435,36393],{"class":2165},[61,36437,36438],{"class":1731,"line":1833},[61,36439,1743],{"emptyLinePlaceholder":1742},[61,36441,36442,36444,36447,36450],{"class":1731,"line":1840},[61,36443,1750],{"class":1749},[61,36445,36446],{"class":3766}," __future__",[61,36448,36449],{"class":1749}," import",[61,36451,36452],{"class":1753}," annotations\n",[61,36454,36455],{"class":1731,"line":1853},[61,36456,1743],{"emptyLinePlaceholder":1742},[61,36458,36459,36461,36463,36465],{"class":1731,"line":1873},[61,36460,1757],{"class":1749},[61,36462,1789],{"class":1753},[61,36464,1792],{"class":1749},[61,36466,1795],{"class":1753},[61,36468,36469,36471,36474,36476],{"class":1731,"line":1889},[61,36470,1757],{"class":1749},[61,36472,36473],{"class":1753}," matplotlib ",[61,36475,1792],{"class":1749},[61,36477,36478],{"class":1753}," mpl\n",[61,36480,36481,36483,36485,36487],{"class":1731,"line":1908},[61,36482,1757],{"class":1749},[61,36484,1803],{"class":1753},[61,36486,1792],{"class":1749},[61,36488,1808],{"class":1753},[61,36490,36491],{"class":1731,"line":1925},[61,36492,1743],{"emptyLinePlaceholder":1742},[61,36494,36495],{"class":1731,"line":1941},[61,36496,36497],{"class":1735},"# Global Matplotlib configuration: DPI = 200 everywhere\n",[61,36499,36500,36503,36506],{"class":1731,"line":1957},[61,36501,36502],{"class":1753},"mpl.rcParams.",[61,36504,36505],{"class":1836},"update",[61,36507,36508],{"class":1753},"({\n",[61,36510,36511,36514,36516,36519,36522],{"class":1731,"line":1974},[61,36512,36513],{"class":2165},"    \"figure.dpi\"",[61,36515,2127],{"class":1753},[61,36517,36518],{"class":1866},"200",[61,36520,36521],{"class":1753},",       ",[61,36523,36524],{"class":1735},"# on-screen DPI\n",[61,36526,36527,36530,36532,36534,36537],{"class":1731,"line":1979},[61,36528,36529],{"class":2165},"    \"savefig.dpi\"",[61,36531,2127],{"class":1753},[61,36533,36518],{"class":1866},[61,36535,36536],{"class":1753},",      ",[61,36538,36539],{"class":1735},"# saved file DPI\n",[61,36541,36542,36545,36548,36551,36553,36556,36559],{"class":1731,"line":1985},[61,36543,36544],{"class":2165},"    \"figure.figsize\"",[61,36546,36547],{"class":1753},": (",[61,36549,36550],{"class":1866},"6.4",[61,36552,1435],{"class":1753},[61,36554,36555],{"class":1866},"3.6",[61,36557,36558],{"class":1753},"),  ",[61,36560,36561],{"class":1735},"# 1280×720 at DPI=200\n",[61,36563,36564,36567,36569,36571],{"class":1731,"line":2019},[61,36565,36566],{"class":2165},"    \"axes.grid\"",[61,36568,2127],{"class":1753},[61,36570,14390],{"class":1866},[61,36572,36573],{"class":1753},",\n",[61,36575,36576],{"class":1731,"line":2033},[61,36577,36578],{"class":1753},"})\n",[1720,36580,36582],{"className":1722,"code":36581,"language":1724,"meta":1725,"style":1725},"# @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",[1727,36583,36584,36589,36599,36611,36622,36629,36633,36643,36653,36665,36677,36681,36685,36689,36698,36703,36707,36712,36717,36722,36727,36732,36737,36742,36747,36752,36757,36762,36767,36772,36777,36811,36823,36856,36868,36880,36892,36904,36916,36941,36953,36965,36977,36981,36985,36990,37008,37029,37034,37044,37082,37120,37158,37163,37193,37208,37251,37255,37259,37264,37292,37297,37316,37330,37372,37385,37427,37440,37480,37493,37497,37501,37527,37538,37543,37561,37574,37586,37598,37612,37681,37698,37714,37757,37761,37765,37770,37797,37802,37820,37844,37869,37892,37913,37927,37947,37951,37955,37974,37979,38044,38048,38052,38057,38088,38093,38107,38121,38159,38172,38206,38219,38253,38266,38298,38330,38334,38338,38352,38363,38378,38394,38411,38416,38428,38440,38451,38478,38527,38532,38536,38540,38544,38566,38571,38581,38593,38618,38630,38670,38681],{"__ignoreMap":1725},[61,36585,36586],{"class":1731,"line":1732},[61,36587,36588],{"class":1735},"# @title Cell 2 — Math & utilities (PEP 8 \u002F PEP 257)\n",[61,36590,36591,36593,36595,36597],{"class":1731,"line":1739},[61,36592,1750],{"class":1749},[61,36594,1754],{"class":1753},[61,36596,1757],{"class":1749},[61,36598,1760],{"class":1753},[61,36600,36601,36603,36606,36608],{"class":1731,"line":1746},[61,36602,1750],{"class":1749},[61,36604,36605],{"class":1753}," pathlib ",[61,36607,1757],{"class":1749},[61,36609,36610],{"class":1753}," Path\n",[61,36612,36613,36615,36617,36619],{"class":1731,"line":1763},[61,36614,1750],{"class":1749},[61,36616,1768],{"class":1753},[61,36618,1757],{"class":1749},[61,36620,36621],{"class":1753}," Tuple, Iterable\n",[61,36623,36624,36626],{"class":1731,"line":1776},[61,36625,1757],{"class":1749},[61,36627,36628],{"class":1753}," textwrap\n",[61,36630,36631],{"class":1731,"line":1784},[61,36632,1743],{"emptyLinePlaceholder":1742},[61,36634,36635,36637,36639,36641],{"class":1731,"line":1798},[61,36636,1757],{"class":1749},[61,36638,1789],{"class":1753},[61,36640,1792],{"class":1749},[61,36642,1795],{"class":1753},[61,36644,36645,36647,36649,36651],{"class":1731,"line":1811},[61,36646,1757],{"class":1749},[61,36648,1803],{"class":1753},[61,36650,1792],{"class":1749},[61,36652,1808],{"class":1753},[61,36654,36655,36657,36660,36662],{"class":1731,"line":1816},[61,36656,1750],{"class":1749},[61,36658,36659],{"class":1753}," matplotlib.animation ",[61,36661,1757],{"class":1749},[61,36663,36664],{"class":1753}," FuncAnimation, PillowWriter\n",[61,36666,36667,36669,36672,36674],{"class":1731,"line":1822},[61,36668,1750],{"class":1749},[61,36670,36671],{"class":1753}," IPython.display ",[61,36673,1757],{"class":1749},[61,36675,36676],{"class":1753}," Image, display\n",[61,36678,36679],{"class":1731,"line":1828},[61,36680,1743],{"emptyLinePlaceholder":1742},[61,36682,36683],{"class":1731,"line":1833},[61,36684,1743],{"emptyLinePlaceholder":1742},[61,36686,36687],{"class":1731,"line":1840},[61,36688,1837],{"class":1836},[61,36690,36691,36693,36696],{"class":1731,"line":1853},[61,36692,1843],{"class":1749},[61,36694,36695],{"class":1846}," Knobs",[61,36697,1850],{"class":1753},[61,36699,36700],{"class":1731,"line":1873},[61,36701,36702],{"class":2165},"    \"\"\"User controls for Wigner animations.\n",[61,36704,36705],{"class":1731,"line":1889},[61,36706,1743],{"emptyLinePlaceholder":1742},[61,36708,36709],{"class":1731,"line":1908},[61,36710,36711],{"class":2165},"    Attributes:\n",[61,36713,36714],{"class":1731,"line":1925},[61,36715,36716],{"class":2165},"        gate_axes: Pauli axes to animate, each in {\"x\", \"y\", \"z\"}.\n",[61,36718,36719],{"class":1731,"line":1941},[61,36720,36721],{"class":2165},"        initial_state: Starting Bloch eigenstate, or \"custom\".\n",[61,36723,36724],{"class":1731,"line":1957},[61,36725,36726],{"class":2165},"        custom_r: Custom Bloch vector if initial_state == \"custom\".\n",[61,36728,36729],{"class":1731,"line":1974},[61,36730,36731],{"class":2165},"        frames: Number of frames (≥ 2).\n",[61,36733,36734],{"class":1731,"line":1979},[61,36735,36736],{"class":2165},"        n_theta: Polar samples (θ ∈ [0, π]).\n",[61,36738,36739],{"class":1731,"line":1985},[61,36740,36741],{"class":2165},"        n_phi: Azimuth samples (φ ∈ [0, 2π]).\n",[61,36743,36744],{"class":1731,"line":2019},[61,36745,36746],{"class":2165},"        fps: GIF framerate.\n",[61,36748,36749],{"class":1731,"line":2033},[61,36750,36751],{"class":2165},"        out_dir: Output directory (e.g., \"\u002Fcontent\" in Colab).\n",[61,36753,36754],{"class":1731,"line":2061},[61,36755,36756],{"class":2165},"        figsize: Figure size in inches (DPI set globally).\n",[61,36758,36759],{"class":1731,"line":2074},[61,36760,36761],{"class":2165},"        info_loc: In-axes anchor (\"ul\", \"ur\", \"ll\", \"lr\").\n",[61,36763,36764],{"class":1731,"line":2079},[61,36765,36766],{"class":2165},"        info_fontsize: Font size for in-axes info text.\n",[61,36768,36769],{"class":1731,"line":2092},[61,36770,36771],{"class":2165},"        wrap_chars: Manual wrap width for info text.\n",[61,36773,36774],{"class":1731,"line":2097},[61,36775,36776],{"class":2165},"    \"\"\"\n",[61,36778,36779,36782,36785,36787,36790,36792,36794,36796,36799,36801,36804,36806,36809],{"class":1731,"line":2102},[61,36780,36781],{"class":1753},"    gate_axes: Tuple[",[61,36783,36784],{"class":1859},"str",[61,36786,1435],{"class":1753},[61,36788,36789],{"class":1866},"...",[61,36791,1997],{"class":1753},[61,36793,2000],{"class":1859},[61,36795,2003],{"class":1753},[61,36797,36798],{"class":2165},"\"x\"",[61,36800,1435],{"class":1753},[61,36802,36803],{"class":2165},"\"y\"",[61,36805,1435],{"class":1753},[61,36807,36808],{"class":2165},"\"z\"",[61,36810,2016],{"class":1753},[61,36812,36813,36816,36818,36820],{"class":1731,"line":2108},[61,36814,36815],{"class":1753},"    initial_state: ",[61,36817,36784],{"class":1859},[61,36819,1863],{"class":1859},[61,36821,36822],{"class":2165}," \"+z\"\n",[61,36824,36825,36828,36830,36832,36834,36836,36838,36840,36842,36844,36846,36848,36850,36852,36854],{"class":1731,"line":2113},[61,36826,36827],{"class":1753},"    custom_r: Tuple[",[61,36829,1860],{"class":1859},[61,36831,1435],{"class":1753},[61,36833,1860],{"class":1859},[61,36835,1435],{"class":1753},[61,36837,1860],{"class":1859},[61,36839,1997],{"class":1753},[61,36841,2000],{"class":1859},[61,36843,2003],{"class":1753},[61,36845,2051],{"class":1866},[61,36847,1435],{"class":1753},[61,36849,2051],{"class":1866},[61,36851,1435],{"class":1753},[61,36853,2319],{"class":1866},[61,36855,2016],{"class":1753},[61,36857,36858,36861,36863,36865],{"class":1731,"line":2162},[61,36859,36860],{"class":1753},"    frames: ",[61,36862,2025],{"class":1859},[61,36864,1863],{"class":1859},[61,36866,36867],{"class":1866}," 24\n",[61,36869,36870,36873,36875,36877],{"class":1731,"line":2169},[61,36871,36872],{"class":1753},"    n_theta: ",[61,36874,2025],{"class":1859},[61,36876,1863],{"class":1859},[61,36878,36879],{"class":1866}," 80\n",[61,36881,36882,36885,36887,36889],{"class":1731,"line":2187},[61,36883,36884],{"class":1753},"    n_phi: ",[61,36886,2025],{"class":1859},[61,36888,1863],{"class":1859},[61,36890,36891],{"class":1866}," 160\n",[61,36893,36894,36897,36899,36901],{"class":1731,"line":2207},[61,36895,36896],{"class":1753},"    fps: ",[61,36898,2025],{"class":1859},[61,36900,1863],{"class":1859},[61,36902,36903],{"class":1866}," 20\n",[61,36905,36906,36909,36911,36913],{"class":1731,"line":2221},[61,36907,36908],{"class":1753},"    out_dir: ",[61,36910,36784],{"class":1859},[61,36912,1863],{"class":1859},[61,36914,36915],{"class":2165}," \"\u002Fcontent\"\n",[61,36917,36918,36921,36923,36925,36927,36929,36931,36933,36935,36937,36939],{"class":1731,"line":2226},[61,36919,36920],{"class":1753},"    figsize: Tuple[",[61,36922,1860],{"class":1859},[61,36924,1435],{"class":1753},[61,36926,1860],{"class":1859},[61,36928,1997],{"class":1753},[61,36930,2000],{"class":1859},[61,36932,2003],{"class":1753},[61,36934,36550],{"class":1866},[61,36936,1435],{"class":1753},[61,36938,36555],{"class":1866},[61,36940,2016],{"class":1753},[61,36942,36943,36946,36948,36950],{"class":1731,"line":2258},[61,36944,36945],{"class":1753},"    info_loc: ",[61,36947,36784],{"class":1859},[61,36949,1863],{"class":1859},[61,36951,36952],{"class":2165}," \"ul\"\n",[61,36954,36955,36958,36960,36962],{"class":1731,"line":2264},[61,36956,36957],{"class":1753},"    info_fontsize: ",[61,36959,2025],{"class":1859},[61,36961,1863],{"class":1859},[61,36963,36964],{"class":1866}," 9\n",[61,36966,36967,36970,36972,36974],{"class":1731,"line":2280},[61,36968,36969],{"class":1753},"    wrap_chars: ",[61,36971,2025],{"class":1859},[61,36973,1863],{"class":1859},[61,36975,36976],{"class":1866}," 42\n",[61,36978,36979],{"class":1731,"line":2304},[61,36980,1743],{"emptyLinePlaceholder":1742},[61,36982,36983],{"class":1731,"line":2326},[61,36984,1743],{"emptyLinePlaceholder":1742},[61,36986,36987],{"class":1731,"line":2342},[61,36988,36989],{"class":1735},"# ------------------------------ Bloch helpers ---------------------------------\n",[61,36991,36992,36994,36997,36999,37002,37004,37006],{"class":1731,"line":2379},[61,36993,2116],{"class":1749},[61,36995,36996],{"class":1836}," bloch_vector",[61,36998,849],{"class":1753},[61,37000,37001],{"class":2124},"initial_state",[61,37003,2127],{"class":1753},[61,37005,36784],{"class":1859},[61,37007,36573],{"class":1753},[61,37009,37010,37013,37016,37018,37020,37022,37024,37026],{"class":1731,"line":2396},[61,37011,37012],{"class":2124},"                 custom_r",[61,37014,37015],{"class":1753},": Tuple[",[61,37017,1860],{"class":1859},[61,37019,1435],{"class":1753},[61,37021,1860],{"class":1859},[61,37023,1435],{"class":1753},[61,37025,1860],{"class":1859},[61,37027,37028],{"class":1753},"]) -> np.ndarray:\n",[61,37030,37031],{"class":1731,"line":2404},[61,37032,37033],{"class":2165},"    \"\"\"Return a unit Bloch vector for a named eigenstate or custom input.\"\"\"\n",[61,37035,37036,37039,37041],{"class":1731,"line":2443},[61,37037,37038],{"class":1753},"    mapping ",[61,37040,2000],{"class":1859},[61,37042,37043],{"class":1753}," {\n",[61,37045,37046,37049,37051,37053,37055,37057,37059,37061,37063,37066,37068,37070,37072,37074,37076,37078,37080],{"class":1731,"line":2475},[61,37047,37048],{"class":2165},"        \"+z\"",[61,37050,36547],{"class":1753},[61,37052,2051],{"class":1866},[61,37054,1435],{"class":1753},[61,37056,2051],{"class":1866},[61,37058,1435],{"class":1753},[61,37060,2319],{"class":1866},[61,37062,3638],{"class":1753},[61,37064,37065],{"class":2165},"\"-z\"",[61,37067,36547],{"class":1753},[61,37069,2051],{"class":1866},[61,37071,1435],{"class":1753},[61,37073,2051],{"class":1866},[61,37075,1435],{"class":1753},[61,37077,2006],{"class":1859},[61,37079,2319],{"class":1866},[61,37081,16975],{"class":1753},[61,37083,37084,37087,37089,37091,37093,37095,37097,37099,37101,37104,37106,37108,37110,37112,37114,37116,37118],{"class":1731,"line":2520},[61,37085,37086],{"class":2165},"        \"+x\"",[61,37088,36547],{"class":1753},[61,37090,2319],{"class":1866},[61,37092,1435],{"class":1753},[61,37094,2051],{"class":1866},[61,37096,1435],{"class":1753},[61,37098,2051],{"class":1866},[61,37100,3638],{"class":1753},[61,37102,37103],{"class":2165},"\"-x\"",[61,37105,36547],{"class":1753},[61,37107,2006],{"class":1859},[61,37109,2319],{"class":1866},[61,37111,1435],{"class":1753},[61,37113,2051],{"class":1866},[61,37115,1435],{"class":1753},[61,37117,2051],{"class":1866},[61,37119,16975],{"class":1753},[61,37121,37122,37125,37127,37129,37131,37133,37135,37137,37139,37142,37144,37146,37148,37150,37152,37154,37156],{"class":1731,"line":2541},[61,37123,37124],{"class":2165},"        \"+y\"",[61,37126,36547],{"class":1753},[61,37128,2051],{"class":1866},[61,37130,1435],{"class":1753},[61,37132,2319],{"class":1866},[61,37134,1435],{"class":1753},[61,37136,2051],{"class":1866},[61,37138,3638],{"class":1753},[61,37140,37141],{"class":2165},"\"-y\"",[61,37143,36547],{"class":1753},[61,37145,2051],{"class":1866},[61,37147,1435],{"class":1753},[61,37149,2006],{"class":1859},[61,37151,2319],{"class":1866},[61,37153,1435],{"class":1753},[61,37155,2051],{"class":1866},[61,37157,16975],{"class":1753},[61,37159,37160],{"class":1731,"line":2546},[61,37161,37162],{"class":1753},"    }\n",[61,37164,37165,37167,37169,37171,37174,37177,37180,37183,37186,37189,37191],{"class":1731,"line":2582},[61,37166,2267],{"class":1753},[61,37168,2000],{"class":1859},[61,37170,2193],{"class":1753},[61,37172,37173],{"class":1836},"asarray",[61,37175,37176],{"class":1753},"(mapping.",[61,37178,37179],{"class":1836},"get",[61,37181,37182],{"class":1753},"(initial_state.",[61,37184,37185],{"class":1836},"lower",[61,37187,37188],{"class":1753},"(), custom_r), ",[61,37190,1860],{"class":1859},[61,37192,2016],{"class":1753},[61,37194,37195,37198,37200,37202,37205],{"class":1731,"line":2588},[61,37196,37197],{"class":1753},"    nrm ",[61,37199,2000],{"class":1859},[61,37201,6895],{"class":1753},[61,37203,37204],{"class":1836},"norm",[61,37206,37207],{"class":1753},"(r)\n",[61,37209,37210,37212,37214,37216,37218,37220,37222,37224,37226,37228,37231,37233,37236,37239,37241,37243,37246,37248],{"class":1731,"line":2601},[61,37211,2210],{"class":1749},[61,37213,2193],{"class":1753},[61,37215,3509],{"class":1836},[61,37217,3512],{"class":1753},[61,37219,2051],{"class":1866},[61,37221,1435],{"class":1753},[61,37223,2051],{"class":1866},[61,37225,1435],{"class":1753},[61,37227,2319],{"class":1866},[61,37229,37230],{"class":1753},"]) ",[61,37232,2498],{"class":1749},[61,37234,37235],{"class":1753}," nrm ",[61,37237,37238],{"class":1859},"==",[61,37240,24134],{"class":1866},[61,37242,4535],{"class":1749},[61,37244,37245],{"class":1753}," r ",[61,37247,602],{"class":1859},[61,37249,37250],{"class":1753}," nrm\n",[61,37252,37253],{"class":1731,"line":2659},[61,37254,1743],{"emptyLinePlaceholder":1742},[61,37256,37257],{"class":1731,"line":2717},[61,37258,1743],{"emptyLinePlaceholder":1742},[61,37260,37261],{"class":1731,"line":2725},[61,37262,37263],{"class":1735},"# ---------------------------- Axis–angle rotations ----------------------------\n",[61,37265,37266,37268,37271,37273,37276,37278,37280,37282,37285,37287,37289],{"class":1731,"line":2730},[61,37267,2116],{"class":1749},[61,37269,37270],{"class":1836}," rotation_matrix",[61,37272,849],{"class":1753},[61,37274,37275],{"class":2124},"axis",[61,37277,2127],{"class":1753},[61,37279,36784],{"class":1859},[61,37281,1435],{"class":1753},[61,37283,37284],{"class":2124},"angle",[61,37286,2127],{"class":1753},[61,37288,1860],{"class":1859},[61,37290,37291],{"class":1753},") -> np.ndarray:\n",[61,37293,37294],{"class":1731,"line":2749},[61,37295,37296],{"class":2165},"    \"\"\"Rodrigues rotation matrix for axis in {'x','y','z'} and angle (radians).\"\"\"\n",[61,37298,37299,37302,37304,37306,37308,37311,37313],{"class":1731,"line":2755},[61,37300,37301],{"class":1753},"    c, s ",[61,37303,2000],{"class":1859},[61,37305,2193],{"class":1753},[61,37307,5051],{"class":1836},[61,37309,37310],{"class":1753},"(angle), np.",[61,37312,5069],{"class":1836},[61,37314,37315],{"class":1753},"(angle)\n",[61,37317,37318,37320,37323,37325,37328],{"class":1731,"line":2778},[61,37319,2172],{"class":1749},[61,37321,37322],{"class":1753}," axis ",[61,37324,37238],{"class":1859},[61,37326,37327],{"class":2165}," \"x\"",[61,37329,1850],{"class":1753},[61,37331,37332,37334,37336,37338,37340,37342,37344,37346,37348,37350,37353,37355,37358,37360,37363,37365,37368,37370],{"class":1731,"line":2783},[61,37333,2190],{"class":1749},[61,37335,2193],{"class":1753},[61,37337,3509],{"class":1836},[61,37339,6541],{"class":1753},[61,37341,57],{"class":1866},[61,37343,1435],{"class":1753},[61,37345,1495],{"class":1866},[61,37347,1435],{"class":1753},[61,37349,1495],{"class":1866},[61,37351,37352],{"class":1753},"], [",[61,37354,1495],{"class":1866},[61,37356,37357],{"class":1753},", c, ",[61,37359,2006],{"class":1859},[61,37361,37362],{"class":1753},"s], [",[61,37364,1495],{"class":1866},[61,37366,37367],{"class":1753},", s, c]], ",[61,37369,1860],{"class":1859},[61,37371,2016],{"class":1753},[61,37373,37374,37376,37378,37380,37383],{"class":1731,"line":2791},[61,37375,2172],{"class":1749},[61,37377,37322],{"class":1753},[61,37379,37238],{"class":1859},[61,37381,37382],{"class":2165}," \"y\"",[61,37384,1850],{"class":1753},[61,37386,37387,37389,37391,37393,37396,37398,37401,37403,37405,37407,37409,37411,37413,37415,37418,37420,37423,37425],{"class":1731,"line":2804},[61,37388,2190],{"class":1749},[61,37390,2193],{"class":1753},[61,37392,3509],{"class":1836},[61,37394,37395],{"class":1753},"([[c, ",[61,37397,1495],{"class":1866},[61,37399,37400],{"class":1753},", s], [",[61,37402,1495],{"class":1866},[61,37404,1435],{"class":1753},[61,37406,57],{"class":1866},[61,37408,1435],{"class":1753},[61,37410,1495],{"class":1866},[61,37412,37352],{"class":1753},[61,37414,2006],{"class":1859},[61,37416,37417],{"class":1753},"s, ",[61,37419,1495],{"class":1866},[61,37421,37422],{"class":1753},", c]], ",[61,37424,1860],{"class":1859},[61,37426,2016],{"class":1753},[61,37428,37429,37431,37433,37435,37438],{"class":1731,"line":2817},[61,37430,2172],{"class":1749},[61,37432,37322],{"class":1753},[61,37434,37238],{"class":1859},[61,37436,37437],{"class":2165}," \"z\"",[61,37439,1850],{"class":1753},[61,37441,37442,37444,37446,37448,37450,37452,37454,37456,37459,37461,37463,37465,37467,37469,37471,37473,37476,37478],{"class":1731,"line":2824},[61,37443,2190],{"class":1749},[61,37445,2193],{"class":1753},[61,37447,3509],{"class":1836},[61,37449,37395],{"class":1753},[61,37451,2006],{"class":1859},[61,37453,37417],{"class":1753},[61,37455,1495],{"class":1866},[61,37457,37458],{"class":1753},"], [s, c, ",[61,37460,1495],{"class":1866},[61,37462,37352],{"class":1753},[61,37464,1495],{"class":1866},[61,37466,1435],{"class":1753},[61,37468,1495],{"class":1866},[61,37470,1435],{"class":1753},[61,37472,57],{"class":1866},[61,37474,37475],{"class":1753},"]], ",[61,37477,1860],{"class":1859},[61,37479,2016],{"class":1753},[61,37481,37482,37485,37488,37491],{"class":1731,"line":2835},[61,37483,37484],{"class":1749},"    raise",[61,37486,37487],{"class":1753}," ValueError(",[61,37489,37490],{"class":2165},"\"axis must be one of {'x','y','z'}\"",[61,37492,2016],{"class":1753},[61,37494,37495],{"class":1731,"line":2846},[61,37496,1743],{"emptyLinePlaceholder":1742},[61,37498,37499],{"class":1731,"line":2851},[61,37500,1743],{"emptyLinePlaceholder":1742},[61,37502,37503,37505,37508,37510,37513,37515,37517,37519,37521,37523,37525],{"class":1731,"line":2856},[61,37504,2116],{"class":1749},[61,37506,37507],{"class":1836}," rotation_matrix_axis",[61,37509,849],{"class":1753},[61,37511,37512],{"class":2124},"axis_vec",[61,37514,37015],{"class":1753},[61,37516,1860],{"class":1859},[61,37518,1435],{"class":1753},[61,37520,1860],{"class":1859},[61,37522,1435],{"class":1753},[61,37524,1860],{"class":1859},[61,37526,6550],{"class":1753},[61,37528,37529,37532,37534,37536],{"class":1731,"line":2862},[61,37530,37531],{"class":2124},"                         angle",[61,37533,2127],{"class":1753},[61,37535,1860],{"class":1859},[61,37537,37291],{"class":1753},[61,37539,37540],{"class":1731,"line":2867},[61,37541,37542],{"class":2165},"    \"\"\"Rodrigues matrix for arbitrary axis vector and angle (radians).\"\"\"\n",[61,37544,37545,37548,37550,37552,37554,37557,37559],{"class":1731,"line":2917},[61,37546,37547],{"class":1753},"    v ",[61,37549,2000],{"class":1859},[61,37551,2193],{"class":1753},[61,37553,37173],{"class":1836},[61,37555,37556],{"class":1753},"(axis_vec, ",[61,37558,1860],{"class":1859},[61,37560,2016],{"class":1753},[61,37562,37563,37565,37567,37569,37571],{"class":1731,"line":2926},[61,37564,37197],{"class":1753},[61,37566,2000],{"class":1859},[61,37568,6895],{"class":1753},[61,37570,37204],{"class":1836},[61,37572,37573],{"class":1753},"(v)\n",[61,37575,37576,37578,37580,37582,37584],{"class":1731,"line":2972},[61,37577,2172],{"class":1749},[61,37579,37235],{"class":1753},[61,37581,37238],{"class":1859},[61,37583,24134],{"class":1866},[61,37585,1850],{"class":1753},[61,37587,37588,37591,37593,37596],{"class":1731,"line":3001},[61,37589,37590],{"class":1749},"        raise",[61,37592,37487],{"class":1753},[61,37594,37595],{"class":2165},"\"axis_vec must be nonzero\"",[61,37597,2016],{"class":1753},[61,37599,37600,37603,37605,37608,37610],{"class":1731,"line":3054},[61,37601,37602],{"class":1753},"    k ",[61,37604,2000],{"class":1859},[61,37606,37607],{"class":1753}," v ",[61,37609,602],{"class":1859},[61,37611,37250],{"class":1753},[61,37613,37614,37617,37619,37621,37623,37625,37627,37629,37631,37634,37636,37639,37641,37644,37646,37648,37650,37652,37654,37656,37658,37661,37663,37665,37667,37669,37671,37673,37675,37677,37679],{"class":1731,"line":3060},[61,37615,37616],{"class":1753},"    K ",[61,37618,2000],{"class":1859},[61,37620,2193],{"class":1753},[61,37622,3509],{"class":1836},[61,37624,6541],{"class":1753},[61,37626,2051],{"class":1866},[61,37628,1435],{"class":1753},[61,37630,2006],{"class":1859},[61,37632,37633],{"class":1753},"k[",[61,37635,337],{"class":1866},[61,37637,37638],{"class":1753},"], k[",[61,37640,57],{"class":1866},[61,37642,37643],{"class":1753},"]], [k[",[61,37645,337],{"class":1866},[61,37647,7056],{"class":1753},[61,37649,2051],{"class":1866},[61,37651,1435],{"class":1753},[61,37653,2006],{"class":1859},[61,37655,37633],{"class":1753},[61,37657,1495],{"class":1866},[61,37659,37660],{"class":1753},"]], [",[61,37662,2006],{"class":1859},[61,37664,37633],{"class":1753},[61,37666,57],{"class":1866},[61,37668,37638],{"class":1753},[61,37670,1495],{"class":1866},[61,37672,7056],{"class":1753},[61,37674,2051],{"class":1866},[61,37676,37475],{"class":1753},[61,37678,1860],{"class":1859},[61,37680,2016],{"class":1753},[61,37682,37683,37686,37688,37690,37692,37694,37696],{"class":1731,"line":3077},[61,37684,37685],{"class":1753},"    I ",[61,37687,2000],{"class":1859},[61,37689,2193],{"class":1753},[61,37691,6521],{"class":1836},[61,37693,849],{"class":1753},[61,37695,570],{"class":1866},[61,37697,2016],{"class":1753},[61,37699,37700,37702,37704,37706,37708,37710,37712],{"class":1731,"line":3082},[61,37701,37301],{"class":1753},[61,37703,2000],{"class":1859},[61,37705,2193],{"class":1753},[61,37707,5051],{"class":1836},[61,37709,37310],{"class":1753},[61,37711,5069],{"class":1836},[61,37713,37315],{"class":1753},[61,37715,37716,37718,37721,37723,37726,37728,37730,37732,37734,37737,37739,37741,37744,37747,37749,37752,37754],{"class":1731,"line":3129},[61,37717,2210],{"class":1749},[61,37719,37720],{"class":1753}," c ",[61,37722,2201],{"class":1859},[61,37724,37725],{"class":1753}," I ",[61,37727,2419],{"class":1859},[61,37729,2003],{"class":1753},[61,37731,57],{"class":1866},[61,37733,1899],{"class":1859},[61,37735,37736],{"class":1753}," c) ",[61,37738,2201],{"class":1859},[61,37740,2193],{"class":1753},[61,37742,37743],{"class":1836},"outer",[61,37745,37746],{"class":1753},"(k, k) ",[61,37748,2419],{"class":1859},[61,37750,37751],{"class":1753}," s ",[61,37753,2201],{"class":1859},[61,37755,37756],{"class":1753}," K\n",[61,37758,37759],{"class":1731,"line":3137},[61,37760,1743],{"emptyLinePlaceholder":1742},[61,37762,37763],{"class":1731,"line":3185},[61,37764,1743],{"emptyLinePlaceholder":1742},[61,37766,37767],{"class":1731,"line":3208},[61,37768,37769],{"class":1735},"# ------------------------------ Wigner on S^2 ---------------------------------\n",[61,37771,37772,37774,37777,37779,37782,37784,37786,37788,37791,37793,37795],{"class":1731,"line":3248},[61,37773,2116],{"class":1749},[61,37775,37776],{"class":1836}," n_grid",[61,37778,849],{"class":1753},[61,37780,37781],{"class":2124},"n_theta",[61,37783,2127],{"class":1753},[61,37785,2025],{"class":1859},[61,37787,1435],{"class":1753},[61,37789,37790],{"class":2124},"n_phi",[61,37792,2127],{"class":1753},[61,37794,2025],{"class":1859},[61,37796,37291],{"class":1753},[61,37798,37799],{"class":1731,"line":3286},[61,37800,37801],{"class":2165},"    \"\"\"Return unit-vector grid n(θ,φ) stacked as (3, n_theta, n_phi).\"\"\"\n",[61,37803,37804,37807,37809,37811,37813,37815,37817],{"class":1731,"line":3291},[61,37805,37806],{"class":1753},"    theta ",[61,37808,2000],{"class":1859},[61,37810,2193],{"class":1753},[61,37812,3453],{"class":1836},[61,37814,849],{"class":1753},[61,37816,2051],{"class":1866},[61,37818,37819],{"class":1753},", np.pi, n_theta)\n",[61,37821,37822,37825,37827,37829,37831,37833,37835,37837,37839,37841],{"class":1731,"line":3306},[61,37823,37824],{"class":1753},"    phi ",[61,37826,2000],{"class":1859},[61,37828,2193],{"class":1753},[61,37830,3453],{"class":1836},[61,37832,849],{"class":1753},[61,37834,2051],{"class":1866},[61,37836,1435],{"class":1753},[61,37838,2435],{"class":1866},[61,37840,2298],{"class":1859},[61,37842,37843],{"class":1753}," np.pi, n_phi)\n",[61,37845,37846,37849,37851,37853,37856,37859,37862,37864,37867],{"class":1731,"line":3311},[61,37847,37848],{"class":1753},"    th, ph ",[61,37850,2000],{"class":1859},[61,37852,2193],{"class":1753},[61,37854,37855],{"class":1836},"meshgrid",[61,37857,37858],{"class":1753},"(theta, phi, ",[61,37860,37861],{"class":3402},"indexing",[61,37863,2000],{"class":1859},[61,37865,37866],{"class":2165},"\"ij\"",[61,37868,2016],{"class":1753},[61,37870,37871,37874,37876,37878,37880,37883,37885,37887,37889],{"class":1731,"line":3317},[61,37872,37873],{"class":1753},"    nx ",[61,37875,2000],{"class":1859},[61,37877,2193],{"class":1753},[61,37879,5069],{"class":1836},[61,37881,37882],{"class":1753},"(th) ",[61,37884,2201],{"class":1859},[61,37886,2193],{"class":1753},[61,37888,5051],{"class":1836},[61,37890,37891],{"class":1753},"(ph)\n",[61,37893,37894,37897,37899,37901,37903,37905,37907,37909,37911],{"class":1731,"line":3358},[61,37895,37896],{"class":1753},"    ny ",[61,37898,2000],{"class":1859},[61,37900,2193],{"class":1753},[61,37902,5069],{"class":1836},[61,37904,37882],{"class":1753},[61,37906,2201],{"class":1859},[61,37908,2193],{"class":1753},[61,37910,5069],{"class":1836},[61,37912,37891],{"class":1753},[61,37914,37915,37918,37920,37922,37924],{"class":1731,"line":3374},[61,37916,37917],{"class":1753},"    nz ",[61,37919,2000],{"class":1859},[61,37921,2193],{"class":1753},[61,37923,5051],{"class":1836},[61,37925,37926],{"class":1753},"(th)\n",[61,37928,37929,37931,37933,37936,37939,37941,37943,37945],{"class":1731,"line":3390},[61,37930,2210],{"class":1749},[61,37932,2193],{"class":1753},[61,37934,37935],{"class":1836},"stack",[61,37937,37938],{"class":1753},"((nx, ny, nz), ",[61,37940,37275],{"class":3402},[61,37942,2000],{"class":1859},[61,37944,1495],{"class":1866},[61,37946,2016],{"class":1753},[61,37948,37949],{"class":1731,"line":3422},[61,37950,1743],{"emptyLinePlaceholder":1742},[61,37952,37953],{"class":1731,"line":3427},[61,37954,1743],{"emptyLinePlaceholder":1742},[61,37956,37957,37959,37962,37964,37967,37969,37972],{"class":1731,"line":3432},[61,37958,2116],{"class":1749},[61,37960,37961],{"class":1836}," spin_wigner_qubit",[61,37963,849],{"class":1753},[61,37965,37966],{"class":2124},"r_vec",[61,37968,6874],{"class":1753},[61,37970,37971],{"class":2124},"ngrid",[61,37973,6885],{"class":1753},[61,37975,37976],{"class":1731,"line":3438},[61,37977,37978],{"class":2165},"    \"\"\"Spin SW Wigner for a qubit: W = 1\u002F2 + (√3\u002F2) * r·n on the (θ,φ) grid.\"\"\"\n",[61,37980,37981,37983,37985,37987,37989,37991,37993,37996,37998,38000,38002,38004,38006,38009,38012,38014,38016,38018,38020,38022,38024,38026,38028,38031,38034,38036,38038,38040,38042],{"class":1731,"line":3443},[61,37982,2210],{"class":1749},[61,37984,2980],{"class":1866},[61,37986,4990],{"class":1859},[61,37988,6984],{"class":1753},[61,37990,2425],{"class":1836},[61,37992,849],{"class":1753},[61,37994,37995],{"class":1866},"3.0",[61,37997,2624],{"class":1753},[61,37999,602],{"class":1859},[61,38001,3029],{"class":1866},[61,38003,2624],{"class":1753},[61,38005,2201],{"class":1859},[61,38007,38008],{"class":1753}," (r_vec.",[61,38010,38011],{"class":1836},"reshape",[61,38013,849],{"class":1753},[61,38015,570],{"class":1866},[61,38017,1435],{"class":1753},[61,38019,57],{"class":1866},[61,38021,1435],{"class":1753},[61,38023,57],{"class":1866},[61,38025,2624],{"class":1753},[61,38027,2201],{"class":1859},[61,38029,38030],{"class":1753}," ngrid).",[61,38032,38033],{"class":1836},"sum",[61,38035,849],{"class":1753},[61,38037,37275],{"class":3402},[61,38039,2000],{"class":1859},[61,38041,1495],{"class":1866},[61,38043,2016],{"class":1753},[61,38045,38046],{"class":1731,"line":3469},[61,38047,1743],{"emptyLinePlaceholder":1742},[61,38049,38050],{"class":1731,"line":3494},[61,38051,1743],{"emptyLinePlaceholder":1742},[61,38053,38054],{"class":1731,"line":3499},[61,38055,38056],{"class":1735},"# ----------------------- In-axes title & info placement -----------------------\n",[61,38058,38059,38061,38064,38066,38069,38071,38073,38075,38077,38079,38081,38083,38086],{"class":1731,"line":3534},[61,38060,2116],{"class":1749},[61,38062,38063],{"class":1836}," _info_xy",[61,38065,849],{"class":1753},[61,38067,38068],{"class":2124},"loc",[61,38070,2127],{"class":1753},[61,38072,36784],{"class":1859},[61,38074,2570],{"class":1753},[61,38076,1860],{"class":1859},[61,38078,1435],{"class":1753},[61,38080,1860],{"class":1859},[61,38082,1435],{"class":1753},[61,38084,38085],{"class":1859},"dict",[61,38087,2579],{"class":1753},[61,38089,38090],{"class":1731,"line":3564},[61,38091,38092],{"class":2165},"    \"\"\"Map location code to (x, y, kwargs) in axes-fraction coordinates.\"\"\"\n",[61,38094,38095,38098,38100,38103,38105],{"class":1731,"line":3602},[61,38096,38097],{"class":1753},"    loc ",[61,38099,2000],{"class":1859},[61,38101,38102],{"class":1753}," loc.",[61,38104,37185],{"class":1836},[61,38106,2089],{"class":1753},[61,38108,38109,38111,38114,38116,38119],{"class":1731,"line":3607},[61,38110,2172],{"class":1749},[61,38112,38113],{"class":1753}," loc ",[61,38115,37238],{"class":1859},[61,38117,38118],{"class":2165}," \"ul\"",[61,38120,1850],{"class":1753},[61,38122,38123,38125,38128,38130,38133,38135,38137,38139,38142,38144,38147,38149,38152,38154,38157],{"class":1731,"line":3613},[61,38124,2190],{"class":1749},[61,38126,38127],{"class":1866}," 0.015",[61,38129,1435],{"class":1753},[61,38131,38132],{"class":1866},"0.985",[61,38134,1435],{"class":1753},[61,38136,38085],{"class":1859},[61,38138,849],{"class":1753},[61,38140,38141],{"class":3402},"ha",[61,38143,2000],{"class":1859},[61,38145,38146],{"class":2165},"\"left\"",[61,38148,1435],{"class":1753},[61,38150,38151],{"class":3402},"va",[61,38153,2000],{"class":1859},[61,38155,38156],{"class":2165},"\"top\"",[61,38158,2016],{"class":1753},[61,38160,38161,38163,38165,38167,38170],{"class":1731,"line":3651},[61,38162,2172],{"class":1749},[61,38164,38113],{"class":1753},[61,38166,37238],{"class":1859},[61,38168,38169],{"class":2165}," \"ur\"",[61,38171,1850],{"class":1753},[61,38173,38174,38176,38179,38181,38183,38185,38187,38189,38191,38193,38196,38198,38200,38202,38204],{"class":1731,"line":3715},[61,38175,2190],{"class":1749},[61,38177,38178],{"class":1866}," 0.985",[61,38180,1435],{"class":1753},[61,38182,38132],{"class":1866},[61,38184,1435],{"class":1753},[61,38186,38085],{"class":1859},[61,38188,849],{"class":1753},[61,38190,38141],{"class":3402},[61,38192,2000],{"class":1859},[61,38194,38195],{"class":2165},"\"right\"",[61,38197,1435],{"class":1753},[61,38199,38151],{"class":3402},[61,38201,2000],{"class":1859},[61,38203,38156],{"class":2165},[61,38205,2016],{"class":1753},[61,38207,38208,38210,38212,38214,38217],{"class":1731,"line":3738},[61,38209,2172],{"class":1749},[61,38211,38113],{"class":1753},[61,38213,37238],{"class":1859},[61,38215,38216],{"class":2165}," \"ll\"",[61,38218,1850],{"class":1753},[61,38220,38221,38223,38225,38227,38230,38232,38234,38236,38238,38240,38242,38244,38246,38248,38251],{"class":1731,"line":3753},[61,38222,2190],{"class":1749},[61,38224,38127],{"class":1866},[61,38226,1435],{"class":1753},[61,38228,38229],{"class":1866},"0.015",[61,38231,1435],{"class":1753},[61,38233,38085],{"class":1859},[61,38235,849],{"class":1753},[61,38237,38141],{"class":3402},[61,38239,2000],{"class":1859},[61,38241,38146],{"class":2165},[61,38243,1435],{"class":1753},[61,38245,38151],{"class":3402},[61,38247,2000],{"class":1859},[61,38249,38250],{"class":2165},"\"bottom\"",[61,38252,2016],{"class":1753},[61,38254,38255,38257,38259,38261,38264],{"class":1731,"line":3782},[61,38256,2172],{"class":1749},[61,38258,38113],{"class":1753},[61,38260,37238],{"class":1859},[61,38262,38263],{"class":2165}," \"lr\"",[61,38265,1850],{"class":1753},[61,38267,38268,38270,38272,38274,38276,38278,38280,38282,38284,38286,38288,38290,38292,38294,38296],{"class":1731,"line":3813},[61,38269,2190],{"class":1749},[61,38271,38178],{"class":1866},[61,38273,1435],{"class":1753},[61,38275,38229],{"class":1866},[61,38277,1435],{"class":1753},[61,38279,38085],{"class":1859},[61,38281,849],{"class":1753},[61,38283,38141],{"class":3402},[61,38285,2000],{"class":1859},[61,38287,38195],{"class":2165},[61,38289,1435],{"class":1753},[61,38291,38151],{"class":3402},[61,38293,2000],{"class":1859},[61,38295,38250],{"class":2165},[61,38297,2016],{"class":1753},[61,38299,38300,38302,38304,38306,38308,38310,38312,38314,38316,38318,38320,38322,38324,38326,38328],{"class":1731,"line":3835},[61,38301,2210],{"class":1749},[61,38303,38127],{"class":1866},[61,38305,1435],{"class":1753},[61,38307,38132],{"class":1866},[61,38309,1435],{"class":1753},[61,38311,38085],{"class":1859},[61,38313,849],{"class":1753},[61,38315,38141],{"class":3402},[61,38317,2000],{"class":1859},[61,38319,38146],{"class":2165},[61,38321,1435],{"class":1753},[61,38323,38151],{"class":3402},[61,38325,2000],{"class":1859},[61,38327,38156],{"class":2165},[61,38329,2016],{"class":1753},[61,38331,38332],{"class":1731,"line":3845},[61,38333,1743],{"emptyLinePlaceholder":1742},[61,38335,38336],{"class":1731,"line":3855},[61,38337,1743],{"emptyLinePlaceholder":1742},[61,38339,38340,38342,38345,38347,38349],{"class":1731,"line":3860},[61,38341,2116],{"class":1749},[61,38343,38344],{"class":1836}," add_info_box",[61,38346,849],{"class":1753},[61,38348,3368],{"class":2124},[61,38350,38351],{"class":1753},": plt.Axes,\n",[61,38353,38354,38357,38359,38361],{"class":1731,"line":3866},[61,38355,38356],{"class":2124},"                 text",[61,38358,2127],{"class":1753},[61,38360,36784],{"class":1859},[61,38362,36573],{"class":1753},[61,38364,38365,38368,38370,38372,38374,38376],{"class":1731,"line":3897},[61,38366,38367],{"class":2124},"                 loc",[61,38369,2127],{"class":1753},[61,38371,36784],{"class":1859},[61,38373,1863],{"class":1859},[61,38375,38118],{"class":2165},[61,38377,36573],{"class":1753},[61,38379,38380,38383,38385,38387,38389,38392],{"class":1731,"line":3943},[61,38381,38382],{"class":2124},"                 fontsize",[61,38384,2127],{"class":1753},[61,38386,2025],{"class":1859},[61,38388,1863],{"class":1859},[61,38390,38391],{"class":1866}," 9",[61,38393,36573],{"class":1753},[61,38395,38396,38399,38401,38403,38405,38408],{"class":1731,"line":3964},[61,38397,38398],{"class":2124},"                 wrap",[61,38400,2127],{"class":1753},[61,38402,2025],{"class":1859},[61,38404,1863],{"class":1859},[61,38406,38407],{"class":1866}," 42",[61,38409,38410],{"class":1753},") -> plt.Text:\n",[61,38412,38413],{"class":1731,"line":3975},[61,38414,38415],{"class":2165},"    \"\"\"Create a wrapped, padded text box inside the axes (never clips).\"\"\"\n",[61,38417,38418,38421,38423,38425],{"class":1731,"line":3998},[61,38419,38420],{"class":1753},"    x, y, kw ",[61,38422,2000],{"class":1859},[61,38424,38063],{"class":1836},[61,38426,38427],{"class":1753},"(loc)\n",[61,38429,38430,38432,38435,38437],{"class":1731,"line":4025},[61,38431,2210],{"class":1749},[61,38433,38434],{"class":1753}," ax.",[61,38436,29075],{"class":1836},[61,38438,38439],{"class":1753},"(\n",[61,38441,38442,38445,38448],{"class":1731,"line":4046},[61,38443,38444],{"class":1753},"        x, y, textwrap.",[61,38446,38447],{"class":1836},"fill",[61,38449,38450],{"class":1753},"(text, wrap),\n",[61,38452,38453,38456,38458,38461,38464,38466,38469,38472,38474,38476],{"class":1731,"line":4055},[61,38454,38455],{"class":3402},"        transform",[61,38457,2000],{"class":1859},[61,38459,38460],{"class":1753},"ax.transAxes, ",[61,38462,38463],{"class":3402},"fontsize",[61,38465,2000],{"class":1859},[61,38467,38468],{"class":1753},"fontsize, ",[61,38470,38471],{"class":3402},"zorder",[61,38473,2000],{"class":1859},[61,38475,1026],{"class":1866},[61,38477,36573],{"class":1753},[61,38479,38480,38483,38486,38488,38491,38493,38496,38498,38501,38503,38506,38508,38510,38512,38515,38517,38520,38522,38525],{"class":1731,"line":4064},[61,38481,38482],{"class":3402},"        bbox",[61,38484,38485],{"class":1859},"=dict",[61,38487,849],{"class":1753},[61,38489,38490],{"class":3402},"boxstyle",[61,38492,2000],{"class":1859},[61,38494,38495],{"class":2165},"\"round,pad=0.3\"",[61,38497,1435],{"class":1753},[61,38499,38500],{"class":3402},"fc",[61,38502,2000],{"class":1859},[61,38504,38505],{"class":2165},"\"white\"",[61,38507,1435],{"class":1753},[61,38509,16926],{"class":3402},[61,38511,2000],{"class":1859},[61,38513,38514],{"class":1866},"0.75",[61,38516,1435],{"class":1753},[61,38518,38519],{"class":3402},"ec",[61,38521,2000],{"class":1859},[61,38523,38524],{"class":2165},"\"none\"",[61,38526,16975],{"class":1753},[61,38528,38529],{"class":1731,"line":4069},[61,38530,38531],{"class":1753},"        **kw,\n",[61,38533,38534],{"class":1731,"line":4075},[61,38535,3057],{"class":1753},[61,38537,38538],{"class":1731,"line":4101},[61,38539,1743],{"emptyLinePlaceholder":1742},[61,38541,38542],{"class":1731,"line":4120},[61,38543,1743],{"emptyLinePlaceholder":1742},[61,38545,38546,38548,38551,38553,38555,38558,38560,38562,38564],{"class":1731,"line":4138},[61,38547,2116],{"class":1749},[61,38549,38550],{"class":1836}," add_title_inside",[61,38552,849],{"class":1753},[61,38554,3368],{"class":2124},[61,38556,38557],{"class":1753},": plt.Axes, ",[61,38559,29075],{"class":2124},[61,38561,2127],{"class":1753},[61,38563,36784],{"class":1859},[61,38565,38410],{"class":1753},[61,38567,38568],{"class":1731,"line":4151},[61,38569,38570],{"class":2165},"    \"\"\"Draw a title inside the Axes at the very top center (never clips).\"\"\"\n",[61,38572,38573,38575,38577,38579],{"class":1731,"line":4156},[61,38574,2210],{"class":1749},[61,38576,38434],{"class":1753},[61,38578,29075],{"class":1836},[61,38580,38439],{"class":1753},[61,38582,38583,38586,38588,38590],{"class":1731,"line":4187},[61,38584,38585],{"class":1866},"        0.5",[61,38587,1435],{"class":1753},[61,38589,38132],{"class":1866},[61,38591,38592],{"class":1753},", text,\n",[61,38594,38595,38597,38599,38601,38603,38605,38608,38610,38612,38614,38616],{"class":1731,"line":4234},[61,38596,38455],{"class":3402},[61,38598,2000],{"class":1859},[61,38600,38460],{"class":1753},[61,38602,38141],{"class":3402},[61,38604,2000],{"class":1859},[61,38606,38607],{"class":2165},"\"center\"",[61,38609,1435],{"class":1753},[61,38611,38151],{"class":3402},[61,38613,2000],{"class":1859},[61,38615,38156],{"class":2165},[61,38617,36573],{"class":1753},[61,38619,38620,38623,38625,38628],{"class":1731,"line":4283},[61,38621,38622],{"class":3402},"        fontsize",[61,38624,2000],{"class":1859},[61,38626,38627],{"class":1866},"11",[61,38629,36573],{"class":1753},[61,38631,38632,38634,38636,38638,38640,38642,38644,38646,38648,38650,38652,38654,38656,38658,38660,38662,38664,38666,38668],{"class":1731,"line":4315},[61,38633,38482],{"class":3402},[61,38635,38485],{"class":1859},[61,38637,849],{"class":1753},[61,38639,38490],{"class":3402},[61,38641,2000],{"class":1859},[61,38643,38495],{"class":2165},[61,38645,1435],{"class":1753},[61,38647,38500],{"class":3402},[61,38649,2000],{"class":1859},[61,38651,38505],{"class":2165},[61,38653,1435],{"class":1753},[61,38655,16926],{"class":3402},[61,38657,2000],{"class":1859},[61,38659,38514],{"class":1866},[61,38661,1435],{"class":1753},[61,38663,38519],{"class":3402},[61,38665,2000],{"class":1859},[61,38667,38524],{"class":2165},[61,38669,16975],{"class":1753},[61,38671,38672,38675,38677,38679],{"class":1731,"line":4344},[61,38673,38674],{"class":3402},"        zorder",[61,38676,2000],{"class":1859},[61,38678,1300],{"class":1866},[61,38680,36573],{"class":1753},[61,38682,38683],{"class":1731,"line":4367},[61,38684,3057],{"class":1753},[1720,38686,38688],{"className":1722,"code":38687,"language":1724,"meta":1725,"style":1725},"# @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",[1727,38689,38690,38695,38705,38709,38726,38734,38745,38756,38767,38778,38793,38804,38815,38826,38834,38839,38856,38868,38895,38922,38947,38951,38979,39013,39039,39081,39096,39100,39113,39126,39157,39161,39191,39201,39205,39224,39253,39275,39291,39347,39354,39358,39397,39425,39459,39477,39488,39495,39499,39503,39508,39523,39544,39556,39572,39597,39608,39620,39637,39668,39672,39676,39688,39698,39710,39741,39751,39759,39764,39769,39774,39779,39783,39787,39792,39804,39825],{"__ignoreMap":1725},[61,38691,38692],{"class":1731,"line":1732},[61,38693,38694],{"class":1735},"# @title Cell 3 — Pauli X\u002FY\u002FZ Wigner animations (titles *inside* the plot)\n",[61,38696,38697,38699,38701,38703],{"class":1731,"line":1739},[61,38698,1750],{"class":1749},[61,38700,36605],{"class":1753},[61,38702,1757],{"class":1749},[61,38704,36610],{"class":1753},[61,38706,38707],{"class":1731,"line":1746},[61,38708,1743],{"emptyLinePlaceholder":1742},[61,38710,38711,38713,38716,38718,38720,38722,38724],{"class":1731,"line":1763},[61,38712,2116],{"class":1749},[61,38714,38715],{"class":1836}," animate_wigner_pauli",[61,38717,849],{"class":1753},[61,38719,37275],{"class":2124},[61,38721,2127],{"class":1753},[61,38723,36784],{"class":1859},[61,38725,36573],{"class":1753},[61,38727,38728,38731],{"class":1731,"line":1776},[61,38729,38730],{"class":2124},"                         r0",[61,38732,38733],{"class":1753},": np.ndarray,\n",[61,38735,38736,38739,38741,38743],{"class":1731,"line":1784},[61,38737,38738],{"class":2124},"                         frames",[61,38740,2127],{"class":1753},[61,38742,2025],{"class":1859},[61,38744,36573],{"class":1753},[61,38746,38747,38750,38752,38754],{"class":1731,"line":1798},[61,38748,38749],{"class":2124},"                         n_theta",[61,38751,2127],{"class":1753},[61,38753,2025],{"class":1859},[61,38755,36573],{"class":1753},[61,38757,38758,38761,38763,38765],{"class":1731,"line":1811},[61,38759,38760],{"class":2124},"                         n_phi",[61,38762,2127],{"class":1753},[61,38764,2025],{"class":1859},[61,38766,36573],{"class":1753},[61,38768,38769,38772,38774,38776],{"class":1731,"line":1816},[61,38770,38771],{"class":2124},"                         fps",[61,38773,2127],{"class":1753},[61,38775,2025],{"class":1859},[61,38777,36573],{"class":1753},[61,38779,38780,38783,38785,38787,38789,38791],{"class":1731,"line":1822},[61,38781,38782],{"class":2124},"                         figsize",[61,38784,37015],{"class":1753},[61,38786,1860],{"class":1859},[61,38788,1435],{"class":1753},[61,38790,1860],{"class":1859},[61,38792,6550],{"class":1753},[61,38794,38795,38798,38800,38802],{"class":1731,"line":1828},[61,38796,38797],{"class":2124},"                         info_loc",[61,38799,2127],{"class":1753},[61,38801,36784],{"class":1859},[61,38803,36573],{"class":1753},[61,38805,38806,38809,38811,38813],{"class":1731,"line":1833},[61,38807,38808],{"class":2124},"                         info_fontsize",[61,38810,2127],{"class":1753},[61,38812,2025],{"class":1859},[61,38814,36573],{"class":1753},[61,38816,38817,38820,38822,38824],{"class":1731,"line":1840},[61,38818,38819],{"class":2124},"                         wrap_chars",[61,38821,2127],{"class":1753},[61,38823,2025],{"class":1859},[61,38825,36573],{"class":1753},[61,38827,38828,38831],{"class":1731,"line":1853},[61,38829,38830],{"class":2124},"                         out_path",[61,38832,38833],{"class":1753},": Path) -> Path:\n",[61,38835,38836],{"class":1731,"line":1873},[61,38837,38838],{"class":2165},"    \"\"\"Animate Wigner under a Pauli rotation U = exp(-i σ_axis * π * t \u002F 2).\"\"\"\n",[61,38840,38841,38844,38847,38849,38851,38853],{"class":1731,"line":1889},[61,38842,38843],{"class":1749},"    assert",[61,38845,38846],{"class":1753}," frames ",[61,38848,6163],{"class":1859},[61,38850,2654],{"class":1866},[61,38852,1435],{"class":1753},[61,38854,38855],{"class":2165},"\"frames must be ≥ 2\"\n",[61,38857,38858,38861,38863,38865],{"class":1731,"line":1908},[61,38859,38860],{"class":1753},"    ngrid ",[61,38862,2000],{"class":1859},[61,38864,37776],{"class":1836},[61,38866,38867],{"class":1753},"(n_theta, n_phi)\n",[61,38869,38870,38873,38875,38877,38879,38881,38883,38885,38887,38889,38891,38893],{"class":1731,"line":1925},[61,38871,38872],{"class":1753},"    w_min 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wrap",[61,39197,2000],{"class":1859},[61,39199,39200],{"class":1753},"wrap_chars)\n",[61,39202,39203],{"class":1731,"line":2162},[61,39204,1743],{"emptyLinePlaceholder":1742},[61,39206,39207,39210,39213,39215,39218,39220,39222],{"class":1731,"line":2169},[61,39208,39209],{"class":1749},"    def",[61,39211,39212],{"class":1836}," update",[61,39214,849],{"class":1753},[61,39216,39217],{"class":2124},"k",[61,39219,2127],{"class":1753},[61,39221,2025],{"class":1859},[61,39223,3371],{"class":1753},[61,39225,39226,39229,39231,39234,39236,39239,39241,39243,39245,39247,39250],{"class":1731,"line":2187},[61,39227,39228],{"class":1753},"        ang ",[61,39230,2000],{"class":1859},[61,39232,39233],{"class":1753}," (k ",[61,39235,602],{"class":1859},[61,39237,39238],{"class":1753}," (frames ",[61,39240,2006],{"class":1859},[61,39242,5266],{"class":1866},[61,39244,2702],{"class":1753},[61,39246,2201],{"class":1859},[61,39248,39249],{"class":1753}," np.pi  ",[61,39251,39252],{"class":1735},"# 0 → π\n",[61,39254,39255,39258,39260,39262,39265,39267,39270,39272],{"class":1731,"line":2207},[61,39256,39257],{"class":1753},"        r ",[61,39259,2000],{"class":1859},[61,39261,37270],{"class":1836},[61,39263,39264],{"class":1753},"(axis.",[61,39266,37185],{"class":1836},[61,39268,39269],{"class":1753},"(), ang) ",[61,39271,6930],{"class":1859},[61,39273,39274],{"class":1753}," r0\n",[61,39276,39277,39280,39283,39285,39288],{"class":1731,"line":2221},[61,39278,39279],{"class":1753},"        im.",[61,39281,39282],{"class":1836},"set_data",[61,39284,849],{"class":1753},[61,39286,39287],{"class":1836},"spin_wigner_qubit",[61,39289,39290],{"class":1753},"(r, ngrid))\n",[61,39292,39293,39296,39299,39301,39303,39306,39308,39311,39314,39316,39319,39322,39325,39327,39329,39331,39334,39336,39338,39341,39343,39345],{"class":1731,"line":2226},[61,39294,39295],{"class":1753},"        info.",[61,39297,39298],{"class":1836},"set_text",[61,39300,849],{"class":1753},[61,39302,19797],{"class":1749},[61,39304,39305],{"class":2165},"\"θ = ",[61,39307,3688],{"class":1866},[61,39309,39310],{"class":1753},"ang",[61,39312,39313],{"class":1749},":.3f",[61,39315,3697],{"class":1866},[61,39317,39318],{"class":2165}," rad",[61,39320,39321],{"class":1859},"\\n",[61,39323,39324],{"class":2165},"frame ",[61,39326,3688],{"class":1866},[61,39328,39217],{"class":1753},[61,39330,2419],{"class":1859},[61,39332,39333],{"class":1866},"1}",[61,39335,602],{"class":2165},[61,39337,3688],{"class":1866},[61,39339,39340],{"class":1753},"frames",[61,39342,3697],{"class":1866},[61,39344,26088],{"class":2165},[61,39346,2016],{"class":1753},[61,39348,39349,39351],{"class":1731,"line":2258},[61,39350,2190],{"class":1749},[61,39352,39353],{"class":1753}," (im, info)\n",[61,39355,39356],{"class":1731,"line":2264},[61,39357,1743],{"emptyLinePlaceholder":1742},[61,39359,39360,39363,39365,39368,39371,39373,39375,39378,39381,39383,39386,39388,39391,39393,39395],{"class":1731,"line":2280},[61,39361,39362],{"class":1753},"    anim ",[61,39364,2000],{"class":1859},[61,39366,39367],{"class":1836}," FuncAnimation",[61,39369,39370],{"class":1753},"(fig, update, ",[61,39372,39340],{"class":3402},[61,39374,2000],{"class":1859},[61,39376,39377],{"class":1753},"frames, ",[61,39379,39380],{"class":3402},"interval",[61,39382,2000],{"class":1859},[61,39384,39385],{"class":1866},"60",[61,39387,1435],{"class":1753},[61,39389,39390],{"class":3402},"blit",[61,39392,2000],{"class":1859},[61,39394,14390],{"class":1866},[61,39396,2016],{"class":1753},[61,39398,39399,39402,39405,39407,39410,39412,39414,39416,39419,39421,39423],{"class":1731,"line":2304},[61,39400,39401],{"class":1753},"    out_path.parent.",[61,39403,39404],{"class":1836},"mkdir",[61,39406,849],{"class":1753},[61,39408,39409],{"class":3402},"parents",[61,39411,2000],{"class":1859},[61,39413,24385],{"class":1866},[61,39415,1435],{"class":1753},[61,39417,39418],{"class":3402},"exist_ok",[61,39420,2000],{"class":1859},[61,39422,24385],{"class":1866},[61,39424,2016],{"class":1753},[61,39426,39427,39430,39433,39436,39439,39441,39444,39446,39449,39451,39454,39456],{"class":1731,"line":2326},[61,39428,39429],{"class":1753},"    anim.",[61,39431,39432],{"class":1836},"save",[61,39434,39435],{"class":1753},"(out_path.",[61,39437,39438],{"class":1836},"as_posix",[61,39440,4205],{"class":1753},[61,39442,39443],{"class":3402},"writer",[61,39445,2000],{"class":1859},[61,39447,39448],{"class":1836},"PillowWriter",[61,39450,849],{"class":1753},[61,39452,39453],{"class":3402},"fps",[61,39455,2000],{"class":1859},[61,39457,39458],{"class":1753},"fps),\n",[61,39460,39461,39464,39466,39468,39471,39473,39475],{"class":1731,"line":2342},[61,39462,39463],{"class":3402},"              savefig_kwargs",[61,39465,38485],{"class":1859},[61,39467,849],{"class":1753},[61,39469,39470],{"class":3402},"facecolor",[61,39472,2000],{"class":1859},[61,39474,38505],{"class":2165},[61,39476,5546],{"class":1753},[61,39478,39479,39482,39485],{"class":1731,"line":2379},[61,39480,39481],{"class":1753},"    plt.",[61,39483,39484],{"class":1836},"close",[61,39486,39487],{"class":1753},"(fig)\n",[61,39489,39490,39492],{"class":1731,"line":2396},[61,39491,2210],{"class":1749},[61,39493,39494],{"class":1753}," out_path\n",[61,39496,39497],{"class":1731,"line":2404},[61,39498,1743],{"emptyLinePlaceholder":1742},[61,39500,39501],{"class":1731,"line":2443},[61,39502,1743],{"emptyLinePlaceholder":1742},[61,39504,39505],{"class":1731,"line":2475},[61,39506,39507],{"class":1735},"# ---------------------------- Run X, Y, Z animations --------------------------\n",[61,39509,39510,39513,39515,39517,39520],{"class":1731,"line":2520},[61,39511,39512],{"class":1753},"knobs ",[61,39514,2000],{"class":1859},[61,39516,36695],{"class":1836},[61,39518,39519],{"class":1753},"(  ",[61,39521,39522],{"class":1735},"# tweak if desired\n",[61,39524,39525,39528,39530,39532,39534,39536,39538,39540,39542],{"class":1731,"line":2541},[61,39526,39527],{"class":3402},"    gate_axes",[61,39529,2000],{"class":1859},[61,39531,849],{"class":1753},[61,39533,36798],{"class":2165},[61,39535,1435],{"class":1753},[61,39537,36803],{"class":2165},[61,39539,1435],{"class":1753},[61,39541,36808],{"class":2165},[61,39543,16975],{"class":1753},[61,39545,39546,39549,39551,39554],{"class":1731,"line":2546},[61,39547,39548],{"class":3402},"    initial_state",[61,39550,2000],{"class":1859},[61,39552,39553],{"class":2165},"\"+z\"",[61,39555,36573],{"class":1753},[61,39557,39558,39561,39563,39566,39569],{"class":1731,"line":2582},[61,39559,39560],{"class":3402},"    frames",[61,39562,2000],{"class":1859},[61,39564,39565],{"class":1866},"24",[61,39567,39568],{"class":1753},",                ",[61,39570,39571],{"class":1735},"# raise to 48 for smoother motion\n",[61,39573,39574,39577,39579,39582,39584,39586,39588,39591,39594],{"class":1731,"line":2588},[61,39575,39576],{"class":3402},"    n_theta",[61,39578,2000],{"class":1859},[61,39580,39581],{"class":1866},"80",[61,39583,1435],{"class":1753},[61,39585,37790],{"class":3402},[61,39587,2000],{"class":1859},[61,39589,39590],{"class":1866},"160",[61,39592,39593],{"class":1753},",    ",[61,39595,39596],{"class":1735},"# raise to 120×240 for more detail\n",[61,39598,39599,39602,39604,39606],{"class":1731,"line":2601},[61,39600,39601],{"class":3402},"    fps",[61,39603,2000],{"class":1859},[61,39605,14756],{"class":1866},[61,39607,36573],{"class":1753},[61,39609,39610,39613,39615,39618],{"class":1731,"line":2659},[61,39611,39612],{"class":3402},"    out_dir",[61,39614,2000],{"class":1859},[61,39616,39617],{"class":2165},"\"\u002Fcontent\"",[61,39619,36573],{"class":1753},[61,39621,39622,39625,39627,39629,39631,39633,39635],{"class":1731,"line":2717},[61,39623,39624],{"class":3402},"    figsize",[61,39626,2000],{"class":1859},[61,39628,849],{"class":1753},[61,39630,36550],{"class":1866},[61,39632,1435],{"class":1753},[61,39634,36555],{"class":1866},[61,39636,16975],{"class":1753},[61,39638,39639,39642,39644,39647,39649,39652,39654,39656,39658,39661,39663,39666],{"class":1731,"line":2725},[61,39640,39641],{"class":3402},"    info_loc",[61,39643,2000],{"class":1859},[61,39645,39646],{"class":2165},"\"ul\"",[61,39648,1435],{"class":1753},[61,39650,39651],{"class":3402},"info_fontsize",[61,39653,2000],{"class":1859},[61,39655,17092],{"class":1866},[61,39657,1435],{"class":1753},[61,39659,39660],{"class":3402},"wrap_chars",[61,39662,2000],{"class":1859},[61,39664,39665],{"class":1866},"42",[61,39667,36573],{"class":1753},[61,39669,39670],{"class":1731,"line":2730},[61,39671,2016],{"class":1753},[61,39673,39674],{"class":1731,"line":2749},[61,39675,1743],{"emptyLinePlaceholder":1742},[61,39677,39678,39681,39683,39685],{"class":1731,"line":2755},[61,39679,39680],{"class":1753},"r0 ",[61,39682,2000],{"class":1859},[61,39684,36996],{"class":1836},[61,39686,39687],{"class":1753},"(knobs.initial_state, knobs.custom_r)\n",[61,39689,39690,39693,39695],{"class":1731,"line":2778},[61,39691,39692],{"class":1753},"outs ",[61,39694,2000],{"class":1859},[61,39696,39697],{"class":1753}," []\n",[61,39699,39700,39702,39705,39707],{"class":1731,"line":2783},[61,39701,2486],{"class":1749},[61,39703,39704],{"class":1753}," axname ",[61,39706,2492],{"class":1749},[61,39708,39709],{"class":1753}," knobs.gate_axes:\n",[61,39711,39712,39715,39717,39720,39723,39725,39728,39731,39733,39736,39738],{"class":1731,"line":2791},[61,39713,39714],{"class":1753},"    path ",[61,39716,2000],{"class":1859},[61,39718,39719],{"class":1836}," Path",[61,39721,39722],{"class":1753},"(knobs.out_dir) ",[61,39724,602],{"class":1859},[61,39726,39727],{"class":1749}," f",[61,39729,39730],{"class":2165},"\"wigner_pauli_",[61,39732,3688],{"class":1866},[61,39734,39735],{"class":1753},"axname",[61,39737,3697],{"class":1866},[61,39739,39740],{"class":2165},".gif\"\n",[61,39742,39743,39746,39749],{"class":1731,"line":2804},[61,39744,39745],{"class":1753},"    outs.",[61,39747,39748],{"class":1836},"append",[61,39750,38439],{"class":1753},[61,39752,39753,39756],{"class":1731,"line":2817},[61,39754,39755],{"class":1836},"        animate_wigner_pauli",[61,39757,39758],{"class":1753},"(axname, r0, knobs.frames,\n",[61,39760,39761],{"class":1731,"line":2824},[61,39762,39763],{"class":1753},"                             knobs.n_theta, knobs.n_phi, knobs.fps,\n",[61,39765,39766],{"class":1731,"line":2835},[61,39767,39768],{"class":1753},"                             knobs.figsize, knobs.info_loc,\n",[61,39770,39771],{"class":1731,"line":2846},[61,39772,39773],{"class":1753},"                             knobs.info_fontsize, knobs.wrap_chars,\n",[61,39775,39776],{"class":1731,"line":2851},[61,39777,39778],{"class":1753},"                             path)\n",[61,39780,39781],{"class":1731,"line":2856},[61,39782,3057],{"class":1753},[61,39784,39785],{"class":1731,"line":2862},[61,39786,1743],{"emptyLinePlaceholder":1742},[61,39788,39789],{"class":1731,"line":2867},[61,39790,39791],{"class":1735},"# Display inline\n",[61,39793,39794,39796,39799,39801],{"class":1731,"line":2917},[61,39795,2486],{"class":1749},[61,39797,39798],{"class":1753}," p ",[61,39800,2492],{"class":1749},[61,39802,39803],{"class":1753}," outs:\n",[61,39805,39806,39809,39811,39814,39816,39819,39822],{"class":1731,"line":2926},[61,39807,39808],{"class":1836},"    display",[61,39810,849],{"class":1753},[61,39812,39813],{"class":1836},"Image",[61,39815,849],{"class":1753},[61,39817,39818],{"class":3402},"filename",[61,39820,39821],{"class":1859},"=str",[61,39823,39824],{"class":1753},"(p)))\n",[61,39826,39827,39830,39832,39835],{"class":1731,"line":2972},[61,39828,39829],{"class":1859},"    print",[61,39831,849],{"class":1753},[61,39833,39834],{"class":2165},"\"Saved:\"",[61,39836,39837],{"class":1753},", p)\n",[1720,39839,39842],{"className":39840,"code":39841,"language":29075,"meta":1725},[36200],"Saved: \u002Fcontent\u002Fwigner_pauli_x.gif\n",[1727,39843,39841],{"__ignoreMap":1725},[1720,39845,39848],{"className":39846,"code":39847,"language":29075,"meta":1725},[36200],"Saved: \u002Fcontent\u002Fwigner_pauli_y.gif\n",[1727,39849,39847],{"__ignoreMap":1725},[1720,39851,39854],{"className":39852,"code":39853,"language":29075,"meta":1725},[36200],"Saved: \u002Fcontent\u002Fwigner_pauli_z.gif\n",[1727,39855,39853],{"__ignoreMap":1725},[4797,39857],{"alt":4799,"src":39858},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-01.webp",[4797,39860],{"alt":4803,"src":39861},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-02.webp",[4797,39863],{"alt":4807,"src":39864},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-03.webp",[1720,39866,39868],{"className":1722,"code":39867,"language":1724,"meta":1725,"style":1725},"# @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",[1727,39869,39870,39875,39885,39889,39914,39925,39932,39943,39954,39965,39976,39991,40002,40013,40024,40031,40036,40050,40060,40086,40112,40134,40138,40164,40194,40216,40250,40262,40266,40278,40290,40301,40305,40331,40339,40343,40359,40387,40399,40412,40424,40432,40489,40525,40530,40536,40540,40572,40596,40622,40638,40646,40652,40656,40660,40665,40681,40688,40709,40721,40731,40745,40763,40773,40782,40798,40808,40818,40822,40826,40844],{"__ignoreMap":1725},[61,39871,39872],{"class":1731,"line":1732},[61,39873,39874],{"class":1735},"# @title Cell 4 — Custom axis–angle gate (example: 45° about y)\n",[61,39876,39877,39879,39881,39883],{"class":1731,"line":1739},[61,39878,1750],{"class":1749},[61,39880,36605],{"class":1753},[61,39882,1757],{"class":1749},[61,39884,36610],{"class":1753},[61,39886,39887],{"class":1731,"line":1746},[61,39888,1743],{"emptyLinePlaceholder":1742},[61,39890,39891,39893,39896,39898,39900,39902,39904,39906,39908,39910,39912],{"class":1731,"line":1763},[61,39892,2116],{"class":1749},[61,39894,39895],{"class":1836}," animate_wigner_custom",[61,39897,849],{"class":1753},[61,39899,37512],{"class":2124},[61,39901,37015],{"class":1753},[61,39903,1860],{"class":1859},[61,39905,1435],{"class":1753},[61,39907,1860],{"class":1859},[61,39909,1435],{"class":1753},[61,39911,1860],{"class":1859},[61,39913,6550],{"class":1753},[61,39915,39916,39919,39921,39923],{"class":1731,"line":1776},[61,39917,39918],{"class":2124},"                          angle_deg",[61,39920,2127],{"class":1753},[61,39922,1860],{"class":1859},[61,39924,36573],{"class":1753},[61,39926,39927,39930],{"class":1731,"line":1784},[61,39928,39929],{"class":2124},"                          r0",[61,39931,38733],{"class":1753},[61,39933,39934,39937,39939,39941],{"class":1731,"line":1798},[61,39935,39936],{"class":2124},"                          frames",[61,39938,2127],{"class":1753},[61,39940,2025],{"class":1859},[61,39942,36573],{"class":1753},[61,39944,39945,39948,39950,39952],{"class":1731,"line":1811},[61,39946,39947],{"class":2124},"                          n_theta",[61,39949,2127],{"class":1753},[61,39951,2025],{"class":1859},[61,39953,36573],{"class":1753},[61,39955,39956,39959,39961,39963],{"class":1731,"line":1816},[61,39957,39958],{"class":2124},"                          n_phi",[61,39960,2127],{"class":1753},[61,39962,2025],{"class":1859},[61,39964,36573],{"class":1753},[61,39966,39967,39970,39972,39974],{"class":1731,"line":1822},[61,39968,39969],{"class":2124},"                          fps",[61,39971,2127],{"class":1753},[61,39973,2025],{"class":1859},[61,39975,36573],{"class":1753},[61,39977,39978,39981,39983,39985,39987,39989],{"class":1731,"line":1828},[61,39979,39980],{"class":2124},"                          figsize",[61,39982,37015],{"class":1753},[61,39984,1860],{"class":1859},[61,39986,1435],{"class":1753},[61,39988,1860],{"class":1859},[61,39990,6550],{"class":1753},[61,39992,39993,39996,39998,40000],{"class":1731,"line":1833},[61,39994,39995],{"class":2124},"                          info_loc",[61,39997,2127],{"class":1753},[61,39999,36784],{"class":1859},[61,40001,36573],{"class":1753},[61,40003,40004,40007,40009,40011],{"class":1731,"line":1840},[61,40005,40006],{"class":2124},"                          info_fontsize",[61,40008,2127],{"class":1753},[61,40010,2025],{"class":1859},[61,40012,36573],{"class":1753},[61,40014,40015,40018,40020,40022],{"class":1731,"line":1853},[61,40016,40017],{"class":2124},"                          wrap_chars",[61,40019,2127],{"class":1753},[61,40021,2025],{"class":1859},[61,40023,36573],{"class":1753},[61,40025,40026,40029],{"class":1731,"line":1873},[61,40027,40028],{"class":2124},"                          out_path",[61,40030,38833],{"class":1753},[61,40032,40033],{"class":1731,"line":1889},[61,40034,40035],{"class":2165},"    \"\"\"Animate Wigner under an arbitrary axis–angle rotation 0 → angle_deg.\"\"\"\n",[61,40037,40038,40040,40042,40044,40046,40048],{"class":1731,"line":1908},[61,40039,38843],{"class":1749},[61,40041,38846],{"class":1753},[61,40043,6163],{"class":1859},[61,40045,2654],{"class":1866},[61,40047,1435],{"class":1753},[61,40049,38855],{"class":2165},[61,40051,40052,40054,40056,40058],{"class":1731,"line":1925},[61,40053,38860],{"class":1753},[61,40055,2000],{"class":1859},[61,40057,37776],{"class":1836},[61,40059,38867],{"class":1753},[61,40061,40062,40064,40066,40068,40070,40072,40074,40076,40078,40080,40082,40084],{"class":1731,"line":1941},[61,40063,38872],{"class":1753},[61,40065,2000],{"class":1859},[61,40067,2980],{"class":1866},[61,40069,1899],{"class":1859},[61,40071,6984],{"class":1753},[61,40073,2425],{"class":1836},[61,40075,849],{"class":1753},[61,40077,37995],{"class":1866},[61,40079,2624],{"class":1753},[61,40081,602],{"class":1859},[61,40083,3029],{"class":1866},[61,40085,2016],{"class":1753},[61,40087,40088,40090,40092,40094,40096,40098,40100,40102,40104,40106,40108,40110],{"class":1731,"line":1957},[61,40089,38899],{"class":1753},[61,40091,2000],{"class":1859},[61,40093,2980],{"class":1866},[61,40095,4990],{"class":1859},[61,40097,6984],{"class":1753},[61,40099,2425],{"class":1836},[61,40101,849],{"class":1753},[61,40103,37995],{"class":1866},[61,40105,2624],{"class":1753},[61,40107,602],{"class":1859},[61,40109,3029],{"class":1866},[61,40111,2016],{"class":1753},[61,40113,40114,40116,40118,40120,40122,40124,40126,40128,40130,40132],{"class":1731,"line":1974},[61,40115,38926],{"class":1753},[61,40117,2000],{"class":1859},[61,40119,2003],{"class":1753},[61,40121,2051],{"class":1866},[61,40123,1435],{"class":1753},[61,40125,2435],{"class":1866},[61,40127,2298],{"class":1859},[61,40129,38941],{"class":1753},[61,40131,2051],{"class":1866},[61,40133,38946],{"class":1753},[61,40135,40136],{"class":1731,"line":1979},[61,40137,1743],{"emptyLinePlaceholder":1742},[61,40139,40140,40142,40144,40146,40148,40150,40152,40154,40156,40158,40160,40162],{"class":1731,"line":1985},[61,40141,38955],{"class":1753},[61,40143,2000],{"class":1859},[61,40145,17065],{"class":1753},[61,40147,17068],{"class":1836},[61,40149,849],{"class":1753},[61,40151,3624],{"class":3402},[61,40153,2000],{"class":1859},[61,40155,38970],{"class":1753},[61,40157,24380],{"class":3402},[61,40159,2000],{"class":1859},[61,40161,24385],{"class":1866},[61,40163,2016],{"class":1753},[61,40165,40166,40168,40170,40172,40174,40176,40178,40180,40182,40184,40186,40188,40190,40192],{"class":1731,"line":2019},[61,40167,38983],{"class":1753},[61,40169,2000],{"class":1859},[61,40171,38434],{"class":1753},[61,40173,4239],{"class":1836},[61,40175,3577],{"class":1753},[61,40177,38994],{"class":1836},[61,40179,38997],{"class":1753},[61,40181,4245],{"class":3402},[61,40183,2000],{"class":1859},[61,40185,4250],{"class":2165},[61,40187,1435],{"class":1753},[61,40189,4265],{"class":3402},[61,40191,2000],{"class":1859},[61,40193,39012],{"class":1753},[61,40195,40196,40198,40200,40202,40204,40206,40208,40210,40212,40214],{"class":1731,"line":2033},[61,40197,39017],{"class":3402},[61,40199,2000],{"class":1859},[61,40201,39022],{"class":1753},[61,40203,39025],{"class":3402},[61,40205,2000],{"class":1859},[61,40207,39030],{"class":1753},[61,40209,4255],{"class":3402},[61,40211,2000],{"class":1859},[61,40213,4260],{"class":2165},[61,40215,2016],{"class":1753},[61,40217,40218,40220,40222,40224,40226,40228,40230,40232,40234,40236,40238,40240,40242,40244,40246,40248],{"class":1731,"line":2061},[61,40219,39043],{"class":1753},[61,40221,2000],{"class":1859},[61,40223,39048],{"class":1753},[61,40225,4288],{"class":1836},[61,40227,39053],{"class":1753},[61,40229,3368],{"class":3402},[61,40231,2000],{"class":1859},[61,40233,39060],{"class":1753},[61,40235,39063],{"class":3402},[61,40237,2000],{"class":1859},[61,40239,39068],{"class":1866},[61,40241,1435],{"class":1753},[61,40243,39073],{"class":3402},[61,40245,2000],{"class":1859},[61,40247,39078],{"class":1866},[61,40249,2016],{"class":1753},[61,40251,40252,40254,40256,40258,40260],{"class":1731,"line":2074},[61,40253,39085],{"class":1753},[61,40255,39088],{"class":1836},[61,40257,849],{"class":1753},[61,40259,39093],{"class":2165},[61,40261,2016],{"class":1753},[61,40263,40264],{"class":1731,"line":2079},[61,40265,1743],{"emptyLinePlaceholder":1742},[61,40267,40268,40270,40272,40274,40276],{"class":1731,"line":2092},[61,40269,24430],{"class":1753},[61,40271,17159],{"class":1836},[61,40273,849],{"class":1753},[61,40275,39110],{"class":2165},[61,40277,2016],{"class":1753},[61,40279,40280,40282,40284,40286,40288],{"class":1731,"line":2097},[61,40281,24430],{"class":1753},[61,40283,17173],{"class":1836},[61,40285,849],{"class":1753},[61,40287,39123],{"class":2165},[61,40289,2016],{"class":1753},[61,40291,40292,40294,40296,40299],{"class":1731,"line":2102},[61,40293,39130],{"class":1836},[61,40295,39133],{"class":1753},[61,40297,40298],{"class":2165},"\"Custom rotation\"",[61,40300,2016],{"class":1753},[61,40302,40303],{"class":1731,"line":2108},[61,40304,1743],{"emptyLinePlaceholder":1742},[61,40306,40307,40309,40311,40313,40315,40317,40319,40321,40323,40325,40327,40329],{"class":1731,"line":2113},[61,40308,39165],{"class":1753},[61,40310,2000],{"class":1859},[61,40312,38344],{"class":1836},[61,40314,39133],{"class":1753},[61,40316,39174],{"class":2165},[61,40318,1435],{"class":1753},[61,40320,38068],{"class":3402},[61,40322,2000],{"class":1859},[61,40324,39183],{"class":1753},[61,40326,38463],{"class":3402},[61,40328,2000],{"class":1859},[61,40330,39190],{"class":1753},[61,40332,40333,40335,40337],{"class":1731,"line":2162},[61,40334,39195],{"class":3402},[61,40336,2000],{"class":1859},[61,40338,39200],{"class":1753},[61,40340,40341],{"class":1731,"line":2169},[61,40342,1743],{"emptyLinePlaceholder":1742},[61,40344,40345,40347,40349,40351,40353,40355,40357],{"class":1731,"line":2187},[61,40346,39209],{"class":1749},[61,40348,39212],{"class":1836},[61,40350,849],{"class":1753},[61,40352,39217],{"class":2124},[61,40354,2127],{"class":1753},[61,40356,2025],{"class":1859},[61,40358,3371],{"class":1753},[61,40360,40361,40363,40365,40367,40369,40371,40373,40375,40377,40379,40381,40384],{"class":1731,"line":2207},[61,40362,39228],{"class":1753},[61,40364,2000],{"class":1859},[61,40366,39233],{"class":1753},[61,40368,602],{"class":1859},[61,40370,39238],{"class":1753},[61,40372,2006],{"class":1859},[61,40374,5266],{"class":1866},[61,40376,2702],{"class":1753},[61,40378,2201],{"class":1859},[61,40380,2193],{"class":1753},[61,40382,40383],{"class":1836},"deg2rad",[61,40385,40386],{"class":1753},"(angle_deg)\n",[61,40388,40389,40392,40394,40396],{"class":1731,"line":2221},[61,40390,40391],{"class":1753},"        R ",[61,40393,2000],{"class":1859},[61,40395,37507],{"class":1836},[61,40397,40398],{"class":1753},"(axis_vec, ang)\n",[61,40400,40401,40403,40405,40408,40410],{"class":1731,"line":2226},[61,40402,39257],{"class":1753},[61,40404,2000],{"class":1859},[61,40406,40407],{"class":1753}," R ",[61,40409,6930],{"class":1859},[61,40411,39274],{"class":1753},[61,40413,40414,40416,40418,40420,40422],{"class":1731,"line":2258},[61,40415,39279],{"class":1753},[61,40417,39282],{"class":1836},[61,40419,849],{"class":1753},[61,40421,39287],{"class":1836},[61,40423,39290],{"class":1753},[61,40425,40426,40428,40430],{"class":1731,"line":2264},[61,40427,39295],{"class":1753},[61,40429,39298],{"class":1836},[61,40431,38439],{"class":1753},[61,40433,40434,40437,40440,40442,40445,40447,40449,40452,40454,40456,40458,40460,40462,40464,40466,40468,40470,40472,40474,40476,40478,40480,40482,40484,40486],{"class":1731,"line":2280},[61,40435,40436],{"class":1749},"            f",[61,40438,40439],{"class":2165},"\"axis = (",[61,40441,3688],{"class":1866},[61,40443,40444],{"class":1753},"axis_vec[",[61,40446,1495],{"class":1866},[61,40448,27817],{"class":1753},[61,40450,40451],{"class":1749},":.2f",[61,40453,3697],{"class":1866},[61,40455,1435],{"class":2165},[61,40457,3688],{"class":1866},[61,40459,40444],{"class":1753},[61,40461,57],{"class":1866},[61,40463,27817],{"class":1753},[61,40465,40451],{"class":1749},[61,40467,3697],{"class":1866},[61,40469,1435],{"class":2165},[61,40471,3688],{"class":1866},[61,40473,40444],{"class":1753},[61,40475,337],{"class":1866},[61,40477,27817],{"class":1753},[61,40479,40451],{"class":1749},[61,40481,3697],{"class":1866},[61,40483,300],{"class":2165},[61,40485,39321],{"class":1859},[61,40487,40488],{"class":2165},"\"\n",[61,40490,40491,40493,40495,40497,40500,40503,40506,40508,40510,40513,40515,40518,40520,40522],{"class":1731,"line":2304},[61,40492,40436],{"class":1749},[61,40494,39305],{"class":2165},[61,40496,3688],{"class":1866},[61,40498,40499],{"class":1753},"np.",[61,40501,40502],{"class":1836},"rad2deg",[61,40504,40505],{"class":1753},"(ang)",[61,40507,3694],{"class":1749},[61,40509,3697],{"class":1866},[61,40511,40512],{"class":2165},"° \u002F ",[61,40514,3688],{"class":1866},[61,40516,40517],{"class":1753},"angle_deg",[61,40519,3694],{"class":1749},[61,40521,3697],{"class":1866},[61,40523,40524],{"class":2165},"°\"\n",[61,40526,40527],{"class":1731,"line":2326},[61,40528,40529],{"class":1753},"        )\n",[61,40531,40532,40534],{"class":1731,"line":2342},[61,40533,2190],{"class":1749},[61,40535,39353],{"class":1753},[61,40537,40538],{"class":1731,"line":2379},[61,40539,1743],{"emptyLinePlaceholder":1742},[61,40541,40542,40544,40546,40548,40550,40552,40554,40556,40558,40560,40562,40564,40566,40568,40570],{"class":1731,"line":2396},[61,40543,39362],{"class":1753},[61,40545,2000],{"class":1859},[61,40547,39367],{"class":1836},[61,40549,39370],{"class":1753},[61,40551,39340],{"class":3402},[61,40553,2000],{"class":1859},[61,40555,39377],{"class":1753},[61,40557,39380],{"class":3402},[61,40559,2000],{"class":1859},[61,40561,39385],{"class":1866},[61,40563,1435],{"class":1753},[61,40565,39390],{"class":3402},[61,40567,2000],{"class":1859},[61,40569,14390],{"class":1866},[61,40571,2016],{"class":1753},[61,40573,40574,40576,40578,40580,40582,40584,40586,40588,40590,40592,40594],{"class":1731,"line":2404},[61,40575,39401],{"class":1753},[61,40577,39404],{"class":1836},[61,40579,849],{"class":1753},[61,40581,39409],{"class":3402},[61,40583,2000],{"class":1859},[61,40585,24385],{"class":1866},[61,40587,1435],{"class":1753},[61,40589,39418],{"class":3402},[61,40591,2000],{"class":1859},[61,40593,24385],{"class":1866},[61,40595,2016],{"class":1753},[61,40597,40598,40600,40602,40604,40606,40608,40610,40612,40614,40616,40618,40620],{"class":1731,"line":2443},[61,40599,39429],{"class":1753},[61,40601,39432],{"class":1836},[61,40603,39435],{"class":1753},[61,40605,39438],{"class":1836},[61,40607,4205],{"class":1753},[61,40609,39443],{"class":3402},[61,40611,2000],{"class":1859},[61,40613,39448],{"class":1836},[61,40615,849],{"class":1753},[61,40617,39453],{"class":3402},[61,40619,2000],{"class":1859},[61,40621,39458],{"class":1753},[61,40623,40624,40626,40628,40630,40632,40634,40636],{"class":1731,"line":2475},[61,40625,39463],{"class":3402},[61,40627,38485],{"class":1859},[61,40629,849],{"class":1753},[61,40631,39470],{"class":3402},[61,40633,2000],{"class":1859},[61,40635,38505],{"class":2165},[61,40637,5546],{"class":1753},[61,40639,40640,40642,40644],{"class":1731,"line":2520},[61,40641,39481],{"class":1753},[61,40643,39484],{"class":1836},[61,40645,39487],{"class":1753},[61,40647,40648,40650],{"class":1731,"line":2541},[61,40649,2210],{"class":1749},[61,40651,39494],{"class":1753},[61,40653,40654],{"class":1731,"line":2546},[61,40655,1743],{"emptyLinePlaceholder":1742},[61,40657,40658],{"class":1731,"line":2582},[61,40659,1743],{"emptyLinePlaceholder":1742},[61,40661,40662],{"class":1731,"line":2588},[61,40663,40664],{"class":1735},"# Example: 45° about the y-axis, starting from |+z⟩\n",[61,40666,40667,40670,40672,40674,40676,40678],{"class":1731,"line":2601},[61,40668,40669],{"class":1753},"custom_out ",[61,40671,2000],{"class":1859},[61,40673,39719],{"class":1836},[61,40675,39722],{"class":1753},[61,40677,602],{"class":1859},[61,40679,40680],{"class":2165}," \"wigner_custom_y45.gif\"\n",[61,40682,40683,40686],{"class":1731,"line":2659},[61,40684,40685],{"class":1836},"animate_wigner_custom",[61,40687,38439],{"class":1753},[61,40689,40690,40693,40695,40697,40699,40701,40703,40705,40707],{"class":1731,"line":2717},[61,40691,40692],{"class":3402},"    axis_vec",[61,40694,2000],{"class":1859},[61,40696,849],{"class":1753},[61,40698,2051],{"class":1866},[61,40700,1435],{"class":1753},[61,40702,2319],{"class":1866},[61,40704,1435],{"class":1753},[61,40706,2051],{"class":1866},[61,40708,16975],{"class":1753},[61,40710,40711,40714,40716,40719],{"class":1731,"line":2725},[61,40712,40713],{"class":3402},"    angle_deg",[61,40715,2000],{"class":1859},[61,40717,40718],{"class":1866},"45.0",[61,40720,36573],{"class":1753},[61,40722,40723,40726,40728],{"class":1731,"line":2730},[61,40724,40725],{"class":3402},"    r0",[61,40727,2000],{"class":1859},[61,40729,40730],{"class":1753},"r0,\n",[61,40732,40733,40735,40737,40739,40742],{"class":1731,"line":2749},[61,40734,39560],{"class":3402},[61,40736,2000],{"class":1859},[61,40738,39565],{"class":1866},[61,40740,40741],{"class":1753},",                 ",[61,40743,40744],{"class":1735},"# raise to 48 if you prefer\n",[61,40746,40747,40749,40751,40753,40755,40757,40759,40761],{"class":1731,"line":2755},[61,40748,39576],{"class":3402},[61,40750,2000],{"class":1859},[61,40752,39581],{"class":1866},[61,40754,1435],{"class":1753},[61,40756,37790],{"class":3402},[61,40758,2000],{"class":1859},[61,40760,39590],{"class":1866},[61,40762,36573],{"class":1753},[61,40764,40765,40767,40769,40771],{"class":1731,"line":2778},[61,40766,39601],{"class":3402},[61,40768,2000],{"class":1859},[61,40770,14756],{"class":1866},[61,40772,36573],{"class":1753},[61,40774,40775,40777,40779],{"class":1731,"line":2783},[61,40776,39624],{"class":3402},[61,40778,2000],{"class":1859},[61,40780,40781],{"class":1753},"knobs.figsize,\n",[61,40783,40784,40786,40788,40791,40793,40795],{"class":1731,"line":2791},[61,40785,39641],{"class":3402},[61,40787,2000],{"class":1859},[61,40789,40790],{"class":1753},"knobs.info_loc, ",[61,40792,39651],{"class":3402},[61,40794,2000],{"class":1859},[61,40796,40797],{"class":1753},"knobs.info_fontsize,\n",[61,40799,40800,40803,40805],{"class":1731,"line":2804},[61,40801,40802],{"class":3402},"    wrap_chars",[61,40804,2000],{"class":1859},[61,40806,40807],{"class":1753},"knobs.wrap_chars,\n",[61,40809,40810,40813,40815],{"class":1731,"line":2817},[61,40811,40812],{"class":3402},"    out_path",[61,40814,2000],{"class":1859},[61,40816,40817],{"class":1753},"custom_out,\n",[61,40819,40820],{"class":1731,"line":2824},[61,40821,2016],{"class":1753},[61,40823,40824],{"class":1731,"line":2835},[61,40825,1743],{"emptyLinePlaceholder":1742},[61,40827,40828,40831,40833,40835,40837,40839,40841],{"class":1731,"line":2846},[61,40829,40830],{"class":1836},"display",[61,40832,849],{"class":1753},[61,40834,39813],{"class":1836},[61,40836,849],{"class":1753},[61,40838,39818],{"class":3402},[61,40840,39821],{"class":1859},[61,40842,40843],{"class":1753},"(custom_out)))\n",[61,40845,40846,40849,40851,40853],{"class":1731,"line":2851},[61,40847,40848],{"class":1859},"print",[61,40850,849],{"class":1753},[61,40852,39834],{"class":2165},[61,40854,40855],{"class":1753},", custom_out)\n",[1720,40857,40860],{"className":40858,"code":40859,"language":29075,"meta":1725},[36200],"Saved: \u002Fcontent\u002Fwigner_custom_y45.gif\n",[1727,40861,40859],{"__ignoreMap":1725},[4797,40863],{"alt":4811,"src":40864},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-04.webp",[12,40866,40867],{},"The next following scripts are for the 3D Wigner plots.",[1720,40869,40871],{"className":1722,"code":40870,"language":1724,"meta":1725,"style":1725},"# @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",[1727,40872,40873,40878,40883,40887,40897,40901,40911,40921,40931,40935,40945,40955,40965,40977,40989,40999,41003,41007,41011,41020,41025,41029,41033,41038,41042,41047,41052,41057,41061,41066,41071,41076,41081,41086,41090,41100,41132,41142,41153,41164,41174,41186,41200,41210,41234,41244,41248,41252,41257,41293,41301,41337,41373,41409,41413,41439,41458,41468,41472,41476,41501,41517,41529,41567,41579,41617,41629,41667,41678,41682,41686,41711,41716,41732,41754,41774,41794,41814,41826,41847,41854,41858,41862,41879,41884,41945,41949,41953,41975,41980,41992,42004,42008,42013,42034,42055,42068,42072,42077,42103,42129,42155,42169,42195,42209,42213,42241,42270,42295,42309,42330,42334,42339,42365,42399,42403,42417,42422,42434,42454,42473,42488,42492,42496,42501,42536,42548,42552,42557,42569,42581,42594,42608,42638,42670,42710,42720,42724,42737,42776,42785,42825,42835,42839,42843,42860,42887,42903,42915,42947,42980,43028,43035,43039,43072,43098,43123,43151,43167,43175,43181,43185,43190,43205,43238,43244,43256,43274,43293,43316,43325,43330,43335,43341,43356,43367,43404,43426,43437,43454,43466,43471,43476],{"__ignoreMap":1725},[61,40874,40875],{"class":1731,"line":1732},[61,40876,40877],{"class":1735},"# @title Cell 5 — 3D Wigner animations for ALL Pauli gates (X, Y, Z)\n",[61,40879,40880],{"class":1731,"line":1739},[61,40881,40882],{"class":1735},"# PEP 8\u002F257 compliant. Stable: updates facecolors in place; no deleting artists.\n",[61,40884,40885],{"class":1731,"line":1746},[61,40886,1743],{"emptyLinePlaceholder":1742},[61,40888,40889,40891,40893,40895],{"class":1731,"line":1763},[61,40890,1750],{"class":1749},[61,40892,36446],{"class":3766},[61,40894,36449],{"class":1749},[61,40896,36452],{"class":1753},[61,40898,40899],{"class":1731,"line":1776},[61,40900,1743],{"emptyLinePlaceholder":1742},[61,40902,40903,40905,40907,40909],{"class":1731,"line":1784},[61,40904,1750],{"class":1749},[61,40906,1754],{"class":1753},[61,40908,1757],{"class":1749},[61,40910,1760],{"class":1753},[61,40912,40913,40915,40917,40919],{"class":1731,"line":1798},[61,40914,1750],{"class":1749},[61,40916,36605],{"class":1753},[61,40918,1757],{"class":1749},[61,40920,36610],{"class":1753},[61,40922,40923,40925,40927,40929],{"class":1731,"line":1811},[61,40924,1750],{"class":1749},[61,40926,1768],{"class":1753},[61,40928,1757],{"class":1749},[61,40930,36621],{"class":1753},[61,40932,40933],{"class":1731,"line":1816},[61,40934,1743],{"emptyLinePlaceholder":1742},[61,40936,40937,40939,40941,40943],{"class":1731,"line":1822},[61,40938,1757],{"class":1749},[61,40940,1789],{"class":1753},[61,40942,1792],{"class":1749},[61,40944,1795],{"class":1753},[61,40946,40947,40949,40951,40953],{"class":1731,"line":1828},[61,40948,1757],{"class":1749},[61,40950,1803],{"class":1753},[61,40952,1792],{"class":1749},[61,40954,1808],{"class":1753},[61,40956,40957,40959,40961,40963],{"class":1731,"line":1833},[61,40958,1750],{"class":1749},[61,40960,36659],{"class":1753},[61,40962,1757],{"class":1749},[61,40964,36664],{"class":1753},[61,40966,40967,40969,40972,40974],{"class":1731,"line":1840},[61,40968,1750],{"class":1749},[61,40970,40971],{"class":1753}," matplotlib.colors ",[61,40973,1757],{"class":1749},[61,40975,40976],{"class":1753}," Normalize\n",[61,40978,40979,40981,40984,40986],{"class":1731,"line":1853},[61,40980,1750],{"class":1749},[61,40982,40983],{"class":1753}," matplotlib.cm ",[61,40985,1757],{"class":1749},[61,40987,40988],{"class":1753}," ScalarMappable\n",[61,40990,40991,40993,40995,40997],{"class":1731,"line":1873},[61,40992,1750],{"class":1749},[61,40994,36671],{"class":1753},[61,40996,1757],{"class":1749},[61,40998,36676],{"class":1753},[61,41000,41001],{"class":1731,"line":1889},[61,41002,1743],{"emptyLinePlaceholder":1742},[61,41004,41005],{"class":1731,"line":1908},[61,41006,1743],{"emptyLinePlaceholder":1742},[61,41008,41009],{"class":1731,"line":1925},[61,41010,1837],{"class":1836},[61,41012,41013,41015,41018],{"class":1731,"line":1941},[61,41014,1843],{"class":1749},[61,41016,41017],{"class":1846}," Knobs3D",[61,41019,1850],{"class":1753},[61,41021,41022],{"class":1731,"line":1957},[61,41023,41024],{"class":2165},"    \"\"\"User controls for 3D Pauli-gate Wigner animations on a colored sphere.\n",[61,41026,41027],{"class":1731,"line":1974},[61,41028,1743],{"emptyLinePlaceholder":1742},[61,41030,41031],{"class":1731,"line":1979},[61,41032,36711],{"class":2165},[61,41034,41035],{"class":1731,"line":1985},[61,41036,41037],{"class":2165},"        initial_state: Starting Bloch eigenstate label or \"custom\".\n",[61,41039,41040],{"class":1731,"line":2019},[61,41041,36726],{"class":2165},[61,41043,41044],{"class":1731,"line":2033},[61,41045,41046],{"class":2165},"        frames: Number of frames (≥ 2) for 0 → π rotation.\n",[61,41048,41049],{"class":1731,"line":2061},[61,41050,41051],{"class":2165},"        n_theta: Polar grid size for the sphere.\n",[61,41053,41054],{"class":1731,"line":2074},[61,41055,41056],{"class":2165},"        n_phi: Azimuth grid size for the sphere.\n",[61,41058,41059],{"class":1731,"line":2079},[61,41060,36746],{"class":2165},[61,41062,41063],{"class":1731,"line":2092},[61,41064,41065],{"class":2165},"        elev: Camera elevation in degrees.\n",[61,41067,41068],{"class":1731,"line":2097},[61,41069,41070],{"class":2165},"        azim: Camera azimuth in degrees.\n",[61,41072,41073],{"class":1731,"line":2102},[61,41074,41075],{"class":2165},"        out_dir: Output directory (e.g., \"\u002Fcontent\" on Colab).\n",[61,41077,41078],{"class":1731,"line":2108},[61,41079,41080],{"class":2165},"        figsize: Figure size in inches (DPI set globally by rcParams).\n",[61,41082,41083],{"class":1731,"line":2113},[61,41084,41085],{"class":2165},"        info_fontsize: In-axes info text font size.\n",[61,41087,41088],{"class":1731,"line":2162},[61,41089,36776],{"class":2165},[61,41091,41092,41094,41096,41098],{"class":1731,"line":2169},[61,41093,36815],{"class":1753},[61,41095,36784],{"class":1859},[61,41097,1863],{"class":1859},[61,41099,36822],{"class":2165},[61,41101,41102,41104,41106,41108,41110,41112,41114,41116,41118,41120,41122,41124,41126,41128,41130],{"class":1731,"line":2187},[61,41103,36827],{"class":1753},[61,41105,1860],{"class":1859},[61,41107,1435],{"class":1753},[61,41109,1860],{"class":1859},[61,41111,1435],{"class":1753},[61,41113,1860],{"class":1859},[61,41115,1997],{"class":1753},[61,41117,2000],{"class":1859},[61,41119,2003],{"class":1753},[61,41121,2051],{"class":1866},[61,41123,1435],{"class":1753},[61,41125,2051],{"class":1866},[61,41127,1435],{"class":1753},[61,41129,2319],{"class":1866},[61,41131,2016],{"class":1753},[61,41133,41134,41136,41138,41140],{"class":1731,"line":2207},[61,41135,36860],{"class":1753},[61,41137,2025],{"class":1859},[61,41139,1863],{"class":1859},[61,41141,36867],{"class":1866},[61,41143,41144,41146,41148,41150],{"class":1731,"line":2221},[61,41145,36872],{"class":1753},[61,41147,2025],{"class":1859},[61,41149,1863],{"class":1859},[61,41151,41152],{"class":1866}," 64\n",[61,41154,41155,41157,41159,41161],{"class":1731,"line":2226},[61,41156,36884],{"class":1753},[61,41158,2025],{"class":1859},[61,41160,1863],{"class":1859},[61,41162,41163],{"class":1866}," 128\n",[61,41165,41166,41168,41170,41172],{"class":1731,"line":2258},[61,41167,36896],{"class":1753},[61,41169,2025],{"class":1859},[61,41171,1863],{"class":1859},[61,41173,36903],{"class":1866},[61,41175,41176,41179,41181,41183],{"class":1731,"line":2264},[61,41177,41178],{"class":1753},"    elev: ",[61,41180,1860],{"class":1859},[61,41182,1863],{"class":1859},[61,41184,41185],{"class":1866}," 25.0\n",[61,41187,41188,41191,41193,41195,41197],{"class":1731,"line":2280},[61,41189,41190],{"class":1753},"    azim: ",[61,41192,1860],{"class":1859},[61,41194,1863],{"class":1859},[61,41196,1899],{"class":1859},[61,41198,41199],{"class":1866},"60.0\n",[61,41201,41202,41204,41206,41208],{"class":1731,"line":2304},[61,41203,36908],{"class":1753},[61,41205,36784],{"class":1859},[61,41207,1863],{"class":1859},[61,41209,36915],{"class":2165},[61,41211,41212,41214,41216,41218,41220,41222,41224,41226,41228,41230,41232],{"class":1731,"line":2326},[61,41213,36920],{"class":1753},[61,41215,1860],{"class":1859},[61,41217,1435],{"class":1753},[61,41219,1860],{"class":1859},[61,41221,1997],{"class":1753},[61,41223,2000],{"class":1859},[61,41225,2003],{"class":1753},[61,41227,36550],{"class":1866},[61,41229,1435],{"class":1753},[61,41231,5636],{"class":1866},[61,41233,2016],{"class":1753},[61,41235,41236,41238,41240,41242],{"class":1731,"line":2342},[61,41237,36957],{"class":1753},[61,41239,2025],{"class":1859},[61,41241,1863],{"class":1859},[61,41243,36964],{"class":1866},[61,41245,41246],{"class":1731,"line":2379},[61,41247,1743],{"emptyLinePlaceholder":1742},[61,41249,41250],{"class":1731,"line":2396},[61,41251,1743],{"emptyLinePlaceholder":1742},[61,41253,41254],{"class":1731,"line":2404},[61,41255,41256],{"class":1735},"# ----------------------------- Helpers (self-contained) -----------------------------\n",[61,41258,41259,41261,41264,41266,41268,41270,41272,41274,41277,41280,41282,41284,41286,41288,41290],{"class":1731,"line":2443},[61,41260,2116],{"class":1749},[61,41262,41263],{"class":1836}," _bloch_vector",[61,41265,849],{"class":1753},[61,41267,3666],{"class":2124},[61,41269,2127],{"class":1753},[61,41271,36784],{"class":1859},[61,41273,1435],{"class":1753},[61,41275,41276],{"class":2124},"custom",[61,41278,41279],{"class":1753},"=(",[61,41281,2051],{"class":1866},[61,41283,1435],{"class":1753},[61,41285,2051],{"class":1866},[61,41287,1435],{"class":1753},[61,41289,2319],{"class":1866},[61,41291,41292],{"class":1753},")) -> np.ndarray:\n",[61,41294,41295,41297,41299],{"class":1731,"line":2475},[61,41296,37038],{"class":1753},[61,41298,2000],{"class":1859},[61,41300,37043],{"class":1753},[61,41302,41303,41305,41307,41309,41311,41313,41315,41317,41319,41321,41323,41325,41327,41329,41331,41333,41335],{"class":1731,"line":2520},[61,41304,37048],{"class":2165},[61,41306,36547],{"class":1753},[61,41308,2051],{"class":1866},[61,41310,1435],{"class":1753},[61,41312,2051],{"class":1866},[61,41314,1435],{"class":1753},[61,41316,2319],{"class":1866},[61,41318,3638],{"class":1753},[61,41320,37065],{"class":2165},[61,41322,36547],{"class":1753},[61,41324,2051],{"class":1866},[61,41326,1435],{"class":1753},[61,41328,2051],{"class":1866},[61,41330,1435],{"class":1753},[61,41332,2006],{"class":1859},[61,41334,2319],{"class":1866},[61,41336,16975],{"class":1753},[61,41338,41339,41341,41343,41345,41347,41349,41351,41353,41355,41357,41359,41361,41363,41365,41367,41369,41371],{"class":1731,"line":2541},[61,41340,37086],{"class":2165},[61,41342,36547],{"class":1753},[61,41344,2319],{"class":1866},[61,41346,1435],{"class":1753},[61,41348,2051],{"class":1866},[61,41350,1435],{"class":1753},[61,41352,2051],{"class":1866},[61,41354,3638],{"class":1753},[61,41356,37103],{"class":2165},[61,41358,36547],{"class":1753},[61,41360,2006],{"class":1859},[61,41362,2319],{"class":1866},[61,41364,1435],{"class":1753},[61,41366,2051],{"class":1866},[61,41368,1435],{"class":1753},[61,41370,2051],{"class":1866},[61,41372,16975],{"class":1753},[61,41374,41375,41377,41379,41381,41383,41385,41387,41389,41391,41393,41395,41397,41399,41401,41403,41405,41407],{"class":1731,"line":2546},[61,41376,37124],{"class":2165},[61,41378,36547],{"class":1753},[61,41380,2051],{"class":1866},[61,41382,1435],{"class":1753},[61,41384,2319],{"class":1866},[61,41386,1435],{"class":1753},[61,41388,2051],{"class":1866},[61,41390,3638],{"class":1753},[61,41392,37141],{"class":2165},[61,41394,36547],{"class":1753},[61,41396,2051],{"class":1866},[61,41398,1435],{"class":1753},[61,41400,2006],{"class":1859},[61,41402,2319],{"class":1866},[61,41404,1435],{"class":1753},[61,41406,2051],{"class":1866},[61,41408,16975],{"class":1753},[61,41410,41411],{"class":1731,"line":2582},[61,41412,37162],{"class":1753},[61,41414,41415,41417,41419,41421,41423,41425,41427,41430,41432,41435,41437],{"class":1731,"line":2588},[61,41416,2267],{"class":1753},[61,41418,2000],{"class":1859},[61,41420,2193],{"class":1753},[61,41422,37173],{"class":1836},[61,41424,37176],{"class":1753},[61,41426,37179],{"class":1836},[61,41428,41429],{"class":1753},"(label.",[61,41431,37185],{"class":1836},[61,41433,41434],{"class":1753},"(), custom), ",[61,41436,1860],{"class":1859},[61,41438,2016],{"class":1753},[61,41440,41441,41443,41445,41447,41449,41452,41455],{"class":1731,"line":2601},[61,41442,37197],{"class":1753},[61,41444,2000],{"class":1859},[61,41446,6895],{"class":1753},[61,41448,37204],{"class":1836},[61,41450,41451],{"class":1753},"(r) ",[61,41453,41454],{"class":1749},"or",[61,41456,41457],{"class":1866}," 1.0\n",[61,41459,41460,41462,41464,41466],{"class":1731,"line":2659},[61,41461,2210],{"class":1749},[61,41463,37245],{"class":1753},[61,41465,602],{"class":1859},[61,41467,37250],{"class":1753},[61,41469,41470],{"class":1731,"line":2717},[61,41471,1743],{"emptyLinePlaceholder":1742},[61,41473,41474],{"class":1731,"line":2725},[61,41475,1743],{"emptyLinePlaceholder":1742},[61,41477,41478,41480,41483,41485,41487,41489,41491,41493,41495,41497,41499],{"class":1731,"line":2730},[61,41479,2116],{"class":1749},[61,41481,41482],{"class":1836}," _rotation_matrix",[61,41484,849],{"class":1753},[61,41486,37275],{"class":2124},[61,41488,2127],{"class":1753},[61,41490,36784],{"class":1859},[61,41492,1435],{"class":1753},[61,41494,37284],{"class":2124},[61,41496,2127],{"class":1753},[61,41498,1860],{"class":1859},[61,41500,37291],{"class":1753},[61,41502,41503,41505,41507,41509,41511,41513,41515],{"class":1731,"line":2749},[61,41504,37301],{"class":1753},[61,41506,2000],{"class":1859},[61,41508,2193],{"class":1753},[61,41510,5051],{"class":1836},[61,41512,37310],{"class":1753},[61,41514,5069],{"class":1836},[61,41516,37315],{"class":1753},[61,41518,41519,41521,41523,41525,41527],{"class":1731,"line":2755},[61,41520,2172],{"class":1749},[61,41522,37322],{"class":1753},[61,41524,37238],{"class":1859},[61,41526,37327],{"class":2165},[61,41528,1850],{"class":1753},[61,41530,41531,41533,41535,41537,41539,41541,41543,41545,41547,41549,41551,41553,41555,41557,41559,41561,41563,41565],{"class":1731,"line":2778},[61,41532,2190],{"class":1749},[61,41534,2193],{"class":1753},[61,41536,3509],{"class":1836},[61,41538,6541],{"class":1753},[61,41540,57],{"class":1866},[61,41542,1435],{"class":1753},[61,41544,1495],{"class":1866},[61,41546,1435],{"class":1753},[61,41548,1495],{"class":1866},[61,41550,37352],{"class":1753},[61,41552,1495],{"class":1866},[61,41554,37357],{"class":1753},[61,41556,2006],{"class":1859},[61,41558,37362],{"class":1753},[61,41560,1495],{"class":1866},[61,41562,37367],{"class":1753},[61,41564,1860],{"class":1859},[61,41566,2016],{"class":1753},[61,41568,41569,41571,41573,41575,41577],{"class":1731,"line":2783},[61,41570,2172],{"class":1749},[61,41572,37322],{"class":1753},[61,41574,37238],{"class":1859},[61,41576,37382],{"class":2165},[61,41578,1850],{"class":1753},[61,41580,41581,41583,41585,41587,41589,41591,41593,41595,41597,41599,41601,41603,41605,41607,41609,41611,41613,41615],{"class":1731,"line":2791},[61,41582,2190],{"class":1749},[61,41584,2193],{"class":1753},[61,41586,3509],{"class":1836},[61,41588,37395],{"class":1753},[61,41590,1495],{"class":1866},[61,41592,37400],{"class":1753},[61,41594,1495],{"class":1866},[61,41596,1435],{"class":1753},[61,41598,57],{"class":1866},[61,41600,1435],{"class":1753},[61,41602,1495],{"class":1866},[61,41604,37352],{"class":1753},[61,41606,2006],{"class":1859},[61,41608,37417],{"class":1753},[61,41610,1495],{"class":1866},[61,41612,37422],{"class":1753},[61,41614,1860],{"class":1859},[61,41616,2016],{"class":1753},[61,41618,41619,41621,41623,41625,41627],{"class":1731,"line":2804},[61,41620,2172],{"class":1749},[61,41622,37322],{"class":1753},[61,41624,37238],{"class":1859},[61,41626,37437],{"class":2165},[61,41628,1850],{"class":1753},[61,41630,41631,41633,41635,41637,41639,41641,41643,41645,41647,41649,41651,41653,41655,41657,41659,41661,41663,41665],{"class":1731,"line":2817},[61,41632,2190],{"class":1749},[61,41634,2193],{"class":1753},[61,41636,3509],{"class":1836},[61,41638,37395],{"class":1753},[61,41640,2006],{"class":1859},[61,41642,37417],{"class":1753},[61,41644,1495],{"class":1866},[61,41646,37458],{"class":1753},[61,41648,1495],{"class":1866},[61,41650,37352],{"class":1753},[61,41652,1495],{"class":1866},[61,41654,1435],{"class":1753},[61,41656,1495],{"class":1866},[61,41658,1435],{"class":1753},[61,41660,57],{"class":1866},[61,41662,37475],{"class":1753},[61,41664,1860],{"class":1859},[61,41666,2016],{"class":1753},[61,41668,41669,41671,41673,41676],{"class":1731,"line":2824},[61,41670,37484],{"class":1749},[61,41672,37487],{"class":1753},[61,41674,41675],{"class":2165},"\"axis must be 'x', 'y', or 'z'\"",[61,41677,2016],{"class":1753},[61,41679,41680],{"class":1731,"line":2835},[61,41681,1743],{"emptyLinePlaceholder":1742},[61,41683,41684],{"class":1731,"line":2846},[61,41685,1743],{"emptyLinePlaceholder":1742},[61,41687,41688,41690,41693,41695,41697,41699,41701,41703,41705,41707,41709],{"class":1731,"line":2851},[61,41689,2116],{"class":1749},[61,41691,41692],{"class":1836}," _n_grid",[61,41694,849],{"class":1753},[61,41696,37781],{"class":2124},[61,41698,2127],{"class":1753},[61,41700,2025],{"class":1859},[61,41702,1435],{"class":1753},[61,41704,37790],{"class":2124},[61,41706,2127],{"class":1753},[61,41708,2025],{"class":1859},[61,41710,3371],{"class":1753},[61,41712,41713],{"class":1731,"line":2856},[61,41714,41715],{"class":2165},"    \"\"\"Return (θ, φ, n) with n stacked as (3, n_theta, n_phi).\"\"\"\n",[61,41717,41718,41720,41722,41724,41726,41728,41730],{"class":1731,"line":2862},[61,41719,37806],{"class":1753},[61,41721,2000],{"class":1859},[61,41723,2193],{"class":1753},[61,41725,3453],{"class":1836},[61,41727,849],{"class":1753},[61,41729,2051],{"class":1866},[61,41731,37819],{"class":1753},[61,41733,41734,41736,41738,41740,41742,41744,41746,41748,41750,41752],{"class":1731,"line":2867},[61,41735,37824],{"class":1753},[61,41737,2000],{"class":1859},[61,41739,2193],{"class":1753},[61,41741,3453],{"class":1836},[61,41743,849],{"class":1753},[61,41745,2051],{"class":1866},[61,41747,1435],{"class":1753},[61,41749,2435],{"class":1866},[61,41751,2298],{"class":1859},[61,41753,37843],{"class":1753},[61,41755,41756,41758,41760,41762,41764,41766,41768,41770,41772],{"class":1731,"line":2917},[61,41757,37848],{"class":1753},[61,41759,2000],{"class":1859},[61,41761,2193],{"class":1753},[61,41763,37855],{"class":1836},[61,41765,37858],{"class":1753},[61,41767,37861],{"class":3402},[61,41769,2000],{"class":1859},[61,41771,37866],{"class":2165},[61,41773,2016],{"class":1753},[61,41775,41776,41778,41780,41782,41784,41786,41788,41790,41792],{"class":1731,"line":2926},[61,41777,37873],{"class":1753},[61,41779,2000],{"class":1859},[61,41781,2193],{"class":1753},[61,41783,5069],{"class":1836},[61,41785,37882],{"class":1753},[61,41787,2201],{"class":1859},[61,41789,2193],{"class":1753},[61,41791,5051],{"class":1836},[61,41793,37891],{"class":1753},[61,41795,41796,41798,41800,41802,41804,41806,41808,41810,41812],{"class":1731,"line":2972},[61,41797,37896],{"class":1753},[61,41799,2000],{"class":1859},[61,41801,2193],{"class":1753},[61,41803,5069],{"class":1836},[61,41805,37882],{"class":1753},[61,41807,2201],{"class":1859},[61,41809,2193],{"class":1753},[61,41811,5069],{"class":1836},[61,41813,37891],{"class":1753},[61,41815,41816,41818,41820,41822,41824],{"class":1731,"line":3001},[61,41817,37917],{"class":1753},[61,41819,2000],{"class":1859},[61,41821,2193],{"class":1753},[61,41823,5051],{"class":1836},[61,41825,37926],{"class":1753},[61,41827,41828,41831,41833,41835,41837,41839,41841,41843,41845],{"class":1731,"line":3054},[61,41829,41830],{"class":1753},"    n ",[61,41832,2000],{"class":1859},[61,41834,2193],{"class":1753},[61,41836,37935],{"class":1836},[61,41838,37938],{"class":1753},[61,41840,37275],{"class":3402},[61,41842,2000],{"class":1859},[61,41844,1495],{"class":1866},[61,41846,2016],{"class":1753},[61,41848,41849,41851],{"class":1731,"line":3060},[61,41850,2210],{"class":1749},[61,41852,41853],{"class":1753}," th, ph, n\n",[61,41855,41856],{"class":1731,"line":3077},[61,41857,1743],{"emptyLinePlaceholder":1742},[61,41859,41860],{"class":1731,"line":3082},[61,41861,1743],{"emptyLinePlaceholder":1742},[61,41863,41864,41866,41869,41871,41873,41875,41877],{"class":1731,"line":3129},[61,41865,2116],{"class":1749},[61,41867,41868],{"class":1836}," _wigner_qubit",[61,41870,849],{"class":1753},[61,41872,37966],{"class":2124},[61,41874,6874],{"class":1753},[61,41876,27948],{"class":2124},[61,41878,6885],{"class":1753},[61,41880,41881],{"class":1731,"line":3137},[61,41882,41883],{"class":2165},"    \"\"\"W(θ, φ) = 1\u002F2 + (√3\u002F2) * r·n on the (θ, φ) grid.\"\"\"\n",[61,41885,41886,41888,41890,41892,41894,41896,41898,41900,41902,41904,41906,41908,41910,41912,41914,41916,41918,41920,41922,41924,41926,41928,41930,41933,41935,41937,41939,41941,41943],{"class":1731,"line":3185},[61,41887,2210],{"class":1749},[61,41889,2980],{"class":1866},[61,41891,4990],{"class":1859},[61,41893,6984],{"class":1753},[61,41895,2425],{"class":1836},[61,41897,849],{"class":1753},[61,41899,37995],{"class":1866},[61,41901,2624],{"class":1753},[61,41903,602],{"class":1859},[61,41905,3029],{"class":1866},[61,41907,2624],{"class":1753},[61,41909,2201],{"class":1859},[61,41911,38008],{"class":1753},[61,41913,38011],{"class":1836},[61,41915,849],{"class":1753},[61,41917,570],{"class":1866},[61,41919,1435],{"class":1753},[61,41921,57],{"class":1866},[61,41923,1435],{"class":1753},[61,41925,57],{"class":1866},[61,41927,2624],{"class":1753},[61,41929,2201],{"class":1859},[61,41931,41932],{"class":1753}," n).",[61,41934,38033],{"class":1836},[61,41936,849],{"class":1753},[61,41938,37275],{"class":3402},[61,41940,2000],{"class":1859},[61,41942,1495],{"class":1866},[61,41944,2016],{"class":1753},[61,41946,41947],{"class":1731,"line":3208},[61,41948,1743],{"emptyLinePlaceholder":1742},[61,41950,41951],{"class":1731,"line":3248},[61,41952,1743],{"emptyLinePlaceholder":1742},[61,41954,41955,41957,41960,41962,41964,41966,41968,41970,41972],{"class":1731,"line":3286},[61,41956,2116],{"class":1749},[61,41958,41959],{"class":1836}," _animate_one_pauli",[61,41961,849],{"class":1753},[61,41963,37275],{"class":2124},[61,41965,2127],{"class":1753},[61,41967,36784],{"class":1859},[61,41969,1435],{"class":1753},[61,41971,39217],{"class":2124},[61,41973,41974],{"class":1753},": Knobs3D) -> Path:\n",[61,41976,41977],{"class":1731,"line":3291},[61,41978,41979],{"class":2165},"    \"\"\"Render one 3D Wigner animation for a chosen Pauli axis; return output path.\"\"\"\n",[61,41981,41982,41985,41987,41989],{"class":1731,"line":3306},[61,41983,41984],{"class":1753},"    r0 ",[61,41986,2000],{"class":1859},[61,41988,41263],{"class":1836},[61,41990,41991],{"class":1753},"(k.initial_state, k.custom_r)\n",[61,41993,41994,41997,41999,42001],{"class":1731,"line":3311},[61,41995,41996],{"class":1753},"    th, ph, n ",[61,41998,2000],{"class":1859},[61,42000,41692],{"class":1836},[61,42002,42003],{"class":1753},"(k.n_theta, k.n_phi)\n",[61,42005,42006],{"class":1731,"line":3317},[61,42007,1743],{"emptyLinePlaceholder":1742},[61,42009,42010],{"class":1731,"line":3358},[61,42011,42012],{"class":1735},"    # Unit sphere geometry\n",[61,42014,42015,42018,42020,42022,42024,42026,42028,42030,42032],{"class":1731,"line":3374},[61,42016,42017],{"class":1753},"    X ",[61,42019,2000],{"class":1859},[61,42021,2193],{"class":1753},[61,42023,5069],{"class":1836},[61,42025,37882],{"class":1753},[61,42027,2201],{"class":1859},[61,42029,2193],{"class":1753},[61,42031,5051],{"class":1836},[61,42033,37891],{"class":1753},[61,42035,42036,42039,42041,42043,42045,42047,42049,42051,42053],{"class":1731,"line":3390},[61,42037,42038],{"class":1753},"    Y ",[61,42040,2000],{"class":1859},[61,42042,2193],{"class":1753},[61,42044,5069],{"class":1836},[61,42046,37882],{"class":1753},[61,42048,2201],{"class":1859},[61,42050,2193],{"class":1753},[61,42052,5069],{"class":1836},[61,42054,37891],{"class":1753},[61,42056,42057,42060,42062,42064,42066],{"class":1731,"line":3422},[61,42058,42059],{"class":1753},"    Z ",[61,42061,2000],{"class":1859},[61,42063,2193],{"class":1753},[61,42065,5051],{"class":1836},[61,42067,37926],{"class":1753},[61,42069,42070],{"class":1731,"line":3427},[61,42071,1743],{"emptyLinePlaceholder":1742},[61,42073,42074],{"class":1731,"line":3432},[61,42075,42076],{"class":1735},"    # Fixed normalization (pure-state bounds) for constant color scale\n",[61,42078,42079,42081,42083,42085,42087,42089,42091,42093,42095,42097,42099,42101],{"class":1731,"line":3438},[61,42080,38872],{"class":1753},[61,42082,2000],{"class":1859},[61,42084,2980],{"class":1866},[61,42086,1899],{"class":1859},[61,42088,6984],{"class":1753},[61,42090,2425],{"class":1836},[61,42092,849],{"class":1753},[61,42094,37995],{"class":1866},[61,42096,2624],{"class":1753},[61,42098,602],{"class":1859},[61,42100,3029],{"class":1866},[61,42102,2016],{"class":1753},[61,42104,42105,42107,42109,42111,42113,42115,42117,42119,42121,42123,42125,42127],{"class":1731,"line":3443},[61,42106,38899],{"class":1753},[61,42108,2000],{"class":1859},[61,42110,2980],{"class":1866},[61,42112,4990],{"class":1859},[61,42114,6984],{"class":1753},[61,42116,2425],{"class":1836},[61,42118,849],{"class":1753},[61,42120,37995],{"class":1866},[61,42122,2624],{"class":1753},[61,42124,602],{"class":1859},[61,42126,3029],{"class":1866},[61,42128,2016],{"class":1753},[61,42130,42131,42134,42136,42139,42141,42144,42146,42148,42150,42152],{"class":1731,"line":3469},[61,42132,42133],{"class":1753},"    norm ",[61,42135,2000],{"class":1859},[61,42137,42138],{"class":1836}," Normalize",[61,42140,849],{"class":1753},[61,42142,42143],{"class":3402},"vmin",[61,42145,2000],{"class":1859},[61,42147,39022],{"class":1753},[61,42149,39025],{"class":3402},[61,42151,2000],{"class":1859},[61,42153,42154],{"class":1753},"w_max)\n",[61,42156,42157,42160,42162,42164,42167],{"class":1731,"line":3494},[61,42158,42159],{"class":1753},"    cmap ",[61,42161,2000],{"class":1859},[61,42163,17065],{"class":1753},[61,42165,42166],{"class":1836},"get_cmap",[61,42168,2089],{"class":1753},[61,42170,42171,42174,42176,42179,42181,42183,42185,42188,42190,42192],{"class":1731,"line":3499},[61,42172,42173],{"class":1753},"    sm ",[61,42175,2000],{"class":1859},[61,42177,42178],{"class":1836}," ScalarMappable",[61,42180,849],{"class":1753},[61,42182,37204],{"class":3402},[61,42184,2000],{"class":1859},[61,42186,42187],{"class":1753},"norm, ",[61,42189,4273],{"class":3402},[61,42191,2000],{"class":1859},[61,42193,42194],{"class":1753},"cmap)\n",[61,42196,42197,42200,42203,42206],{"class":1731,"line":3534},[61,42198,42199],{"class":1753},"    sm.",[61,42201,42202],{"class":1836},"set_array",[61,42204,42205],{"class":1753},"([])  ",[61,42207,42208],{"class":1735},"# required for colorbar from a ScalarMappable\n",[61,42210,42211],{"class":1731,"line":3564},[61,42212,1743],{"emptyLinePlaceholder":1742},[61,42214,42215,42218,42220,42222,42224,42226,42228,42230,42233,42235,42237,42239],{"class":1731,"line":3602},[61,42216,42217],{"class":1753},"    fig ",[61,42219,2000],{"class":1859},[61,42221,17065],{"class":1753},[61,42223,3619],{"class":1836},[61,42225,849],{"class":1753},[61,42227,3624],{"class":3402},[61,42229,2000],{"class":1859},[61,42231,42232],{"class":1753},"k.figsize, ",[61,42234,24380],{"class":3402},[61,42236,2000],{"class":1859},[61,42238,24385],{"class":1866},[61,42240,2016],{"class":1753},[61,42242,42243,42246,42248,42250,42253,42255,42258,42260,42263,42265,42268],{"class":1731,"line":3607},[61,42244,42245],{"class":1753},"    ax ",[61,42247,2000],{"class":1859},[61,42249,39048],{"class":1753},[61,42251,42252],{"class":1836},"add_subplot",[61,42254,849],{"class":1753},[61,42256,42257],{"class":1866},"111",[61,42259,1435],{"class":1753},[61,42261,42262],{"class":3402},"projection",[61,42264,2000],{"class":1859},[61,42266,42267],{"class":2165},"\"3d\"",[61,42269,2016],{"class":1753},[61,42271,42272,42274,42277,42279,42282,42284,42287,42290,42292],{"class":1731,"line":3613},[61,42273,24430],{"class":1753},[61,42275,42276],{"class":1836},"view_init",[61,42278,849],{"class":1753},[61,42280,42281],{"class":3402},"elev",[61,42283,2000],{"class":1859},[61,42285,42286],{"class":1753},"k.elev, ",[61,42288,42289],{"class":3402},"azim",[61,42291,2000],{"class":1859},[61,42293,42294],{"class":1753},"k.azim)\n",[61,42296,42297,42299,42302,42304,42307],{"class":1731,"line":3651},[61,42298,2172],{"class":1749},[61,42300,42301],{"class":1859}," hasattr",[61,42303,39133],{"class":1753},[61,42305,42306],{"class":2165},"\"set_box_aspect\"",[61,42308,3371],{"class":1753},[61,42310,42311,42313,42316,42318,42320,42322,42324,42326,42328],{"class":1731,"line":3715},[61,42312,3393],{"class":1753},[61,42314,42315],{"class":1836},"set_box_aspect",[61,42317,4088],{"class":1753},[61,42319,57],{"class":1866},[61,42321,1435],{"class":1753},[61,42323,57],{"class":1866},[61,42325,1435],{"class":1753},[61,42327,57],{"class":1866},[61,42329,5546],{"class":1753},[61,42331,42332],{"class":1731,"line":3738},[61,42333,1743],{"emptyLinePlaceholder":1742},[61,42335,42336],{"class":1731,"line":3753},[61,42337,42338],{"class":1735},"    # Initial facecolors (use cell-centered (M-1, N-1) colors for quads)\n",[61,42340,42341,42344,42346,42348,42350,42353,42356,42358,42360,42362],{"class":1731,"line":3782},[61,42342,42343],{"class":1753},"    W0 ",[61,42345,2000],{"class":1859},[61,42347,41868],{"class":1836},[61,42349,849],{"class":1753},[61,42351,42352],{"class":1836},"_rotation_matrix",[61,42354,42355],{"class":1753},"(axis, ",[61,42357,2051],{"class":1866},[61,42359,2624],{"class":1753},[61,42361,6930],{"class":1859},[61,42363,42364],{"class":1753}," r0, n)\n",[61,42366,42367,42370,42372,42375,42377,42379,42382,42384,42386,42389,42391,42393,42396],{"class":1731,"line":3813},[61,42368,42369],{"class":1866},"    FC0",[61,42371,1863],{"class":1859},[61,42373,42374],{"class":1836}," cmap",[61,42376,849],{"class":1753},[61,42378,37204],{"class":1836},[61,42380,42381],{"class":1753},"(W0[:",[61,42383,2006],{"class":1859},[61,42385,57],{"class":1866},[61,42387,42388],{"class":1753},", :",[61,42390,2006],{"class":1859},[61,42392,57],{"class":1866},[61,42394,42395],{"class":1753},"]))  ",[61,42397,42398],{"class":1735},"# shape: (n_theta-1, n_phi-1, 4)\n",[61,42400,42401],{"class":1731,"line":3835},[61,42402,1743],{"emptyLinePlaceholder":1742},[61,42404,42405,42408,42410,42412,42415],{"class":1731,"line":3845},[61,42406,42407],{"class":1753},"    surf ",[61,42409,2000],{"class":1859},[61,42411,38434],{"class":1753},[61,42413,42414],{"class":1836},"plot_surface",[61,42416,38439],{"class":1753},[61,42418,42419],{"class":1731,"line":3855},[61,42420,42421],{"class":1753},"        X, Y, Z,\n",[61,42423,42424,42427,42429,42432],{"class":1731,"line":3860},[61,42425,42426],{"class":3402},"        facecolors",[61,42428,2000],{"class":1859},[61,42430,42431],{"class":1866},"FC0",[61,42433,36573],{"class":1753},[61,42435,42436,42439,42441,42443,42445,42448,42450,42452],{"class":1731,"line":3866},[61,42437,42438],{"class":3402},"        rstride",[61,42440,2000],{"class":1859},[61,42442,57],{"class":1866},[61,42444,1435],{"class":1753},[61,42446,42447],{"class":3402},"cstride",[61,42449,2000],{"class":1859},[61,42451,57],{"class":1866},[61,42453,36573],{"class":1753},[61,42455,42456,42459,42461,42463,42465,42467,42469,42471],{"class":1731,"line":3897},[61,42457,42458],{"class":3402},"        antialiased",[61,42460,2000],{"class":1859},[61,42462,14390],{"class":1866},[61,42464,1435],{"class":1753},[61,42466,3413],{"class":3402},[61,42468,2000],{"class":1859},[61,42470,1495],{"class":1866},[61,42472,36573],{"class":1753},[61,42474,42475,42478,42480,42482,42485],{"class":1731,"line":3943},[61,42476,42477],{"class":3402},"        shade",[61,42479,2000],{"class":1859},[61,42481,14390],{"class":1866},[61,42483,42484],{"class":1753},",  ",[61,42486,42487],{"class":1735},"# use given facecolors directly\n",[61,42489,42490],{"class":1731,"line":3964},[61,42491,3057],{"class":1753},[61,42493,42494],{"class":1731,"line":3975},[61,42495,1743],{"emptyLinePlaceholder":1742},[61,42497,42498],{"class":1731,"line":3998},[61,42499,42500],{"class":1735},"    # Colorbar (bound to ScalarMappable using same norm+cmap)\n",[61,42502,42503,42505,42507,42509,42511,42514,42516,42518,42520,42522,42524,42526,42528,42530,42532,42534],{"class":1731,"line":4025},[61,42504,39043],{"class":1753},[61,42506,2000],{"class":1859},[61,42508,39048],{"class":1753},[61,42510,4288],{"class":1836},[61,42512,42513],{"class":1753},"(sm, ",[61,42515,3368],{"class":3402},[61,42517,2000],{"class":1859},[61,42519,39060],{"class":1753},[61,42521,39063],{"class":3402},[61,42523,2000],{"class":1859},[61,42525,39068],{"class":1866},[61,42527,1435],{"class":1753},[61,42529,39073],{"class":3402},[61,42531,2000],{"class":1859},[61,42533,39078],{"class":1866},[61,42535,2016],{"class":1753},[61,42537,42538,42540,42542,42544,42546],{"class":1731,"line":4046},[61,42539,39085],{"class":1753},[61,42541,39088],{"class":1836},[61,42543,849],{"class":1753},[61,42545,39093],{"class":2165},[61,42547,2016],{"class":1753},[61,42549,42550],{"class":1731,"line":4055},[61,42551,1743],{"emptyLinePlaceholder":1742},[61,42553,42554],{"class":1731,"line":4064},[61,42555,42556],{"class":1735},"    # Labels and in-axes title and info\n",[61,42558,42559,42561,42563,42565,42567],{"class":1731,"line":4069},[61,42560,24430],{"class":1753},[61,42562,17159],{"class":1836},[61,42564,849],{"class":1753},[61,42566,36798],{"class":2165},[61,42568,2016],{"class":1753},[61,42570,42571,42573,42575,42577,42579],{"class":1731,"line":4075},[61,42572,24430],{"class":1753},[61,42574,17173],{"class":1836},[61,42576,849],{"class":1753},[61,42578,36803],{"class":2165},[61,42580,2016],{"class":1753},[61,42582,42583,42585,42588,42590,42592],{"class":1731,"line":4101},[61,42584,24430],{"class":1753},[61,42586,42587],{"class":1836},"set_zlabel",[61,42589,849],{"class":1753},[61,42591,36808],{"class":2165},[61,42593,2016],{"class":1753},[61,42595,42596,42599,42601,42603,42606],{"class":1731,"line":4120},[61,42597,42598],{"class":1753},"    title_txt ",[61,42600,2000],{"class":1859},[61,42602,38434],{"class":1753},[61,42604,42605],{"class":1836},"text2D",[61,42607,38439],{"class":1753},[61,42609,42610,42612,42614,42617,42619,42621,42623,42625,42627,42629,42631,42633,42636],{"class":1731,"line":4138},[61,42611,38585],{"class":1866},[61,42613,1435],{"class":1753},[61,42615,42616],{"class":1866},"0.98",[61,42618,1435],{"class":1753},[61,42620,19797],{"class":1749},[61,42622,39138],{"class":2165},[61,42624,3688],{"class":1866},[61,42626,39143],{"class":1753},[61,42628,39146],{"class":1836},[61,42630,39149],{"class":1753},[61,42632,3697],{"class":1866},[61,42634,42635],{"class":2165}," rotation (3D sphere)\"",[61,42637,36573],{"class":1753},[61,42639,42640,42642,42644,42646,42648,42650,42652,42654,42656,42658,42660,42662,42664,42666,42668],{"class":1731,"line":4151},[61,42641,38455],{"class":3402},[61,42643,2000],{"class":1859},[61,42645,38460],{"class":1753},[61,42647,38141],{"class":3402},[61,42649,2000],{"class":1859},[61,42651,38607],{"class":2165},[61,42653,1435],{"class":1753},[61,42655,38151],{"class":3402},[61,42657,2000],{"class":1859},[61,42659,38156],{"class":2165},[61,42661,1435],{"class":1753},[61,42663,38463],{"class":3402},[61,42665,2000],{"class":1859},[61,42667,38627],{"class":1866},[61,42669,36573],{"class":1753},[61,42671,42672,42674,42676,42678,42680,42682,42684,42686,42688,42690,42692,42694,42696,42698,42700,42702,42704,42706,42708],{"class":1731,"line":4156},[61,42673,38482],{"class":3402},[61,42675,38485],{"class":1859},[61,42677,849],{"class":1753},[61,42679,38490],{"class":3402},[61,42681,2000],{"class":1859},[61,42683,38495],{"class":2165},[61,42685,1435],{"class":1753},[61,42687,38500],{"class":3402},[61,42689,2000],{"class":1859},[61,42691,38505],{"class":2165},[61,42693,1435],{"class":1753},[61,42695,16926],{"class":3402},[61,42697,2000],{"class":1859},[61,42699,38514],{"class":1866},[61,42701,1435],{"class":1753},[61,42703,38519],{"class":3402},[61,42705,2000],{"class":1859},[61,42707,38524],{"class":2165},[61,42709,16975],{"class":1753},[61,42711,42712,42714,42716,42718],{"class":1731,"line":4187},[61,42713,38674],{"class":3402},[61,42715,2000],{"class":1859},[61,42717,1300],{"class":1866},[61,42719,36573],{"class":1753},[61,42721,42722],{"class":1731,"line":4234},[61,42723,3057],{"class":1753},[61,42725,42726,42729,42731,42733,42735],{"class":1731,"line":4283},[61,42727,42728],{"class":1753},"    info_txt ",[61,42730,2000],{"class":1859},[61,42732,38434],{"class":1753},[61,42734,42605],{"class":1836},[61,42736,38439],{"class":1753},[61,42738,42739,42742,42744,42747,42749,42751,42753,42756,42758,42760,42762,42764,42766,42768,42770,42772,42774],{"class":1731,"line":4315},[61,42740,42741],{"class":1866},"        0.02",[61,42743,1435],{"class":1753},[61,42745,42746],{"class":1866},"0.96",[61,42748,1435],{"class":1753},[61,42750,39174],{"class":2165},[61,42752,1435],{"class":1753},[61,42754,42755],{"class":3402},"transform",[61,42757,2000],{"class":1859},[61,42759,38460],{"class":1753},[61,42761,38141],{"class":3402},[61,42763,2000],{"class":1859},[61,42765,38146],{"class":2165},[61,42767,1435],{"class":1753},[61,42769,38151],{"class":3402},[61,42771,2000],{"class":1859},[61,42773,38156],{"class":2165},[61,42775,36573],{"class":1753},[61,42777,42778,42780,42782],{"class":1731,"line":4344},[61,42779,38622],{"class":3402},[61,42781,2000],{"class":1859},[61,42783,42784],{"class":1753},"k.info_fontsize,\n",[61,42786,42787,42789,42791,42793,42795,42797,42799,42801,42803,42805,42807,42809,42811,42813,42815,42817,42819,42821,42823],{"class":1731,"line":4367},[61,42788,38482],{"class":3402},[61,42790,38485],{"class":1859},[61,42792,849],{"class":1753},[61,42794,38490],{"class":3402},[61,42796,2000],{"class":1859},[61,42798,38495],{"class":2165},[61,42800,1435],{"class":1753},[61,42802,38500],{"class":3402},[61,42804,2000],{"class":1859},[61,42806,38505],{"class":2165},[61,42808,1435],{"class":1753},[61,42810,16926],{"class":3402},[61,42812,2000],{"class":1859},[61,42814,38514],{"class":1866},[61,42816,1435],{"class":1753},[61,42818,38519],{"class":3402},[61,42820,2000],{"class":1859},[61,42822,38524],{"class":2165},[61,42824,16975],{"class":1753},[61,42826,42827,42829,42831,42833],{"class":1731,"line":4411},[61,42828,38674],{"class":3402},[61,42830,2000],{"class":1859},[61,42832,1300],{"class":1866},[61,42834,36573],{"class":1753},[61,42836,42837],{"class":1731,"line":4420},[61,42838,3057],{"class":1753},[61,42840,42841],{"class":1731,"line":4425},[61,42842,1743],{"emptyLinePlaceholder":1742},[61,42844,42845,42847,42849,42851,42854,42856,42858],{"class":1731,"line":4431},[61,42846,39209],{"class":1749},[61,42848,39212],{"class":1836},[61,42850,849],{"class":1753},[61,42852,42853],{"class":2124},"frame",[61,42855,2127],{"class":1753},[61,42857,2025],{"class":1859},[61,42859,3371],{"class":1753},[61,42861,42862,42865,42867,42870,42872,42875,42877,42879,42881,42883,42885],{"class":1731,"line":4447},[61,42863,42864],{"class":1753},"        angle ",[61,42866,2000],{"class":1859},[61,42868,42869],{"class":1753}," (frame ",[61,42871,602],{"class":1859},[61,42873,42874],{"class":1753}," (k.frames ",[61,42876,2006],{"class":1859},[61,42878,5266],{"class":1866},[61,42880,2702],{"class":1753},[61,42882,2201],{"class":1859},[61,42884,39249],{"class":1753},[61,42886,39252],{"class":1735},[61,42888,42889,42892,42894,42896,42899,42901],{"class":1731,"line":4464},[61,42890,42891],{"class":1753},"        r_now ",[61,42893,2000],{"class":1859},[61,42895,41482],{"class":1836},[61,42897,42898],{"class":1753},"(axis, angle) ",[61,42900,6930],{"class":1859},[61,42902,39274],{"class":1753},[61,42904,42905,42908,42910,42912],{"class":1731,"line":4477},[61,42906,42907],{"class":1753},"        W_now ",[61,42909,2000],{"class":1859},[61,42911,41868],{"class":1836},[61,42913,42914],{"class":1753},"(r_now, n)\n",[61,42916,42917,42920,42922,42924,42926,42928,42931,42933,42935,42937,42939,42941,42944],{"class":1731,"line":4494},[61,42918,42919],{"class":1866},"        FC_now",[61,42921,1863],{"class":1859},[61,42923,42374],{"class":1836},[61,42925,849],{"class":1753},[61,42927,37204],{"class":1836},[61,42929,42930],{"class":1753},"(W_now[:",[61,42932,2006],{"class":1859},[61,42934,57],{"class":1866},[61,42936,42388],{"class":1753},[61,42938,2006],{"class":1859},[61,42940,57],{"class":1866},[61,42942,42943],{"class":1753},"]))        ",[61,42945,42946],{"class":1735},"# (M-1, N-1, 4)\n",[61,42948,42949,42952,42955,42957,42960,42962,42964,42966,42968,42970,42972,42974,42977],{"class":1731,"line":4506},[61,42950,42951],{"class":1753},"        surf.",[61,42953,42954],{"class":1836},"set_facecolors",[61,42956,849],{"class":1753},[61,42958,42959],{"class":1866},"FC_now",[61,42961,332],{"class":1753},[61,42963,38011],{"class":1836},[61,42965,849],{"class":1753},[61,42967,2006],{"class":1859},[61,42969,57],{"class":1866},[61,42971,1435],{"class":1753},[61,42973,822],{"class":1866},[61,42975,42976],{"class":1753},"))  ",[61,42978,42979],{"class":1735},"# update in place\n",[61,42981,42982,42985,42987,42989,42991,42993,42995,42997,42999,43001,43003,43005,43007,43009,43011,43013,43015,43017,43019,43022,43024,43026],{"class":1731,"line":4541},[61,42983,42984],{"class":1753},"        info_txt.",[61,42986,39298],{"class":1836},[61,42988,849],{"class":1753},[61,42990,19797],{"class":1749},[61,42992,39305],{"class":2165},[61,42994,3688],{"class":1866},[61,42996,37284],{"class":1753},[61,42998,39313],{"class":1749},[61,43000,3697],{"class":1866},[61,43002,39318],{"class":2165},[61,43004,39321],{"class":1859},[61,43006,39324],{"class":2165},[61,43008,3688],{"class":1866},[61,43010,42853],{"class":1753},[61,43012,2419],{"class":1859},[61,43014,39333],{"class":1866},[61,43016,602],{"class":2165},[61,43018,3688],{"class":1866},[61,43020,43021],{"class":1753},"k.frames",[61,43023,3697],{"class":1866},[61,43025,26088],{"class":2165},[61,43027,2016],{"class":1753},[61,43029,43030,43032],{"class":1731,"line":4546},[61,43031,2190],{"class":1749},[61,43033,43034],{"class":1753}," (surf, info_txt, title_txt)\n",[61,43036,43037],{"class":1731,"line":4572},[61,43038,1743],{"emptyLinePlaceholder":1742},[61,43040,43041,43043,43045,43047,43049,43051,43053,43056,43058,43060,43062,43064,43066,43068,43070],{"class":1731,"line":4593},[61,43042,39362],{"class":1753},[61,43044,2000],{"class":1859},[61,43046,39367],{"class":1836},[61,43048,39370],{"class":1753},[61,43050,39340],{"class":3402},[61,43052,2000],{"class":1859},[61,43054,43055],{"class":1753},"k.frames, ",[61,43057,39380],{"class":3402},[61,43059,2000],{"class":1859},[61,43061,39385],{"class":1866},[61,43063,1435],{"class":1753},[61,43065,39390],{"class":3402},[61,43067,2000],{"class":1859},[61,43069,14390],{"class":1866},[61,43071,2016],{"class":1753},[61,43073,43074,43076,43078,43080,43083,43085,43087,43090,43092,43094,43096],{"class":1731,"line":4598},[61,43075,6938],{"class":1753},[61,43077,2000],{"class":1859},[61,43079,39719],{"class":1836},[61,43081,43082],{"class":1753},"(k.out_dir) ",[61,43084,602],{"class":1859},[61,43086,39727],{"class":1749},[61,43088,43089],{"class":2165},"\"wigner3d_pauli_",[61,43091,3688],{"class":1866},[61,43093,37275],{"class":1753},[61,43095,3697],{"class":1866},[61,43097,39740],{"class":2165},[61,43099,43100,43103,43105,43107,43109,43111,43113,43115,43117,43119,43121],{"class":1731,"line":4629},[61,43101,43102],{"class":1753},"    out.parent.",[61,43104,39404],{"class":1836},[61,43106,849],{"class":1753},[61,43108,39409],{"class":3402},[61,43110,2000],{"class":1859},[61,43112,24385],{"class":1866},[61,43114,1435],{"class":1753},[61,43116,39418],{"class":3402},[61,43118,2000],{"class":1859},[61,43120,24385],{"class":1866},[61,43122,2016],{"class":1753},[61,43124,43125,43127,43129,43132,43134,43136,43138,43140,43142,43144,43146,43148],{"class":1731,"line":4662},[61,43126,39429],{"class":1753},[61,43128,39432],{"class":1836},[61,43130,43131],{"class":1753},"(out.",[61,43133,39438],{"class":1836},[61,43135,4205],{"class":1753},[61,43137,39443],{"class":3402},[61,43139,2000],{"class":1859},[61,43141,39448],{"class":1836},[61,43143,849],{"class":1753},[61,43145,39453],{"class":3402},[61,43147,2000],{"class":1859},[61,43149,43150],{"class":1753},"k.fps),\n",[61,43152,43153,43155,43157,43159,43161,43163,43165],{"class":1731,"line":4699},[61,43154,39463],{"class":3402},[61,43156,38485],{"class":1859},[61,43158,849],{"class":1753},[61,43160,39470],{"class":3402},[61,43162,2000],{"class":1859},[61,43164,38505],{"class":2165},[61,43166,5546],{"class":1753},[61,43168,43169,43171,43173],{"class":1731,"line":4726},[61,43170,39481],{"class":1753},[61,43172,39484],{"class":1836},[61,43174,39487],{"class":1753},[61,43176,43177,43179],{"class":1731,"line":4749},[61,43178,2210],{"class":1749},[61,43180,7017],{"class":1753},[61,43182,43183],{"class":1731,"line":4789},[61,43184,1743],{"emptyLinePlaceholder":1742},[61,43186,43188],{"class":1731,"line":43187},162,[61,43189,1743],{"emptyLinePlaceholder":1742},[61,43191,43193,43195,43198,43200,43202],{"class":1731,"line":43192},163,[61,43194,2116],{"class":1749},[61,43196,43197],{"class":1836}," animate_all_paulis_3d",[61,43199,849],{"class":1753},[61,43201,39217],{"class":2124},[61,43203,43204],{"class":1753},": Knobs3D,\n",[61,43206,43208,43211,43213,43215,43217,43219,43221,43223,43225,43227,43229,43231,43233,43235],{"class":1731,"line":43207},164,[61,43209,43210],{"class":2124},"                          axes",[61,43212,37015],{"class":1753},[61,43214,36784],{"class":1859},[61,43216,1435],{"class":1753},[61,43218,36789],{"class":1866},[61,43220,1997],{"class":1753},[61,43222,2000],{"class":1859},[61,43224,2003],{"class":1753},[61,43226,36798],{"class":2165},[61,43228,1435],{"class":1753},[61,43230,36803],{"class":2165},[61,43232,1435],{"class":1753},[61,43234,36808],{"class":2165},[61,43236,43237],{"class":1753},")) -> Iterable[Path]:\n",[61,43239,43241],{"class":1731,"line":43240},165,[61,43242,43243],{"class":2165},"    \"\"\"Generate and display GIFs for all requested Pauli axes; yield file paths.\"\"\"\n",[61,43245,43247,43249,43251,43253],{"class":1731,"line":43246},166,[61,43248,3377],{"class":1749},[61,43250,39704],{"class":1753},[61,43252,2492],{"class":1749},[61,43254,43255],{"class":1753}," axes:\n",[61,43257,43259,43262,43264,43266,43269,43271],{"class":1731,"line":43258},167,[61,43260,43261],{"class":1753},"        path ",[61,43263,2000],{"class":1859},[61,43265,41959],{"class":1836},[61,43267,43268],{"class":1753},"(axname.",[61,43270,37185],{"class":1836},[61,43272,43273],{"class":1753},"(), k)\n",[61,43275,43277,43280,43282,43284,43286,43288,43290],{"class":1731,"line":43276},168,[61,43278,43279],{"class":1836},"        display",[61,43281,849],{"class":1753},[61,43283,39813],{"class":1836},[61,43285,849],{"class":1753},[61,43287,39818],{"class":3402},[61,43289,39821],{"class":1859},[61,43291,43292],{"class":1753},"(path)))\n",[61,43294,43296,43299,43301,43303,43306,43308,43310,43312,43314],{"class":1731,"line":43295},169,[61,43297,43298],{"class":1859},"        print",[61,43300,849],{"class":1753},[61,43302,19797],{"class":1749},[61,43304,43305],{"class":2165},"\"Saved: ",[61,43307,3688],{"class":1866},[61,43309,28410],{"class":1753},[61,43311,3697],{"class":1866},[61,43313,26088],{"class":2165},[61,43315,2016],{"class":1753},[61,43317,43319,43322],{"class":1731,"line":43318},170,[61,43320,43321],{"class":1749},"        yield",[61,43323,43324],{"class":1753}," path\n",[61,43326,43328],{"class":1731,"line":43327},171,[61,43329,1743],{"emptyLinePlaceholder":1742},[61,43331,43333],{"class":1731,"line":43332},172,[61,43334,1743],{"emptyLinePlaceholder":1742},[61,43336,43338],{"class":1731,"line":43337},173,[61,43339,43340],{"class":1735},"# ------------------------------ Run all three gates ------------------------------\n",[61,43342,43344,43347,43349,43351,43353],{"class":1731,"line":43343},174,[61,43345,43346],{"class":1866},"K3D",[61,43348,1863],{"class":1859},[61,43350,41017],{"class":1836},[61,43352,39519],{"class":1753},[61,43354,43355],{"class":1735},"# adjust as desired\n",[61,43357,43359,43361,43363,43365],{"class":1731,"line":43358},175,[61,43360,39548],{"class":3402},[61,43362,2000],{"class":1859},[61,43364,39553],{"class":2165},[61,43366,36573],{"class":1753},[61,43368,43370,43372,43374,43376,43378,43380,43382,43385,43387,43389,43391,43394,43396,43398,43400,43402],{"class":1731,"line":43369},176,[61,43371,39560],{"class":3402},[61,43373,2000],{"class":1859},[61,43375,39565],{"class":1866},[61,43377,1435],{"class":1753},[61,43379,37781],{"class":3402},[61,43381,2000],{"class":1859},[61,43383,43384],{"class":1866},"64",[61,43386,1435],{"class":1753},[61,43388,37790],{"class":3402},[61,43390,2000],{"class":1859},[61,43392,43393],{"class":1866},"128",[61,43395,1435],{"class":1753},[61,43397,39453],{"class":3402},[61,43399,2000],{"class":1859},[61,43401,14756],{"class":1866},[61,43403,36573],{"class":1753},[61,43405,43407,43410,43412,43415,43417,43419,43422,43424],{"class":1731,"line":43406},177,[61,43408,43409],{"class":3402},"    elev",[61,43411,2000],{"class":1859},[61,43413,43414],{"class":1866},"25.0",[61,43416,1435],{"class":1753},[61,43418,42289],{"class":3402},[61,43420,43421],{"class":1859},"=-",[61,43423,2056],{"class":1866},[61,43425,36573],{"class":1753},[61,43427,43429,43431,43433,43435],{"class":1731,"line":43428},178,[61,43430,39612],{"class":3402},[61,43432,2000],{"class":1859},[61,43434,39617],{"class":2165},[61,43436,36573],{"class":1753},[61,43438,43440,43442,43444,43446,43448,43450,43452],{"class":1731,"line":43439},179,[61,43441,39624],{"class":3402},[61,43443,2000],{"class":1859},[61,43445,849],{"class":1753},[61,43447,36550],{"class":1866},[61,43449,1435],{"class":1753},[61,43451,5636],{"class":1866},[61,43453,16975],{"class":1753},[61,43455,43457,43460,43462,43464],{"class":1731,"line":43456},180,[61,43458,43459],{"class":3402},"    info_fontsize",[61,43461,2000],{"class":1859},[61,43463,17092],{"class":1866},[61,43465,36573],{"class":1753},[61,43467,43469],{"class":1731,"line":43468},181,[61,43470,2016],{"class":1753},[61,43472,43474],{"class":1731,"line":43473},182,[61,43475,1743],{"emptyLinePlaceholder":1742},[61,43477,43479,43482,43484,43487,43489,43491,43493,43496,43498,43500,43502,43504,43506,43508,43510],{"class":1731,"line":43478},183,[61,43480,43481],{"class":1859},"list",[61,43483,849],{"class":1753},[61,43485,43486],{"class":1836},"animate_all_paulis_3d",[61,43488,849],{"class":1753},[61,43490,43346],{"class":1866},[61,43492,1435],{"class":1753},[61,43494,43495],{"class":3402},"axes",[61,43497,2000],{"class":1859},[61,43499,849],{"class":1753},[61,43501,36798],{"class":2165},[61,43503,1435],{"class":1753},[61,43505,36803],{"class":2165},[61,43507,1435],{"class":1753},[61,43509,36808],{"class":2165},[61,43511,43512],{"class":1753},")))\n",[1720,43514,43517],{"className":43515,"code":43516,"language":29075,"meta":1725},[36200],"Saved: \u002Fcontent\u002Fwigner3d_pauli_x.gif\n",[1727,43518,43516],{"__ignoreMap":1725},[1720,43520,43523],{"className":43521,"code":43522,"language":29075,"meta":1725},[36200],"Saved: \u002Fcontent\u002Fwigner3d_pauli_y.gif\n",[1727,43524,43522],{"__ignoreMap":1725},[1720,43526,43529],{"className":43527,"code":43528,"language":29075,"meta":1725},[36200],"Saved: \u002Fcontent\u002Fwigner3d_pauli_z.gif\n",[1727,43530,43528],{"__ignoreMap":1725},[4797,43532],{"alt":5802,"src":43533},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-05.webp",[4797,43535],{"alt":7511,"src":43536},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-06.webp",[4797,43538],{"alt":16263,"src":43539},"\u002F_content\u002Fimages\u002Fwigner-functions-single-qubit-gates\u002Foutput-07.webp",[1720,43541,43543],{"className":1722,"code":43542,"language":1724,"meta":1725,"style":1725},"# @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",[1727,43544,43545,43550,43555,43559,43569,43573,43583,43593,43603,43607,43617,43627,43637,43647,43657,43667,43671,43675,43679,43688,43693,43729,43741,43751,43783,43793,43803,43813,43823,43833,43845,43857,43881,43891,43895,43899,43904,43936,43944,43980,44016,44052,44056,44080,44096,44106,44110,44114,44147,44163,44175,44187,44197,44209,44273,44289,44305,44341,44345,44349,44373,44389,44411,44431,44451,44471,44483,44503,44509,44513,44517,44533,44593,44597,44601,44615,44620,44630,44640,44644,44664,44684,44696,44700,44726,44752,44774,44786,44808,44817,44821,44847,44871,44891,44903,44923,44927,44932,44942,44958,44969,44998,45002,45014,45018,45028,45046,45064,45074,45078,45082,45116,45128,45132,45144,45156,45168,45180,45195,45227,45267,45277,45281,45293,45329,45337,45377,45387,45391,45395,45411,45438,45453,45463,45492,45522,45530,45583,45614,45618,45624,45628,45660,45671,45695,45721,45737,45745,45751,45755,45759,45764,45775,45803,45823,45827,45842,45864],{"__ignoreMap":1725},[61,43546,43547],{"class":1731,"line":1732},[61,43548,43549],{"class":1735},"# @title Cell 6 — 3D Wigner animation (custom axis–angle; colors-only, no geometry edits)\n",[61,43551,43552],{"class":1731,"line":1739},[61,43553,43554],{"class":1735},"# PEP 8\u002F257 compliant. Rotates Bloch vector about an arbitrary axis by a target angle.\n",[61,43556,43557],{"class":1731,"line":1746},[61,43558,1743],{"emptyLinePlaceholder":1742},[61,43560,43561,43563,43565,43567],{"class":1731,"line":1763},[61,43562,1750],{"class":1749},[61,43564,36446],{"class":3766},[61,43566,36449],{"class":1749},[61,43568,36452],{"class":1753},[61,43570,43571],{"class":1731,"line":1776},[61,43572,1743],{"emptyLinePlaceholder":1742},[61,43574,43575,43577,43579,43581],{"class":1731,"line":1784},[61,43576,1750],{"class":1749},[61,43578,1754],{"class":1753},[61,43580,1757],{"class":1749},[61,43582,1760],{"class":1753},[61,43584,43585,43587,43589,43591],{"class":1731,"line":1798},[61,43586,1750],{"class":1749},[61,43588,36605],{"class":1753},[61,43590,1757],{"class":1749},[61,43592,36610],{"class":1753},[61,43594,43595,43597,43599,43601],{"class":1731,"line":1811},[61,43596,1750],{"class":1749},[61,43598,1768],{"class":1753},[61,43600,1757],{"class":1749},[61,43602,1773],{"class":1753},[61,43604,43605],{"class":1731,"line":1816},[61,43606,1743],{"emptyLinePlaceholder":1742},[61,43608,43609,43611,43613,43615],{"class":1731,"line":1822},[61,43610,1757],{"class":1749},[61,43612,1789],{"class":1753},[61,43614,1792],{"class":1749},[61,43616,1795],{"class":1753},[61,43618,43619,43621,43623,43625],{"class":1731,"line":1828},[61,43620,1757],{"class":1749},[61,43622,1803],{"class":1753},[61,43624,1792],{"class":1749},[61,43626,1808],{"class":1753},[61,43628,43629,43631,43633,43635],{"class":1731,"line":1833},[61,43630,1750],{"class":1749},[61,43632,36659],{"class":1753},[61,43634,1757],{"class":1749},[61,43636,36664],{"class":1753},[61,43638,43639,43641,43643,43645],{"class":1731,"line":1840},[61,43640,1750],{"class":1749},[61,43642,40971],{"class":1753},[61,43644,1757],{"class":1749},[61,43646,40976],{"class":1753},[61,43648,43649,43651,43653,43655],{"class":1731,"line":1853},[61,43650,1750],{"class":1749},[61,43652,40983],{"class":1753},[61,43654,1757],{"class":1749},[61,43656,40988],{"class":1753},[61,43658,43659,43661,43663,43665],{"class":1731,"line":1873},[61,43660,1750],{"class":1749},[61,43662,36671],{"class":1753},[61,43664,1757],{"class":1749},[61,43666,36676],{"class":1753},[61,43668,43669],{"class":1731,"line":1889},[61,43670,1743],{"emptyLinePlaceholder":1742},[61,43672,43673],{"class":1731,"line":1908},[61,43674,1743],{"emptyLinePlaceholder":1742},[61,43676,43677],{"class":1731,"line":1925},[61,43678,1837],{"class":1836},[61,43680,43681,43683,43686],{"class":1731,"line":1941},[61,43682,1843],{"class":1749},[61,43684,43685],{"class":1846}," Knobs3DCustom",[61,43687,1850],{"class":1753},[61,43689,43690],{"class":1731,"line":1957},[61,43691,43692],{"class":2165},"    \"\"\"User controls for 3D custom axis–angle Wigner animation.\"\"\"\n",[61,43694,43695,43698,43700,43702,43704,43706,43708,43710,43712,43714,43716,43718,43720,43722,43724,43726],{"class":1731,"line":1974},[61,43696,43697],{"class":1753},"    axis_vec: Tuple[",[61,43699,1860],{"class":1859},[61,43701,1435],{"class":1753},[61,43703,1860],{"class":1859},[61,43705,1435],{"class":1753},[61,43707,1860],{"class":1859},[61,43709,1997],{"class":1753},[61,43711,2000],{"class":1859},[61,43713,2003],{"class":1753},[61,43715,2051],{"class":1866},[61,43717,1435],{"class":1753},[61,43719,2319],{"class":1866},[61,43721,1435],{"class":1753},[61,43723,2051],{"class":1866},[61,43725,14257],{"class":1753},[61,43727,43728],{"class":1735},"# default: y-axis\n",[61,43730,43731,43734,43736,43738],{"class":1731,"line":1979},[61,43732,43733],{"class":1753},"    angle_deg: ",[61,43735,1860],{"class":1859},[61,43737,1863],{"class":1859},[61,43739,43740],{"class":1866}," 45.0\n",[61,43742,43743,43745,43747,43749],{"class":1731,"line":1985},[61,43744,36815],{"class":1753},[61,43746,36784],{"class":1859},[61,43748,1863],{"class":1859},[61,43750,36822],{"class":2165},[61,43752,43753,43755,43757,43759,43761,43763,43765,43767,43769,43771,43773,43775,43777,43779,43781],{"class":1731,"line":2019},[61,43754,36827],{"class":1753},[61,43756,1860],{"class":1859},[61,43758,1435],{"class":1753},[61,43760,1860],{"class":1859},[61,43762,1435],{"class":1753},[61,43764,1860],{"class":1859},[61,43766,1997],{"class":1753},[61,43768,2000],{"class":1859},[61,43770,2003],{"class":1753},[61,43772,2051],{"class":1866},[61,43774,1435],{"class":1753},[61,43776,2051],{"class":1866},[61,43778,1435],{"class":1753},[61,43780,2319],{"class":1866},[61,43782,2016],{"class":1753},[61,43784,43785,43787,43789,43791],{"class":1731,"line":2033},[61,43786,36860],{"class":1753},[61,43788,2025],{"class":1859},[61,43790,1863],{"class":1859},[61,43792,36867],{"class":1866},[61,43794,43795,43797,43799,43801],{"class":1731,"line":2061},[61,43796,36872],{"class":1753},[61,43798,2025],{"class":1859},[61,43800,1863],{"class":1859},[61,43802,41152],{"class":1866},[61,43804,43805,43807,43809,43811],{"class":1731,"line":2074},[61,43806,36884],{"class":1753},[61,43808,2025],{"class":1859},[61,43810,1863],{"class":1859},[61,43812,41163],{"class":1866},[61,43814,43815,43817,43819,43821],{"class":1731,"line":2079},[61,43816,36896],{"class":1753},[61,43818,2025],{"class":1859},[61,43820,1863],{"class":1859},[61,43822,36903],{"class":1866},[61,43824,43825,43827,43829,43831],{"class":1731,"line":2092},[61,43826,41178],{"class":1753},[61,43828,1860],{"class":1859},[61,43830,1863],{"class":1859},[61,43832,41185],{"class":1866},[61,43834,43835,43837,43839,43841,43843],{"class":1731,"line":2097},[61,43836,41190],{"class":1753},[61,43838,1860],{"class":1859},[61,43840,1863],{"class":1859},[61,43842,1899],{"class":1859},[61,43844,41199],{"class":1866},[61,43846,43847,43850,43852,43854],{"class":1731,"line":2102},[61,43848,43849],{"class":1753},"    out_path: ",[61,43851,36784],{"class":1859},[61,43853,1863],{"class":1859},[61,43855,43856],{"class":2165}," \"\u002Fcontent\u002Fwigner3d_custom_y45.gif\"\n",[61,43858,43859,43861,43863,43865,43867,43869,43871,43873,43875,43877,43879],{"class":1731,"line":2108},[61,43860,36920],{"class":1753},[61,43862,1860],{"class":1859},[61,43864,1435],{"class":1753},[61,43866,1860],{"class":1859},[61,43868,1997],{"class":1753},[61,43870,2000],{"class":1859},[61,43872,2003],{"class":1753},[61,43874,36550],{"class":1866},[61,43876,1435],{"class":1753},[61,43878,5636],{"class":1866},[61,43880,2016],{"class":1753},[61,43882,43883,43885,43887,43889],{"class":1731,"line":2113},[61,43884,36957],{"class":1753},[61,43886,2025],{"class":1859},[61,43888,1863],{"class":1859},[61,43890,36964],{"class":1866},[61,43892,43893],{"class":1731,"line":2162},[61,43894,1743],{"emptyLinePlaceholder":1742},[61,43896,43897],{"class":1731,"line":2169},[61,43898,1743],{"emptyLinePlaceholder":1742},[61,43900,43901],{"class":1731,"line":2187},[61,43902,43903],{"class":1735},"# ------------- helpers (self-contained; safe for this 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animate_wigner_custom_3d",[61,44609,849],{"class":1753},[61,44611,39217],{"class":2124},[61,44613,44614],{"class":1753},": Knobs3DCustom) -> Path:\n",[61,44616,44617],{"class":1731,"line":2926},[61,44618,44619],{"class":2165},"    \"\"\"Animate a 3D sphere with colors driven by Wigner under axis–angle rotation.\"\"\"\n",[61,44621,44622,44624,44626,44628],{"class":1731,"line":2972},[61,44623,41984],{"class":1753},[61,44625,2000],{"class":1859},[61,44627,41263],{"class":1836},[61,44629,41991],{"class":1753},[61,44631,44632,44634,44636,44638],{"class":1731,"line":3001},[61,44633,41996],{"class":1753},[61,44635,2000],{"class":1859},[61,44637,41692],{"class":1836},[61,44639,42003],{"class":1753},[61,44641,44642],{"class":1731,"line":3054},[61,44643,1743],{"emptyLinePlaceholder":1742},[61,44645,44646,44648,44650,44652,44654,44656,44658,44660,44662],{"class":1731,"line":3060},[61,44647,42017],{"class":1753},[61,44649,2000],{"class":1859},[61,44651,2193],{"class":1753},[61,44653,5069],{"class":1836},[61,44655,37882],{"class":1753},[61,44657,2201],{"class":1859},[61,44659,2193],{"class":1753},[61,44661,5051],{"class":1836},[61,44663,37891],{"class":1753},[61,44665,44666,44668,44670,44672,44674,44676,44678,44680,44682],{"class":1731,"line":3077},[61,44667,42038],{"class":1753},[61,44669,2000],{"class":1859},[61,44671,2193],{"class":1753},[61,44673,5069],{"class":1836},[61,44675,37882],{"class":1753},[61,44677,2201],{"class":1859},[61,44679,2193],{"class":1753},[61,44681,5069],{"class":1836},[61,44683,37891],{"class":1753},[61,44685,44686,44688,44690,44692,44694],{"class":1731,"line":3082},[61,44687,42059],{"class":1753},[61,44689,2000],{"class":1859},[61,44691,2193],{"class":1753},[61,44693,5051],{"class":1836},[61,44695,37926],{"class":1753},[61,44697,44698],{"class":1731,"line":3129},[61,44699,1743],{"emptyLinePlaceholder":1742},[61,44701,44702,44704,44706,44708,44710,44712,44714,44716,44718,44720,44722,44724],{"class":1731,"line":3137},[61,44703,38872],{"class":1753},[61,44705,2000],{"class":1859},[61,44707,2980],{"class":1866},[61,44709,1899],{"class":1859},[61,44711,6984],{"class":1753},[61,44713,2425],{"class":1836},[61,44715,849],{"class":1753},[61,44717,37995],{"class":1866},[61,44719,2624],{"class":1753},[61,44721,602],{"class":1859},[61,44723,3029],{"class":1866},[61,44725,2016],{"class":1753},[61,44727,44728,44730,44732,44734,44736,44738,44740,44742,44744,44746,44748,44750],{"class":1731,"line":3185},[61,44729,38899],{"class":1753},[61,44731,2000],{"class":1859},[61,44733,2980],{"class":1866},[61,44735,4990],{"class":1859},[61,44737,6984],{"class":1753},[61,44739,2425],{"class":1836},[61,44741,849],{"class":1753},[61,44743,37995],{"class":1866},[61,44745,2624],{"class":1753},[61,44747,602],{"class":1859},[61,44749,3029],{"class":1866},[61,44751,2016],{"class":1753},[61,44753,44754,44756,44758,44760,44762,44764,44766,44768,44770,44772],{"class":1731,"line":3208},[61,44755,42133],{"class":1753},[61,44757,2000],{"class":1859},[61,44759,42138],{"class":1836},[61,44761,849],{"class":1753},[61,44763,42143],{"class":3402},[61,44765,2000],{"class":1859},[61,44767,39022],{"class":1753},[61,44769,39025],{"class":3402},[61,44771,2000],{"class":1859},[61,44773,42154],{"class":1753},[61,44775,44776,44778,44780,44782,44784],{"class":1731,"line":3248},[61,44777,42159],{"class":1753},[61,44779,2000],{"class":1859},[61,44781,17065],{"class":1753},[61,44783,42166],{"class":1836},[61,44785,2089],{"class":1753},[61,44787,44788,44790,44792,44794,44796,44798,44800,44802,44804,44806],{"class":1731,"line":3286},[61,44789,42173],{"class":1753},[61,44791,2000],{"class":1859},[61,44793,42178],{"class":1836},[61,44795,849],{"class":1753},[61,44797,37204],{"class":3402},[61,44799,2000],{"class":1859},[61,44801,42187],{"class":1753},[61,44803,4273],{"class":3402},[61,44805,2000],{"class":1859},[61,44807,42194],{"class":1753},[61,44809,44810,44812,44814],{"class":1731,"line":3291},[61,44811,42199],{"class":1753},[61,44813,42202],{"class":1836},[61,44815,44816],{"class":1753},"([])\n",[61,44818,44819],{"class":1731,"line":3306},[61,44820,1743],{"emptyLinePlaceholder":1742},[61,44822,44823,44825,44827,44829,44831,44833,44835,44837,44839,44841,44843,44845],{"class":1731,"line":3311},[61,44824,42217],{"class":1753},[61,44826,2000],{"class":1859},[61,44828,17065],{"class":1753},[61,44830,3619],{"class":1836},[61,44832,849],{"class":1753},[61,44834,3624],{"class":3402},[61,44836,2000],{"class":1859},[61,44838,42232],{"class":1753},[61,44840,24380],{"class":3402},[61,44842,2000],{"class":1859},[61,44844,24385],{"class":1866},[61,44846,2016],{"class":1753},[61,44848,44849,44851,44853,44855,44857,44859,44861,44863,44865,44867,44869],{"class":1731,"line":3317},[61,44850,42245],{"class":1753},[61,44852,2000],{"class":1859},[61,44854,39048],{"class":1753},[61,44856,42252],{"class":1836},[61,44858,849],{"class":1753},[61,44860,42257],{"class":1866},[61,44862,1435],{"class":1753},[61,44864,42262],{"class":3402},[61,44866,2000],{"class":1859},[61,44868,42267],{"class":2165},[61,44870,2016],{"class":1753},[61,44872,44873,44875,44877,44879,44881,44883,44885,44887,44889],{"class":1731,"line":3358},[61,44874,24430],{"class":1753},[61,44876,42276],{"class":1836},[61,44878,849],{"class":1753},[61,44880,42281],{"class":3402},[61,44882,2000],{"class":1859},[61,44884,42286],{"class":1753},[61,44886,42289],{"class":3402},[61,44888,2000],{"class":1859},[61,44890,42294],{"class":1753},[61,44892,44893,44895,44897,44899,44901],{"class":1731,"line":3374},[61,44894,2172],{"class":1749},[61,44896,42301],{"class":1859},[61,44898,39133],{"class":1753},[61,44900,42306],{"class":2165},[61,44902,3371],{"class":1753},[61,44904,44905,44907,44909,44911,44913,44915,44917,44919,44921],{"class":1731,"line":3390},[61,44906,3393],{"class":1753},[61,44908,42315],{"class":1836},[61,44910,4088],{"class":1753},[61,44912,57],{"class":1866},[61,44914,1435],{"class":1753},[61,44916,57],{"class":1866},[61,44918,1435],{"class":1753},[61,44920,57],{"class":1866},[61,44922,5546],{"class":1753},[61,44924,44925],{"class":1731,"line":3422},[61,44926,1743],{"emptyLinePlaceholder":1742},[61,44928,44929],{"class":1731,"line":3427},[61,44930,44931],{"class":1735},"    # Initial facecolors\n",[61,44933,44934,44937,44939],{"class":1731,"line":3432},[61,44935,44936],{"class":1753},"    angle0 ",[61,44938,2000],{"class":1859},[61,44940,44941],{"class":1866}," 0.0\n",[61,44943,44944,44947,44949,44951,44954,44956],{"class":1731,"line":3438},[61,44945,44946],{"class":1753},"    r_init ",[61,44948,2000],{"class":1859},[61,44950,44120],{"class":1836},[61,44952,44953],{"class":1753},"(k.axis_vec, angle0) ",[61,44955,6930],{"class":1859},[61,44957,39274],{"class":1753},[61,44959,44960,44962,44964,44966],{"class":1731,"line":3443},[61,44961,42343],{"class":1753},[61,44963,2000],{"class":1859},[61,44965,41868],{"class":1836},[61,44967,44968],{"class":1753},"(r_init, n)\n",[61,44970,44971,44973,44975,44977,44979,44981,44983,44985,44987,44989,44991,44993,44995],{"class":1731,"line":3469},[61,44972,42369],{"class":1866},[61,44974,1863],{"class":1859},[61,44976,42374],{"class":1836},[61,44978,849],{"class":1753},[61,44980,37204],{"class":1836},[61,44982,42381],{"class":1753},[61,44984,2006],{"class":1859},[61,44986,57],{"class":1866},[61,44988,42388],{"class":1753},[61,44990,2006],{"class":1859},[61,44992,57],{"class":1866},[61,44994,42395],{"class":1753},[61,44996,44997],{"class":1735},"# (M-1,N-1,4)\n",[61,44999,45000],{"class":1731,"line":3494},[61,45001,1743],{"emptyLinePlaceholder":1742},[61,45003,45004,45006,45008,45010,45012],{"class":1731,"line":3499},[61,45005,42407],{"class":1753},[61,45007,2000],{"class":1859},[61,45009,38434],{"class":1753},[61,45011,42414],{"class":1836},[61,45013,38439],{"class":1753},[61,45015,45016],{"class":1731,"line":3534},[61,45017,42421],{"class":1753},[61,45019,45020,45022,45024,45026],{"class":1731,"line":3564},[61,45021,42426],{"class":3402},[61,45023,2000],{"class":1859},[61,45025,42431],{"class":1866},[61,45027,36573],{"class":1753},[61,45029,45030,45032,45034,45036,45038,45040,45042,45044],{"class":1731,"line":3602},[61,45031,42438],{"class":3402},[61,45033,2000],{"class":1859},[61,45035,57],{"class":1866},[61,45037,1435],{"class":1753},[61,45039,42447],{"class":3402},[61,45041,2000],{"class":1859},[61,45043,57],{"class":1866},[61,45045,36573],{"class":1753},[61,45047,45048,45050,45052,45054,45056,45058,45060,45062],{"class":1731,"line":3607},[61,45049,42458],{"class":3402},[61,45051,2000],{"class":1859},[61,45053,14390],{"class":1866},[61,45055,1435],{"class":1753},[61,45057,3413],{"class":3402},[61,45059,2000],{"class":1859},[61,45061,1495],{"class":1866},[61,45063,36573],{"class":1753},[61,45065,45066,45068,45070,45072],{"class":1731,"line":3613},[61,45067,42477],{"class":3402},[61,45069,2000],{"class":1859},[61,45071,14390],{"class":1866},[61,45073,36573],{"class":1753},[61,45075,45076],{"class":1731,"line":3651},[61,45077,3057],{"class":1753},[61,45079,45080],{"class":1731,"line":3715},[61,45081,1743],{"emptyLinePlaceholder":1742},[61,45083,45084,45086,45088,45090,45092,45094,45096,45098,45100,45102,45104,45106,45108,45110,45112,45114],{"class":1731,"line":3738},[61,45085,39043],{"class":1753},[61,45087,2000],{"class":1859},[61,45089,39048],{"class":1753},[61,45091,4288],{"class":1836},[61,45093,42513],{"class":1753},[61,45095,3368],{"class":3402},[61,45097,2000],{"class":1859},[61,45099,39060],{"class":1753},[61,45101,39063],{"class":3402},[61,45103,2000],{"class":1859},[61,45105,39068],{"class":1866},[61,45107,1435],{"class":1753},[61,45109,39073],{"class":3402},[61,45111,2000],{"class":1859},[61,45113,39078],{"class":1866},[61,45115,2016],{"class":1753},[61,45117,45118,45120,45122,45124,45126],{"class":1731,"line":3753},[61,45119,39085],{"class":1753},[61,45121,39088],{"class":1836},[61,45123,849],{"class":1753},[61,45125,39093],{"class":2165},[61,45127,2016],{"class":1753},[61,45129,45130],{"class":1731,"line":3782},[61,45131,1743],{"emptyLinePlaceholder":1742},[61,45133,45134,45136,45138,45140,45142],{"class":1731,"line":3813},[61,45135,24430],{"class":1753},[61,45137,17159],{"class":1836},[61,45139,849],{"class":1753},[61,45141,36798],{"class":2165},[61,45143,2016],{"class":1753},[61,45145,45146,45148,45150,45152,45154],{"class":1731,"line":3835},[61,45147,24430],{"class":1753},[61,45149,17173],{"class":1836},[61,45151,849],{"class":1753},[61,45153,36803],{"class":2165},[61,45155,2016],{"class":1753},[61,45157,45158,45160,45162,45164,45166],{"class":1731,"line":3845},[61,45159,24430],{"class":1753},[61,45161,42587],{"class":1836},[61,45163,849],{"class":1753},[61,45165,36808],{"class":2165},[61,45167,2016],{"class":1753},[61,45169,45170,45172,45174,45176,45178],{"class":1731,"line":3855},[61,45171,42598],{"class":1753},[61,45173,2000],{"class":1859},[61,45175,38434],{"class":1753},[61,45177,42605],{"class":1836},[61,45179,38439],{"class":1753},[61,45181,45182,45184,45186,45188,45190,45193],{"class":1731,"line":3860},[61,45183,38585],{"class":1866},[61,45185,1435],{"class":1753},[61,45187,42616],{"class":1866},[61,45189,1435],{"class":1753},[61,45191,45192],{"class":2165},"\"Wigner - custom 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ang) ",[61,45450,6930],{"class":1859},[61,45452,39274],{"class":1753},[61,45454,45455,45457,45459,45461],{"class":1731,"line":4138},[61,45456,42907],{"class":1753},[61,45458,2000],{"class":1859},[61,45460,41868],{"class":1836},[61,45462,42914],{"class":1753},[61,45464,45465,45467,45469,45471,45473,45475,45477,45479,45481,45483,45485,45487,45490],{"class":1731,"line":4151},[61,45466,42919],{"class":1866},[61,45468,1863],{"class":1859},[61,45470,42374],{"class":1836},[61,45472,849],{"class":1753},[61,45474,37204],{"class":1836},[61,45476,42930],{"class":1753},[61,45478,2006],{"class":1859},[61,45480,57],{"class":1866},[61,45482,42388],{"class":1753},[61,45484,2006],{"class":1859},[61,45486,57],{"class":1866},[61,45488,45489],{"class":1753},"]))          ",[61,45491,44997],{"class":1735},[61,45493,45494,45496,45498,45500,45502,45504,45506,45508,45510,45512,45514,45516,45519],{"class":1731,"line":4156},[61,45495,42951],{"class":1753},[61,45497,42954],{"class":1836},[61,45499,849],{"class":1753},[61,45501,42959],{"class":1866},[61,45503,332],{"class":1753},[61,45505,38011],{"class":1836},[61,45507,849],{"class":1753},[61,45509,2006],{"class":1859},[61,45511,57],{"class":1866},[61,45513,1435],{"class":1753},[61,45515,822],{"class":1866},[61,45517,45518],{"class":1753},"))    ",[61,45520,45521],{"class":1735},"# in-place update\n",[61,45523,45524,45526,45528],{"class":1731,"line":4187},[61,45525,42984],{"class":1753},[61,45527,39298],{"class":1836},[61,45529,38439],{"class":1753},[61,45531,45532,45534,45536,45538,45541,45543,45545,45547,45549,45551,45553,45555,45557,45559,45561,45563,45565,45567,45569,45571,45573,45575,45577,45579,45581],{"class":1731,"line":4234},[61,45533,40436],{"class":1749},[61,45535,40439],{"class":2165},[61,45537,3688],{"class":1866},[61,45539,45540],{"class":1753},"k.axis_vec[",[61,45542,1495],{"class":1866},[61,45544,27817],{"class":1753},[61,45546,40451],{"class":1749},[61,45548,3697],{"class":1866},[61,45550,1435],{"class":2165},[61,45552,3688],{"class":1866},[61,45554,45540],{"class":1753},[61,45556,57],{"class":1866},[61,45558,27817],{"class":1753},[61,45560,40451],{"class":1749},[61,45562,3697],{"class":1866},[61,45564,1435],{"class":2165},[61,45566,3688],{"class":1866},[61,45568,45540],{"class":1753},[61,45570,337],{"class":1866},[61,45572,27817],{"class":1753},[61,45574,40451],{"class":1749},[61,45576,3697],{"class":1866},[61,45578,300],{"class":2165},[61,45580,39321],{"class":1859},[61,45582,40488],{"class":2165},[61,45584,45585,45587,45589,45591,45593,45595,45597,45599,45601,45603,45605,45608,45610,45612],{"class":1731,"line":4283},[61,45586,40436],{"class":1749},[61,45588,39305],{"class":2165},[61,45590,3688],{"class":1866},[61,45592,40499],{"class":1753},[61,45594,40502],{"class":1836},[61,45596,40505],{"class":1753},[61,45598,3694],{"class":1749},[61,45600,3697],{"class":1866},[61,45602,40512],{"class":2165},[61,45604,3688],{"class":1866},[61,45606,45607],{"class":1753},"k.angle_deg",[61,45609,3694],{"class":1749},[61,45611,3697],{"class":1866},[61,45613,40524],{"class":2165},[61,45615,45616],{"class":1731,"line":4315},[61,45617,40529],{"class":1753},[61,45619,45620,45622],{"class":1731,"line":4344},[61,45621,2190],{"class":1749},[61,45623,43034],{"class":1753},[61,45625,45626],{"class":1731,"line":4367},[61,45627,1743],{"emptyLinePlaceholder":1742},[61,45629,45630,45632,45634,45636,45638,45640,45642,45644,45646,45648,45650,45652,45654,45656,45658],{"class":1731,"line":4411},[61,45631,39362],{"class":1753},[61,45633,2000],{"class":1859},[61,45635,39367],{"class":1836},[61,45637,39370],{"class":1753},[61,45639,39340],{"class":3402},[61,45641,2000],{"class":1859},[61,45643,43055],{"class":1753},[61,45645,39380],{"class":3402},[61,45647,2000],{"class":1859},[61,45649,39385],{"class":1866},[61,45651,1435],{"class":1753},[61,45653,39390],{"class":3402},[61,45655,2000],{"class":1859},[61,45657,14390],{"class":1866},[61,45659,2016],{"class":1753},[61,45661,45662,45664,45666,45668],{"class":1731,"line":4420},[61,45663,6938],{"class":1753},[61,45665,2000],{"class":1859},[61,45667,39719],{"class":1836},[61,45669,45670],{"class":1753},"(k.out_path)\n",[61,45672,45673,45675,45677,45679,45681,45683,45685,45687,45689,45691,45693],{"class":1731,"line":4425},[61,45674,43102],{"class":1753},[61,45676,39404],{"class":1836},[61,45678,849],{"class":1753},[61,45680,39409],{"class":3402},[61,45682,2000],{"class":1859},[61,45684,24385],{"class":1866},[61,45686,1435],{"class":1753},[61,45688,39418],{"class":3402},[61,45690,2000],{"class":1859},[61,45692,24385],{"class":1866},[61,45694,2016],{"class":1753},[61,45696,45697,45699,45701,45703,45705,45707,45709,45711,45713,45715,45717,45719],{"class":1731,"line":4431},[61,45698,39429],{"class":1753},[61,45700,39432],{"class":1836},[61,45702,43131],{"class":1753},[61,45704,39438],{"class":1836},[61,45706,4205],{"class":1753},[61,45708,39443],{"class":3402},[61,45710,2000],{"class":1859},[61,45712,39448],{"class":1836},[61,45714,849],{"class":1753},[61,45716,39453],{"class":3402},[61,45718,2000],{"class":1859},[61,45720,43150],{"class":1753},[61,45722,45723,45725,45727,45729,45731,45733,45735],{"class":1731,"line":4447},[61,45724,39463],{"class":3402},[61,45726,38485],{"class":1859},[61,45728,849],{"class":1753},[61,45730,39470],{"class":3402},[61,45732,2000],{"class":1859},[61,45734,38505],{"class":2165},[61,45736,5546],{"class":1753},[61,45738,45739,45741,45743],{"class":1731,"line":4464},[61,45740,39481],{"class":1753},[61,45742,39484],{"class":1836},[61,45744,39487],{"class":1753},[61,45746,45747,45749],{"class":1731,"line":4477},[61,45748,2210],{"class":1749},[61,45750,7017],{"class":1753},[61,45752,45753],{"class":1731,"line":4494},[61,45754,1743],{"emptyLinePlaceholder":1742},[61,45756,45757],{"class":1731,"line":4506},[61,45758,1743],{"emptyLinePlaceholder":1742},[61,45760,45761],{"class":1731,"line":4541},[61,45762,45763],{"class":1735},"# ----------------------------- Example run 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Pauli Gates as (π\\piπ)-Rotations",{"id":31254,"depth":1739,"text":31255},{"id":32060,"depth":1739,"text":45895},"5. Rodrigues’ Formula (Axis–Angle (→\\to→) Rotation Matrix)",{"id":34199,"depth":1739,"text":45897},"6. Optional: Discrete (2×22\\times22×2) (Finite‑State) Wigner",{"id":35111,"depth":1739,"text":35112},{"id":35859,"depth":1739,"text":35860},{"id":36192,"depth":1739,"text":36193},{"id":36208,"depth":1739,"text":36209},[7523,7524,7525,45903],"Wigner functions",[45905],{"username":7,"name":7529,"role":7530,"bio":7531,"links":45906},[45907,45908],{"label":7534,"href":7535},{"label":7537,"href":26272},{"username":7,"name":7529,"role":7539},"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":26262,"description":45910},"blog\u002Fexpert-notes\u002Fwigner-functions-single-qubit-gates",[7556,45920],"quantum","F21gsgnqd_6pE29RCUocKz6jo6vSs6rJlQM17vqwdBU",{"id":45923,"title":45924,"authors":45925,"body":45926,"breadcrumb":71761,"builders":71763,"byline":71768,"challenge":7540,"courseAuthor":7540,"courseLead":7540,"dek":7540,"description":71769,"draft":7542,"extension":7543,"eyebrow":7544,"featured":1742,"finish":7540,"fork":7540,"hero":7540,"heroAlt":7540,"heroCta":7540,"heroImage":7540,"homepageFeatured":7542,"kind":7545,"lessonCount":7540,"meta":71770,"navigation":1742,"newsItems":7540,"next":7540,"ogImage":71771,"order":7540,"outcomes":7540,"path":71772,"publishDate":71773,"readingTime":71774,"related":71775,"relatedProjects":7540,"seo":71776,"stem":71777,"tags":71778,"track":7540,"trackName":7540,"__hash__":71779},"blog\u002Fblog\u002Fexpert-notes\u002Frabi-oscillation-visualization.md","Rabi Oscillations: Visualizing Excited-State Probability",[7],{"type":9,"value":45927,"toc":71744},[45928,45935,45938,46047,46110,46114,46117,47582,47586,47667,47879,47882,48033,48036,48159,48268,48368,48371,48375,48378,48499,48559,48717,48720,48942,48945,49674,49711,49715,49718,50052,50182,50492,50495,50722,50840,50844,50914,51384,51386,51618,51621,51800,51855,52056,52059,52062,52313,52345,52625,52628,52778,52810,52930,52933,53200,53203,53321,53325,53328,53694,53697,53831,53838,53842,53845,53987,53990,54532,54535,54761,54937,54941,55015,55286,55288,55353,55385,55552,55621,55846,55849,55852,56213,56215,56539,56542,56546,56549,56656,56688,56872,56915,56919,57050,57296,57299,57692,57695,57906,57909,57913,58386,58846,58965,59084,59087,59091,59192,59195,59294,59377,59562,59565,59765,59768,59869,60266,60270,60273,60758,60761,60765,60768,61027,61030,61243,61246,61348,61351,61381,61692,61698,62178,62269,62275,62451,62457,62610,62614,67438,67442,67448,68150,68154,68800,68804,68808,69963,69969,70169,70450,70454,71259,71263,71719,71723,71727,71731,71741],[12,45929,45930],{},[15,45931,17,45932,23],{},[19,45933,22],{"href":45934},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FQC-Hardware-How-To\u002Fblob\u002Fmain\u002FJupyter%20Notebook%20Scripts\u002FRabi_Oscillation_Visualization_for_Excited_State_Probability.ipynb",[12,45936,45937],{},"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",[61,45939,45941],{"className":45940},[26354],[61,45942,45944,45976],{"className":45943},[64],[61,45945,45947],{"className":45946},[68],[70,45948,45949],{"xmlns":72,"display":26363},[74,45950,45951,45973],{},[77,45952,45953,45960,45962,45964,45966,45969,45971],{},[80,45954,45955,45958],{},[83,45956,45957],{},"P",[83,45959,21378],{},[586,45961,849],{"stretchy":588},[83,45963,19797],{},[586,45965,862],{"separator":103},[83,45967,45968],{},"τ",[586,45970,300],{"stretchy":588},[586,45972,862],{"separator":103},[95,45974,45975],{"encoding":97},"P_e(f,\\tau),",[61,45977,45979],{"className":45978,"ariaHidden":103},[102],[61,45980,45982,45985,46025,46028,46031,46034,46037,46041,46044],{"className":45981},[107],[61,45983],{"className":45984,"style":627},[111],[61,45986,45988,45991],{"className":45987},[116],[61,45989,45957],{"className":45990,"style":23700},[116,120],[61,45992,45994],{"className":45993},[125],[61,45995,45997,46017],{"className":45996},[129,130],[61,45998,46000,46014],{"className":45999},[134],[61,46001,46003],{"className":46002,"style":20164},[138],[61,46004,46005,46008],{"style":28699},[61,46006],{"className":46007,"style":147},[146],[61,46009,46011],{"className":46010},[151,152,153,154],[61,46012,21378],{"className":46013},[116,120,154],[61,46015,170],{"className":46016},[169],[61,46018,46020],{"className":46019},[134],[61,46021,46023],{"className":46022,"style":177},[138],[61,46024],{},[61,46026,849],{"className":46027},[631],[61,46029,19797],{"className":46030,"style":20151},[116,120],[61,46032,862],{"className":46033},[977],[61,46035],{"className":46036,"style":982},[981],[61,46038,45968],{"className":46039,"style":46040},[116,120],"margin-right:0.1132em;",[61,46042,300],{"className":46043},[730],[61,46045,862],{"className":46046},[977],[12,46048,46049,46050,46079,46080,46109],{},"where ",[61,46051,46053,46066],{"className":46052},[64],[61,46054,46056],{"className":46055},[68],[70,46057,46058],{"xmlns":72},[74,46059,46060,46064],{},[77,46061,46062],{},[83,46063,19797],{},[95,46065,19797],{"encoding":97},[61,46067,46069],{"className":46068,"ariaHidden":103},[102],[61,46070,46072,46076],{"className":46071},[107],[61,46073],{"className":46074,"style":46075},[111],"height:0.8889em;vertical-align:-0.1944em;",[61,46077,19797],{"className":46078,"style":20151},[116,120]," denotes the applied drive frequency and ",[61,46081,46083,46097],{"className":46082},[64],[61,46084,46086],{"className":46085},[68],[70,46087,46088],{"xmlns":72},[74,46089,46090,46094],{},[77,46091,46092],{},[83,46093,45968],{},[95,46095,46096],{"encoding":97},"\\tau",[61,46098,46100],{"className":46099,"ariaHidden":103},[102],[61,46101,46103,46106],{"className":46102},[107],[61,46104],{"className":46105,"style":30277},[111],[61,46107,45968],{"className":46108,"style":46040},[116,120]," 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.",[25,46111,46113],{"id":46112},"nomenclature-and-definitions","Nomenclature and Definitions",[12,46115,46116],{},"The following table defines the physical and mathematical quantities that recur throughout the visualization workflow.",[30,46118,46119,46129],{},[33,46120,46121],{},[36,46122,46123,46126],{},[39,46124,46125],{},"Symbol or Term",[39,46127,46128],{},"Meaning",[50,46130,46131,46178,46224,46270,46397,46474,46551,46586,46806,46903,46938,46984,47134,47211,47436,47471,47566,47574],{},[36,46132,46133,46175],{},[55,46134,46135],{},[61,46136,46138,46157],{"className":46137},[64],[61,46139,46141],{"className":46140},[68],[70,46142,46143],{"xmlns":72},[74,46144,46145,46154],{},[77,46146,46147,46149,46152],{},[586,46148,589],{"stretchy":588},[83,46150,46151],{},"g",[586,46153,1498],{"stretchy":588},[95,46155,46156],{"encoding":97},"\\lvert g\\rangle",[61,46158,46160],{"className":46159,"ariaHidden":103},[102],[61,46161,46163,46166,46169,46172],{"className":46162},[107],[61,46164],{"className":46165,"style":627},[111],[61,46167,589],{"className":46168},[631],[61,46170,46151],{"className":46171,"style":121},[116,120],[61,46173,1498],{"className":46174},[730],[55,46176,46177],{},"Ground state of the qubit",[36,46179,46180,46221],{},[55,46181,46182],{},[61,46183,46185,46203],{"className":46184},[64],[61,46186,46188],{"className":46187},[68],[70,46189,46190],{"xmlns":72},[74,46191,46192,46200],{},[77,46193,46194,46196,46198],{},[586,46195,589],{"stretchy":588},[83,46197,21378],{},[586,46199,1498],{"stretchy":588},[95,46201,46202],{"encoding":97},"\\lvert e\\rangle",[61,46204,46206],{"className":46205,"ariaHidden":103},[102],[61,46207,46209,46212,46215,46218],{"className":46208},[107],[61,46210],{"className":46211,"style":627},[111],[61,46213,589],{"className":46214},[631],[61,46216,21378],{"className":46217},[116,120],[61,46219,1498],{"className":46220},[730],[55,46222,46223],{},"Excited state of the qubit",[36,46225,46226,46267],{},[55,46227,46228],{},[61,46229,46231,46249],{"className":46230},[64],[61,46232,46234],{"className":46233},[68],[70,46235,46236],{"xmlns":72},[74,46237,46238,46246],{},[77,46239,46240,46242,46244],{},[586,46241,589],{"stretchy":588},[83,46243,27662],{},[586,46245,1498],{"stretchy":588},[95,46247,46248],{"encoding":97},"\\lvert \\psi\\rangle",[61,46250,46252],{"className":46251,"ariaHidden":103},[102],[61,46253,46255,46258,46261,46264],{"className":46254},[107],[61,46256],{"className":46257,"style":627},[111],[61,46259,589],{"className":46260},[631],[61,46262,27662],{"className":46263,"style":121},[116,120],[61,46265,1498],{"className":46266},[730],[55,46268,46269],{},"Quantum state vector in a two-dimensional Hilbert space",[36,46271,46272,46318],{},[55,46273,46274],{},[61,46275,46277,46296],{"className":46276},[64],[61,46278,46280],{"className":46279},[68],[70,46281,46282],{"xmlns":72},[74,46283,46284,46293],{},[77,46285,46286,46288,46290],{},[83,46287,31343],{},[586,46289,862],{"separator":103},[83,46291,46292],{},"β",[95,46294,46295],{"encoding":97},"\\alpha,\\beta",[61,46297,46299],{"className":46298,"ariaHidden":103},[102],[61,46300,46302,46305,46308,46311,46314],{"className":46301},[107],[61,46303],{"className":46304,"style":46075},[111],[61,46306,31343],{"className":46307,"style":31359},[116,120],[61,46309,862],{"className":46310},[977],[61,46312],{"className":46313,"style":982},[981],[61,46315,46292],{"className":46316,"style":46317},[116,120],"margin-right:0.0528em;",[55,46319,46320,46321,27098,46359],{},"Complex probability amplitudes for ",[61,46322,46324,46341],{"className":46323},[64],[61,46325,46327],{"className":46326},[68],[70,46328,46329],{"xmlns":72},[74,46330,46331,46339],{},[77,46332,46333,46335,46337],{},[586,46334,589],{"stretchy":588},[83,46336,46151],{},[586,46338,1498],{"stretchy":588},[95,46340,46156],{"encoding":97},[61,46342,46344],{"className":46343,"ariaHidden":103},[102],[61,46345,46347,46350,46353,46356],{"className":46346},[107],[61,46348],{"className":46349,"style":627},[111],[61,46351,589],{"className":46352},[631],[61,46354,46151],{"className":46355,"style":121},[116,120],[61,46357,1498],{"className":46358},[730],[61,46360,46362,46379],{"className":46361},[64],[61,46363,46365],{"className":46364},[68],[70,46366,46367],{"xmlns":72},[74,46368,46369,46377],{},[77,46370,46371,46373,46375],{},[586,46372,589],{"stretchy":588},[83,46374,21378],{},[586,46376,1498],{"stretchy":588},[95,46378,46202],{"encoding":97},[61,46380,46382],{"className":46381,"ariaHidden":103},[102],[61,46383,46385,46388,46391,46394],{"className":46384},[107],[61,46386],{"className":46387,"style":627},[111],[61,46389,589],{"className":46390},[631],[61,46392,21378],{"className":46393},[116,120],[61,46395,1498],{"className":46396},[730],[36,46398,46399,46471],{},[55,46400,46401],{},[61,46402,46404,46422],{"className":46403},[64],[61,46405,46407],{"className":46406},[68],[70,46408,46409],{"xmlns":72},[74,46410,46411,46419],{},[77,46412,46413],{},[80,46414,46415,46417],{},[83,46416,45957],{},[83,46418,21378],{},[95,46420,46421],{"encoding":97},"P_e",[61,46423,46425],{"className":46424,"ariaHidden":103},[102],[61,46426,46428,46431],{"className":46427},[107],[61,46429],{"className":46430,"style":219},[111],[61,46432,46434,46437],{"className":46433},[116],[61,46435,45957],{"className":46436,"style":23700},[116,120],[61,46438,46440],{"className":46439},[125],[61,46441,46443,46463],{"className":46442},[129,130],[61,46444,46446,46460],{"className":46445},[134],[61,46447,46449],{"className":46448,"style":20164},[138],[61,46450,46451,46454],{"style":28699},[61,46452],{"className":46453,"style":147},[146],[61,46455,46457],{"className":46456},[151,152,153,154],[61,46458,21378],{"className":46459},[116,120,154],[61,46461,170],{"className":46462},[169],[61,46464,46466],{"className":46465},[134],[61,46467,46469],{"className":46468,"style":177},[138],[61,46470],{},[55,46472,46473],{},"Probability of measuring the qubit in the excited 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transition frequency of the qubit",[36,46552,46553,46583],{},[55,46554,46555],{},[61,46556,46558,46571],{"className":46557},[64],[61,46559,46561],{"className":46560},[68],[70,46562,46563],{"xmlns":72},[74,46564,46565,46569],{},[77,46566,46567],{},[83,46568,19797],{},[95,46570,19797],{"encoding":97},[61,46572,46574],{"className":46573,"ariaHidden":103},[102],[61,46575,46577,46580],{"className":46576},[107],[61,46578],{"className":46579,"style":46075},[111],[61,46581,19797],{"className":46582,"style":20151},[116,120],[55,46584,46585],{},"Applied drive 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transition frequency, ",[61,46664,46666,46696],{"className":46665},[64],[61,46667,46669],{"className":46668},[68],[70,46670,46671],{"xmlns":72},[74,46672,46673,46693],{},[77,46674,46675,46681,46683,46685,46687],{},[80,46676,46677,46679],{},[83,46678,85],{},[1493,46680,1495],{},[586,46682,2000],{},[1493,46684,337],{},[83,46686,19792],{},[80,46688,46689,46691],{},[83,46690,19797],{},[1493,46692,1495],{},[95,46694,46695],{"encoding":97},"\\omega_0=2\\pi 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drive frequency, ",[61,46843,46845,46867],{"className":46844},[64],[61,46846,46848],{"className":46847},[68],[70,46849,46850],{"xmlns":72},[74,46851,46852,46864],{},[77,46853,46854,46856,46858,46860,46862],{},[83,46855,85],{},[586,46857,2000],{},[1493,46859,337],{},[83,46861,19792],{},[83,46863,19797],{},[95,46865,46866],{"encoding":97},"\\omega=2\\pi 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Rabi angular frequency",[36,46939,46940,46981],{},[55,46941,46942],{},[61,46943,46945,46965],{"className":46944},[64],[61,46946,46948],{"className":46947},[68],[70,46949,46950],{"xmlns":72},[74,46951,46952,46962],{},[77,46953,46954,46956,46958,46960],{},[83,46955,283],{"mathvariant":197},[83,46957,602],{"mathvariant":197},[1493,46959,337],{},[83,46961,19792],{},[95,46963,46964],{"encoding":97},"\\Omega\u002F2\\pi",[61,46966,46968],{"className":46967,"ariaHidden":103},[102],[61,46969,46971,46974,46978],{"className":46970},[107],[61,46972],{"className":46973,"style":627},[111],[61,46975,46977],{"className":46976},[116],"Ω\u002F2",[61,46979,19792],{"className":46980,"style":121},[116,120],[55,46982,46983],{},"On-resonance Rabi frequency in ordinary frequency units",[36,46985,46986,47016],{},[55,46987,46988],{},[61,46989,46991,47004],{"className":46990},[64],[61,46992,46994],{"className":46993},[68],[70,46995,46996],{"xmlns":72},[74,46997,46998,47002],{},[77,46999,47000],{},[83,47001,198],{"mathvariant":197},[95,47003,36231],{"encoding":97},[61,47005,47007],{"className":47006,"ariaHidden":103},[102],[61,47008,47010,47013],{"className":47009},[107],[61,47011],{"className":47012,"style":296},[111],[61,47014,198],{"className":47015},[116],[55,47017,47018,47019],{},"Drive detuning, 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Rabi angular frequency",[36,47212,47213,47288],{},[55,47214,47215],{},[61,47216,47218,47237],{"className":47217},[64],[61,47219,47221],{"className":47220},[68],[70,47222,47223],{"xmlns":72},[74,47224,47225,47234],{},[77,47226,47227],{},[80,47228,47229,47232],{},[83,47230,47231],{},"ν",[83,47233,27125],{},[95,47235,47236],{"encoding":97},"\\nu_R",[61,47238,47240],{"className":47239,"ariaHidden":103},[102],[61,47241,47243,47246],{"className":47242},[107],[61,47244],{"className":47245,"style":112},[111],[61,47247,47249,47253],{"className":47248},[116],[61,47250,47231],{"className":47251,"style":47252},[116,120],"margin-right:0.0637em;",[61,47254,47256],{"className":47255},[125],[61,47257,47259,47280],{"className":47258},[129,130],[61,47260,47262,47277],{"className":47261},[134],[61,47263,47265],{"className":47264,"style":139},[138],[61,47266,47268,47271],{"style":47267},"top:-2.55em;margin-left:-0.0637em;margin-right:0.05em;",[61,47269],{"className":47270,"style":147},[146],[61,47272,47274],{"className":47273},[151,152,153,154],[61,47275,27125],{"className":47276,"style":30082},[116,120,154],[61,47278,170],{"className":47279},[169],[61,47281,47283],{"className":47282},[134],[61,47284,47286],{"className":47285,"style":177},[138],[61,47287],{},[55,47289,47290,47291],{},"Generalized Rabi frequency, ",[61,47292,47294,47326],{"className":47293},[64],[61,47295,47297],{"className":47296},[68],[70,47298,47299],{"xmlns":72},[74,47300,47301,47323],{},[77,47302,47303,47309,47311,47317,47319,47321],{},[80,47304,47305,47307],{},[83,47306,47231],{},[83,47308,27125],{},[586,47310,2000],{},[80,47312,47313,47315],{},[83,47314,283],{"mathvariant":197},[83,47316,27125],{},[83,47318,602],{"mathvariant":197},[1493,47320,337],{},[83,47322,19792],{},[95,47324,47325],{"encoding":97},"\\nu_R=\\Omega_R\u002F2\\pi",[61,47327,47329,47384],{"className":47328,"ariaHidden":103},[102],[61,47330,47332,47335,47375,47378,47381],{"className":47331},[107],[61,47333],{"className":47334,"style":112},[111],[61,47336,47338,47341],{"className":47337},[116],[61,47339,47231],{"className":47340,"style":47252},[116,120],[61,47342,47344],{"className":47343},[125],[61,47345,47347,47367],{"className":47346},[129,130],[61,47348,47350,47364],{"className":47349},[134],[61,47351,47353],{"className":47352,"style":139},[138],[61,47354,47355,47358],{"style":47267},[61,47356],{"className":47357,"style":147},[146],[61,47359,47361],{"className":47360},[151,152,153,154],[61,47362,27125],{"className":47363,"style":30082},[116,120,154],[61,47365,170],{"className":47366},[169],[61,47368,47370],{"className":47369},[134],[61,47371,47373],{"className":47372,"style":177},[138],[61,47374],{},[61,47376],{"className":47377,"style":19197},[981],[61,47379,2000],{"className":47380},[19201],[61,47382],{"className":47383,"style":19197},[981],[61,47385,47387,47390,47430,47433],{"className":47386},[107],[61,47388],{"className":47389,"style":627},[111],[61,47391,47393,47396],{"className":47392},[116],[61,47394,283],{"className":47395},[116],[61,47397,47399],{"className":47398},[125],[61,47400,47402,47422],{"className":47401},[129,130],[61,47403,47405,47419],{"className":47404},[134],[61,47406,47408],{"className":47407,"style":139},[138],[61,47409,47410,47413],{"style":241},[61,47411],{"className":47412,"style":147},[146],[61,47414,47416],{"className":47415},[151,152,153,154],[61,47417,27125],{"className":47418,"style":30082},[116,120,154],[61,47420,170],{"className":47421},[169],[61,47423,47425],{"className":47424},[134],[61,47426,47428],{"className":47427,"style":177},[138],[61,47429],{},[61,47431,20506],{"className":47432},[116],[61,47434,19792],{"className":47435,"style":121},[116,120],[36,47437,47438,47468],{},[55,47439,47440],{},[61,47441,47443,47456],{"className":47442},[64],[61,47444,47446],{"className":47445},[68],[70,47447,47448],{"xmlns":72},[74,47449,47450,47454],{},[77,47451,47452],{},[83,47453,45968],{},[95,47455,46096],{"encoding":97},[61,47457,47459],{"className":47458,"ariaHidden":103},[102],[61,47460,47462,47465],{"className":47461},[107],[61,47463],{"className":47464,"style":30277},[111],[61,47466,45968],{"className":47467,"style":46040},[116,120],[55,47469,47470],{},"Pulse duration",[36,47472,47473,47563],{},[55,47474,47475],{},[61,47476,47478,47499],{"className":47477},[64],[61,47479,47481],{"className":47480},[68],[70,47482,47483],{"xmlns":72},[74,47484,47485,47496],{},[77,47486,47487],{},[34480,47488,47489,47491,47493],{},[83,47490,23559],{},[1493,47492,337],{},[586,47494,47495],{},"∗",[95,47497,47498],{"encoding":97},"T_2^\\ast",[61,47500,47502],{"className":47501,"ariaHidden":103},[102],[61,47503,47505,47509],{"className":47504},[107],[61,47506],{"className":47507,"style":47508},[111],"height:0.9368em;vertical-align:-0.2481em;",[61,47510,47512,47515],{"className":47511},[116],[61,47513,23559],{"className":47514,"style":23700},[116,120],[61,47516,47518],{"className":47517},[125],[61,47519,47521,47554],{"className":47520},[129,130],[61,47522,47524,47551],{"className":47523},[134],[61,47525,47528,47540],{"className":47526,"style":47527},[138],"height:0.6887em;",[61,47529,47531,47534],{"style":47530},"top:-2.4519em;margin-left:-0.1389em;margin-right:0.05em;",[61,47532],{"className":47533,"style":147},[146],[61,47535,47537],{"className":47536},[151,152,153,154],[61,47538,337],{"className":47539},[116,154],[61,47541,47542,47545],{"style":21487},[61,47543],{"className":47544,"style":147},[146],[61,47546,47548],{"className":47547},[151,152,153,154],[61,47549,47495],{"className":47550},[19382,154],[61,47552,170],{"className":47553},[169],[61,47555,47557],{"className":47556},[134],[61,47558,47561],{"className":47559,"style":47560},[138],"height:0.2481em;",[61,47562],{},[55,47564,47565],{},"Phenomenological dephasing time used as an optional damping parameter",[36,47567,47568,47571],{},[55,47569,47570],{},"RWA",[55,47572,47573],{},"Rotating-wave approximation",[36,47575,47576,47579],{},[55,47577,47578],{},"FFT",[55,47580,47581],{},"Fast Fourier transform",[25,47583,47585],{"id":47584},"physical-model","Physical Model",[12,47587,47588,47589,47627,47628,47666],{},"A qubit can be modeled as a two-level quantum system with a ground state ",[61,47590,47592,47609],{"className":47591},[64],[61,47593,47595],{"className":47594},[68],[70,47596,47597],{"xmlns":72},[74,47598,47599,47607],{},[77,47600,47601,47603,47605],{},[586,47602,589],{"stretchy":588},[83,47604,46151],{},[586,47606,1498],{"stretchy":588},[95,47608,46156],{"encoding":97},[61,47610,47612],{"className":47611,"ariaHidden":103},[102],[61,47613,47615,47618,47621,47624],{"className":47614},[107],[61,47616],{"className":47617,"style":627},[111],[61,47619,589],{"className":47620},[631],[61,47622,46151],{"className":47623,"style":121},[116,120],[61,47625,1498],{"className":47626},[730]," and an excited state ",[61,47629,47631,47648],{"className":47630},[64],[61,47632,47634],{"className":47633},[68],[70,47635,47636],{"xmlns":72},[74,47637,47638,47646],{},[77,47639,47640,47642,47644],{},[586,47641,589],{"stretchy":588},[83,47643,21378],{},[586,47645,1498],{"stretchy":588},[95,47647,46202],{"encoding":97},[61,47649,47651],{"className":47650,"ariaHidden":103},[102],[61,47652,47654,47657,47660,47663],{"className":47653},[107],[61,47655],{"className":47656,"style":627},[111],[61,47658,589],{"className":47659},[631],[61,47661,21378],{"className":47662},[116,120],[61,47664,1498],{"className":47665},[730],". In the absence of a microwave drive, the energy splitting is",[61,47668,47670],{"className":47669},[26354],[61,47671,47673,47712],{"className":47672},[64],[61,47674,47676],{"className":47675},[68],[70,47677,47678],{"xmlns":72,"display":26363},[74,47679,47680,47709],{},[77,47681,47682,47688,47690,47696,47698,47701,47707],{},[80,47683,47684,47686],{},[83,47685,19721],{},[83,47687,21378],{},[586,47689,19132],{},[80,47691,47692,47694],{},[83,47693,19721],{},[83,47695,46151],{},[586,47697,2000],{},[83,47699,47700],{"mathvariant":197},"ℏ",[80,47702,47703,47705],{},[83,47704,85],{},[1493,47706,1495],{},[586,47708,862],{"separator":103},[95,47710,47711],{"encoding":97},"E_e-E_g=\\hbar\\omega_0,",[61,47713,47715,47770,47826],{"className":47714,"ariaHidden":103},[102],[61,47716,47718,47721,47761,47764,47767],{"className":47717},[107],[61,47719],{"className":47720,"style":219},[111],[61,47722,47724,47727],{"className":47723},[116],[61,47725,19721],{"className":47726,"style":19831},[116,120],[61,47728,47730],{"className":47729},[125],[61,47731,47733,47753],{"className":47732},[129,130],[61,47734,47736,47750],{"className":47735},[134],[61,47737,47739],{"className":47738,"style":20164},[138],[61,47740,47741,47744],{"style":19846},[61,47742],{"className":47743,"style":147},[146],[61,47745,47747],{"className":47746},[151,152,153,154],[61,47748,21378],{"className":47749},[116,120,154],[61,47751,170],{"className":47752},[169],[61,47754,47756],{"className":47755},[134],[61,47757,47759],{"className":47758,"style":177},[138],[61,47760],{},[61,47762],{"className":47763,"style":20194},[981],[61,47765,19132],{"className":47766},[19382],[61,47768],{"className":47769,"style":20194},[981],[61,47771,47773,47777,47817,47820,47823],{"className":47772},[107],[61,47774],{"className":47775,"style":47776},[111],"height:0.9694em;vertical-align:-0.2861em;",[61,47778,47780,47783],{"className":47779},[116],[61,47781,19721],{"className":47782,"style":19831},[116,120],[61,47784,47786],{"className":47785},[125],[61,47787,47789,47809],{"className":47788},[129,130],[61,47790,47792,47806],{"className":47791},[134],[61,47793,47795],{"className":47794,"style":20164},[138],[61,47796,47797,47800],{"style":19846},[61,47798],{"className":47799,"style":147},[146],[61,47801,47803],{"className":47802},[151,152,153,154],[61,47804,46151],{"className":47805,"style":121},[116,120,154],[61,47807,170],{"className":47808},[169],[61,47810,47812],{"className":47811},[134],[61,47813,47815],{"className":47814,"style":26767},[138],[61,47816],{},[61,47818],{"className":47819,"style":19197},[981],[61,47821,2000],{"className":47822},[19201],[61,47824],{"className":47825,"style":19197},[981],[61,47827,47829,47833,47836,47876],{"className":47828},[107],[61,47830],{"className":47831,"style":47832},[111],"height:0.8833em;vertical-align:-0.1944em;",[61,47834,47700],{"className":47835},[116],[61,47837,47839,47842],{"className":47838},[116],[61,47840,85],{"className":47841,"style":121},[116,120],[61,47843,47845],{"className":47844},[125],[61,47846,47848,47868],{"className":47847},[129,130],[61,47849,47851,47865],{"className":47850},[134],[61,47852,47854],{"className":47853,"style":19230},[138],[61,47855,47856,47859],{"style":142},[61,47857],{"className":47858,"style":147},[146],[61,47860,47862],{"className":47861},[151,152,153,154],[61,47863,1495],{"className":47864},[116,154],[61,47866,170],{"className":47867},[169],[61,47869,47871],{"className":47870},[134],[61,47872,47874],{"className":47873,"style":177},[138],[61,47875],{},[61,47877,862],{"className":47878},[977],[12,47880,47881],{},"where the angular transition frequency is",[61,47883,47885],{"className":47884},[26354],[61,47886,47888,47920],{"className":47887},[64],[61,47889,47891],{"className":47890},[68],[70,47892,47893],{"xmlns":72,"display":26363},[74,47894,47895,47917],{},[77,47896,47897,47903,47905,47907,47909,47915],{},[80,47898,47899,47901],{},[83,47900,85],{},[1493,47902,1495],{},[586,47904,2000],{},[1493,47906,337],{},[83,47908,19792],{},[80,47910,47911,47913],{},[83,47912,19797],{},[1493,47914,1495],{},[83,47916,332],{"mathvariant":197},[95,47918,47919],{"encoding":97},"\\omega_0=2\\pi f_0.",[61,47921,47923,47978],{"className":47922,"ariaHidden":103},[102],[61,47924,47926,47929,47969,47972,47975],{"className":47925},[107],[61,47927],{"className":47928,"style":112},[111],[61,47930,47932,47935],{"className":47931},[116],[61,47933,85],{"className":47934,"style":121},[116,120],[61,47936,47938],{"className":47937},[125],[61,47939,47941,47961],{"className":47940},[129,130],[61,47942,47944,47958],{"className":47943},[134],[61,47945,47947],{"className":47946,"style":19230},[138],[61,47948,47949,47952],{"style":142},[61,47950],{"className":47951,"style":147},[146],[61,47953,47955],{"className":47954},[151,152,153,154],[61,47956,1495],{"className":47957},[116,154],[61,47959,170],{"className":47960},[169],[61,47962,47964],{"className":47963},[134],[61,47965,47967],{"className":47966,"style":177},[138],[61,47968],{},[61,47970],{"className":47971,"style":19197},[981],[61,47973,2000],{"className":47974},[19201],[61,47976],{"className":47977,"style":19197},[981],[61,47979,47981,47984,47987,47990,48030],{"className":47980},[107],[61,47982],{"className":47983,"style":46075},[111],[61,47985,337],{"className":47986},[116],[61,47988,19792],{"className":47989,"style":121},[116,120],[61,47991,47993,47996],{"className":47992},[116],[61,47994,19797],{"className":47995,"style":20151},[116,120],[61,47997,47999],{"className":47998},[125],[61,48000,48002,48022],{"className":48001},[129,130],[61,48003,48005,48019],{"className":48004},[134],[61,48006,48008],{"className":48007,"style":19230},[138],[61,48009,48010,48013],{"style":20167},[61,48011],{"className":48012,"style":147},[146],[61,48014,48016],{"className":48015},[151,152,153,154],[61,48017,1495],{"className":48018},[116,154],[61,48020,170],{"className":48021},[169],[61,48023,48025],{"className":48024},[134],[61,48026,48028],{"className":48027,"style":177},[138],[61,48029],{},[61,48031,332],{"className":48032},[116],[12,48034,48035],{},"The default configuration uses a transition frequency of",[61,48037,48039],{"className":48038},[26354],[61,48040,48042,48078],{"className":48041},[64],[61,48043,48045],{"className":48044},[68],[70,48046,48047],{"xmlns":72,"display":26363},[74,48048,48049,48075],{},[77,48050,48051,48057,48059,48062,48064,48073],{},[80,48052,48053,48055],{},[83,48054,19797],{},[1493,48056,1495],{},[586,48058,2000],{},[1493,48060,48061],{},"5.000",[27272,48063,27274],{},[77,48065,48066,48069,48071],{},[83,48067,48068],{"mathvariant":197},"G",[83,48070,6871],{"mathvariant":197},[83,48072,26482],{"mathvariant":197},[83,48074,332],{"mathvariant":197},[95,48076,48077],{"encoding":97},"f_0=5.000~\\mathrm{GHz}.",[61,48079,48081,48136],{"className":48080,"ariaHidden":103},[102],[61,48082,48084,48087,48127,48130,48133],{"className":48083},[107],[61,48085],{"className":48086,"style":46075},[111],[61,48088,48090,48093],{"className":48089},[116],[61,48091,19797],{"className":48092,"style":20151},[116,120],[61,48094,48096],{"className":48095},[125],[61,48097,48099,48119],{"className":48098},[129,130],[61,48100,48102,48116],{"className":48101},[134],[61,48103,48105],{"className":48104,"style":19230},[138],[61,48106,48107,48110],{"style":20167},[61,48108],{"className":48109,"style":147},[146],[61,48111,48113],{"className":48112},[151,152,153,154],[61,48114,1495],{"className":48115},[116,154],[61,48117,170],{"className":48118},[169],[61,48120,48122],{"className":48121},[134],[61,48123,48125],{"className":48124,"style":177},[138],[61,48126],{},[61,48128],{"className":48129,"style":19197},[981],[61,48131,2000],{"className":48132},[19201],[61,48134],{"className":48135,"style":19197},[981],[61,48137,48139,48142,48145,48149,48156],{"className":48138},[107],[61,48140],{"className":48141,"style":296},[111],[61,48143,48061],{"className":48144},[116],[61,48146,27274],{"className":48147},[981,48148],"nobreak",[61,48150,48152],{"className":48151},[116],[61,48153,48155],{"className":48154},[116,27529],"GHz",[61,48157,332],{"className":48158},[116],[12,48160,48161,48162,48190,48191,27098,48229,48267],{},"Applying a microwave drive with frequency ",[61,48163,48165,48178],{"className":48164},[64],[61,48166,48168],{"className":48167},[68],[70,48169,48170],{"xmlns":72},[74,48171,48172,48176],{},[77,48173,48174],{},[83,48175,19797],{},[95,48177,19797],{"encoding":97},[61,48179,48181],{"className":48180,"ariaHidden":103},[102],[61,48182,48184,48187],{"className":48183},[107],[61,48185],{"className":48186,"style":46075},[111],[61,48188,19797],{"className":48189,"style":20151},[116,120]," induces coherent rotations between ",[61,48192,48194,48211],{"className":48193},[64],[61,48195,48197],{"className":48196},[68],[70,48198,48199],{"xmlns":72},[74,48200,48201,48209],{},[77,48202,48203,48205,48207],{},[586,48204,589],{"stretchy":588},[83,48206,46151],{},[586,48208,1498],{"stretchy":588},[95,48210,46156],{"encoding":97},[61,48212,48214],{"className":48213,"ariaHidden":103},[102],[61,48215,48217,48220,48223,48226],{"className":48216},[107],[61,48218],{"className":48219,"style":627},[111],[61,48221,589],{"className":48222},[631],[61,48224,46151],{"className":48225,"style":121},[116,120],[61,48227,1498],{"className":48228},[730],[61,48230,48232,48249],{"className":48231},[64],[61,48233,48235],{"className":48234},[68],[70,48236,48237],{"xmlns":72},[74,48238,48239,48247],{},[77,48240,48241,48243,48245],{},[586,48242,589],{"stretchy":588},[83,48244,21378],{},[586,48246,1498],{"stretchy":588},[95,48248,46202],{"encoding":97},[61,48250,48252],{"className":48251,"ariaHidden":103},[102],[61,48253,48255,48258,48261,48264],{"className":48254},[107],[61,48256],{"className":48257,"style":627},[111],[61,48259,589],{"className":48260},[631],[61,48262,21378],{"className":48263},[116,120],[61,48265,1498],{"className":48266},[730],". Exact resonance occurs when",[61,48269,48271],{"className":48270},[26354],[61,48272,48274,48298],{"className":48273},[64],[61,48275,48277],{"className":48276},[68],[70,48278,48279],{"xmlns":72,"display":26363},[74,48280,48281,48295],{},[77,48282,48283,48285,48287,48293],{},[83,48284,19797],{},[586,48286,2000],{},[80,48288,48289,48291],{},[83,48290,19797],{},[1493,48292,1495],{},[83,48294,332],{"mathvariant":197},[95,48296,48297],{"encoding":97},"f=f_0.",[61,48299,48301,48319],{"className":48300,"ariaHidden":103},[102],[61,48302,48304,48307,48310,48313,48316],{"className":48303},[107],[61,48305],{"className":48306,"style":46075},[111],[61,48308,19797],{"className":48309,"style":20151},[116,120],[61,48311],{"className":48312,"style":19197},[981],[61,48314,2000],{"className":48315},[19201],[61,48317],{"className":48318,"style":19197},[981],[61,48320,48322,48325,48365],{"className":48321},[107],[61,48323],{"className":48324,"style":46075},[111],[61,48326,48328,48331],{"className":48327},[116],[61,48329,19797],{"className":48330,"style":20151},[116,120],[61,48332,48334],{"className":48333},[125],[61,48335,48337,48357],{"className":48336},[129,130],[61,48338,48340,48354],{"className":48339},[134],[61,48341,48343],{"className":48342,"style":19230},[138],[61,48344,48345,48348],{"style":20167},[61,48346],{"className":48347,"style":147},[146],[61,48349,48351],{"className":48350},[151,152,153,154],[61,48352,1495],{"className":48353},[116,154],[61,48355,170],{"className":48356},[169],[61,48358,48360],{"className":48359},[134],[61,48361,48363],{"className":48362,"style":177},[138],[61,48364],{},[61,48366,332],{"className":48367},[116],[12,48369,48370],{},"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.",[25,48372,48374],{"id":48373},"mathematical-foundation","Mathematical Foundation",[12,48376,48377],{},"The quantum state is a normalized vector in a two-dimensional Hilbert space,",[61,48379,48381],{"className":48380},[26354],[61,48382,48384,48424],{"className":48383},[64],[61,48385,48387],{"className":48386},[68],[70,48388,48389],{"xmlns":72,"display":26363},[74,48390,48391,48421],{},[77,48392,48393,48395,48397,48399,48401,48403,48405,48407,48409,48411,48413,48415,48417,48419],{},[586,48394,589],{"stretchy":588},[83,48396,27662],{},[586,48398,1498],{"stretchy":588},[586,48400,2000],{},[83,48402,31343],{},[586,48404,589],{"stretchy":588},[83,48406,46151],{},[586,48408,1498],{"stretchy":588},[586,48410,2419],{},[83,48412,46292],{},[586,48414,589],{"stretchy":588},[83,48416,21378],{},[586,48418,1498],{"stretchy":588},[586,48420,862],{"separator":103},[95,48422,48423],{"encoding":97},"\\lvert \\psi\\rangle=\\alpha\\lvert g\\rangle+\\beta\\lvert e\\rangle,",[61,48425,48427,48451,48478],{"className":48426,"ariaHidden":103},[102],[61,48428,48430,48433,48436,48439,48442,48445,48448],{"className":48429},[107],[61,48431],{"className":48432,"style":627},[111],[61,48434,589],{"className":48435},[631],[61,48437,27662],{"className":48438,"style":121},[116,120],[61,48440,1498],{"className":48441},[730],[61,48443],{"className":48444,"style":19197},[981],[61,48446,2000],{"className":48447},[19201],[61,48449],{"className":48450,"style":19197},[981],[61,48452,48454,48457,48460,48463,48466,48469,48472,48475],{"className":48453},[107],[61,48455],{"className":48456,"style":627},[111],[61,48458,31343],{"className":48459,"style":31359},[116,120],[61,48461,589],{"className":48462},[631],[61,48464,46151],{"className":48465,"style":121},[116,120],[61,48467,1498],{"className":48468},[730],[61,48470],{"className":48471,"style":20194},[981],[61,48473,2419],{"className":48474},[19382],[61,48476],{"className":48477,"style":20194},[981],[61,48479,48481,48484,48487,48490,48493,48496],{"className":48480},[107],[61,48482],{"className":48483,"style":627},[111],[61,48485,46292],{"className":48486,"style":46317},[116,120],[61,48488,589],{"className":48489},[631],[61,48491,21378],{"className":48492},[116,120],[61,48494,1498],{"className":48495},[730],[61,48497,862],{"className":48498},[977],[12,48500,46049,48501,27098,48529,48558],{},[61,48502,48504,48517],{"className":48503},[64],[61,48505,48507],{"className":48506},[68],[70,48508,48509],{"xmlns":72},[74,48510,48511,48515],{},[77,48512,48513],{},[83,48514,31343],{},[95,48516,31346],{"encoding":97},[61,48518,48520],{"className":48519,"ariaHidden":103},[102],[61,48521,48523,48526],{"className":48522},[107],[61,48524],{"className":48525,"style":30277},[111],[61,48527,31343],{"className":48528,"style":31359},[116,120],[61,48530,48532,48546],{"className":48531},[64],[61,48533,48535],{"className":48534},[68],[70,48536,48537],{"xmlns":72},[74,48538,48539,48543],{},[77,48540,48541],{},[83,48542,46292],{},[95,48544,48545],{"encoding":97},"\\beta",[61,48547,48549],{"className":48548,"ariaHidden":103},[102],[61,48550,48552,48555],{"className":48551},[107],[61,48553],{"className":48554,"style":46075},[111],[61,48556,46292],{"className":48557,"style":46317},[116,120]," are complex probability amplitudes. Normalization requires",[61,48560,48562],{"className":48561},[26354],[61,48563,48565,48604],{"className":48564},[64],[61,48566,48568],{"className":48567},[68],[70,48569,48570],{"xmlns":72,"display":26363},[74,48571,48572,48601],{},[77,48573,48574,48576,48578,48584,48586,48588,48590,48596,48598],{},[83,48575,589],{"mathvariant":197},[83,48577,31343],{},[19151,48579,48580,48582],{},[83,48581,589],{"mathvariant":197},[1493,48583,337],{},[586,48585,2419],{},[83,48587,589],{"mathvariant":197},[83,48589,46292],{},[19151,48591,48592,48594],{},[83,48593,589],{"mathvariant":197},[1493,48595,337],{},[586,48597,2000],{},[1493,48599,48600],{},"1.",[95,48602,48603],{"encoding":97},"|\\alpha|^2+|\\beta|^2=1.",[61,48605,48607,48658,48708],{"className":48606,"ariaHidden":103},[102],[61,48608,48610,48614,48617,48620,48649,48652,48655],{"className":48609},[107],[61,48611],{"className":48612,"style":48613},[111],"height:1.1141em;vertical-align:-0.25em;",[61,48615,589],{"className":48616},[116],[61,48618,31343],{"className":48619,"style":31359},[116,120],[61,48621,48623,48626],{"className":48622},[116],[61,48624,589],{"className":48625},[116],[61,48627,48629],{"className":48628},[125],[61,48630,48632],{"className":48631},[129],[61,48633,48635],{"className":48634},[134],[61,48636,48638],{"className":48637,"style":30095},[138],[61,48639,48640,48643],{"style":30098},[61,48641],{"className":48642,"style":147},[146],[61,48644,48646],{"className":48645},[151,152,153,154],[61,48647,337],{"className":48648},[116,154],[61,48650],{"className":48651,"style":20194},[981],[61,48653,2419],{"className":48654},[19382],[61,48656],{"className":48657,"style":20194},[981],[61,48659,48661,48664,48667,48670,48699,48702,48705],{"className":48660},[107],[61,48662],{"className":48663,"style":48613},[111],[61,48665,589],{"className":48666},[116],[61,48668,46292],{"className":48669,"style":46317},[116,120],[61,48671,48673,48676],{"className":48672},[116],[61,48674,589],{"className":48675},[116],[61,48677,48679],{"className":48678},[125],[61,48680,48682],{"className":48681},[129],[61,48683,48685],{"className":48684},[134],[61,48686,48688],{"className":48687,"style":30095},[138],[61,48689,48690,48693],{"style":30098},[61,48691],{"className":48692,"style":147},[146],[61,48694,48696],{"className":48695},[151,152,153,154],[61,48697,337],{"className":48698},[116,154],[61,48700],{"className":48701,"style":19197},[981],[61,48703,2000],{"className":48704},[19201],[61,48706],{"className":48707,"style":19197},[981],[61,48709,48711,48714],{"className":48710},[107],[61,48712],{"className":48713,"style":27780},[111],[61,48715,48600],{"className":48716},[116],[12,48718,48719],{},"The Born rule gives the probability of measuring the excited state,",[61,48721,48723],{"className":48722},[26354],[61,48724,48726,48778],{"className":48725},[64],[61,48727,48729],{"className":48728},[68],[70,48730,48731],{"xmlns":72,"display":26363},[74,48732,48733,48775],{},[77,48734,48735,48741,48743,48745,48747,48749,48751,48753,48755,48761,48763,48765,48767,48773],{},[80,48736,48737,48739],{},[83,48738,45957],{},[83,48740,21378],{},[586,48742,2000],{},[83,48744,589],{"mathvariant":197},[586,48746,27667],{"stretchy":588},[83,48748,21378],{},[586,48750,589],{"stretchy":588},[83,48752,27662],{},[586,48754,1498],{"stretchy":588},[19151,48756,48757,48759],{},[83,48758,589],{"mathvariant":197},[1493,48760,337],{},[586,48762,2000],{},[83,48764,589],{"mathvariant":197},[83,48766,46292],{},[19151,48768,48769,48771],{},[83,48770,589],{"mathvariant":197},[1493,48772,337],{},[83,48774,332],{"mathvariant":197},[95,48776,48777],{"encoding":97},"P_e=|\\langle e\\lvert\\psi\\rangle|^2=|\\beta|^2.",[61,48779,48781,48836,48898],{"className":48780,"ariaHidden":103},[102],[61,48782,48784,48787,48827,48830,48833],{"className":48783},[107],[61,48785],{"className":48786,"style":219},[111],[61,48788,48790,48793],{"className":48789},[116],[61,48791,45957],{"className":48792,"style":23700},[116,120],[61,48794,48796],{"className":48795},[125],[61,48797,48799,48819],{"className":48798},[129,130],[61,48800,48802,48816],{"className":48801},[134],[61,48803,48805],{"className":48804,"style":20164},[138],[61,48806,48807,48810],{"style":28699},[61,48808],{"className":48809,"style":147},[146],[61,48811,48813],{"className":48812},[151,152,153,154],[61,48814,21378],{"className":48815},[116,120,154],[61,48817,170],{"className":48818},[169],[61,48820,48822],{"className":48821},[134],[61,48823,48825],{"className":48824,"style":177},[138],[61,48826],{},[61,48828],{"className":48829,"style":19197},[981],[61,48831,2000],{"className":48832},[19201],[61,48834],{"className":48835,"style":19197},[981],[61,48837,48839,48842,48845,48848,48851,48854,48857,48860,48889,48892,48895],{"className":48838},[107],[61,48840],{"className":48841,"style":48613},[111],[61,48843,589],{"className":48844},[116],[61,48846,27667],{"className":48847},[631],[61,48849,21378],{"className":48850},[116,120],[61,48852,589],{"className":48853},[631],[61,48855,27662],{"className":48856,"style":121},[116,120],[61,48858,1498],{"className":48859},[730],[61,48861,48863,48866],{"className":48862},[116],[61,48864,589],{"className":48865},[116],[61,48867,48869],{"className":48868},[125],[61,48870,48872],{"className":48871},[129],[61,48873,48875],{"className":48874},[134],[61,48876,48878],{"className":48877,"style":30095},[138],[61,48879,48880,48883],{"style":30098},[61,48881],{"className":48882,"style":147},[146],[61,48884,48886],{"className":48885},[151,152,153,154],[61,48887,337],{"className":48888},[116,154],[61,48890],{"className":48891,"style":19197},[981],[61,48893,2000],{"className":48894},[19201],[61,48896],{"className":48897,"style":19197},[981],[61,48899,48901,48904,48907,48910,48939],{"className":48900},[107],[61,48902],{"className":48903,"style":48613},[111],[61,48905,589],{"className":48906},[116],[61,48908,46292],{"className":48909,"style":46317},[116,120],[61,48911,48913,48916],{"className":48912},[116],[61,48914,589],{"className":48915},[116],[61,48917,48919],{"className":48918},[125],[61,48920,48922],{"className":48921},[129],[61,48923,48925],{"className":48924},[134],[61,48926,48928],{"className":48927,"style":30095},[138],[61,48929,48930,48933],{"style":30098},[61,48931],{"className":48932,"style":147},[146],[61,48934,48936],{"className":48935},[151,152,153,154],[61,48937,337],{"className":48938},[116,154],[61,48940,332],{"className":48941},[116],[12,48943,48944],{},"Pauli matrices then provide a natural operator basis for the two-level system,",[61,48946,48948],{"className":48947},[26354],[61,48949,48951,49113],{"className":48950},[64],[61,48952,48954],{"className":48953},[68],[70,48955,48956],{"xmlns":72,"display":26363},[74,48957,48958,49110],{},[77,48959,48960,48966,48968,49004,49006,49008,49014,49016,49056,49058,49060,49066,49068,49108],{},[80,48961,48962,48964],{},[83,48963,19165],{},[83,48965,26374],{},[586,48967,2000],{},[77,48969,48970,48972,49002],{},[586,48971,27808],{"fence":103},[26382,48973,48974,48988],{"rowspacing":26384,"columnalign":26385,"columnspacing":21365},[26387,48975,48976,48982],{},[26390,48977,48978],{},[26393,48979,48980],{"scriptlevel":1495,"displaystyle":588},[1493,48981,1495],{},[26390,48983,48984],{},[26393,48985,48986],{"scriptlevel":1495,"displaystyle":588},[1493,48987,57],{},[26387,48989,48990,48996],{},[26390,48991,48992],{},[26393,48993,48994],{"scriptlevel":1495,"displaystyle":588},[1493,48995,57],{},[26390,48997,48998],{},[26393,48999,49000],{"scriptlevel":1495,"displaystyle":588},[1493,49001,1495],{},[586,49003,27817],{"fence":103},[586,49005,862],{"separator":103},[981,49007],{"width":27241},[80,49009,49010,49012],{},[83,49011,19165],{},[83,49013,26429],{},[586,49015,2000],{},[77,49017,49018,49020,49054],{},[586,49019,27808],{"fence":103},[26382,49021,49022,49040],{"rowspacing":26384,"columnalign":26385,"columnspacing":21365},[26387,49023,49024,49030],{},[26390,49025,49026],{},[26393,49027,49028],{"scriptlevel":1495,"displaystyle":588},[1493,49029,1495],{},[26390,49031,49032],{},[26393,49033,49034],{"scriptlevel":1495,"displaystyle":588},[77,49035,49036,49038],{},[586,49037,19132],{},[83,49039,21383],{},[26387,49041,49042,49048],{},[26390,49043,49044],{},[26393,49045,49046],{"scriptlevel":1495,"displaystyle":588},[83,49047,21383],{},[26390,49049,49050],{},[26393,49051,49052],{"scriptlevel":1495,"displaystyle":588},[1493,49053,1495],{},[586,49055,27817],{"fence":103},[586,49057,862],{"separator":103},[981,49059],{"width":27241},[80,49061,49062,49064],{},[83,49063,19165],{},[83,49065,26482],{},[586,49067,2000],{},[77,49069,49070,49072,49106],{},[586,49071,27808],{"fence":103},[26382,49073,49074,49088],{"rowspacing":26384,"columnalign":26385,"columnspacing":21365},[26387,49075,49076,49082],{},[26390,49077,49078],{},[26393,49079,49080],{"scriptlevel":1495,"displaystyle":588},[1493,49081,57],{},[26390,49083,49084],{},[26393,49085,49086],{"scriptlevel":1495,"displaystyle":588},[1493,49087,1495],{},[26387,49089,49090,49096],{},[26390,49091,49092],{},[26393,49093,49094],{"scriptlevel":1495,"displaystyle":588},[1493,49095,1495],{},[26390,49097,49098],{},[26393,49099,49100],{"scriptlevel":1495,"displaystyle":588},[77,49101,49102,49104],{},[586,49103,19132],{},[1493,49105,57],{},[586,49107,27817],{"fence":103},[83,49109,332],{"mathvariant":197},[95,49111,49112],{"encoding":97},"\\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}.",[61,49114,49116,49171,49355,49542],{"className":49115,"ariaHidden":103},[102],[61,49117,49119,49122,49162,49165,49168],{"className":49118},[107],[61,49120],{"className":49121,"style":112},[111],[61,49123,49125,49128],{"className":49124},[116],[61,49126,19165],{"className":49127,"style":121},[116,120],[61,49129,49131],{"className":49130},[125],[61,49132,49134,49154],{"className":49133},[129,130],[61,49135,49137,49151],{"className":49136},[134],[61,49138,49140],{"className":49139,"style":20164},[138],[61,49141,49142,49145],{"style":142},[61,49143],{"className":49144,"style":147},[146],[61,49146,49148],{"className":49147},[151,152,153,154],[61,49149,26374],{"className":49150},[116,120,154],[61,49152,170],{"className":49153},[169],[61,49155,49157],{"className":49156},[134],[61,49158,49160],{"className":49159,"style":177},[138],[61,49161],{},[61,49163],{"className":49164,"style":19197},[981],[61,49166,2000],{"className":49167},[19201],[61,49169],{"className":49170,"style":19197},[981],[61,49172,49174,49177,49294,49297,49300,49303,49306,49346,49349,49352],{"className":49173},[107],[61,49175],{"className":49176,"style":26592},[111],[61,49178,49180,49186,49288],{"className":49179},[19266],[61,49181,49183],{"className":49182,"style":19271},[631,19270],[61,49184,27808],{"className":49185},[19275,153],[61,49187,49189],{"className":49188},[116],[61,49190,49192,49237,49240,49243],{"className":49191},[26382],[61,49193,49195],{"className":49194},[26611],[61,49196,49198,49229],{"className":49197},[129,130],[61,49199,49201,49226],{"className":49200},[134],[61,49202,49204,49215],{"className":49203,"style":26621},[138],[61,49205,49206,49209],{"style":26624},[61,49207],{"className":49208,"style":19306},[146],[61,49210,49212],{"className":49211},[116],[61,49213,1495],{"className":49214},[116],[61,49216,49217,49220],{"style":26636},[61,49218],{"className":49219,"style":19306},[146],[61,49221,49223],{"className":49222},[116],[61,49224,57],{"className":49225},[116],[61,49227,170],{"className":49228},[169],[61,49230,49232],{"className":49231},[134],[61,49233,49235],{"className":49234,"style":26655},[138],[61,49236],{},[61,49238],{"className":49239,"style":26662},[26661],[61,49241],{"className":49242,"style":26662},[26661],[61,49244,49246],{"className":49245},[26611],[61,49247,49249,49280],{"className":49248},[129,130],[61,49250,49252,49277],{"className":49251},[134],[61,49253,49255,49266],{"className":49254,"style":26621},[138],[61,49256,49257,49260],{"style":26624},[61,49258],{"className":49259,"style":19306},[146],[61,49261,49263],{"className":49262},[116],[61,49264,57],{"className":49265},[116],[61,49267,49268,49271],{"style":26636},[61,49269],{"className":49270,"style":19306},[146],[61,49272,49274],{"className":49273},[116],[61,49275,1495],{"className":49276},[116],[61,49278,170],{"className":49279},[169],[61,49281,49283],{"className":49282},[134],[61,49284,49286],{"className":49285,"style":26655},[138],[61,49287],{},[61,49289,49291],{"className":49290,"style":19271},[730,19270],[61,49292,27817],{"className":49293},[19275,153],[61,49295],{"className":49296,"style":982},[981],[61,49298,862],{"className":49299},[977],[61,49301],{"className":49302,"style":27460},[981],[61,49304],{"className":49305,"style":982},[981],[61,49307,49309,49312],{"className":49308},[116],[61,49310,19165],{"className":49311,"style":121},[116,120],[61,49313,49315],{"className":49314},[125],[61,49316,49318,49338],{"className":49317},[129,130],[61,49319,49321,49335],{"className":49320},[134],[61,49322,49324],{"className":49323,"style":20164},[138],[61,49325,49326,49329],{"style":142},[61,49327],{"className":49328,"style":147},[146],[61,49330,49332],{"className":49331},[151,152,153,154],[61,49333,26429],{"className":49334,"style":121},[116,120,154],[61,49336,170],{"className":49337},[169],[61,49339,49341],{"className":49340},[134],[61,49342,49344],{"className":49343,"style":26767},[138],[61,49345],{},[61,49347],{"className":49348,"style":19197},[981],[61,49350,2000],{"className":49351},[19201],[61,49353],{"className":49354,"style":19197},[981],[61,49356,49358,49361,49481,49484,49487,49490,49493,49533,49536,49539],{"className":49357},[107],[61,49359],{"className":49360,"style":26592},[111],[61,49362,49364,49370,49475],{"className":49363},[19266],[61,49365,49367],{"className":49366,"style":19271},[631,19270],[61,49368,27808],{"className":49369},[19275,153],[61,49371,49373],{"className":49372},[116],[61,49374,49376,49421,49424,49427],{"className":49375},[26382],[61,49377,49379],{"className":49378},[26611],[61,49380,49382,49413],{"className":49381},[129,130],[61,49383,49385,49410],{"className":49384},[134],[61,49386,49388,49399],{"className":49387,"style":26621},[138],[61,49389,49390,49393],{"style":26624},[61,49391],{"className":49392,"style":19306},[146],[61,49394,49396],{"className":49395},[116],[61,49397,1495],{"className":49398},[116],[61,49400,49401,49404],{"style":26636},[61,49402],{"className":49403,"style":19306},[146],[61,49405,49407]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this representation, a driven two-level system behaves like a spin-",[61,49678,49680,49698],{"className":49679},[64],[61,49681,49683],{"className":49682},[68],[70,49684,49685],{"xmlns":72},[74,49686,49687,49695],{},[77,49688,49689,49691,49693],{},[1493,49690,57],{},[83,49692,602],{"mathvariant":197},[1493,49694,337],{},[95,49696,49697],{"encoding":97},"1\u002F2",[61,49699,49701],{"className":49700,"ariaHidden":103},[102],[61,49702,49704,49707],{"className":49703},[107],[61,49705],{"className":49706,"style":627},[111],[61,49708,49697],{"className":49709},[116]," particle precessing around an effective magnetic field. The drive strength supplies the transverse control component. Applied detuning supplies the longitudinal control component.",[25,49712,49714],{"id":49713},"rotating-frame-hamiltonian","Rotating-Frame Hamiltonian",[12,49716,49717],{},"In the laboratory frame, a minimal semiclassical Hamiltonian for the driven qubit is",[61,49719,49721],{"className":49720},[26354],[61,49722,49724,49792],{"className":49723},[64],[61,49725,49727],{"className":49726},[68],[70,49728,49729],{"xmlns":72,"display":26363},[74,49730,49731,49789],{},[77,49732,49733,49735,49737,49739,49741,49743,49757,49763,49765,49767,49769,49771,49773,49775,49777,49779,49781,49787],{},[83,49734,6871],{},[586,49736,849],{"stretchy":588},[83,49738,206],{},[586,49740,300],{"stretchy":588},[586,49742,2000],{},[19134,49744,49745,49755],{},[77,49746,49747,49749],{},[83,49748,47700],{"mathvariant":197},[80,49750,49751,49753],{},[83,49752,85],{},[1493,49754,1495],{},[1493,49756,337],{},[80,49758,49759,49761],{},[83,49760,19165],{},[83,49762,26482],{},[586,49764,2419],{},[83,49766,47700],{"mathvariant":197},[83,49768,283],{"mathvariant":197},[83,49770,5051],{},[586,49772,19125],{},[586,49774,849],{"stretchy":588},[83,49776,85],{},[83,49778,206],{},[586,49780,300],{"stretchy":588},[80,49782,49783,49785],{},[83,49784,19165],{},[83,49786,26374],{},[586,49788,862],{"separator":103},[95,49790,49791],{"encoding":97},"H(t)=\\frac{\\hbar\\omega_0}{2}\\sigma_z+\\hbar\\Omega\\cos(\\omega t)\\sigma_x,",[61,49793,49795,49822,49981],{"className":49794,"ariaHidden":103},[102],[61,49796,49798,49801,49804,49807,49810,49813,49816,49819],{"className":49797},[107],[61,49799],{"className":49800,"style":627},[111],[61,49802,6871],{"className":49803,"style":23130},[116,120],[61,49805,849],{"className":49806},[631],[61,49808,206],{"className":49809},[116,120],[61,49811,300],{"className":49812},[730],[61,49814],{"className":49815,"style":19197},[981],[61,49817,2000],{"className":49818},[19201],[61,49820],{"className":49821,"style":19197},[981],[61,49823,49825,49829,49932,49972,49975,49978],{"className":49824},[107],[61,49826],{"className":49827,"style":49828},[111],"height:2.0519em;vertical-align:-0.686em;",[61,49830,49832,49835,49929],{"className":49831},[116],[61,49833],{"className":49834},[631,19286],[61,49836,49838],{"className":49837},[19134],[61,49839,49841,49921],{"className":49840},[129,130],[61,49842,49844,49918],{"className":49843},[134],[61,49845,49848,49859,49867],{"className":49846,"style":49847},[138],"height:1.3659em;",[61,49849,49850,49853],{"style":28925},[61,49851],{"className":49852,"style":19306},[146],[61,49854,49856],{"className":49855},[116],[61,49857,337],{"className":49858},[116],[61,49860,49861,49864],{"style":19352},[61,49862],{"className":49863,"style":19306},[146],[61,49865],{"className":49866,"style":19360},[19359],[61,49868,49869,49872],{"style":28945},[61,49870],{"className":49871,"style":19306},[146],[61,49873,49875,49878],{"className":49874},[116],[61,49876,47700],{"className":49877},[116],[61,49879,49881,49884],{"className":49880},[116],[61,49882,85],{"className":49883,"style":121},[116,120],[61,49885,49887],{"className":49886},[125],[61,49888,49890,49910],{"className":49889},[129,130],[61,49891,49893,49907],{"className":49892},[134],[61,49894,49896],{"className":49895,"style":19230},[138],[61,49897,49898,49901],{"style":142},[61,49899],{"className":49900,"style":147},[146],[61,49902,49904],{"className":49903},[151,152,153,154],[61,49905,1495],{"className":49906},[116,154],[61,49908,170],{"className":49909},[169],[61,49911,49913],{"className":49912},[134],[61,49914,49916],{"className":49915,"style":177},[138],[61,49917],{},[61,49919,170],{"className":49920},[169],[61,49922,49924],{"className":49923},[134],[61,49925,49927],{"className":49926,"style":29011},[138],[61,49928],{},[61,49930],{"className":49931},[730,19286],[61,49933,49935,49938],{"className":49934},[116],[61,49936,19165],{"className":49937,"style":121},[116,120],[61,49939,49941],{"className":49940},[125],[61,49942,49944,49964],{"className":49943},[129,130],[61,49945,49947,49961],{"className":49946},[134],[61,49948,49950],{"className":49949,"style":20164},[138],[61,49951,49952,49955],{"style":142},[61,49953],{"className":49954,"style":147},[146],[61,49956,49958],{"className":49957},[151,152,153,154],[61,49959,26482],{"className":49960,"style":26946},[116,120,154],[61,49962,170],{"className":49963},[169],[61,49965,49967],{"className":49966},[134],[61,49968,49970],{"className":49969,"style":177},[138],[61,49971],{},[61,49973],{"className":49974,"style":20194},[981],[61,49976,2419],{"className":49977},[19382],[61,49979],{"className":49980,"style":20194},[981],[61,49982,49984,49987,49991,49994,49997,50000,50003,50006,50009,50049],{"className":49983},[107],[61,49985],{"className":49986,"style":627},[111],[61,49988,49990],{"className":49989},[116],"ℏΩ",[61,49992],{"className":49993,"style":982},[981],[61,49995,5051],{"className":49996},[19259],[61,49998,849],{"className":49999},[631],[61,50001,85],{"className":50002,"style":121},[116,120],[61,50004,206],{"className":50005},[116,120],[61,50007,300],{"className":50008},[730],[61,50010,50012,50015],{"className":50011},[116],[61,50013,19165],{"className":50014,"style":121},[116,120],[61,50016,50018],{"className":50017},[125],[61,50019,50021,50041],{"className":50020},[129,130],[61,50022,50024,50038],{"className":50023},[134],[61,50025,50027],{"className":50026,"style":20164},[138],[61,50028,50029,50032],{"style":142},[61,50030],{"className":50031,"style":147},[146],[61,50033,50035],{"className":50034},[151,152,153,154],[61,50036,26374],{"className":50037},[116,120,154],[61,50039,170],{"className":50040},[169],[61,50042,50044],{"className":50043},[134],[61,50045,50047],{"className":50046,"style":177},[138],[61,50048],{},[61,50050,862],{"className":50051},[977],[12,50053,46049,50054,50114,50115,50143,50144,50181],{},[61,50055,50057,50078],{"className":50056},[64],[61,50058,50060],{"className":50059},[68],[70,50061,50062],{"xmlns":72},[74,50063,50064,50076],{},[77,50065,50066,50068,50070,50072,50074],{},[83,50067,85],{},[586,50069,2000],{},[1493,50071,337],{},[83,50073,19792],{},[83,50075,19797],{},[95,50077,46866],{"encoding":97},[61,50079,50081,50099],{"className":50080,"ariaHidden":103},[102],[61,50082,50084,50087,50090,50093,50096],{"className":50083},[107],[61,50085],{"className":50086,"style":30277},[111],[61,50088,85],{"className":50089,"style":121},[116,120],[61,50091],{"className":50092,"style":19197},[981],[61,50094,2000],{"className":50095},[19201],[61,50097],{"className":50098,"style":19197},[981],[61,50100,50102,50105,50108,50111],{"className":50101},[107],[61,50103],{"className":50104,"style":46075},[111],[61,50106,337],{"className":50107},[116],[61,50109,19792],{"className":50110,"style":121},[116,120],[61,50112,19797],{"className":50113,"style":20151},[116,120]," is the drive angular frequency and ",[61,50116,50118,50131],{"className":50117},[64],[61,50119,50121],{"className":50120},[68],[70,50122,50123],{"xmlns":72},[74,50124,50125,50129],{},[77,50126,50127],{},[83,50128,283],{"mathvariant":197},[95,50130,286],{"encoding":97},[61,50132,50134],{"className":50133,"ariaHidden":103},[102],[61,50135,50137,50140],{"className":50136},[107],[61,50138],{"className":50139,"style":296},[111],[61,50141,283],{"className":50142},[116]," is the on-resonance Rabi angular frequency. The computational model specifies the drive strength using ",[61,50145,50147,50166],{"className":50146},[64],[61,50148,50150],{"className":50149},[68],[70,50151,50152],{"xmlns":72},[74,50153,50154,50164],{},[77,50155,50156,50158,50160,50162],{},[83,50157,283],{"mathvariant":197},[83,50159,602],{"mathvariant":197},[1493,50161,337],{},[83,50163,19792],{},[95,50165,46964],{"encoding":97},[61,50167,50169],{"className":50168,"ariaHidden":103},[102],[61,50170,50172,50175,50178],{"className":50171},[107],[61,50173],{"className":50174,"style":627},[111],[61,50176,46977],{"className":50177},[116],[61,50179,19792],{"className":50180,"style":121},[116,120]," in MHz. Transforming into a frame rotating at the drive frequency and applying the rotating-wave approximation gives the effective time-independent Hamiltonian",[61,50183,50185],{"className":50184},[26354],[61,50186,50188,50246],{"className":50187},[64],[61,50189,50191],{"className":50190},[68],[70,50192,50193],{"xmlns":72,"display":26363},[74,50194,50195,50243],{},[77,50196,50197,50209,50211,50217,50241],{},[80,50198,50199,50201],{},[83,50200,6871],{},[77,50202,50203,50205,50207],{},[83,50204,27125],{"mathvariant":197},[83,50206,28581],{"mathvariant":197},[83,50208,19106],{"mathvariant":197},[586,50210,2000],{},[19134,50212,50213,50215],{},[83,50214,47700],{"mathvariant":197},[1493,50216,337],{},[77,50218,50219,50221,50223,50229,50231,50233,50239],{},[586,50220,849],{"fence":103},[83,50222,198],{"mathvariant":197},[80,50224,50225,50227],{},[83,50226,19165],{},[83,50228,26482],{},[586,50230,2419],{},[83,50232,283],{"mathvariant":197},[80,50234,50235,50237],{},[83,50236,19165],{},[83,50238,26374],{},[586,50240,300],{"fence":103},[586,50242,862],{"separator":103},[95,50244,50245],{"encoding":97},"H_{\\mathrm{RWA}}=\\frac{\\hbar}{2}\\left(\\Delta\\sigma_z+\\Omega\\sigma_x\\right),",[61,50247,50249,50311],{"className":50248,"ariaHidden":103},[102],[61,50250,50252,50255,50302,50305,50308],{"className":50251},[107],[61,50253],{"className":50254,"style":219},[111],[61,50256,50258,50261],{"className":50257},[116],[61,50259,6871],{"className":50260,"style":23130},[116,120],[61,50262,50264],{"className":50263},[125],[61,50265,50267,50294],{"className":50266},[129,130],[61,50268,50270,50291],{"className":50269},[134],[61,50271,50273],{"className":50272,"style":139},[138],[61,50274,50276,50279],{"style":50275},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[61,50277],{"className":50278,"style":147},[146],[61,50280,50282],{"className":50281},[151,152,153,154],[61,50283,50285],{"className":50284},[116,154],[61,50286,50288],{"className":50287},[116,154],[61,50289,47570],{"className":50290},[116,27529,154],[61,50292,170],{"className":50293},[169],[61,50295,50297],{"className":50296},[134],[61,50298,50300],{"className":50299,"style":177},[138],[61,50301],{},[61,50303],{"className":50304,"style":19197},[981],[61,50306,2000],{"className":50307},[19201],[61,50309],{"className":50310,"style":19197},[981],[61,50312,50314,50317,50379,50382,50486,50489],{"className":50313},[107],[61,50315],{"className":50316,"style":49828},[111],[61,50318,50320,50323,50376],{"className":50319},[116],[61,50321],{"className":50322},[631,19286],[61,50324,50326],{"className":50325},[19134],[61,50327,50329,50368],{"className":50328},[129,130],[61,50330,50332,50365],{"className":50331},[134],[61,50333,50335,50346,50354],{"className":50334,"style":49847},[138],[61,50336,50337,50340],{"style":28925},[61,50338],{"className":50339,"style":19306},[146],[61,50341,50343],{"className":50342},[116],[61,50344,337],{"className":50345},[116],[61,50347,50348,50351],{"style":19352},[61,50349],{"className":50350,"style":19306},[146],[61,50352],{"className":50353,"style":19360},[19359],[61,50355,50356,50359],{"style":28945},[61,50357],{"className":50358,"style":19306},[146],[61,50360,50362],{"className":50361},[116],[61,50363,47700],{"className":50364},[116],[61,50366,170],{"className":50367},[169],[61,50369,50371],{"className":50370},[134],[61,50372,50374],{"className":50373,"style":29011},[138],[61,50375],{},[61,50377],{"className":50378},[730,19286],[61,50380],{"className":50381,"style":982},[981],[61,50383,50385,50388,50391,50431,50434,50437,50440,50443,50483],{"className":50384},[19266],[61,50386,849],{"className":50387,"style":19271},[631,19270],[61,50389,198],{"className":50390},[116],[61,50392,50394,50397],{"className":50393},[116],[61,50395,19165],{"className":50396,"style":121},[116,120],[61,50398,50400],{"className":50399},[125],[61,50401,50403,50423],{"className":50402},[129,130],[61,50404,50406,50420],{"className":50405},[134],[61,50407,50409],{"className":50408,"style":20164},[138],[61,50410,50411,50414],{"style":142},[61,50412],{"className":50413,"style":147},[146],[61,50415,50417],{"className":50416},[151,152,153,154],[61,50418,26482],{"className":50419,"style":26946},[116,120,154],[61,50421,170],{"className":50422},[169],[61,50424,50426],{"className":50425},[134],[61,50427,50429],{"className":50428,"style":177},[138],[61,50430],{},[61,50432],{"className":50433,"style":20194},[981],[61,50435,2419],{"className":50436},[19382],[61,50438],{"className":50439,"style":20194},[981],[61,50441,283],{"className":50442},[116],[61,50444,50446,50449],{"className":50445},[116],[61,50447,19165],{"className":50448,"style":121},[116,120],[61,50450,50452],{"className":50451},[125],[61,50453,50455,50475],{"className":50454},[129,130],[61,50456,50458,50472],{"className":50457},[134],[61,50459,50461],{"className":50460,"style":20164},[138],[61,50462,50463,50466],{"style":142},[61,50464],{"className":50465,"style":147},[146],[61,50467,50469],{"className":50468},[151,152,153,154],[61,50470,26374],{"className":50471},[116,120,154],[61,50473,170],{"className":50474},[169],[61,50476,50478],{"className":50477},[134],[61,50479,50481],{"className":50480,"style":177},[138],[61,50482],{},[61,50484,300],{"className":50485,"style":19271},[730,19270],[61,50487],{"className":50488,"style":982},[981],[61,50490,862],{"className":50491},[977],[12,50493,50494],{},"with detuning",[61,50496,50498],{"className":50497},[26354],[61,50499,50501,50549],{"className":50500},[64],[61,50502,50504],{"className":50503},[68],[70,50505,50506],{"xmlns":72,"display":26363},[74,50507,50508,50546],{},[77,50509,50510,50512,50514,50516,50518,50524,50526,50528,50530,50532,50534,50536,50542,50544],{},[83,50511,198],{"mathvariant":197},[586,50513,2000],{},[83,50515,85],{},[586,50517,19132],{},[80,50519,50520,50522],{},[83,50521,85],{},[1493,50523,1495],{},[586,50525,2000],{},[1493,50527,337],{},[83,50529,19792],{},[586,50531,849],{"stretchy":588},[83,50533,19797],{},[586,50535,19132],{},[80,50537,50538,50540],{},[83,50539,19797],{},[1493,50541,1495],{},[586,50543,300],{"stretchy":588},[83,50545,332],{"mathvariant":197},[95,50547,50548],{"encoding":97},"\\Delta=\\omega-\\omega_0=2\\pi(f-f_0).",[61,50550,50552,50570,50588,50643,50670],{"className":50551,"ariaHidden":103},[102],[61,50553,50555,50558,50561,50564,50567],{"className":50554},[107],[61,50556],{"className":50557,"style":296},[111],[61,50559,198],{"className":50560},[116],[61,50562],{"className":50563,"style":19197},[981],[61,50565,2000],{"className":50566},[19201],[61,50568],{"className":50569,"style":19197},[981],[61,50571,50573,50576,50579,50582,50585],{"className":50572},[107],[61,50574],{"className":50575,"style":47075},[111],[61,50577,85],{"className":50578,"style":121},[116,120],[61,50580],{"className":50581,"style":20194},[981],[61,50583,19132],{"className":50584},[19382],[61,50586],{"className":50587,"style":20194},[981],[61,50589,50591,50594,50634,50637,50640],{"className":50590},[107],[61,50592],{"className":50593,"style":112},[111],[61,50595,50597,50600],{"className":50596},[116],[61,50598,85],{"className":50599,"style":121},[116,120],[61,50601,50603],{"className":50602},[125],[61,50604,50606,50626],{"className":50605},[129,130],[61,50607,50609,50623],{"className":50608},[134],[61,50610,50612],{"className":50611,"style":19230},[138],[61,50613,50614,50617],{"style":142},[61,50615],{"className":50616,"style":147},[146],[61,50618,50620],{"className":50619},[151,152,153,154],[61,50621,1495],{"className":50622},[116,154],[61,50624,170],{"className":50625},[169],[61,50627,50629],{"className":50628},[134],[61,50630,50632],{"className":50631,"style":177},[138],[61,50633],{},[61,50635],{"className":50636,"style":19197},[981],[61,50638,2000],{"className":50639},[19201],[61,50641],{"className":50642,"style":19197},[981],[61,50644,50646,50649,50652,50655,50658,50661,50664,50667],{"className":50645},[107],[61,50647],{"className":50648,"style":627},[111],[61,50650,337],{"className":50651},[116],[61,50653,19792],{"className":50654,"style":121},[116,120],[61,50656,849],{"className":50657},[631],[61,50659,19797],{"className":50660,"style":20151},[116,120],[61,50662],{"className":50663,"style":20194},[981],[61,50665,19132],{"className":50666},[19382],[61,50668],{"className":50669,"style":20194},[981],[61,50671,50673,50676,50716,50719],{"className":50672},[107],[61,50674],{"className":50675,"style":627},[111],[61,50677,50679,50682],{"className":50678},[116],[61,50680,19797],{"className":50681,"style":20151},[116,120],[61,50683,50685],{"className":50684},[125],[61,50686,50688,50708],{"className":50687},[129,130],[61,50689,50691,50705],{"className":50690},[134],[61,50692,50694],{"className":50693,"style":19230},[138],[61,50695,50696,50699],{"style":20167},[61,50697],{"className":50698,"style":147},[146],[61,50700,50702],{"className":50701},[151,152,153,154],[61,50703,1495],{"className":50704},[116,154],[61,50706,170],{"className":50707},[169],[61,50709,50711],{"className":50710},[134],[61,50712,50714],{"className":50713,"style":177},[138],[61,50715],{},[61,50717,300],{"className":50718},[730],[61,50720,332],{"className":50721},[116],[12,50723,50724,50725,27098,50753,50781,50782,50810,50811,50839],{},"This Hamiltonian describes precession around an effective control axis with components proportional to ",[61,50726,50728,50741],{"className":50727},[64],[61,50729,50731],{"className":50730},[68],[70,50732,50733],{"xmlns":72},[74,50734,50735,50739],{},[77,50736,50737],{},[83,50738,283],{"mathvariant":197},[95,50740,286],{"encoding":97},[61,50742,50744],{"className":50743,"ariaHidden":103},[102],[61,50745,50747,50750],{"className":50746},[107],[61,50748],{"className":50749,"style":296},[111],[61,50751,283],{"className":50752},[116],[61,50754,50756,50769],{"className":50755},[64],[61,50757,50759],{"className":50758},[68],[70,50760,50761],{"xmlns":72},[74,50762,50763,50767],{},[77,50764,50765],{},[83,50766,198],{"mathvariant":197},[95,50768,36231],{"encoding":97},[61,50770,50772],{"className":50771,"ariaHidden":103},[102],[61,50773,50775,50778],{"className":50774},[107],[61,50776],{"className":50777,"style":296},[111],[61,50779,198],{"className":50780},[116],". The transverse component ",[61,50783,50785,50798],{"className":50784},[64],[61,50786,50788],{"className":50787},[68],[70,50789,50790],{"xmlns":72},[74,50791,50792,50796],{},[77,50793,50794],{},[83,50795,283],{"mathvariant":197},[95,50797,286],{"encoding":97},[61,50799,50801],{"className":50800,"ariaHidden":103},[102],[61,50802,50804,50807],{"className":50803},[107],[61,50805],{"className":50806,"style":296},[111],[61,50808,283],{"className":50809},[116]," drives population transfer. A longitudinal component ",[61,50812,50814,50827],{"className":50813},[64],[61,50815,50817],{"className":50816},[68],[70,50818,50819],{"xmlns":72},[74,50820,50821,50825],{},[77,50822,50823],{},[83,50824,198],{"mathvariant":197},[95,50826,36231],{"encoding":97},[61,50828,50830],{"className":50829,"ariaHidden":103},[102],[61,50831,50833,50836],{"className":50832},[107],[61,50834],{"className":50835,"style":296},[111],[61,50837,198],{"className":50838},[116]," tilts the rotation axis away from the resonant direction.",[25,50841,50843],{"id":50842},"excited-state-probability","Excited-State Probability",[12,50845,50846,50847,50885,50886,34830],{},"The simulation assumes that the qubit begins in the ground state ",[61,50848,50850,50867],{"className":50849},[64],[61,50851,50853],{"className":50852},[68],[70,50854,50855],{"xmlns":72},[74,50856,50857,50865],{},[77,50858,50859,50861,50863],{},[586,50860,589],{"stretchy":588},[83,50862,46151],{},[586,50864,1498],{"stretchy":588},[95,50866,46156],{"encoding":97},[61,50868,50870],{"className":50869,"ariaHidden":103},[102],[61,50871,50873,50876,50879,50882],{"className":50872},[107],[61,50874],{"className":50875,"style":627},[111],[61,50877,589],{"className":50878},[631],[61,50880,46151],{"className":50881,"style":121},[116,120],[61,50883,1498],{"className":50884},[730],". Evolving under the RWA Hamiltonian, the probability of occupying the excited state after pulse duration ",[61,50887,50889,50902],{"className":50888},[64],[61,50890,50892],{"className":50891},[68],[70,50893,50894],{"xmlns":72},[74,50895,50896,50900],{},[77,50897,50898],{},[83,50899,45968],{},[95,50901,46096],{"encoding":97},[61,50903,50905],{"className":50904,"ariaHidden":103},[102],[61,50906,50908,50911],{"className":50907},[107],[61,50909],{"className":50910,"style":30277},[111],[61,50912,45968],{"className":50913,"style":46040},[116,120],[61,50915,50917],{"className":50916},[26354],[61,50918,50920,50998],{"className":50919},[64],[61,50921,50923],{"className":50922},[68],[70,50924,50925],{"xmlns":72,"display":26363},[74,50926,50927,50995],{},[77,50928,50929,50935,50937,50939,50941,50943,50945,50947,50963,50973,50993],{},[80,50930,50931,50933],{},[83,50932,45957],{},[83,50934,21378],{},[586,50936,849],{"stretchy":588},[83,50938,19797],{},[586,50940,862],{"separator":103},[83,50942,45968],{},[586,50944,300],{"stretchy":588},[586,50946,2000],{},[19134,50948,50949,50955],{},[19151,50950,50951,50953],{},[83,50952,283],{"mathvariant":197},[1493,50954,337],{},[34480,50956,50957,50959,50961],{},[83,50958,283],{"mathvariant":197},[83,50960,27125],{},[1493,50962,337],{},[19151,50964,50965,50971],{},[77,50966,50967,50969],{},[83,50968,5069],{},[586,50970,19125],{},[1493,50972,337],{},[77,50974,50975,50977,50991],{},[586,50976,849],{"fence":103},[19134,50978,50979,50989],{},[77,50980,50981,50987],{},[80,50982,50983,50985],{},[83,50984,283],{"mathvariant":197},[83,50986,27125],{},[83,50988,45968],{},[1493,50990,337],{},[586,50992,300],{"fence":103},[586,50994,862],{"separator":103},[95,50996,50997],{"encoding":97},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right),",[61,50999,51001,51074],{"className":51000,"ariaHidden":103},[102],[61,51002,51004,51007,51047,51050,51053,51056,51059,51062,51065,51068,51071],{"className":51003},[107],[61,51005],{"className":51006,"style":627},[111],[61,51008,51010,51013],{"className":51009},[116],[61,51011,45957],{"className":51012,"style":23700},[116,120],[61,51014,51016],{"className":51015},[125],[61,51017,51019,51039],{"className":51018},[129,130],[61,51020,51022,51036],{"className":51021},[134],[61,51023,51025],{"className":51024,"style":20164},[138],[61,51026,51027,51030],{"style":28699},[61,51028],{"className":51029,"style":147},[146],[61,51031,51033],{"className":51032},[151,152,153,154],[61,51034,21378],{"className":51035},[116,120,154],[61,51037,170],{"className":51038},[169],[61,51040,51042],{"className":51041},[134],[61,51043,51045],{"className":51044,"style":177},[138],[61,51046],{},[61,51048,849],{"className":51049},[631],[61,51051,19797],{"className":51052,"style":20151},[116,120],[61,51054,862],{"className":51055},[977],[61,51057],{"className":51058,"style":982},[981],[61,51060,45968],{"className":51061,"style":46040},[116,120],[61,51063,300],{"className":51064},[730],[61,51066],{"className":51067,"style":19197},[981],[61,51069,2000],{"className":51070},[19201],[61,51072],{"className":51073,"style":19197},[981],[61,51075,51077,51081,51223,51226,51257,51260,51378,51381],{"className":51076},[107],[61,51078],{"className":51079,"style":51080},[111],"height:2.4706em;vertical-align:-0.9795em;",[61,51082,51084,51087,51220],{"className":51083},[116],[61,51085],{"className":51086},[631,19286],[61,51088,51090],{"className":51089},[19134],[61,51091,51093,51211],{"className":51092},[129,130],[61,51094,51096,51208],{"className":51095},[134],[61,51097,51100,51163,51171],{"className":51098,"style":51099},[138],"height:1.4911em;",[61,51101,51102,51105],{"style":28925},[61,51103],{"className":51104,"style":19306},[146],[61,51106,51108],{"className":51107},[116],[61,51109,51111,51114],{"className":51110},[116],[61,51112,283],{"className":51113},[116],[61,51115,51117],{"className":51116},[125],[61,51118,51120,51154],{"className":51119},[129,130],[61,51121,51123,51151],{"className":51122},[134],[61,51124,51127,51139],{"className":51125,"style":51126},[138],"height:0.7959em;",[61,51128,51130,51133],{"style":51129},"top:-2.4065em;margin-left:0em;margin-right:0.05em;",[61,51131],{"className":51132,"style":147},[146],[61,51134,51136],{"className":51135},[151,152,153,154],[61,51137,27125],{"className":51138,"style":30082},[116,120,154],[61,51140,51142,51145],{"style":51141},"top:-3.0448em;margin-right:0.05em;",[61,51143],{"className":51144,"style":147},[146],[61,51146,51148],{"className":51147},[151,152,153,154],[61,51149,337],{"className":51150},[116,154],[61,51152,170],{"className":51153},[169],[61,51155,51157],{"className":51156},[134],[61,51158,51161],{"className":51159,"style":51160},[138],"height:0.2935em;",[61,51162],{},[61,51164,51165,51168],{"style":19352},[61,51166],{"className":51167,"style":19306},[146],[61,51169],{"className":51170,"style":19360},[19359],[61,51172,51173,51176],{"style":28945},[61,51174],{"className":51175,"style":19306},[146],[61,51177,51179],{"className":51178},[116],[61,51180,51182,51185],{"className":51181},[116],[61,51183,283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,[107],[61,51492],{"className":51493,"style":51494},[111],"height:1.24em;vertical-align:-0.1777em;",[61,51496,51498],{"className":51497},[116,2425],[61,51499,51501,51609],{"className":51500},[129,130],[61,51502,51504,51606],{"className":51503},[134],[61,51505,51508,51589],{"className":51506,"style":51507},[138],"height:1.0623em;",[61,51509,51512,51516],{"className":51510,"style":51511},[28378],"top:-3.2em;",[61,51513],{"className":51514,"style":51515},[146],"height:3.2em;",[61,51517,51520,51551,51554,51557,51560],{"className":51518,"style":51519},[116],"padding-left:1em;",[61,51521,51523,51526],{"className":51522},[116],[61,51524,283],{"className":51525},[116],[61,51527,51529],{"className":51528},[125],[61,51530,51532],{"className":51531},[129],[61,51533,51535],{"className":51534},[134],[61,51536,51539],{"className":51537,"style":51538},[138],"height:0.7401em;",[61,51540,51542,51545],{"style":51541},"top:-2.989em;margin-right:0.05em;",[61,51543],{"className":51544,"style":147},[146],[61,51546,51548],{"className":51547},[151,152,153,154],[61,51549,337],{"className":51550},[116,154],[61,51552],{"className":51553,"style":20194},[981],[61,51555,2419],{"className":51556},[19382],[61,51558],{"className":51559,"style":20194},[981],[61,51561,51563,51566],{"className":51562},[116],[61,51564,198],{"className":51565},[116],[61,51567,51569],{"className":51568},[125],[61,51570,51572],{"className":51571},[129],[61,51573,51575],{"className":51574},[134],[61,51576,51578],{"className":51577,"style":51538},[138],[61,51579,51580,51583],{"style":51541},[61,51581],{"className":51582,"style":147},[146],[61,51584,51586],{"className":51585},[151,152,153,154],[61,51587,337],{"className":51588},[116,154],[61,51590,51592,51595],{"style":51591},"top:-3.0223em;",[61,51593],{"className":51594,"style":51515},[146],[61,51596,51599],{"className":51597,"style":51598},[28399],"min-width:1.02em;height:1.28em;",[28402,51600,51603],{"xmlns":28404,"width":28405,"height":51601,"viewBox":51602,"preserveAspectRatio":28408},"1.28em","0 0 400000 1296",[28410,51604],{"d":51605},"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",[61,51607,170],{"className":51608},[169],[61,51610,51612],{"className":51611},[134],[61,51613,51616],{"className":51614,"style":51615},[138],"height:0.1777em;",[61,51617],{},[12,51619,51620],{},"is the generalized Rabi angular frequency. The amplitude coefficient",[61,51622,51624],{"className":51623},[26354],[61,51625,51627,51655],{"className":51626},[64],[61,51628,51630],{"className":51629},[68],[70,51631,51632],{"xmlns":72,"display":26363},[74,51633,51634,51652],{},[77,51635,51636],{},[19134,51637,51638,51644],{},[19151,51639,51640,51642],{},[83,51641,283],{"mathvariant":197},[1493,51643,337],{},[34480,51645,51646,51648,51650],{},[83,51647,283],{"mathvariant":197},[83,51649,27125],{},[1493,51651,337],{},[95,51653,51654],{"encoding":97},"\\frac{\\Omega^2}{\\Omega_R^2}",[61,51656,51658],{"className":51657,"ariaHidden":103},[102],[61,51659,51661,51664],{"className":51660},[107],[61,51662],{"className":51663,"style":51080},[111],[61,51665,51667,51670,51797],{"className":51666},[116],[61,51668],{"className":51669},[631,19286],[61,51671,51673],{"className":51672},[19134],[61,51674,51676,51789],{"className":51675},[129,130],[61,51677,51679,51786],{"className":51678},[134],[61,51680,51682,51741,51749],{"className":51681,"style":51099},[138],[61,51683,51684,51687],{"style":28925},[61,51685],{"className":51686,"style":19306},[146],[61,51688,51690],{"className":51689},[116],[61,51691,51693,51696],{"className":51692},[116],[61,51694,283],{"className":51695},[116],[61,51697,51699],{"className":51698},[125],[61,51700,51702,51733],{"className":51701},[129,130],[61,51703,51705,51730],{"className":51704},[134],[61,51706,51708,51719],{"className":51707,"style":51126},[138],[61,51709,51710,51713],{"style":51129},[61,51711],{"className":51712,"style":147},[146],[61,51714,51716],{"className":51715},[151,152,153,154],[61,51717,27125],{"className":51718,"style":30082},[116,120,154],[61,51720,51721,51724],{"style":51141},[61,51722],{"className":51723,"style":147},[146],[61,51725,51727],{"className":51726},[151,152,153,154],[61,51728,337],{"className":51729},[116,154],[61,51731,170],{"className":51732},[169],[61,51734,51736],{"className":51735},[134],[61,51737,51739],{"className":51738,"style":51160},[138],[61,51740],{},[61,51742,51743,51746],{"style":19352},[61,51744],{"className":51745,"style":19306},[146],[61,51747],{"className":51748,"style":19360},[19359],[61,51750,51751,51754],{"style":28945},[61,51752],{"className":51753,"style":19306},[146],[61,51755,51757],{"className":51756},[116],[61,51758,51760,51763],{"className":51759},[116],[61,51761,283],{"className":51762},[116],[61,51764,51766],{"className":51765},[125],[61,51767,51769],{"className":51768},[129],[61,51770,51772],{"className":51771},[134],[61,51773,51775],{"className":51774,"style":27160},[138],[61,51776,51777,51780],{"style":21487},[61,51778],{"className":51779,"style":147},[146],[61,51781,51783],{"className":51782},[151,152,153,154],[61,51784,337],{"className":51785},[116,154],[61,51787,170],{"className":51788},[169],[61,51790,51792],{"className":51791},[134],[61,51793,51795],{"className":51794,"style":51217},[138],[61,51796],{},[61,51798],{"className":51799},[730,19286],[12,51801,51802,51803,51854],{},"sets the maximum possible population transfer. At exact resonance, ",[61,51804,51806,51824],{"className":51805},[64],[61,51807,51809],{"className":51808},[68],[70,51810,51811],{"xmlns":72},[74,51812,51813,51821],{},[77,51814,51815,51817,51819],{},[83,51816,198],{"mathvariant":197},[586,51818,2000],{},[1493,51820,1495],{},[95,51822,51823],{"encoding":97},"\\Delta=0",[61,51825,51827,51845],{"className":51826,"ariaHidden":103},[102],[61,51828,51830,51833,51836,51839,51842],{"className":51829},[107],[61,51831],{"className":51832,"style":296},[111],[61,51834,198],{"className":51835},[116],[61,51837],{"className":51838,"style":19197},[981],[61,51840,2000],{"className":51841},[19201],[61,51843],{"className":51844,"style":19197},[981],[61,51846,51848,51851],{"className":51847},[107],[61,51849],{"className":51850,"style":27780},[111],[61,51852,1495],{"className":51853},[116],", so",[61,51856,51858],{"className":51857},[26354],[61,51859,51861,51893],{"className":51860},[64],[61,51862,51864],{"className":51863},[68],[70,51865,51866],{"xmlns":72,"display":26363},[74,51867,51868,51890],{},[77,51869,51870,51886,51888],{},[19134,51871,51872,51878],{},[19151,51873,51874,51876],{},[83,51875,283],{"mathvariant":197},[1493,51877,337],{},[34480,51879,51880,51882,51884],{},[83,51881,283],{"mathvariant":197},[83,51883,27125],{},[1493,51885,337],{},[586,51887,2000],{},[1493,51889,48600],{},[95,51891,51892],{"encoding":97},"\\frac{\\Omega^2}{\\Omega_R^2}=1.",[61,51894,51896,52047],{"className":51895,"ariaHidden":103},[102],[61,51897,51899,51902,52038,52041,52044],{"className":51898},[107],[61,51900],{"className":51901,"style":51080},[111],[61,51903,51905,51908,52035],{"className":51904},[116],[61,51906],{"className":51907},[631,19286],[61,51909,51911],{"className":51910},[19134],[61,51912,51914,52027],{"className":51913},[129,130],[61,51915,51917,52024],{"className":51916},[134],[61,51918,51920,51979,51987],{"className":51919,"style":51099},[138],[61,51921,51922,51925],{"style":28925},[61,51923],{"className":51924,"style":19306},[146],[61,51926,51928],{"className":51927},[116],[61,51929,51931,51934],{"className":51930},[116],[61,51932,283],{"className":51933},[116],[61,51935,51937],{"className":51936},[125],[61,51938,51940,51971],{"className":51939},[129,130],[61,51941,51943,51968],{"className":51942},[134],[61,51944,51946,51957],{"className":51945,"style":51126},[138],[61,51947,51948,51951],{"style":51129},[61,51949],{"className":51950,"style":147},[146],[61,51952,51954],{"className":51953},[151,152,153,154],[61,51955,27125],{"className":51956,"style":30082},[116,120,154],[61,51958,51959,51962],{"style":51141},[61,51960],{"className":51961,"style":147},[146],[61,51963,51965],{"className":51964},[151,152,153,154],[61,51966,337],{"className":51967},[116,154],[61,51969,170],{"className":51970},[169],[61,51972,51974],{"className":51973},[134],[61,51975,51977],{"className":51976,"style":51160},[138],[61,51978],{},[61,51980,51981,51984],{"style":19352},[61,51982],{"className":51983,"style":19306},[146],[61,51985],{"className":51986,"style":19360},[19359],[61,51988,51989,51992],{"style":28945},[61,51990],{"className":51991,"style":19306},[146],[61,51993,51995],{"className":51994},[116],[61,51996,51998,52001],{"className":51997},[116],[61,51999,283],{"className":52000},[116],[61,52002,52004],{"className":52003},[125],[61,52005,52007],{"className":52006},[129],[61,52008,52010],{"className":52009},[134],[61,52011,52013],{"className":52012,"style":27160},[138],[61,52014,52015,52018],{"style":21487},[61,52016],{"className":52017,"style":147},[146],[61,52019,52021],{"className":52020},[151,152,153,154],[61,52022,337],{"className":52023},[116,154],[61,52025,170],{"className":52026},[169],[61,52028,52030],{"className":52029},[134],[61,52031,52033],{"className":52032,"style":51217},[138],[61,52034],{},[61,52036],{"className":52037},[730,19286],[61,52039],{"className":52040,"style":19197},[981],[61,52042,2000],{"className":52043},[19201],[61,52045],{"className":52046,"style":19197},[981],[61,52048,52050,52053],{"className":52049},[107],[61,52051],{"className":52052,"style":27780},[111],[61,52054,48600],{"className":52055},[116],[12,52057,52058],{},"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.",[12,52060,52061],{},"At resonance, the excited-state probability simplifies to",[61,52063,52065],{"className":52064},[26354],[61,52066,52068,52122],{"className":52067},[64],[61,52069,52071],{"className":52070},[68],[70,52072,52073],{"xmlns":72,"display":26363},[74,52074,52075,52119],{},[77,52076,52077,52083,52085,52087,52089,52091,52101,52117],{},[80,52078,52079,52081],{},[83,52080,45957],{},[83,52082,21378],{},[586,52084,849],{"stretchy":588},[83,52086,45968],{},[586,52088,300],{"stretchy":588},[586,52090,2000],{},[19151,52092,52093,52099],{},[77,52094,52095,52097],{},[83,52096,5069],{},[586,52098,19125],{},[1493,52100,337],{},[77,52102,52103,52105,52115],{},[586,52104,849],{"fence":103},[19134,52106,52107,52113],{},[77,52108,52109,52111],{},[83,52110,283],{"mathvariant":197},[83,52112,45968],{},[1493,52114,337],{},[586,52116,300],{"fence":103},[83,52118,332],{"mathvariant":197},[95,52120,52121],{"encoding":97},"P_e(\\tau)=\\sin^2\\left(\\frac{\\Omega\\tau}{2}\\right).",[61,52123,52125,52189],{"className":52124,"ariaHidden":103},[102],[61,52126,52128,52131,52171,52174,52177,52180,52183,52186],{"className":52127},[107],[61,52129],{"className":52130,"style":627},[111],[61,52132,52134,52137],{"className":52133},[116],[61,52135,45957],{"className":52136,"style":23700},[116,120],[61,52138,52140],{"className":52139},[125],[61,52141,52143,52163],{"className":52142},[129,130],[61,52144,52146,52160],{"className":52145},[134],[61,52147,52149],{"className":52148,"style":20164},[138],[61,52150,52151,52154],{"style":28699},[61,52152],{"className":52153,"style":147},[146],[61,52155,52157],{"className":52156},[151,152,153,154],[61,52158,21378],{"className":52159},[116,120,154],[61,52161,170],{"className":52162},[169],[61,52164,52166],{"className":52165},[134],[61,52167,52169],{"className":52168,"style":177},[138],[61,52170],{},[61,52172,849],{"className":52173},[631],[61,52175,45968],{"className":52176,"style":46040},[116,120],[61,52178,300],{"className":52179},[730],[61,52181],{"className":52182,"style":19197},[981],[61,52184,2000],{"className":52185},[19201],[61,52187],{"className":52188,"style":19197},[981],[61,52190,52192,52195,52224,52227,52307,52310],{"className":52191},[107],[61,52193],{"className":52194,"style":26592},[111],[61,52196,52198,52201],{"className":52197},[19259],[61,52199,5069],{"className":52200},[19259],[61,52202,52204],{"className":52203},[125],[61,52205,52207],{"className":52206},[129],[61,52208,52210],{"className":52209},[134],[61,52211,52213],{"className":52212,"style":51244},[138],[61,52214,52215,52218],{"style":51247},[61,52216],{"className":52217,"style":147},[146],[61,52219,52221],{"className":52220},[151,152,153,154],[61,52222,337],{"className":52223},[116,154],[61,52225],{"className":52226,"style":982},[981],[61,52228,52230,52236,52301],{"className":52229},[19266],[61,52231,52233],{"className":52232,"style":19271},[631,19270],[61,52234,849],{"className":52235},[19275,153],[61,52237,52239,52242,52298],{"className":52238},[116],[61,52240],{"className":52241},[631,19286],[61,52243,52245],{"className":52244},[19134],[61,52246,52248,52290],{"className":52247},[129,130],[61,52249,52251,52287],{"className":52250},[134],[61,52252,52254,52265,52273],{"className":52253,"style":51287},[138],[61,52255,52256,52259],{"style":28925},[61,52257],{"className":52258,"style":19306},[146],[61,52260,52262],{"className":52261},[116],[61,52263,337],{"className":52264},[116],[61,52266,52267,52270],{"style":19352},[61,52268],{"className":52269,"style":19306},[146],[61,52271],{"className":52272,"style":19360},[19359],[61,52274,52275,52278],{"style":28945},[61,52276],{"className":52277,"style":19306},[146],[61,52279,52281,52284],{"className":52280},[116],[61,52282,283],{"className":52283},[116],[61,52285,45968],{"className":52286,"style":46040},[116,120],[61,52288,170],{"className":52289},[169],[61,52291,52293],{"className":52292},[134],[61,52294,52296],{"className":52295,"style":29011},[138],[61,52297],{},[61,52299],{"className":52300},[730,19286],[61,52302,52304],{"className":52303,"style":19271},[730,19270],[61,52305,300],{"className":52306},[19275,153],[61,52308],{"className":52309,"style":982},[981],[61,52311,332],{"className":52312},[116],[12,52314,52315,52316,52344],{},"The pulse duration required for ideal population inversion defines the standard ",[61,52317,52319,52332],{"className":52318},[64],[61,52320,52322],{"className":52321},[68],[70,52323,52324],{"xmlns":72},[74,52325,52326,52330],{},[77,52327,52328],{},[83,52329,19792],{},[95,52331,30267],{"encoding":97},[61,52333,52335],{"className":52334,"ariaHidden":103},[102],[61,52336,52338,52341],{"className":52337},[107],[61,52339],{"className":52340,"style":30277},[111],[61,52342,19792],{"className":52343,"style":121},[116,120],"-pulse time,",[61,52346,52348],{"className":52347},[26354],[61,52349,52351,52401],{"className":52350},[64],[61,52352,52354],{"className":52353},[68],[70,52355,52356],{"xmlns":72,"display":26363},[74,52357,52358,52398],{},[77,52359,52360,52366,52368,52374,52376,52396],{},[80,52361,52362,52364],{},[83,52363,45968],{},[83,52365,19792],{},[586,52367,2000],{},[19134,52369,52370,52372],{},[83,52371,19792],{},[83,52373,283],{"mathvariant":197},[586,52375,2000],{},[19134,52377,52378,52380],{},[1493,52379,57],{},[77,52381,52382,52384,52386,52388,52390,52392,52394],{},[1493,52383,337],{},[586,52385,849],{"stretchy":588},[83,52387,283],{"mathvariant":197},[83,52389,602],{"mathvariant":197},[1493,52391,337],{},[83,52393,19792],{},[586,52395,300],{"stretchy":588},[83,52397,332],{"mathvariant":197},[95,52399,52400],{"encoding":97},"\\tau_\\pi=\\frac{\\pi}{\\Omega}=\\frac{1}{2(\\Omega\u002F2\\pi)}.",[61,52402,52404,52460,52539],{"className":52403,"ariaHidden":103},[102],[61,52405,52407,52410,52451,52454,52457],{"className":52406},[107],[61,52408],{"className":52409,"style":112},[111],[61,52411,52413,52416],{"className":52412},[116],[61,52414,45968],{"className":52415,"style":46040},[116,120],[61,52417,52419],{"className":52418},[125],[61,52420,52422,52443],{"className":52421},[129,130],[61,52423,52425,52440],{"className":52424},[134],[61,52426,52428],{"className":52427,"style":20164},[138],[61,52429,52431,52434],{"style":52430},"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;",[61,52432],{"className":52433,"style":147},[146],[61,52435,52437],{"className":52436},[151,152,153,154],[61,52438,19792],{"className":52439,"style":121},[116,120,154],[61,52441,170],{"className":52442},[169],[61,52444,52446],{"className":52445},[134],[61,52447,52449],{"className":52448,"style":177},[138],[61,52450],{},[61,52452],{"className":52453,"style":19197},[981],[61,52455,2000],{"className":52456},[19201],[61,52458],{"className":52459,"style":19197},[981],[61,52461,52463,52467,52530,52533,52536],{"className":52462},[107],[61,52464],{"className":52465,"style":52466},[111],"height:1.7936em;vertical-align:-0.686em;",[61,52468,52470,52473,52527],{"className":52469},[116],[61,52471],{"className":52472},[631,19286],[61,52474,52476],{"className":52475},[19134],[61,52477,52479,52519],{"className":52478},[129,130],[61,52480,52482,52516],{"className":52481},[134],[61,52483,52486,52497,52505],{"className":52484,"style":52485},[138],"height:1.1076em;",[61,52487,52488,52491],{"style":28925},[61,52489],{"className":52490,"style":19306},[146],[61,52492,52494],{"className":52493},[116],[61,52495,283],{"className":52496},[116],[61,52498,52499,52502],{"style":19352},[61,52500],{"className":52501,"style":19306},[146],[61,52503],{"className":52504,"style":19360},[19359],[61,52506,52507,52510],{"style":28945},[61,52508],{"className":52509,"style":19306},[146],[61,52511,52513],{"className":52512},[116],[61,52514,19792],{"className":52515,"style":121},[116,120],[61,52517,170],{"className":52518},[169],[61,52520,52522],{"className":52521},[134],[61,52523,52525],{"className":52524,"style":29011},[138],[61,52526],{},[61,52528],{"className":52529},[730,19286],[61,52531],{"className":52532,"style":19197},[981],[61,52534,2000],{"className":52535},[19201],[61,52537],{"className":52538,"style":19197},[981],[61,52540,52542,52546,52622],{"className":52541},[107],[61,52543],{"className":52544,"style":52545},[111],"height:2.2574em;vertical-align:-0.936em;",[61,52547,52549,52552,52619],{"className":52548},[116],[61,52550],{"className":52551},[631,19286],[61,52553,52555],{"className":52554},[19134],[61,52556,52558,52610],{"className":52557},[129,130],[61,52559,52561,52607],{"className":52560},[134],[61,52562,52565,52588,52596],{"className":52563,"style":52564},[138],"height:1.3214em;",[61,52566,52567,52570],{"style":28925},[61,52568],{"className":52569,"style":19306},[146],[61,52571,52573,52576,52579,52582,52585],{"className":52572},[116],[61,52574,337],{"className":52575},[116],[61,52577,849],{"className":52578},[631],[61,52580,46977],{"className":52581},[116],[61,52583,19792],{"className":52584,"style":121},[116,120],[61,52586,300],{"className":52587},[730],[61,52589,52590,52593],{"style":19352},[61,52591],{"className":52592,"style":19306},[146],[61,52594],{"className":52595,"style":19360},[19359],[61,52597,52598,52601],{"style":28945},[61,52599],{"className":52600,"style":19306},[146],[61,52602,52604],{"className":52603},[116],[61,52605,57],{"className":52606},[116],[61,52608,170],{"className":52609},[169],[61,52611,52613],{"className":52612},[134],[61,52614,52617],{"className":52615,"style":52616},[138],"height:0.936em;",[61,52618],{},[61,52620],{"className":52621},[730,19286],[61,52623,332],{"className":52624},[116],[12,52626,52627],{},"Given the default drive strength",[61,52629,52631],{"className":52630},[26354],[61,52632,52634,52673],{"className":52633},[64],[61,52635,52637],{"className":52636},[68],[70,52638,52639],{"xmlns":72,"display":26363},[74,52640,52641,52670],{},[77,52642,52643,52653,52655,52657,52659,52668],{},[19134,52644,52645,52647],{},[83,52646,283],{"mathvariant":197},[77,52648,52649,52651],{},[1493,52650,337],{},[83,52652,19792],{},[586,52654,2000],{},[1493,52656,14756],{},[27272,52658,27274],{},[77,52660,52661,52664,52666],{},[83,52662,52663],{"mathvariant":197},"M",[83,52665,6871],{"mathvariant":197},[83,52667,26482],{"mathvariant":197},[586,52669,862],{"separator":103},[95,52671,52672],{"encoding":97},"\\frac{\\Omega}{2\\pi}=20~\\mathrm{MHz},",[61,52674,52676,52757],{"className":52675,"ariaHidden":103},[102],[61,52677,52679,52683,52748,52751,52754],{"className":52678},[107],[61,52680],{"className":52681,"style":52682},[111],"height:2.0463em;vertical-align:-0.686em;",[61,52684,52686,52689,52745],{"className":52685},[116],[61,52687],{"className":52688},[631,19286],[61,52690,52692],{"className":52691},[19134],[61,52693,52695,52737],{"className":52694},[129,130],[61,52696,52698,52734],{"className":52697},[134],[61,52699,52701,52715,52723],{"className":52700,"style":51287},[138],[61,52702,52703,52706],{"style":28925},[61,52704],{"className":52705,"style":19306},[146],[61,52707,52709,52712],{"className":52708},[116],[61,52710,337],{"className":52711},[116],[61,52713,19792],{"className":52714,"style":121},[116,120],[61,52716,52717,52720],{"style":19352},[61,52718],{"className":52719,"style":19306},[146],[61,52721],{"className":52722,"style":19360},[19359],[61,52724,52725,52728],{"style":28945},[61,52726],{"className":52727,"style":19306},[146],[61,52729,52731],{"className":52730},[116],[61,52732,283],{"className":52733},[116],[61,52735,170],{"className":52736},[169],[61,52738,52740],{"className":52739},[134],[61,52741,52743],{"className":52742,"style":29011},[138],[61,52744],{},[61,52746],{"className":52747},[730,19286],[61,52749],{"className":52750,"style":19197},[981],[61,52752,2000],{"className":52753},[19201],[61,52755],{"className":52756,"style":19197},[981],[61,52758,52760,52763,52766,52769,52775],{"className":52759},[107],[61,52761],{"className":52762,"style":33289},[111],[61,52764,14756],{"className":52765},[116],[61,52767,27274],{"className":52768},[981,48148],[61,52770,52772],{"className":52771},[116],[61,52773,3772],{"className":52774},[116,27529],[61,52776,862],{"className":52777},[977],[12,52779,52780,52781,52809],{},"the ideal ",[61,52782,52784,52797],{"className":52783},[64],[61,52785,52787],{"className":52786},[68],[70,52788,52789],{"xmlns":72},[74,52790,52791,52795],{},[77,52792,52793],{},[83,52794,19792],{},[95,52796,30267],{"encoding":97},[61,52798,52800],{"className":52799,"ariaHidden":103},[102],[61,52801,52803,52806],{"className":52802},[107],[61,52804],{"className":52805,"style":30277},[111],[61,52807,19792],{"className":52808,"style":121},[116,120],"-pulse time is",[61,52811,52813],{"className":52812},[26354],[61,52814,52816,52850],{"className":52815},[64],[61,52817,52819],{"className":52818},[68],[70,52820,52821],{"xmlns":72,"display":26363},[74,52822,52823,52847],{},[77,52824,52825,52831,52833,52836,52838,52845],{},[80,52826,52827,52829],{},[83,52828,45968],{},[83,52830,19792],{},[586,52832,2000],{},[1493,52834,52835],{},"25",[27272,52837,27274],{},[77,52839,52840,52842],{},[83,52841,27948],{"mathvariant":197},[83,52843,52844],{"mathvariant":197},"s",[83,52846,332],{"mathvariant":197},[95,52848,52849],{"encoding":97},"\\tau_\\pi=25~\\mathrm{ns}.",[61,52851,52853,52908],{"className":52852,"ariaHidden":103},[102],[61,52854,52856,52859,52899,52902,52905],{"className":52855},[107],[61,52857],{"className":52858,"style":112},[111],[61,52860,52862,52865],{"className":52861},[116],[61,52863,45968],{"className":52864,"style":46040},[116,120],[61,52866,52868],{"className":52867},[125],[61,52869,52871,52891],{"className":52870},[129,130],[61,52872,52874,52888],{"className":52873},[134],[61,52875,52877],{"className":52876,"style":20164},[138],[61,52878,52879,52882],{"style":52430},[61,52880],{"className":52881,"style":147},[146],[61,52883,52885],{"className":52884},[151,152,153,154],[61,52886,19792],{"className":52887,"style":121},[116,120,154],[61,52889,170],{"className":52890},[169],[61,52892,52894],{"className":52893},[134],[61,52895,52897],{"className":52896,"style":177},[138],[61,52898],{},[61,52900],{"className":52901,"style":19197},[981],[61,52903,2000],{"className":52904},[19201],[61,52906],{"className":52907,"style":19197},[981],[61,52909,52911,52914,52917,52920,52927],{"className":52910},[107],[61,52912],{"className":52913,"style":27780},[111],[61,52915,52835],{"className":52916},[116],[61,52918,27274],{"className":52919},[981,48148],[61,52921,52923],{"className":52922},[116],[61,52924,52926],{"className":52925},[116,27529],"ns",[61,52928,332],{"className":52929},[116],[12,52931,52932],{},"A full period of the resonant excited-state probability oscillation is",[61,52934,52936],{"className":52935},[26354],[61,52937,52939,52987],{"className":52938},[64],[61,52940,52942],{"className":52941},[68],[70,52943,52944],{"xmlns":72,"display":26363},[74,52945,52946,52984],{},[77,52947,52948,52954,52956,52966,52968,52982],{},[80,52949,52950,52952],{},[83,52951,23559],{},[83,52953,27125],{},[586,52955,2000],{},[19134,52957,52958,52964],{},[77,52959,52960,52962],{},[1493,52961,337],{},[83,52963,19792],{},[83,52965,283],{"mathvariant":197},[586,52967,2000],{},[19134,52969,52970,52972],{},[1493,52971,57],{},[77,52973,52974,52976,52978,52980],{},[83,52975,283],{"mathvariant":197},[83,52977,602],{"mathvariant":197},[1493,52979,337],{},[83,52981,19792],{},[83,52983,332],{"mathvariant":197},[95,52985,52986],{"encoding":97},"T_R=\\frac{2\\pi}{\\Omega}=\\frac{1}{\\Omega\u002F2\\pi}.",[61,52988,52990,53045,53126],{"className":52989,"ariaHidden":103},[102],[61,52991,52993,52996,53036,53039,53042],{"className":52992},[107],[61,52994],{"className":52995,"style":219},[111],[61,52997,52999,53002],{"className":52998},[116],[61,53000,23559],{"className":53001,"style":23700},[116,120],[61,53003,53005],{"className":53004},[125],[61,53006,53008,53028],{"className":53007},[129,130],[61,53009,53011,53025],{"className":53010},[134],[61,53012,53014],{"className":53013,"style":139},[138],[61,53015,53016,53019],{"style":28699},[61,53017],{"className":53018,"style":147},[146],[61,53020,53022],{"className":53021},[151,152,153,154],[61,53023,27125],{"className":53024,"style":30082},[116,120,154],[61,53026,170],{"className":53027},[169],[61,53029,53031],{"className":53030},[134],[61,53032,53034],{"className":53033,"style":177},[138],[61,53035],{},[61,53037],{"className":53038,"style":19197},[981],[61,53040,2000],{"className":53041},[19201],[61,53043],{"className":53044,"style":19197},[981],[61,53046,53048,53052,53117,53120,53123],{"className":53047},[107],[61,53049],{"className":53050,"style":53051},[111],"height:2.0074em;vertical-align:-0.686em;",[61,53053,53055,53058,53114],{"className":53054},[116],[61,53056],{"className":53057},[631,19286],[61,53059,53061],{"className":53060},[19134],[61,53062,53064,53106],{"className":53063},[129,130],[61,53065,53067,53103],{"className":53066},[134],[61,53068,53070,53081,53089],{"className":53069,"style":52564},[138],[61,53071,53072,53075],{"style":28925},[61,53073],{"className":53074,"style":19306},[146],[61,53076,53078],{"className":53077},[116],[61,53079,283],{"className":53080},[116],[61,53082,53083,53086],{"style":19352},[61,53084],{"className":53085,"style":19306},[146],[61,53087],{"className":53088,"style":19360},[19359],[61,53090,53091,53094],{"style":28945},[61,53092],{"className":53093,"style":19306},[146],[61,53095,53097,53100],{"className":53096},[116],[61,53098,337],{"className":53099},[116],[61,53101,19792],{"className":53102,"style":121},[116,120],[61,53104,170],{"className":53105},[169],[61,53107,53109],{"className":53108},[134],[61,53110,53112],{"className":53111,"style":29011},[138],[61,53113],{},[61,53115],{"className":53116},[730,19286],[61,53118],{"className":53119,"style":19197},[981],[61,53121,2000],{"className":53122},[19201],[61,53124],{"className":53125,"style":19197},[981],[61,53127,53129,53132,53197],{"className":53128},[107],[61,53130],{"className":53131,"style":52545},[111],[61,53133,53135,53138,53194],{"className":53134},[116],[61,53136],{"className":53137},[631,19286],[61,53139,53141],{"className":53140},[19134],[61,53142,53144,53186],{"className":53143},[129,130],[61,53145,53147,53183],{"className":53146},[134],[61,53148,53150,53164,53172],{"className":53149,"style":52564},[138],[61,53151,53152,53155],{"style":28925},[61,53153],{"className":53154,"style":19306},[146],[61,53156,53158,53161],{"className":53157},[116],[61,53159,46977],{"className":53160},[116],[61,53162,19792],{"className":53163,"style":121},[116,120],[61,53165,53166,53169],{"style":19352},[61,53167],{"className":53168,"style":19306},[146],[61,53170],{"className":53171,"style":19360},[19359],[61,53173,53174,53177],{"style":28945},[61,53175],{"className":53176,"style":19306},[146],[61,53178,53180],{"className":53179},[116],[61,53181,57],{"className":53182},[116],[61,53184,170],{"className":53185},[169],[61,53187,53189],{"className":53188},[134],[61,53190,53192],{"className":53191,"style":52616},[138],[61,53193],{},[61,53195],{"className":53196},[730,19286],[61,53198,332],{"className":53199},[116],[12,53201,53202],{},"For the same default drive strength, the full temporal period is",[61,53204,53206],{"className":53205},[26354],[61,53207,53209,53242],{"className":53208},[64],[61,53210,53212],{"className":53211},[68],[70,53213,53214],{"xmlns":72,"display":26363},[74,53215,53216,53239],{},[77,53217,53218,53224,53226,53229,53231,53237],{},[80,53219,53220,53222],{},[83,53221,23559],{},[83,53223,27125],{},[586,53225,2000],{},[1493,53227,53228],{},"50",[27272,53230,27274],{},[77,53232,53233,53235],{},[83,53234,27948],{"mathvariant":197},[83,53236,52844],{"mathvariant":197},[83,53238,332],{"mathvariant":197},[95,53240,53241],{"encoding":97},"T_R=50~\\mathrm{ns}.",[61,53243,53245,53300],{"className":53244,"ariaHidden":103},[102],[61,53246,53248,53251,53291,53294,53297],{"className":53247},[107],[61,53249],{"className":53250,"style":219},[111],[61,53252,53254,53257],{"className":53253},[116],[61,53255,23559],{"className":53256,"style":23700},[116,120],[61,53258,53260],{"className":53259},[125],[61,53261,53263,53283],{"className":53262},[129,130],[61,53264,53266,53280],{"className":53265},[134],[61,53267,53269],{"className":53268,"style":139},[138],[61,53270,53271,53274],{"style":28699},[61,53272],{"className":53273,"style":147},[146],[61,53275,53277],{"className":53276},[151,152,153,154],[61,53278,27125],{"className":53279,"style":30082},[116,120,154],[61,53281,170],{"className":53282},[169],[61,53284,53286],{"className":53285},[134],[61,53287,53289],{"className":53288,"style":177},[138],[61,53290],{},[61,53292],{"className":53293,"style":19197},[981],[61,53295,2000],{"className":53296},[19201],[61,53298],{"className":53299,"style":19197},[981],[61,53301,53303,53306,53309,53312,53318],{"className":53302},[107],[61,53304],{"className":53305,"style":27780},[111],[61,53307,53228],{"className":53308},[116],[61,53310,27274],{"className":53311},[981,48148],[61,53313,53315],{"className":53314},[116],[61,53316,52926],{"className":53317},[116,27529],[61,53319,332],{"className":53320},[116],[25,53322,53324],{"id":53323},"phenomenological-dephasing","Phenomenological Dephasing",[12,53326,53327],{},"The computational model includes an optional phenomenological damping factor,",[61,53329,53331],{"className":53330},[26354],[61,53332,53334,53406],{"className":53333},[64],[61,53335,53337],{"className":53336},[68],[70,53338,53339],{"xmlns":72,"display":26363},[74,53340,53341,53403],{},[77,53342,53343,53349,53351,53353,53355,53357,53359,53361,53367,53369,53371,53373,53375,53377,53379,53381,53401],{},[80,53344,53345,53347],{},[83,53346,45957],{},[83,53348,21378],{},[586,53350,849],{"stretchy":588},[83,53352,19797],{},[586,53354,862],{"separator":103},[83,53356,45968],{},[586,53358,300],{"stretchy":588},[586,53360,21354],{},[80,53362,53363,53365],{},[83,53364,45957],{},[83,53366,21378],{},[586,53368,849],{"stretchy":588},[83,53370,19797],{},[586,53372,862],{"separator":103},[83,53374,45968],{},[586,53376,300],{"stretchy":588},[83,53378,6987],{},[586,53380,19125],{},[77,53382,53383,53385,53387,53399],{},[586,53384,849],{"fence":103},[586,53386,19132],{},[19134,53388,53389,53391],{},[83,53390,45968],{},[34480,53392,53393,53395,53397],{},[83,53394,23559],{},[1493,53396,337],{},[586,53398,47495],{},[586,53400,300],{"fence":103},[83,53402,332],{"mathvariant":197},[95,53404,53405],{"encoding":97},"P_e(f,\\tau)\\rightarrow P_e(f,\\tau)\\exp\\left(-\\frac{\\tau}{T_2^\\ast}\\right).",[61,53407,53409,53482],{"className":53408,"ariaHidden":103},[102],[61,53410,53412,53415,53455,53458,53461,53464,53467,53470,53473,53476,53479],{"className":53411},[107],[61,53413],{"className":53414,"style":627},[111],[61,53416,53418,53421],{"className":53417},[116],[61,53419,45957],{"className":53420,"style":23700},[116,120],[61,53422,53424],{"className":53423},[125],[61,53425,53427,53447],{"className":53426},[129,130],[61,53428,53430,53444],{"className":53429},[134],[61,53431,53433],{"className":53432,"style":20164},[138],[61,53434,53435,53438],{"style":28699},[61,53436],{"className":53437,"style":147},[146],[61,53439,53441],{"className":53440},[151,152,153,154],[61,53442,21378],{"className":53443},[116,120,154],[61,53445,170],{"className":53446},[169],[61,53448,53450],{"className":53449},[134],[61,53451,53453],{"className":53452,"style":177},[138],[61,53454],{},[61,53456,849],{"className":53457},[631],[61,53459,19797],{"className":53460,"style":20151},[116,120],[61,53462,862],{"className":53463},[977],[61,53465],{"className":53466,"style":982},[981],[61,53468,45968],{"className":53469,"style":46040},[116,120],[61,53471,300],{"className":53472},[730],[61,53474],{"className":53475,"style":19197},[981],[61,53477,21354],{"className":53478},[19201],[61,53480],{"className":53481,"style":19197},[981],[61,53483,53485,53489,53529,53532,53535,53538,53541,53544,53547,53550,53553,53556,53688,53691],{"className":53484},[107],[61,53486],{"className":53487,"style":53488},[111],"height:2.4023em;vertical-align:-0.9523em;",[61,53490,53492,53495],{"className":53491},[116],[61,53493,45957],{"className":53494,"style":23700},[116,120],[61,53496,53498],{"className":53497},[125],[61,53499,53501,53521],{"className":53500},[129,130],[61,53502,53504,53518],{"className":53503},[134],[61,53505,53507],{"className":53506,"style":20164},[138],[61,53508,53509,53512],{"style":28699},[61,53510],{"className":53511,"style":147},[146],[61,53513,53515],{"className":53514},[151,152,153,154],[61,53516,21378],{"className":53517},[116,120,154],[61,53519,170],{"className":53520},[169],[61,53522,53524],{"className":53523},[134],[61,53525,53527],{"className":53526,"style":177},[138],[61,53528],{},[61,53530,849],{"className":53531},[631],[61,53533,19797],{"className":53534,"style":20151},[116,120],[61,53536,862],{"className":53537},[977],[61,53539],{"className":53540,"style":982},[981],[61,53542,45968],{"className":53543,"style":46040},[116,120],[61,53545,300],{"className":53546},[730],[61,53548],{"className":53549,"style":982},[981],[61,53551,6987],{"className":53552},[19259],[61,53554],{"className":53555,"style":982},[981],[61,53557,53559,53565,53568,53682],{"className":53558},[19266],[61,53560,53562],{"className":53561,"style":19271},[631,19270],[61,53563,849],{"className":53564},[19275,153],[61,53566,19132],{"className":53567},[116],[61,53569,53571,53574,53679],{"className":53570},[116],[61,53572],{"className":53573},[631,19286],[61,53575,53577],{"className":53576},[19134],[61,53578,53580,53670],{"className":53579},[129,130],[61,53581,53583,53667],{"className":53582},[134],[61,53584,53586,53648,53656],{"className":53585,"style":52485},[138],[61,53587,53588,53591],{"style":28925},[61,53589],{"className":53590,"style":19306},[146],[61,53592,53594],{"className":53593},[116],[61,53595,53597,53600],{"className":53596},[116],[61,53598,23559],{"className":53599,"style":23700},[116,120],[61,53601,53603],{"className":53602},[125],[61,53604,53606,53639],{"className":53605},[129,130],[61,53607,53609,53636],{"className":53608},[134],[61,53610,53613,53625],{"className":53611,"style":53612},[138],"height:0.6705em;",[61,53614,53616,53619],{"style":53615},"top:-2.4337em;margin-left:-0.1389em;margin-right:0.05em;",[61,53617],{"className":53618,"style":147},[146],[61,53620,53622],{"className":53621},[151,152,153,154],[61,53623,337],{"className":53624},[116,154],[61,53626,53627,53630],{"style":51141},[61,53628],{"className":53629,"style":147},[146],[61,53631,53633],{"className":53632},[151,152,153,154],[61,53634,47495],{"className":53635},[19382,154],[61,53637,170],{"className":53638},[169],[61,53640,53642],{"className":53641},[134],[61,53643,53646],{"className":53644,"style":53645},[138],"height:0.2663em;",[61,53647],{},[61,53649,53650,53653],{"style":19352},[61,53651],{"className":53652,"style":19306},[146],[61,53654],{"className":53655,"style":19360},[19359],[61,53657,53658,53661],{"style":28945},[61,53659],{"className":53660,"style":19306},[146],[61,53662,53664],{"className":53663},[116],[61,53665,45968],{"className":53666,"style":46040},[116,120],[61,53668,170],{"className":53669},[169],[61,53671,53673],{"className":53672},[134],[61,53674,53677],{"className":53675,"style":53676},[138],"height:0.9523em;",[61,53678],{},[61,53680],{"className":53681},[730,19286],[61,53683,53685],{"className":53684,"style":19271},[730,19270],[61,53686,300],{"className":53687},[19275,153],[61,53689],{"className":53690,"sty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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 is",[61,53698,53700],{"className":53699},[26354],[61,53701,53703,53738],{"className":53702},[64],[61,53704,53706],{"className":53705},[68],[70,53707,53708],{"xmlns":72,"display":26363},[74,53709,53710,53735],{},[77,53711,53712,53720,53722,53725,53727,53733],{},[34480,53713,53714,53716,53718],{},[83,53715,23559],{},[1493,53717,337],{},[586,53719,47495],{},[586,53721,2000],{},[1493,53723,53724],{},"500",[27272,53726,27274],{},[77,53728,53729,53731],{},[83,53730,27948],{"mathvariant":197},[83,53732,52844],{"mathvariant":197},[83,53734,332],{"mathvariant":197},[95,53736,53737],{"encoding":97},"T_2^\\ast=500~\\mathrm{ns}.",[61,53739,53741,53810],{"className":53740,"ariaHidden":103},[102],[61,53742,53744,53748,53801,53804,53807],{"className":53743},[107],[61,53745],{"className":53746,"style":53747},[111],"height:0.9857em;vertical-align:-0.247em;",[61,53749,53751,53754],{"className":53750},[116],[61,53752,23559],{"className":53753,"style":23700},[116,120],[61,53755,53757],{"className":53756},[125],[61,53758,53760,53793],{"className":53759},[129,130],[61,53761,53763,53790],{"className":53762},[134],[61,53764,53767,53779],{"className":53765,"style":53766},[138],"height:0.7387em;",[61,53768,53770,53773],{"style":53769},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[61,53771],{"className":53772,"style":147},[146],[61,53774,53776],{"className":53775},[151,152,153,154],[61,53777,337],{"className":53778},[116,154],[61,53780,53781,53784],{"style":30098},[61,53782],{"className":53783,"style":147},[146],[61,53785,53787],{"className":53786},[151,152,153,154],[61,53788,47495],{"className":53789},[19382,154],[61,53791,170],{"className":53792},[169],[61,53794,53796],{"className":53795},[134],[61,53797,53799],{"className":53798,"style":34819},[138],[61,53800],{},[61,53802],{"className":53803,"style":19197},[981],[61,53805,2000],{"className":53806},[19201],[61,53808],{"className":53809,"style":19197},[981],[61,53811,53813,53816,53819,53822,53828],{"className":53812},[107],[61,53814],{"className":53815,"style":27780},[111],[61,53817,53724],{"className":53818},[116],[61,53820,27274],{"className":53821},[981,48148],[61,53823,53825],{"className":53824},[116],[61,53826,52926],{"className":53827},[116,27529],[61,53829,332],{"className":53830},[116],[12,53832,53833,53834,53837],{},"Setting ",[1727,53835,53836],{},"T2STAR_NS = None"," disables the phenomenological damping factor.",[25,53839,53841],{"id":53840},"detuning-and-generalized-rabi-frequency","Detuning and Generalized Rabi Frequency",[12,53843,53844],{},"Detuning quantifies the spectral mismatch between the applied drive and the natural qubit 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in ordinary frequency units, the generalized Rabi frequency is",[61,53991,53993],{"className":53992},[26354],[61,53994,53996,54084],{"className":53995},[64],[61,53997,53999],{"className":53998},[68],[70,54000,54001],{"xmlns":72,"display":26363},[74,54002,54003,54081],{},[77,54004,54005,54011,54013,54015,54017,54019,54033,54035,54079],{},[80,54006,54007,54009],{},[83,54008,47231],{},[83,54010,27125],{},[586,54012,849],{"stretchy":588},[83,54014,19797],{},[586,54016,300],{"stretchy":588},[586,54018,2000],{},[19134,54020,54021,54027],{},[80,54022,54023,54025],{},[83,54024,283],{"mathvariant":197},[83,54026,27125],{},[77,54028,54029,54031],{},[1493,54030,337],{},[83,54032,19792],{},[586,54034,2000],{},[28194,54036,54037],{},[77,54038,54039,54059,54061,54063,54065,54067,54073],{},[19151,54040,54041,54057],{},[77,54042,54043,54045,54055],{},[586,54044,849],{"fence":103},[19134,54046,54047,54049],{},[83,54048,283],{"mathvariant":197},[77,54050,54051,54053],{},[1493,54052,337],{},[83,54054,19792],{},[586,54056,300],{"fence":103},[1493,54058,337],{},[586,54060,2419],{},[586,54062,849],{"stretchy":588},[83,54064,19797],{},[586,54066,19132],{},[80,54068,54069,54071],{},[83,54070,19797],{},[1493,54072,1495],{},[19151,54074,54075,54077],{},[586,54076,300],{"stretchy":588},[1493,54078,337],{},[83,54080,332],{"mathvariant":197},[95,54082,54083],{"encoding":97},"\\nu_R(f)=\\frac{\\Omega_R}{2\\pi}=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[61,54085,54087,54151,54268],{"className":54086,"ariaHidden":103},[102],[61,54088,54090,54093,54133,54136,54139,54142,54145,54148],{"className":54089},[107],[61,54091],{"className":54092,"style":627},[111],[61,54094,54096,54099],{"className":54095},[116],[61,54097,47231],{"className":54098,"style":47252},[116,120],[61,54100,54102],{"className":54101},[125],[61,54103,54105,54125],{"className":54104},[129,130],[61,54106,54108,54122],{"className":54107},[134],[61,54109,54111],{"className":54110,"style":139},[138],[61,54112,54113,54116],{"style":47267},[61,54114],{"className":54115,"style":147},[146],[61,54117,54119],{"className":54118},[151,152,153,154],[61,54120,27125],{"className":54121,"style":30082},[116,120,154],[61,54123,170],{"className":54124},[169],[61,54126,54128],{"className":54127},[134],[61,54129,54131],{"className":54130,"style":177},[138],[61,54132],{},[61,54134,849],{"className":54135},[631],[61,54137,19797],{"className":54138,"style":20151},[116,120],[61,54140,300],{"className":54141},[730],[61,54143],{"className":54144,"style":19197},[981],[61,54146,2000],{"className":54147},[19201],[61,54149],{"className":54150,"style":19197},[981],[61,54152,54154,54157,54259,54262,54265],{"className":54153},[107],[61,54155],{"className":54156,"style":52682},[111],[61,54158,54160,54163,54256],{"className":54159},[116],[61,54161],{"className":54162},[631,19286],[61,54164,54166],{"className":54165},[19134],[61,54167,54169,54248],{"className":54168},[129,130],[61,54170,54172,54245],{"className":54171},[134],[61,54173,54175,54189,54197],{"className":54174,"style":51287},[138],[61,54176,54177,54180],{"style":28925},[61,54178],{"className":54179,"style":19306},[146],[61,54181,54183,54186],{"className":54182},[116],[61,54184,337],{"className":54185},[116],[61,54187,19792],{"className":54188,"style":121},[116,120],[61,54190,54191,54194],{"style":19352},[61,54192],{"className":54193,"style":19306},[146],[61,54195],{"className":54196,"style":19360},[19359],[61,54198,54199,54202],{"style":28945},[61,54200],{"className":54201,"style":19306},[146],[61,54203,54205],{"className":54204},[116],[61,54206,54208,54211],{"className":54207},[116],[61,54209,283],{"className":54210},[116],[61,54212,54214],{"className":54213},[125],[61,54215,54217,54237],{"className":54216},[129,130],[61,54218,54220,54234],{"className":54219},[134],[61,54221,54223],{"className":54222,"style":139},[138],[61,54224,54225,54228],{"style":241},[61,54226],{"className":54227,"style":147},[146],[61,54229,54231],{"className":54230},[151,152,153,154],[61,54232,27125],{"className":54233,"style":30082},[116,120,154],[61,54235,170],{"className":54236},[169],[61,54238,54240],{"className":54239},[134],[61,54241,54243],{"className":54242,"style":177},[138],[61,54244],{},[61,54246,170],{"className":54247},[169],[61,54249,54251],{"className":54250},[134],[61,54252,54254],{"className":54253,"style":29011},[138],[61,54255],{},[61,54257],{"className":54258},[730,19286],[61,54260],{"className":54261,"style":19197},[981],[61,54263,2000],{"className":54264},[19201],[61,54266],{"className":54267,"style":19197},[981],[61,54269,54271,54275,54529],{"className":54270},[107],[61,54272],{"className":54273,"style":54274},[111],"height:3.04em;vertical-align:-1.0541em;",[61,54276,54278],{"className":54277},[116,2425],[61,54279,54281,54520],{"className":54280},[129,130],[61,54282,54284,54517],{"className":54283},[134],[61,54285,54288,54500],{"className":54286,"style":54287},[138],"height:1.9859em;",[61,54289,54292,54296],{"className":54290,"style":54291},[28378],"top:-5em;",[61,54293],{"className":54294,"st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0 400000 3240",[28410,54515],{"d":54516},"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",[61,54518,170],{"className":54519},[169],[61,54521,54523],{"className":54522},[134],[61,54524,54527],{"className":54525,"style":54526},[138],"height:1.0541em;",[61,54528],{},[61,54530,332],{"className":54531},[116],[12,54533,54534],{},"This expression explains the ridge visible in the Fourier-domain heatmap. At resonance, the generalized frequency equals the applied drive strength,",[61,54536,54538],{"className":54537},[26354],[61,54539,54541,54583],{"className":54540},[64],[61,54542,54544],{"className":54543},[68],[70,54545,54546],{"xmlns":72,"display":26363},[74,54547,54548,54580],{},[77,54549,54550,54556,54558,54564,54566,54568,54578],{},[80,54551,54552,54554],{},[83,54553,47231],{},[83,54555,27125],{},[586,54557,849],{"stretchy":588},[80,54559,54560,54562],{},[83,54561,19797],{},[1493,54563,1495],{},[586,54565,300],{"stretchy":588},[586,54567,2000],{},[19134,54569,54570,54572],{},[83,54571,283],{"mathvariant":197},[77,54573,54574,54576],{},[1493,54575,337],{},[83,54577,19792],{},[83,54579,332],{"mathvariant":197},[95,54581,54582],{"encoding":97},"\\nu_R(f_0)=\\frac{\\Omega}{2\\pi}.",[61,54584,54586,54687],{"className":54585,"ariaHidden":103},[102],[61,54587,54589,54592,54632,54635,54675,54678,54681,54684],{"className":54588},[107],[61,54590],{"className":54591,"style":627},[111],[61,54593,54595,54598],{"className":54594},[116],[61,54596,47231],{"className":54597,"style":47252},[116,120],[61,54599,54601],{"className":54600},[125],[61,54602,54604,54624],{"className":54603},[129,130],[61,54605,54607,54621],{"className":54606},[134],[61,54608,54610],{"className":54609,"style":139},[138],[61,54611,54612,54615],{"style":47267},[61,54613],{"className":54614,"style":147},[146],[61,54616,54618],{"className":54617},[151,152,153,154],[61,54619,27125],{"className":54620,"style":30082},[116,120,154],[61,54622,170],{"className":54623},[169],[61,54625,54627],{"className":54626},[134],[61,54628,54630],{"className":54629,"style":177},[138],[61,54631],{},[61,54633,849],{"className":54634},[631],[61,54636,54638,54641],{"className":54637},[116],[61,54639,19797],{"className":54640,"style":20151},[116,120],[61,54642,54644],{"className":54643},[125],[61,54645,54647,54667],{"className":54646},[129,130],[61,54648,54650,54664],{"className":54649},[134],[61,54651,54653],{"className":54652,"style":19230},[138],[61,54654,54655,54658],{"style":20167},[61,54656],{"className":54657,"style":147},[146],[61,54659,54661],{"className":54660},[151,152,153,154],[61,54662,1495],{"className":54663},[116,154],[61,54665,170],{"className":54666},[169],[61,54668,54670],{"className":54669},[134],[61,54671,54673],{"className":54672,"style":177},[138],[61,54674],{},[61,54676,300],{"className":54677},[730],[61,54679],{"className":54680,"style":19197},[981],[61,54682,2000],{"className":54683},[19201],[61,54685],{"className":54686,"style":19197},[981],[61,54688,54690,54693,54758],{"className":54689},[107],[61,54691],{"className":54692,"style":52682},[111],[61,54694,54696,54699,54755],{"className":54695},[116],[61,54697],{"className":54698},[631,19286],[61,54700,54702],{"className":54701},[19134],[61,54703,54705,54747],{"className":54704},[129,130],[61,54706,54708,54744],{"className":54707},[134],[61,54709,54711,54725,54733],{"className":54710,"style":51287},[138],[61,54712,54713,54716],{"style":28925},[61,54714],{"className":54715,"style":19306},[146],[61,54717,54719,54722],{"className":54718},[116],[61,54720,337],{"className":54721},[116],[61,54723,19792],{"className":54724,"style":121},[116,120],[61,54726,54727,54730],{"style":19352},[61,54728],{"className":54729,"style":19306},[146],[61,54731],{"className":54732,"style":19360},[19359],[61,54734,54735,54738],{"style":28945},[61,54736],{"className":54737,"style":19306},[146],[61,54739,54741],{"className":54740},[116],[61,54742,283],{"className":54743},[116],[61,54745,170],{"className":54746},[169],[61,54748,54750],{"className":54749},[134],[61,54751,54753],{"className":54752,"style":29011},[138],[61,54754],{},[61,54756],{"className":54757},[730,19286],[61,54759,332],{"className":54760},[116],[12,54762,54763,54764,54833,54834,54936],{},"As the system moves away from resonance, ",[61,54765,54767,54784],{"className":54766},[64],[61,54768,54770],{"className":54769},[68],[70,54771,54772],{"xmlns":72},[74,54773,54774,54782],{},[77,54775,54776],{},[80,54777,54778,54780],{},[83,54779,47231],{},[83,54781,27125],{},[95,54783,47236],{"encoding":97},[61,54785,54787],{"className":54786,"ariaHidden":103},[102],[61,54788,54790,54793],{"className":54789},[107],[61,54791],{"className":54792,"style":112},[111],[61,54794,54796,54799],{"className":54795},[116],[61,54797,47231],{"className":54798,"style":47252},[116,120],[61,54800,54802],{"className":54801},[125],[61,54803,54805,54825],{"className":54804},[129,130],[61,54806,54808,54822],{"className":54807},[134],[61,54809,54811],{"className":54810,"style":139},[138],[61,54812,54813,54816],{"style":47267},[61,54814],{"className":54815,"style":147},[146],[61,54817,54819],{"className":54818},[151,152,153,154],[61,54820,27125],{"className":54821,"style":30082},[116,120,154],[61,54823,170],{"className":54824},[169],[61,54826,54828],{"className":54827},[134],[61,54829,54831],{"className":54830,"style":177},[138],[61,54832],{}," increases symmetrically with ",[61,54835,54837,54863],{"className":54836},[64],[61,54838,54840],{"className":54839},[68],[70,54841,54842],{"xmlns":72},[74,54843,54844,54860],{},[77,54845,54846,54848,54850,54852,54858],{},[586,54847,589],{"stretchy":588},[83,54849,19797],{},[586,54851,19132],{},[80,54853,54854,54856],{},[83,54855,19797],{},[1493,54857,1495],{},[586,54859,589],{"stretchy":588},[95,54861,54862],{"encoding":97},"\\lvert f-f_0\\rvert",[61,54864,54866,54887],{"className":54865,"ariaHidden":103},[102],[61,54867,54869,54872,54875,54878,54881,54884],{"className":54868},[107],[61,54870],{"className":54871,"style":627},[111],[61,54873,589],{"className":54874},[631],[61,54876,19797],{"className":54877,"style":20151},[116,120],[61,54879],{"className":54880,"style":20194},[981],[61,54882,19132],{"className":54883},[19382],[61,54885],{"className":54886,"style":20194},[981],[61,54888,54890,54893,54933],{"className":54889},[107],[61,54891],{"className":54892,"style":627},[111],[61,54894,54896,54899],{"className":54895},[116],[61,54897,19797],{"className":54898,"style":20151},[116,120],[61,54900,54902],{"className":54901},[125],[61,54903,54905,54925],{"className":54904},[129,130],[61,54906,54908,54922],{"className":54907},[134],[61,54909,54911],{"className":54910,"style":19230},[138],[61,54912,54913,54916],{"style":20167},[61,54914],{"className":54915,"style":147},[146],[61,54917,54919],{"className":54918},[151,152,153,154],[61,54920,1495],{"className":54921},[116,154],[61,54923,170],{"className":54924},[169],[61,54926,54928],{"className":54927},[134],[61,54929,54931],{"className":54930,"style":177},[138],[61,54932],{},[61,54934,589],{"className":54935},[730],". 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.",[25,54938,54940],{"id":54939},"quantum-information-interpretation","Quantum Information Interpretation",[12,54942,54943,54944,55014],{},"In quantum information language, a calibrated resonant Rabi pulse implements a single-qubit rotation. At exact resonance, the effective Hamiltonian is proportional to ",[61,54945,54947,54965],{"className":54946},[64],[61,54948,54950],{"className":54949},[68],[70,54951,54952],{"xmlns":72},[74,54953,54954,54962],{},[77,54955,54956],{},[80,54957,54958,54960],{},[83,54959,19165],{},[83,54961,26374],{},[95,54963,54964],{"encoding":97},"\\sigma_x",[61,54966,54968],{"className":54967,"ariaHidden":103},[102],[61,54969,54971,54974],{"className":54970},[107],[61,54972],{"className":54973,"style":112},[111],[61,54975,54977,54980],{"className":54976},[116],[61,54978,19165],{"className":54979,"style":121},[116,120],[61,54981,54983],{"className":54982},[125],[61,54984,54986,55006],{"className":54985},[129,130],[61,54987,54989,55003],{"className":54988},[134],[61,54990,54992],{"className":54991,"style":20164},[138],[61,54993,54994,54997],{"style":142},[61,54995],{"className":54996,"style":147},[146],[61,54998,55000],{"className":54999},[151,152,153,154],[61,55001,26374],{"className":55002},[116,120,154],[61,55004,170],{"className":55005},[169],[61,55007,55009],{"className":55008},[134],[61,55010,55012],{"className":55011,"style":177},[138],[61,55013],{},", so the microwave pulse implements",[61,55016,55018],{"className":55017},[26354],[61,55019,55021,55077],{"className":55020},[64],[61,55022,55024],{"className":55023},[68],[70,55025,55026],{"xmlns":72,"display":26363},[74,55027,55028,55074],{},[77,55029,55030,55036,55038,55040,55042,55044,55046,55048,55072],{},[80,55031,55032,55034],{},[83,55033,27125],{},[83,55035,26374],{},[586,55037,849],{"stretchy":588},[83,55039,21386],{},[586,55041,300],{"stretchy":588},[586,55043,2000],{},[83,55045,6987],{},[586,55047,19125],{},[77,55049,55050,55052,55054,55070],{},[586,55051,849],{"fence":103},[586,55053,19132],{},[19134,55055,55056,55068],{},[77,55057,55058,55060,55062],{},[83,55059,21383],{},[83,55061,21386],{},[80,55063,55064,55066],{},[83,55065,19165],{},[83,55067,26374],{},[1493,55069,337],{},[586,55071,300],{"fence":103},[586,55073,862],{"separator":103},[95,55075,55076],{"encoding":97},"R_x(\\theta)=\\exp\\left(-\\frac{i\\theta\\sigma_x}{2}\\right),",[61,55078,55080,55144],{"className":55079,"ariaHidden":103},[102],[61,55081,55083,55086,55126,55129,55132,55135,55138,55141],{"className":55082},[107],[61,55084],{"className":55085,"style":627},[111],[61,55087,55089,55092],{"className":55088},[116],[61,55090,27125],{"className":55091,"style":30082},[116,120],[61,55093,55095],{"className":55094},[125],[61,55096,55098,55118],{"className":55097},[129,130],[61,55099,55101,55115],{"className":55100},[134],[61,55102,55104],{"className":55103,"style":20164},[138],[61,55105,55106,55109],{"style":31048},[61,55107],{"className":55108,"style":147},[146],[61,55110,55112],{"className":55111},[151,152,153,154],[61,55113,26374],{"className":55114},[116,120,154],[61,55116,170],{"className":55117},[169],[61,55119,55121],{"className":55120},[134],[61,55122,55124],{"className":55123,"style":177},[138],[61,55125],{},[61,55127,849],{"className":55128},[631],[61,55130,21386],{"className":55131,"style":21503},[116,120],[61,55133,300],{"className":55134},[730],[61,55136],{"className":55137,"style":19197},[981],[61,55139,2000],{"className":55140},[19201],[61,55142],{"className":55143,"style":19197},[981],[61,55145,55147,55150,55153,55156,55280,55283],{"className":55146},[107],[61,55148],{"className":55149,"style":26592},[111],[61,55151,6987],{"className":55152},[19259],[61,55154],{"className":55155,"style":982},[981],[61,55157,55159,55165,55168,55274],{"className":55158},[19266],[61,55160,55162],{"className":55161,"style":19271},[631,19270],[61,55163,849],{"className":55164},[19275,153],[61,55166,19132],{"className":55167},[116],[61,55169,55171,55174,55271],{"className":55170},[116],[61,55172],{"className":55173},[631,19286],[61,55175,55177],{"className":55176},[19134],[61,55178,55180,55263],{"className":55179},[129,130],[61,55181,55183,55260],{"className":55182},[134],[61,55184,55187,55198,55206],{"className":55185,"style":55186},[138],"height:1.3714em;",[61,55188,55189,55192],{"style":28925},[61,55190],{"className":55191,"style":19306},[146],[61,55193,55195],{"className":55194},[116],[61,55196,337],{"className":55197},[116],[61,55199,55200,55203],{"style":19352},[61,55201],{"className":55202,"style":19306},[146],[61,55204],{"className":55205,"style":19360},[19359],[61,55207,55208,55211],{"style":28945},[61,55209],{"className":55210,"style":19306},[146],[61,55212,55214,55217,55220],{"className":55213},[116],[61,55215,21383],{"className":55216},[116,120],[61,55218,21386],{"className":55219,"style":21503},[116,120],[61,55221,55223,55226],{"className":55222},[116],[61,55224,19165],{"className":55225,"style":121},[116,120],[61,55227,55229],{"className":55228},[125],[61,55230,55232,55252],{"className":55231},[129,130],[61,55233,55235,55249],{"className":55234},[134],[61,55236,55238],{"className":55237,"style":20164},[138],[61,55239,55240,55243],{"style":142},[61,55241],{"className":55242,"style":147},[146],[61,55244,55246],{"className":55245},[151,152,153,154],[61,55247,26374],{"className":55248},[116,120,154],[61,55250,170],{"className":55251},[169],[61,55253,55255],{"className":55254},[134],[61,55256,55258],{"className":55257,"style":177},[138],[61,55259],{},[61,55261,170],{"className":55262},[169],[61,55264,55266],{"className":55265},[134],[61,55267,55269],{"className":55268,"style":29011},[138],[61,55270],{},[61,55272],{"className":55273},[730,19286],[61,55275,55277],{"className":55276,"style":19271},[730,19270],[61,55278,300],{"className":55279},[19275,153],[61,55281],{"className":55282,"style":982},[981],[61,55284,862],{"className":55285},[977],[12,55287,3580],{},[61,55289,55291],{"className":55290},[26354],[61,55292,55294,55316],{"className":55293},[64],[61,55295,55297],{"className":55296},[68],[70,55298,55299],{"xmlns":72,"display":26363},[74,55300,55301,55313],{},[77,55302,55303,55305,55307,55309,55311],{},[83,55304,21386],{},[586,55306,2000],{},[83,55308,283],{"mathvariant":197},[83,55310,45968],{},[83,55312,332],{"mathvariant":197},[95,55314,55315],{"encoding":97},"\\theta=\\Omega\\tau.",[61,55317,55319,55338],{"className":55318,"ariaHidden":103},[102],[61,55320,55322,55326,55329,55332,55335],{"className":55321},[107],[61,55323],{"className":55324,"style":55325},[111],"height:0.6944em;",[61,55327,21386],{"className":55328,"style":21503},[116,120],[61,55330],{"className":55331,"style":19197},[981],[61,55333,2000],{"className":55334},[19201],[61,55336],{"className":55337,"style":19197},[981],[61,55339,55341,55344,55347,55350],{"className":55340},[107],[61,55342],{"className":55343,"style":296},[111],[61,55345,283],{"className":55346},[116],[61,55348,45968],{"className":55349,"style":46040},[116,120],[61,55351,332],{"className":55352},[116],[12,55354,55355,55356,55384],{},"A calibrated resonant ",[61,55357,55359,55372],{"className":55358},[64],[61,55360,55362],{"className":55361},[68],[70,55363,55364],{"xmlns":72},[74,55365,55366,55370],{},[77,55367,55368],{},[83,55369,19792],{},[95,55371,30267],{"encoding":97},[61,55373,55375],{"className":55374,"ariaHidden":103},[102],[61,55376,55378,55381],{"className":55377},[107],[61,55379],{"className":55380,"style":30277},[111],[61,55382,19792],{"className":55383,"style":121},[116,120],"-pulse enacts",[61,55386,55388],{"className":55387},[26354],[61,55389,55391,55429],{"className":55390},[64],[61,55392,55394],{"className":55393},[68],[70,55395,55396],{"xmlns":72,"display":26363},[74,55397,55398,55426],{},[77,55399,55400,55406,55408,55410,55412,55414,55416,55418,55424],{},[80,55401,55402,55404],{},[83,55403,27125],{},[83,55405,26374],{},[586,55407,849],{"stretchy":588},[83,55409,19792],{},[586,55411,300],{"stretchy":588},[586,55413,2000],{},[586,55415,19132],{},[83,55417,21383],{},[80,55419,55420,55422],{},[83,55421,19165],{},[83,55423,26374],{},[586,55425,862],{"separator":103},[95,55427,55428],{"encoding":97},"R_x(\\pi)=-i\\sigma_x,",[61,55430,55432,55496],{"className":55431,"ariaHidden":103},[102],[61,55433,55435,55438,55478,55481,55484,55487,55490,55493],{"className":55434},[107],[61,55436],{"className":55437,"style":627},[111],[61,55439,55441,55444],{"className":55440},[116],[61,55442,27125],{"className":55443,"style":30082},[116,120],[61,55445,55447],{"className":55446},[125],[61,55448,55450,55470],{"className":55449},[129,130],[61,55451,55453,55467],{"className":55452},[134],[61,55454,55456],{"className":55455,"style":20164},[138],[61,55457,55458,55461],{"style":31048},[61,55459],{"className":55460,"style":147},[146],[61,55462,55464],{"className":55463},[151,152,153,154],[61,55465,26374],{"className":55466},[116,120,154],[61,55468,170],{"className":55469},[169],[61,55471,55473],{"className":55472},[134],[61,55474,55476],{"className":55475,"style":177},[138],[61,55477],{},[61,55479,849],{"className":55480},[631],[61,55482,19792],{"className":55483,"style":121},[116,120],[61,55485,300],{"className":55486},[730],[61,55488],{"className":55489,"style":19197},[981],[61,55491,2000],{"className":55492},[19201],[61,55494],{"className":55495,"style":19197},[981],[61,55497,55499,55503,55506,55509,55549],{"className":55498},[107],[61,55500],{"className":55501,"style":55502},[111],"height:0.854em;vertical-align:-0.1944em;",[61,55504,19132],{"className":55505},[116],[61,55507,21383],{"className":55508},[116,120],[61,55510,55512,55515],{"className":55511},[116],[61,55513,19165],{"className":55514,"style":121},[116,120],[61,55516,55518],{"className":55517},[125],[61,55519,55521,55541],{"className":55520},[129,130],[61,55522,55524,55538],{"className":55523},[134],[61,55525,55527],{"className":55526,"style":20164},[138],[61,55528,55529,55532],{"style":142},[61,55530],{"className":55531,"style":147},[146],[61,55533,55535],{"className":55534},[151,152,153,154],[61,55536,26374],{"className":55537},[116,120,154],[61,55539,170],{"className":55540},[169],[61,55542,55544],{"className":55543},[134],[61,55545,55547],{"className":55546,"style":177},[138],[61,55548],{},[61,55550,862],{"className":55551},[977],[12,55553,55554,55555,55583,55584,55620],{},"which corresponds to a quantum ",[61,55556,55558,55571],{"className":55557},[64],[61,55559,55561],{"className":55560},[68],[70,55562,55563],{"xmlns":72},[74,55564,55565,55569],{},[77,55566,55567],{},[83,55568,93],{},[95,55570,93],{"encoding":97},[61,55572,55574],{"className":55573,"ariaHidden":103},[102],[61,55575,55577,55580],{"className":55576},[107],[61,55578],{"className":55579,"style":296},[111],[61,55581,93],{"className":55582,"style":165},[116,120]," gate up to a global phase. A resonant ",[61,55585,55587,55605],{"className":55586},[64],[61,55588,55590],{"className":55589},[68],[70,55591,55592],{"xmlns":72},[74,55593,55594,55602],{},[77,55595,55596,55598,55600],{},[83,55597,19792],{},[83,55599,602],{"mathvariant":197},[1493,55601,337],{},[95,55603,55604],{"encoding":97},"\\pi\u002F2",[61,55606,55608],{"className":55607,"ariaHidden":103},[102],[61,55609,55611,55614,55617],{"className":55610},[107],[61,55612],{"className":55613,"style":627},[111],[61,55615,19792],{"className":55616,"style":121},[116,120],[61,55618,20506],{"className":55619},[116],"-pulse prepares a balanced coherent superposition,",[61,55622,55624],{"className":55623},[26354],[61,55625,55627,55673],{"className":55626},[64],[61,55628,55630],{"className":55629},[68],[70,55631,55632],{"xmlns":72,"display":26363},[74,55633,55634,55670],{},[77,55635,55636,55638,55640,55642,55644,55668],{},[586,55637,589],{"stretchy":588},[83,55639,46151],{},[586,55641,1498],{"stretchy":588},[586,55643,21354],{},[19134,55645,55646,55664],{},[77,55647,55648,55650,55652,55654,55656,55658,55660,55662],{},[586,55649,589],{"stretchy":588},[83,55651,46151],{},[586,55653,1498],{"stretchy":588},[586,55655,19132],{},[83,55657,21383],{},[586,55659,589],{"stretchy":588},[83,55661,21378],{},[586,55663,1498],{"stretchy":588},[28194,55665,55666],{},[1493,55667,337],{},[586,55669,862],{"separator":103},[95,55671,55672],{"encoding":97},"\\lvert g\\rangle\\rightarrow\\frac{\\lvert g\\rangle-i\\lvert e\\rangle}{\\sqrt{2}},",[61,55674,55676,55700],{"className":55675,"ariaHidden":103},[102],[61,55677,55679,55682,55685,55688,55691,55694,55697],{"className":55678},[107],[61,55680],{"className":55681,"style":627},[111],[61,55683,589],{"className":55684},[631],[61,55686,46151],{"className":55687,"style":121},[116,120],[61,55689,1498],{"className":55690},[730],[61,55692],{"className":55693,"style":19197},[981],[61,55695,21354],{"className":55696},[19201],[61,55698],{"className":55699,"style":19197},[981],[61,55701,55703,55707,55843],{"className":55702},[107],[61,55704],{"className":55705,"style":55706},[111],"height:2.357em;vertical-align:-0.93em;",[61,55708,55710,55713,55840],{"className":55709},[116],[61,55711],{"className":55712},[631,19286],[61,55714,55716],{"className":55715},[19134],[61,55717,55719,55831],{"className":55718},[129,130],[61,55720,55722,55828],{"className":55721},[134],[61,55723,55726,55782,55790],{"className":55724,"style":55725},[138],"height:1.427em;",[61,55727,55729,55732],{"style":55728},"top:-2.2028em;",[61,55730],{"className":55731,"style":19306},[146],[61,55733,55735],{"className":55734},[116],[61,55736,55738],{"className":55737},[116,2425],[61,55739,55741,55774],{"className":55740},[129,130],[61,55742,55744,55771],{"className":55743},[134],[61,55745,55747,55759],{"className":55746,"style":28964},[138],[61,55748,55750,55753],{"className":55749,"style":28379},[28378],[61,55751],{"className":55752,"style":19306},[146],[61,55754,55756],{"className":55755,"style":28386},[116],[61,55757,337],{"className":55758},[116],[61,55760,55761,55764],{"style":28979},[61,55762],{"className":55763,"style":19306},[146],[61,55765,55767],{"className":55766,"style":28400},[28399],[28402,55768,55769],{"xmlns":28404,"width":28405,"height":28406,"viewBox":28407,"preserveAspectRatio":28408},[28410,55770],{"d":28412},[61,55772,170],{"className":55773},[169],[61,55775,55777],{"className":55776},[134],[61,55778,55780],{"className":55779,"style":28999},[138],[61,55781],{},[61,55783,55784,55787],{"style":19352},[61,55785],{"className":55786,"style":19306},[146],[61,55788],{"className":55789,"style":19360},[19359],[61,55791,55792,55795],{"style":28945},[61,55793],{"className":55794,"style":19306},[146],[61,55796,55798,55801,55804,55807,55810,55813,55816,55819,55822,55825],{"className":55797},[116],[61,55799,589],{"className":55800},[631],[61,55802,46151],{"className":55803,"style":121},[116,120],[61,55805,1498],{"className":55806},[730],[61,55808],{"className":55809,"style":20194},[981],[61,55811,19132],{"className":55812},[19382],[61,55814],{"className":55815,"style":20194},[981],[61,55817,21383],{"className":55818},[116,120],[61,55820,589],{"className":55821},[631],[61,55823,21378],{"className":55824},[116,120],[61,55826,1498],{"className":55827},[730],[61,55829,170],{"className":55830},[169],[61,55832,55834],{"className":55833},[134],[61,55835,55838],{"className":55836,"style":55837},[138],"height:0.93em;",[61,55839],{},[61,55841],{"className":55842},[730,19286],[61,55844,862],{"className":55845},[977],[12,55847,55848],{},"with a phase convention set by the microwave drive phase.",[12,55850,55851],{},"Detuning changes the rotation axis so it is no longer purely transverse. The unitary can be written as",[61,55853,55855],{"className":55854},[26354],[61,55856,55858,55927],{"className":55857},[64],[61,55859,55861],{"className":55860},[68],[70,55862,55863],{"xmlns":72,"display":26363},[74,55864,55865,55924],{},[77,55866,55867,55877,55879,55881,55883,55885,55887,55889,55922],{},[80,55868,55869,55871],{},[83,55870,27125],{},[31272,55872,55873,55875],{"accent":103},[83,55874,27948],{},[586,55876,31279],{},[586,55878,849],{"stretchy":588},[83,55880,21386],{},[586,55882,300],{"stretchy":588},[586,55884,2000],{},[83,55886,6987],{},[586,55888,19125],{},[77,55890,55891,55893,55895,55905,55911,55913,55920],{},[586,55892,849],{"fence":103},[586,55894,19132],{},[19134,55896,55897,55903],{},[77,55898,55899,55901],{},[83,55900,21383],{},[83,55902,21386],{},[1493,55904,337],{},[31272,55906,55907,55909],{"accent":103},[83,55908,27948],{},[586,55910,31279],{},[586,55912,27231],{},[31272,55914,55915,55917],{"accent":103},[83,55916,19165],{},[586,55918,55919],{},"⃗",[586,55921,300],{"fence":103},[586,55923,862],{"separator":103},[95,55925,55926],{"encoding":97},"R_{\\hat{n}}(\\theta)=\\exp\\left(-\\frac{i\\theta}{2}\\hat{n}\\cdot\\vec{\\sigma}\\right),",[61,55928,55930,56027],{"className":55929,"ariaHidden":103},[102],[61,55931,55933,55936,56009,56012,56015,56018,56021,56024],{"className":55932},[107],[61,55934],{"className":55935,"style":627},[111],[61,55937,55939,55942],{"className":55938},[116],[61,55940,27125],{"className":55941,"style":30082},[116,120],[61,55943,55945],{"className":55944},[125],[61,55946,55948,56001],{"className":55947},[129,130],[61,55949,55951,55998],{"className":55950},[134],[61,55952,55955],{"className":55953,"style":55954},[138],"height:0.3361em;",[61,55956,55957,55960],{"style":31048},[61,55958],{"className":55959,"style":147},[146],[61,55961,55963],{"className":55962},[151,152,153,154],[61,55964,55966],{"className":55965},[116,154],[61,55967,55969],{"className":55968},[116,31296,154],[61,55970,55972],{"className":55971},[129],[61,55973,55975],{"className":55974},[134],[61,55976,55978,55987],{"className":55977,"style":55325},[138],[61,55979,55981,55984],{"style":55980},"top:-2.7em;",[61,55982],{"className":55983,"style":147},[146],[61,55985,27948],{"className":55986},[116,120,154],[61,55988,55989,55992],{"style":55980},[61,55990],{"className":55991,"style":147},[146],[61,55993,55995],{"className":55994,"style":31324},[31323],[61,55996,31279],{"className":55997},[116,154],[61,55999,170],{"className":56000},[169],[61,56002,56004],{"className":56003},[134],[61,56005,56007],{"className":56006,"style":177},[138],[61,56008],{},[61,56010,849],{"className":56011},[631],[61,56013,21386],{"className":56014,"style":21503},[116,120],[61,56016,300],{"className":56017},[730],[61,56019],{"className":56020,"style":19197},[981],[61,56022,2000],{"className":56023},[19201],[61,56025],{"className":56026,"style":19197},[981],[61,56028,56030,56033,56036,56039,56207,56210],{"className":56029},[107],[61,56031],{"className":56032,"style":26592},[111],[61,56034,6987],{"className":56035},[19259],[61,56037],{"className":56038,"style":982},[981],[61,56040,56042,56048,56051,56116,56147,56150,56153,56156,56201],{"className":56041},[19266],[61,56043,56045],{"className":56044,"style":19271},[631,19270],[61,56046,849],{"className":56047},[19275,153],[61,56049,19132],{"className":56050},[116],[61,56052,56054,56057,56113],{"className":56053},[116],[61,56055],{"className":56056},[631,19286],[61,56058,56060],{"className":56059},[19134],[61,56061,56063,56105],{"className":56062},[129,130],[61,56064,56066,56102],{"className":56065},[134],[61,56067,56069,56080,56088],{"className":56068,"style":55186},[138],[61,56070,56071,56074],{"style":28925},[61,56072],{"className":56073,"style":19306},[146],[61,56075,56077],{"className":56076},[116],[61,56078,337],{"className":56079},[116],[61,56081,56082,56085],{"style":19352},[61,56083],{"className":56084,"style":19306},[146],[61,56086],{"className":56087,"style":19360},[19359],[61,56089,56090,56093],{"style":28945},[61,56091],{"className":56092,"style":19306},[146],[61,56094,56096,56099],{"className":56095},[116],[61,56097,21383],{"className":56098},[116,120],[61,56100,21386],{"className":56101,"style":21503},[116,120],[61,56103,170],{"className":56104},[169],[61,56106,56108],{"className":56107},[134],[61,56109,56111],{"className":56110,"style":29011},[138],[61,56112],{},[61,56114],{"className":56115},[730,19286],[61,56117,56119],{"className":56118},[116,31296],[61,56120,56122],{"className":56121},[129],[61,56123,56125],{"className":56124},[134],[61,56126,56128,56136],{"className":56127,"style":55325},[138],[61,56129,56130,56133],{"style":28379},[61,56131],{"className":56132,"style":19306},[146],[61,56134,27948],{"className":56135},[116,120],[61,56137,56138,56141],{"style":28379},[61,56139],{"className":56140,"style":19306},[146],[61,56142,56144],{"className":56143,"style":31324},[31323],[61,56145,31279],{"className":56146},[116],[61,56148],{"className":56149,"style":20194},[981],[61,56151,27231],{"className":56152},[19382],[61,56154],{"className":56155,"style":20194},[981],[61,56157,56159],{"className":56158},[116,31296],[61,56160,56162],{"className":56161},[129],[61,56163,56165],{"className":56164},[134],[61,56166,56169,56177],{"className":56167,"style":56168},[138],"height:0.714em;",[61,56170,56171,56174],{"style":28379},[61,56172],{"className":56173,"style":19306},[146],[61,56175,19165],{"className":56176,"style":121},[116,120],[61,56178,56179,56182],{"style":28379},[61,56180],{"className":56181,"style":19306},[146],[61,56183,56186],{"className":56184,"style":56185},[31323],"left:-0.2355em;",[61,56187,56191],{"className":56188,"style":56190},[56189],"overlay","height:0.714em;width:0.471em;",[28402,56192,56198],{"xmlns":28404,"width":56193,"height":56194,"style":56195,"viewBox":56196,"preserveAspectRatio":56197},"0.471em","0.714em","width:0.471em","0 0 471 714","xMinYMin",[28410,56199],{"d":56200},"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",[61,56202,56204],{"className":56203,"style":19271},[730,19270],[61,56205,300],{"className":56206},[19275,153],[61,56208],{"className":56209,"style":982},[981],[61,56211,862],{"className":56212},[977],[12,56214,3580],{},[61,56216,56218],{"className":56217},[26354],[61,56219,56221,56283],{"className":56220},[64],[61,56222,56224],{"className":56223},[68],[70,56225,56226],{"xmlns":72,"display":26363},[74,56227,56228,56280],{},[77,56229,56230,56236,56238,56248,56250,56252,56254,56256,56258,56260,56262,56264,56266,56268,56270,56276,56278],{},[31272,56231,56232,56234],{"accent":103},[83,56233,27948],{},[586,56235,31279],{},[586,56237,2000],{},[19134,56239,56240,56242],{},[1493,56241,57],{},[80,56243,56244,56246],{},[83,56245,283],{"mathvariant":197},[83,56247,27125],{},[586,56249,849],{"stretchy":588},[83,56251,283],{"mathvariant":197},[586,56253,862],{"separator":103},[1493,56255,1495],{},[586,56257,862],{"separator":103},[83,56259,198],{"mathvariant":197},[586,56261,300],{"stretchy":588},[586,56263,862],{"separator":103},[981,56265],{"width":27241},[83,56267,21386],{},[586,56269,2000],{},[80,56271,56272,56274],{},[83,56273,283],{"mathvariant":197},[83,56275,27125],{},[83,56277,45968],{},[83,56279,332],{"mathvariant":197},[95,56281,56282],{"encoding":97},"\\hat{n}=\\frac{1}{\\Omega_R}(\\Omega,0,\\Delta), \\qquad \\theta=\\Omega_R\\tau.",[61,56284,56286,56332,56487],{"className":56285,"ariaHidden":103},[102],[61,56287,56289,56292,56323,56326,56329],{"className":56288},[107],[61,56290],{"className":56291,"style":55325},[111],[61,56293,56295],{"className":56294},[116,31296],[61,56296,56298],{"className":56297},[129],[61,56299,56301],{"className":56300},[134],[61,56302,56304,56312],{"className":56303,"style":55325},[138],[61,56305,56306,56309],{"style":28379},[61,56307],{"className":56308,"style":19306},[146],[61,56310,27948],{"className":56311},[116,120],[61,56313,56314,56317],{"style":28379},[61,56315],{"className":56316,"style":19306},[146],[61,56318,56320],{"className":56319,"style":31324},[31323],[61,56321,31279],{"className":56322},[116],[61,56324],{"className":56325,"style":19197},[981],[61,56327,2000],{"className":56328},[19201],[61,56330],{"className":56331,"style":19197},[981],[61,56333,56335,56339,56439,56442,56445,56448,56451,56454,56457,56460,56463,56466,56469,56472,56475,56478,56481,56484],{"className":56334},[107],[61,56336],{"className":56337,"style":56338},[111],"height:2.1574em;vertical-align:-0.836em;",[61,56340,56342,56345,56436],{"className":56341},[116],[61,56343],{"className":56344},[631,19286],[61,56346,56348],{"className":56347},[19134],[61,56349,56351,56427],{"className":56350},[129,130],[61,56352,56354,56424],{"className":56353},[134],[61,56355,56357,56405,56413],{"className":56356,"style":52564},[138],[61,56358,56359,56362],{"style":28925},[61,56360],{"className":56361,"style":19306},[146],[61,56363,56365],{"className":56364},[116],[61,56366,56368,56371],{"className":56367},[116],[61,56369,283],{"className":56370},[116],[61,56372,56374],{"className":56373},[125],[61,56375,56377,56397],{"className":56376},[129,130],[61,56378,56380,56394],{"className":56379},[134],[61,56381,56383],{"className":56382,"style":139},[138],[61,56384,56385,56388],{"style":241},[61,56386],{"className":56387,"style":147},[146],[61,56389,56391],{"className":56390},[151,152,153,154],[61,56392,27125],{"className":56393,"style":30082},[116,120,154],[61,56395,170],{"className":56396},[169],[61,56398,56400],{"className":56399},[134],[61,56401,56403],{"className":56402,"style":177},[138],[61,56404],{},[61,56406,56407,56410],{"style":19352},[61,56408],{"className":56409,"style":19306},[146],[61,56411],{"className":56412,"style":19360},[19359],[61,56414,56415,56418],{"style":28945},[61,56416],{"className":56417,"style":19306},[146],[61,56419,56421],{"className":56420},[116],[61,56422,57],{"className":56423},[116],[61,56425,170],{"className":56426},[169],[61,56428,56430],{"className":56429},[134],[61,56431,56434],{"className":56432,"style":56433},[138],"height:0.836em;",[61,56435],{},[61,56437],{"className":56438},[730,19286],[61,56440,849],{"className":56441},[631],[61,56443,283],{"className":56444},[116],[61,56446,862],{"className":56447},[977],[61,56449],{"className":56450,"style":982},[981],[61,56452,1495],{"className":56453},[116],[61,56455,862],{"className":56456},[977],[61,56458],{"className":56459,"style":982},[981],[61,56461,198],{"className":56462},[116],[61,56464,300],{"className":56465},[730],[61,56467,862],{"className":56468},[977],[61,56470],{"className":56471,"style":27460},[981],[61,56473],{"className":56474,"style":982},[981],[61,56476,21386],{"className":56477,"style":21503},[116,120],[61,56479],{"className":56480,"style":19197},[981],[61,56482,2000],{"className":56483},[19201],[61,56485],{"className":56486,"style":19197},[981],[61,56488,56490,56493,56533,56536],{"className":56489},[107],[61,56491],{"className":56492,"style":219},[111],[61,56494,56496,56499],{"className":56495},[116],[61,56497,283],{"className":56498},[116],[61,56500,56502],{"className":56501},[125],[61,56503,56505,56525],{"className":56504},[129,130],[61,56506,56508,56522],{"className":56507},[134],[61,56509,56511],{"className":56510,"style":139},[138],[61,56512,56513,56516],{"style":241},[61,56514],{"className":56515,"style":147},[146],[61,56517,56519],{"className":56518},[151,152,153,154],[61,56520,27125],{"className":56521,"style":30082},[116,120,154],[61,56523,170],{"className":56524},[169],[61,56526,56528],{"className":56527},[134],[61,56529,56531],{"className":56530,"style":177},[138],[61,56532],{},[61,56534,45968],{"className":56535,"style":46040},[116,120],[61,56537,332],{"className":56538},[116],[12,56540,56541],{},"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.",[25,56543,56545],{"id":56544},"bloch-sphere-interpretation","Bloch-Sphere Interpretation",[12,56547,56548],{},"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.",[12,56550,56551,56552,56590,56591,56619,56620,56655],{},"A resonant microwave drive rotates the Bloch vector around a transverse axis. Starting from ",[61,56553,56555,56572],{"className":56554},[64],[61,56556,56558],{"className":56557},[68],[70,56559,56560],{"xmlns":72},[74,56561,56562,56570],{},[77,56563,56564,56566,56568],{},[586,56565,589],{"stretchy":588},[83,56567,46151],{},[586,56569,1498],{"stretchy":588},[95,56571,46156],{"encoding":97},[61,56573,56575],{"className":56574,"ariaHidden":103},[102],[61,56576,56578,56581,56584,56587],{"className":56577},[107],[61,56579],{"className":56580,"style":627},[111],[61,56582,589],{"className":56583},[631],[61,56585,46151],{"className":56586,"style":121},[116,120],[61,56588,1498],{"className":56589},[730],", a properly timed ",[61,56592,56594,56607],{"className":56593},[64],[61,56595,56597],{"className":56596},[68],[70,56598,56599],{"xmlns":72},[74,56600,56601,56605],{},[77,56602,56603],{},[83,56604,19792],{},[95,56606,30267],{"encoding":97},[61,56608,56610],{"className":56609,"ariaHidden":103},[102],[61,56611,56613,56616],{"className":56612},[107],[61,56614],{"className":56615,"style":30277},[111],[61,56617,19792],{"className":56618,"style":121},[116,120],"-pulse moves the state to the excited-state pole. A ",[61,56621,56623,56640],{"className":56622},[64],[61,56624,56626],{"className":56625},[68],[70,56627,56628],{"xmlns":72},[74,56629,56630,56638],{},[77,56631,56632,56634,56636],{},[83,56633,19792],{},[83,56635,602],{"mathvariant":197},[1493,56637,337],{},[95,56639,55604],{"encoding":97},[61,56641,56643],{"className":56642,"ariaHidden":103},[102],[61,56644,56646,56649,56652],{"className":56645},[107],[61,56647],{"className":56648,"style":627},[111],[61,56650,19792],{"className":56651,"style":121},[116,120],[61,56653,20506],{"className":56654},[116],"-pulse moves the state to the equator, where the ground-state and excited-state measurement probabilities are equal.",[12,56657,56658,56659,56687],{},"Detuning tilts the rotation axis toward the ",[61,56660,56662,56675],{"className":56661},[64],[61,56663,56665],{"className":56664},[68],[70,56666,56667],{"xmlns":72},[74,56668,56669,56673],{},[77,56670,56671],{},[83,56672,26482],{},[95,56674,26482],{"encoding":97},[61,56676,56678],{"className":56677,"ariaHidden":103},[102],[61,56679,56681,56684],{"className":56680},[107],[61,56682],{"className":56683,"style":30277},[111],[61,56685,26482],{"className":56686,"style":26946},[116,120],"-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",[61,56689,56691],{"className":56690},[26354],[61,56692,56694,56724],{"className":56693},[64],[61,56695,56697],{"className":56696},[68],[70,56698,56699],{"xmlns":72,"display":26363},[74,56700,56701,56721],{},[77,56702,56703,56719],{},[19134,56704,56705,56711],{},[19151,56706,56707,56709],{},[83,56708,283],{"mathvariant":197},[1493,56710,337],{},[34480,56712,56713,56715,56717],{},[83,56714,283],{"mathvariant":197},[83,56716,27125],{},[1493,56718,337],{},[83,56720,332],{"mathvariant":197},[95,56722,56723],{"encoding":97},"\\frac{\\Omega^2}{\\Omega_R^2}.",[61,56725,56727],{"className":56726,"ariaHidden":103},[102],[61,56728,56730,56733,56869],{"className":56729},[107],[61,56731],{"className":56732,"style":51080},[111],[61,56734,56736,56739,56866],{"className":56735},[116],[61,56737],{"className":56738},[631,19286],[61,56740,56742],{"className":56741},[19134],[61,56743,56745,56858],{"className":56744},[129,130],[61,56746,56748,56855],{"className":56747},[134],[61,56749,56751,56810,56818],{"className":56750,"style":51099},[138],[61,56752,56753,56756],{"style":28925},[61,56754],{"className":56755,"style":19306},[146],[61,56757,56759],{"className":56758},[116],[61,56760,56762,56765],{"className":56761},[116],[61,56763,283],{"className":56764},[116],[61,56766,56768],{"className":56767},[125],[61,56769,56771,56802],{"className":56770},[129,130],[61,56772,56774,56799],{"className":56773},[134],[61,56775,56777,56788],{"className":56776,"style":51126},[138],[61,56778,56779,56782],{"style":51129},[61,56780],{"className":56781,"style":147},[146],[61,56783,56785],{"className":56784},[151,152,153,154],[61,56786,27125],{"className":56787,"style":30082},[116,120,154],[61,56789,56790,56793],{"style":51141},[61,56791],{"className":56792,"style":147},[146],[61,56794,56796],{"className":56795},[151,152,153,154],[61,56797,337],{"className":56798},[116,154],[61,56800,170],{"className":56801},[169],[61,56803,56805],{"className":56804},[134],[61,56806,56808],{"className":56807,"style":51160},[138],[61,56809],{},[61,56811,56812,56815],{"style":19352},[61,56813],{"className":56814,"style":19306},[146],[61,56816],{"className":56817,"style":19360},[19359],[61,56819,56820,56823],{"style":28945},[61,56821],{"className":56822,"style":19306},[146],[61,56824,56826],{"className":56825},[116],[61,56827,56829,56832],{"className":56828},[116],[61,56830,283],{"className":56831},[116],[61,56833,56835],{"className":56834},[125],[61,56836,56838],{"className":56837},[129],[61,56839,56841],{"className":56840},[134],[61,56842,56844],{"className":56843,"style":27160},[138],[61,56845,56846,56849],{"style":21487},[61,56847],{"className":56848,"style":147},[146],[61,56850,56852],{"className":56851},[151,152,153,154],[61,56853,337],{"className":56854},[116,154],[61,56856,170],{"className":56857},[169],[61,56859,56861],{"className":56860},[134],[61,56862,56864],{"className":56863,"style":51217},[138],[61,56865],{},[61,56867],{"className":56868},[730,19286],[61,56870,332],{"className":56871},[116],[12,56873,56874,56875,56914],{},"As ",[61,56876,56878,56896],{"className":56877},[64],[61,56879,56881],{"className":56880},[68],[70,56882,56883],{"xmlns":72},[74,56884,56885,56893],{},[77,56886,56887,56889,56891],{},[586,56888,589],{"stretchy":588},[83,56890,198],{"mathvariant":197},[586,56892,589],{"stretchy":588},[95,56894,56895],{"encoding":97},"\\lvert\\Delta\\rvert",[61,56897,56899],{"className":56898,"ariaHidden":103},[102],[61,56900,56902,56905,56908,56911],{"className":56901},[107],[61,56903],{"className":56904,"style":627},[111],[61,56906,589],{"className":56907},[631],[61,56909,198],{"className":56910},[116],[61,56912,589],{"className":56913},[730]," grows, the rotation axis becomes more longitudinal, the oscillation frequency increases, and the maximum excited-state population decreases.",[25,56916,56918],{"id":56917},"fourier-analysis-dynamics","Fourier Analysis Dynamics",[12,56920,56921,56922,56950,56951,57049],{},"The spectral analysis computes a real Fast Fourier transform along the pulse-duration axis. For each sampled drive frequency ",[61,56923,56925,56938],{"className":56924},[64],[61,56926,56928],{"className":56927},[68],[70,56929,56930],{"xmlns":72},[74,56931,56932,56936],{},[77,56933,56934],{},[83,56935,19797],{},[95,56937,19797],{"encoding":97},[61,56939,56941],{"className":56940,"ariaHidden":103},[102],[61,56942,56944,56947],{"className":56943},[107],[61,56945],{"className":56946,"style":46075},[111],[61,56948,19797],{"className":56949,"style":20151},[116,120],", the trace ",[61,56952,56954,56982],{"className":56953},[64],[61,56955,56957],{"className":56956},[68],[70,56958,56959],{"xmlns":72},[74,56960,56961,56979],{},[77,56962,56963,56969,56971,56973,56975,56977],{},[80,56964,56965,56967],{},[83,56966,45957],{},[83,56968,21378],{},[586,56970,849],{"stretchy":588},[83,56972,45968],{},[586,56974,862],{"separator":103},[83,56976,19797],{},[586,56978,300],{"stretchy":588},[95,56980,56981],{"encoding":97},"P_e(\\tau,f)",[61,56983,56985],{"className":56984,"ariaHidden":103},[102],[61,56986,56988,56991,57031,57034,57037,57040,57043,57046],{"className":56987},[107],[61,56989],{"className":56990,"style":627},[111],[61,56992,56994,56997],{"className":56993},[116],[61,56995,45957],{"className":56996,"style":23700},[116,120],[61,56998,57000],{"className":56999},[125],[61,57001,57003,57023],{"className":57002},[129,130],[61,57004,57006,57020],{"className":57005},[134],[61,57007,57009],{"className":57008,"style":20164},[138],[61,57010,57011,57014],{"style":28699},[61,57012],{"className":57013,"style":147},[146],[61,57015,57017],{"className":57016},[151,152,153,154],[61,57018,21378],{"className":57019},[116,120,154],[61,57021,170],{"className":57022},[169],[61,57024,57026],{"className":57025},[134],[61,57027,57029],{"className":57028,"style":177},[138],[61,57030],{},[61,57032,849],{"className":57033},[631],[61,57035,45968],{"className":57036,"style":46040},[116,120],[61,57038,862],{"className":57039},[977],[61,57041],{"className":57042,"style":982},[981],[61,57044,19797],{"className":57045,"style":20151},[116,120],[61,57047,300],{"className":57048},[730]," is treated as a time-domain signal. The spectral amplitude is",[61,57051,57053],{"className":57052},[26354],[61,57054,57056,57120],{"className":57055},[64],[61,57057,57059],{"className":57058},[68],[70,57060,57061],{"xmlns":72,"display":26363},[74,57062,57063,57117],{},[77,57064,57065,57067,57069,57071,57073,57075,57077,57079,57115],{},[83,57066,28508],{},[586,57068,849],{"stretchy":588},[83,57070,47231],{},[586,57072,862],{"separator":103},[83,57074,19797],{},[586,57076,300],{"stretchy":588},[586,57078,2000],{},[77,57080,57081,57083,57091,57113],{},[586,57082,589],{"fence":103},[80,57084,57085,57089],{},[83,57086,57088],{"mathvariant":57087},"script","F",[83,57090,45968],{},[77,57092,57093,57095,57101,57103,57105,57107,57109,57111],{},[586,57094,3688],{"fence":103},[80,57096,57097,57099],{},[83,57098,45957],{},[83,57100,21378],{},[586,57102,849],{"stretchy":588},[83,57104,45968],{},[586,57106,862],{"separator":103},[83,57108,19797],{},[586,57110,300],{"stretchy":588},[586,57112,3697],{"fence":103},[586,57114,589],{"fence":103},[83,57116,332],{"mathvariant":197},[95,57118,57119],{"encoding":97},"S(\\nu,f)=\\left|\\mathcal{F}_{\\tau}\\left\\{P_e(\\tau,f)\\right\\}\\right|.",[61,57121,57123,57159],{"className":57122,"ariaHidden":103},[102],[61,57124,57126,57129,57132,57135,57138,57141,57144,57147,57150,57153,57156],{"className":57125},[107],[61,57127],{"className":57128,"style":627},[111],[61,57130,28508],{"className":57131,"style":19831},[116,120],[61,57133,849],{"className":57134},[631],[61,57136,47231],{"className":57137,"style":47252},[116,120],[61,57139,862],{"className":57140},[977],[61,57142],{"className":57143,"style":982},[981],[61,57145,19797],{"className":57146,"style":20151},[116,120],[61,57148,300],{"className":57149},[730],[61,57151],{"className":57152,"style":19197},[981],[61,57154,2000],{"className":57155},[19201],[61,57157],{"className":57158,"style":19197},[981],[61,57160,57162,57165,57290,57293],{"className":57161},[107],[61,57163],{"className":57164,"style":627},[111],[61,57166,57168,57171,57217,57220,57287],{"className":57167},[19266],[61,57169,589],{"className":57170,"style":19271},[631,19270],[61,57172,57174,57179],{"className":57173},[116],[61,57175,57088],{"className":57176,"style":57178},[116,57177],"mathcal","margin-right:0.0993em;",[61,57180,57182],{"className":57181},[125],[61,57183,57185,57209],{"className":57184},[129,130],[61,57186,57188,57206],{"className":57187},[134],[61,57189,57191],{"className":57190,"style":20164},[138],[61,57192,57194,57197],{"style":57193},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[61,57195],{"className":57196,"style":147},[146],[61,57198,57200],{"className":57199},[151,152,153,154],[61,57201,57203],{"className":57202},[116,154],[61,57204,45968],{"className":57205,"style":46040},[116,120,154],[61,57207,170],{"className":57208},[169],[61,57210,57212],{"className":57211},[134],[61,57213,57215],{"className":57214,"style":177},[138],[61,57216],{},[61,57218],{"className":57219,"style":982},[981],[61,57221,57223,57226,57266,57269,57272,57275,57278,57281,57284],{"className":57222},[19266],[61,57224,3688],{"className":57225,"style":19271},[631,19270],[61,57227,57229,57232],{"className":57228},[116],[61,57230,45957],{"className":57231,"style":23700},[116,120],[61,57233,57235],{"className":57234},[125],[61,57236,57238,57258],{"className":57237},[129,130],[61,57239,57241,57255],{"className":57240},[134],[61,57242,57244],{"className":57243,"style":20164},[138],[61,57245,57246,57249],{"style":28699},[61,57247],{"className":57248,"style":147},[146],[61,57250,57252],{"className":57251},[151,152,153,154],[61,57253,21378],{"className":57254},[116,120,154],[61,57256,170],{"className":57257},[169],[61,57259,57261],{"className":57260},[134],[61,57262,57264],{"className":57263,"style":177},[138],[61,57265],{},[61,57267,849],{"className":57268},[631],[61,57270,45968],{"className":57271,"style":46040},[116,120],[61,57273,862],{"className":57274},[977],[61,57276],{"className":57277,"style":982},[981],[61,57279,19797],{"className":57280,"style":20151},[116,120],[61,57282,300],{"className":57283},[730],[61,57285,3697],{"className":57286,"style":19271},[730,19270],[61,57288,589],{"className":57289,"style":19271},[730,19270],[61,57291],{"className":57292,"style":982},[981],[61,57294,332],{"className":57295},[116],[12,57297,57298],{},"This probability expression contains a squared sinusoid. Using the identity",[61,57300,57302],{"className":57301},[26354],[61,57303,57305,57381],{"className":57304},[64],[61,57306,57308],{"className":57307},[68],[70,57309,57310],{"xmlns":72,"display":26363},[74,57311,57312,57378],{},[77,57313,57314,57324,57344,57346,57352],{},[19151,57315,57316,57322],{},[77,57317,57318,57320],{},[83,57319,5069],{},[586,57321,19125],{},[1493,57323,337],{},[77,57325,57326,57328,57342],{},[586,57327,849],{"fence":103},[19134,57329,57330,57340],{},[77,57331,57332,57338],{},[80,57333,57334,57336],{},[83,57335,283],{"mathvariant":197},[83,57337,27125],{},[83,57339,45968],{},[1493,57341,337],{},[586,57343,300],{"fence":103},[586,57345,2000],{},[19134,57347,57348,57350],{},[1493,57349,57],{},[1493,57351,337],{},[77,57353,57354,57356,57358,57360,57362,57364,57366,57372,57374,57376],{},[586,57355,27808],{"fence":103},[1493,57357,57],{},[586,57359,19132],{},[83,57361,5051],{},[586,57363,19125],{},[586,57365,849],{"stretchy":588},[80,57367,57368,57370],{},[83,57369,283],{"mathvariant":197},[83,57371,27125],{},[83,57373,45968],{},[586,57375,300],{"stretchy":588},[586,57377,27817],{"fence":103},[95,57379,57380],{"encoding":97},"\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right)=\\frac{1}{2}\\left[1-\\cos(\\Omega_R\\tau)\\right]",[61,57382,57384,57548],{"className":57383,"ariaHidden":103},[102],[61,57385,57387,57390,57419,57422,57539,57542,57545],{"className":57386},[107],[61,57388],{"className":57389,"style":26592},[111],[61,57391,57393,57396],{"className":57392},[19259],[61,57394,5069],{"className":57395},[19259],[61,57397,57399],{"className":57398},[125],[61,57400,57402],{"className":57401},[129],[61,57403,57405],{"className":57404},[134],[61,57406,57408],{"className":57407,"style":51244},[138],[61,57409,57410,57413],{"style":51247},[61,57411],{"className":57412,"style":147},[146],[61,57414,57416],{"className":57415},[151,152,153,154],[61,57417,337],{"className":57418},[116,154],[61,57420],{"className":57421,"style":982},[981],[61,57423,57425,57431,57533],{"className":57424},[19266],[61,57426,57428],{"className":57427,"style":19271},[631,19270],[61,57429,849],{"className":57430},[19275,153],[61,57432,57434,57437,57530],{"className":57433},[116],[61,57435],{"className":57436},[631,19286],[61,57438,57440],{"className":57439},[19134],[61,57441,57443,57522],{"className":57442},[129,130],[61,57444,57446,57519],{"className":57445},[134],[61,57447,57449,57460,57468],{"className":57448,"style":51287},[138],[61,57450,57451,57454],{"style":28925},[61,57452],{"className":57453,"style":19306},[146],[61,57455,57457],{"className":57456},[116],[61,57458,337],{"className":57459},[116],[61,57461,57462,57465],{"style":19352},[61,57463],{"className":57464,"style":19306},[146],[61,57466],{"className":57467,"style":19360},[19359],[61,57469,57470,57473],{"style":28945},[61,57471],{"className":57472,"style":19306},[146],[61,57474,57476,57516],{"className":57475},[116],[61,57477,57479,57482],{"className":57478},[116],[61,57480,283],{"className":57481},[116],[61,57483,57485],{"className":57484},[125],[61,57486,57488,57508],{"className":57487},[129,130],[61,57489,57491,57505],{"className":57490},[134],[61,57492,57494],{"className":57493,"style":139},[138],[61,57495,57496,57499],{"style":241},[61,57497],{"className":57498,"style":147},[146],[61,57500,57502],{"className":57501},[151,152,153,154],[61,57503,27125],{"className":57504,"style":30082},[116,120,154],[61,57506,170],{"className":57507},[169],[61,57509,57511],{"className":57510},[134],[61,57512,57514],{"className":57513,"style":177},[138],[61,57515],{},[61,57517,45968],{"className":57518,"style":46040},[116,120],[61,57520,170],{"className":57521},[169],[61,57523,57525],{"className":57524},[134],[61,57526,57528],{"className":57527,"style":29011},[138],[61,57529],{},[61,57531],{"className":57532},[730,19286],[61,57534,57536],{"className":57535,"style":19271},[730,19270],[61,57537,300],{"className":57538},[19275,153],[61,57540],{"className":57541,"style":19197},[981],[61,57543,2000],{"className":57544},[19201],[61,57546],{"className":57547,"style":19197},[981],[61,57549,57551,57554,57616,57619],{"className":57550},[107],[61,57552],{"className":57553,"style":53051},[111],[61,57555,57557,57560,57613],{"className":57556},[116],[61,57558],{"className":57559},[631,19286],[61,57561,57563],{"className":57562},[19134],[61,57564,57566,57605],{"className":57565},[129,130],[61,57567,57569,57602],{"className":57568},[134],[61,57570,57572,57583,57591],{"className":57571,"style":52564},[138],[61,57573,57574,57577],{"style":28925},[61,57575],{"className":57576,"style":19306},[146],[61,57578,57580],{"className":57579},[116],[61,57581,337],{"className":57582},[116],[61,57584,57585,57588],{"style":19352},[61,57586],{"className":57587,"style":19306},[146],[61,57589],{"className":57590,"style":19360},[19359],[61,57592,57593,57596],{"style":28945},[61,57594],{"className":57595,"style":19306},[146],[61,57597,57599],{"className":57598},[116],[61,57600,57],{"className":57601},[116],[61,57603,170],{"c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that the dominant oscillatory spectral component appears at the generalized frequency",[61,57696,57698],{"className":57697},[26354],[61,57699,57701,57737],{"className":57700},[64],[61,57702,57704],{"className":57703},[68],[70,57705,57706],{"xmlns":72,"display":26363},[74,57707,57708,57734],{},[77,57709,57710,57716,57718,57732],{},[80,57711,57712,57714],{},[83,57713,47231],{},[83,57715,27125],{},[586,57717,2000],{},[19134,57719,57720,57726],{},[80,57721,57722,57724],{},[83,57723,283],{"mathvariant":197},[83,57725,27125],{},[77,57727,57728,57730],{},[1493,57729,337],{},[83,57731,19792],{},[83,57733,332],{"mathvariant":197},[95,57735,57736],{"encoding":97},"\\nu_R=\\frac{\\Omega_R}{2\\pi}.",[61,57738,57740,57795],{"className":57739,"ariaHidden":103},[102],[61,57741,57743,57746,57786,57789,57792],{"className":57742},[107],[61,57744],{"className":57745,"style":112},[111],[61,57747,57749,57752],{"className":57748},[116],[61,57750,47231],{"className":57751,"style":47252},[116,120],[61,57753,57755],{"className":57754},[125],[61,57756,57758,57778],{"className":57757},[129,130],[61,57759,57761,57775],{"className":57760},[134],[61,57762,57764],{"className":57763,"style":139},[138],[61,57765,57766,57769],{"style":47267},[61,57767],{"className":57768,"style":147},[146],[61,57770,57772],{"className":57771},[151,152,153,154],[61,57773,27125],{"className":57774,"style":30082},[116,120,154],[61,57776,170],{"className":57777},[169],[61,57779,57781],{"className":57780},[134],[61,57782,57784],{"className":57783,"style":177},[138],[61,57785],{},[61,57787],{"className":57788,"style":19197},[981],[61,57790,2000],{"className":57791},[19201],[61,57793],{"className":57794,"style":19197},[981],[61,57796,57798,57801,57903],{"className":57797},[107],[61,57799],{"className":57800,"style":52682},[111],[61,57802,57804,57807,57900],{"className":57803},[116],[61,57805],{"className":57806},[631,19286],[61,57808,57810],{"className":57809},[19134],[61,57811,57813,57892],{"className":57812},[129,130],[61,57814,57816,57889],{"className":57815},[134],[61,57817,57819,57833,57841],{"className":57818,"style":51287},[138],[61,57820,57821,57824],{"style":28925},[61,57822],{"className":57823,"style":19306},[146],[61,57825,57827,57830],{"className":57826},[116],[61,57828,337],{"className":57829},[116],[61,57831,19792],{"className":57832,"style":121},[116,120],[61,57834,57835,57838],{"style":19352},[61,57836],{"className":57837,"style":19306},[146],[61,57839],{"className":57840,"style":19360},[19359],[61,57842,57843,57846],{"style":28945},[61,57844],{"className":57845,"style":19306},[146],[61,57847,57849],{"className":57848},[116],[61,57850,57852,57855],{"className":57851},[116],[61,57853,283],{"className":57854},[116],[61,57856,57858],{"className":57857},[125],[61,57859,57861,57881],{"className":57860},[129,130],[61,57862,57864,57878],{"className":57863},[134],[61,57865,57867],{"className":57866,"style":139},[138],[61,57868,57869,57872],{"style":241},[61,57870],{"className":57871,"style":147},[146],[61,57873,57875],{"className":57874},[151,152,153,154],[61,57876,27125],{"className":57877,"style":30082},[116,120,154],[61,57879,170],{"className":57880},[169],[61,57882,57884],{"className":57883},[134],[61,57885,57887],{"className":57886,"style":177},[138],[61,57888],{},[61,57890,170],{"className":57891},[169],[61,57893,57895],{"className":57894},[134],[61,57896,57898],{"className":57897,"style":29011},[138],[61,57899],{},[61,57901],{"className":57902},[730,19286],[61,57904,332],{"className":57905},[116],[12,57907,57908],{},"The computational workflow subtracts the mean signal along the pulse-duration axis, applies a Hann window to reduce spectral leakage, uses zero-padding to smooth the displayed Fourier grid, and overlays the exact generalized Rabi ridge on the Fourier heatmap. Zero-padding improves visual interpolation of the plotted spectrum, although the physical frequency resolution remains set by the sampled temporal window.",[25,57910,57912],{"id":57911},"computational-methodology","Computational Methodology",[12,57914,57915,57916,57944,57945,57973,57974,27098,58002,58030,58031,58165,58166,58385],{},"The notebook uses a deterministic closed-form workflow, so it does not require numerical time stepping of the Schrödinger equation. It defines a drive-frequency grid ",[61,57917,57919,57932],{"className":57918},[64],[61,57920,57922],{"className":57921},[68],[70,57923,57924],{"xmlns":72},[74,57925,57926,57930],{},[77,57927,57928],{},[83,57929,19797],{},[95,57931,19797],{"encoding":97},[61,57933,57935],{"className":57934,"ariaHidden":103},[102],[61,57936,57938,57941],{"className":57937},[107],[61,57939],{"className":57940,"style":46075},[111],[61,57942,19797],{"className":57943,"style":20151},[116,120]," in GHz and a pulse-duration grid ",[61,57946,57948,57961],{"className":57947},[64],[61,57949,57951],{"className":57950},[68],[70,57952,57953],{"xmlns":72},[74,57954,57955,57959],{},[77,57956,57957],{},[83,57958,45968],{},[95,57960,46096],{"encoding":97},[61,57962,57964],{"className":57963,"ariaHidden":103},[102],[61,57965,57967,57970],{"className":57966},[107],[61,57968],{"className":57969,"style":30277},[111],[61,57971,45968],{"className":57972,"style":46040},[116,120]," in ns, then converts both to SI units. From a dense two-dimensional mesh over ",[61,57975,57977,57990],{"className":57976},[64],[61,57978,57980],{"className":57979},[68],[70,57981,57982],{"xmlns":72},[74,57983,57984,57988],{},[77,57985,57986],{},[83,57987,19797],{},[95,57989,19797],{"encoding":97},[61,57991,57993],{"className":57992,"ariaHidden":103},[102],[61,57994,57996,57999],{"className":57995},[107],[61,57997],{"className":57998,"style":46075},[111],[61,58000,19797],{"className":58001,"style":20151},[116,120],[61,58003,58005,58018],{"className":58004},[64],[61,58006,58008],{"className":58007},[68],[70,58009,58010],{"xmlns":72},[74,58011,58012,58016],{},[77,58013,58014],{},[83,58015,45968],{},[95,58017,46096],{"encoding":97},[61,58019,58021],{"className":58020,"ariaHidden":103},[102],[61,58022,58024,58027],{"className":58023},[107],[61,58025],{"className":58026,"style":30277},[111],[61,58028,45968],{"className":58029,"style":46040},[116,120],", the workflow computes the detuning ",[61,58032,58034,58068],{"className":58033},[64],[61,58035,58037],{"className":58036},[68],[70,58038,58039],{"xmlns":72},[74,58040,58041,58065],{},[77,58042,58043,58045,58047,58049,58051,58053,58055,58057,58063],{},[83,58044,198],{"mathvariant":197},[586,58046,2000],{},[1493,58048,337],{},[83,58050,19792],{},[586,58052,849],{"stretchy":588},[83,58054,19797],{},[586,58056,19132],{},[80,58058,58059,58061],{},[83,58060,19797],{},[1493,58062,1495],{},[586,58064,300],{"stretchy":588},[95,58066,58067],{"encoding":97},"\\Delta=2\\pi(f-f_0)",[61,58069,58071,58089,58116],{"className":58070,"ariaHidden":103},[102],[61,58072,58074,58077,58080,58083,58086],{"className":58073},[107],[61,58075],{"className":58076,"style":296},[111],[61,58078,198],{"className":58079},[116],[61,58081],{"className":58082,"style":19197},[981],[61,58084,2000],{"className":58085},[19201],[61,58087],{"className":58088,"style":19197},[981],[61,58090,58092,58095,58098,58101,58104,58107,58110,58113],{"className":58091},[107],[61,58093],{"className":58094,"style":627},[111],[61,58096,337],{"className":58097},[116],[61,58099,19792],{"className":58100,"style":121},[116,120],[61,58102,849],{"className":58103},[631],[61,58105,19797],{"className":58106,"style":20151},[116,120],[61,58108],{"className":58109,"style":20194},[981],[61,58111,19132],{"className":58112},[19382],[61,58114],{"className":58115,"style":20194},[981],[61,58117,58119,58122,58162],{"className":58118},[107],[61,58120],{"className":58121,"style":627},[111],[61,58123,58125,58128],{"className":58124},[116],[61,58126,19797],{"className":58127,"style":20151},[116,120],[61,58129,58131],{"className":58130},[125],[61,58132,58134,58154],{"className":58133},[129,130],[61,58135,58137,58151],{"className":58136},[134],[61,58138,58140],{"className":58139,"style":19230},[138],[61,58141,58142,58145],{"style":20167},[61,58143],{"className":58144,"style":147},[146],[61,58146,58148],{"className":58147},[151,152,153,154],[61,58149,1495],{"className":58150},[116,154],[61,58152,170],{"className":58153},[169],[61,58155,58157],{"className":58156},[134],[61,58158,58160],{"className":58159,"style":177},[138],[61,58161],{},[61,58163,300],{"className":58164},[730],", evaluates the generalized Rabi angular frequency ",[61,58167,58169,58206],{"className":58168},[64],[61,58170,58172],{"className":58171},[68],[70,58173,58174],{"xmlns":72},[74,58175,58176,58204],{},[77,58177,58178,58184,58186],{},[80,58179,58180,58182],{},[83,58181,283],{"mathvariant":197},[83,58183,27125],{},[586,58185,2000],{},[28194,58187,58188],{},[77,58189,58190,58196,58198],{},[19151,58191,58192,58194],{},[83,58193,283],{"mathvariant":197},[1493,58195,337],{},[586,58197,2419],{},[19151,58199,58200,58202],{},[83,58201,198],{"mathvariant":197},[1493,58203,337],{},[95,58205,51429],{"encoding":97},[61,58207,58209,58264],{"className":58208,"ariaHidden":103},[102],[61,58210,58212,58215,58255,58258,58261],{"className":58211},[107],[61,58213],{"className":58214,"style":219},[111],[61,58216,58218,58221],{"className":58217},[116],[61,58219,283],{"className":58220},[116],[61,58222,58224],{"className":58223},[125],[61,58225,58227,58247],{"className":58226},[129,130],[61,58228,58230,58244],{"className":58229},[134],[61,58231,58233],{"className":58232,"style":139},[138],[61,58234,58235,58238],{"style":241},[61,58236],{"className":58237,"style":147},[146],[61,58239,58241],{"className":58240},[151,152,153,154],[61,58242,27125],{"className":58243,"style":30082},[116,120,154],[61,58245,170],{"className":58246},[169],[61,58248,58250],{"className":58249},[134],[61,58251,58253],{"className":58252,"style":177},[138],[61,58254],{},[61,58256],{"className":58257,"style":19197},[981],[61,58259,2000],{"className":58260},[19201],[61,58262],{"className":58263,"style":19197},[981],[61,58265,58267,58271],{"className":58266},[107],[61,58268],{"className":58269,"style":58270},[111],"height:1.04em;vertical-align:-0.1266em;",[61,58272,58274],{"className":58273},[116,2425],[61,58275,58277,58376],{"className":58276},[129,130],[61,58278,58280,58373],{"className":58279},[134],[61,58281,58284,58360],{"className":58282,"style":58283},[138],"height:0.9134em;",[61,58285,58287,58290],{"className":58286,"style":28379},[28378],[61,58288],{"className":58289,"style":19306},[146],[61,58291,58293,58322,58325,58328,58331],{"className":58292,"style":28386},[116],[61,58294,58296,58299],{"className":58295},[116],[61,58297,283],{"className":58298},[116],[61,58300,58302],{"className":58301},[125],[61,58303,58305],{"className":58304},[129],[61,58306,58308],{"className":58307},[134],[61,58309,58311],{"className":58310,"style":51538},[138],[61,58312,58313,58316],{"style":51541},[61,58314],{"className":58315,"style":147},[146],[61,58317,58319],{"className":58318},[151,152,153,154],[61,58320,337],{"className":58321},[116,154],[61,58323],{"className":58324,"style":20194},[981],[61,58326,2419],{"className":58327},[19382],[61,58329],{"className":58330,"style":20194},[981],[61,58332,58334,58337],{"className":58333},[116],[61,58335,198],{"className":58336},[116],[61,58338,58340],{"className":58339},[125],[61,58341,58343],{"className":58342},[129],[61,58344,58346],{"className":58345},[134],[61,58347,58349],{"className":58348,"style":51538},[138],[61,58350,58351,58354],{"style":51541},[61,58352],{"className":58353,"style":147},[146],[61,58355,58357],{"className":58356},[151,152,153,154],[61,58358,337],{"className":58359},[116,154],[61,58361,58363,58366],{"style":58362},"top:-2.8734em;",[61,58364],{"className":58365,"style":19306},[146],[61,58367,58369],{"className":58368,"style":28400},[28399],[28402,58370,58371],{"xmlns":28404,"width":28405,"height":28406,"viewBox":28407,"preserveAspectRatio":28408},[28410,58372],{"d":28412},[61,58374,170],{"className":58375},[169],[61,58377,58379],{"className":58378},[134],[61,58380,58383],{"className":58381,"style":58382},[138],"height:0.1266em;",[61,58384],{},", and calculates the analytic excited-state probability",[61,58387,58389],{"className":58388},[26354],[61,58390,58392,58470],{"className":58391},[64],[61,58393,58395],{"className":58394},[68],[70,58396,58397],{"xmlns":72,"display":26363},[74,58398,58399,58467],{},[77,58400,58401,58407,58409,58411,58413,58415,58417,58419,58435,58445,58465],{},[80,58402,58403,58405],{},[83,58404,45957],{},[83,58406,21378],{},[586,58408,849],{"stretchy":588},[83,58410,19797],{},[586,58412,862],{"separator":103},[83,58414,45968],{},[586,58416,300],{"stretchy":588},[586,58418,2000],{},[19134,58420,58421,58427],{},[19151,58422,58423,58425],{},[83,58424,283],{"mathvariant":197},[1493,58426,337],{},[34480,58428,58429,58431,58433],{},[83,58430,283],{"mathvariant":197},[83,58432,27125],{},[1493,58434,337],{},[19151,58436,58437,58443],{},[77,58438,58439,58441],{},[83,58440,5069],{},[586,58442,19125],{},[1493,58444,337],{},[77,58446,58447,58449,58463],{},[586,58448,849],{"fence":103},[19134,58450,58451,58461],{},[77,58452,58453,58459],{},[80,58454,58455,58457],{},[83,58456,283],{"mathvariant":197},[83,58458,27125],{},[83,58460,45968],{},[1493,58462,337],{},[586,58464,300],{"fence":103},[83,58466,332],{"mathvariant":197},[95,58468,58469],{"encoding":97},"P_e(f,\\tau)=\\frac{\\Omega^2}{\\Omega_R^2}\\sin^2\\left(\\frac{\\Omega_R\\tau}{2}\\right).",[61,58471,58473,58546],{"className":58472,"ariaHidden":103},[102],[61,58474,58476,58479,58519,58522,58525,58528,58531,58534,58537,58540,58543],{"className":58475},[107],[61,58477],{"className":58478,"style":627},[111],[61,58480,58482,58485],{"className":58481},[116],[61,58483,45957],{"className":58484,"style":23700},[116,120],[61,58486,58488],{"className":58487},[125],[61,58489,58491,58511],{"className":58490},[129,130],[61,58492,58494,58508],{"className":58493},[134],[61,58495,58497],{"className":58496,"style":20164},[138],[61,58498,58499,58502],{"style":28699},[61,58500],{"className":58501,"style":147},[146],[61,58503,58505],{"className":58504},[151,152,153,154],[61,58506,21378],{"className":58507},[116,120,154],[61,58509,170],{"className":58510},[169],[61,58512,58514],{"className":58513},[134],[61,58515,58517],{"className":58516,"style":177},[138],[61,58518],{},[61,58520,849],{"className":58521},[631],[61,58523,19797],{"className":58524,"style":20151},[116,120],[61,58526,862],{"className":58527},[977],[61,58529],{"className":58530,"style":982},[981],[61,58532,45968],{"className":58533,"style":46040},[116,120],[61,58535,300],{"className":58536},[730],[61,58538],{"className":58539,"style":19197},[981],[61,58541,2000],{"className":58542},[19201],[61,58544],{"className":58545,"style":19197},[981],[61,58547,58549,58552,58688,58691,58720,58723,58840,58843],{"className":58548},[107],[61,58550],{"className":58551,"style":51080},[111],[61,58553,58555,58558,58685],{"className":58554},[116],[61,58556],{"className":58557},[631,19286],[61,58559,58561],{"className":58560},[19134],[61,58562,58564,58677],{"className":58563},[129,130],[61,58565,58567,58674],{"className":58566},[134],[61,58568,58570,58629,58637],{"className":58569,"style":51099},[138],[61,58571,58572,58575],{"style":28925},[61,58573],{"className":58574,"style":19306},[146],[61,58576,58578],{"className":58577},[116],[61,58579,58581,58584],{"className":58580},[116],[61,58582,283],{"className":58583},[116],[61,58585,58587],{"className":58586},[125],[61,58588,58590,58621],{"className":58589},[129,130],[61,58591,58593,58618],{"className":58592},[134],[61,58594,58596,58607],{"className":58595,"style":51126},[138],[61,58597,58598,58601],{"style":51129},[61,58599],{"className":58600,"style":147},[146],[61,58602,58604],{"className":58603},[151,152,153,154],[61,58605,27125],{"className":58606,"style":30082},[116,120,154],[61,58608,58609,58612],{"style":51141},[61,58610],{"className":58611,"style":147},[146],[61,58613,58615],{"className":58614},[151,152,153,154],[61,58616,337],{"className":58617},[116,154],[61,58619,170],{"className":58620},[169],[61,58622,58624],{"className":58623},[134],[61,58625,58627],{"className":58626,"style":51160},[138],[61,58628],{},[61,58630,58631,58634],{"style":19352},[61,58632],{"className":58633,"style":19306},[146],[61,58635],{"className":58636,"style":19360},[19359],[61,58638,58639,58642],{"style":28945},[61,58640],{"className":58641,"style":19306},[146],[61,58643,58645],{"className":58644},[116],[61,58646,58648,58651],{"className":58647},[116],[61,58649,283],{"className":58650},[116],[61,58652,58654],{"className":58653},[125],[61,58655,58657],{"className":58656},[129],[61,58658,58660],{"className":58659},[134],[61,58661,58663],{"className":58662,"style":27160},[138],[61,58664,58665,58668],{"style":21487},[61,58666],{"className":58667,"style":147},[146],[61,58669,58671],{"className":58670},[151,152,153,154],[61,58672,337],{"className":58673},[116,154],[61,58675,170],{"className":58676},[169],[61,58678,58680],{"className":58679},[134],[61,58681,58683],{"className":58682,"style":51217},[138],[61,58684],{},[61,58686],{"className":58687},[730,19286],[61,58689],{"className":58690,"style":982},[981],[61,58692,58694,58697],{"className":58693},[19259],[61,58695,5069],{"className":58696},[19259],[61,58698,58700],{"className":58699},[125],[61,58701,58703],{"className":58702},[129],[61,58704,58706],{"className":58705},[134],[61,58707,58709],{"className":58708,"style":51244},[138],[61,58710,58711,58714],{"style":51247},[61,58712],{"className":58713,"style":147},[146],[61,58715,58717],{"className":58716},[151,152,153,154],[61,58718,337],{"className":58719},[116,154],[61,58721],{"className":58722,"style":982},[981],[61,58724,58726,58732,58834],{"className":58725},[19266],[61,58727,58729],{"className":58728,"style":19271},[631,19270],[61,58730,849],{"className":58731},[19275,153],[61,58733,58735,58738,58831],{"className":58734},[116],[61,58736],{"className":58737},[631,19286],[61,58739,58741],{"className":58740},[19134],[61,58742,58744,58823],{"className":58743},[129,130],[61,58745,58747,58820],{"className":58746},[134],[61,58748,58750,58761,58769],{"className":58749,"style":51287},[138],[61,58751,58752,58755],{"style":28925},[61,58753],{"className":58754,"style":19306},[146],[61,58756,58758],{"className":58757},[116],[61,58759,337],{"className":58760},[116],[61,58762,58763,58766],{"style":19352},[61,58764],{"className":58765,"style":19306},[146],[61,58767],{"className":58768,"style":19360},[19359],[61,58770,58771,58774],{"style":28945},[61,58772],{"className":58773,"style":19306},[146],[61,58775,58777,58817],{"className":58776},[116],[61,58778,58780,58783],{"className":58779},[116],[61,58781,283],{"className":58782},[116],[61,58784,58786],{"className":58785},[125],[61,58787,58789,58809],{"className":58788},[129,130],[61,58790,58792,58806],{"className":58791},[134],[61,58793,58795],{"className":58794,"style":139},[138],[61,58796,58797,58800],{"style":241},[61,58798],{"className":58799,"style":147},[146],[61,58801,58803],{"className":58802},[151,152,153,154],[61,58804,27125],{"className":58805,"style":30082},[116,120,154],[61,58807,170],{"className":58808},[169],[61,58810,58812],{"className":58811},[134],[61,58813,58815],{"className":58814,"style":177},[138],[61,58816],{},[61,58818,45968],{"className":58819,"style":46040},[116,120],[61,58821,170],{"className":58822},[169],[61,58824,58826],{"className":58825},[134],[61,58827,58829],{"className":58828,"style":29011},[138],[61,58830],{},[61,58832],{"className":58833},[730,19286],[61,58835,58837],{"className":58836,"style":19271},[730,19270],[61,58838,300],{"className":58839},[19275,153],[61,58841],{"className":58842,"style":982},[981],[61,58844,332],{"className":58845},[116],[12,58847,58848,58849,58964],{},"The routine optionally applies the damping envelope ",[61,58850,58852,58886],{"className":58851},[64],[61,58853,58855],{"className":58854},[68],[70,58856,58857],{"xmlns":72},[74,58858,58859,58883],{},[77,58860,58861,58863,58865,58867,58869,58871,58873,58881],{},[83,58862,6987],{},[586,58864,19125],{},[586,58866,849],{"stretchy":588},[586,58868,19132],{},[83,58870,45968],{},[83,58872,602],{"mathvariant":197},[34480,58874,58875,58877,58879],{},[83,58876,23559],{},[1493,58878,337],{},[586,58880,47495],{},[586,58882,300],{"stretchy":588},[95,58884,58885],{"encoding":97},"\\exp(-\\tau\u002FT_2^\\ast)",[61,58887,58889],{"className":58888,"ariaHidden":103},[102],[61,58890,58892,58895,58898,58901,58904,58907,58910,58961],{"className":58891},[107],[61,58893],{"className":58894,"style":627},[111],[61,58896,6987],{"className":58897},[19259],[61,58899,849],{"className":58900},[631],[61,58902,19132],{"className":58903},[116],[61,58905,45968],{"className":58906,"style":46040},[116,120],[61,58908,602],{"className":58909},[116],[61,58911,58913,58916],{"className":58912},[116],[61,58914,23559],{"className":58915,"style":23700},[116,120],[61,58917,58919],{"className":58918},[125],[61,58920,58922,58953],{"className":58921},[129,130],[61,58923,58925,58950],{"className":58924},[134],[61,58926,58928,58939],{"className":58927,"style":47527},[138],[61,58929,58930,58933],{"style":47530},[61,58931],{"className":58932,"style":147},[146],[61,58934,58936],{"className":58935},[151,152,153,154],[61,58937,337],{"className":58938},[116,154],[61,58940,58941,58944],{"style":21487},[61,58942],{"className":58943,"style":147},[146],[61,58945,58947],{"className":58946},[151,152,153,154],[61,58948,47495],{"className":58949},[19382,154],[61,58951,170],{"className":58952},[169],[61,58954,58956],{"className":58955},[134],[61,58957,58959],{"className":58958,"style":47560},[138],[61,58960],{},[61,58962,300],{"className":58963},[730]," and clips the result to the physical interval",[61,58966,58968],{"className":58967},[26354],[61,58969,58971,58998],{"className":58970},[64],[61,58972,58974],{"className":58973},[68],[70,58975,58976],{"xmlns":72,"display":26363},[74,58977,58978,58995],{},[77,58979,58980,58982,58985,58991,58993],{},[1493,58981,1495],{},[586,58983,58984],{},"≤",[80,58986,58987,58989],{},[83,58988,45957],{},[83,58990,21378],{},[586,58992,58984],{},[1493,58994,48600],{},[95,58996,58997],{"encoding":97},"0\\le P_e\\le 1.",[61,58999,59001,59020,59075],{"className":59000,"ariaHidden":103},[102],[61,59002,59004,59008,59011,59014,59017],{"className":59003},[107],[61,59005],{"className":59006,"style":59007},[111],"height:0.7804em;vertical-align:-0.136em;",[61,59009,1495],{"className":59010},[116],[61,59012],{"className":59013,"style":19197},[981],[61,59015,58984],{"className":59016},[19201],[61,59018],{"className":59019,"style":19197},[981],[61,59021,59023,59026,59066,59069,59072],{"className":59022},[107],[61,59024],{"className":59025,"style":219},[111],[61,59027,59029,59032],{"className":59028},[116],[61,59030,45957],{"className":59031,"style":23700},[116,120],[61,59033,59035],{"className":59034},[125],[61,59036,59038,59058],{"className":59037},[129,130],[61,59039,59041,59055],{"className":59040},[134],[61,59042,59044],{"className":59043,"style":20164},[138],[61,59045,59046,59049],{"style":28699},[61,59047],{"className":59048,"style":147},[146],[61,59050,59052],{"className":59051},[151,152,153,154],[61,59053,21378],{"className":59054},[116,120,154],[61,59056,170],{"className":59057},[169],[61,59059,59061],{"className":59060},[134],[61,59062,59064],{"className":59063,"style":177},[138],[61,59065],{},[61,59067],{"className":59068,"style":19197},[981],[61,59070,58984],{"className":59071},[19201],[61,59073],{"className":59074,"style":19197},[981],[61,59076,59078,59081],{"className":59077},[107],[61,59079],{"className":59080,"style":27780},[111],[61,59082,48600],{"className":59083},[116],[12,59085,59086],{},"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.",[25,59088,59090],{"id":59089},"graphical-interpretation","Graphical Interpretation",[12,59092,19697,59093,59191],{},[61,59094,59096,59124],{"className":59095},[64],[61,59097,59099],{"className":59098},[68],[70,59100,59101],{"xmlns":72},[74,59102,59103,59121],{},[77,59104,59105,59111,59113,59115,59117,59119],{},[80,59106,59107,59109],{},[83,59108,45957],{},[83,59110,21378],{},[586,59112,849],{"stretchy":588},[83,59114,19797],{},[586,59116,862],{"separator":103},[83,59118,45968],{},[586,59120,300],{"stretchy":588},[95,59122,59123],{"encoding":97},"P_e(f,\\tau)",[61,59125,59127],{"className":59126,"ariaHidden":103},[102],[61,59128,59130,59133,59173,59176,59179,59182,59185,59188],{"className":59129},[107],[61,59131],{"className":59132,"style":627},[111],[61,59134,59136,59139],{"className":59135},[116],[61,59137,45957],{"className":59138,"style":23700},[116,120],[61,59140,59142],{"className":59141},[125],[61,59143,59145,59165],{"className":59144},[129,130],[61,59146,59148,59162],{"className":59147},[134],[61,59149,59151],{"className":59150,"style":20164},[138],[61,59152,59153,59156],{"style":28699},[61,59154],{"className":59155,"style":147},[146],[61,59157,59159],{"className":59158},[151,152,153,154],[61,59160,21378],{"className":59161},[116,120,154],[61,59163,170],{"className":59164},[169],[61,59166,59168],{"className":59167},[134],[61,59169,59171],{"className":59170,"style":177},[138],[61,59172],{},[61,59174,849],{"className":59175},[631],[61,59177,19797],{"className":59178,"style":20151},[116,120],[61,59180,862],{"className":59181},[977],[61,59183],{"className":59184,"style":982},[981],[61,59186,45968],{"className":59187,"style":46040},[116,120],[61,59189,300],{"className":59190},[730]," 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.",[12,59193,59194],{},"The resonant interaction region appears near",[61,59196,59198],{"className":59197},[26354],[61,59199,59201,59224],{"className":59200},[64],[61,59202,59204],{"className":59203},[68],[70,59205,59206],{"xmlns":72,"display":26363},[74,59207,59208,59222],{},[77,59209,59210,59212,59214,59220],{},[83,59211,19797],{},[586,59213,2000],{},[80,59215,59216,59218],{},[83,59217,19797],{},[1493,59219,1495],{},[83,59221,332],{"mathvariant":197},[95,59223,48297],{"encoding":97},[61,59225,59227,59245],{"className":59226,"ariaHidden":103},[102],[61,59228,59230,59233,59236,59239,59242],{"className":59229},[107],[61,59231],{"className":59232,"style":46075},[111],[61,59234,19797],{"className":59235,"style":20151},[116,120],[61,59237],{"className":59238,"style":19197},[981],[61,59240,2000],{"className":59241},[19201],[61,59243],{"className":59244,"style":19197},[981],[61,59246,59248,59251,59291],{"className":59247},[107],[61,59249],{"className":59250,"style":46075},[111],[61,59252,59254,59257],{"className":59253},[116],[61,59255,19797],{"className":59256,"style":20151},[116,120],[61,59258,59260],{"className":59259},[125],[61,59261,59263,59283],{"className":59262},[129,130],[61,59264,59266,59280],{"className":59265},[134],[61,59267,59269],{"className":59268,"style":19230},[138],[61,59270,59271,59274],{"style":20167},[61,59272],{"className":59273,"style":147},[146],[61,59275,59277],{"className":59276},[151,152,153,154],[61,59278,1495],{"className":59279},[116,154],[61,59281,170],{"className":59282},[169],[61,59284,59286],{"className":59285},[134],[61,59287,59289],{"className":59288,"style":177},[138],[61,59290],{},[61,59292,332],{"className":59293},[116],[12,59295,59296,59297,59376],{},"At resonance, contrast is maximal and the pulse duration alone sets the rotation angle. With the default drive strength ",[61,59298,59300,59334],{"className":59299},[64],[61,59301,59303],{"className":59302},[68],[70,59304,59305],{"xmlns":72},[74,59306,59307,59331],{},[77,59308,59309,59311,59313,59315,59317,59319,59321,59323],{},[83,59310,283],{"mathvariant":197},[83,59312,602],{"mathvariant":197},[1493,59314,337],{},[83,59316,19792],{},[586,59318,2000],{},[1493,59320,14756],{},[27272,59322,27274],{},[77,59324,59325,59327,59329],{},[83,59326,52663],{"mathvariant":197},[83,59328,6871],{"mathvariant":197},[83,59330,26482],{"mathvariant":197},[95,59332,59333],{"encoding":97},"\\Omega\u002F2\\pi=20~\\mathrm{MHz}",[61,59335,59337,59358],{"className":59336,"ariaHidden":103},[102],[61,59338,59340,59343,59346,59349,59352,59355],{"className":59339},[107],[61,59341],{"className":59342,"style":627},[111],[61,59344,46977],{"className":59345},[116],[61,59347,19792],{"className":59348,"style":121},[116,120],[61,59350],{"className":59351,"style":19197},[981],[61,59353,2000],{"className":59354},[19201],[61,59356],{"className":59357,"style":19197},[981],[61,59359,59361,59364,59367,59370],{"className":59360},[107],[61,59362],{"className":59363,"style":296},[111],[61,59365,14756],{"className":59366},[116],[61,59368,27274],{"className":59369},[981,48148],[61,59371,59373],{"className":59372},[116],[61,59374,3772],{"className":59375},[116,27529],", the ideal resonant maxima appear near",[61,59378,59380],{"className":59379},[26354],[61,59381,59383,59456],{"className":59382},[64],[61,59384,59386],{"className":59385},[68],[70,59387,59388],{"xmlns":72,"display":26363},[74,59389,59390,59453],{},[77,59391,59392,59394,59396,59398,59400,59406,59408,59410,59413,59415,59421,59423,59425,59428,59430,59436,59438,59440,59443,59445,59451],{},[83,59393,45968],{},[586,59395,2000],{},[1493,59397,52835],{},[27272,59399,27274],{},[77,59401,59402,59404],{},[83,59403,27948],{"mathvariant":197},[83,59405,52844],{"mathvariant":197},[586,59407,862],{"separator":103},[27272,59409,27274],{},[1493,59411,59412],{},"75",[27272,59414,27274],{},[77,59416,59417,59419],{},[83,59418,27948],{"mathvariant":197},[83,59420,52844],{"mathvariant":197},[586,59422,862],{"separator":103},[27272,59424,27274],{},[1493,59426,59427],{},"125",[27272,59429,27274],{},[77,59431,59432,59434],{},[83,59433,27948],{"mathvariant":197},[83,59435,52844],{"mathvariant":197},[586,59437,862],{"separator":103},[27272,59439,27274],{},[1493,59441,59442],{},"175",[27272,59444,27274],{},[77,59446,59447,59449],{},[83,59448,27948],{"mathvariant":197},[83,59450,52844],{"mathvariant":197},[586,59452,862],{"separator":103},[95,59454,59455],{"encoding":97},"\\tau=25~\\mathrm{ns},\\ 75~\\mathrm{ns},\\ 125~\\mathrm{ns},\\ 175~\\mathrm{ns},",[61,59457,59459,59477],{"className":59458,"ariaHidden":103},[102],[61,59460,59462,59465,59468,59471,59474],{"className":59461},[107],[61,59463],{"className":59464,"style":30277},[111],[61,59466,45968],{"className":59467,"style":46040},[116,120],[61,59469],{"className":59470,"style":19197},[981],[61,59472,2000],{"className":59473},[19201],[61,59475],{"className":59476,"style":19197},[981],[61,59478,59480,59484,59487,59490,59496,59499,59502,59505,59508,59511,59517,59520,59523,59526,59529,59532,59538,59541,59544,59547,59550,59553,59559],{"className":59479},[107],[61,59481],{"className":59482,"style":59483},[111],"height:0.8389em;vertical-align:-0.1944em;",[61,59485,52835],{"className":59486},[116],[61,59488,27274],{"className":59489},[981,48148],[61,59491,59493],{"className":59492},[116],[61,59494,52926],{"className":59495},[116,27529],[61,59497,862],{"className":59498},[977],[61,59500,27274],{"className":59501},[981],[61,59503],{"className":59504,"style":982},[981],[61,59506,59412],{"className":59507},[116],[61,59509,27274],{"className":59510},[981,48148],[61,59512,59514],{"className":59513},[116],[61,59515,52926],{"className":59516},[116,27529],[61,59518,862],{"className":59519},[977],[61,59521,27274],{"className":59522},[981],[61,59524],{"className":59525,"style":982},[981],[61,59527,59427],{"className":59528},[116],[61,59530,27274],{"className":59531},[981,48148],[61,59533,59535],{"className":59534},[116],[61,59536,52926],{"className":59537},[116,27529],[61,59539,862],{"className":59540},[977],[61,59542,27274],{"className":59543},[981],[61,59545],{"className":59546,"style":982},[981],[61,59548,59442],{"className":59549},[116],[61,59551,27274],{"className":59552},[981,48148],[61,59554,59556],{"className":59555},[116],[61,59557,52926],{"className":59558},[116,27529],[61,59560,862],{"className":59561},[977],[12,59563,59564],{},"before the optional phenomenological damping factor is applied.",[12,59566,59567,59568,59637,59638,59764],{},"Off resonance, the fringes show lower contrast and faster temporal oscillations. This behavior follows from the generalized Rabi frequency ",[61,59569,59571,59588],{"className":59570},[64],[61,59572,59574],{"className":59573},[68],[70,59575,59576],{"xmlns":72},[74,59577,59578,59586],{},[77,59579,59580],{},[80,59581,59582,59584],{},[83,59583,283],{"mathvariant":197},[83,59585,27125],{},[95,59587,47158],{"encoding":97},[61,59589,59591],{"className":59590,"ariaHidden":103},[102],[61,59592,59594,59597],{"className":59593},[107],[61,59595],{"className":59596,"style":219},[111],[61,59598,59600,59603],{"className":59599},[116],[61,59601,283],{"className":59602},[116],[61,59604,59606],{"className":59605},[125],[61,59607,59609,59629],{"className":59608},[129,130],[61,59610,59612,59626],{"className":59611},[134],[61,59613,59615],{"className":59614,"style":139},[138],[61,59616,59617,59620],{"style":241},[61,59618],{"className":59619,"style":147},[146],[61,59621,59623],{"className":59622},[151,152,153,154],[61,59624,27125],{"className":59625,"style":30082},[116,120,154],[61,59627,170],{"className":59628},[169],[61,59630,59632],{"className":59631},[134],[61,59633,59635],{"className":59634,"style":177},[138],[61,59636],{},", which increases with detuning, and from the amplitude coefficient ",[61,59639,59641,59669],{"className":59640},[64],[61,59642,59644],{"className":59643},[68],[70,59645,59646],{"xmlns":72},[74,59647,59648,59666],{},[77,59649,59650,59656,59658],{},[19151,59651,59652,59654],{},[83,59653,283],{"mathvariant":197},[1493,59655,337],{},[83,59657,602],{"mathvariant":197},[34480,59659,59660,59662,59664],{},[83,59661,283],{"mathvariant":197},[83,59663,27125],{},[1493,59665,337],{},[95,59667,59668],{"encoding":97},"\\Omega^2\u002F\\Omega_R^2",[61,59670,59672],{"className":59671,"ariaHidden":103},[102],[61,59673,59675,59679,59708,59711],{"className":59674},[107],[61,59676],{"className":59677,"style":59678},[111],"height:1.0894em;vertical-align:-0.2753em;",[61,59680,59682,59685],{"className":59681},[116],[61,59683,283],{"className":59684},[116],[61,59686,59688],{"className":59687},[125],[61,59689,59691],{"className":59690},[129],[61,59692,59694],{"className":59693},[134],[61,59695,59697],{"className":59696,"style":27160},[138],[61,59698,59699,59702],{"style":21487},[61,59700],{"className":59701,"style":147},[146],[61,59703,59705],{"className":59704},[151,152,153,154],[61,59706,337],{"className":59707},[116,154],[61,59709,602],{"className":59710},[116],[61,59712,59714,59717],{"className":59713},[116],[61,59715,283],{"className":59716},[116],[61,59718,59720],{"className":59719},[125],[61,59721,59723,59755],{"className":59722},[129,130],[61,59724,59726,59752],{"className":59725},[134],[61,59727,59729,59741],{"className":59728,"style":27160},[138],[61,59730,59732,59735],{"style":59731},"top:-2.4247em;margin-left:0em;margin-right:0.05em;",[61,59733],{"className":59734,"style":147},[146],[61,59736,59738],{"className":59737},[151,152,153,154],[61,59739,27125],{"className":59740,"style":30082},[116,120,154],[61,59742,59743,59746],{"style":21487},[61,59744],{"className":59745,"style":147},[146],[61,59747,59749],{"className":59748},[151,152,153,154],[61,59750,337],{"className":59751},[116,154],[61,59753,170],{"className":59754},[169],[61,59756,59758],{"className":59757},[134],[61,59759,59762],{"className":59760,"style":59761},[138],"height:0.2753em;",[61,59763],{},", which decreases with detuning.",[12,59766,59767],{},"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.",[12,59769,59770,59771,59868],{},"The FFT heatmap characterizes the oscillation-frequency content of ",[61,59772,59774,59801],{"className":59773},[64],[61,59775,59777],{"className":59776},[68],[70,59778,59779],{"xmlns":72},[74,59780,59781,59799],{},[77,59782,59783,59789,59791,59793,59795,59797],{},[80,59784,59785,59787],{},[83,59786,45957],{},[83,59788,21378],{},[586,59790,849],{"stretchy":588},[83,59792,45968],{},[586,59794,862],{"separator":103},[83,59796,19797],{},[586,59798,300],{"stretchy":588},[95,59800,56981],{"encoding":97},[61,59802,59804],{"className":59803,"ariaHidden":103},[102],[61,59805,59807,59810,59850,59853,59856,59859,59862,59865],{"className":59806},[107],[61,59808],{"className":59809,"style":627},[111],[61,59811,59813,59816],{"className":59812},[116],[61,59814,45957],{"className":59815,"style":23700},[116,120],[61,59817,59819],{"className":59818},[125],[61,59820,59822,59842],{"className":59821},[129,130],[61,59823,59825,59839],{"className":59824},[134],[61,59826,59828],{"className":59827,"style":20164},[138],[61,59829,59830,59833],{"style":28699},[61,59831],{"className":59832,"style":147},[146],[61,59834,59836],{"className":59835},[151,152,153,154],[61,59837,21378],{"className":59838},[116,120,154],[61,59840,170],{"className":59841},[169],[61,59843,59845],{"className":59844},[134],[61,59846,59848],{"className":59847,"style":177},[138],[61,59849],{},[61,59851,849],{"className":59852},[631],[61,59854,45968],{"className":59855,"style":46040},[116,120],[61,59857,862],{"className":59858},[977],[61,59860],{"className":59861,"style":982},[981],[61,59863,19797],{"className":59864,"style":20151},[116,120],[61,59866,300],{"className":59867},[730],". An overlaid theoretical curve follows",[61,59870,59872],{"className":59871},[26354],[61,59873,59875,59947],{"className":59874},[64],[61,59876,59878],{"className":59877},[68],[70,59879,59880],{"xmlns":72,"display":26363},[74,59881,59882,59944],{},[77,59883,59884,59890,59892,59894,59896,59898,59942],{},[80,59885,59886,59888],{},[83,59887,47231],{},[83,59889,27125],{},[586,59891,849],{"stretchy":588},[83,59893,19797],{},[586,59895,300],{"stretchy":588},[586,59897,2000],{},[28194,59899,59900],{},[77,59901,59902,59922,59924,59926,59928,59930,59936],{},[19151,59903,59904,59920],{},[77,59905,59906,59908,59918],{},[586,59907,849],{"fence":103},[19134,59909,59910,59912],{},[83,59911,283],{"mathvariant":197},[77,59913,59914,59916],{},[1493,59915,337],{},[83,59917,19792],{},[586,59919,300],{"fence":103},[1493,59921,337],{},[586,59923,2419],{},[586,59925,849],{"stretchy":588},[83,59927,19797],{},[586,59929,19132],{},[80,59931,59932,59934],{},[83,59933,19797],{},[1493,59935,1495],{},[19151,59937,59938,59940],{},[586,59939,300],{"stretchy":588},[1493,59941,337],{},[83,59943,332],{"mathvariant":197},[95,59945,59946],{"encoding":97},"\\nu_R(f)=\\sqrt{\\left(\\frac{\\Omega}{2\\pi}\\right)^2+(f-f_0)^2}.",[61,59948,59950,60014],{"className":59949,"ariaHidden":103},[102],[61,59951,59953,59956,59996,59999,60002,60005,60008,60011],{"className":59952},[107],[61,59954],{"className":59955,"style":627},[111],[61,59957,59959,59962],{"className":59958},[116],[61,59960,47231],{"className":59961,"style":47252},[116,120],[61,59963,59965],{"className":59964},[125],[61,59966,59968,59988],{"className":59967},[129,130],[61,59969,59971,59985],{"className":59970},[134],[61,59972,59974],{"className":59973,"style":139},[138],[61,59975,59976,59979],{"style":47267},[61,59977],{"className":59978,"style":147},[146],[61,59980,59982],{"className":59981},[151,152,153,154],[61,59983,27125],{"className":59984,"style":30082},[116,120,154],[61,59986,170],{"className":59987},[169],[61,59989,59991],{"className":59990},[134],[61,59992,59994],{"className":59993,"style":177},[138],[61,59995],{},[61,59997,849],{"className":59998},[631],[61,60000,19797],{"className":60001,"style":20151},[116,120],[61,60003,300],{"className":60004},[730],[61,60006],{"className":60007,"style":19197},[981],[61,60009,2000],{"className":60010},[19201],[61,60012],{"className":60013,"style":19197},[981],[61,60015,60017,60020,60263],{"className":60016},[107],[61,60018],{"className":60019,"style":54274},[111],[61,60021,60023],{"className":60022},[116,2425],[61,60024,60026,60255],{"className":60025},[129,130],[61,60027,60029,60252],{"className":60028},[134],[61,60030,60032,60240],{"className":60031,"style":54287},[138],[61,60033,60035,60038],{"className":60034,"style":54291},[28378],[61,60036],{"className":60037,"style":54295},[146],[61,60039,60041,60147,60150,60153,60156,60159,60162,60165,60168,60171,60211],{"className":60040,"style":51519},[116],[61,60042,60044,60124],{"className":60043},[19266],[61,60045,60047,60053,60118],{"className":60046},[19266],[61,60048,60050],{"className":60049,"style":19271},[631,19270],[61,60051,849],{"className":60052},[19275,153],[61,60054,60056,60059,60115],{"className":60055},[116],[61,60057],{"className":60058},[631,19286],[61,60060,60062],{"className":60061},[19134],[61,60063,60065,60107],{"className":60064},[129,130],[61,60066,60068,60104],{"className":60067},[134],[61,60069,60071,60085,60093],{"className":60070,"style":51287},[138],[61,60072,60073,60076],{"style":28925},[61,60074],{"className":60075,"style":19306},[146],[61,60077,60079,60082],{"className":60078},[116],[61,60080,337],{"className":60081},[116],[61,60083,19792],{"className":60084,"style":121},[116,120],[61,60086,60087,60090],{"style":19352},[61,60088],{"className":60089,"style":19306},[146],[61,60091],{"className":60092,"style":19360},[19359],[61,60094,60095,60098],{"style":28945},[61,60096],{"className":60097,"style":19306},[146],[61,60099,60101],{"className":60100},[116],[61,60102,283],{"className":60103},[116],[61,60105,170],{"className":60106},[169],[61,60108,60110],{"className":60109},[134],[61,60111,60113],{"className":60112,"style":29011},[138],[61,60114],{},[61,60116],{"className":60117},[730,19286],[61,60119,60121],{"className":60120,"style":19271},[730,19270],[61,60122,300],{"className":60123},[19275,153],[61,60125,60127],{"className":60126},[125],[61,60128,60130],{"className":60129},[129],[61,60131,60133],{"className":60132},[134],[61,60134,60136],{"className":60135,"style":54394},[138],[61,60137,60138,60141],{"style":54397},[61,60139],{"className":60140,"style":147},[146],[61,60142,60144],{"className":60143},[151,152,153,154],[61,60145,337],{"className":60146},[116,154],[61,60148],{"className":60149,"style":20194},[981],[61,60151,2419],{"className":60152},[19382],[61,60154],{"className":60155,"style":20194},[981],[61,60157,849],{"className":60158},[631],[61,60160,19797],{"className":60161,"style":20151},[116,120],[61,60163],{"className":60164,"style":20194},[981],[61,60166,19132],{"className":60167},[19382],[61,60169],{"className":60170,"style":20194},[981],[61,60172,60174,60177],{"className":60173},[116],[61,60175,19797],{"className":60176,"style":20151},[116,120],[61,60178,60180],{"className":60179},[125],[61,60181,60183,60203],{"className":60182},[129,130],[61,60184,60186,60200],{"className":60185},[134],[61,60187,60189],{"className":60188,"style":19230},[138],[61,60190,60191,60194],{"style":20167},[61,60192],{"className":60193,"style":147},[146],[61,60195,60197],{"className":60196},[151,152,153,154],[61,60198,1495],{"className":60199},[116,154],[61,60201,170],{"className":60202},[169],[61,60204,60206],{"className":60205},[134],[61,60207,60209],{"className":60208,"style":177},[138],[61,60210],{},[61,60212,60214,60217],{"className":60213},[730],[61,60215,300],{"className":60216},[730],[61,60218,60220],{"className":60219},[125],[61,60221,60223],{"className":60222},[129],[61,60224,60226],{"className":60225},[134],[61,60227,60229],{"className":60228,"style":51538},[138],[61,60230,60231,60234],{"style":51541},[61,60232],{"className":60233,"style":147},[146],[61,60235,60237],{"className":60236},[151,152,153,154],[61,60238,337],{"className":60239},[116,154],[61,60241,60242,60245],{"style":54502},[61,60243],{"className":60244,"style":54295},[146],[61,60246,60248],{"className":60247,"style":54509},[28399],[28402,60249,60250],{"xmlns":28404,"width":28405,"height":54512,"viewBox":54513,"preserveAspectRatio":28408},[28410,60251],{"d":54516},[61,60253,170],{"className":60254},[169],[61,60256,60258],{"className":60257},[134],[61,60259,60261],{"className":60260,"style":54526},[138],[61,60262],{},[61,60264,332],{"className":60265},[116],[25,60267,60269],{"id":60268},"primary-modeling-assumptions","Primary Modeling Assumptions",[12,60271,60272],{},"This notebook is an idealized mathematical physics visualization designed for conceptual clarity and calibration intuition.",[30,60274,60275,60285],{},[33,60276,60277],{},[36,60278,60279,60282],{},[39,60280,60281],{},"Assumption",[39,60283,60284],{},"Implication",[50,60286,60287,60371,60379,60386,60394,60431,60521,60678,60726,60734,60742,60750],{},[36,60288,60289,60292],{},[55,60290,60291],{},"Two-level truncation",[55,60293,60294,60295,27098,60333,332],{},"The simulated model includes only ",[61,60296,60298,60315],{"className":60297},[64],[61,60299,60301],{"className":60300},[68],[70,60302,60303],{"xmlns":72},[74,60304,60305,60313],{},[77,60306,60307,60309,60311],{},[586,60308,589],{"stretchy":588},[83,60310,46151],{},[586,60312,1498],{"stretchy":588},[95,60314,46156],{"encoding":97},[61,60316,60318],{"className":60317,"ariaHidden":103},[102],[61,60319,60321,60324,60327,60330],{"className":60320},[107],[61,60322],{"className":60323,"style":627},[111],[61,60325,589],{"className":60326},[631],[61,60328,46151],{"className":60329,"style":121},[116,120],[61,60331,1498],{"className":60332},[730],[61,60334,60336,60353],{"className":60335},[64],[61,60337,60339],{"className":60338},[68],[70,60340,60341],{"xmlns":72},[74,60342,60343,60351],{},[77,60344,60345,60347,60349],{},[586,60346,589],{"stretchy":588},[83,60348,21378],{},[586,60350,1498],{"stretchy":588},[95,60352,46202],{"encoding":97},[61,60354,60356],{"className":60355,"ariaHidden":103},[102],[61,60357,60359,60362,60365,60368],{"className":60358},[107],[61,60360],{"className":60361,"style":627},[111],[61,60363,589],{"className":60364},[631],[61,60366,21378],{"className":60367},[116,120],[61,60369,1498],{"className":60370},[730],[36,60372,60373,60376],{},[55,60374,60375],{},"Classical drive",[55,60377,60378],{},"The microwave field is prescribed externally instead of quantized as a field mode.",[36,60380,60381,60383],{},[55,60382,47573],{},[55,60384,60385],{},"Fast counter-rotating terms are omitted.",[36,60387,60388,60391],{},[55,60389,60390],{},"Square pulse",[55,60392,60393],{},"The drive is assumed to turn on and off instantaneously.",[36,60395,60396,60399],{},[55,60397,60398],{},"Uniform drive amplitude",[55,60400,60401,60402,60430],{},"The parameter ",[61,60403,60405,60418],{"className":60404},[64],[61,60406,60408],{"className":60407},[68],[70,60409,60410],{"xmlns":72},[74,60411,60412,60416],{},[77,60413,60414],{},[83,60415,283],{"mathvariant":197},[95,60417,286],{"encoding":97},[61,60419,60421],{"className":60420,"ariaHidden":103},[102],[61,60422,60424,60427],{"className":60423},[107],[61,60425],{"className":60426,"style":296},[111],[61,60428,283],{"className":60429},[116]," remains constant across the frequency sweep.",[36,60432,60433,60436],{},[55,60434,60435],{},"Optional damping envelope",[55,60437,60401,60438,60520],{},[61,60439,60441,60460],{"className":60440},[64],[61,60442,60444],{"className":60443},[68],[70,60445,60446],{"xmlns":72},[74,60447,60448,60458],{},[77,60449,60450],{},[34480,60451,60452,60454,60456],{},[83,60453,23559],{},[1493,60455,337],{},[586,60457,47495],{},[95,60459,47498],{"encoding":97},[61,60461,60463],{"className":60462,"ariaHidden":103},[102],[61,60464,60466,60469],{"className":60465},[107],[61,60467],{"className":60468,"style":47508},[111],[61,60470,60472,60475],{"className":60471},[116],[61,60473,23559],{"className":60474,"style":23700},[116,120],[61,60476,60478],{"className":60477},[125],[61,60479,60481,60512],{"className":60480},[129,130],[61,60482,60484,60509],{"className":60483},[134],[61,60485,60487,60498],{"className":60486,"style":47527},[138],[61,60488,60489,60492],{"style":47530},[61,60490],{"className":60491,"style":147},[146],[61,60493,60495],{"className":60494},[151,152,153,154],[61,60496,337],{"className":60497},[116,154],[61,60499,60500,60503],{"style":21487},[61,60501],{"className":60502,"style":147},[146],[61,60504,60506],{"className":60505},[151,152,153,154],[61,60507,47495],{"className":60508},[19382,154],[61,60510,170],{"className":60511},[169],[61,60513,60515],{"className":60514},[134],[61,60516,60518],{"className":60517,"style":47560},[138],[61,60519],{}," enters as a simple exponential contrast envelope.",[36,60522,60523,60597],{},[55,60524,60525,60526,60596],{},"No explicit ",[61,60527,60529,60547],{"className":60528},[64],[61,60530,60532],{"className":60531},[68],[70,60533,60534],{"xmlns":72},[74,60535,60536,60544],{},[77,60537,60538],{},[80,60539,60540,60542],{},[83,60541,23559],{},[1493,60543,57],{},[95,60545,60546],{"encoding":97},"T_1",[61,60548,60550],{"className":60549,"ariaHidden":103},[102],[61,60551,60553,60556],{"className":60552},[107],[61,60554],{"className":60555,"style":219},[111],[61,60557,60559,60562],{"className":60558},[116],[61,60560,23559],{"className":60561,"style":23700},[116,120],[61,60563,60565],{"className":60564},[125],[61,60566,60568,60588],{"className":60567},[129,130],[61,60569,60571,60585],{"className":60570},[134],[61,60572,60574],{"className":60573,"style":19230},[138],[61,60575,60576,60579],{"style":28699},[61,60577],{"className":60578,"style":147},[146],[61,60580,60582],{"className":60581},[151,152,153,154],[61,60583,57],{"className":60584},[116,154],[61,60586,170],{"className":60587},[169],[61,60589,60591],{"className":60590},[134],[61,60592,60594],{"className":60593,"style":177},[138],[61,60595],{}," relaxation",[55,60598,60599,60600,60638,60639,60677],{},"Energy decay from ",[61,60601,60603,60620],{"className":60602},[64],[61,60604,60606],{"className":60605},[68],[70,60607,60608],{"xmlns":72},[74,60609,60610,60618],{},[77,60611,60612,60614,60616],{},[586,60613,589],{"stretchy":588},[83,60615,21378],{},[586,60617,1498],{"stretchy":588},[95,60619,46202],{"encoding":97},[61,60621,60623],{"className":60622,"ariaHidden":103},[102],[61,60624,60626,60629,60632,60635],{"className":60625},[107],[61,60627],{"className":60628,"style":627},[111],[61,60630,589],{"className":60631},[631],[61,60633,21378],{"className":60634},[116,120],[61,60636,1498],{"className":60637},[730]," to ",[61,60640,60642,60659],{"className":60641},[64],[61,60643,60645],{"className":60644},[68],[70,60646,60647],{"xmlns":72},[74,60648,60649,60657],{},[77,60650,60651,60653,60655],{},[586,60652,589],{"stretchy":588},[83,60654,46151],{},[586,60656,1498],{"stretchy":588},[95,60658,46156],{"encoding":97},[61,60660,60662],{"className":60661,"ariaHidden":103},[102],[61,60663,60665,60668,60671,60674],{"className":60664},[107],[61,60666],{"className":60667,"style":627},[111],[61,60669,589],{"className":60670},[631],[61,60672,46151],{"className":60673,"style":121},[116,120],[61,60675,1498],{"className":60676},[730]," is omitted.",[36,60679,60680,60683],{},[55,60681,60682],{},"No higher-state leakage",[55,60684,60685,60686,60725],{},"Higher levels, including a transmon ",[61,60687,60689,60707],{"className":60688},[64],[61,60690,60692],{"className":60691},[68],[70,60693,60694],{"xmlns":72},[74,60695,60696,60704],{},[77,60697,60698,60700,60702],{},[586,60699,589],{"stretchy":588},[83,60701,19797],{},[586,60703,1498],{"stretchy":588},[95,60705,60706],{"encoding":97},"\\lvert f\\rangle",[61,60708,60710],{"className":60709,"ariaHidden":103},[102],[61,60711,60713,60716,60719,60722],{"className":60712},[107],[61,60714],{"className":60715,"style":627},[111],[61,60717,589],{"className":60718},[631],[61,60720,19797],{"className":60721,"style":20151},[116,120],[61,60723,1498],{"className":60724},[730]," state, are excluded.",[36,60727,60728,60731],{},[55,60729,60730],{},"No AC Stark shift",[55,60732,60733],{},"Drive-induced transition-frequency shifts are excluded.",[36,60735,60736,60739],{},[55,60737,60738],{},"No Bloch-Siegert shift",[55,60740,60741],{},"Counter-rotating corrections are excluded.",[36,60743,60744,60747],{},[55,60745,60746],{},"No experimental readout model",[55,60748,60749],{},"Assignment errors and finite signal-to-noise effects are omitted.",[36,60751,60752,60755],{},[55,60753,60754],{},"No pulse-envelope shaping",[55,60756,60757],{},"Gaussian, DRAG, cosine, and hardware-specific envelopes are omitted.",[12,60759,60760],{},"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.",[25,60762,60764],{"id":60763},"notebook-control-knobs","Notebook Control Knobs",[12,60766,60767],{},"The first code section centralizes the main physical and numerical parameters for direct adjustment.",[30,60769,60770,60782],{},[33,60771,60772],{},[36,60773,60774,60777,60780],{},[39,60775,60776],{},"Parameter",[39,60778,60779],{"align":41},"Default",[39,60781,46128],{},[50,60783,60784,60868,60921,60936,60951,60966,60980,60995,61009],{},[36,60785,60786,60791,60795],{},[55,60787,60788],{},[1727,60789,60790],{},"F0_GHZ",[55,60792,60793],{"align":41},[1727,60794,48061],{},[55,60796,60797,60798,60867],{},"Qubit transition frequency ",[61,60799,60801,60818],{"className":60800},[64],[61,60802,60804],{"className":60803},[68],[70,60805,60806],{"xmlns":72},[74,60807,60808,60816],{},[77,60809,60810],{},[80,60811,60812,60814],{},[83,60813,19797],{},[1493,60815,1495],{},[95,60817,46498],{"encoding":97},[61,60819,60821],{"className":60820,"ariaHidden":103},[102],[61,60822,60824,60827],{"className":60823},[107],[61,60825],{"className":60826,"style":46075},[111],[61,60828,60830,60833],{"className":60829},[116],[61,60831,19797],{"className":60832,"style":20151},[116,120],[61,60834,60836],{"className":60835},[125],[61,60837,60839,60859],{"className":60838},[129,130],[61,60840,60842,60856],{"className":60841},[134],[61,60843,60845],{"className":60844,"style":19230},[138],[61,60846,60847,60850],{"style":20167},[61,60848],{"className":60849,"style":147},[146],[61,60851,60853],{"className":60852},[151,152,153,154],[61,60854,1495],{"className":60855},[116,154],[61,60857,170],{"className":60858},[169],[61,60860,60862],{"className":60861},[134],[61,60863,60865],{"className":60864,"style":177},[138],[61,60866],{}," in GHz",[36,60869,60870,60875,60880],{},[55,60871,60872],{},[1727,60873,60874],{},"OMEGA_ONRESONANCE_MHZ",[55,60876,60877],{"align":41},[1727,60878,60879],{},"20.0",[55,60881,60882,60883,60920],{},"On-resonance Rabi rate ",[61,60884,60886,60905],{"className":60885},[64],[61,60887,60889],{"className":60888},[68],[70,60890,60891],{"xmlns":72},[74,60892,60893,60903],{},[77,60894,60895,60897,60899,60901],{},[83,60896,283],{"mathvariant":197},[83,60898,602],{"mathvariant":197},[1493,60900,337],{},[83,60902,19792],{},[95,60904,46964],{"encoding":97},[61,60906,60908],{"className":60907,"ariaHidden":103},[102],[61,60909,60911,60914,60917],{"className":60910},[107],[61,60912],{"className":60913,"style":627},[111],[61,60915,46977],{"className":60916},[116],[61,60918,19792],{"className":60919,"style":121},[116,120]," in MHz",[36,60922,60923,60928,60933],{},[55,60924,60925],{},[1727,60926,60927],{},"FREQ_MIN_GHZ",[55,60929,60930],{"align":41},[1727,60931,60932],{},"4.90",[55,60934,60935],{},"Minimum swept drive frequency in GHz",[36,60937,60938,60943,60948],{},[55,60939,60940],{},[1727,60941,60942],{},"FREQ_MAX_GHZ",[55,60944,60945],{"align":41},[1727,60946,60947],{},"5.10",[55,60949,60950],{},"Maximum swept drive frequency in GHz",[36,60952,60953,60958,60963],{},[55,60954,60955],{},[1727,60956,60957],{},"N_FREQ",[55,60959,60960],{"align":41},[1727,60961,60962],{},"401",[55,60964,60965],{},"Number of drive-frequency samples",[36,60967,60968,60973,60977],{},[55,60969,60970],{},[1727,60971,60972],{},"DUR_MIN_NS",[55,60974,60975],{"align":41},[1727,60976,2051],{},[55,60978,60979],{},"Minimum pulse duration in ns",[36,60981,60982,60987,60992],{},[55,60983,60984],{},[1727,60985,60986],{},"DUR_MAX_NS",[55,60988,60989],{"align":41},[1727,60990,60991],{},"200.0",[55,60993,60994],{},"Maximum pulse duration in ns",[36,60996,60997,61002,61006],{},[55,60998,60999],{},[1727,61000,61001],{},"N_DUR",[55,61003,61004],{"align":41},[1727,61005,60962],{},[55,61007,61008],{},"Number of pulse-duration samples",[36,61010,61011,61016,61020],{},[55,61012,61013],{},[1727,61014,61015],{},"T2STAR_NS",[55,61017,61018],{"align":41},[1727,61019,2009],{},[55,61021,61022,61023,61026],{},"Optional damping time in ns, with ",[1727,61024,61025],{},"None"," disabling damping",[12,61028,61029],{},"Additional cross-section plotting cells expose localized slice controls.",[30,61031,61032,61042],{},[33,61033,61034],{},[36,61035,61036,61038,61040],{},[39,61037,60776],{},[39,61039,60779],{"align":41},[39,61041,46128],{},[50,61043,61044,61143],{},[36,61045,61046,61051,61055],{},[55,61047,61048],{},[1727,61049,61050],{},"F_CROSS_GHZ",[55,61052,61053],{"align":41},[1727,61054,48061],{},[55,61056,61057,61058],{},"Drive-frequency slice used to plot ",[61,61059,61061,61085],{"className":61060},[64],[61,61062,61064],{"className":61063},[68],[70,61065,61066],{"xmlns":72},[74,61067,61068,61082],{},[77,61069,61070,61076,61078,61080],{},[80,61071,61072,61074],{},[83,61073,45957],{},[83,61075,21378],{},[586,61077,849],{"stretchy":588},[83,61079,45968],{},[586,61081,300],{"stretchy":588},[95,61083,61084],{"encoding":97},"P_e(\\tau)",[61,61086,61088],{"className":61087,"ariaHidden":103},[102],[61,61089,61091,61094,61134,61137,61140],{"className":61090},[107],[61,61092],{"className":61093,"style":627},[111],[61,61095,61097,61100],{"className":61096},[116],[61,61098,45957],{"className":61099,"style":23700},[116,120],[61,61101,61103],{"className":61102},[125],[61,61104,61106,61126],{"className":61105},[129,130],[61,61107,61109,61123],{"className":61108},[134],[61,61110,61112],{"className":61111,"style":20164},[138],[61,61113,61114,61117],{"style":28699},[61,61115],{"className":61116,"style":147},[146],[61,61118,61120],{"className":61119},[151,152,153,154],[61,61121,21378],{"className":61122},[116,120,154],[61,61124,170],{"className":61125},[169],[61,61127,61129],{"className":61128},[134],[61,61130,61132],{"className":61131,"style":177},[138],[61,61133],{},[61,61135,849],{"className":61136},[631],[61,61138,45968],{"className":61139,"style":46040},[116,120],[61,61141,300],{"className":61142},[730],[36,61144,61145,61150,61155],{},[55,61146,61147],{},[1727,61148,61149],{},"T_CROSS_NS",[55,61151,61152],{"align":41},[1727,61153,61154],{},"100.0",[55,61156,61157,61158],{},"Pulse-duration slice used to plot ",[61,61159,61161,61185],{"className":61160},[64],[61,61162,61164],{"className":61163},[68],[70,61165,61166],{"xmlns":72},[74,61167,61168,61182],{},[77,61169,61170,61176,61178,61180],{},[80,61171,61172,61174],{},[83,61173,45957],{},[83,61175,21378],{},[586,61177,849],{"stretchy":588},[83,61179,19797],{},[586,61181,300],{"stretchy":588},[95,61183,61184],{"encoding":97},"P_e(f)",[61,61186,61188],{"className":61187,"ariaHidden":103},[102],[61,61189,61191,61194,61234,61237,61240],{"className":61190},[107],[61,61192],{"className":61193,"style":627},[111],[61,61195,61197,61200],{"className":61196},[116],[61,61198,45957],{"className":61199,"style":23700},[116,120],[61,61201,61203],{"className":61202},[125],[61,61204,61206,61226],{"className":61205},[129,130],[61,61207,61209,61223],{"className":61208},[134],[61,61210,61212],{"className":61211,"style":20164},[138],[61,61213,61214,61217],{"style":28699},[61,61215],{"className":61216,"style":147},[146],[61,61218,61220],{"className":61219},[151,152,153,154],[61,61221,21378],{"className":61222},[116,120,154],[61,61224,170],{"className":61225},[169],[61,61227,61229],{"className":61228},[134],[61,61230,61232],{"className":61231,"style":177},[138],[61,61233],{},[61,61235,849],{"className":61236},[631],[61,61238,19797],{"className":61239,"style":20151},[116,120],[61,61241,300],{"className":61242},[730],[12,61244,61245],{},"Separate FFT processing controls support spectral analysis.",[30,61247,61248,61258],{},[33,61249,61250],{},[36,61251,61252,61254,61256],{},[39,61253,60776],{},[39,61255,60779],{"align":41},[39,61257,46128],{},[50,61259,61260,61274,61289,61303,61318,61333],{},[36,61261,61262,61267,61271],{},[55,61263,61264],{},[1727,61265,61266],{},"FFT_PAD",[55,61268,61269],{"align":41},[1727,61270,3631],{},[55,61272,61273],{},"Zero-padding factor used in the Fourier transform",[36,61275,61276,61281,61286],{},[55,61277,61278],{},[1727,61279,61280],{},"WINDOW",[55,61282,61283],{"align":41},[1727,61284,61285],{},"\"hann\"",[55,61287,61288],{},"Window function applied along the pulse-duration axis",[36,61290,61291,61296,61300],{},[55,61292,61293],{},[1727,61294,61295],{},"DETREND_MEAN",[55,61297,61298],{"align":41},[1727,61299,24385],{},[55,61301,61302],{},"Flag that subtracts the mean before the FFT",[36,61304,61305,61310,61315],{},[55,61306,61307],{},[1727,61308,61309],{},"AMP_MODE",[55,61311,61312],{"align":41},[1727,61313,61314],{},"\"magnitude\"",[55,61316,61317],{},"Display mode for magnitude, power, or dB-scaled amplitude",[36,61319,61320,61325,61330],{},[55,61321,61322],{},[1727,61323,61324],{},"NORM_MODE",[55,61326,61327],{"align":41},[1727,61328,61329],{},"\"global\"",[55,61331,61332],{},"Normalization mode applied globally or per drive-frequency column",[36,61334,61335,61340,61345],{},[55,61336,61337],{},[1727,61338,61339],{},"FREQ_UNITS",[55,61341,61342],{"align":41},[1727,61343,61344],{},"\"MHz\"",[55,61346,61347],{},"Frequency units used on the Fourier-frequency axis",[12,61349,61350],{},"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.",[1720,61352,61354],{"className":1722,"code":61353,"language":1724,"meta":1725,"style":1725},"# @title Install dependencies\nimport sys\n%pip -q install pyvista pythreejs trame\n",[1727,61355,61356,61361,61368],{"__ignoreMap":1725},[61,61357,61358],{"class":1731,"line":1732},[61,61359,61360],{"class":1735},"# @title Install dependencies\n",[61,61362,61363,61365],{"class":1731,"line":1739},[61,61364,1757],{"class":1749},[61,61366,61367],{"class":1753}," sys\n",[61,61369,61370,61373,61376,61378],{"class":1731,"line":1746},[61,61371,61372],{"class":1859},"%",[61,61374,61375],{"class":1753},"pip ",[61,61377,2006],{"class":1859},[61,61379,61380],{"class":1753},"q install pyvista pythreejs trame\n",[1720,61382,61384],{"className":1722,"code":61383,"language":1724,"meta":1725,"style":1725},"# @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",[1727,61385,61386,61391,61397,61408,61418,61428,61440,61444,61451,61467,61477,61485,61494,61498,61511,61524,61528,61533,61549,61564,61580,61596,61612,61627,61643,61658,61677,61681],{"__ignoreMap":1725},[61,61387,61388],{"class":1731,"line":1732},[61,61389,61390],{"class":1735},"# @title Controls & imports\n",[61,61392,61393,61395],{"class":1731,"line":1739},[61,61394,1757],{"class":1749},[61,61396,1781],{"class":1753},[61,61398,61399,61401,61403,61405],{"class":1731,"line":1746},[61,61400,1750],{"class":1749},[61,61402,1768],{"class":1753},[61,61404,1757],{"class":1749},[61,61406,61407],{"class":1753}," Optional, Tuple\n",[61,61409,61410,61412,61414,61416],{"class":1731,"line":1763},[61,61411,1757],{"class":1749},[61,61413,1789],{"class":1753},[61,61415,1792],{"class":1749},[61,61417,1795],{"class":1753},[61,61419,61420,61422,61424,61426],{"class":1731,"line":1776},[61,61421,1757],{"class":1749},[61,61423,1803],{"class":1753},[61,61425,1792],{"class":1749},[61,61427,1808],{"class":1753},[61,61429,61430,61432,61435,61437],{"class":1731,"line":1784},[61,61431,1757],{"class":1749},[61,61433,61434],{"class":1753}," pandas ",[61,61436,1792],{"class":1749},[61,61438,61439],{"class":1753}," pd\n",[61,61441,61442],{"class":1731,"line":1798},[61,61443,1743],{"emptyLinePlaceholder":1742},[61,61445,61446,61449],{"class":1731,"line":1811},[61,61447,61448],{"class":1749},"try",[61,61450,1850],{"class":1753},[61,61452,61453,61456,61459,61461,61464],{"class":1731,"line":1816},[61,61454,61455],{"class":1749},"    import",[61,61457,61458],{"class":1753}," pyvista ",[61,61460,1792],{"class":1749},[61,61462,61463],{"class":1753}," pv  ",[61,61465,61466],{"class":1735},"# optional; only for 3D interactive\n",[61,61468,61469,61472,61474],{"class":1731,"line":1822},[61,61470,61471],{"class":1866},"    PV_AVAILABLE",[61,61473,1863],{"class":1859},[61,61475,61476],{"class":1866}," True\n",[61,61478,61479,61482],{"class":1731,"line":1828},[61,61480,61481],{"class":1749},"except",[61,61483,61484],{"class":1753}," Exception:\n",[61,61486,61487,61489,61491],{"class":1731,"line":1833},[61,61488,61471],{"class":1866},[61,61490,1863],{"class":1859},[61,61492,61493],{"class":1866}," False\n",[61,61495,61496],{"class":1731,"line":1840},[61,61497,1743],{"emptyLinePlaceholder":1742},[61,61499,61500,61502,61504,61506,61508],{"class":1731,"line":1853},[61,61501,14189],{"class":1753},[61,61503,14192],{"class":2165},[61,61505,1997],{"class":1753},[61,61507,2000],{"class":1859},[61,61509,61510],{"class":1866}," 250\n",[61,61512,61513,61515,61518,61520,61522],{"class":1731,"line":1873},[61,61514,14189],{"class":1753},[61,61516,61517],{"class":2165},"'axes.grid'",[61,61519,1997],{"class":1753},[61,61521,2000],{"class":1859},[61,61523,61493],{"class":1866},[61,61525,61526],{"class":1731,"line":1889},[61,61527,1743],{"emptyLinePlaceholder":1742},[61,61529,61530],{"class":1731,"line":1908},[61,61531,61532],{"class":1735},"# ---------- Control knobs ----------\n",[61,61534,61535,61537,61539,61541,61543,61546],{"class":1731,"line":1925},[61,61536,60790],{"class":1866},[61,61538,2127],{"class":1753},[61,61540,1860],{"class":1859},[61,61542,1863],{"class":1859},[61,61544,61545],{"class":1866}," 5.000",[61,61547,61548],{"class":1735},"        # Qubit transition freq f0 [GHz]\n",[61,61550,61551,61553,61555,61557,61559,61561],{"class":1731,"line":1941},[61,61552,60874],{"class":1866},[61,61554,2127],{"class":1753},[61,61556,1860],{"class":1859},[61,61558,1863],{"class":1859},[61,61560,6006],{"class":1866},[61,61562,61563],{"class":1735},"  # Ω\u002F2π on-resonance [MHz]\n",[61,61565,61566,61568,61570,61572,61574,61577],{"class":1731,"line":1957},[61,61567,60927],{"class":1866},[61,61569,2127],{"class":1753},[61,61571,1860],{"class":1859},[61,61573,1863],{"class":1859},[61,61575,61576],{"class":1866}," 4.90",[61,61578,61579],{"class":1735},"   # Sweep min f [GHz]\n",[61,61581,61582,61584,61586,61588,61590,61593],{"class":1731,"line":1974},[61,61583,60942],{"class":1866},[61,61585,2127],{"class":1753},[61,61587,1860],{"class":1859},[61,61589,1863],{"class":1859},[61,61591,61592],{"class":1866}," 5.10",[61,61594,61595],{"class":1735},"   # Sweep max f [GHz]\n",[61,61597,61598,61600,61602,61604,61606,61609],{"class":1731,"line":1979},[61,61599,60957],{"class":1866},[61,61601,2127],{"class":1753},[61,61603,2025],{"class":1859},[61,61605,1863],{"class":1859},[61,61607,61608],{"class":1866}," 401",[61,61610,61611],{"class":1735},"            # # of frequency points\n",[61,61613,61614,61616,61618,61620,61622,61624],{"class":1731,"line":1985},[61,61615,60972],{"class":1866},[61,61617,2127],{"class":1753},[61,61619,1860],{"class":1859},[61,61621,1863],{"class":1859},[61,61623,2506],{"class":1866},[61,61625,61626],{"class":1735},"      # Sweep min duration τ [ns]\n",[61,61628,61629,61631,61633,61635,61637,61640],{"class":1731,"line":2019},[61,61630,60986],{"class":1866},[61,61632,2127],{"class":1753},[61,61634,1860],{"class":1859},[61,61636,1863],{"class":1859},[61,61638,61639],{"class":1866}," 200.0",[61,61641,61642],{"class":1735},"    # Sweep max duration τ [ns]\n",[61,61644,61645,61647,61649,61651,61653,61655],{"class":1731,"line":2033},[61,61646,61001],{"class":1866},[61,61648,2127],{"class":1753},[61,61650,2025],{"class":1859},[61,61652,1863],{"class":1859},[61,61654,61608],{"class":1866},[61,61656,61657],{"class":1735},"             # # of duration points\n",[61,61659,61660,61662,61665,61667,61669,61671,61674],{"class":1731,"line":2061},[61,61661,61015],{"class":1866},[61,61663,61664],{"class":1753},": Optional[",[61,61666,1860],{"class":1859},[61,61668,1997],{"class":1753},[61,61670,2000],{"class":1859},[61,61672,61673],{"class":1866}," 500.0",[61,61675,61676],{"class":1735},"  # None to disable exp(-τ\u002FT2*)\n",[61,61678,61679],{"class":1731,"line":2074},[61,61680,1743],{"emptyLinePlaceholder":1742},[61,61682,61683,61685,61687,61690],{"class":1731,"line":2079},[61,61684,40848],{"class":1859},[61,61686,849],{"class":1753},[61,61688,61689],{"class":2165},"'Controls loaded.'",[61,61691,2016],{"class":1753},[1720,61693,61696],{"className":61694,"code":61695,"language":29075,"meta":1725},[36200],"Controls loaded.\n",[1727,61697,61695],{"__ignoreMap":1725},[1720,61699,61701],{"className":1722,"code":61700,"language":1724,"meta":1725,"style":1725},"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",[1727,61702,61703,61712,61719,61726,61737,61748,61766,61770,61775,61779,61784,61789,61794,61799,61804,61809,61813,61818,61823,61827,61841,61856,61870,61884,61906,61910,61936,61958,61987,61991,62022,62042,62061,62084,62088,62113,62141,62155,62159],{"__ignoreMap":1725},[61,61704,61705,61707,61710],{"class":1731,"line":1732},[61,61706,2116],{"class":1749},[61,61708,61709],{"class":1836}," rabi_probability",[61,61711,38439],{"class":1753},[61,61713,61714,61717],{"class":1731,"line":1739},[61,61715,61716],{"class":2124},"    f_ghz",[61,61718,38733],{"class":1753},[61,61720,61721,61724],{"class":1731,"line":1746},[61,61722,61723],{"class":2124},"    tau_ns",[61,61725,38733],{"class":1753},[61,61727,61728,61731,61733,61735],{"class":1731,"line":1763},[61,61729,61730],{"class":2124},"    f0_ghz",[61,61732,2127],{"class":1753},[61,61734,1860],{"class":1859},[61,61736,36573],{"class":1753},[61,61738,61739,61742,61744,61746],{"class":1731,"line":1776},[61,61740,61741],{"class":2124},"    omega_onres_mhz",[61,61743,2127],{"class":1753},[61,61745,1860],{"class":1859},[61,61747,36573],{"class":1753},[61,61749,61750,61753,61755,61757,61759,61761,61764],{"class":1731,"line":1784},[61,61751,61752],{"class":2124},"    t2star_ns",[61,61754,61664],{"class":1753},[61,61756,1860],{"class":1859},[61,61758,1997],{"class":1753},[61,61760,2000],{"class":1859},[61,61762,61763],{"class":1866}," None",[61,61765,36573],{"class":1753},[61,61767,61768],{"class":1731,"line":1798},[61,61769,37291],{"class":1753},[61,61771,61772],{"class":1731,"line":1811},[61,61773,61774],{"class":2165},"    \"\"\"Excited-state probability on a 2D grid (f, τ), under RWA.\n",[61,61776,61777],{"class":1731,"line":1816},[61,61778,1743],{"emptyLinePlaceholder":1742},[61,61780,61781],{"class":1731,"line":1822},[61,61782,61783],{"class":2165},"    Args:\n",[61,61785,61786],{"class":1731,"line":1828},[61,61787,61788],{"class":2165},"        f_ghz: 1D array of drive frequencies [GHz].\n",[61,61790,61791],{"class":1731,"line":1833},[61,61792,61793],{"class":2165},"        tau_ns: 1D array of pulse durations [ns].\n",[61,61795,61796],{"class":1731,"line":1840},[61,61797,61798],{"class":2165},"        f0_ghz: Qubit frequency on resonance [GHz].\n",[61,61800,61801],{"class":1731,"line":1853},[61,61802,61803],{"class":2165},"        omega_onres_mhz: On-resonance Rabi rate Ω\u002F2π [MHz].\n",[61,61805,61806],{"class":1731,"line":1873},[61,61807,61808],{"class":2165},"        t2star_ns: Optional T2* [ns]; multiplies exp(−τ\u002FT2*).\n",[61,61810,61811],{"class":1731,"line":1889},[61,61812,1743],{"emptyLinePlaceholder":1742},[61,61814,61815],{"class":1731,"line":1908},[61,61816,61817],{"class":2165},"    Returns:\n",[61,61819,61820],{"class":1731,"line":1925},[61,61821,61822],{"class":2165},"        2D array P_e with shape (len(tau_ns), len(f_ghz)).\n",[61,61824,61825],{"class":1731,"line":1941},[61,61826,36776],{"class":2165},[61,61828,61829,61832,61834,61836,61838],{"class":1731,"line":1957},[61,61830,61831],{"class":1753},"    two_pi ",[61,61833,2000],{"class":1859},[61,61835,3029],{"class":1866},[61,61837,2298],{"class":1859},[61,61839,61840],{"class":1753}," math.pi\n",[61,61842,61843,61846,61848,61851,61853],{"class":1731,"line":1974},[61,61844,61845],{"class":1753},"    f ",[61,61847,2000],{"class":1859},[61,61849,61850],{"class":1753}," f_ghz ",[61,61852,2201],{"class":1859},[61,61854,61855],{"class":1866}," 1e9\n",[61,61857,61858,61861,61863,61866,61868],{"class":1731,"line":1979},[61,61859,61860],{"class":1753},"    f0 ",[61,61862,2000],{"class":1859},[61,61864,61865],{"class":1753}," f0_ghz ",[61,61867,2201],{"class":1859},[61,61869,61855],{"class":1866},[61,61871,61872,61875,61877,61880,61882],{"class":1731,"line":1985},[61,61873,61874],{"class":1753},"    tau ",[61,61876,2000],{"class":1859},[61,61878,61879],{"class":1753}," tau_ns ",[61,61881,2201],{"class":1859},[61,61883,6224],{"class":1866},[61,61885,61886,61889,61891,61894,61896,61899,61901,61903],{"class":1731,"line":2019},[61,61887,61888],{"class":1753},"    omega ",[61,61890,2000],{"class":1859},[61,61892,61893],{"class":1753}," two_pi ",[61,61895,2201],{"class":1859},[61,61897,61898],{"class":1753}," omega_onres_mhz ",[61,61900,2201],{"class":1859},[61,61902,6264],{"class":1866},[61,61904,61905],{"class":1735},"  # rad\u002Fs\n",[61,61907,61908],{"class":1731,"line":2033},[61,61909,1743],{"emptyLinePlaceholder":1742},[61,61911,61912,61915,61918,61920,61922,61924,61927,61929,61931,61934],{"class":1731,"line":2061},[61,61913,61914],{"class":1753},"    F, ",[61,61916,61917],{"class":1866},"TAU",[61,61919,1863],{"class":1859},[61,61921,2193],{"class":1753},[61,61923,37855],{"class":1836},[61,61925,61926],{"class":1753},"(f, tau, ",[61,61928,37861],{"class":3402},[61,61930,2000],{"class":1859},[61,61932,61933],{"class":2165},"'xy'",[61,61935,2016],{"class":1753},[61,61937,61938,61941,61943,61945,61947,61950,61952,61955],{"class":1731,"line":2074},[61,61939,61940],{"class":1753},"    Delta ",[61,61942,2000],{"class":1859},[61,61944,61893],{"class":1753},[61,61946,2201],{"class":1859},[61,61948,61949],{"class":1753}," (F ",[61,61951,2006],{"class":1859},[61,61953,61954],{"class":1753}," f0)               ",[61,61956,61957],{"class":1735},"# rad\u002Fs\n",[61,61959,61960,61963,61965,61967,61969,61972,61974,61976,61978,61981,61983,61985],{"class":1731,"line":2079},[61,61961,61962],{"class":1753},"    Omega_R ",[61,61964,2000],{"class":1859},[61,61966,2193],{"class":1753},[61,61968,2425],{"class":1836},[61,61970,61971],{"class":1753},"(omega",[61,61973,2651],{"class":1859},[61,61975,337],{"class":1866},[61,61977,4990],{"class":1859},[61,61979,61980],{"class":1753}," Delta",[61,61982,2651],{"class":1859},[61,61984,337],{"class":1866},[61,61986,2016],{"class":1753},[61,61988,61989],{"class":1731,"line":2092},[61,61990,1743],{"emptyLinePlaceholder":1742},[61,61992,61993,61996,61998,62001,62003,62006,62008,62011,62013,62016,62018,62020],{"class":1731,"line":2097},[61,61994,61995],{"class":1749},"    with",[61,61997,2193],{"class":1753},[61,61999,62000],{"class":1836},"errstate",[61,62002,849],{"class":1753},[61,62004,62005],{"class":3402},"divide",[61,62007,2000],{"class":1859},[61,62009,62010],{"class":2165},"'ignore'",[61,62012,1435],{"class":1753},[61,62014,62015],{"class":3402},"invalid",[61,62017,2000],{"class":1859},[61,62019,62010],{"class":2165},[61,62021,3371],{"class":1753},[61,62023,62024,62027,62029,62032,62034,62037,62039],{"class":1731,"line":2102},[61,62025,62026],{"class":1753},"        amp ",[61,62028,2000],{"class":1859},[61,62030,62031],{"class":1753}," (omega ",[61,62033,602],{"class":1859},[61,62035,62036],{"class":1753}," Omega_R) ",[61,62038,2651],{"class":1859},[61,62040,62041],{"class":1866}," 2\n",[61,62043,62044,62047,62049,62051,62053,62056,62058],{"class":1731,"line":2108},[61,62045,62046],{"class":1753},"        phase ",[61,62048,2000],{"class":1859},[61,62050,2980],{"class":1866},[61,62052,2298],{"class":1859},[61,62054,62055],{"class":1753}," Omega_R ",[61,62057,2201],{"class":1859},[61,62059,62060],{"class":1866}," TAU\n",[61,62062,62063,62066,62068,62071,62073,62075,62077,62080,62082],{"class":1731,"line":2113},[61,62064,62065],{"class":1753},"        Pe ",[61,62067,2000],{"class":1859},[61,62069,62070],{"class":1753}," amp ",[61,62072,2201],{"class":1859},[61,62074,2193],{"class":1753},[61,62076,5069],{"class":1836},[61,62078,62079],{"class":1753},"(phase) ",[61,62081,2651],{"class":1859},[61,62083,62041],{"class":1866},[61,62085,62086],{"class":1731,"line":2162},[61,62087,1743],{"emptyLinePlaceholder":1742},[61,62089,62090,62092,62095,62098,62101,62103,62105,62107,62109,62111],{"class":1731,"line":2169},[61,62091,2172],{"class":1749},[61,62093,62094],{"class":1753}," t2star_ns ",[61,62096,62097],{"class":1749},"is",[61,62099,62100],{"class":1749}," not",[61,62102,61763],{"class":1866},[61,62104,2509],{"class":1749},[61,62106,62094],{"class":1753},[61,62108,2503],{"class":1859},[61,62110,2506],{"class":1866},[61,62112,1850],{"class":1753},[61,62114,62115,62118,62120,62122,62124,62126,62128,62130,62132,62135,62137,62139],{"class":1731,"line":2187},[61,62116,62117],{"class":1753},"        decay ",[61,62119,2000],{"class":1859},[61,62121,2193],{"class":1753},[61,62123,6987],{"class":1836},[61,62125,849],{"class":1753},[61,62127,2006],{"class":1859},[61,62129,61917],{"class":1866},[61,62131,2618],{"class":1859},[61,62133,62134],{"class":1753}," (t2star_ns ",[61,62136,2201],{"class":1859},[61,62138,25775],{"class":1866},[61,62140,5546],{"class":1753},[61,62142,62143,62145,62147,62150,62152],{"class":1731,"line":2207},[61,62144,62065],{"class":1753},[61,62146,2000],{"class":1859},[61,62148,62149],{"class":1753}," Pe ",[61,62151,2201],{"class":1859},[61,62153,62154],{"class":1753}," decay\n",[61,62156,62157],{"class":1731,"line":2221},[61,62158,1743],{"emptyLinePlaceholder":1742},[61,62160,62161,62163,62165,62167,62170,62172,62174,62176],{"class":1731,"line":2226},[61,62162,2210],{"class":1749},[61,62164,2193],{"class":1753},[61,62166,4582],{"class":1836},[61,62168,62169],{"class":1753},"(Pe, ",[61,62171,2051],{"class":1866},[61,62173,1435],{"class":1753},[61,62175,2319],{"class":1866},[61,62177,2016],{"class":1753},[1720,62179,62181],{"className":1722,"code":62180,"language":1724,"meta":1725,"style":1725},"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",[1727,62182,62183,62208,62233,62257],{"__ignoreMap":1725},[61,62184,62185,62188,62190,62192,62194,62196,62198,62200,62202,62204,62206],{"class":1731,"line":1732},[61,62186,62187],{"class":1753},"freq_ghz ",[61,62189,2000],{"class":1859},[61,62191,2193],{"class":1753},[61,62193,3453],{"class":1836},[61,62195,849],{"class":1753},[61,62197,60927],{"class":1866},[61,62199,1435],{"class":1753},[61,62201,60942],{"class":1866},[61,62203,1435],{"class":1753},[61,62205,60957],{"class":1866},[61,62207,2016],{"class":1753},[61,62209,62210,62213,62215,62217,62219,62221,62223,62225,62227,62229,62231],{"class":1731,"line":1739},[61,62211,62212],{"class":1753},"dur_ns ",[61,62214,2000],{"class":1859},[61,62216,2193],{"class":1753},[61,62218,3453],{"class":1836},[61,62220,849],{"class":1753},[61,62222,60972],{"class":1866},[61,62224,1435],{"class":1753},[61,62226,60986],{"class":1866},[61,62228,1435],{"class":1753},[61,62230,61001],{"class":1866},[61,62232,2016],{"class":1753},[61,62234,62235,62238,62240,62242,62245,62247,62249,62251,62253,62255],{"class":1731,"line":1746},[61,62236,62237],{"class":1753},"Pe ",[61,62239,2000],{"class":1859},[61,62241,61709],{"class":1836},[61,62243,62244],{"class":1753},"(freq_ghz, dur_ns, ",[61,62246,60790],{"class":1866},[61,62248,1435],{"class":1753},[61,62250,60874],{"class":1866},[61,62252,1435],{"class":1753},[61,62254,61015],{"class":1866},[61,62256,2016],{"class":1753},[61,62258,62259,62261,62263,62266],{"class":1731,"line":1763},[61,62260,40848],{"class":1859},[61,62262,849],{"class":1753},[61,62264,62265],{"class":2165},"'Grid shape (τ × f):'",[61,62267,62268],{"class":1753},", Pe.shape)\n",[1720,62270,62273],{"className":62271,"code":62272,"language":29075,"meta":1725},[36200],"Grid shape (τ × f): (401, 401)\n",[1727,62274,62272],{"__ignoreMap":1725},[1720,62276,62278],{"className":1722,"code":62277,"language":1724,"meta":1725,"style":1725},"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",[1727,62279,62280,62295,62300,62314,62327,62340,62353,62366,62379,62392,62405,62418,62423,62442,62446],{"__ignoreMap":1725},[61,62281,62282,62285,62287,62290,62293],{"class":1731,"line":1732},[61,62283,62284],{"class":1753},"controls_table ",[61,62286,2000],{"class":1859},[61,62288,62289],{"class":1753}," pd.",[61,62291,62292],{"class":1836},"DataFrame",[61,62294,38439],{"class":1753},[61,62296,62297],{"class":1731,"line":1739},[61,62298,62299],{"class":1753},"    [\n",[61,62301,62302,62305,62308,62310,62312],{"class":1731,"line":1746},[61,62303,62304],{"class":1753},"        [",[61,62306,62307],{"class":2165},"'f0 (GHz)'",[61,62309,1435],{"class":1753},[61,62311,60790],{"class":1866},[61,62313,6550],{"class":1753},[61,62315,62316,62318,62321,62323,62325],{"class":1731,"line":1763},[61,62317,62304],{"class":1753},[61,62319,62320],{"class":2165},"'Ω\u002F2π on-res (MHz)'",[61,62322,1435],{"class":1753},[61,62324,60874],{"class":1866},[61,62326,6550],{"class":1753},[61,62328,62329,62331,62334,62336,62338],{"class":1731,"line":1776},[61,62330,62304],{"class":1753},[61,62332,62333],{"class":2165},"'f sweep min (GHz)'",[61,62335,1435],{"class":1753},[61,62337,60927],{"class":1866},[61,62339,6550],{"class":1753},[61,62341,62342,62344,62347,62349,62351],{"class":1731,"line":1784},[61,62343,62304],{"class":1753},[61,62345,62346],{"class":2165},"'f sweep max (GHz)'",[61,62348,1435],{"class":1753},[61,62350,60942],{"class":1866},[61,62352,6550],{"class":1753},[61,62354,62355,62357,62360,62362,62364],{"class":1731,"line":1798},[61,62356,62304],{"class":1753},[61,62358,62359],{"class":2165},"'N_f'",[61,62361,1435],{"class":1753},[61,62363,60957],{"class":1866},[61,62365,6550],{"class":1753},[61,62367,62368,62370,62373,62375,62377],{"class":1731,"line":1811},[61,62369,62304],{"class":1753},[61,62371,62372],{"class":2165},"'τ sweep min (ns)'",[61,62374,1435],{"class":1753},[61,62376,60972],{"class":1866},[61,62378,6550],{"class":1753},[61,62380,62381,62383,62386,62388,62390],{"class":1731,"line":1816},[61,62382,62304],{"class":1753},[61,62384,62385],{"class":2165},"'τ sweep max (ns)'",[61,62387,1435],{"class":1753},[61,62389,60986],{"class":1866},[61,62391,6550],{"class":1753},[61,62393,62394,62396,62399,62401,62403],{"class":1731,"line":1822},[61,62395,62304],{"class":1753},[61,62397,62398],{"class":2165},"'N_τ'",[61,62400,1435],{"class":1753},[61,62402,61001],{"class":1866},[61,62404,6550],{"class":1753},[61,62406,62407,62409,62412,62414,62416],{"class":1731,"line":1828},[61,62408,62304],{"class":1753},[61,62410,62411],{"class":2165},"'T2* (ns; None=off)'",[61,62413,1435],{"class":1753},[61,62415,61015],{"class":1866},[61,62417,6550],{"class":1753},[61,62419,62420],{"class":1731,"line":1833},[61,62421,62422],{"class":1753},"    ],\n",[61,62424,62425,62428,62430,62432,62435,62437,62440],{"class":1731,"line":1840},[61,62426,62427],{"class":3402},"    columns",[61,62429,2000],{"class":1859},[61,62431,27808],{"class":1753},[61,62433,62434],{"class":2165},"'Parameter'",[61,62436,1435],{"class":1753},[61,62438,62439],{"class":2165},"'Value'",[61,62441,5701],{"class":1753},[61,62443,62444],{"class":1731,"line":1853},[61,62445,2016],{"class":1753},[61,62447,62448],{"class":1731,"line":1873},[61,62449,62450],{"class":1753},"controls_table\n",[1720,62452,62455],{"className":62453,"code":62454,"language":29075,"meta":1725},[36200],"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",[1727,62456,62454],{"__ignoreMap":1725},[1720,62458,62460],{"className":1722,"code":62459,"language":1724,"meta":1725,"style":1725},"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",[1727,62461,62462,62488,62513,62544,62557,62570,62593,62602],{"__ignoreMap":1725},[61,62463,62464,62466,62468,62470,62472,62474,62476,62478,62480,62482,62484,62486],{"class":1731,"line":1732},[61,62465,24952],{"class":1753},[61,62467,2000],{"class":1859},[61,62469,17065],{"class":1753},[61,62471,17068],{"class":1836},[61,62473,849],{"class":1753},[61,62475,3624],{"class":3402},[61,62477,2000],{"class":1859},[61,62479,849],{"class":1753},[61,62481,1474],{"class":1866},[61,62483,1435],{"class":1753},[61,62485,1026],{"class":1866},[61,62487,5546],{"class":1753},[61,62489,62490,62493,62495,62497,62500,62503,62506,62508,62511],{"class":1731,"line":1739},[61,62491,62492],{"class":1753},"c ",[61,62494,2000],{"class":1859},[61,62496,38434],{"class":1753},[61,62498,62499],{"class":1836},"pcolormesh",[61,62501,62502],{"class":1753},"(freq_ghz, dur_ns, Pe, ",[61,62504,62505],{"class":3402},"shading",[61,62507,2000],{"class":1859},[61,62509,62510],{"class":2165},"'auto'",[61,62512,2016],{"class":1753},[61,62514,62515,62518,62520,62522,62524,62527,62529,62531,62533,62535,62537,62539,62542],{"class":1731,"line":1746},[61,62516,62517],{"class":1753},"cb ",[61,62519,2000],{"class":1859},[61,62521,39048],{"class":1753},[61,62523,4288],{"class":1836},[61,62525,62526],{"class":1753},"(c, ",[61,62528,3368],{"class":3402},[61,62530,2000],{"class":1859},[61,62532,39060],{"class":1753},[61,62534,3666],{"class":3402},[61,62536,2000],{"class":1859},[61,62538,3763],{"class":1749},[61,62540,62541],{"class":3766},"'$P_e$'",[61,62543,2016],{"class":1753},[61,62545,62546,62548,62550,62552,62555],{"class":1731,"line":1763},[61,62547,24987],{"class":1753},[61,62549,17159],{"class":1836},[61,62551,849],{"class":1753},[61,62553,62554],{"class":2165},"'Drive frequency f (GHz)'",[61,62556,2016],{"class":1753},[61,62558,62559,62561,62563,62565,62568],{"class":1731,"line":1776},[61,62560,24987],{"class":1753},[61,62562,17173],{"class":1836},[61,62564,849],{"class":1753},[61,62566,62567],{"class":2165},"'Pulse duration τ (ns)'",[61,62569,2016],{"class":1753},[61,62571,62572,62574,62576,62578,62581,62584,62586,62588,62591],{"class":1731,"line":1784},[61,62573,24987],{"class":1753},[61,62575,17149],{"class":1836},[61,62577,849],{"class":1753},[61,62579,62580],{"class":2165},"'Rabi excited-state probability $P_e(f,",[61,62582,62583],{"class":1859},"\\\\",[61,62585,862],{"class":2165},[61,62587,62583],{"class":1859},[61,62589,62590],{"class":2165},"tau)$'",[61,62592,2016],{"class":1753},[61,62594,62595,62598,62600],{"class":1731,"line":1798},[61,62596,62597],{"class":1753},"fig.",[61,62599,17242],{"class":1836},[61,62601,2089],{"class":1753},[61,62603,62604,62606,62608],{"class":1731,"line":1811},[61,62605,3616],{"class":1753},[61,62607,3850],{"class":1836},[61,62609,2089],{"class":1753},[4797,62611],{"alt":62612,"src":62613},"Output 1 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-01.webp",[1720,62615,62617],{"className":1722,"code":62616,"language":1724,"meta":1725,"style":1725},"# @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",[1727,62618,62619,62624,62629,62633,62640,62647,62653,62657,62662,62667,62671,62680,62684,62693,62705,62717,62729,62741,62746,62750,62776,62791,62798,62805,62809,62813,62822,62832,62836,62841,62851,62855,62865,62884,62893,62912,62916,62925,62934,62943,62952,62961,62965,62975,62985,62995,63005,63015,63019,63029,63039,63049,63059,63069,63073,63083,63101,63105,63109,63131,63136,63155,63159,63163,63172,63188,63196,63201,63212,63217,63221,63230,63235,63253,63265,63270,63274,63283,63287,63291,63298,63308,63338,63349,63353,63361,63366,63371,63378,63385,63392,63399,63406,63413,63421,63426,63436,63440,63448,63456,63460,63464,63470,63476,63482,63488,63494,63498,63508,63512,63516,63520,63531,63535,63545,63555,63567,63579,63583,63589,63602,63606,63618,63633,63652,63659,63664,63668,63677,63682,63693,63697,63701,63705,63709,63731,63736,63754,63758,63762,63771,63782,63793,63804,63821,63826,63849,63871,63886,63890,63916,63920,63943,63964,63984,63988,64001,64015,64032,64036,64040,64054,64059,64064,64069,64073,64077,64106,64132,64157,64161,64174,64188,64192,64196,64210,64215,64220,64225,64230,64235,64239,64243,64256,64261,64272,64280,64290,64295,64300,64305,64312,64328,64337,64346,64355,64364,64387,64393,64398,64403,64408,64418,64427,64436,64441,64447,64462,64473,64488,64498,64506,64515,64520,64542,64547,64563,64580,64585,64598,64611,64616,64624,64629,64634,64644,64652,64660,64672,64684,64693,64699,64709,64719,64728,64737,64748,64760,64771,64776,64781,64786,64796,64803,64811,64822,64833,64842,64848,64858,64868,64877,64886,64895,64907,64919,64928,64933,64938,64943,64953,64960,64967,64978,64989,64998,65004,65014,65022,65029,65036,65045,65054,65063,65072,65083,65095,65107,65116,65121,65126,65131,65141,65148,65155,65167,65178,65187,65193,65209,65214,65223,65233,65242,65251,65257,65266,65278,65290,65302,65307,65312,65317,65335,65341,65350,65357,65364,65372,65383,65394,65405,65410,65419,65426,65433,65441,65462,65473,65484,65495,65506,65511,65516,65521,65535,65547,65553,65558,65565,65575,65580,65592,65603,65613,65624,65629,65634,65646,65657,65666,65677,65682,65687,65697,65703,65717,65722,65732,65737,65749,65754,65759,65781,65786,65809,65820,65859,65869,65879,65898,65916,65921,65926,65938,65948,65958,65963,65968,65976,65986,66001,66012,66024,66029,66036,66045,66059,66070,66082,66087,66092,66102,66110,66120,66141,66152,66164,66169,66176,66185,66204,66215,66227,66232,66241,66246,66256,66264,66273,66287,66298,66324,66329,66336,66345,66358,66369,66392,66397,66402,66416,66424,66433,66446,66468,66479,66484,66491,66500,66513,66534,66545,66550,66555,66565,66575,66583,66593,66602,66614,66626,66638,66643,66650,66659,66668,66679,66690,66701,66706,66713,66722,66731,66742,66753,66774,66779,66786,66795,66804,66815,66826,66847,66852,66857,66862,66872,66883,66895,66907,66919,66931,66936,66941,66950,66955,66965,66980,66995,67000,67012,67037,67062,67087,67111,67137,67144,67150,67175,67196,67218,67223,67230,67236,67258,67279,67300,67305,67312,67318,67347,67352,67359,67365,67392,67397,67404,67410,67433],{"__ignoreMap":1725},[61,62620,62621],{"class":1731,"line":1732},[61,62622,62623],{"class":1735},"# @title Cell: QuTiP Bloch sphere with labeled Rabi-control axes\n",[61,62625,62626],{"class":1731,"line":1739},[61,62627,62628],{"class":2165},"\"\"\"Render labeled Bloch-sphere trajectories for resonant and detuned drives.\"\"\"\n",[61,62630,62631],{"class":1731,"line":1746},[61,62632,1743],{"emptyLinePlaceholder":1742},[61,62634,62635,62637],{"class":1731,"line":1763},[61,62636,1757],{"class":1749},[61,62638,62639],{"class":1753}," importlib.util\n",[61,62641,62642,62644],{"class":1731,"line":1776},[61,62643,1757],{"class":1749},[61,62645,62646],{"class":1753}," subprocess\n",[61,62648,62649,62651],{"class":1731,"line":1784},[61,62650,1757],{"class":1749},[61,62652,61367],{"class":1753},[61,62654,62655],{"class":1731,"line":1798},[61,62656,1743],{"emptyLinePlaceholder":1742},[61,62658,62659],{"class":1731,"line":1811},[61,62660,62661],{"class":1735},"# -------------------------------------------------------------------------\n",[61,62663,62664],{"class":1731,"line":1816},[61,62665,62666],{"class":1735},"# Control knobs\n",[61,62668,62669],{"class":1731,"line":1822},[61,62670,62661],{"class":1735},[61,62672,62673,62676,62678],{"class":1731,"line":1828},[61,62674,62675],{"class":1866},"INSTALL_BLOCH_DEPENDENCIES",[61,62677,1863],{"class":1859},[61,62679,61476],{"class":1866},[61,62681,62682],{"class":1731,"line":1833},[61,62683,1743],{"emptyLinePlaceholder":1742},[61,62685,62686,62689,62691],{"class":1731,"line":1840},[61,62687,62688],{"class":1866},"BLOCH_REQUIRED_PACKAGES",[61,62690,1863],{"class":1859},[61,62692,37043],{"class":1753},[61,62694,62695,62698,62700,62703],{"class":1731,"line":1853},[61,62696,62697],{"class":2165},"    \"numpy\"",[61,62699,2127],{"class":1753},[61,62701,62702],{"class":2165},"\"numpy\"",[61,62704,36573],{"class":1753},[61,62706,62707,62710,62712,62715],{"class":1731,"line":1873},[61,62708,62709],{"class":2165},"    \"matplotlib\"",[61,62711,2127],{"class":1753},[61,62713,62714],{"class":2165},"\"matplotlib\"",[61,62716,36573],{"class":1753},[61,62718,62719,62722,62724,62727],{"class":1731,"line":1889},[61,62720,62721],{"class":2165},"    \"scipy\"",[61,62723,2127],{"class":1753},[61,62725,62726],{"class":2165},"\"scipy\"",[61,62728,36573],{"class":1753},[61,62730,62731,62734,62736,62739],{"class":1731,"line":1908},[61,62732,62733],{"class":2165},"    \"qutip\"",[61,62735,2127],{"class":1753},[61,62737,62738],{"class":2165},"\"qutip\"",[61,62740,36573],{"class":1753},[61,62742,62743],{"class":1731,"line":1925},[61,62744,62745],{"class":1753},"}\n",[61,62747,62748],{"class":1731,"line":1941},[61,62749,1743],{"emptyLinePlaceholder":1742},[61,62751,62752,62755,62757,62760,62763,62765,62767,62770,62772,62774],{"class":1731,"line":1957},[61,62753,62754],{"class":1866},"BLOCH_F0_GHZ",[61,62756,1863],{"class":1859},[61,62758,62759],{"class":1859}," globals",[61,62761,62762],{"class":1753},"().",[61,62764,37179],{"class":1836},[61,62766,849],{"class":1753},[61,62768,62769],{"class":2165},"\"F0_GHZ\"",[61,62771,1435],{"class":1753},[61,62773,48061],{"class":1866},[61,62775,2016],{"class":1753},[61,62777,62778,62781,62783,62785,62787,62789],{"class":1731,"line":1974},[61,62779,62780],{"class":1866},"BLOCH_OMEGA_ONRESONANCE_MHZ",[61,62782,1863],{"class":1859},[61,62784,62759],{"class":1859},[61,62786,62762],{"class":1753},[61,62788,37179],{"class":1836},[61,62790,38439],{"class":1753},[61,62792,62793,62796],{"class":1731,"line":1979},[61,62794,62795],{"class":2165},"    \"OMEGA_ONRESONANCE_MHZ\"",[61,62797,36573],{"class":1753},[61,62799,62800,62803],{"class":1731,"line":1985},[61,62801,62802],{"class":1866},"    20.0",[61,62804,36573],{"class":1753},[61,62806,62807],{"class":1731,"line":2019},[61,62808,2016],{"class":1753},[61,62810,62811],{"class":1731,"line":2033},[61,62812,1743],{"emptyLinePlaceholder":1742},[61,62814,62815,62818,62820],{"class":1731,"line":2061},[61,62816,62817],{"class":1866},"BLOCH_RESONANT_DETUNING_MHZ",[61,62819,1863],{"class":1859},[61,62821,44941],{"class":1866},[61,62823,62824,62827,62829],{"class":1731,"line":2074},[61,62825,62826],{"class":1866},"BLOCH_DETUNED_DETUNING_MHZ",[61,62828,1863],{"class":1859},[61,62830,62831],{"class":1866}," 40.0\n",[61,62833,62834],{"class":1731,"line":2079},[61,62835,1743],{"emptyLinePlaceholder":1742},[61,62837,62838],{"class":1731,"line":2092},[61,62839,62840],{"class":1735},"# Use None to automatically plot one resonant pi pulse.\n",[61,62842,62843,62846,62848],{"class":1731,"line":2097},[61,62844,62845],{"class":1866},"BLOCH_DURATION_NS",[61,62847,1863],{"class":1859},[61,62849,62850],{"class":1866}," None\n",[61,62852,62853],{"class":1731,"line":2102},[61,62854,1743],{"emptyLinePlaceholder":1742},[61,62856,62857,62860,62862],{"class":1731,"line":2108},[61,62858,62859],{"class":1866},"BLOCH_NUM_TIME_POINTS",[61,62861,1863],{"class":1859},[61,62863,62864],{"class":1866}," 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\"\n",[61,67176,67178,67180,67182,67184,67186,67188,67190,67192,67194],{"class":1731,"line":67177},480,[61,67179,67155],{"class":1749},[61,67181,26088],{"class":2165},[61,67183,3688],{"class":1866},[61,67185,67163],{"class":1753},[61,67187,57],{"class":1866},[61,67189,27817],{"class":1753},[61,67191,67029],{"class":1749},[61,67193,3697],{"class":1866},[61,67195,67174],{"class":2165},[61,67197,67199,67201,67203,67205,67207,67209,67211,67213,67215],{"class":1731,"line":67198},481,[61,67200,67155],{"class":1749},[61,67202,26088],{"class":2165},[61,67204,3688],{"class":1866},[61,67206,67163],{"class":1753},[61,67208,337],{"class":1866},[61,67210,27817],{"class":1753},[61,67212,67029],{"class":1749},[61,67214,3697],{"class":1866},[61,67216,67217],{"class":2165},")\"\n",[61,67219,67221],{"class":1731,"line":67220},482,[61,67222,3057],{"class":1753},[61,67224,67226,67228],{"class":1731,"line":67225},483,[61,67227,39829],{"class":1859},[61,67229,38439],{"class":1753},[61,67231,67233],{"class":1731,"line":67232},484,[61,67234,67235],{"class":2165},"        \"Detuned control axis n_det: \"\n",[61,67237,67239,67241,67243,67245,67248,67250,67252,67254,67256],{"class":1731,"line":67238},485,[61,67240,67155],{"class":1749},[61,67242,67158],{"class":2165},[61,67244,3688],{"class":1866},[61,67246,67247],{"class":1753},"detuned_axis[",[61,67249,1495],{"class":1866},[61,67251,27817],{"class":1753},[61,67253,67029],{"class":1749},[61,67255,3697],{"class":1866},[61,67257,67174],{"class":2165},[61,67259,67261,67263,67265,67267,67269,67271,67273,67275,67277],{"class":1731,"line":67260},486,[61,67262,67155],{"class":1749},[61,67264,26088],{"class":2165},[61,67266,3688],{"class":1866},[61,67268,67247],{"class":1753},[61,67270,57],{"class":1866},[61,67272,27817],{"class":1753},[61,67274,67029],{"class":1749},[61,67276,3697],{"class":1866},[61,67278,67174],{"class":2165},[61,67280,67282,67284,67286,67288,67290,67292,67294,67296,67298],{"class":1731,"line":67281},487,[61,67283,67155],{"class":1749},[61,67285,26088],{"class":2165},[61,67287,3688],{"class":1866},[61,67289,67247],{"class":1753},[61,67291,337],{"class":1866},[61,67293,27817],{"class":1753},[61,67295,67029],{"class":1749},[61,67297,3697],{"class":1866},[61,67299,67217],{"class":2165},[61,67301,67303],{"class":1731,"line":67302},488,[61,67304,3057],{"class":1753},[61,67306,67308,67310],{"class":1731,"line":67307},489,[61,67309,39829],{"class":1859},[61,67311,38439],{"class":1753},[61,67313,67315],{"class":1731,"line":67314},490,[61,67316,67317],{"class":2165},"        \"Resonant final excited-state probability: \"\n",[61,67319,67321,67323,67325,67327,67329,67332,67335,67337,67339,67341,67343,67345],{"class":1731,"line":67320},491,[61,67322,67155],{"class":1749},[61,67324,26088],{"class":2165},[61,67326,3688],{"class":1866},[61,67328,65995],{"class":1753},[61,67330,67331],{"class":2165},"'excited_probability'",[61,67333,67334],{"class":1753},"][",[61,67336,2006],{"class":1859},[61,67338,57],{"class":1866},[61,67340,27817],{"class":1753},[61,67342,67029],{"class":1749},[61,67344,3697],{"class":1866},[61,67346,40488],{"class":2165},[61,67348,67350],{"class":1731,"line":67349},492,[61,67351,3057],{"class":1753},[61,67353,67355,67357],{"class":1731,"line":67354},493,[61,67356,39829],{"class":1859},[61,67358,38439],{"class":1753},[61,67360,67362],{"class":1731,"line":67361},494,[61,67363,67364],{"class":2165},"        \"Detuned final excited-state probability: \"\n",[61,67366,67368,67370,67372,67374,67376,67378,67380,67382,67384,67386,67388,67390],{"class":1731,"line":67367},495,[61,67369,67155],{"class":1749},[61,67371,26088],{"class":2165},[61,67373,3688],{"class":1866},[61,67375,66054],{"class":1753},[61,67377,67331],{"class":2165},[61,67379,67334],{"class":1753},[61,67381,2006],{"class":1859},[61,67383,57],{"class":1866},[61,67385,27817],{"class":1753},[61,67387,67029],{"class":1749},[61,67389,3697],{"class":1866},[61,67391,40488],{"class":2165},[61,67393,67395],{"class":1731,"line":67394},496,[61,67396,3057],{"class":1753},[61,67398,67400,67402],{"class":1731,"line":67399},497,[61,67401,39829],{"class":1859},[61,67403,38439],{"class":1753},[61,67405,67407],{"class":1731,"line":67406},498,[61,67408,67409],{"class":2165},"        \"Detuned generalized Rabi frequency: \"\n",[61,67411,67413,67415,67417,67419,67421,67424,67426,67428,67430],{"class":1731,"line":67412},499,[61,67414,67155],{"class":1749},[61,67416,26088],{"class":2165},[61,67418,3688],{"class":1866},[61,67420,66054],{"class":1753},[61,67422,67423],{"class":2165},"'generalized_rabi_mhz'",[61,67425,27817],{"class":1753},[61,67427,67029],{"class":1749},[61,67429,3697],{"class":1866},[61,67431,67432],{"class":2165}," MHz\"\n",[61,67434,67436],{"class":1731,"line":67435},500,[61,67437,3057],{"class":1753},[4797,67439],{"alt":67440,"src":67441},"Output 2 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-02.webp",[1720,67443,67446],{"className":67444,"code":67445,"language":29075,"meta":1725},[36200],"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",[1727,67447,67445],{"__ignoreMap":1725},[1720,67449,67451],{"className":1722,"code":67450,"language":1724,"meta":1725,"style":1725},"# --- 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",[1727,67452,67453,67458,67470,67480,67490,67494,67499,67515,67532,67553,67557,67562,67567,67591,67617,67647,67665,67680,67684,67689,67712,67738,67766,67783,67797,67801,67806,67828,67832,67837,67872,67898,67902,67915,67920,67943,67963,67974,67993,67997,68001,68014,68042,68056,68069,68073,68078,68115,68130,68134,68142],{"__ignoreMap":1725},[61,67454,67455],{"class":1731,"line":1732},[61,67456,67457],{"class":1735},"# --- High-resolution 3D surface with colormap & legend  ---\n",[61,67459,67460,67462,67465,67467],{"class":1731,"line":1739},[61,67461,1750],{"class":1749},[61,67463,67464],{"class":1753}," mpl_toolkits.mplot3d ",[61,67466,1757],{"class":1749},[61,67468,67469],{"class":1753}," Axes3D\n",[61,67471,67472,67474,67476,67478],{"class":1731,"line":1746},[61,67473,1757],{"class":1749},[61,67475,1789],{"class":1753},[61,67477,1792],{"class":1749},[61,67479,1795],{"class":1753},[61,67481,67482,67484,67486,67488],{"class":1731,"line":1763},[61,67483,1757],{"class":1749},[61,67485,1803],{"class":1753},[61,67487,1792],{"class":1749},[61,67489,1808],{"class":1753},[61,67491,67492],{"class":1731,"line":1776},[61,67493,1743],{"emptyLinePlaceholder":1742},[61,67495,67496],{"class":1731,"line":1784},[61,67497,67498],{"class":1735},"# Controls for surface quality and appearance\n",[61,67500,67501,67504,67506,67508,67510,67512],{"class":1731,"line":1798},[61,67502,67503],{"class":1866},"UPSAMPLE_FACTOR",[61,67505,2127],{"class":1753},[61,67507,2025],{"class":1859},[61,67509,1863],{"class":1859},[61,67511,2654],{"class":1866},[61,67513,67514],{"class":1735},"           # 2=~4× more faces; try 3 for even smoother\n",[61,67516,67517,67520,67522,67524,67526,67529],{"class":1731,"line":1811},[61,67518,67519],{"class":1866},"SURFACE_CMAP",[61,67521,2127],{"class":1753},[61,67523,36784],{"class":1859},[61,67525,1863],{"class":1859},[61,67527,67528],{"class":2165}," 'inferno'",[61,67530,67531],{"class":1735},"      # e.g., 'viridis', 'plasma', 'magma', 'inferno'\n",[61,67533,67534,67537,67539,67542,67544,67546,67548,67550],{"class":1731,"line":1816},[61,67535,67536],{"class":1866},"Z_MIN",[61,67538,1435],{"class":1753},[61,67540,67541],{"class":1866},"Z_MAX",[61,67543,1863],{"class":1859},[61,67545,2506],{"class":1866},[61,67547,1435],{"class":1753},[61,67549,2319],{"class":1866},[61,67551,67552],{"class":1735},"            # clamp\u002Fnormalize expected Pe range\n",[61,67554,67555],{"class":1731,"line":1822},[61,67556,1743],{"emptyLinePlaceholder":1742},[61,67558,67559],{"class":1731,"line":1828},[61,67560,67561],{"class":1735},"# --- Bilinear upsampling (SciPy-free) ---\n",[61,67563,67564],{"class":1731,"line":1833},[61,67565,67566],{"class":1735},"# 1) Upsample frequency axis for each fixed τ\n",[61,67568,67569,67572,67574,67576,67578,67581,67583,67586,67588],{"class":1731,"line":1840},[61,67570,67571],{"class":1753},"freq_hi ",[61,67573,2000],{"class":1859},[61,67575,2193],{"class":1753},[61,67577,3453],{"class":1836},[61,67579,67580],{"class":1753},"(freq_ghz.",[61,67582,4202],{"class":1836},[61,67584,67585],{"class":1753},"(), freq_ghz.",[61,67587,4212],{"class":1836},[61,67589,67590],{"class":1753},"(),\n",[61,67592,67593,67596,67598,67600,67602,67605,67607,67609,67611,67613,67615],{"class":1731,"line":1853},[61,67594,67595],{"class":1866},"                      UPSAMPLE_FACTOR",[61,67597,2298],{"class":1859},[61,67599,2003],{"class":1753},[61,67601,5742],{"class":1859},[61,67603,67604],{"class":1753},"(freq_ghz) ",[61,67606,2006],{"class":1859},[61,67608,5266],{"class":1866},[61,67610,2624],{"class":1753},[61,67612,2419],{"class":1859},[61,67614,5266],{"class":1866},[61,67616,2016],{"class":1753},[61,67618,67619,67622,67624,67626,67628,67630,67632,67635,67637,67640,67642,67645],{"class":1731,"line":1873},[61,67620,67621],{"class":1753},"Pe_tau_f ",[61,67623,2000],{"class":1859},[61,67625,2193],{"class":1753},[61,67627,4085],{"class":1836},[61,67629,4088],{"class":1753},[61,67631,5742],{"class":1859},[61,67633,67634],{"class":1753},"(dur_ns), ",[61,67636,5742],{"class":1859},[61,67638,67639],{"class":1753},"(freq_hi)), ",[61,67641,6950],{"class":3402},[61,67643,67644],{"class":1859},"=float",[61,67646,2016],{"class":1753},[61,67648,67649,67651,67654,67656,67658,67660,67662],{"class":1731,"line":1889},[61,67650,2486],{"class":1749},[61,67652,67653],{"class":1753}," i ",[61,67655,2492],{"class":1749},[61,67657,17206],{"class":1859},[61,67659,849],{"class":1753},[61,67661,5742],{"class":1859},[61,67663,67664],{"class":1753},"(dur_ns)):\n",[61,67666,67667,67670,67672,67674,67677],{"class":1731,"line":1908},[61,67668,67669],{"class":1753},"    Pe_tau_f[i, :] ",[61,67671,2000],{"class":1859},[61,67673,2193],{"class":1753},[61,67675,67676],{"class":1836},"interp",[61,67678,67679],{"class":1753},"(freq_hi, freq_ghz, Pe[i, :])\n",[61,67681,67682],{"class":1731,"line":1925},[61,67683,1743],{"emptyLinePlaceholder":1742},[61,67685,67686],{"class":1731,"line":1941},[61,67687,67688],{"class":1735},"# 2) Upsample τ axis for each fixed frequency\n",[61,67690,67691,67694,67696,67698,67700,67703,67705,67708,67710],{"class":1731,"line":1957},[61,67692,67693],{"class":1753},"dur_hi ",[61,67695,2000],{"class":1859},[61,67697,2193],{"class":1753},[61,67699,3453],{"class":1836},[61,67701,67702],{"class":1753},"(dur_ns.",[61,67704,4202],{"class":1836},[61,67706,67707],{"class":1753},"(), dur_ns.",[61,67709,4212],{"class":1836},[61,67711,67590],{"class":1753},[61,67713,67714,67717,67719,67721,67723,67726,67728,67730,67732,67734,67736],{"class":1731,"line":1974},[61,67715,67716],{"class":1866},"                     UPSAMPLE_FACTOR",[61,67718,2298],{"class":1859},[61,67720,2003],{"class":1753},[61,67722,5742],{"class":1859},[61,67724,67725],{"class":1753},"(dur_ns) ",[61,67727,2006],{"class":1859},[61,67729,5266],{"class":1866},[61,67731,2624],{"class":1753},[61,67733,2419],{"class":1859},[61,67735,5266],{"class":1866},[61,67737,2016],{"class":1753},[61,67739,67740,67743,67745,67747,67749,67751,67753,67756,67758,67760,67762,67764],{"class":1731,"line":1979},[61,67741,67742],{"class":1753},"Pe_hi ",[61,67744,2000],{"class":1859},[61,67746,2193],{"class":1753},[61,67748,4085],{"class":1836},[61,67750,4088],{"class":1753},[61,67752,5742],{"class":1859},[61,67754,67755],{"class":1753},"(dur_hi), ",[61,67757,5742],{"class":1859},[61,67759,67639],{"class":1753},[61,67761,6950],{"class":3402},[61,67763,67644],{"class":1859},[61,67765,2016],{"class":1753},[61,67767,67768,67770,67772,67774,67776,67778,67780],{"class":1731,"line":1985},[61,67769,2486],{"class":1749},[61,67771,17201],{"class":1753},[61,67773,2492],{"class":1749},[61,67775,17206],{"class":1859},[61,67777,849],{"class":1753},[61,67779,5742],{"class":1859},[61,67781,67782],{"class":1753},"(freq_hi)):\n",[61,67784,67785,67788,67790,67792,67794],{"class":1731,"line":2019},[61,67786,67787],{"class":1753},"    Pe_hi[:, j] ",[61,67789,2000],{"class":1859},[61,67791,2193],{"class":1753},[61,67793,67676],{"class":1836},[61,67795,67796],{"class":1753},"(dur_hi, dur_ns, Pe_tau_f[:, j])\n",[61,67798,67799],{"class":1731,"line":2033},[61,67800,1743],{"emptyLinePlaceholder":1742},[61,67802,67803],{"class":1731,"line":2061},[61,67804,67805],{"class":1735},"# Mesh for plotting\n",[61,67807,67808,67811,67813,67815,67817,67820,67822,67824,67826],{"class":1731,"line":2074},[61,67809,67810],{"class":1753},"F_hi, T_hi ",[61,67812,2000],{"class":1859},[61,67814,2193],{"class":1753},[61,67816,37855],{"class":1836},[61,67818,67819],{"class":1753},"(freq_hi, dur_hi, ",[61,67821,37861],{"class":3402},[61,67823,2000],{"class":1859},[61,67825,61933],{"class":2165},[61,67827,2016],{"class":1753},[61,67829,67830],{"class":1731,"line":2079},[61,67831,1743],{"emptyLinePlaceholder":1742},[61,67833,67834],{"class":1731,"line":2092},[61,67835,67836],{"class":1735},"# --- Plot ---\n",[61,67838,67839,67842,67844,67846,67848,67850,67852,67854,67856,67858,67860,67862,67864,67866,67868,67870],{"class":1731,"line":2097},[61,67840,67841],{"class":1753},"fig ",[61,67843,2000],{"class":1859},[61,67845,17065],{"class":1753},[61,67847,3619],{"class":1836},[61,67849,849],{"class":1753},[61,67851,3624],{"class":3402},[61,67853,2000],{"class":1859},[61,67855,849],{"class":1753},[61,67857,3631],{"class":1866},[61,67859,1435],{"class":1753},[61,67861,1300],{"class":1866},[61,67863,3638],{"class":1753},[61,67865,3641],{"class":3402},[61,67867,2000],{"class":1859},[61,67869,36518],{"class":1866},[61,67871,2016],{"class":1753},[61,67873,67874,67877,67879,67881,67883,67885,67887,67889,67891,67893,67896],{"class":1731,"line":2102},[61,67875,67876],{"class":1753},"ax ",[61,67878,2000],{"class":1859},[61,67880,39048],{"class":1753},[61,67882,42252],{"class":1836},[61,67884,849],{"class":1753},[61,67886,42257],{"class":1866},[61,67888,1435],{"class":1753},[61,67890,42262],{"class":3402},[61,67892,2000],{"class":1859},[61,67894,67895],{"class":2165},"'3d'",[61,67897,2016],{"class":1753},[61,67899,67900],{"class":1731,"line":2108},[61,67901,1743],{"emptyLinePlaceholder":1742},[61,67903,67904,67907,67909,67911,67913],{"class":1731,"line":2113},[61,67905,67906],{"class":1753},"surf ",[61,67908,2000],{"class":1859},[61,67910,38434],{"class":1753},[61,67912,42414],{"class":1836},[61,67914,38439],{"class":1753},[61,67916,67917],{"class":1731,"line":2162},[61,67918,67919],{"class":1753},"    F_hi, T_hi, Pe_hi,\n",[61,67921,67922,67925,67927,67929,67931,67933,67935,67937,67940],{"class":1731,"line":2169},[61,67923,67924],{"class":3402},"    rstride",[61,67926,2000],{"class":1859},[61,67928,57],{"class":1866},[61,67930,1435],{"class":1753},[61,67932,42447],{"class":3402},[61,67934,2000],{"class":1859},[61,67936,57],{"class":1866},[61,67938,67939],{"class":1753},",             ",[61,67941,67942],{"class":1735},"# use every row\u002Fcolumn (after upsampling)\n",[61,67944,67945,67948,67950,67952,67954,67957,67959,67961],{"class":1731,"line":2187},[61,67946,67947],{"class":3402},"    linewidth",[61,67949,2000],{"class":1859},[61,67951,1495],{"class":1866},[61,67953,1435],{"class":1753},[61,67955,67956],{"class":3402},"antialiased",[61,67958,2000],{"class":1859},[61,67960,24385],{"class":1866},[61,67962,36573],{"class":1753},[61,67964,67965,67968,67970,67972],{"class":1731,"line":2207},[61,67966,67967],{"class":3402},"    cmap",[61,67969,2000],{"class":1859},[61,67971,67519],{"class":1866},[61,67973,36573],{"class":1753},[61,67975,67976,67979,67981,67983,67985,67987,67989,67991],{"class":1731,"line":2221},[61,67977,67978],{"class":3402},"    vmin",[61,67980,2000],{"class":1859},[61,67982,67536],{"class":1866},[61,67984,1435],{"class":1753},[61,67986,39025],{"class":3402},[61,67988,2000],{"class":1859},[61,67990,67541],{"class":1866},[61,67992,36573],{"class":1753},[61,67994,67995],{"class":1731,"line":2226},[61,67996,2016],{"class":1753},[61,67998,67999],{"class":1731,"line":2258},[61,68000,1743],{"emptyLinePlaceholder":1742},[61,68002,68003,68005,68007,68009,68012],{"class":1731,"line":2264},[61,68004,24987],{"class":1753},[61,68006,17159],{"class":1836},[61,68008,849],{"class":1753},[61,68010,68011],{"class":2165},"'f (GHz)'",[61,68013,2016],{"class":1753},[61,68015,68016,68018,68020,68022,68024,68026,68029,68032,68034,68036,68038,68040],{"class":1731,"line":2280},[61,68017,24987],{"class":1753},[61,68019,17173],{"class":1836},[61,68021,849],{"class":1753},[61,68023,3763],{"class":1749},[61,68025,3794],{"class":3766},[61,68027,68028],{"class":1859},"\\t",[61,68030,68031],{"class":3766},"au$ ",[61,68033,849],{"class":1866},[61,68035,52926],{"class":3766},[61,68037,300],{"class":1866},[61,68039,3777],{"class":3766},[61,68041,2016],{"class":1753},[61,68043,68044,68046,68048,68050,68052,68054],{"class":1731,"line":2304},[61,68045,24987],{"class":1753},[61,68047,42587],{"class":1836},[61,68049,849],{"class":1753},[61,68051,3763],{"class":1749},[61,68053,62541],{"class":3766},[61,68055,2016],{"class":1753},[61,68057,68058,68060,68062,68064,68067],{"class":1731,"line":2326},[61,68059,24987],{"class":1753},[61,68061,17149],{"class":1836},[61,68063,849],{"class":1753},[61,68065,68066],{"class":2165},"'Rabi $P_e$ surface'",[61,68068,2016],{"class":1753},[61,68070,68071],{"class":1731,"line":2342},[61,68072,1743],{"emptyLinePlaceholder":1742},[61,68074,68075],{"class":1731,"line":2379},[61,68076,68077],{"class":1735},"# Colorbar (legend for the colormap)\n",[61,68079,68080,68082,68084,68086,68088,68091,68093,68095,68097,68100,68102,68104,68106,68108,68110,68113],{"class":1731,"line":2396},[61,68081,62517],{"class":1753},[61,68083,2000],{"class":1859},[61,68085,39048],{"class":1753},[61,68087,4288],{"class":1836},[61,68089,68090],{"class":1753},"(surf, ",[61,68092,3368],{"class":3402},[61,68094,2000],{"class":1859},[61,68096,39060],{"class":1753},[61,68098,68099],{"class":3402},"shrink",[61,68101,2000],{"class":1859},[61,68103,25004],{"class":1866},[61,68105,1435],{"class":1753},[61,68107,39073],{"class":3402},[61,68109,2000],{"class":1859},[61,68111,68112],{"class":1866},"0.08",[61,68114,2016],{"class":1753},[61,68116,68117,68120,68122,68124,68126,68128],{"class":1731,"line":2404},[61,68118,68119],{"class":1753},"cb.",[61,68121,39088],{"class":1836},[61,68123,849],{"class":1753},[61,68125,3763],{"class":1749},[61,68127,62541],{"class":3766},[61,68129,2016],{"class":1753},[61,68131,68132],{"class":1731,"line":2443},[61,68133,1743],{"emptyLinePlaceholder":1742},[61,68135,68136,68138,68140],{"class":1731,"line":2475},[61,68137,62597],{"class":1753},[61,68139,17242],{"class":1836},[61,68141,2089],{"class":1753},[61,68143,68144,68146,68148],{"class":1731,"line":2520},[61,68145,3616],{"class":1753},[61,68147,3850],{"class":1836},[61,68149,2089],{"class":1753},[4797,68151],{"alt":68152,"src":68153},"Output 3 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-03.webp",[1720,68155,68157],{"className":1722,"code":68156,"language":1724,"meta":1725,"style":1725},"# @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",[1727,68158,68159,68164,68169,68184,68199,68203,68213,68223,68227,68263,68268,68293,68314,68361,68375,68398,68402,68407,68428,68438,68442,68469,68486,68512,68526,68531,68576,68593,68601,68609,68613,68618,68639,68649,68653,68679,68696,68708,68722,68768,68784,68792],{"__ignoreMap":1725},[61,68160,68161],{"class":1731,"line":1732},[61,68162,68163],{"class":1735},"# @title Cross-sections\n",[61,68165,68166],{"class":1731,"line":1739},[61,68167,68168],{"class":1735},"# Slicing controls\n",[61,68170,68171,68173,68175,68177,68179,68181],{"class":1731,"line":1746},[61,68172,61050],{"class":1866},[61,68174,2127],{"class":1753},[61,68176,1860],{"class":1859},[61,68178,1863],{"class":1859},[61,68180,61545],{"class":1866},[61,68182,68183],{"class":1735},"  # drive frequency slice [GHz]\n",[61,68185,68186,68188,68190,68192,68194,68196],{"class":1731,"line":1763},[61,68187,61149],{"class":1866},[61,68189,2127],{"class":1753},[61,68191,1860],{"class":1859},[61,68193,1863],{"class":1859},[61,68195,6036],{"class":1866},[61,68197,68198],{"class":1735},"   # duration slice [ns]\n",[61,68200,68201],{"class":1731,"line":1776},[61,68202,1743],{"emptyLinePlaceholder":1742},[61,68204,68205,68207,68209,68211],{"class":1731,"line":1784},[61,68206,1757],{"class":1749},[61,68208,1789],{"class":1753},[61,68210,1792],{"class":1749},[61,68212,1795],{"class":1753},[61,68214,68215,68217,68219,68221],{"class":1731,"line":1798},[61,68216,1757],{"class":1749},[61,68218,1803],{"class":1753},[61,68220,1792],{"class":1749},[61,68222,1808],{"class":1753},[61,68224,68225],{"class":1731,"line":1811},[61,68226,1743],{"emptyLinePlaceholder":1742},[61,68228,68229,68231,68234,68236,68239,68241,68244,68246,68248,68250,68253,68255,68257,68259,68261],{"class":1731,"line":1816},[61,68230,2116],{"class":1749},[61,68232,68233],{"class":1836}," _nearest_index",[61,68235,849],{"class":1753},[61,68237,68238],{"class":2124},"vec",[61,68240,6874],{"class":1753},[61,68242,68243],{"class":2124},"value",[61,68245,2127],{"class":1753},[61,68247,1860],{"class":1859},[61,68249,1435],{"class":1753},[61,68251,68252],{"class":2124},"name",[61,68254,2127],{"class":1753},[61,68256,36784],{"class":1859},[61,68258,2155],{"class":1753},[61,68260,2025],{"class":1859},[61,68262,1850],{"class":1753},[61,68264,68265],{"class":1731,"line":1822},[61,68266,68267],{"class":2165},"    \"\"\"Return nearest index to 'value' in 'vec'; clamp and notify if out of range.\"\"\"\n",[61,68269,68270,68273,68275,68277,68280,68282,68285,68287,68289,68291],{"class":1731,"line":1828},[61,68271,68272],{"class":1753},"    vmin, vmax ",[61,68274,2000],{"class":1859},[61,68276,63818],{"class":1859},[61,68278,68279],{"class":1753},"(vec.",[61,68281,4202],{"class":1836},[61,68283,68284],{"class":1753},"()), ",[61,68286,1860],{"class":1859},[61,68288,68279],{"class":1753},[61,68290,4212],{"class":1836},[61,68292,3750],{"class":1753},[61,68294,68295,68297,68300,68302,68305,68307,68309,68311],{"class":1731,"line":1833},[61,68296,2172],{"class":1749},[61,68298,68299],{"class":1753}," value ",[61,68301,2181],{"class":1859},[61,68303,68304],{"class":1753}," vmin ",[61,68306,41454],{"class":1749},[61,68308,68299],{"class":1753},[61,68310,2503],{"class":1859},[61,68312,68313],{"class":1753}," vmax:\n",[61,68315,68316,68318,68320,68322,68325,68327,68329,68331,68333,68335,68337,68339,68342,68344,68346,68348,68350,68352,68354,68356,68359],{"class":1731,"line":1840},[61,68317,43298],{"class":1859},[61,68319,849],{"class":1753},[61,68321,19797],{"class":1749},[61,68323,68324],{"class":2165},"\"[Note] ",[61,68326,3688],{"class":1866},[61,68328,68252],{"class":1753},[61,68330,3697],{"class":1866},[61,68332,2000],{"class":2165},[61,68334,3688],{"class":1866},[61,68336,68243],{"class":1753},[61,68338,3697],{"class":1866},[61,68340,68341],{"class":2165}," is outside sweep; clamping to [",[61,68343,3688],{"class":1866},[61,68345,42143],{"class":1753},[61,68347,3697],{"class":1866},[61,68349,1435],{"class":2165},[61,68351,3688],{"class":1866},[61,68353,39025],{"class":1753},[61,68355,3697],{"class":1866},[61,68357,68358],{"class":2165},"].\"",[61,68360,2016],{"class":1753},[61,68362,68363,68366,68368,68370,68372],{"class":1731,"line":1853},[61,68364,68365],{"class":1753},"        value ",[61,68367,2000],{"class":1859},[61,68369,2193],{"class":1753},[61,68371,4582],{"class":1836},[61,68373,68374],{"class":1753},"(value, vmin, vmax)\n",[61,68376,68377,68379,68381,68383,68386,68388,68390,68393,68395],{"class":1731,"line":1873},[61,68378,2210],{"class":1749},[61,68380,24180],{"class":1859},[61,68382,3577],{"class":1753},[61,68384,68385],{"class":1836},"argmin",[61,68387,3577],{"class":1753},[61,68389,3574],{"class":1836},[61,68391,68392],{"class":1753},"(vec ",[61,68394,2006],{"class":1859},[61,68396,68397],{"class":1753}," value)))\n",[61,68399,68400],{"class":1731,"line":1889},[61,68401,1743],{"emptyLinePlaceholder":1742},[61,68403,68404],{"class":1731,"line":1908},[61,68405,68406],{"class":1735},"# --- Slice at f = F_CROSS_GHZ ---\n",[61,68408,68409,68412,68414,68416,68419,68421,68423,68426],{"class":1731,"line":1925},[61,68410,68411],{"class":1753},"j ",[61,68413,2000],{"class":1859},[61,68415,68233],{"class":1836},[61,68417,68418],{"class":1753},"(freq_ghz, ",[61,68420,61050],{"class":1866},[61,68422,1435],{"class":1753},[61,68424,68425],{"class":2165},"\"F_CROSS_GHZ\"",[61,68427,2016],{"class":1753},[61,68429,68430,68433,68435],{"class":1731,"line":1941},[61,68431,68432],{"class":1753},"Pe_vs_tau ",[61,68434,2000],{"class":1859},[61,68436,68437],{"class":1753}," Pe[:, j]\n",[61,68439,68440],{"class":1731,"line":1957},[61,68441,1743],{"emptyLinePlaceholder":1742},[61,68443,68444,68446,68448,68450,68452,68454,68456,68458,68460,68462,68464,68467],{"class":1731,"line":1974},[61,68445,24952],{"class":1753},[61,68447,2000],{"class":1859},[61,68449,17065],{"class":1753},[61,68451,17068],{"class":1836},[61,68453,849],{"class":1753},[61,68455,3624],{"class":3402},[61,68457,2000],{"class":1859},[61,68459,849],{"class":1753},[61,68461,1300],{"class":1866},[61,68463,1435],{"class":1753},[61,68465,68466],{"class":1866},"3.5",[61,68468,5546],{"class":1753},[61,68470,68471,68473,68475,68478,68480,68482,68484],{"class":1731,"line":1979},[61,68472,24987],{"class":1753},[61,68474,3656],{"class":1836},[61,68476,68477],{"class":1753},"(dur_ns, Pe_vs_tau, ",[61,68479,3705],{"class":3402},[61,68481,2000],{"class":1859},[61,68483,3710],{"class":2165},[61,68485,2016],{"class":1753},[61,68487,68488,68490,68492,68494,68496,68498,68500,68502,68504,68506,68508,68510],{"class":1731,"line":1985},[61,68489,24987],{"class":1753},[61,68491,17159],{"class":1836},[61,68493,849],{"class":1753},[61,68495,3763],{"class":1749},[61,68497,3794],{"class":3766},[61,68499,68028],{"class":1859},[61,68501,68031],{"class":3766},[61,68503,849],{"class":1866},[61,68505,52926],{"class":3766},[61,68507,300],{"class":1866},[61,68509,3777],{"class":3766},[61,68511,2016],{"class":1753},[61,68513,68514,68516,68518,68520,68522,68524],{"class":1731,"line":2019},[61,68515,24987],{"class":1753},[61,68517,17173],{"class":1836},[61,68519,849],{"class":1753},[61,68521,3763],{"class":1749},[61,68523,62541],{"class":3766},[61,68525,2016],{"class":1753},[61,68527,68528],{"class":1731,"line":2033},[61,68529,68530],{"class":1735},"# Avoid \\text{}; mix math for symbols and plain text for units\u002Fvalues\n",[61,68532,68533,68535,68537,68539,68541,68544,68546,68548,68551,68553,68556,68558,68560,68562,68564,68567,68569,68571,68574],{"class":1731,"line":2061},[61,68534,24987],{"class":1753},[61,68536,17149],{"class":1836},[61,68538,849],{"class":1753},[61,68540,3763],{"class":1749},[61,68542,68543],{"class":3766},"'$P_e",[61,68545,849],{"class":1866},[61,68547,68028],{"class":1859},[61,68549,68550],{"class":3766},"au",[61,68552,300],{"class":1866},[61,68554,68555],{"class":3766},"$ at f = '",[61,68557,4990],{"class":1859},[61,68559,39727],{"class":1749},[61,68561,3777],{"class":2165},[61,68563,3688],{"class":1866},[61,68565,68566],{"class":1753},"freq_ghz[j]",[61,68568,67029],{"class":1749},[61,68570,3697],{"class":1866},[61,68572,68573],{"class":2165}," GHz'",[61,68575,2016],{"class":1753},[61,68577,68578,68580,68583,68585,68587,68589,68591],{"class":1731,"line":2074},[61,68579,24987],{"class":1753},[61,68581,68582],{"class":1836},"set_ylim",[61,68584,849],{"class":1753},[61,68586,2051],{"class":1866},[61,68588,1435],{"class":1753},[61,68590,2319],{"class":1866},[61,68592,2016],{"class":1753},[61,68594,68595,68597,68599],{"class":1731,"line":2079},[61,68596,62597],{"class":1753},[61,68598,17242],{"class":1836},[61,68600,2089],{"class":1753},[61,68602,68603,68605,68607],{"class":1731,"line":2092},[61,68604,3616],{"class":1753},[61,68606,3850],{"class":1836},[61,68608,2089],{"class":1753},[61,68610,68611],{"class":1731,"line":2097},[61,68612,1743],{"emptyLinePlaceholder":1742},[61,68614,68615],{"class":1731,"line":2102},[61,68616,68617],{"class":1735},"# --- Slice at τ = T_CROSS_NS ---\n",[61,68619,68620,68623,68625,68627,68630,68632,68634,68637],{"class":1731,"line":2108},[61,68621,68622],{"class":1753},"i ",[61,68624,2000],{"class":1859},[61,68626,68233],{"class":1836},[61,68628,68629],{"class":1753},"(dur_ns, ",[61,68631,61149],{"class":1866},[61,68633,1435],{"class":1753},[61,68635,68636],{"class":2165},"\"T_CROSS_NS\"",[61,68638,2016],{"class":1753},[61,68640,68641,68644,68646],{"class":1731,"line":2113},[61,68642,68643],{"class":1753},"Pe_vs_f ",[61,68645,2000],{"class":1859},[61,68647,68648],{"class":1753}," Pe[i, :]\n",[61,68650,68651],{"class":1731,"line":2162},[61,68652,1743],{"emptyLinePlaceholder":1742},[61,68654,68655,68657,68659,68661,68663,68665,68667,68669,68671,68673,68675,68677],{"class":1731,"line":2169},[61,68656,24952],{"class":1753},[61,68658,2000],{"class":1859},[61,68660,17065],{"class":1753},[61,68662,17068],{"class":1836},[61,68664,849],{"class":1753},[61,68666,3624],{"class":3402},[61,68668,2000],{"class":1859},[61,68670,849],{"class":1753},[61,68672,1300],{"class":1866},[61,68674,1435],{"class":1753},[61,68676,68466],{"class":1866},[61,68678,5546],{"class":1753},[61,68680,68681,68683,68685,68688,68690,68692,68694],{"class":1731,"line":2187},[61,68682,24987],{"class":1753},[61,68684,3656],{"class":1836},[61,68686,68687],{"class":1753},"(freq_ghz, Pe_vs_f, ",[61,68689,3705],{"class":3402},[61,68691,2000],{"class":1859},[61,68693,3710],{"class":2165},[61,68695,2016],{"class":1753},[61,68697,68698,68700,68702,68704,68706],{"class":1731,"line":2207},[61,68699,24987],{"class":1753},[61,68701,17159],{"class":1836},[61,68703,849],{"class":1753},[61,68705,68011],{"class":2165},[61,68707,2016],{"class":1753},[61,68709,68710,68712,68714,68716,68718,68720],{"class":1731,"line":2221},[61,68711,24987],{"class":1753},[61,68713,17173],{"class":1836},[61,68715,849],{"class":1753},[61,68717,3763],{"class":1749},[61,68719,62541],{"class":3766},[61,68721,2016],{"class":1753},[61,68723,68724,68726,68728,68730,68732,68734,68736,68738,68740,68743,68745,68748,68750,68752,68754,68756,68759,68761,68763,68766],{"class":1731,"line":2226},[61,68725,24987],{"class":1753},[61,68727,17149],{"class":1836},[61,68729,849],{"class":1753},[61,68731,3763],{"class":1749},[61,68733,68543],{"class":3766},[61,68735,849],{"class":1866},[61,68737,19797],{"class":3766},[61,68739,300],{"class":1866},[61,68741,68742],{"class":3766},"$ at $",[61,68744,68028],{"class":1859},[61,68746,68747],{"class":3766},"au$ = '",[61,68749,4990],{"class":1859},[61,68751,39727],{"class":1749},[61,68753,3777],{"class":2165},[61,68755,3688],{"class":1866},[61,68757,68758],{"class":1753},"dur_ns[i]",[61,68760,39313],{"class":1749},[61,68762,3697],{"class":1866},[61,68764,68765],{"class":2165}," ns'",[61,68767,2016],{"class":1753},[61,68769,68770,68772,68774,68776,68778,68780,68782],{"class":1731,"line":2258},[61,68771,24987],{"class":1753},[61,68773,68582],{"class":1836},[61,68775,849],{"class":1753},[61,68777,2051],{"class":1866},[61,68779,1435],{"class":1753},[61,68781,2319],{"class":1866},[61,68783,2016],{"class":1753},[61,68785,68786,68788,68790],{"class":1731,"line":2264},[61,68787,62597],{"class":1753},[61,68789,17242],{"class":1836},[61,68791,2089],{"class":1753},[61,68793,68794,68796,68798],{"class":1731,"line":2280},[61,68795,3616],{"class":1753},[61,68797,3850],{"class":1836},[61,68799,2089],{"class":1753},[4797,68801],{"alt":68802,"src":68803},"Output 4 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-04.webp",[4797,68805],{"alt":68806,"src":68807},"Output 5 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-05.webp",[1720,68809,68811],{"className":1722,"code":68810,"language":1724,"meta":1725,"style":1725},"# @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",[1727,68812,68813,68818,68828,68838,68842,68847,68863,68879,68895,68911,68927,68943,68947,68952,68978,68995,69007,69028,69032,69041,69048,69055,69070,69085,69100,69115,69130,69145,69150,69155,69159,69163,69168,69173,69178,69183,69188,69193,69198,69203,69207,69211,69216,69221,69225,69242,69259,69278,69282,69287,69319,69329,69371,69383,69387,69392,69401,69415,69422,69459,69473,69477,69482,69517,69543,69547,69552,69570,69588,69594,69607,69611,69616,69634,69674,69688,69694,69716,69720,69737,69766,69770,69775,69793,69807,69813,69826,69830,69837,69841,69846,69857,69862,69872,69882,69892,69902,69912,69922,69926],{"__ignoreMap":1725},[61,68814,68815],{"class":1731,"line":1732},[61,68816,68817],{"class":1735},"# @title FFT of Rabi oscillations along tau (controls + computation)\n",[61,68819,68820,68822,68824,68826],{"class":1731,"line":1739},[61,68821,1757],{"class":1749},[61,68823,1789],{"class":1753},[61,68825,1792],{"class":1749},[61,68827,1795],{"class":1753},[61,68829,68830,68832,68834,68836],{"class":1731,"line":1746},[61,68831,1757],{"class":1749},[61,68833,1803],{"class":1753},[61,68835,1792],{"class":1749},[61,68837,1808],{"class":1753},[61,68839,68840],{"class":1731,"line":1763},[61,68841,1743],{"emptyLinePlaceholder":1742},[61,68843,68844],{"class":1731,"line":1776},[61,68845,68846],{"class":1735},"# -------- Control knobs (FFT & display) --------\n",[61,68848,68849,68851,68853,68855,68857,68860],{"class":1731,"line":1784},[61,68850,61266],{"class":1866},[61,68852,2127],{"class":1753},[61,68854,2025],{"class":1859},[61,68856,1863],{"class":1859},[61,68858,68859],{"class":1866}," 8",[61,68861,68862],{"class":1735},"                 # zero-padding factor (2, 4, 8, ...)\n",[61,68864,68865,68867,68869,68871,68873,68876],{"class":1731,"line":1798},[61,68866,61280],{"class":1866},[61,68868,2127],{"class":1753},[61,68870,36784],{"class":1859},[61,68872,1863],{"class":1859},[61,68874,68875],{"class":2165}," \"hann\"",[61,68877,68878],{"class":1735},"             # \"hann\", \"boxcar\"\n",[61,68880,68881,68883,68885,68887,68889,68892],{"class":1731,"line":1811},[61,68882,61295],{"class":1866},[61,68884,2127],{"class":1753},[61,68886,1963],{"class":1859},[61,68888,1863],{"class":1859},[61,68890,68891],{"class":1866}," True",[61,68893,68894],{"class":1735},"        # subtract mean along τ before FFT\n",[61,68896,68897,68899,68901,68903,68905,68908],{"class":1731,"line":1816},[61,68898,61309],{"class":1866},[61,68900,2127],{"class":1753},[61,68902,36784],{"class":1859},[61,68904,1863],{"class":1859},[61,68906,68907],{"class":2165}," \"magnitude\"",[61,68909,68910],{"class":1735},"      # \"magnitude\" | \"power\" | \"db\"\n",[61,68912,68913,68915,68917,68919,68921,68924],{"class":1731,"line":1822},[61,68914,61324],{"class":1866},[61,68916,2127],{"class":1753},[61,68918,36784],{"class":1859},[61,68920,1863],{"class":1859},[61,68922,68923],{"class":2165}," \"global\"",[61,68925,68926],{"class":1735},"        # \"global\" | \"per_f\" (column-wise)\n",[61,68928,68929,68931,68933,68935,68937,68940],{"class":1731,"line":1828},[61,68930,61339],{"class":1866},[61,68932,2127],{"class":1753},[61,68934,36784],{"class":1859},[61,68936,1863],{"class":1859},[61,68938,68939],{"class":2165}," \"MHz\"",[61,68941,68942],{"class":1735},"          # \"MHz\" or \"GHz\" for Fourier axis\n",[61,68944,68945],{"class":1731,"line":1833},[61,68946,1743],{"emptyLinePlaceholder":1742},[61,68948,68949],{"class":1731,"line":1840},[61,68950,68951],{"class":1735},"# -------- Helpers --------\n",[61,68953,68954,68956,68959,68961,68963,68965,68967,68969,68972,68974,68976],{"class":1731,"line":1853},[61,68955,2116],{"class":1749},[61,68957,68958],{"class":1836}," _make_window",[61,68960,849],{"class":1753},[61,68962,27948],{"class":2124},[61,68964,2127],{"class":1753},[61,68966,2025],{"class":1859},[61,68968,1435],{"class":1753},[61,68970,68971],{"class":2124},"kind",[61,68973,2127],{"class":1753},[61,68975,36784],{"class":1859},[61,68977,37291],{"class":1753},[61,68979,68980,68982,68985,68987,68989,68991,68993],{"class":1731,"line":1873},[61,68981,2172],{"class":1749},[61,68983,68984],{"class":1753}," kind.",[61,68986,37185],{"class":1836},[61,68988,7275],{"class":1753},[61,68990,37238],{"class":1859},[61,68992,68875],{"class":2165},[61,68994,1850],{"class":1753},[61,68996,68997,68999,69001,69004],{"class":1731,"line":1889},[61,68998,2190],{"class":1749},[61,69000,2193],{"class":1753},[61,69002,69003],{"class":1836},"hanning",[61,69005,69006],{"class":1753},"(n)\n",[61,69008,69009,69011,69013,69016,69019,69021,69023,69025],{"class":1731,"line":1908},[61,69010,2210],{"class":1749},[61,69012,2193],{"class":1753},[61,69014,69015],{"class":1836},"ones",[61,69017,69018],{"class":1753},"(n, ",[61,69020,6950],{"class":3402},[61,69022,67644],{"class":1859},[61,69024,14257],{"class":1753},[61,69026,69027],{"class":1735},"# boxcar\n",[61,69029,69030],{"class":1731,"line":1925},[61,69031,1743],{"emptyLinePlaceholder":1742},[61,69033,69034,69036,69039],{"class":1731,"line":1941},[61,69035,2116],{"class":1749},[61,69037,69038],{"class":1836}," rfft_spectrogram",[61,69040,38439],{"class":1753},[61,69042,69043,69046],{"class":1731,"line":1957},[61,69044,69045],{"class":2124},"    Pe",[61,69047,38733],{"class":1753},[61,69049,69050,69053],{"class":1731,"line":1974},[61,69051,69052],{"class":2124},"    dur_ns",[61,69054,38733],{"class":1753},[61,69056,69057,69060,69062,69064,69066,69068],{"class":1731,"line":1979},[61,69058,69059],{"class":2124},"    pad",[61,69061,2127],{"class":1753},[61,69063,2025],{"class":1859},[61,69065,1863],{"class":1859},[61,69067,68859],{"class":1866},[61,69069,36573],{"class":1753},[61,69071,69072,69075,69077,69079,69081,69083],{"class":1731,"line":1985},[61,69073,69074],{"class":2124},"    window",[61,69076,2127],{"class":1753},[61,69078,36784],{"class":1859},[61,69080,1863],{"class":1859},[61,69082,68875],{"class":2165},[61,69084,36573],{"class":1753},[61,69086,69087,69090,69092,69094,69096,69098],{"class":1731,"line":2019},[61,69088,69089],{"class":2124},"    detrend_mean",[61,69091,2127],{"class":1753},[61,69093,1963],{"class":1859},[61,69095,1863],{"class":1859},[61,69097,68891],{"class":1866},[61,69099,36573],{"class":1753},[61,69101,69102,69105,69107,69109,69111,69113],{"class":1731,"line":2033},[61,69103,69104],{"class":2124},"    amp_mode",[61,69106,2127],{"class":1753},[61,69108,36784],{"class":1859},[61,69110,1863],{"class":1859},[61,69112,68907],{"class":2165},[61,69114,36573],{"class":1753},[61,69116,69117,69120,69122,69124,69126,69128],{"class":1731,"line":2061},[61,69118,69119],{"class":2124},"    norm_mode",[61,69121,2127],{"class":1753},[61,69123,36784],{"class":1859},[61,69125,1863],{"class":1859},[61,69127,68923],{"class":2165},[61,69129,36573],{"class":1753},[61,69131,69132,69135,69137,69139,69141,69143],{"class":1731,"line":2074},[61,69133,69134],{"class":2124},"    freq_units",[61,69136,2127],{"class":1753},[61,69138,36784],{"class":1859},[61,69140,1863],{"class":1859},[61,69142,68939],{"class":2165},[61,69144,36573],{"class":1753},[61,69146,69147],{"class":1731,"line":2079},[61,69148,69149],{"class":1753},") -> tuple[np.ndarray, np.ndarray]:\n",[61,69151,69152],{"class":1731,"line":2092},[61,69153,69154],{"class":2165},"    \"\"\"Compute RFFT along τ for each drive frequency column of Pe(τ, f).\n",[61,69156,69157],{"class":1731,"line":2097},[61,69158,1743],{"emptyLinePlaceholder":1742},[61,69160,69161],{"class":1731,"line":2102},[61,69162,61783],{"class":2165},[61,69164,69165],{"class":1731,"line":2108},[61,69166,69167],{"class":2165},"        Pe: array of shape (N_tau, N_f) with excited-state probabilities.\n",[61,69169,69170],{"class":1731,"line":2113},[61,69171,69172],{"class":2165},"        dur_ns: τ grid [ns], length N_tau, evenly spaced.\n",[61,69174,69175],{"class":1731,"line":2162},[61,69176,69177],{"class":2165},"        pad: zero-padding factor (≥1).\n",[61,69179,69180],{"class":1731,"line":2169},[61,69181,69182],{"class":2165},"        window: \"hann\" or \"boxcar\".\n",[61,69184,69185],{"class":1731,"line":2187},[61,69186,69187],{"class":2165},"        detrend_mean: subtract column-wise mean before FFT.\n",[61,69189,69190],{"class":1731,"line":2207},[61,69191,69192],{"class":2165},"        amp_mode: \"magnitude\", \"power\", or \"db\".\n",[61,69194,69195],{"class":1731,"line":2221},[61,69196,69197],{"class":2165},"        norm_mode: \"global\" (single max) or \"per_f\" (each column scaled to its max).\n",[61,69199,69200],{"class":1731,"line":2226},[61,69201,69202],{"class":2165},"        freq_units: \"MHz\" or \"GHz\" for the Fourier axis.\n",[61,69204,69205],{"class":1731,"line":2258},[61,69206,1743],{"emptyLinePlaceholder":1742},[61,69208,69209],{"class":1731,"line":2264},[61,69210,61817],{"class":2165},[61,69212,69213],{"class":1731,"line":2280},[61,69214,69215],{"class":2165},"        f_fft: Fourier frequency axis [MHz or GHz] (length N_w),\n",[61,69217,69218],{"class":1731,"line":2304},[61,69219,69220],{"class":2165},"        S:     2D nonnegative spectrogram (N_w, N_f).\n",[61,69222,69223],{"class":1731,"line":2326},[61,69224,36776],{"class":2165},[61,69226,69227,69230,69232,69234,69236,69238,69240],{"class":1731,"line":2342},[61,69228,69229],{"class":1753},"    Pe ",[61,69231,2000],{"class":1859},[61,69233,2193],{"class":1753},[61,69235,37173],{"class":1836},[61,69237,62169],{"class":1753},[61,69239,1860],{"class":1859},[61,69241,2016],{"class":1753},[61,69243,69244,69247,69249,69251,69253,69255,69257],{"class":1731,"line":2379},[61,69245,69246],{"class":1753},"    dur_ns ",[61,69248,2000],{"class":1859},[61,69250,2193],{"class":1753},[61,69252,37173],{"class":1836},[61,69254,68629],{"class":1753},[61,69256,1860],{"class":1859},[61,69258,2016],{"class":1753},[61,69260,69261,69263,69266,69268,69270,69272,69275],{"class":1731,"line":2396},[61,69262,38843],{"class":1749},[61,69264,69265],{"class":1753}," Pe.shape[",[61,69267,1495],{"class":1866},[61,69269,1997],{"class":1753},[61,69271,37238],{"class":1859},[61,69273,69274],{"class":1753}," dur_ns.size, ",[61,69276,69277],{"class":2165},"\"Pe must be shaped (N_tau, N_f).\"\n",[61,69279,69280],{"class":1731,"line":2404},[61,69281,1743],{"emptyLinePlaceholder":1742},[61,69283,69284],{"class":1731,"line":2443},[61,69285,69286],{"class":1735},"    # Uniform sampling assumption (true for np.linspace)\n",[61,69288,69289,69292,69294,69296,69299,69301,69303,69305,69308,69310,69312,69314,69316],{"class":1731,"line":2475},[61,69290,69291],{"class":1753},"    dt ",[61,69293,2000],{"class":1859},[61,69295,63818],{"class":1859},[61,69297,69298],{"class":1753},"(dur_ns[",[61,69300,57],{"class":1866},[61,69302,1997],{"class":1753},[61,69304,2006],{"class":1859},[61,69306,69307],{"class":1753}," dur_ns[",[61,69309,1495],{"class":1866},[61,69311,37230],{"class":1753},[61,69313,2201],{"class":1859},[61,69315,25775],{"class":1866},[61,69317,69318],{"class":1735},"               # [s]\n",[61,69320,69321,69324,69326],{"class":1731,"line":2520},[61,69322,69323],{"class":1753},"    n_tau ",[61,69325,2000],{"class":1859},[61,69327,69328],{"class":1753}," dur_ns.size\n",[61,69330,69331,69334,69336,69338,69340,69342,69345,69347,69349,69352,69354,69357,69360,69362,69364,69366,69368],{"class":1731,"line":2541},[61,69332,69333],{"class":1753},"    nfft ",[61,69335,2000],{"class":1859},[61,69337,24180],{"class":1859},[61,69339,849],{"class":1753},[61,69341,337],{"class":1866},[61,69343,69344],{"class":1859}," **",[61,69346,24180],{"class":1859},[61,69348,3577],{"class":1753},[61,69350,69351],{"class":1836},"ceil",[61,69353,3577],{"class":1753},[61,69355,69356],{"class":1836},"log2",[61,69358,69359],{"class":1753},"(n_tau))) ",[61,69361,2201],{"class":1859},[61,69363,2350],{"class":1859},[61,69365,849],{"class":1753},[61,69367,57],{"class":1866},[61,69369,69370],{"class":1753},", pad))\n",[61,69372,69373,69376,69378,69380],{"class":1731,"line":2546},[61,69374,69375],{"class":1753},"    w ",[61,69377,2000],{"class":1859},[61,69379,68958],{"class":1836},[61,69381,69382],{"class":1753},"(n_tau, window)\n",[61,69384,69385],{"class":1731,"line":2582},[61,69386,1743],{"emptyLinePlaceholder":1742},[61,69388,69389],{"class":1731,"line":2588},[61,69390,69391],{"class":1735},"    # Prepare output\n",[61,69393,69394,69397,69399],{"class":1731,"line":2601},[61,69395,69396],{"class":1753},"    S_list ",[61,69398,2000],{"class":1859},[61,69400,39697],{"class":1753},[61,69402,69403,69405,69407,69410,69413],{"class":1731,"line":2659},[61,69404,42017],{"class":1753},[61,69406,2000],{"class":1859},[61,69408,69409],{"class":1753}," Pe.",[61,69411,69412],{"class":1836},"copy",[61,69414,2089],{"class":1753},[61,69416,69417,69419],{"class":1731,"line":2717},[61,69418,2172],{"class":1749},[61,69420,69421],{"class":1753}," detrend_mean:\n",[61,69423,69424,69427,69429,69432,69434,69437,69440,69442,69444,69446,69448,69450,69453,69455,69457],{"class":1731,"line":2725},[61,69425,69426],{"class":1753},"        X ",[61,69428,2000],{"class":1859},[61,69430,69431],{"class":1753}," X ",[61,69433,2006],{"class":1859},[61,69435,69436],{"class":1753}," X.",[61,69438,69439],{"class":1836},"mean",[61,69441,849],{"class":1753},[61,69443,37275],{"class":3402},[61,69445,2000],{"class":1859},[61,69447,1495],{"class":1866},[61,69449,1435],{"class":1753},[61,69451,69452],{"class":3402},"keepdims",[61,69454,2000],{"class":1859},[61,69456,24385],{"class":1866},[61,69458,2016],{"class":1753},[61,69460,69461,69463,69466,69469,69471],{"class":1731,"line":2730},[61,69462,42017],{"class":1753},[61,69464,69465],{"class":1859},"*=",[61,69467,69468],{"class":1753}," w[:, ",[61,69470,61025],{"class":1866},[61,69472,5701],{"class":1753},[61,69474,69475],{"class":1731,"line":2749},[61,69476,1743],{"emptyLinePlaceholder":1742},[61,69478,69479],{"class":1731,"line":2755},[61,69480,69481],{"class":1735},"    # Real FFT along τ (axis=0)\n",[61,69483,69484,69487,69489,69492,69495,69498,69500,69502,69505,69507,69509,69511,69514],{"class":1731,"line":2778},[61,69485,69486],{"class":1753},"    Xf ",[61,69488,2000],{"class":1859},[61,69490,69491],{"class":1753}," np.fft.",[61,69493,69494],{"class":1836},"rfft",[61,69496,69497],{"class":1753},"(X, ",[61,69499,27948],{"class":3402},[61,69501,2000],{"class":1859},[61,69503,69504],{"class":1753},"nfft, ",[61,69506,37275],{"class":3402},[61,69508,2000],{"class":1859},[61,69510,1495],{"class":1866},[61,69512,69513],{"class":1753},")                     ",[61,69515,69516],{"class":1735},"# shape (N_w, N_f)\n",[61,69518,69519,69522,69524,69526,69529,69532,69535,69537,69540],{"class":1731,"line":2783},[61,69520,69521],{"class":1753},"    f_fft_hz ",[61,69523,2000],{"class":1859},[61,69525,69491],{"class":1753},[61,69527,69528],{"class":1836},"rfftfreq",[61,69530,69531],{"class":1753},"(nfft, ",[61,69533,69534],{"class":3402},"d",[61,69536,2000],{"class":1859},[61,69538,69539],{"class":1753},"dt)                  ",[61,69541,69542],{"class":1735},"# [Hz]\n",[61,69544,69545],{"class":1731,"line":2791},[61,69546,1743],{"emptyLinePlaceholder":1742},[61,69548,69549],{"class":1731,"line":2804},[61,69550,69551],{"class":1735},"    # Amplitude selection\n",[61,69553,69554,69556,69559,69561,69563,69565,69568],{"class":1731,"line":2817},[61,69555,2172],{"class":1749},[61,69557,69558],{"class":1753}," amp_mode.",[61,69560,37185],{"class":1836},[61,69562,7275],{"class":1753},[61,69564,37238],{"class":1859},[61,69566,69567],{"class":2165}," \"power\"",[61,69569,1850],{"class":1753},[61,69571,69572,69575,69577,69579,69581,69584,69586],{"class":1731,"line":2824},[61,69573,69574],{"class":1753},"        S ",[61,69576,2000],{"class":1859},[61,69578,2193],{"class":1753},[61,69580,3574],{"class":1836},[61,69582,69583],{"class":1753},"(Xf) ",[61,69585,2651],{"class":1859},[61,69587,62041],{"class":1866},[61,69589,69590,69592],{"class":1731,"line":2835},[61,69591,64503],{"class":1749},[61,69593,1850],{"class":1753},[61,69595,69596,69598,69600,69602,69604],{"class":1731,"line":2846},[61,69597,69574],{"class":1753},[61,69599,2000],{"class":1859},[61,69601,2193],{"class":1753},[61,69603,3574],{"class":1836},[61,69605,69606],{"class":1753},"(Xf)\n",[61,69608,69609],{"class":1731,"line":2851},[61,69610,1743],{"emptyLinePlaceholder":1742},[61,69612,69613],{"class":1731,"line":2856},[61,69614,69615],{"class":1735},"    # Normalization\n",[61,69617,69618,69620,69623,69625,69627,69629,69632],{"class":1731,"line":2862},[61,69619,2172],{"class":1749},[61,69621,69622],{"class":1753}," norm_mode.",[61,69624,37185],{"class":1836},[61,69626,7275],{"class":1753},[61,69628,37238],{"class":1859},[61,69630,69631],{"class":2165}," \"per_f\"",[61,69633,1850],{"class":1753},[61,69635,69636,69639,69641,69643,69646,69649,69651,69653,69655,69657,69659,69661,69663,69665,69667,69669,69672],{"class":1731,"line":2867},[61,69637,69638],{"class":1753},"        col_max ",[61,69640,2000],{"class":1859},[61,69642,2193],{"class":1753},[61,69644,69645],{"class":1836},"maximum",[61,69647,69648],{"class":1753},"(S.",[61,69650,4212],{"class":1836},[61,69652,849],{"class":1753},[61,69654,37275],{"class":3402},[61,69656,2000],{"class":1859},[61,69658,1495],{"class":1866},[61,69660,1435],{"class":1753},[61,69662,69452],{"class":3402},[61,69664,2000],{"class":1859},[61,69666,24385],{"class":1866},[61,69668,3638],{"class":1753},[61,69670,69671],{"class":1866},"1e-15",[61,69673,2016],{"class":1753},[61,69675,69676,69678,69680,69683,69685],{"class":1731,"line":2917},[61,69677,69574],{"class":1753},[61,69679,2000],{"class":1859},[61,69681,69682],{"class":1753}," S ",[61,69684,602],{"class":1859},[61,69686,69687],{"class":1753}," col_max\n",[61,69689,69690,69692],{"class":1731,"line":2926},[61,69691,64503],{"class":1749},[61,69693,1850],{"class":1753},[61,69695,69696,69698,69700,69702,69704,69706,69708,69710,69712,69714],{"class":1731,"line":2972},[61,69697,69574],{"class":1753},[61,69699,2000],{"class":1859},[61,69701,69682],{"class":1753},[61,69703,602],{"class":1859},[61,69705,2350],{"class":1859},[61,69707,69648],{"class":1753},[61,69709,4212],{"class":1836},[61,69711,4205],{"class":1753},[61,69713,69671],{"class":1866},[61,69715,2016],{"class":1753},[61,69717,69718],{"class":1731,"line":3001},[61,69719,1743],{"emptyLinePlaceholder":1742},[61,69721,69722,69724,69726,69728,69730,69732,69735],{"class":1731,"line":3054},[61,69723,2172],{"class":1749},[61,69725,69558],{"class":1753},[61,69727,37185],{"class":1836},[61,69729,7275],{"class":1753},[61,69731,37238],{"class":1859},[61,69733,69734],{"class":2165}," \"db\"",[61,69736,1850],{"class":1753},[61,69738,69739,69741,69743,69745,69747,69749,69752,69754,69756,69759,69761,69763],{"class":1731,"line":3060},[61,69740,69574],{"class":1753},[61,69742,2000],{"class":1859},[61,69744,6006],{"class":1866},[61,69746,2298],{"class":1859},[61,69748,2193],{"class":1753},[61,69750,69751],{"class":1836},"log10",[61,69753,3577],{"class":1753},[61,69755,69645],{"class":1836},[61,69757,69758],{"class":1753},"(S, ",[61,69760,5543],{"class":1866},[61,69762,42976],{"class":1753},[61,69764,69765],{"class":1735},"# dB, clipped floor\n",[61,69767,69768],{"class":1731,"line":3077},[61,69769,1743],{"emptyLinePlaceholder":1742},[61,69771,69772],{"class":1731,"line":3082},[61,69773,69774],{"class":1735},"    # Units\n",[61,69776,69777,69779,69782,69784,69786,69788,69791],{"class":1731,"line":3129},[61,69778,2172],{"class":1749},[61,69780,69781],{"class":1753}," freq_units.",[61,69783,39146],{"class":1836},[61,69785,7275],{"class":1753},[61,69787,37238],{"class":1859},[61,69789,69790],{"class":2165}," \"GHz\"",[61,69792,1850],{"class":1753},[61,69794,69795,69798,69800,69803,69805],{"class":1731,"line":3137},[61,69796,69797],{"class":1753},"        f_fft ",[61,69799,2000],{"class":1859},[61,69801,69802],{"class":1753}," f_fft_hz ",[61,69804,2201],{"class":1859},[61,69806,6224],{"class":1866},[61,69808,69809,69811],{"class":1731,"line":3185},[61,69810,64503],{"class":1749},[61,69812,1850],{"class":1753},[61,69814,69815,69817,69819,69821,69823],{"class":1731,"line":3208},[61,69816,69797],{"class":1753},[61,69818,2000],{"class":1859},[61,69820,69802],{"class":1753},[61,69822,2201],{"class":1859},[61,69824,69825],{"class":1866}," 1e-6\n",[61,69827,69828],{"class":1731,"line":3248},[61,69829,1743],{"emptyLinePlaceholder":1742},[61,69831,69832,69834],{"class":1731,"line":3286},[61,69833,2210],{"class":1749},[61,69835,69836],{"class":1753}," f_fft, S\n",[61,69838,69839],{"class":1731,"line":3291},[61,69840,1743],{"emptyLinePlaceholder":1742},[61,69842,69843],{"class":1731,"line":3306},[61,69844,69845],{"class":1735},"# Compute spectrogram with current grid\n",[61,69847,69848,69851,69853,69855],{"class":1731,"line":3311},[61,69849,69850],{"class":1753},"f_fft_axis, S_fft ",[61,69852,2000],{"class":1859},[61,69854,69038],{"class":1836},[61,69856,38439],{"class":1753},[61,69858,69859],{"class":1731,"line":3317},[61,69860,69861],{"class":1753},"    Pe, dur_ns,\n",[61,69863,69864,69866,69868,69870],{"class":1731,"line":3358},[61,69865,69059],{"class":3402},[61,69867,2000],{"class":1859},[61,69869,61266],{"class":1866},[61,69871,36573],{"class":1753},[61,69873,69874,69876,69878,69880],{"class":1731,"line":3374},[61,69875,69074],{"class":3402},[61,69877,2000],{"class":1859},[61,69879,61280],{"class":1866},[61,69881,36573],{"class":1753},[61,69883,69884,69886,69888,69890],{"class":1731,"line":3390},[61,69885,69089],{"class":3402},[61,69887,2000],{"class":1859},[61,69889,61295],{"class":1866},[61,69891,36573],{"class":1753},[61,69893,69894,69896,69898,69900],{"class":1731,"line":3422},[61,69895,69104],{"class":3402},[61,69897,2000],{"class":1859},[61,69899,61309],{"class":1866},[61,69901,36573],{"class":1753},[61,69903,69904,69906,69908,69910],{"class":1731,"line":3427},[61,69905,69119],{"class":3402},[61,69907,2000],{"class":1859},[61,69909,61324],{"class":1866},[61,69911,36573],{"class":1753},[61,69913,69914,69916,69918,69920],{"class":1731,"line":3432},[61,69915,69134],{"class":3402},[61,69917,2000],{"class":1859},[61,69919,61339],{"class":1866},[61,69921,36573],{"class":1753},[61,69923,69924],{"class":1731,"line":3438},[61,69925,2016],{"class":1753},[61,69927,69928,69930,69932,69935,69938,69941,69944,69946,69948,69951,69953,69955,69957,69959,69961],{"class":1731,"line":3443},[61,69929,40848],{"class":1859},[61,69931,849],{"class":1753},[61,69933,69934],{"class":2165},"\"FFT grid:\"",[61,69936,69937],{"class":1753},", S_fft.shape, ",[61,69939,69940],{"class":2165},"\"| Fourier freq range:\"",[61,69942,69943],{"class":1753},", f_fft_axis[",[61,69945,1495],{"class":1866},[61,69947,7056],{"class":1753},[61,69949,69950],{"class":2165},"\"→\"",[61,69952,69943],{"class":1753},[61,69954,2006],{"class":1859},[61,69956,57],{"class":1866},[61,69958,7056],{"class":1753},[61,69960,61339],{"class":1866},[61,69962,2016],{"class":1753},[1720,69964,69967],{"className":69965,"code":69966,"language":29075,"meta":1725},[36200],"FFT grid: (2049, 401) | Fourier freq range: 0.0 → 999.9999999999999 MHz\n",[1727,69968,69966],{"__ignoreMap":1725},[1720,69970,69972],{"className":1722,"code":69971,"language":1724,"meta":1725,"style":1725},"# @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",[1727,69973,69974,69979,69985,69998,70012,70039,70059,70091,70114,70133,70147,70153],{"__ignoreMap":1725},[61,69975,69976],{"class":1731,"line":1732},[61,69977,69978],{"class":1735},"# @title Theoretical Rabi ridge\n",[61,69980,69981,69983],{"class":1731,"line":1739},[61,69982,1757],{"class":1749},[61,69984,1781],{"class":1753},[61,69986,69987,69990,69992,69994,69996],{"class":1731,"line":1746},[61,69988,69989],{"class":1753},"two_pi ",[61,69991,2000],{"class":1859},[61,69993,3029],{"class":1866},[61,69995,2298],{"class":1859},[61,69997,61840],{"class":1753},[61,69999,70000,70003,70005,70008,70010],{"class":1731,"line":1763},[61,70001,70002],{"class":1753},"f0 ",[61,70004,2000],{"class":1859},[61,70006,70007],{"class":1866}," F0_GHZ",[61,70009,2298],{"class":1859},[61,70011,61855],{"class":1866},[61,70013,70014,70017,70019,70021,70023,70026,70028,70031,70033,70036],{"class":1731,"line":1776},[61,70015,70016],{"class":1753},"delta_hz ",[61,70018,2000],{"class":1859},[61,70020,61893],{"class":1753},[61,70022,2201],{"class":1859},[61,70024,70025],{"class":1753}," (freq_ghz",[61,70027,2201],{"class":1859},[61,70029,70030],{"class":1866},"1e9",[61,70032,1899],{"class":1859},[61,70034,70035],{"class":1753}," f0)   ",[61,70037,70038],{"class":1735},"# rad\u002Fs detuning\n",[61,70040,70041,70044,70046,70048,70050,70053,70055,70057],{"class":1731,"line":1784},[61,70042,70043],{"class":1753},"omega_onres ",[61,70045,2000],{"class":1859},[61,70047,61893],{"class":1753},[61,70049,2201],{"class":1859},[61,70051,70052],{"class":1866}," OMEGA_ONRESONANCE_MHZ",[61,70054,2298],{"class":1859},[61,70056,6264],{"class":1866},[61,70058,61905],{"class":1735},[61,70060,70061,70064,70066,70068,70070,70073,70075,70077,70079,70082,70084,70086,70089],{"class":1731,"line":1798},[61,70062,70063],{"class":1753},"Omega_R ",[61,70065,2000],{"class":1859},[61,70067,2193],{"class":1753},[61,70069,2425],{"class":1836},[61,70071,70072],{"class":1753},"(omega_onres",[61,70074,2651],{"class":1859},[61,70076,337],{"class":1866},[61,70078,4990],{"class":1859},[61,70080,70081],{"class":1753}," delta_hz",[61,70083,2651],{"class":1859},[61,70085,337],{"class":1866},[61,70087,70088],{"class":1753},")     ",[61,70090,61957],{"class":1735},[61,70092,70093,70096,70098,70100,70102,70104,70106,70108,70111],{"class":1731,"line":1811},[61,70094,70095],{"class":1753},"ridge_hz ",[61,70097,2000],{"class":1859},[61,70099,62055],{"class":1753},[61,70101,602],{"class":1859},[61,70103,2003],{"class":1753},[61,70105,2435],{"class":1866},[61,70107,2298],{"class":1859},[61,70109,70110],{"class":1753}," math.pi)                ",[61,70112,70113],{"class":1735},"# Hz\n",[61,70115,70116,70118,70121,70123,70125,70127,70129,70131],{"class":1731,"line":1816},[61,70117,2498],{"class":1749},[61,70119,70120],{"class":1866}," FREQ_UNITS",[61,70122,332],{"class":1753},[61,70124,39146],{"class":1836},[61,70126,7275],{"class":1753},[61,70128,37238],{"class":1859},[61,70130,69790],{"class":2165},[61,70132,1850],{"class":1753},[61,70134,70135,70138,70140,70143,70145],{"class":1731,"line":1822},[61,70136,70137],{"class":1753},"    ridge_axis ",[61,70139,2000],{"class":1859},[61,70141,70142],{"class":1753}," ridge_hz ",[61,70144,2201],{"class":1859},[61,70146,6224],{"class":1866},[61,70148,70149,70151],{"class":1731,"line":1828},[61,70150,2536],{"class":1749},[61,70152,1850],{"class":1753},[61,70154,70155,70157,70159,70161,70163,70166],{"class":1731,"line":1833},[61,70156,70137],{"class":1753},[61,70158,2000],{"class":1859},[61,70160,70142],{"class":1753},[61,70162,2201],{"class":1859},[61,70164,70165],{"class":1866}," 1e-6",[61,70167,70168],{"class":1735},"  # MHz\n",[1720,70170,70172],{"className":1722,"code":70171,"language":1724,"meta":1725,"style":1725},"# @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",[1727,70173,70174,70179,70206,70237,70256,70298,70302,70345,70358,70379,70417,70434,70442],{"__ignoreMap":1725},[61,70175,70176],{"class":1731,"line":1732},[61,70177,70178],{"class":1735},"# @title 2D heatmap\n",[61,70180,70181,70183,70185,70187,70189,70191,70193,70195,70197,70199,70201,70204],{"class":1731,"line":1739},[61,70182,24952],{"class":1753},[61,70184,2000],{"class":1859},[61,70186,17065],{"class":1753},[61,70188,17068],{"class":1836},[61,70190,849],{"class":1753},[61,70192,3624],{"class":3402},[61,70194,2000],{"class":1859},[61,70196,849],{"class":1753},[61,70198,5631],{"class":1866},[61,70200,1435],{"class":1753},[61,70202,70203],{"class":1866},"5.5",[61,70205,5546],{"class":1753},[61,70207,70208,70211,70213,70215,70217,70220,70222,70224,70226,70228,70230,70232,70235],{"class":1731,"line":1746},[61,70209,70210],{"class":1753},"im ",[61,70212,2000],{"class":1859},[61,70214,38434],{"class":1753},[61,70216,62499],{"class":1836},[61,70218,70219],{"class":1753},"(freq_ghz, f_fft_axis, S_fft, ",[61,70221,62505],{"class":3402},[61,70223,2000],{"class":1859},[61,70225,4260],{"class":2165},[61,70227,1435],{"class":1753},[61,70229,4273],{"class":3402},[61,70231,2000],{"class":1859},[61,70233,70234],{"class":2165},"\"inferno\"",[61,70236,2016],{"class":1753},[61,70238,70239,70241,70243,70245,70247,70249,70251,70253],{"class":1731,"line":1763},[61,70240,62517],{"class":1753},[61,70242,2000],{"class":1859},[61,70244,39048],{"class":1753},[61,70246,4288],{"class":1836},[61,70248,39053],{"class":1753},[61,70250,3368],{"class":3402},[61,70252,2000],{"class":1859},[61,70254,70255],{"class":1753},"ax)\n",[61,70257,70258,70260,70262,70264,70267,70269,70271,70274,70277,70280,70282,70284,70286,70288,70291,70293,70296],{"class":1731,"line":1776},[61,70259,68119],{"class":1753},[61,70261,39088],{"class":1836},[61,70263,849],{"class":1753},[61,70265,70266],{"class":2165},"\"FFT amplitude\"",[61,70268,4990],{"class":1859},[61,70270,2003],{"class":1753},[61,70272,70273],{"class":2165},"\" (dB)\"",[61,70275,70276],{"class":1749}," if",[61,70278,70279],{"class":1866}," AMP_MODE",[61,70281,332],{"class":1753},[61,70283,37185],{"class":1836},[61,70285,39149],{"class":1753},[61,70287,37238],{"class":1859},[61,70289,70290],{"class":2165},"\"db\"",[61,70292,4535],{"class":1749},[61,70294,70295],{"class":2165}," \" (arb.)\"",[61,70297,5546],{"class":1753},[61,70299,70300],{"class":1731,"line":1784},[61,70301,1743],{"emptyLinePlaceholder":1742},[61,70303,70304,70306,70308,70311,70313,70315,70318,70320,70322,70324,70326,70328,70330,70332,70334,70337,70340,70343],{"class":1731,"line":1798},[61,70305,24987],{"class":1753},[61,70307,3656],{"class":1836},[61,70309,70310],{"class":1753},"(freq_ghz, ridge_axis, ",[61,70312,3408],{"class":2165},[61,70314,1435],{"class":1753},[61,70316,70317],{"class":3402},"lw",[61,70319,2000],{"class":1859},[61,70321,3350],{"class":1866},[61,70323,1435],{"class":1753},[61,70325,3666],{"class":3402},[61,70327,2000],{"class":1859},[61,70329,3763],{"class":1749},[61,70331,65834],{"class":3766},[61,70333,4328],{"class":1859},[61,70335,70336],{"class":3766},"mega_R\u002F2",[61,70338,70339],{"class":1859},"\\p",[61,70341,70342],{"class":3766},"i$\"",[61,70344,2016],{"class":1753},[61,70346,70347,70349,70351,70353,70356],{"class":1731,"line":1811},[61,70348,24987],{"class":1753},[61,70350,17159],{"class":1836},[61,70352,849],{"class":1753},[61,70354,70355],{"class":2165},"\"Drive frequency f (GHz)\"",[61,70357,2016],{"class":1753},[61,70359,70360,70362,70364,70366,70368,70371,70374,70377],{"class":1731,"line":1816},[61,70361,24987],{"class":1753},[61,70363,17173],{"class":1836},[61,70365,849],{"class":1753},[61,70367,19797],{"class":1749},[61,70369,70370],{"class":2165},"\"Fourier frequency (",[61,70372,70373],{"class":1866},"{FREQ_UNITS}",[61,70375,70376],{"class":2165},")\"",[61,70378,2016],{"class":1753},[61,70380,70381,70383,70385,70387,70390,70392,70394,70397,70400,70402,70405,70407,70410,70412,70415],{"class":1731,"line":1822},[61,70382,24987],{"class":1753},[61,70384,17149],{"class":1836},[61,70386,849],{"class":1753},[61,70388,70389],{"class":2165},"\"Fourier transform along pulse length: $|",[61,70391,62583],{"class":1859},[61,70393,57177],{"class":2165},[61,70395,70396],{"class":1866},"{F}",[61,70398,70399],{"class":2165},"_",[61,70401,62583],{"class":1859},[61,70403,70404],{"class":2165},"tau",[61,70406,62583],{"class":1859},[61,70408,70409],{"class":2165},"{P_e",[61,70411,62583],{"class":1859},[61,70413,70414],{"class":2165},"}|$\"",[61,70416,2016],{"class":1753},[61,70418,70419,70421,70423,70425,70427,70429,70432],{"class":1731,"line":1828},[61,70420,24987],{"class":1753},[61,70422,3840],{"class":1836},[61,70424,849],{"class":1753},[61,70426,38068],{"class":3402},[61,70428,2000],{"class":1859},[61,70430,70431],{"class":2165},"\"upper right\"",[61,70433,2016],{"class":1753},[61,70435,70436,70438,70440],{"class":1731,"line":1833},[61,70437,62597],{"class":1753},[61,70439,17242],{"class":1836},[61,70441,2089],{"class":1753},[61,70443,70444,70446,70448],{"class":1731,"line":1840},[61,70445,3616],{"class":1753},[61,70447,3850],{"class":1836},[61,70449,2089],{"class":1753},[4797,70451],{"alt":70452,"src":70453},"Output 6 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-06.webp",[1720,70455,70457],{"className":1722,"code":70456,"language":1724,"meta":1725,"style":1725},"# --- 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",[1727,70458,70459,70464,70474,70484,70494,70498,70503,70520,70537,70554,70571,70575,70580,70597,70613,70617,70622,70635,70639,70644,70682,70703,70707,70727,70740,70754,70758,70763,70768,70802,70831,70835,70840,70874,70884,70894,70898,70902,70928,70932,70937,70971,70995,71007,71017,71035,71053,71063,71079,71083,71087,71100,71119,71155,71168,71206,71210,71215,71239,71243,71251],{"__ignoreMap":1725},[61,70460,70461],{"class":1731,"line":1732},[61,70462,70463],{"class":1735},"# --- Adaptive high-res 3D surface ---\n",[61,70465,70466,70468,70470,70472],{"class":1731,"line":1739},[61,70467,1750],{"class":1749},[61,70469,67464],{"class":1753},[61,70471,1757],{"class":1749},[61,70473,67469],{"class":1753},[61,70475,70476,70478,70480,70482],{"class":1731,"line":1746},[61,70477,1757],{"class":1749},[61,70479,1789],{"class":1753},[61,70481,1792],{"class":1749},[61,70483,1795],{"class":1753},[61,70485,70486,70488,70490,70492],{"class":1731,"line":1763},[61,70487,1757],{"class":1749},[61,70489,1803],{"class":1753},[61,70491,1792],{"class":1749},[61,70493,1808],{"class":1753},[61,70495,70496],{"class":1731,"line":1776},[61,70497,1743],{"emptyLinePlaceholder":1742},[61,70499,70500],{"class":1731,"line":1784},[61,70501,70502],{"class":1735},"# ------ Control knobs ------\n",[61,70504,70505,70508,70510,70512,70514,70517],{"class":1731,"line":1798},[61,70506,70507],{"class":1866},"UPSAMPLE_F_FREQ",[61,70509,2127],{"class":1753},[61,70511,2025],{"class":1859},[61,70513,1863],{"class":1859},[61,70515,70516],{"class":1866}," 3",[61,70518,70519],{"class":1735},"       # try 2–3; upsample the drive-frequency axis only\n",[61,70521,70522,70525,70527,70529,70531,70534],{"class":1731,"line":1811},[61,70523,70524],{"class":1866},"MAX_FACES",[61,70526,2127],{"class":1753},[61,70528,2025],{"class":1859},[61,70530,1863],{"class":1859},[61,70532,70533],{"class":1866}," 300_000",[61,70535,70536],{"class":1735},"       # cap on triangles to avoid OOM \u002F crashes (≈300k–500k is OK)\n",[61,70538,70539,70542,70544,70546,70548,70551],{"class":1731,"line":1816},[61,70540,70541],{"class":1866},"MIN_ROWS",[61,70543,2127],{"class":1753},[61,70545,2025],{"class":1859},[61,70547,1863],{"class":1859},[61,70549,70550],{"class":1866}," 128",[61,70552,70553],{"class":1735},"            # keep at least this many Fourier-frequency rows\n",[61,70555,70556,70559,70561,70563,70565,70568],{"class":1731,"line":1822},[61,70557,70558],{"class":1866},"MAX_ROWS",[61,70560,2127],{"class":1753},[61,70562,2025],{"class":1859},[61,70564,1863],{"class":1859},[61,70566,70567],{"class":1866}," 1024",[61,70569,70570],{"class":1735},"           # hard cap for rows to keep plotting fast\n",[61,70572,70573],{"class":1731,"line":1828},[61,70574,1743],{"emptyLinePlaceholder":1742},[61,70576,70577],{"class":1731,"line":1833},[61,70578,70579],{"class":1735},"# Preserve global color limits so the legend stays consistent\n",[61,70581,70582,70585,70587,70589,70591,70594],{"class":1731,"line":1840},[61,70583,70584],{"class":1753},"vmin ",[61,70586,2000],{"class":1859},[61,70588,63818],{"class":1859},[61,70590,3577],{"class":1753},[61,70592,70593],{"class":1836},"nanmin",[61,70595,70596],{"class":1753},"(S_fft))\n",[61,70598,70599,70602,70604,70606,70608,70611],{"class":1731,"line":1853},[61,70600,70601],{"class":1753},"vmax ",[61,70603,2000],{"class":1859},[61,70605,63818],{"class":1859},[61,70607,3577],{"class":1753},[61,70609,70610],{"class":1836},"nanmax",[61,70612,70596],{"class":1753},[61,70614,70615],{"class":1731,"line":1873},[61,70616,1743],{"emptyLinePlaceholder":1742},[61,70618,70619],{"class":1731,"line":1889},[61,70620,70621],{"class":1735},"# Shapes\n",[61,70623,70624,70627,70629,70632],{"class":1731,"line":1908},[61,70625,70626],{"class":1753},"n_rows, n_cols ",[61,70628,2000],{"class":1859},[61,70630,70631],{"class":1753}," S_fft.shape        ",[61,70633,70634],{"class":1735},"# rows=Fourier freq (ν), cols=drive freq (f)\n",[61,70636,70637],{"class":1731,"line":1925},[61,70638,1743],{"emptyLinePlaceholder":1742},[61,70640,70641],{"class":1731,"line":1941},[61,70642,70643],{"class":1735},"# 1) Upsample along the typically smaller axis (drive frequency) only\n",[61,70645,70646,70649,70651,70653,70655,70657,70659,70662,70664,70666,70668,70670,70672,70674,70677,70679],{"class":1731,"line":1957},[61,70647,70648],{"class":1753},"n_cols_up ",[61,70650,2000],{"class":1859},[61,70652,2525],{"class":1859},[61,70654,849],{"class":1753},[61,70656,70507],{"class":1866},[61,70658,2298],{"class":1859},[61,70660,70661],{"class":1753}," (n_cols ",[61,70663,2006],{"class":1859},[61,70665,5266],{"class":1866},[61,70667,2624],{"class":1753},[61,70669,2419],{"class":1859},[61,70671,5266],{"class":1866},[61,70673,1435],{"class":1753},[61,70675,70676],{"class":1866},"1200",[61,70678,14257],{"class":1753},[61,70680,70681],{"class":1735},"# sensible upper bound\n",[61,70683,70684,70686,70688,70690,70692,70694,70696,70698,70700],{"class":1731,"line":1974},[61,70685,67571],{"class":1753},[61,70687,2000],{"class":1859},[61,70689,2193],{"class":1753},[61,70691,3453],{"class":1836},[61,70693,67580],{"class":1753},[61,70695,4202],{"class":1836},[61,70697,67585],{"class":1753},[61,70699,4212],{"class":1836},[61,70701,70702],{"class":1753},"(), n_cols_up)\n",[61,70704,70705],{"class":1731,"line":1979},[61,70706,1743],{"emptyLinePlaceholder":1742},[61,70708,70709,70712,70714,70716,70718,70721,70723,70725],{"class":1731,"line":1985},[61,70710,70711],{"class":1753},"S_up_f ",[61,70713,2000],{"class":1859},[61,70715,2193],{"class":1753},[61,70717,4085],{"class":1836},[61,70719,70720],{"class":1753},"((n_rows, n_cols_up), ",[61,70722,6950],{"class":3402},[61,70724,67644],{"class":1859},[61,70726,2016],{"class":1753},[61,70728,70729,70731,70733,70735,70737],{"class":1731,"line":2019},[61,70730,2486],{"class":1749},[61,70732,67653],{"class":1753},[61,70734,2492],{"class":1749},[61,70736,17206],{"class":1859},[61,70738,70739],{"class":1753},"(n_rows):\n",[61,70741,70742,70745,70747,70749,70751],{"class":1731,"line":2033},[61,70743,70744],{"class":1753},"    S_up_f[i, :] ",[61,70746,2000],{"class":1859},[61,70748,2193],{"class":1753},[61,70750,67676],{"class":1836},[61,70752,70753],{"class":1753},"(freq_hi, freq_ghz, S_fft[i, :])\n",[61,70755,70756],{"class":1731,"line":2061},[61,70757,1743],{"emptyLinePlaceholder":1742},[61,70759,70760],{"class":1731,"line":2074},[61,70761,70762],{"class":1735},"# 2) Determine how many Fourier rows we can afford for MAX_FACES\n",[61,70764,70765],{"class":1731,"line":2079},[61,70766,70767],{"class":1735},"# faces ≈ (n_rows_plot-1) * (n_cols_up-1)\n",[61,70769,70770,70773,70775,70777,70779,70781,70783,70785,70788,70790,70792,70794,70796,70798,70800],{"class":1731,"line":2092},[61,70771,70772],{"class":1753},"n_rows_cap ",[61,70774,2000],{"class":1859},[61,70776,24180],{"class":1859},[61,70778,849],{"class":1753},[61,70780,70524],{"class":1866},[61,70782,2618],{"class":1859},[61,70784,2350],{"class":1859},[61,70786,70787],{"class":1753},"(n_cols_up ",[61,70789,2006],{"class":1859},[61,70791,5266],{"class":1866},[61,70793,1435],{"class":1753},[61,70795,57],{"class":1866},[61,70797,2702],{"class":1753},[61,70799,2419],{"class":1859},[61,70801,14228],{"class":1866},[61,70803,70804,70807,70809,70811,70813,70815,70818,70820,70822,70824,70826,70828],{"class":1731,"line":2097},[61,70805,70806],{"class":1753},"n_rows_plot ",[61,70808,2000],{"class":1859},[61,70810,24180],{"class":1859},[61,70812,3577],{"class":1753},[61,70814,4582],{"class":1836},[61,70816,70817],{"class":1753},"(n_rows_cap, ",[61,70819,70541],{"class":1866},[61,70821,1435],{"class":1753},[61,70823,4202],{"class":1859},[61,70825,849],{"class":1753},[61,70827,70558],{"class":1866},[61,70829,70830],{"class":1753},", n_rows)))\n",[61,70832,70833],{"class":1731,"line":2102},[61,70834,1743],{"emptyLinePlaceholder":1742},[61,70836,70837],{"class":1731,"line":2108},[61,70838,70839],{"class":1735},"# Down-sample ν-axis evenly to n_rows_plot\n",[61,70841,70842,70845,70847,70849,70851,70853,70855,70858,70860,70862,70865,70868,70870,70872],{"class":1731,"line":2113},[61,70843,70844],{"class":1753},"row_idx ",[61,70846,2000],{"class":1859},[61,70848,2193],{"class":1753},[61,70850,3453],{"class":1836},[61,70852,849],{"class":1753},[61,70854,1495],{"class":1866},[61,70856,70857],{"class":1753},", n_rows ",[61,70859,2006],{"class":1859},[61,70861,5266],{"class":1866},[61,70863,70864],{"class":1753},", n_rows_plot).",[61,70866,70867],{"class":1836},"astype",[61,70869,849],{"class":1753},[61,70871,2025],{"class":1859},[61,70873,2016],{"class":1753},[61,70875,70876,70879,70881],{"class":1731,"line":2162},[61,70877,70878],{"class":1753},"f_fft_plot ",[61,70880,2000],{"class":1859},[61,70882,70883],{"class":1753}," f_fft_axis[row_idx]\n",[61,70885,70886,70889,70891],{"class":1731,"line":2169},[61,70887,70888],{"class":1753},"S_final ",[61,70890,2000],{"class":1859},[61,70892,70893],{"class":1753}," S_up_f[row_idx, :]\n",[61,70895,70896],{"class":1731,"line":2187},[61,70897,1743],{"emptyLinePlaceholder":1742},[61,70899,70900],{"class":1731,"line":2207},[61,70901,67805],{"class":1735},[61,70903,70904,70907,70910,70912,70914,70916,70919,70921,70923,70926],{"class":1731,"line":2221},[61,70905,70906],{"class":1753},"F2_hi, ",[61,70908,70909],{"class":1866},"FFOUR_hi",[61,70911,1863],{"class":1859},[61,70913,2193],{"class":1753},[61,70915,37855],{"class":1836},[61,70917,70918],{"class":1753},"(freq_hi, f_fft_plot, ",[61,70920,37861],{"class":3402},[61,70922,2000],{"class":1859},[61,70924,70925],{"class":2165},"\"xy\"",[61,70927,2016],{"class":1753},[61,70929,70930],{"class":1731,"line":2226},[61,70931,1743],{"emptyLinePlaceholder":1742},[61,70933,70934],{"class":1731,"line":2258},[61,70935,70936],{"class":1735},"# 3) Plot (keeps 'inferno')\n",[61,70938,70939,70941,70943,70945,70947,70949,70951,70953,70955,70957,70959,70961,70963,70965,70967,70969],{"class":1731,"line":2264},[61,70940,67841],{"class":1753},[61,70942,2000],{"class":1859},[61,70944,17065],{"class":1753},[61,70946,3619],{"class":1836},[61,70948,849],{"class":1753},[61,70950,3624],{"class":3402},[61,70952,2000],{"class":1859},[61,70954,849],{"class":1753},[61,70956,3631],{"class":1866},[61,70958,1435],{"class":1753},[61,70960,1300],{"class":1866},[61,70962,3638],{"class":1753},[61,70964,3641],{"class":3402},[61,70966,2000],{"class":1859},[61,70968,36518],{"class":1866},[61,70970,2016],{"class":1753},[61,70972,70973,70975,70977,70979,70981,70983,70985,70987,70989,70991,70993],{"class":1731,"line":2280},[61,70974,67876],{"class":1753},[61,70976,2000],{"class":1859},[61,70978,39048],{"class":1753},[61,70980,42252],{"class":1836},[61,70982,849],{"class":1753},[61,70984,42257],{"class":1866},[61,70986,1435],{"class":1753},[61,70988,42262],{"class":3402},[61,70990,2000],{"class":1859},[61,70992,42267],{"class":2165},[61,70994,2016],{"class":1753},[61,70996,70997,70999,71001,71003,71005],{"class":1731,"line":2304},[61,70998,67906],{"class":1753},[61,71000,2000],{"class":1859},[61,71002,38434],{"class":1753},[61,71004,42414],{"class":1836},[61,71006,38439],{"class":1753},[61,71008,71009,71012,71014],{"class":1731,"line":2326},[61,71010,71011],{"class":1753},"    F2_hi, ",[61,71013,70909],{"class":1866},[61,71015,71016],{"class":1753},", S_final,\n",[61,71018,71019,71021,71023,71025,71027,71029,71031,71033],{"class":1731,"line":2342},[61,71020,67924],{"class":3402},[61,71022,2000],{"class":1859},[61,71024,57],{"class":1866},[61,71026,1435],{"class":1753},[61,71028,42447],{"class":3402},[61,71030,2000],{"class":1859},[61,71032,57],{"class":1866},[61,71034,36573],{"class":1753},[61,71036,71037,71039,71041,71043,71045,71047,71049,71051],{"class":1731,"line":2379},[61,71038,67947],{"class":3402},[61,71040,2000],{"class":1859},[61,71042,1495],{"class":1866},[61,71044,1435],{"class":1753},[61,71046,67956],{"class":3402},[61,71048,2000],{"class":1859},[61,71050,24385],{"class":1866},[61,71052,36573],{"class":1753},[61,71054,71055,71057,71059,71061],{"class":1731,"line":2396},[61,71056,67967],{"class":3402},[61,71058,2000],{"class":1859},[61,71060,70234],{"class":2165},[61,71062,36573],{"class":1753},[61,71064,71065,71067,71069,71072,71074,71076],{"class":1731,"line":2404},[61,71066,67978],{"class":3402},[61,71068,2000],{"class":1859},[61,71070,71071],{"class":1753},"vmin, ",[61,71073,39025],{"class":3402},[61,71075,2000],{"class":1859},[61,71077,71078],{"class":1753},"vmax\n",[61,71080,71081],{"class":1731,"line":2443},[61,71082,2016],{"class":1753},[61,71084,71085],{"class":1731,"line":2475},[61,71086,1743],{"emptyLinePlaceholder":1742},[61,71088,71089,71091,71093,71095,71098],{"class":1731,"line":2520},[61,71090,24987],{"class":1753},[61,71092,17159],{"class":1836},[61,71094,849],{"class":1753},[61,71096,71097],{"class":2165},"\"f (GHz)\"",[61,71099,2016],{"class":1753},[61,71101,71102,71104,71106,71108,71110,71113,71115,71117],{"class":1731,"line":2541},[61,71103,24987],{"class":1753},[61,71105,17173],{"class":1836},[61,71107,849],{"class":1753},[61,71109,19797],{"class":1749},[61,71111,71112],{"class":2165},"\"Fourier freq (",[61,71114,70373],{"class":1866},[61,71116,70376],{"class":2165},[61,71118,2016],{"class":1753},[61,71120,71121,71123,71125,71127,71129,71131,71133,71135,71137,71139,71141,71143,71145,71147,71149,71151,71153],{"class":1731,"line":2546},[61,71122,24987],{"class":1753},[61,71124,42587],{"class":1836},[61,71126,849],{"class":1753},[61,71128,70266],{"class":2165},[61,71130,4990],{"class":1859},[61,71132,2003],{"class":1753},[61,71134,70273],{"class":2165},[61,71136,70276],{"class":1749},[61,71138,70279],{"class":1866},[61,71140,332],{"class":1753},[61,71142,37185],{"class":1836},[61,71144,7275],{"class":1753},[61,71146,37238],{"class":1859},[61,71148,69734],{"class":2165},[61,71150,4535],{"class":1749},[61,71152,70295],{"class":2165},[61,71154,5546],{"class":1753},[61,71156,71157,71159,71161,71163,71166],{"class":1731,"line":2582},[61,71158,24987],{"class":1753},[61,71160,17149],{"class":1836},[61,71162,849],{"class":1753},[61,71164,71165],{"class":2165},"\"Rabi FFT surface (adaptive resolution)\"",[61,71167,2016],{"class":1753},[61,71169,71170,71172,71174,71176,71178,71180,71182,71184,71186,71188,71190,71192,71194,71196,71198,71200,71202,71204],{"class":1731,"line":2588},[61,71171,62597],{"class":1753},[61,71173,4288],{"class":1836},[61,71175,68090],{"class":1753},[61,71177,3368],{"class":3402},[61,71179,2000],{"class":1859},[61,71181,39060],{"class":1753},[61,71183,68099],{"class":3402},[61,71185,2000],{"class":1859},[61,71187,25004],{"class":1866},[61,71189,1435],{"class":1753},[61,71191,39073],{"class":3402},[61,71193,2000],{"class":1859},[61,71195,68112],{"class":1866},[61,71197,1435],{"class":1753},[61,71199,3666],{"class":3402},[61,71201,2000],{"class":1859},[61,71203,17178],{"class":2165},[61,71205,2016],{"class":1753},[61,71207,71208],{"class":1731,"line":2601},[61,71209,1743],{"emptyLinePlaceholder":1742},[61,71211,71212],{"class":1731,"line":2659},[61,71213,71214],{"class":1735},"# A clearer default view\n",[61,71216,71217,71219,71221,71223,71225,71227,71229,71231,71233,71235,71237],{"class":1731,"line":2717},[61,71218,24987],{"class":1753},[61,71220,42276],{"class":1836},[61,71222,849],{"class":1753},[61,71224,42281],{"class":3402},[61,71226,2000],{"class":1859},[61,71228,52835],{"class":1866},[61,71230,1435],{"class":1753},[61,71232,42289],{"class":3402},[61,71234,43421],{"class":1859},[61,71236,39385],{"class":1866},[61,71238,2016],{"class":1753},[61,71240,71241],{"class":1731,"line":2725},[61,71242,1743],{"emptyLinePlaceholder":1742},[61,71244,71245,71247,71249],{"class":1731,"line":2730},[61,71246,62597],{"class":1753},[61,71248,17242],{"class":1836},[61,71250,2089],{"class":1753},[61,71252,71253,71255,71257],{"class":1731,"line":2749},[61,71254,3616],{"class":1753},[61,71256,3850],{"class":1836},[61,71258,2089],{"class":1753},[4797,71260],{"alt":71261,"src":71262},"Output 7 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-07.webp",[1720,71264,71266],{"className":1722,"code":71265,"language":1724,"meta":1725,"style":1725},"# @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",[1727,71267,71268,71273,71296,71328,71332,71337,71361,71389,71406,71450,71468,71504,71531,71539,71552,71556,71561,71587,71613,71630,71642,71678,71707],{"__ignoreMap":1725},[61,71269,71270],{"class":1731,"line":1732},[61,71271,71272],{"class":1735},"# @title Slices: amplitude vs Fourier freq (at f*) and vs f (at ν*)\n",[61,71274,71275,71278,71280,71282,71284,71286,71288,71290,71293],{"class":1731,"line":1739},[61,71276,71277],{"class":1866},"F_STAR_GHZ",[61,71279,2127],{"class":1753},[61,71281,1860],{"class":1859},[61,71283,1863],{"class":1859},[61,71285,63818],{"class":1859},[61,71287,849],{"class":1753},[61,71289,60790],{"class":1866},[61,71291,71292],{"class":1753},")   ",[61,71294,71295],{"class":1735},"# slice near resonance\n",[61,71297,71298,71301,71303,71305,71308,71310,71312,71314,71317,71319,71322,71325],{"class":1731,"line":1746},[61,71299,71300],{"class":1866},"NU_STAR",[61,71302,1863],{"class":1859},[61,71304,63818],{"class":1859},[61,71306,71307],{"class":1753},"(ridge_axis[np.",[61,71309,68385],{"class":1836},[61,71311,3577],{"class":1753},[61,71313,3574],{"class":1836},[61,71315,71316],{"class":1753},"(freq_ghz ",[61,71318,2006],{"class":1859},[61,71320,71321],{"class":1866}," F_STAR_GHZ",[61,71323,71324],{"class":1753},"))])  ",[61,71326,71327],{"class":1735},"# near Ω_R\u002F2π\n",[61,71329,71330],{"class":1731,"line":1763},[61,71331,1743],{"emptyLinePlaceholder":1742},[61,71333,71334],{"class":1731,"line":1776},[61,71335,71336],{"class":1735},"# Slice at f = F_STAR_GHZ\n",[61,71338,71339,71341,71343,71345,71347,71349,71351,71353,71355,71357,71359],{"class":1731,"line":1784},[61,71340,68411],{"class":1753},[61,71342,2000],{"class":1859},[61,71344,24180],{"class":1859},[61,71346,3577],{"class":1753},[61,71348,68385],{"class":1836},[61,71350,3577],{"class":1753},[61,71352,3574],{"class":1836},[61,71354,71316],{"class":1753},[61,71356,2006],{"class":1859},[61,71358,71321],{"class":1866},[61,71360,43512],{"class":1753},[61,71362,71363,71365,71367,71369,71371,71373,71375,71377,71379,71382,71384,71387],{"class":1731,"line":1798},[61,71364,24952],{"class":1753},[61,71366,2000],{"class":1859},[61,71368,17065],{"class":1753},[61,71370,17068],{"class":1836},[61,71372,849],{"class":1753},[61,71374,3624],{"class":3402},[61,71376,2000],{"class":1859},[61,71378,849],{"class":1753},[61,71380,71381],{"class":1866},"6.0",[61,71383,1435],{"class":1753},[61,71385,71386],{"class":1866},"3.8",[61,71388,5546],{"class":1753},[61,71390,71391,71393,71395,71398,71400,71402,71404],{"class":1731,"line":1811},[61,71392,24987],{"class":1753},[61,71394,3656],{"class":1836},[61,71396,71397],{"class":1753},"(f_fft_axis, S_fft[:, j], ",[61,71399,3705],{"class":3402},[61,71401,2000],{"class":1859},[61,71403,3710],{"class":2165},[61,71405,2016],{"class":1753},[61,71407,71408,71410,71412,71415,71418,71420,71422,71424,71426,71428,71430,71432,71434,71436,71438,71440,71442,71444,71446,71448],{"class":1731,"line":1816},[61,71409,24987],{"class":1753},[61,71411,3396],{"class":1836},[61,71413,71414],{"class":1753},"(ridge_axis[j], ",[61,71416,71417],{"class":3402},"ls",[61,71419,2000],{"class":1859},[61,71421,3408],{"class":2165},[61,71423,1435],{"class":1753},[61,71425,70317],{"class":3402},[61,71427,2000],{"class":1859},[61,71429,2319],{"class":1866},[61,71431,1435],{"class":1753},[61,71433,3666],{"class":3402},[61,71435,2000],{"class":1859},[61,71437,3763],{"class":1749},[61,71439,65834],{"class":3766},[61,71441,4328],{"class":1859},[61,71443,70336],{"class":3766},[61,71445,70339],{"class":1859},[61,71447,70342],{"class":3766},[61,71449,2016],{"class":1753},[61,71451,71452,71454,71456,71458,71460,71462,71464,71466],{"class":1731,"line":1822},[61,71453,24987],{"class":1753},[61,71455,17159],{"class":1836},[61,71457,849],{"class":1753},[61,71459,19797],{"class":1749},[61,71461,70370],{"class":2165},[61,71463,70373],{"class":1866},[61,71465,70376],{"class":2165},[61,71467,2016],{"class":1753},[61,71469,71470,71472,71474,71476,71478,71480,71482,71484,71486,71488,71490,71492,71494,71496,71498,71500,71502],{"class":1731,"line":1828},[61,71471,24987],{"class":1753},[61,71473,17173],{"class":1836},[61,71475,849],{"class":1753},[61,71477,70266],{"class":2165},[61,71479,4990],{"class":1859},[61,71481,2003],{"class":1753},[61,71483,70273],{"class":2165},[61,71485,70276],{"class":1749},[61,71487,70279],{"class":1866},[61,71489,332],{"class":1753},[61,71491,37185],{"class":1836},[61,71493,39149],{"class":1753},[61,71495,37238],{"class":1859},[61,71497,70290],{"class":2165},[61,71499,4535],{"class":1749},[61,71501,70295],{"class":2165},[61,71503,5546],{"class":1753},[61,71505,71506,71508,71510,71512,71515,71518,71520,71522,71524,71526,71529],{"class":1731,"line":1833},[61,71507,24987],{"class":1753},[61,71509,17149],{"class":1836},[61,71511,849],{"class":1753},[61,71513,71514],{"class":1749},"fr",[61,71516,71517],{"class":2165},"\"Slice at $f=",[61,71519,3688],{"class":1866},[61,71521,68566],{"class":1753},[61,71523,67029],{"class":1749},[61,71525,3697],{"class":1866},[61,71527,71528],{"class":2165},"$ GHz\"",[61,71530,2016],{"class":1753},[61,71532,71533,71535,71537],{"class":1731,"line":1840},[61,71534,24987],{"class":1753},[61,71536,3840],{"class":1836},[61,71538,2089],{"class":1753},[61,71540,71541,71543,71545,71548,71550],{"class":1731,"line":1853},[61,71542,62597],{"class":1753},[61,71544,17242],{"class":1836},[61,71546,71547],{"class":1753},"(); plt.",[61,71549,3850],{"class":1836},[61,71551,2089],{"class":1753},[61,71553,71554],{"class":1731,"line":1873},[61,71555,1743],{"emptyLinePlaceholder":1742},[61,71557,71558],{"class":1731,"line":1889},[61,71559,71560],{"class":1735},"# Slice at ν = NU_STAR\n",[61,71562,71563,71565,71567,71569,71571,71573,71575,71577,71580,71582,71585],{"class":1731,"line":1908},[61,71564,68622],{"class":1753},[61,71566,2000],{"class":1859},[61,71568,24180],{"class":1859},[61,71570,3577],{"class":1753},[61,71572,68385],{"class":1836},[61,71574,3577],{"class":1753},[61,71576,3574],{"class":1836},[61,71578,71579],{"class":1753},"(f_fft_axis ",[61,71581,2006],{"class":1859},[61,71583,71584],{"class":1866}," NU_STAR",[61,71586,43512],{"class":1753},[61,71588,71589,71591,71593,71595,71597,71599,71601,71603,71605,71607,71609,71611],{"class":1731,"line":1925},[61,71590,24952],{"class":1753},[61,71592,2000],{"class":1859},[61,71594,17065],{"class":1753},[61,71596,17068],{"class":1836},[61,71598,849],{"class":1753},[61,71600,3624],{"class":3402},[61,71602,2000],{"class":1859},[61,71604,849],{"class":1753},[61,71606,71381],{"class":1866},[61,71608,1435],{"class":1753},[61,71610,71386],{"class":1866},[61,71612,5546],{"class":1753},[61,71614,71615,71617,71619,71622,71624,71626,71628],{"class":1731,"line":1941},[61,71616,24987],{"class":1753},[61,71618,3656],{"class":1836},[61,71620,71621],{"class":1753},"(freq_ghz, S_fft[i, :], ",[61,71623,3705],{"class":3402},[61,71625,2000],{"class":1859},[61,71627,3710],{"class":2165},[61,71629,2016],{"class":1753},[61,71631,71632,71634,71636,71638,71640],{"class":1731,"line":1957},[61,71633,24987],{"class":1753},[61,71635,17159],{"class":1836},[61,71637,849],{"class":1753},[61,71639,70355],{"class":2165},[61,71641,2016],{"class":1753},[61,71643,71644,71646,71648,71650,71652,71654,71656,71658,71660,71662,71664,71666,71668,71670,71672,71674,71676],{"class":1731,"line":1974},[61,71645,24987],{"class":1753},[61,71647,17173],{"class":1836},[61,71649,849],{"class":1753},[61,71651,70266],{"class":2165},[61,71653,4990],{"class":1859},[61,71655,2003],{"class":1753},[61,71657,70273],{"class":2165},[61,71659,70276],{"class":1749},[61,71661,70279],{"class":1866},[61,71663,332],{"class":1753},[61,71665,37185],{"class":1836},[61,71667,39149],{"class":1753},[61,71669,37238],{"class":1859},[61,71671,70290],{"class":2165},[61,71673,4535],{"class":1749},[61,71675,70295],{"class":2165},[61,71677,5546],{"class":1753},[61,71679,71680,71682,71684,71686,71688,71691,71693,71696,71698,71700,71703,71705],{"class":1731,"line":1979},[61,71681,24987],{"class":1753},[61,71683,17149],{"class":1836},[61,71685,849],{"class":1753},[61,71687,71514],{"class":1749},[61,71689,71690],{"class":2165},"\"Slice at Fourier freq ≈ ",[61,71692,3688],{"class":1866},[61,71694,71695],{"class":1753},"f_fft_axis[i]",[61,71697,39313],{"class":1749},[61,71699,3697],{"class":1866},[61,71701,71702],{"class":1866}," {FREQ_UNITS}",[61,71704,26088],{"class":2165},[61,71706,2016],{"class":1753},[61,71708,71709,71711,71713,71715,71717],{"class":1731,"line":1985},[61,71710,62597],{"class":1753},[61,71712,17242],{"class":1836},[61,71714,71547],{"class":1753},[61,71716,3850],{"class":1836},[61,71718,2089],{"class":1753},[4797,71720],{"alt":71721,"src":71722},"Output 8 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-08.webp",[4797,71724],{"alt":71725,"src":71726},"Output 9 of the notebook code above","\u002F_content\u002Fimages\u002Frabi-oscillation-visualization\u002Foutput-09.webp",[25,71728,71730],{"id":71729},"run-it-yourself","Run it yourself",[12,71732,71733,71734,71736,71737,332],{},"Get ",[19,71735,22],{"href":45934}," or 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.sVyAn{--shiki-default:#E06C75}",{"title":1725,"searchDepth":1739,"depth":1739,"links":71745},[71746,71747,71748,71749,71750,71751,71752,71753,71754,71755,71756,71757,71758,71759,71760],{"id":46112,"depth":1739,"text":46113},{"id":47584,"depth":1739,"text":47585},{"id":48373,"depth":1739,"text":48374},{"id":49713,"depth":1739,"text":49714},{"id":50842,"depth":1739,"text":50843},{"id":53323,"depth":1739,"text":53324},{"id":53840,"depth":1739,"text":53841},{"id":54939,"depth":1739,"text":54940},{"id":56544,"depth":1739,"text":56545},{"id":56917,"depth":1739,"text":56918},{"id":57911,"depth":1739,"text":57912},{"id":59089,"depth":1739,"text":59090},{"id":60268,"depth":1739,"text":60269},{"id":60763,"depth":1739,"text":60764},{"id":71729,"depth":1739,"text":71730},[7523,7524,7525,71762],"Rabi oscillations",[71764],{"username":7,"name":7529,"role":7530,"bio":7531,"links":71765},[71766,71767],{"label":7534,"href":7535},{"label":7537,"href":45934},{"username":7,"name":7529,"role":7539},"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":45924,"description":71769},"blog\u002Fexpert-notes\u002Frabi-oscillation-visualization",[7555,7556],"IAzB0ZaRHqWwL_egvokIhbtpcSM2VZ_XYnWDf-q-QyM",{"id":71781,"title":71782,"authors":71783,"body":71784,"breadcrumb":76352,"builders":76354,"byline":76359,"challenge":7540,"courseAuthor":7540,"courseLead":7540,"dek":7540,"description":76360,"draft":7542,"extension":7543,"eyebrow":7544,"featured":7542,"finish":7540,"fork":7540,"hero":7540,"heroAlt":7540,"heroCta":7540,"heroImage":7540,"homepageFeatured":7542,"kind":7545,"lessonCount":7540,"meta":76361,"navigation":1742,"newsItems":7540,"next":7540,"ogImage":76362,"order":7540,"outcomes":7540,"path":76363,"publishDate":71773,"readingTime":76364,"related":76365,"relatedProjects":7540,"seo":76366,"stem":76367,"tags":76368,"track":7540,"trackName":7540,"__hash__":76369},"blog\u002Fblog\u002Fexpert-notes\u002Fsampling-budget-qubit-classifications.md","A Sampling-Budget & Bit-String Analogy for Qubit Classifications",[7],{"type":9,"value":71785,"toc":76348},[71786,71793,76189,76195,76198,76201,76207,76213,76215,76217,76336,76338,76345],[12,71787,71788],{},[15,71789,17,71790,23],{},[19,71791,22],{"href":71792},"https:\u002F\u002Fgithub.com\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[1720,71794,71796],{"className":1722,"code":71795,"language":1724,"meta":1725,"style":1725},"# 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",[1727,71797,71798,71803,71808,71813,71818,71823,71828,71833,71838,71843,71848,71852,71859,71865,71875,71886,71890,71900,71910,71920,71931,71943,71947,71959,71963,71968,71972,72018,72022,72027,72032,72036,72049,72063,72078,72101,72137,72202,72235,72239,72243,72290,72294,72299,72304,72308,72332,72352,72365,72371,72400,72422,72426,72430,72435,72439,72455,72464,72472,72482,72489,72493,72497,72509,72516,72523,72530,72535,72540,72546,72553,72560,72565,72569,72575,72582,72589,72594,72598,72604,72611,72618,72623,72627,72633,72640,72647,72652,72656,72660,72664,72669,72673,72685,72697,72709,72736,72744,72756,72766,72771,72778,72788,72792,72796,72800,72824,72828,72832,72837,72841,72873,72877,72882,72886,72919,72923,72927,72958,72962,72967,72971,72982,72998,73015,73048,73060,73064,73073,73081,73089,73097,73105,73113,73121,73129,73143,73156,73169,73173,73184,73222,73229,73233,73237,73276,73280,73285,73289,73306,73310,73314,73319,73323,73338,73347,73351,73362,73376,73398,73408,73433,73437,73451,73455,73460,73482,73502,73525,73539,73557,73574,73578,73582,73587,73591,73596,73613,73650,73681,73685,73695,73713,73717,73722,73747,73771,73775,73786,73790,73795,73799,73804,73819,73841,73858,73879,73883,73888,73893,73898,73925,73951,73975,73980,73984,74012,74065,74101,74114,74136,74144,74154,74182,74190,74194,74199,74221,74230,74250,74274,74278,74300,74325,74380,74384,74423,74436,74457,74471,74488,74496,74506,74531,74539,74543,74554,74558,74563,74567,74583,74595,74614,74634,74657,74679,74698,74715,74720,74724,74728,74733,74737,74749,74754,74758,74789,74793,74874,74878,74882,74893,74898,74903,74908,74913,74917,74925,74932,74943,74950,74957,74961,74972,74977,74982,74987,74992,74996,75010,75015,75020,75025,75030,75035,75039,75043,75055,75059,75063,75074,75079,75084,75089,75094,75098,75103,75108,75112,75132,75144,75154,75164,75184,75244,75275,75279,75294,75310,75360,75364,75379,75390,75401,75413,75418,75433,75448,75453,75457,75462,75480,75485,75489,75494,75502,75514,75519,75543,75547,75562,75589,75593,75598,75609,75636,75640,75645,75658,75666,75677,75685,75689,75707,75729,75741,75745,75750,75755,75771,75779,75793,75801,75827,75831,75848,75869,75880,75892,75903,75915,75919,75926,75933,75937,75941,75956,75970,75981,75991,76000,76004,76008,76018,76045,76056,76060,76065,76088,76099,76114,76136,76158],{"__ignoreMap":1725},[61,71799,71800],{"class":1731,"line":1732},[61,71801,71802],{"class":1735},"# This script builds a small simulation toolkit, runs it with sensible defaults,\n",[61,71804,71805],{"class":1731,"line":1739},[61,71806,71807],{"class":1735},"# generates tables and figures, so you can iterate further.\n",[61,71809,71810],{"class":1731,"line":1746},[61,71811,71812],{"class":1735},"#\n",[61,71814,71815],{"class":1731,"line":1763},[61,71816,71817],{"class":1735},"# Contents created in the current directory:\n",[61,71819,71820],{"class":1731,"line":1776},[61,71821,71822],{"class":1735},"#  - Sampling_Budget_Qubit_Bitstrings.ipynb      (Colab notebook)\n",[61,71824,71825],{"class":1731,"line":1784},[61,71826,71827],{"class":1735},"#  - summary.csv                                 (summary table)\n",[61,71829,71830],{"class":1731,"line":1798},[61,71831,71832],{"class":1735},"#  - bitstrings_\u003CMODEL>.csv                      (raw bit-strings per model)\n",[61,71834,71835],{"class":1731,"line":1811},[61,71836,71837],{"class":1735},"#  - fig_ones_fraction.png                       (bar chart with Wilson CI)\n",[61,71839,71840],{"class":1731,"line":1816},[61,71841,71842],{"class":1735},"#  - fig_required_shots.png                      (required shots for ±ε)\n",[61,71844,71845],{"class":1731,"line":1822},[61,71846,71847],{"class":1735},"#  - simulation_config.json                      (run parameters)\n",[61,71849,71850],{"class":1731,"line":1828},[61,71851,1743],{"emptyLinePlaceholder":1742},[61,71853,71854,71856],{"class":1731,"line":1833},[61,71855,1757],{"class":1749},[61,71857,71858],{"class":1753}," json\n",[61,71860,71861,71863],{"class":1731,"line":1840},[61,71862,1757],{"class":1749},[61,71864,1781],{"class":1753},[61,71866,71867,71869,71871,71873],{"class":1731,"line":1853},[61,71868,1750],{"class":1749},[61,71870,1754],{"class":1753},[61,71872,1757],{"class":1749},[61,71874,1760],{"class":1753},[61,71876,71877,71879,71881,71883],{"class":1731,"line":1873},[61,71878,1750],{"class":1749},[61,71880,1768],{"class":1753},[61,71882,1757],{"class":1749},[61,71884,71885],{"class":1753}," List, Dict, Tuple\n",[61,71887,71888],{"class":1731,"line":1889},[61,71889,1743],{"emptyLinePlaceholder":1742},[61,71891,71892,71894,71896,71898],{"class":1731,"line":1908},[61,71893,1757],{"class":1749},[61,71895,1789],{"class":1753},[61,71897,1792],{"class":1749},[61,71899,1795],{"class":1753},[61,71901,71902,71904,71906,71908],{"class":1731,"line":1925},[61,71903,1757],{"class":1749},[61,71905,61434],{"class":1753},[61,71907,1792],{"class":1749},[61,71909,61439],{"class":1753},[61,71911,71912,71914,71916,71918],{"class":1731,"line":1941},[61,71913,1757],{"class":1749},[61,71915,1803],{"class":1753},[61,71917,1792],{"class":1749},[61,71919,1808],{"class":1753},[61,71921,71922,71924,71926,71928],{"class":1731,"line":1957},[61,71923,1750],{"class":1749},[61,71925,36671],{"class":1753},[61,71927,1757],{"class":1749},[61,71929,71930],{"class":1753}," display, Markdown\n",[61,71932,71933,71935,71938,71940],{"class":1731,"line":1974},[61,71934,1757],{"class":1749},[61,71936,71937],{"class":1753}," nbformat ",[61,71939,1792],{"class":1749},[61,71941,71942],{"class":1753}," nbf\n",[61,71944,71945],{"class":1731,"line":1979},[61,71946,1743],{"emptyLinePlaceholder":1742},[61,71948,71949,71951,71953,71955,71957],{"class":1731,"line":1985},[61,71950,14189],{"class":1753},[61,71952,14192],{"class":2165},[61,71954,1997],{"class":1753},[61,71956,2000],{"class":1859},[61,71958,14199],{"class":1866},[61,71960,71961],{"class":1731,"line":2019},[61,71962,1743],{"emptyLinePlaceholder":1742},[61,71964,71965],{"class":1731,"line":2033},[61,71966,71967],{"class":1735},"# ---------- Helper math ----------\n",[61,71969,71970],{"class":1731,"line":2061},[61,71971,1743],{"emptyLinePlaceholder":1742},[61,71973,71974,71976,71979,71981,71983,71985,71987,71989,71991,71993,71995,71997,71999,72001,72003,72005,72008,72010,72012,72014,72016],{"class":1731,"line":2074},[61,71975,2116],{"class":1749},[61,71977,71978],{"class":1836}," wilson_ci",[61,71980,849],{"class":1753},[61,71982,39217],{"class":2124},[61,71984,2127],{"class":1753},[61,71986,2025],{"class":1859},[61,71988,1435],{"class":1753},[61,71990,27948],{"class":2124},[61,71992,2127],{"class":1753},[61,71994,2025],{"class":1859},[61,71996,1435],{"class":1753},[61,71998,26482],{"class":2124},[61,72000,2127],{"class":1753},[61,72002,1860],{"class":1859},[61,72004,1863],{"class":1859},[61,72006,72007],{"class":1866}," 1.96",[61,72009,2570],{"class":1753},[61,72011,1860],{"class":1859},[61,72013,1435],{"class":1753},[61,72015,1860],{"class":1859},[61,72017,2579],{"class":1753},[61,72019,72020],{"class":1731,"line":2079},[61,72021,36776],{"class":2165},[61,72023,72024],{"class":1731,"line":2092},[61,72025,72026],{"class":2165},"    Wilson score interval for binomial proportion with confidence ~95% at z=1.96.\n",[61,72028,72029],{"class":1731,"line":2097},[61,72030,72031],{"class":2165},"    Returns (low, high). Works for edge cases k=0 or k=n.\n",[61,72033,72034],{"class":1731,"line":2102},[61,72035,36776],{"class":2165},[61,72037,72038,72040,72043,72045,72047],{"class":1731,"line":2108},[61,72039,2172],{"class":1749},[61,72041,72042],{"class":1753}," n ",[61,72044,37238],{"class":1859},[61,72046,24134],{"class":1866},[61,72048,1850],{"class":1753},[61,72050,72051,72053,72055,72057,72059,72061],{"class":1731,"line":2113},[61,72052,2190],{"class":1749},[61,72054,2003],{"class":1753},[61,72056,2051],{"class":1866},[61,72058,1435],{"class":1753},[61,72060,2319],{"class":1866},[61,72062,2016],{"class":1753},[61,72064,72065,72068,72070,72073,72075],{"class":1731,"line":2162},[61,72066,72067],{"class":1753},"    p_hat ",[61,72069,2000],{"class":1859},[61,72071,72072],{"class":1753}," k ",[61,72074,602],{"class":1859},[61,72076,72077],{"class":1753}," n\n",[61,72079,72080,72082,72084,72086,72088,72091,72093,72095,72097,72099],{"class":1731,"line":2169},[61,72081,5125],{"class":1753},[61,72083,2000],{"class":1859},[61,72085,2609],{"class":1866},[61,72087,4990],{"class":1859},[61,72089,72090],{"class":1753}," (z ",[61,72092,2651],{"class":1859},[61,72094,2654],{"class":1866},[61,72096,2624],{"class":1753},[61,72098,602],{"class":1859},[61,72100,72077],{"class":1753},[61,72102,72103,72106,72108,72111,72113,72115,72117,72119,72121,72123,72125,72127,72129,72132,72134],{"class":1731,"line":2187},[61,72104,72105],{"class":1753},"    center ",[61,72107,2000],{"class":1859},[61,72109,72110],{"class":1753}," (p_hat ",[61,72112,2419],{"class":1859},[61,72114,72090],{"class":1753},[61,72116,2651],{"class":1859},[61,72118,2654],{"class":1866},[61,72120,2624],{"class":1753},[61,72122,602],{"class":1859},[61,72124,2003],{"class":1753},[61,72126,337],{"class":1866},[61,72128,2298],{"class":1859},[61,72130,72131],{"class":1753}," n)) ",[61,72133,602],{"class":1859},[61,72135,72136],{"class":1753}," denom\n",[61,72138,72139,72142,72144,72146,72148,72150,72152,72155,72157,72159,72161,72163,72166,72168,72171,72173,72175,72177,72179,72181,72183,72185,72187,72189,72191,72193,72195,72198,72200],{"class":1731,"line":2207},[61,72140,72141],{"class":1753},"    span ",[61,72143,2000],{"class":1859},[61,72145,72090],{"class":1753},[61,72147,2201],{"class":1859},[61,72149,2422],{"class":1753},[61,72151,2425],{"class":1836},[61,72153,72154],{"class":1753},"((p_hat ",[61,72156,2201],{"class":1859},[61,72158,2003],{"class":1753},[61,72160,57],{"class":1866},[61,72162,1899],{"class":1859},[61,72164,72165],{"class":1753}," p_hat) ",[61,72167,602],{"class":1859},[61,72169,72170],{"class":1753}," n) ",[61,72172,2419],{"class":1859},[61,72174,72090],{"class":1753},[61,72176,2651],{"class":1859},[61,72178,2654],{"class":1866},[61,72180,2624],{"class":1753},[61,72182,602],{"class":1859},[61,72184,2003],{"class":1753},[61,72186,822],{"class":1866},[61,72188,2298],{"class":1859},[61,72190,72042],{"class":1753},[61,72192,2651],{"class":1859},[61,72194,2654],{"class":1866},[61,72196,72197],{"class":1753},"))) ",[61,72199,602],{"class":1859},[61,72201,72136],{"class":1753},[61,72203,72204,72206,72208,72210,72212,72214,72217,72219,72222,72224,72226,72228,72230,72232],{"class":1731,"line":2221},[61,72205,2210],{"class":1749},[61,72207,2003],{"class":1753},[61,72209,4212],{"class":1859},[61,72211,849],{"class":1753},[61,72213,2051],{"class":1866},[61,72215,72216],{"class":1753},", center ",[61,72218,2006],{"class":1859},[61,72220,72221],{"class":1753}," span), ",[61,72223,4202],{"class":1859},[61,72225,849],{"class":1753},[61,72227,2319],{"class":1866},[61,72229,72216],{"class":1753},[61,72231,2419],{"class":1859},[61,72233,72234],{"class":1753}," span))\n",[61,72236,72237],{"class":1731,"line":2226},[61,72238,1743],{"emptyLinePlaceholder":1742},[61,72240,72241],{"class":1731,"line":2258},[61,72242,1743],{"emptyLinePlaceholder":1742},[61,72244,72245,72247,72250,72252,72254,72256,72258,72260,72263,72265,72267,72269,72272,72274,72276,72278,72280,72282,72284,72286,72288],{"class":1731,"line":2264},[61,72246,2116],{"class":1749},[61,72248,72249],{"class":1836}," shots_for_epsilon",[61,72251,849],{"class":1753},[61,72253,12],{"class":2124},[61,72255,2127],{"class":1753},[61,72257,1860],{"class":1859},[61,72259,1435],{"class":1753},[61,72261,72262],{"class":2124},"epsilon",[61,72264,2127],{"class":1753},[61,72266,1860],{"class":1859},[61,72268,1863],{"class":1859},[61,72270,72271],{"class":1866}," 0.01",[61,72273,1435],{"class":1753},[61,72275,26482],{"class":2124},[61,72277,2127],{"class":1753},[61,72279,1860],{"class":1859},[61,72281,1863],{"class":1859},[61,72283,72007],{"class":1866},[61,72285,2155],{"class":1753},[61,72287,2025],{"class":1859},[61,72289,1850],{"class":1753},[61,72291,72292],{"class":1731,"line":2280},[61,72293,36776],{"class":2165},[61,72295,72296],{"class":1731,"line":2304},[61,72297,72298],{"class":2165},"    Approximate shots needed to estimate a Bernoulli mean p within ±epsilon (absolute error)\n",[61,72300,72301],{"class":1731,"line":2326},[61,72302,72303],{"class":2165},"    at ~95% confidence using normal approximation. Clamps to at least 1 shot.\n",[61,72305,72306],{"class":1731,"line":2342},[61,72307,36776],{"class":2165},[61,72309,72310,72313,72315,72317,72319,72321,72324,72326,72328,72330],{"class":1731,"line":2379},[61,72311,72312],{"class":1753},"    p ",[61,72314,2000],{"class":1859},[61,72316,2525],{"class":1859},[61,72318,849],{"class":1753},[61,72320,4212],{"class":1859},[61,72322,72323],{"class":1753},"(p, ",[61,72325,2051],{"class":1866},[61,72327,3638],{"class":1753},[61,72329,2319],{"class":1866},[61,72331,2016],{"class":1753},[61,72333,72334,72337,72339,72341,72343,72345,72347,72349],{"class":1731,"line":2396},[61,72335,72336],{"class":1753},"    var ",[61,72338,2000],{"class":1859},[61,72340,39798],{"class":1753},[61,72342,2201],{"class":1859},[61,72344,2003],{"class":1753},[61,72346,2319],{"class":1866},[61,72348,1899],{"class":1859},[61,72350,72351],{"class":1753}," p)\n",[61,72353,72354,72356,72359,72361,72363],{"class":1731,"line":2404},[61,72355,2172],{"class":1749},[61,72357,72358],{"class":1753}," var ",[61,72360,37238],{"class":1859},[61,72362,2506],{"class":1866},[61,72364,1850],{"class":1753},[61,72366,72367,72369],{"class":1731,"line":2443},[61,72368,2190],{"class":1749},[61,72370,14228],{"class":1866},[61,72372,72373,72375,72377,72379,72381,72383,72385,72387,72389,72391,72394,72396,72398],{"class":1731,"line":2475},[61,72374,41830],{"class":1753},[61,72376,2000],{"class":1859},[61,72378,72090],{"class":1753},[61,72380,2651],{"class":1859},[61,72382,2654],{"class":1866},[61,72384,2624],{"class":1753},[61,72386,2201],{"class":1859},[61,72388,72358],{"class":1753},[61,72390,602],{"class":1859},[61,72392,72393],{"class":1753}," (epsilon ",[61,72395,2651],{"class":1859},[61,72397,2654],{"class":1866},[61,72399,2016],{"class":1753},[61,72401,72402,72404,72406,72408,72410,72412,72414,72417,72419],{"class":1731,"line":2520},[61,72403,2210],{"class":1749},[61,72405,2350],{"class":1859},[61,72407,849],{"class":1753},[61,72409,57],{"class":1866},[61,72411,1435],{"class":1753},[61,72413,2025],{"class":1859},[61,72415,72416],{"class":1753},"(math.",[61,72418,69351],{"class":1836},[61,72420,72421],{"class":1753},"(n)))\n",[61,72423,72424],{"class":1731,"line":2541},[61,72425,1743],{"emptyLinePlaceholder":1742},[61,72427,72428],{"class":1731,"line":2546},[61,72429,1743],{"emptyLinePlaceholder":1742},[61,72431,72432],{"class":1731,"line":2582},[61,72433,72434],{"class":1735},"# ---------- Domain model ----------\n",[61,72436,72437],{"class":1731,"line":2588},[61,72438,1743],{"emptyLinePlaceholder":1742},[61,72440,72441,72444,72446,72449,72451,72453],{"class":1731,"line":2601},[61,72442,72443],{"class":1836},"@dataclass",[61,72445,849],{"class":1753},[61,72447,72448],{"class":3402},"frozen",[61,72450,2000],{"class":1859},[61,72452,24385],{"class":1866},[61,72454,2016],{"class":1753},[61,72456,72457,72459,72462],{"class":1731,"line":2659},[61,72458,1843],{"class":1749},[61,72460,72461],{"class":1846}," QubitModel",[61,72463,1850],{"class":1753},[61,72465,72466,72469],{"class":1731,"line":2717},[61,72467,72468],{"class":1753},"    name: ",[61,72470,72471],{"class":1859},"str\n",[61,72473,72474,72477,72479],{"class":1731,"line":2725},[61,72475,72476],{"class":1753},"    satisfaction_prob: ",[61,72478,1860],{"class":1859},[61,72480,72481],{"class":1735},"  # Probability a single shot meets the analytical criterion (bit = 1)\n",[61,72483,72484,72487],{"class":1731,"line":2730},[61,72485,72486],{"class":1753},"    description: ",[61,72488,72471],{"class":1859},[61,72490,72491],{"class":1731,"line":2749},[61,72492,1743],{"emptyLinePlaceholder":1742},[61,72494,72495],{"class":1731,"line":2755},[61,72496,1743],{"emptyLinePlaceholder":1742},[61,72498,72499,72502,72505,72507],{"class":1731,"line":2778},[61,72500,72501],{"class":1866},"DEFAULT_MODELS",[61,72503,72504],{"class":1753},": List[QubitModel] ",[61,72506,2000],{"class":1859},[61,72508,24292],{"class":1753},[61,72510,72511,72514],{"class":1731,"line":2783},[61,72512,72513],{"class":1836},"    QubitModel",[61,72515,38439],{"class":1753},[61,72517,72518,72521],{"class":1731,"line":2791},[61,72519,72520],{"class":2165},"        \"Toy working qubit\"",[61,72522,36573],{"class":1753},[61,72524,72525,72528],{"class":1731,"line":2804},[61,72526,72527],{"class":1866},"        0.50",[61,72529,36573],{"class":1753},[61,72531,72532],{"class":1731,"line":2817},[61,72533,72534],{"class":2165},"        \"Simple demonstrator; about half shots meet the criterion.\"\n",[61,72536,72537],{"class":1731,"line":2824},[61,72538,72539],{"class":1753},"    ),\n",[61,72541,72542,72544],{"class":1731,"line":2835},[61,72543,72513],{"class":1836},[61,72545,38439],{"class":1753},[61,72547,72548,72551],{"class":1731,"line":2846},[61,72549,72550],{"class":2165},"        \"NISQ qubit\"",[61,72552,36573],{"class":1753},[61,72554,72555,72558],{"class":1731,"line":2851},[61,72556,72557],{"class":1866},"        0.60",[61,72559,36573],{"class":1753},[61,72561,72562],{"class":1731,"line":2856},[61,72563,72564],{"class":2165},"        \"Noisy Intermediate-Scale Quantum device; modest success rate above chance.\"\n",[61,72566,72567],{"class":1731,"line":2862},[61,72568,72539],{"class":1753},[61,72570,72571,72573],{"class":1731,"line":2867},[61,72572,72513],{"class":1836},[61,72574,38439],{"class":1753},[61,72576,72577,72580],{"class":1731,"line":2917},[61,72578,72579],{"class":2165},"        \"Fault-tolerant (logical)\"",[61,72581,36573],{"class":1753},[61,72583,72584,72587],{"class":1731,"line":2926},[61,72585,72586],{"class":1866},"        0.90",[61,72588,36573],{"class":1753},[61,72590,72591],{"class":1731,"line":2972},[61,72592,72593],{"class":2165},"        \"Error-corrected logical qubit; high single-shot success probability.\"\n",[61,72595,72596],{"class":1731,"line":3001},[61,72597,72539],{"class":1753},[61,72599,72600,72602],{"class":1731,"line":3054},[61,72601,72513],{"class":1836},[61,72603,38439],{"class":1753},[61,72605,72606,72609],{"class":1731,"line":3060},[61,72607,72608],{"class":2165},"        \"Topologically protected\"",[61,72610,36573],{"class":1753},[61,72612,72613,72616],{"class":1731,"line":3077},[61,72614,72615],{"class":1866},"        0.98",[61,72617,36573],{"class":1753},[61,72619,72620],{"class":1731,"line":3082},[61,72621,72622],{"class":2165},"        \"Intrinsic protection; very high single-shot success probability.\"\n",[61,72624,72625],{"class":1731,"line":3129},[61,72626,72539],{"class":1753},[61,72628,72629,72631],{"class":1731,"line":3137},[61,72630,72513],{"class":1836},[61,72632,38439],{"class":1753},[61,72634,72635,72638],{"class":1731,"line":3185},[61,72636,72637],{"class":2165},"        \"Ideal qubit\"",[61,72639,36573],{"class":1753},[61,72641,72642,72645],{"class":1731,"line":3208},[61,72643,72644],{"class":1866},"        1.00",[61,72646,36573],{"class":1753},[61,72648,72649],{"class":1731,"line":3248},[61,72650,72651],{"class":2165},"        \"Theoretical perfect qubit; always successful in a single shot.\"\n",[61,72653,72654],{"class":1731,"line":3286},[61,72655,72539],{"class":1753},[61,72657,72658],{"class":1731,"line":3291},[61,72659,5701],{"class":1753},[61,72661,72662],{"class":1731,"line":3306},[61,72663,1743],{"emptyLinePlaceholder":1742},[61,72665,72666],{"class":1731,"line":3311},[61,72667,72668],{"class":1735},"# ---------- Simulation configuration ----------\n",[61,72670,72671],{"class":1731,"line":3317},[61,72672,1743],{"emptyLinePlaceholder":1742},[61,72674,72675,72678,72681,72683],{"class":1731,"line":3358},[61,72676,72677],{"class":1866},"CONFIG",[61,72679,72680],{"class":1753},": Dict ",[61,72682,2000],{"class":1859},[61,72684,37043],{"class":1753},[61,72686,72687,72690,72692,72695],{"class":1731,"line":3374},[61,72688,72689],{"class":2165},"    \"random_seed\"",[61,72691,2127],{"class":1753},[61,72693,72694],{"class":1866},"123456",[61,72696,36573],{"class":1753},[61,72698,72699,72702,72704,72707],{"class":1731,"line":3390},[61,72700,72701],{"class":2165},"    \"shots_per_model\"",[61,72703,2127],{"class":1753},[61,72705,72706],{"class":1866},"10000",[61,72708,36573],{"class":1753},[61,72710,72711,72714,72717,72720,72722,72725,72727,72730,72733],{"class":1731,"line":3422},[61,72712,72713],{"class":2165},"    \"epsilons\"",[61,72715,72716],{"class":1753},": [",[61,72718,72719],{"class":1866},"0.02",[61,72721,1435],{"class":1753},[61,72723,72724],{"class":1866},"0.01",[61,72726,1435],{"class":1753},[61,72728,72729],{"class":1866},"0.005",[61,72731,72732],{"class":1753},"],  ",[61,72734,72735],{"class":1735},"# ±2%, ±1%, ±0.5% targets\n",[61,72737,72738,72741],{"class":1731,"line":3427},[61,72739,72740],{"class":2165},"    \"cost_model\"",[61,72742,72743],{"class":1753},": {\n",[61,72745,72746,72749,72751,72754],{"class":1731,"line":3432},[61,72747,72748],{"class":2165},"        \"default_cost_per_shot_usd\"",[61,72750,2127],{"class":1753},[61,72752,72753],{"class":1866},"0.00005",[61,72755,36573],{"class":1753},[61,72757,72758,72761,72763],{"class":1731,"line":3438},[61,72759,72760],{"class":2165},"        \"fixed_overhead_usd\"",[61,72762,2127],{"class":1753},[61,72764,72765],{"class":1866},"0.10\n",[61,72767,72768],{"class":1731,"line":3443},[61,72769,72770],{"class":1753},"    },\n",[61,72772,72773,72776],{"class":1731,"line":3469},[61,72774,72775],{"class":2165},"    \"timing\"",[61,72777,72743],{"class":1753},[61,72779,72780,72783,72785],{"class":1731,"line":3494},[61,72781,72782],{"class":2165},"        \"time_per_shot_seconds\"",[61,72784,2127],{"class":1753},[61,72786,72787],{"class":1866},"2e-6\n",[61,72789,72790],{"class":1731,"line":3499},[61,72791,37162],{"class":1753},[61,72793,72794],{"class":1731,"line":3534},[61,72795,62745],{"class":1753},[61,72797,72798],{"class":1731,"line":3564},[61,72799,1743],{"emptyLinePlaceholder":1742},[61,72801,72802,72805,72807,72810,72813,72815,72817,72819,72822],{"class":1731,"line":3602},[61,72803,72804],{"class":1753},"rng ",[61,72806,2000],{"class":1859},[61,72808,72809],{"class":1753}," np.random.",[61,72811,72812],{"class":1836},"default_rng",[61,72814,849],{"class":1753},[61,72816,72677],{"class":1866},[61,72818,27808],{"class":1753},[61,72820,72821],{"class":2165},"\"random_seed\"",[61,72823,3531],{"class":1753},[61,72825,72826],{"class":1731,"line":3607},[61,72827,1743],{"emptyLinePlaceholder":1742},[61,72829,72830],{"class":1731,"line":3613},[61,72831,1743],{"emptyLinePlaceholder":1742},[61,72833,72834],{"class":1731,"line":3651},[61,72835,72836],{"class":1735},"# ---------- Core simulation ----------\n",[61,72838,72839],{"class":1731,"line":3715},[61,72840,1743],{"emptyLinePlaceholder":1742},[61,72842,72843,72845,72848,72850,72852,72854,72856,72858,72861,72863,72865,72867,72870],{"class":1731,"line":3738},[61,72844,2116],{"class":1749},[61,72846,72847],{"class":1836}," simulate_bitstring",[61,72849,849],{"class":1753},[61,72851,12],{"class":2124},[61,72853,2127],{"class":1753},[61,72855,1860],{"class":1859},[61,72857,1435],{"class":1753},[61,72859,72860],{"class":2124},"shots",[61,72862,2127],{"class":1753},[61,72864,2025],{"class":1859},[61,72866,1435],{"class":1753},[61,72868,72869],{"class":2124},"rng_",[61,72871,72872],{"class":1753},": np.random.Generator) -> np.ndarray:\n",[61,72874,72875],{"class":1731,"line":3753},[61,72876,36776],{"class":2165},[61,72878,72879],{"class":1731,"line":3782},[61,72880,72881],{"class":2165},"    Generate a 0\u002F1 bitstring for given per-shot success probability p and shot count.\n",[61,72883,72884],{"class":1731,"line":3813},[61,72885,36776],{"class":2165},[61,72887,72888,72890,72893,72896,72898,72900,72903,72906,72908,72911,72913,72915,72917],{"class":1731,"line":3835},[61,72889,2210],{"class":1749},[61,72891,72892],{"class":1753}," rng_.",[61,72894,72895],{"class":1836},"binomial",[61,72897,849],{"class":1753},[61,72899,57],{"class":1866},[61,72901,72902],{"class":1753},", p, ",[61,72904,72905],{"class":3402},"size",[61,72907,2000],{"class":1859},[61,72909,72910],{"class":1753},"shots).",[61,72912,70867],{"class":1836},[61,72914,849],{"class":1753},[61,72916,2025],{"class":1859},[61,72918,2016],{"class":1753},[61,72920,72921],{"class":1731,"line":3845},[61,72922,1743],{"emptyLinePlaceholder":1742},[61,72924,72925],{"class":1731,"line":3855},[61,72926,1743],{"emptyLinePlaceholder":1742},[61,72928,72929,72931,72934,72936,72939,72942,72945,72947,72950,72953,72955],{"class":1731,"line":3860},[61,72930,2116],{"class":1749},[61,72932,72933],{"class":1836}," summarize_model",[61,72935,849],{"class":1753},[61,72937,72938],{"class":2124},"model",[61,72940,72941],{"class":1753},": QubitModel, ",[61,72943,72944],{"class":2124},"bits",[61,72946,6874],{"class":1753},[61,72948,72949],{"class":2124},"epsilons",[61,72951,72952],{"class":1753},": List[",[61,72954,1860],{"class":1859},[61,72956,72957],{"class":1753},"]) -> Dict:\n",[61,72959,72960],{"class":1731,"line":3866},[61,72961,36776],{"class":2165},[61,72963,72964],{"class":1731,"line":3897},[61,72965,72966],{"class":2165},"    Calculate summary statistics for a given model and its simulated bitstring.\n",[61,72968,72969],{"class":1731,"line":3943},[61,72970,36776],{"class":2165},[61,72972,72973,72975,72977,72979],{"class":1731,"line":3964},[61,72974,41830],{"class":1753},[61,72976,2000],{"class":1859},[61,72978,24180],{"class":1859},[61,72980,72981],{"class":1753},"(bits.size)\n",[61,72983,72984,72987,72989,72991,72994,72996],{"class":1731,"line":3975},[61,72985,72986],{"class":1753},"    ones ",[61,72988,2000],{"class":1859},[61,72990,24180],{"class":1859},[61,72992,72993],{"class":1753},"(bits.",[61,72995,38033],{"class":1836},[61,72997,3750],{"class":1753},[61,72999,73000,73003,73005,73007,73010,73012],{"class":1731,"line":3998},[61,73001,73002],{"class":1753},"    zeros ",[61,73004,2000],{"class":1859},[61,73006,24180],{"class":1859},[61,73008,73009],{"class":1753},"(n ",[61,73011,2006],{"class":1859},[61,73013,73014],{"class":1753}," ones)\n",[61,73016,73017,73020,73022,73025,73027,73029,73031,73033,73035,73037,73039,73041,73043,73046],{"class":1731,"line":4025},[61,73018,73019],{"class":1753},"    frac ",[61,73021,2000],{"class":1859},[61,73023,73024],{"class":1753}," ones ",[61,73026,602],{"class":1859},[61,73028,72042],{"class":1753},[61,73030,2498],{"class":1749},[61,73032,72042],{"class":1753},[61,73034,2503],{"class":1859},[61,73036,24134],{"class":1866},[61,73038,4535],{"class":1749},[61,73040,63818],{"class":1859},[61,73042,849],{"class":1753},[61,73044,73045],{"class":2165},"\"nan\"",[61,73047,2016],{"class":1753},[61,73049,73050,73053,73055,73057],{"class":1731,"line":4046},[61,73051,73052],{"class":1753},"    lo, hi ",[61,73054,2000],{"class":1859},[61,73056,71978],{"class":1836},[61,73058,73059],{"class":1753},"(ones, n)\n",[61,73061,73062],{"class":1731,"line":4055},[61,73063,1743],{"emptyLinePlaceholder":1742},[61,73065,73066,73069,73071],{"class":1731,"line":4064},[61,73067,73068],{"class":1753},"    row ",[61,73070,2000],{"class":1859},[61,73072,37043],{"class":1753},[61,73074,73075,73078],{"class":1731,"line":4069},[61,73076,73077],{"class":2165},"        \"Model\"",[61,73079,73080],{"class":1753},": model.name,\n",[61,73082,73083,73086],{"class":1731,"line":4075},[61,73084,73085],{"class":2165},"        \"Description\"",[61,73087,73088],{"class":1753},": model.description,\n",[61,73090,73091,73094],{"class":1731,"line":4101},[61,73092,73093],{"class":2165},"        \"Configured p (target)\"",[61,73095,73096],{"class":1753},": model.satisfaction_prob,\n",[61,73098,73099,73102],{"class":1731,"line":4120},[61,73100,73101],{"class":2165},"        \"Shots\"",[61,73103,73104],{"class":1753},": n,\n",[61,73106,73107,73110],{"class":1731,"line":4138},[61,73108,73109],{"class":2165},"        \"Ones (successes)\"",[61,73111,73112],{"class":1753},": ones,\n",[61,73114,73115,73118],{"class":1731,"line":4151},[61,73116,73117],{"class":2165},"        \"Zeros (failures)\"",[61,73119,73120],{"class":1753},": zeros,\n",[61,73122,73123,73126],{"class":1731,"line":4156},[61,73124,73125],{"class":2165},"        \"Ones fraction (observed)\"",[61,73127,73128],{"class":1753},": frac,\n",[61,73130,73131,73134,73137,73140],{"class":1731,"line":4187},[61,73132,73133],{"class":2165},"        \"Wilson CI 95",[61,73135,73136],{"class":1866},"% lo",[61,73138,73139],{"class":2165},"w\"",[61,73141,73142],{"class":1753},": lo,\n",[61,73144,73145,73147,73150,73153],{"class":1731,"line":4234},[61,73146,73133],{"class":2165},[61,73148,73149],{"class":1866},"% hi",[61,73151,73152],{"class":2165},"gh\"",[61,73154,73155],{"class":1753},": hi,\n",[61,73157,73158,73161,73164,73166],{"class":1731,"line":4283},[61,73159,73160],{"class":2165},"        \"CI width (95%)\"",[61,73162,73163],{"class":1753},": hi ",[61,73165,2006],{"class":1859},[61,73167,73168],{"class":1753}," lo,\n",[61,73170,73171],{"class":1731,"line":4315},[61,73172,37162],{"class":1753},[61,73174,73175,73177,73179,73181],{"class":1731,"line":4344},[61,73176,3377],{"class":1749},[61,73178,2629],{"class":1753},[61,73180,2492],{"class":1749},[61,73182,73183],{"class":1753}," epsilons:\n",[61,73185,73186,73189,73191,73194,73196,73198,73201,73203,73206,73208,73210,73213,73215,73217,73219],{"class":1731,"line":4367},[61,73187,73188],{"class":1753},"        row[",[61,73190,19797],{"class":1749},[61,73192,73193],{"class":2165},"\"Required shots for ±",[61,73195,3688],{"class":1866},[61,73197,2025],{"class":1859},[61,73199,73200],{"class":1753},"(eps",[61,73202,2201],{"class":1859},[61,73204,73205],{"class":1866},"100",[61,73207,300],{"class":1753},[61,73209,3697],{"class":1866},[61,73211,73212],{"class":2165},"%\"",[61,73214,1997],{"class":1753},[61,73216,2000],{"class":1859},[61,73218,72249],{"class":1836},[61,73220,73221],{"class":1753},"(model.satisfaction_prob, eps)\n",[61,73223,73224,73226],{"class":1731,"line":4411},[61,73225,2210],{"class":1749},[61,73227,73228],{"class":1753}," row\n",[61,73230,73231],{"class":1731,"line":4420},[61,73232,1743],{"emptyLinePlaceholder":1742},[61,73234,73235],{"class":1731,"line":4425},[61,73236,1743],{"emptyLinePlaceholder":1742},[61,73238,73239,73241,73244,73246,73248,73250,73252,73254,73257,73259,73261,73263,73266,73268,73270,73272,73274],{"class":1731,"line":4431},[61,73240,2116],{"class":1749},[61,73242,73243],{"class":1836}," apply_costs",[61,73245,849],{"class":1753},[61,73247,72860],{"class":2124},[61,73249,2127],{"class":1753},[61,73251,2025],{"class":1859},[61,73253,1435],{"class":1753},[61,73255,73256],{"class":2124},"cost_per_shot",[61,73258,2127],{"class":1753},[61,73260,1860],{"class":1859},[61,73262,1435],{"class":1753},[61,73264,73265],{"class":2124},"fixed_overhead",[61,73267,2127],{"class":1753},[61,73269,1860],{"class":1859},[61,73271,2155],{"class":1753},[61,73273,1860],{"class":1859},[61,73275,1850],{"class":1753},[61,73277,73278],{"class":1731,"line":4447},[61,73279,36776],{"class":2165},[61,73281,73282],{"class":1731,"line":4464},[61,73283,73284],{"class":2165},"    Calculate illustrative cost based on shots and cost model.\n",[61,73286,73287],{"class":1731,"line":4477},[61,73288,36776],{"class":2165},[61,73290,73291,73293,73296,73298,73301,73303],{"class":1731,"line":4494},[61,73292,2210],{"class":1749},[61,73294,73295],{"class":1753}," fixed_overhead ",[61,73297,2419],{"class":1859},[61,73299,73300],{"class":1753}," shots ",[61,73302,2201],{"class":1859},[61,73304,73305],{"class":1753}," cost_per_shot\n",[61,73307,73308],{"class":1731,"line":4506},[61,73309,1743],{"emptyLinePlaceholder":1742},[61,73311,73312],{"class":1731,"line":4541},[61,73313,1743],{"emptyLinePlaceholder":1742},[61,73315,73316],{"class":1731,"line":4546},[61,73317,73318],{"class":1735},"# ---------- Run the simulation ----------\n",[61,73320,73321],{"class":1731,"line":4572},[61,73322,1743],{"emptyLinePlaceholder":1742},[61,73324,73325,73328,73330,73333,73335],{"class":1731,"line":4593},[61,73326,73327],{"class":1753},"all_bitstrings: Dict[",[61,73329,36784],{"class":1859},[61,73331,73332],{"class":1753},", np.ndarray] ",[61,73334,2000],{"class":1859},[61,73336,73337],{"class":1753}," {}\n",[61,73339,73340,73343,73345],{"class":1731,"line":4598},[61,73341,73342],{"class":1753},"summary_rows: List[Dict] ",[61,73344,2000],{"class":1859},[61,73346,39697],{"class":1753},[61,73348,73349],{"class":1731,"line":4629},[61,73350,1743],{"emptyLinePlaceholder":1742},[61,73352,73353,73355,73357,73360],{"class":1731,"line":4662},[61,73354,40848],{"class":1859},[61,73356,849],{"class":1753},[61,73358,73359],{"class":2165},"\"Running simulation...\"",[61,73361,2016],{"class":1753},[61,73363,73364,73366,73369,73371,73374],{"class":1731,"line":4699},[61,73365,2486],{"class":1749},[61,73367,73368],{"class":1753}," m ",[61,73370,2492],{"class":1749},[61,73372,73373],{"class":1866}," DEFAULT_MODELS",[61,73375,1850],{"class":1753},[61,73377,73378,73381,73383,73385,73388,73390,73392,73395],{"class":1731,"line":4726},[61,73379,73380],{"class":1753},"    bits ",[61,73382,2000],{"class":1859},[61,73384,72847],{"class":1836},[61,73386,73387],{"class":1753},"(m.satisfaction_prob, ",[61,73389,72677],{"class":1866},[61,73391,27808],{"class":1753},[61,73393,73394],{"class":2165},"\"shots_per_model\"",[61,73396,73397],{"class":1753},"], rng)\n",[61,73399,73400,73403,73405],{"class":1731,"line":4749},[61,73401,73402],{"class":1753},"    all_bitstrings[m.name] ",[61,73404,2000],{"class":1859},[61,73406,73407],{"class":1753}," bits\n",[61,73409,73410,73413,73415,73417,73420,73423,73425,73427,73430],{"class":1731,"line":4789},[61,73411,73412],{"class":1753},"    summary_rows.",[61,73414,39748],{"class":1836},[61,73416,849],{"class":1753},[61,73418,73419],{"class":1836},"summarize_model",[61,73421,73422],{"class":1753},"(m, bits, ",[61,73424,72677],{"class":1866},[61,73426,27808],{"class":1753},[61,73428,73429],{"class":2165},"\"epsilons\"",[61,73431,73432],{"class":1753},"]))\n",[61,73434,73435],{"class":1731,"line":43187},[61,73436,1743],{"emptyLinePlaceholder":1742},[61,73438,73439,73442,73444,73446,73448],{"class":1731,"line":43192},[61,73440,73441],{"class":1753},"summary_df ",[61,73443,2000],{"class":1859},[61,73445,62289],{"class":1753},[61,73447,62292],{"class":1836},[61,73449,73450],{"class":1753},"(summary_rows)\n",[61,73452,73453],{"class":1731,"line":43207},[61,73454,1743],{"emptyLinePlaceholder":1742},[61,73456,73457],{"class":1731,"line":43240},[61,73458,73459],{"class":1735},"# Add cost estimates\n",[61,73461,73462,73465,73467,73470,73472,73475,73477,73480],{"class":1731,"line":43246},[61,73463,73464],{"class":1753},"cps ",[61,73466,2000],{"class":1859},[61,73468,73469],{"class":1866}," CONFIG",[61,73471,27808],{"class":1753},[61,73473,73474],{"class":2165},"\"cost_model\"",[61,73476,67334],{"class":1753},[61,73478,73479],{"class":2165},"\"default_cost_per_shot_usd\"",[61,73481,5701],{"class":1753},[61,73483,73484,73487,73489,73491,73493,73495,73497,73500],{"class":1731,"line":43258},[61,73485,73486],{"class":1753},"over ",[61,73488,2000],{"class":1859},[61,73490,73469],{"class":1866},[61,73492,27808],{"class":1753},[61,73494,73474],{"class":2165},[61,73496,67334],{"class":1753},[61,73498,73499],{"class":2165},"\"fixed_overhead_usd\"",[61,73501,5701],{"class":1753},[61,73503,73504,73506,73509,73511,73513,73516,73519,73521,73523],{"class":1731,"line":43276},[61,73505,2498],{"class":1749},[61,73507,73508],{"class":1753}," cps ",[61,73510,2503],{"class":1859},[61,73512,24134],{"class":1866},[61,73514,73515],{"class":1749}," or",[61,73517,73518],{"class":1753}," over ",[61,73520,2503],{"class":1859},[61,73522,24134],{"class":1866},[61,73524,1850],{"class":1753},[61,73526,73527,73530,73533,73535,73537],{"class":1731,"line":43295},[61,73528,73529],{"class":1753},"    summary_df[",[61,73531,73532],{"class":2165},"\"Illustrative cost (USD)\"",[61,73534,1997],{"class":1753},[61,73536,2000],{"class":1859},[61,73538,24292],{"class":1753},[61,73540,73541,73544,73546,73548,73551,73554],{"class":1731,"line":43318},[61,73542,73543],{"class":1836},"        apply_costs",[61,73545,849],{"class":1753},[61,73547,2025],{"class":1859},[61,73549,73550],{"class":1753},"(row[",[61,73552,73553],{"class":2165},"\"Shots\"",[61,73555,73556],{"class":1753},"]), cps, over)\n",[61,73558,73559,73561,73564,73566,73569,73572],{"class":1731,"line":43327},[61,73560,63239],{"class":1749},[61,73562,73563],{"class":1753}," _, row ",[61,73565,2492],{"class":1749},[61,73567,73568],{"class":1753}," summary_df.",[61,73570,73571],{"class":1836},"iterrows",[61,73573,2089],{"class":1753},[61,73575,73576],{"class":1731,"line":43332},[61,73577,63269],{"class":1753},[61,73579,73580],{"class":1731,"line":43337},[61,73581,1743],{"emptyLinePlaceholder":1742},[61,73583,73584],{"class":1731,"line":43343},[61,73585,73586],{"class":1735},"# ---------- Save artifacts ----------\n",[61,73588,73589],{"class":1731,"line":43358},[61,73590,1743],{"emptyLinePlaceholder":1742},[61,73592,73593],{"class":1731,"line":43369},[61,73594,73595],{"class":1735},"# Save raw bitstrings and summary CSV\n",[61,73597,73598,73600,73603,73605,73608,73610],{"class":1731,"line":43406},[61,73599,2486],{"class":1749},[61,73601,73602],{"class":1753}," name, arr ",[61,73604,2492],{"class":1749},[61,73606,73607],{"class":1753}," all_bitstrings.",[61,73609,63250],{"class":1836},[61,73611,73612],{"class":1753},"():\n",[61,73614,73615,73618,73620,73622,73625,73627,73630,73633,73635,73638,73640,73643,73645,73647],{"class":1731,"line":43428},[61,73616,73617],{"class":1753},"    out_path ",[61,73619,2000],{"class":1859},[61,73621,39727],{"class":1749},[61,73623,73624],{"class":2165},"\"bitstrings_",[61,73626,3688],{"class":1866},[61,73628,73629],{"class":1753},"name.",[61,73631,73632],{"class":1836},"replace",[61,73634,849],{"class":1753},[61,73636,73637],{"class":2165},"' '",[61,73639,1435],{"class":1753},[61,73641,73642],{"class":2165},"'_'",[61,73644,300],{"class":1753},[61,73646,3697],{"class":1866},[61,73648,73649],{"class":2165},".csv\"\n",[61,73651,73652,73655,73657,73660,73663,73666,73669,73672,73675,73677,73679],{"class":1731,"line":43439},[61,73653,73654],{"class":1753},"    pd.",[61,73656,62292],{"class":1836},[61,73658,73659],{"class":1753},"({",[61,73661,73662],{"class":2165},"\"bit\"",[61,73664,73665],{"class":1753},": arr}).",[61,73667,73668],{"class":1836},"to_csv",[61,73670,73671],{"class":1753},"(out_path, ",[61,73673,73674],{"class":3402},"index",[61,73676,2000],{"class":1859},[61,73678,14390],{"class":1866},[61,73680,2016],{"class":1753},[61,73682,73683],{"class":1731,"line":43456},[61,73684,1743],{"emptyLinePlaceholder":1742},[61,73686,73687,73690,73692],{"class":1731,"line":43468},[61,73688,73689],{"class":1753},"summary_csv_path ",[61,73691,2000],{"class":1859},[61,73693,73694],{"class":2165}," \"summary.csv\"\n",[61,73696,73697,73700,73702,73705,73707,73709,73711],{"class":1731,"line":43473},[61,73698,73699],{"class":1753},"summary_df.",[61,73701,73668],{"class":1836},[61,73703,73704],{"class":1753},"(summary_csv_path, ",[61,73706,73674],{"class":3402},[61,73708,2000],{"class":1859},[61,73710,14390],{"class":1866},[61,73712,2016],{"class":1753},[61,73714,73715],{"class":1731,"line":43478},[61,73716,1743],{"emptyLinePlaceholder":1742},[61,73718,73719],{"class":1731,"line":64258},[61,73720,73721],{"class":1735},"# Save config JSON\n",[61,73723,73724,73727,73730,73732,73735,73737,73740,73742,73744],{"class":1731,"line":64263},[61,73725,73726],{"class":1749},"with",[61,73728,73729],{"class":1859}," open",[61,73731,849],{"class":1753},[61,73733,73734],{"class":2165},"\"simulation_config.json\"",[61,73736,1435],{"class":1753},[61,73738,73739],{"class":2165},"\"w\"",[61,73741,2624],{"class":1753},[61,73743,1792],{"class":1749},[61,73745,73746],{"class":1753}," f:\n",[61,73748,73749,73752,73755,73757,73759,73762,73765,73767,73769],{"class":1731,"line":64274},[61,73750,73751],{"class":1753},"    json.",[61,73753,73754],{"class":1836},"dump",[61,73756,849],{"class":1753},[61,73758,72677],{"class":1866},[61,73760,73761],{"class":1753},", f, ",[61,73763,73764],{"class":3402},"indent",[61,73766,2000],{"class":1859},[61,73768,337],{"class":1866},[61,73770,2016],{"class":1753},[61,73772,73773],{"class":1731,"line":64282},[61,73774,1743],{"emptyLinePlaceholder":1742},[61,73776,73777,73779,73781,73784],{"class":1731,"line":64292},[61,73778,40848],{"class":1859},[61,73780,849],{"class":1753},[61,73782,73783],{"class":2165},"\"Saved data to CSV and JSON files.\"",[61,73785,2016],{"class":1753},[61,73787,73788],{"class":1731,"line":64297},[61,73789,1743],{"emptyLinePlaceholder":1742},[61,73791,73792],{"class":1731,"line":64302},[61,73793,73794],{"class":1735},"# ---------- Visualizations ----------\n",[61,73796,73797],{"class":1731,"line":64307},[61,73798,1743],{"emptyLinePlaceholder":1742},[61,73800,73801],{"class":1731,"line":64314},[61,73802,73803],{"class":1735},"# Figure 1: Observed ones fraction with Wilson CI\n",[61,73805,73806,73809,73812,73814,73817],{"class":1731,"line":64330},[61,73807,73808],{"class":1753},"plt.style.",[61,73810,73811],{"class":1836},"use",[61,73813,849],{"class":1753},[61,73815,73816],{"class":2165},"'seaborn-v0_8-whitegrid'",[61,73818,2016],{"class":1753},[61,73820,73821,73823,73825,73827,73829,73831,73833,73835,73837,73839],{"class":1731,"line":64339},[61,73822,3616],{"class":1753},[61,73824,3619],{"class":1836},[61,73826,849],{"class":1753},[61,73828,3624],{"class":3402},[61,73830,2000],{"class":1859},[61,73832,849],{"class":1753},[61,73834,65275],{"class":1866},[61,73836,1435],{"class":1753},[61,73838,1300],{"class":1866},[61,73840,5546],{"class":1753},[61,73842,73843,73845,73847,73849,73851,73853,73855],{"class":1731,"line":64348},[61,73844,5730],{"class":1753},[61,73846,2000],{"class":1859},[61,73848,2193],{"class":1753},[61,73850,5737],{"class":1836},[61,73852,849],{"class":1753},[61,73854,5742],{"class":1859},[61,73856,73857],{"class":1753},"(summary_df))\n",[61,73859,73860,73863,73865,73868,73871,73874,73877],{"class":1731,"line":64357},[61,73861,73862],{"class":1753},"y ",[61,73864,2000],{"class":1859},[61,73866,73867],{"class":1753}," summary_df[",[61,73869,73870],{"class":2165},"\"Ones fraction (observed)\"",[61,73872,73873],{"class":1753},"].",[61,73875,73876],{"class":1836},"to_numpy",[61,73878,2089],{"class":1753},[61,73880,73881],{"class":1731,"line":64366},[61,73882,1743],{"emptyLinePlaceholder":1742},[61,73884,73885],{"class":1731,"line":64389},[61,73886,73887],{"class":1735},"# --- FIX APPLIED HERE ---\n",[61,73889,73890],{"class":1731,"line":64395},[61,73891,73892],{"class":1735},"# Take the absolute value to prevent tiny negative numbers from floating-point\n",[61,73894,73895],{"class":1731,"line":64400},[61,73896,73897],{"class":1735},"# inaccuracies from causing a ValueError in the errorbar plot.\n",[61,73899,73900,73903,73905,73908,73910,73912,73915,73917,73919,73921,73923],{"class":1731,"line":64405},[61,73901,73902],{"class":1753},"lower_error ",[61,73904,2000],{"class":1859},[61,73906,73907],{"class":1753}," y ",[61,73909,2006],{"class":1859},[61,73911,73867],{"class":1753},[61,73913,73914],{"class":2165},"\"Wilson CI 95",[61,73916,73136],{"class":1866},[61,73918,73139],{"class":2165},[61,73920,73873],{"class":1753},[61,73922,73876],{"class":1836},[61,73924,2089],{"class":1753},[61,73926,73927,73930,73932,73934,73936,73938,73940,73942,73944,73946,73948],{"class":1731,"line":64410},[61,73928,73929],{"class":1753},"upper_error ",[61,73931,2000],{"class":1859},[61,73933,73867],{"class":1753},[61,73935,73914],{"class":2165},[61,73937,73149],{"class":1866},[61,73939,73152],{"class":2165},[61,73941,73873],{"class":1753},[61,73943,73876],{"class":1836},[61,73945,7275],{"class":1753},[61,73947,2006],{"class":1859},[61,73949,73950],{"class":1753}," y\n",[61,73952,73953,73956,73958,73960,73962,73965,73967,73970,73972],{"class":1731,"line":64420},[61,73954,73955],{"class":1753},"yerr ",[61,73957,2000],{"class":1859},[61,73959,2193],{"class":1753},[61,73961,64207],{"class":1836},[61,73963,73964],{"class":1753},"([np.",[61,73966,3574],{"class":1836},[61,73968,73969],{"class":1753},"(lower_error), np.",[61,73971,3574],{"class":1836},[61,73973,73974],{"class":1753},"(upper_error)])\n",[61,73976,73977],{"class":1731,"line":64429},[61,73978,73979],{"class":1735},"# --- END OF FIX ---\n",[61,73981,73982],{"class":1731,"line":64438},[61,73983,1743],{"emptyLinePlaceholder":1742},[61,73985,73986,73988,73990,73993,73995,73997,74000,74002,74005,74007,74010],{"class":1731,"line":64443},[61,73987,3616],{"class":1753},[61,73989,5752],{"class":1836},[61,73991,73992],{"class":1753},"(x, y, ",[61,73994,3705],{"class":3402},[61,73996,2000],{"class":1859},[61,73998,73999],{"class":2165},"'skyblue'",[61,74001,1435],{"class":1753},[61,74003,74004],{"class":3402},"edgecolor",[61,74006,2000],{"class":1859},[61,74008,74009],{"class":2165},"'black'",[61,74011,2016],{"class":1753},[61,74013,74014,74016,74019,74021,74024,74026,74029,74032,74034,74036,74038,74041,74043,74045,74047,74049,74051,74054,74056,74059,74061,74063],{"class":1731,"line":64449},[61,74015,3616],{"class":1753},[61,74017,74018],{"class":1836},"errorbar",[61,74020,73992],{"class":1753},[61,74022,74023],{"class":3402},"yerr",[61,74025,2000],{"class":1859},[61,74027,74028],{"class":1753},"yerr, ",[61,74030,74031],{"class":3402},"fmt",[61,74033,2000],{"class":1859},[61,74035,38524],{"class":2165},[61,74037,1435],{"class":1753},[61,74039,74040],{"class":3402},"capsize",[61,74042,2000],{"class":1859},[61,74044,1026],{"class":1866},[61,74046,1435],{"class":1753},[61,74048,3705],{"class":3402},[61,74050,2000],{"class":1859},[61,74052,74053],{"class":2165},"\"black\"",[61,74055,1435],{"class":1753},[61,74057,74058],{"class":3402},"elinewidth",[61,74060,2000],{"class":1859},[61,74062,3350],{"class":1866},[61,74064,2016],{"class":1753},[61,74066,74067,74069,74071,74074,74077,74079,74082,74084,74087,74089,74091,74093,74095,74097,74099],{"class":1731,"line":64464},[61,74068,3616],{"class":1753},[61,74070,5762],{"class":1836},[61,74072,74073],{"class":1753},"(x, summary_df[",[61,74075,74076],{"class":2165},"\"Model\"",[61,74078,73873],{"class":1753},[61,74080,74081],{"class":1836},"tolist",[61,74083,4205],{"class":1753},[61,74085,74086],{"class":3402},"rotation",[61,74088,2000],{"class":1859},[61,74090,52835],{"class":1866},[61,74092,1435],{"class":1753},[61,74094,38141],{"class":3402},[61,74096,2000],{"class":1859},[61,74098,38195],{"class":2165},[61,74100,2016],{"class":1753},[61,74102,74103,74105,74107,74109,74112],{"class":1731,"line":64475},[61,74104,3616],{"class":1753},[61,74106,3787],{"class":1836},[61,74108,849],{"class":1753},[61,74110,74111],{"class":2165},"\"Observed Ones Fraction\"",[61,74113,2016],{"class":1753},[61,74115,74116,74118,74120,74122,74125,74127,74129,74131,74134],{"class":1731,"line":64490},[61,74117,3616],{"class":1753},[61,74119,3818],{"class":1836},[61,74121,849],{"class":1753},[61,74123,74124],{"class":2165},"\"Observed Single-Shot Success Rates with 95% Wilson CI\"",[61,74126,1435],{"class":1753},[61,74128,38463],{"class":3402},[61,74130,2000],{"class":1859},[61,74132,74133],{"class":1866},"14",[61,74135,2016],{"class":1753},[61,74137,74138,74140,74142],{"class":1731,"line":64500},[61,74139,3616],{"class":1753},[61,74141,17242],{"class":1836},[61,74143,2089],{"class":1753},[61,74145,74146,74149,74151],{"class":1731,"line":64508},[61,74147,74148],{"class":1753},"fig1_path ",[61,74150,2000],{"class":1859},[61,74152,74153],{"class":2165}," \"fig_ones_fraction.png\"\n",[61,74155,74156,74158,74161,74164,74166,74168,74170,74172,74175,74177,74180],{"class":1731,"line":64517},[61,74157,3616],{"class":1753},[61,74159,74160],{"class":1836},"savefig",[61,74162,74163],{"class":1753},"(fig1_path, ",[61,74165,3641],{"class":3402},[61,74167,2000],{"class":1859},[61,74169,36518],{"class":1866},[61,74171,1435],{"class":1753},[61,74173,74174],{"class":3402},"bbox_inches",[61,74176,2000],{"class":1859},[61,74178,74179],{"class":2165},"'tight'",[61,74181,2016],{"class":1753},[61,74183,74184,74186,74188],{"class":1731,"line":64522},[61,74185,3616],{"class":1753},[61,74187,3850],{"class":1836},[61,74189,2089],{"class":1753},[61,74191,74192],{"class":1731,"line":64544},[61,74193,1743],{"emptyLinePlaceholder":1742},[61,74195,74196],{"class":1731,"line":64549},[61,74197,74198],{"class":1735},"# Figure 2: Required shots at several ±epsilon targets\n",[61,74200,74201,74203,74205,74207,74209,74211,74213,74215,74217,74219],{"class":1731,"line":64565},[61,74202,3616],{"class":1753},[61,74204,3619],{"class":1836},[61,74206,849],{"class":1753},[61,74208,3624],{"class":3402},[61,74210,2000],{"class":1859},[61,74212,849],{"class":1753},[61,74214,65275],{"class":1866},[61,74216,1435],{"class":1753},[61,74218,1300],{"class":1866},[61,74220,5546],{"class":1753},[61,74222,74223,74226,74228],{"class":1731,"line":64582},[61,74224,74225],{"class":1753},"width ",[61,74227,2000],{"class":1859},[61,74229,2401],{"class":1866},[61,74231,74232,74234,74236,74238,74240,74242,74244,74246,74248],{"class":1731,"line":64587},[61,74233,5730],{"class":1753},[61,74235,2000],{"class":1859},[61,74237,2193],{"class":1753},[61,74239,5737],{"class":1836},[61,74241,849],{"class":1753},[61,74243,5742],{"class":1859},[61,74245,849],{"class":1753},[61,74247,72501],{"class":1866},[61,74249,5546],{"class":1753},[61,74251,74252,74255,74257,74259,74262,74264,74267,74269,74272],{"class":1731,"line":64600},[61,74253,74254],{"class":1753},"colors 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±ε\"",[61,74448,1435],{"class":1753},[61,74450,38463],{"class":3402},[61,74452,2000],{"class":1859},[61,74454,74133],{"class":1866},[61,74456,2016],{"class":1753},[61,74458,74459,74461,74464,74466,74469],{"class":1731,"line":64674},[61,74460,3616],{"class":1753},[61,74462,74463],{"class":1836},"yscale",[61,74465,849],{"class":1753},[61,74467,74468],{"class":2165},"'log'",[61,74470,2016],{"class":1753},[61,74472,74473,74475,74477,74479,74481,74483,74486],{"class":1731,"line":64686},[61,74474,3616],{"class":1753},[61,74476,3840],{"class":1836},[61,74478,849],{"class":1753},[61,74480,3818],{"class":3402},[61,74482,2000],{"class":1859},[61,74484,74485],{"class":2165},"\"Error Tolerance\"",[61,74487,2016],{"class":1753},[61,74489,74490,74492,74494],{"class":1731,"line":64695},[61,74491,3616],{"class":1753},[61,74493,17242],{"class":1836},[61,74495,2089],{"class":1753},[61,74497,74498,74501,74503],{"class":1731,"line":64701},[61,74499,74500],{"class":1753},"fig2_path ",[61,74502,2000],{"class":1859},[61,74504,74505],{"class":2165}," \"fig_required_shots.png\"\n",[61,74507,74508,74510,74512,74515,74517,74519,74521,74523,74525,74527,74529],{"class":1731,"line":64711},[61,74509,3616],{"class":1753},[61,74511,74160],{"class":1836},[61,74513,74514],{"class":1753},"(fig2_path, ",[61,74516,3641],{"class":3402},[61,74518,2000],{"class":1859},[61,74520,36518],{"class":1866},[61,74522,1435],{"class":1753},[61,74524,74174],{"class":3402},[61,74526,2000],{"class":1859},[61,74528,74179],{"class":2165},[61,74530,2016],{"class":1753},[61,74532,74533,74535,74537],{"class":1731,"line":64721},[61,74534,3616],{"class":1753},[61,74536,3850],{"class":1836},[61,74538,2089],{"class":1753},[61,74540,74541],{"class":1731,"line":64730},[61,74542,1743],{"emptyLinePlaceholder":1742},[61,74544,74545,74547,74549,74552],{"class":1731,"line":64739},[61,74546,40848],{"class":1859},[61,74548,849],{"class":1753},[61,74550,74551],{"class":2165},"\"Generated and saved plots.\"",[61,74553,2016],{"class":1753},[61,74555,74556],{"class":1731,"line":64750},[61,74557,1743],{"emptyLinePlaceholder":1742},[61,74559,74560],{"class":1731,"line":64762},[61,74561,74562],{"class":1735},"# ---------- Display interactive table ----------\n",[61,74564,74565],{"class":1731,"line":64773},[61,74566,1743],{"emptyLinePlaceholder":1742},[61,74568,74569,74571,74573,74576,74578,74581],{"class":1731,"line":64778},[61,74570,40830],{"class":1836},[61,74572,849],{"class":1753},[61,74574,74575],{"class":1836},"Markdown",[61,74577,849],{"class":1753},[61,74579,74580],{"class":2165},"\"##  Qubit Sampling Budget - Summary\"",[61,74582,5546],{"class":1753},[61,74584,74585,74587,74590,74593],{"class":1731,"line":64783},[61,74586,40830],{"class":1836},[61,74588,74589],{"class":1753},"(summary_df.style.",[61,74591,74592],{"class":1836},"format",[61,74594,36508],{"class":1753},[61,74596,74597,74600,74602,74604,74606,74608,74610,74612],{"class":1731,"line":64788},[61,74598,74599],{"class":2165},"    \"Configured p (target)\"",[61,74601,2127],{"class":1753},[61,74603,26088],{"class":2165},[61,74605,3688],{"class":1866},[61,74607,40451],{"class":1749},[61,74609,3697],{"class":1866},[61,74611,26088],{"class":2165},[61,74613,36573],{"class":1753},[61,74615,74616,74619,74621,74623,74625,74628,74630,74632],{"class":1731,"line":64798},[61,74617,74618],{"class":2165},"    \"Ones fraction (observed)\"",[61,74620,2127],{"class":1753},[61,74622,26088],{"class":2165},[61,74624,3688],{"class":1866},[61,74626,74627],{"class":1749},":.4f",[61,74629,3697],{"class":1866},[61,74631,26088],{"class":2165},[61,74633,36573],{"class":1753},[61,74635,74636,74639,74641,74643,74645,74647,74649,74651,74653,74655],{"class":1731,"line":64805},[61,74637,74638],{"class":2165},"    \"Wilson CI 95",[61,74640,73136],{"class":1866},[61,74642,73139],{"class":2165},[61,74644,2127],{"class":1753},[61,74646,26088],{"class":2165},[61,74648,3688],{"class":1866},[61,74650,74627],{"class":1749},[61,74652,3697],{"class":1866},[61,74654,26088],{"class":2165},[61,74656,36573],{"class":1753},[61,74658,74659,74661,74663,74665,74667,74669,74671,74673,74675,74677],{"class":1731,"line":64813},[61,74660,74638],{"class":2165},[61,74662,73149],{"class":1866},[61,74664,73152],{"class":2165},[61,74666,2127],{"class":1753},[61,74668,26088],{"class":2165},[61,74670,3688],{"class":1866},[61,74672,74627],{"class":1749},[61,74674,3697],{"class":1866},[61,74676,26088],{"class":2165},[61,74678,36573],{"class":1753},[61,74680,74681,74684,74686,74688,74690,74692,74694,74696],{"class":1731,"line":64824},[61,74682,74683],{"class":2165},"    \"CI width (95%)\"",[61,74685,2127],{"class":1753},[61,74687,26088],{"class":2165},[61,74689,3688],{"class":1866},[61,74691,74627],{"class":1749},[61,74693,3697],{"class":1866},[61,74695,26088],{"class":2165},[61,74697,36573],{"class":1753},[61,74699,74700,74703,74705,74707,74709,74711,74713],{"class":1731,"line":64835},[61,74701,74702],{"class":2165},"    \"Illustrative cost (USD)\"",[61,74704,2127],{"class":1753},[61,74706,65834],{"class":2165},[61,74708,3688],{"class":1866},[61,74710,40451],{"class":1749},[61,74712,3697],{"class":1866},[61,74714,40488],{"class":2165},[61,74716,74717],{"class":1731,"line":64844},[61,74718,74719],{"class":1753},"}))\n",[61,74721,74722],{"class":1731,"line":64850},[61,74723,1743],{"emptyLinePlaceholder":1742},[61,74725,74726],{"class":1731,"line":64860},[61,74727,1743],{"emptyLinePlaceholder":1742},[61,74729,74730],{"class":1731,"line":64870},[61,74731,74732],{"class":1735},"# ---------- Build a ready-to-run Colab notebook ----------\n",[61,74734,74735],{"class":1731,"line":64879},[61,74736,1743],{"emptyLinePlaceholder":1742},[61,74738,74739,74742,74744,74747],{"class":1731,"line":64888},[61,74740,74741],{"class":1753},"md_intro ",[61,74743,2000],{"class":1859},[61,74745,74746],{"class":1749}," r",[61,74748,36393],{"class":3766},[61,74750,74751],{"class":1731,"line":64897},[61,74752,74753],{"class":1735},"# Sampling Budget & Bit-String Analogy for Qubit Classes\n",[61,74755,74756],{"class":1731,"line":64909},[61,74757,1743],{"emptyLinePlaceholder":1742},[61,74759,74760,74762,74765,74767,74770,74772,74774,74776,74779,74781,74784,74786],{"class":1731,"line":64921},[61,74761,2651],{"class":1866},[61,74763,74764],{"class":3766},"High school level ",[61,74766,849],{"class":1866},[61,74768,74769],{"class":3766},"concise",[61,74771,300],{"class":1866},[61,74773,36027],{"class":3766},[61,74775,2651],{"class":1866},[61,74777,74778],{"class":3766}," We treat each measurement ",[61,74780,849],{"class":1866},[61,74782,74783],{"class":3766},"“single shot”",[61,74785,300],{"class":1866},[61,74787,74788],{"class":3766}," 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",[61,74790,74791],{"class":1731,"line":64930},[61,74792,1743],{"emptyLinePlaceholder":1742},[61,74794,74795,74797,74800,74802,74804,74806,74808,74810,74813,74815,74818,74820,74823,74825,74828,74830,74833,74835,74838,74840,74843,74846,74849,74852,74854,74856,74859,74861,74863,74866,74869,74871],{"class":1731,"line":64935},[61,74796,2651],{"class":1866},[61,74798,74799],{"class":3766},"Graduate level ",[61,74801,849],{"class":1866},[61,74803,74769],{"class":3766},[61,74805,300],{"class":1866},[61,74807,36027],{"class":3766},[61,74809,2651],{"class":1866},[61,74811,74812],{"class":3766}," We model single-shot “threshold of satisfaction” as a Bernoulli process with success probability $p$ per qubit class: toy, NISQ ",[61,74814,849],{"class":1866},[61,74816,74817],{"class":3766},"noisy intermediate-scale quantum",[61,74819,300],{"class":1866},[61,74821,74822],{"class":3766},", fault-tolerant ",[61,74824,849],{"class":1866},[61,74826,74827],{"class":3766},"logical",[61,74829,300],{"class":1866},[61,74831,74832],{"class":3766},", topologically protected, and ideal. We report Wilson score intervals ",[61,74834,849],{"class":1866},[61,74836,74837],{"class":3766},"95%",[61,74839,300],{"class":1866},[61,74841,74842],{"class":3766}," for observed proportions and estimate the required shot budget $n ",[61,74844,74845],{"class":1859},"\\a",[61,74847,74848],{"class":3766},"pprox z^2",[61,74850,74851],{"class":1859},"\\,",[61,74853,12],{"class":3766},[61,74855,849],{"class":1866},[61,74857,74858],{"class":3766},"1-p",[61,74860,300],{"class":1866},[61,74862,602],{"class":3766},[61,74864,74865],{"class":1859},"\\e",[61,74867,74868],{"class":3766},"psilon^2$ to bound absolute error by ±$",[61,74870,74865],{"class":1859},[61,74872,74873],{"class":3766},"psilon$ at ~95% confidence.\n",[61,74875,74876],{"class":1731,"line":64940},[61,74877,36393],{"class":3766},[61,74879,74880],{"class":1731,"line":64945},[61,74881,1743],{"emptyLinePlaceholder":1742},[61,74883,74884,74887,74889,74891],{"class":1731,"line":64955},[61,74885,74886],{"class":1753},"code_config ",[61,74888,2000],{"class":1859},[61,74890,74746],{"class":1749},[61,74892,36393],{"class":3766},[61,74894,74895],{"class":1731,"line":64962},[61,74896,74897],{"class":1735},"# @title Configuration\n",[61,74899,74900],{"class":1731,"line":64969},[61,74901,74902],{"class":3766},"from dataclasses import dataclass\n",[61,74904,74905],{"class":1731,"line":64980},[61,74906,74907],{"class":3766},"from typing import List, Dict\n",[61,74909,74910],{"class":1731,"line":64991},[61,74911,74912],{"class":3766},"import numpy as np\n",[61,74914,74915],{"class":1731,"line":65000},[61,74916,1743],{"emptyLinePlaceholder":1742},[61,74918,74919,74922],{"class":1731,"line":65006},[61,74920,74921],{"class":3766},"RANDOM_SEED = 579345  ",[61,74923,74924],{"class":1735},"# @param {type:\"number\"}\n",[61,74926,74927,74930],{"class":1731,"line":65016},[61,74928,74929],{"class":3766},"SHOTS_PER_MODEL = 10000  ",[61,74931,74924],{"class":1735},[61,74933,74934,74937,74940],{"class":1731,"line":65024},[61,74935,74936],{"class":3766},"EPSILONS = ",[61,74938,74939],{"class":1866},"[0.02, 0.01, 0.005]",[61,74941,74942],{"class":1735},"  # @param\n",[61,74944,74945,74948],{"class":1731,"line":65031},[61,74946,74947],{"class":3766},"COST_PER_SHOT_USD = 0.00005  ",[61,74949,74924],{"class":1735},[61,74951,74952,74955],{"class":1731,"line":65038},[61,74953,74954],{"class":3766},"FIXED_OVERHEAD_USD = 0.10    ",[61,74956,74924],{"class":1735},[61,74958,74959],{"class":1731,"line":65047},[61,74960,1743],{"emptyLinePlaceholder":1742},[61,74962,74963,74965,74967,74970],{"class":1731,"line":65056},[61,74964,72443],{"class":3766},[61,74966,849],{"class":1866},[61,74968,74969],{"class":3766},"frozen=True",[61,74971,2016],{"class":1866},[61,74973,74974],{"class":1731,"line":65065},[61,74975,74976],{"class":3766},"class QubitModel:\n",[61,74978,74979],{"class":1731,"line":65074},[61,74980,74981],{"class":3766},"    name: str\n",[61,74983,74984],{"class":1731,"line":65085},[61,74985,74986],{"class":3766},"    satisfaction_prob: float\n",[61,74988,74989],{"class":1731,"line":65097},[61,74990,74991],{"class":3766},"    description: str\n",[61,74993,74994],{"class":1731,"line":65109},[61,74995,1743],{"emptyLinePlaceholder":1742},[61,74997,74998,75001,75004,75007],{"class":1731,"line":65118},[61,74999,75000],{"class":3766},"MODELS: List",[61,75002,75003],{"class":1866},"[QubitModel]",[61,75005,75006],{"class":3766}," = ",[61,75008,75009],{"class":1866},"[\n",[61,75011,75012],{"class":1731,"line":65123},[61,75013,75014],{"class":1866},"    QubitModel(\"Toy working qubit\", 0.50, \"Simple demonstrator\"),\n",[61,75016,75017],{"class":1731,"line":65128},[61,75018,75019],{"class":1866},"    QubitModel(\"NISQ qubit\", 0.60, \"Noisy Intermediate-Scale Quantum device\"),\n",[61,75021,75022],{"class":1731,"line":65133},[61,75023,75024],{"class":1866},"    QubitModel(\"Fault-tolerant (logical)\", 0.90, \"Error-corrected logical qubit\"),\n",[61,75026,75027],{"class":1731,"line":65143},[61,75028,75029],{"class":1866},"    QubitModel(\"Topologically protected\", 0.98, \"Intrinsic protection\"),\n",[61,75031,75032],{"class":1731,"line":65150},[61,75033,75034],{"class":1866},"    QubitModel(\"Ideal qubit\", 1.00, \"Theoretical perfect qubit\"),\n",[61,75036,75037],{"class":1731,"line":65157},[61,75038,5701],{"class":1866},[61,75040,75041],{"class":1731,"line":65169},[61,75042,1743],{"emptyLinePlaceholder":1742},[61,75044,75045,75048,75050,75053],{"class":1731,"line":65180},[61,75046,75047],{"class":3766},"rng = np.random.default_rng",[61,75049,849],{"class":1866},[61,75051,75052],{"class":3766},"RANDOM_SEED",[61,75054,2016],{"class":1866},[61,75056,75057],{"class":1731,"line":65189},[61,75058,36393],{"class":3766},[61,75060,75061],{"class":1731,"line":65195},[61,75062,1743],{"emptyLinePlaceholder":1742},[61,75064,75065,75068,75070,75072],{"class":1731,"line":65211},[61,75066,75067],{"class":1753},"code_lib_and_run ",[61,75069,2000],{"class":1859},[61,75071,74746],{"class":1749},[61,75073,36393],{"class":3766},[61,75075,75076],{"class":1731,"line":65216},[61,75077,75078],{"class":1735},"# @title Run Simulation & Display Results\n",[61,75080,75081],{"class":1731,"line":65225},[61,75082,75083],{"class":3766},"import math\n",[61,75085,75086],{"class":1731,"line":65235},[61,75087,75088],{"class":3766},"from typing import Tuple, Dict\n",[61,75090,75091],{"class":1731,"line":65244},[61,75092,75093],{"class":3766},"import pandas as pd\n",[61,75095,75096],{"class":1731,"line":65253},[61,75097,74912],{"class":3766},[61,75099,75100],{"class":1731,"line":65259},[61,75101,75102],{"class":3766},"import matplotlib.pyplot as plt\n",[61,75104,75105],{"class":1731,"line":65268},[61,75106,75107],{"class":3766},"from IPython.display import display, Markdown\n",[61,75109,75110],{"class":1731,"line":65280},[61,75111,1743],{"emptyLinePlaceholder":1742},[61,75113,75114,75117,75119,75122,75124,75127,75130],{"class":1731,"line":65292},[61,75115,75116],{"class":3766},"def wilson_ci",[61,75118,849],{"class":1866},[61,75120,75121],{"class":3766},"k: int, n: int, z: float = 1.96",[61,75123,300],{"class":1866},[61,75125,75126],{"class":3766}," -> Tuple",[61,75128,75129],{"class":1866},"[float, float]",[61,75131,1850],{"class":3766},[61,75133,75134,75137,75139,75142],{"class":1731,"line":65304},[61,75135,75136],{"class":3766},"    if n == 0: return ",[61,75138,849],{"class":1866},[61,75140,75141],{"class":3766},"0.0, 1.0",[61,75143,2016],{"class":1866},[61,75145,75146,75149,75151],{"class":1731,"line":65309},[61,75147,75148],{"class":3766},"    p_hat, z2_n = k \u002F n, z",[61,75150,2651],{"class":1866},[61,75152,75153],{"class":3766},"2 \u002F n\n",[61,75155,75156,75159,75161],{"class":1731,"line":65314},[61,75157,75158],{"class":3766},"    denom = 1.0 ",[61,75160,2419],{"class":1866},[61,75162,75163],{"class":3766}," z2_n\n",[61,75165,75166,75169,75171,75174,75176,75179,75181],{"class":1731,"line":65319},[61,75167,75168],{"class":3766},"    center = ",[61,75170,849],{"class":1866},[61,75172,75173],{"class":3766},"p_hat ",[61,75175,2419],{"class":1866},[61,75177,75178],{"class":3766}," z2_n \u002F 2",[61,75180,300],{"class":1866},[61,75182,75183],{"class":3766}," \u002F denom\n",[61,75185,75186,75189,75191,75194,75196,75199,75201,75203,75205,75207,75210,75212,75215,75217,75220,75222,75225,75227,75230,75232,75235,75237,75239,75242],{"class":1731,"line":65337},[61,75187,75188],{"class":3766},"    span = ",[61,75190,849],{"class":1866},[61,75192,75193],{"class":3766},"z ",[61,75195,2201],{"class":1866},[61,75197,75198],{"class":3766}," math.sqrt",[61,75200,849],{"class":1866},[61,75202,75173],{"class":3766},[61,75204,2201],{"class":1866},[61,75206,2003],{"class":1866},[61,75208,75209],{"class":3766},"1 - p_hat",[61,75211,300],{"class":1866},[61,75213,75214],{"class":3766}," \u002F n ",[61,75216,2419],{"class":1866},[61,75218,75219],{"class":3766}," z",[61,75221,2651],{"class":1866},[61,75223,75224],{"class":3766},"2 \u002F ",[61,75226,849],{"class":1866},[61,75228,75229],{"class":3766},"4 ",[61,75231,2201],{"class":1866},[61,75233,75234],{"class":3766}," n",[61,75236,2651],{"class":1866},[61,75238,337],{"class":3766},[61,75240,75241],{"class":1866},")))",[61,75243,75183],{"class":3766},[61,75245,75246,75249,75251,75253,75255,75258,75260,75263,75265,75268,75270,75273],{"class":1731,"line":65343},[61,75247,75248],{"class":3766},"    return ",[61,75250,849],{"class":1866},[61,75252,4212],{"class":3766},[61,75254,849],{"class":1866},[61,75256,75257],{"class":3766},"0.0, center - span",[61,75259,300],{"class":1866},[61,75261,75262],{"class":3766},", min",[61,75264,849],{"class":1866},[61,75266,75267],{"class":3766},"1.0, center ",[61,75269,2419],{"class":1866},[61,75271,75272],{"class":3766}," span",[61,75274,5546],{"class":1866},[61,75276,75277],{"class":1731,"line":65352},[61,75278,1743],{"emptyLinePlaceholder":1742},[61,75280,75281,75284,75286,75289,75291],{"class":1731,"line":65359},[61,75282,75283],{"class":3766},"def shots_for_epsilon",[61,75285,849],{"class":1866},[61,75287,75288],{"class":3766},"p: float, epsilon: float = 0.01, z: float = 1.96",[61,75290,300],{"class":1866},[61,75292,75293],{"class":3766}," -> int:\n",[61,75295,75296,75299,75301,75303,75305,75307],{"class":1731,"line":65366},[61,75297,75298],{"class":3766},"    if p ",[61,75300,2201],{"class":1866},[61,75302,2003],{"class":1866},[61,75304,74858],{"class":3766},[61,75306,300],{"class":1866},[61,75308,75309],{"class":3766}," == 0: return 1\n",[61,75311,75312,75315,75317,75320,75322,75325,75327,75329,75331,75334,75336,75338,75340,75342,75344,75346,75349,75351,75353,75355,75357],{"class":1731,"line":65374},[61,75313,75314],{"class":3766},"    return max",[61,75316,849],{"class":1866},[61,75318,75319],{"class":3766},"1, int",[61,75321,849],{"class":1866},[61,75323,75324],{"class":3766},"math.ceil",[61,75326,4088],{"class":1866},[61,75328,26482],{"class":3766},[61,75330,2651],{"class":1866},[61,75332,75333],{"class":3766},"2 ",[61,75335,2201],{"class":1866},[61,75337,39798],{"class":3766},[61,75339,2201],{"class":1866},[61,75341,2003],{"class":1866},[61,75343,74858],{"class":3766},[61,75345,5290],{"class":1866},[61,75347,75348],{"class":3766}," \u002F ",[61,75350,849],{"class":1866},[61,75352,72262],{"class":3766},[61,75354,2651],{"class":1866},[61,75356,337],{"class":3766},[61,75358,75359],{"class":1866},"))))\n",[61,75361,75362],{"class":1731,"line":65385},[61,75363,1743],{"emptyLinePlaceholder":1742},[61,75365,75366,75369,75371,75374,75376],{"class":1731,"line":65396},[61,75367,75368],{"class":3766},"def summarize_model",[61,75370,849],{"class":1866},[61,75372,75373],{"class":3766},"model, bits: np.ndarray, epsilons",[61,75375,300],{"class":1866},[61,75377,75378],{"class":3766}," -> Dict:\n",[61,75380,75381,75384,75386,75388],{"class":1731,"line":65407},[61,75382,75383],{"class":3766},"    n = len",[61,75385,849],{"class":1866},[61,75387,72944],{"class":3766},[61,75389,2016],{"class":1866},[61,75391,75392,75395,75397,75399],{"class":1731,"line":65412},[61,75393,75394],{"class":3766},"    ones = sum",[61,75396,849],{"class":1866},[61,75398,72944],{"class":3766},[61,75400,2016],{"class":1866},[61,75402,75403,75406,75408,75411],{"class":1731,"line":65421},[61,75404,75405],{"class":3766},"    lo, hi = wilson_ci",[61,75407,849],{"class":1866},[61,75409,75410],{"class":3766},"ones, n",[61,75412,2016],{"class":1866},[61,75414,75415],{"class":1731,"line":65428},[61,75416,75417],{"class":3766},"    row = {\n",[61,75419,75420,75423,75425,75428,75430],{"class":1731,"line":65435},[61,75421,75422],{"class":3766},"        \"Model\": model.name, \"p ",[61,75424,849],{"class":1866},[61,75426,75427],{"class":3766},"target",[61,75429,300],{"class":1866},[61,75431,75432],{"class":3766},"\": model.satisfaction_prob, \"Shots\": n,\n",[61,75434,75435,75438,75440,75443,75445],{"class":1731,"line":65443},[61,75436,75437],{"class":3766},"        \"Ones\": ones, \"p ",[61,75439,849],{"class":1866},[61,75441,75442],{"class":3766},"observed",[61,75444,300],{"class":1866},[61,75446,75447],{"class":3766},"\": ones \u002F n if n > 0 else 0,\n",[61,75449,75450],{"class":1731,"line":65464},[61,75451,75452],{"class":3766},"        \"CI 95% low\": lo, \"CI 95% high\": hi, \"CI width\": hi - lo,\n",[61,75454,75455],{"class":1731,"line":65475},[61,75456,37162],{"class":3766},[61,75458,75459],{"class":1731,"line":65486},[61,75460,75461],{"class":3766},"    for eps in epsilons:\n",[61,75463,75464,75467,75470,75473,75475,75478],{"class":1731,"line":65497},[61,75465,75466],{"class":3766},"        row",[61,75468,75469],{"class":1866},"[f\"Shots for ±{eps:.1%}\"]",[61,75471,75472],{"class":3766}," = shots_for_epsilon",[61,75474,849],{"class":1866},[61,75476,75477],{"class":3766},"model.satisfaction_prob, eps",[61,75479,2016],{"class":1866},[61,75481,75482],{"class":1731,"line":65508},[61,75483,75484],{"class":3766},"    return row\n",[61,75486,75487],{"class":1731,"line":65513},[61,75488,1743],{"emptyLinePlaceholder":1742},[61,75490,75491],{"class":1731,"line":65518},[61,75492,75493],{"class":1735},"# --- Run Simulation ---\n",[61,75495,75496,75499],{"class":1731,"line":65523},[61,75497,75498],{"class":3766},"summary_rows = ",[61,75500,75501],{"class":1866},"[summarize_model(m, rng.binomial(1, m.satisfaction_prob, SHOTS_PER_MODEL), EPSILONS) for m in MODELS]\n",[61,75503,75504,75507,75509,75512],{"class":1731,"line":65537},[61,75505,75506],{"class":3766},"summary_df = pd.DataFrame",[61,75508,849],{"class":1866},[61,75510,75511],{"class":3766},"summary_rows",[61,75513,2016],{"class":1866},[61,75515,75516],{"class":1731,"line":65549},[61,75517,75518],{"class":3766},"if COST_PER_SHOT_USD > 0 or FIXED_OVERHEAD_USD > 0:\n",[61,75520,75521,75524,75527,75530,75532,75535,75538,75540],{"class":1731,"line":65555},[61,75522,75523],{"class":3766},"    summary_df",[61,75525,75526],{"class":1866},"[\"Cost (USD)\"]",[61,75528,75529],{"class":3766}," = FIXED_OVERHEAD_USD ",[61,75531,2419],{"class":1866},[61,75533,75534],{"class":3766}," summary_df",[61,75536,75537],{"class":1866},"[\"Shots\"]",[61,75539,2298],{"class":1866},[61,75541,75542],{"class":3766}," COST_PER_SHOT_USD\n",[61,75544,75545],{"class":1731,"line":65560},[61,75546,1743],{"emptyLinePlaceholder":1742},[61,75548,75549,75551,75553,75555,75557,75559],{"class":1731,"line":65567},[61,75550,40830],{"class":3766},[61,75552,849],{"class":1866},[61,75554,74575],{"class":3766},[61,75556,849],{"class":1866},[61,75558,26088],{"class":3766},[61,75560,75561],{"class":1735},"### Simulation Summary\"))\n",[61,75563,75564,75566,75568,75571,75573,75576,75578,75581,75583,75586],{"class":1731,"line":65577},[61,75565,40830],{"class":3766},[61,75567,849],{"class":1866},[61,75569,75570],{"class":3766},"summary_df.style.format",[61,75572,849],{"class":1866},[61,75574,75575],{"class":3766},"precision=4",[61,75577,300],{"class":1866},[61,75579,75580],{"class":3766},".background_gradient",[61,75582,849],{"class":1866},[61,75584,75585],{"class":3766},"cmap='viridis', subset=",[61,75587,75588],{"class":1866},"['p (observed)', 'CI width']))\n",[61,75590,75591],{"class":1731,"line":65582},[61,75592,1743],{"emptyLinePlaceholder":1742},[61,75594,75595],{"class":1731,"line":65594},[61,75596,75597],{"class":1735},"# --- Plotting ---\n",[61,75599,75600,75603,75605,75607],{"class":1731,"line":65605},[61,75601,75602],{"class":3766},"plt.style.use",[61,75604,849],{"class":1866},[61,75606,73816],{"class":3766},[61,75608,2016],{"class":1866},[61,75610,75611,75614,75616,75619,75621,75624,75626,75629,75631,75634],{"class":1731,"line":65615},[61,75612,75613],{"class":3766},"fig, ",[61,75615,849],{"class":1866},[61,75617,75618],{"class":3766},"ax1, ax2",[61,75620,300],{"class":1866},[61,75622,75623],{"class":3766}," = plt.subplots",[61,75625,849],{"class":1866},[61,75627,75628],{"class":3766},"1, 2, figsize=",[61,75630,849],{"class":1866},[61,75632,75633],{"class":3766},"16, 6",[61,75635,5546],{"class":1866},[61,75637,75638],{"class":1731,"line":65626},[61,75639,1743],{"emptyLinePlaceholder":1742},[61,75641,75642],{"class":1731,"line":65631},[61,75643,75644],{"class":1735},"# Plot 1\n",[61,75646,75647,75650,75653,75656],{"class":1731,"line":65636},[61,75648,75649],{"class":3766},"y = summary_df",[61,75651,75652],{"class":1866},"[\"p (observed)\"]",[61,75654,75655],{"class":3766},".to_numpy",[61,75657,2089],{"class":1866},[61,75659,75660,75663],{"class":1731,"line":65648},[61,75661,75662],{"class":3766},"lower_error = y - summary_df",[61,75664,75665],{"class":1866},"[\"CI 95% low\"]\n",[61,75667,75668,75671,75674],{"class":1731,"line":65659},[61,75669,75670],{"class":3766},"upper_error = summary_df",[61,75672,75673],{"class":1866},"[\"CI 95% high\"]",[61,75675,75676],{"class":3766}," - y\n",[61,75678,75679,75682],{"class":1731,"line":65668},[61,75680,75681],{"class":3766},"yerr = np.vstack",[61,75683,75684],{"class":1866},"([np.abs(lower_error), np.abs(upper_error)])\n",[61,75686,75687],{"class":1731,"line":65679},[61,75688,1743],{"emptyLinePlaceholder":1742},[61,75690,75691,75694,75696,75699,75702,75705],{"class":1731,"line":65684},[61,75692,75693],{"class":3766},"ax1.bar",[61,75695,849],{"class":1866},[61,75697,75698],{"class":3766},"summary_df",[61,75700,75701],{"class":1866},"[\"Model\"]",[61,75703,75704],{"class":3766},", y, yerr=yerr, capsize=5, color='skyblue', edgecolor='black', ecolor='black'",[61,75706,2016],{"class":1866},[61,75708,75709,75712,75714,75717,75719,75722,75724,75727],{"class":1731,"line":65689},[61,75710,75711],{"class":3766},"ax1.set_title",[61,75713,849],{"class":1866},[61,75715,75716],{"class":3766},"\"Observed Success Rate ",[61,75718,849],{"class":1866},[61,75720,75721],{"class":3766},"95% Wilson CI",[61,75723,300],{"class":1866},[61,75725,75726],{"class":3766},"\", fontsize=14",[61,75728,2016],{"class":1866},[61,75730,75731,75734,75736,75739],{"class":1731,"line":65699},[61,75732,75733],{"class":3766},"ax1.tick_params",[61,75735,849],{"class":1866},[61,75737,75738],{"class":3766},"axis='x', rotation=30, labelsize=10",[61,75740,2016],{"class":1866},[61,75742,75743],{"class":1731,"line":65705},[61,75744,1743],{"emptyLinePlaceholder":1742},[61,75746,75747],{"class":1731,"line":65719},[61,75748,75749],{"class":1735},"# Plot 2\n",[61,75751,75752],{"class":1731,"line":65724},[61,75753,75754],{"class":3766},"width = 0.2\n",[61,75756,75757,75760,75762,75764,75766,75769],{"class":1731,"line":65734},[61,75758,75759],{"class":3766},"x = np.arange",[61,75761,849],{"class":1866},[61,75763,5742],{"class":3766},[61,75765,849],{"class":1866},[61,75767,75768],{"class":3766},"MODELS",[61,75770,5546],{"class":1866},[61,75772,75773,75776],{"class":1731,"line":65739},[61,75774,75775],{"class":3766},"colors = ",[61,75777,75778],{"class":1866},"['#ff9999','#66b3ff','#99ff99']\n",[61,75780,75781,75784,75786,75789,75791],{"class":1731,"line":65751},[61,75782,75783],{"class":3766},"for i, eps in enumerate",[61,75785,849],{"class":1866},[61,75787,75788],{"class":3766},"EPSILONS",[61,75790,300],{"class":1866},[61,75792,1850],{"class":3766},[61,75794,75795,75798],{"class":1731,"line":65756},[61,75796,75797],{"class":3766},"    req = summary_df",[61,75799,75800],{"class":1866},"[f\"Shots for ±{eps:.1%}\"]\n",[61,75802,75803,75806,75808,75810,75812,75814,75816,75819,75822,75825],{"class":1731,"line":65761},[61,75804,75805],{"class":3766},"    ax2.bar",[61,75807,849],{"class":1866},[61,75809,5730],{"class":3766},[61,75811,2419],{"class":1866},[61,75813,67653],{"class":3766},[61,75815,2201],{"class":1866},[61,75817,75818],{"class":3766}," width, req, width, label=f\"±{eps:.1%}\", color=colors",[61,75820,75821],{"class":1866},"[i]",[61,75823,75824],{"class":3766},", edgecolor='black'",[61,75826,2016],{"class":1866},[61,75828,75829],{"class":1731,"line":65783},[61,75830,1743],{"emptyLinePlaceholder":1742},[61,75832,75833,75836,75838,75840,75842,75845],{"class":1731,"line":65788},[61,75834,75835],{"class":3766},"ax2.set_xticks",[61,75837,849],{"class":1866},[61,75839,5730],{"class":3766},[61,75841,2419],{"class":1866},[61,75843,75844],{"class":3766}," width, summary_df",[61,75846,75847],{"class":1866},"[\"Model\"])\n",[61,75849,75850,75853,75855,75858,75860,75863,75865,75867],{"class":1731,"line":65811},[61,75851,75852],{"class":3766},"ax2.set_ylabel",[61,75854,849],{"class":1866},[61,75856,75857],{"class":3766},"\"Required Shots ",[61,75859,849],{"class":1866},[61,75861,75862],{"class":3766},"Log Scale",[61,75864,300],{"class":1866},[61,75866,26088],{"class":3766},[61,75868,2016],{"class":1866},[61,75870,75871,75874,75876,75878],{"class":1731,"line":65822},[61,75872,75873],{"class":3766},"ax2.set_yscale",[61,75875,849],{"class":1866},[61,75877,74468],{"class":3766},[61,75879,2016],{"class":1866},[61,75881,75882,75885,75887,75890],{"class":1731,"line":65861},[61,75883,75884],{"class":3766},"ax2.set_title",[61,75886,849],{"class":1866},[61,75888,75889],{"class":3766},"\"Shots Needed for ±ε Tolerance\", fontsize=14",[61,75891,2016],{"class":1866},[61,75893,75894,75897,75899,75901],{"class":1731,"line":65871},[61,75895,75896],{"class":3766},"ax2.tick_params",[61,75898,849],{"class":1866},[61,75900,75738],{"class":3766},[61,75902,2016],{"class":1866},[61,75904,75905,75908,75910,75913],{"class":1731,"line":65881},[61,75906,75907],{"class":3766},"ax2.legend",[61,75909,849],{"class":1866},[61,75911,75912],{"class":3766},"title=\"Tolerance\"",[61,75914,2016],{"class":1866},[61,75916,75917],{"class":1731,"line":65900},[61,75918,1743],{"emptyLinePlaceholder":1742},[61,75920,75921,75924],{"class":1731,"line":65918},[61,75922,75923],{"class":3766},"plt.tight_layout",[61,75925,2089],{"class":1866},[61,75927,75928,75931],{"class":1731,"line":65923},[61,75929,75930],{"class":3766},"plt.show",[61,75932,2089],{"class":1866},[61,75934,75935],{"class":1731,"line":65928},[61,75936,36393],{"class":3766},[61,75938,75939],{"class":1731,"line":65940},[61,75940,1743],{"emptyLinePlaceholder":1742},[61,75942,75943,75946,75948,75951,75954],{"class":1731,"line":65950},[61,75944,75945],{"class":1753},"nb ",[61,75947,2000],{"class":1859},[61,75949,75950],{"class":1753}," nbf.v4.",[61,75952,75953],{"class":1836},"new_notebook",[61,75955,2089],{"class":1753},[61,75957,75958,75961,75964,75966,75968],{"class":1731,"line":65960},[61,75959,75960],{"class":1753},"nb[",[61,75962,75963],{"class":2165},"\"cells\"",[61,75965,1997],{"class":1753},[61,75967,2000],{"class":1859},[61,75969,24292],{"class":1753},[61,75971,75972,75975,75978],{"class":1731,"line":65965},[61,75973,75974],{"class":1753},"    nbf.v4.",[61,75976,75977],{"class":1836},"new_markdown_cell",[61,75979,75980],{"class":1753},"(md_intro),\n",[61,75982,75983,75985,75988],{"class":1731,"line":65970},[61,75984,75974],{"class":1753},[61,75986,75987],{"class":1836},"new_code_cell",[61,75989,75990],{"class":1753},"(code_config),\n",[61,75992,75993,75995,75997],{"class":1731,"line":65978},[61,75994,75974],{"class":1753},[61,75996,75987],{"class":1836},[61,75998,75999],{"class":1753},"(code_lib_and_run),\n",[61,76001,76002],{"class":1731,"line":65988},[61,76003,5701],{"class":1753},[61,76005,76006],{"class":1731,"line":66003},[61,76007,1743],{"emptyLinePlaceholder":1742},[61,76009,76010,76013,76015],{"class":1731,"line":66014},[61,76011,76012],{"class":1753},"nb_path ",[61,76014,2000],{"class":1859},[61,76016,76017],{"class":2165}," \"Sampling_Budget_Qubit_Bitstrings.ipynb\"\n",[61,76019,76020,76022,76024,76027,76029,76031,76034,76036,76039,76041,76043],{"class":1731,"line":66026},[61,76021,73726],{"class":1749},[61,76023,73729],{"class":1859},[61,76025,76026],{"class":1753},"(nb_path, ",[61,76028,73739],{"class":2165},[61,76030,1435],{"class":1753},[61,76032,76033],{"class":3402},"encoding",[61,76035,2000],{"class":1859},[61,76037,76038],{"class":2165},"\"utf-8\"",[61,76040,2624],{"class":1753},[61,76042,1792],{"class":1749},[61,76044,73746],{"class":1753},[61,76046,76047,76050,76053],{"class":1731,"line":66031},[61,76048,76049],{"class":1753},"    nbf.",[61,76051,76052],{"class":1836},"write",[61,76054,76055],{"class":1753},"(nb, f)\n",[61,76057,76058],{"class":1731,"line":66038},[61,76059,1743],{"emptyLinePlaceholder":1742},[61,76061,76062],{"class":1731,"line":66047},[61,76063,76064],{"class":1735},"# ---------- Final confirmation ----------\n",[61,76066,76067,76069,76071,76073,76075,76077,76079,76082,76084,76086],{"class":1731,"line":66061},[61,76068,40848],{"class":1859},[61,76070,849],{"class":1753},[61,76072,26088],{"class":2165},[61,76074,39321],{"class":1859},[61,76076,26088],{"class":2165},[61,76078,4990],{"class":1859},[61,76080,76081],{"class":2165}," \"=\"",[61,76083,2201],{"class":1859},[61,76085,53228],{"class":1866},[61,76087,2016],{"class":1753},[61,76089,76090,76092,76094,76097],{"class":1731,"line":66072},[61,76091,40848],{"class":1859},[61,76093,849],{"class":1753},[61,76095,76096],{"class":2165},"\"SCRIPT COMPLETE\"",[61,76098,2016],{"class":1753},[61,76100,76101,76103,76105,76108,76110,76112],{"class":1731,"line":66084},[61,76102,40848],{"class":1859},[61,76104,849],{"class":1753},[61,76106,76107],{"class":2165},"\"=\"",[61,76109,2201],{"class":1859},[61,76111,53228],{"class":1866},[61,76113,2016],{"class":1753},[61,76115,76116,76118,76120,76122,76125,76127,76130,76132,76134],{"class":1731,"line":66089},[61,76117,40848],{"class":1859},[61,76119,849],{"class":1753},[61,76121,19797],{"class":1749},[61,76123,76124],{"class":2165},"\"Google Colab notebook saved to: ",[61,76126,3688],{"class":1866},[61,76128,76129],{"class":1753},"nb_path",[61,76131,3697],{"class":1866},[61,76133,26088],{"class":2165},[61,76135,2016],{"class":1753},[61,76137,76138,76140,76142,76144,76147,76149,76152,76154,76156],{"class":1731,"line":66094},[61,76139,40848],{"class":1859},[61,76141,849],{"class":1753},[61,76143,19797],{"class":1749},[61,76145,76146],{"class":2165},"\"Summary data saved to: ",[61,76148,3688],{"class":1866},[61,76150,76151],{"class":1753},"summary_csv_path",[61,76153,3697],{"class":1866},[61,76155,26088],{"class":2165},[61,76157,2016],{"class":1753},[61,76159,76160,76162,76164,76166,76169,76171,76174,76176,76178,76180,76183,76185,76187],{"class":1731,"line":66104},[61,76161,40848],{"class":1859},[61,76163,849],{"class":1753},[61,76165,19797],{"class":1749},[61,76167,76168],{"class":2165},"\"Plots saved to: ",[61,76170,3688],{"class":1866},[61,76172,76173],{"class":1753},"fig1_path",[61,76175,3697],{"class":1866},[61,76177,1435],{"class":2165},[61,76179,3688],{"class":1866},[61,76181,76182],{"class":1753},"fig2_path",[61,76184,3697],{"class":1866},[61,76186,26088],{"class":2165},[61,76188,2016],{"class":1753},[1720,76190,76193],{"className":76191,"code":76192,"language":29075,"meta":1725},[36200],"Running simulation...\nSaved data to CSV and JSON files.\n",[1727,76194,76192],{"__ignoreMap":1725},[4797,76196],{"alt":62612,"src":76197},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-01.webp",[4797,76199],{"alt":67440,"src":76200},"\u002F_content\u002Fimages\u002Fsampling-budget-qubit-classifications\u002Foutput-02.webp",[1720,76202,76205],{"className":76203,"code":76204,"language":29075,"meta":1725},[36200],"Generated and saved plots.\n",[1727,76206,76204],{"__ignoreMap":1725},[1720,76208,76211],{"className":76209,"code":76210,"language":29075,"meta":1725},[36200],"==================================================\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",[1727,76212,76210],{"__ignoreMap":1725},[1632,76214],{},[25,76216,1638],{"id":1637},[76218,76219,76220,76227,76234,76241,76248,76255,76262,76269,76276,76283,76290,76297,76304,76315,76322,76329],"ol",{},[1643,76221,76222,76223,76226],{},"Agresti, A. and Coull, B.A. (1998) 'Approximate is better than “exact” for interval estimation of binomial proportions', ",[15,76224,76225],{},"The American Statistician",", 52(2), pp. 119-126.",[1643,76228,76229,76230,76233],{},"Brown, L.D., Cai, T.T. and DasGupta, A. (2001) 'Interval Estimation for a Binomial Proportion', ",[15,76231,76232],{},"Statistical Science",", 16(2), pp. 101-133.",[1643,76235,76236,76237,76240],{},"Cochran, W.G. (1977) ",[15,76238,76239],{},"Sampling Techniques",". 3rd edn. New York: John Wiley & Sons.",[1643,76242,76243,76244,76247],{},"Gottesman, D. (2009) 'An introduction to quantum error correction and fault-tolerant quantum computation', in ",[15,76245,76246],{},"Quantum Information Science and Its Contributions to Mathematics, Proceedings of Symposia in Applied Mathematics",", 72, pp. 13-58.",[1643,76249,76250,76251,76254],{},"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', ",[15,76252,76253],{},"Nature",", 585, pp. 357–362.",[1643,76256,76257,76258,76261],{},"Hunter, J.D. (2007) 'Matplotlib: A 2D Graphics Environment', ",[15,76259,76260],{},"Computing in Science & Engineering",", 9(3), pp. 90-95.",[1643,76263,76264,76265,76268],{},"Kitaev, A.Y. (2003) 'Fault-tolerant quantum computation by anyons', ",[15,76266,76267],{},"Annals of Physics",", 303(1), pp. 2-30.",[1643,76270,76271,76272,76275],{},"McKinney, W. (2010) 'Data Structures for Statistical Computing in Python', in ",[15,76273,76274],{},"Proceedings of the 9th Python in Science Conference",", pp. 56-61.",[1643,76277,76278,76279,76282],{},"Nayak, C., Simon, S.H., Stern, A., Freedman, M. and Das Sarma, S. (2008) 'Non-Abelian anyons and topological quantum computation', ",[15,76280,76281],{},"Reviews of Modern Physics",", 80(3), pp. 1083-1159.",[1643,76284,76285,76286,76289],{},"Newman, M. (2013) ",[15,76287,76288],{},"Computational Physics",". CreateSpace Independent Publishing Platform.",[1643,76291,76292,76293,76296],{},"Nielsen, M.A. and Chuang, I.L. (2010) ",[15,76294,76295],{},"Quantum Computation and Quantum Information: 10th Anniversary Edition",". Cambridge: Cambridge University Press.",[1643,76298,76299,76300,76303],{},"Preskill, J. (2018) 'Quantum Computing in the NISQ era and beyond', ",[15,76301,76302],{},"Quantum",", 2, p. 79.",[1643,76305,76306,76307,76310,76311,332],{},"The pandas development team (2020) ",[15,76308,76309],{},"pandas-dev\u002Fpandas: Pandas",". Zenodo. Available at: ",[19,76312,76314],{"href":76313},"https:\u002F\u002Fwww.google.com\u002Fsearch?q=https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134","https:\u002F\u002Fdoi.org\u002F10.5281\u002Fzenodo.3509134",[1643,76316,76317,76318,76321],{},"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', ",[15,76319,76320],{},"Nature Methods",", 17, pp. 261–272.",[1643,76323,76324,76325,76328],{},"Wilde, M.M. (2017) ",[15,76326,76327],{},"Quantum Information Theory",". 2nd edn. Cambridge: Cambridge University Press.",[1643,76330,76331,76332,76335],{},"Wilson, E.B. (1927) 'Probable inference, the law of succession, and statistical inference', ",[15,76333,76334],{},"Journal of the American Statistical Association",", 22(158), pp. 209-212.",[25,76337,71730],{"id":71729},[12,76339,71733,76340,71736,76342,332],{},[19,76341,22],{"href":71792},[19,76343,71740],{"href":76344},"https:\u002F\u002Fcolab.research.google.com\u002Fgithub\u002FOJB-Quantum\u002FNotebooks-for-Ideas\u002Fblob\u002Fmain\u002FSampling_Budget_Analogy_for_Qubit_Classifications.ipynb",[7514,76346,76347],{},"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":1725,"searchDepth":1739,"depth":1739,"links":76349},[76350,76351],{"id":1637,"depth":1739,"text":1638},{"id":71729,"depth":1739,"text":71730},[7523,7524,7525,76353],"Sampling budget",[76355],{"username":7,"name":7529,"role":7530,"bio":7531,"links":76356},[76357,76358],{"label":7534,"href":7535},{"label":7537,"href":71792},{"username":7,"name":7529,"role":7539},"A Sampling-Budget & Bit-String Analogy for Qubit Classifications. 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