2075:
1668:
2070:{\displaystyle {\begin{aligned}\langle 3\rangle &=\langle 3\rangle _{\mathrm {S} }+\langle 1\rangle \ \Delta \langle 2\rangle +\Delta \langle 3\rangle \,,\\\langle N\rangle &=\langle N\rangle _{\mathrm {S} }\\&\quad +\langle N-2\rangle _{\mathrm {S} }\ \Delta \langle 2\rangle \\&\quad +\langle N-4\rangle _{\mathrm {S} }\ \Delta \langle 2\rangle \ \Delta \langle 2\rangle +\dots \\&\quad +\langle N-3\rangle _{\mathrm {S} }\ \Delta \langle 3\rangle \\&\quad +\langle N-5\rangle _{\mathrm {S} }\ \Delta \langle 3\rangle \ \Delta \langle 2\rangle +\dots \\&\quad +\Delta \langle N\rangle \,,\end{aligned}}}
144:, the many-body wavefunction has an overwhelmingly complicated structure such that the direct wave-function-solution techniques are intractable. The cluster expansion is a variant of the coupled-clusters approach and it solves the dynamical equations of correlations instead of attempting to solve the quantum dynamics of an approximated wavefunction or density matrix. It is equally well suited to treat properties of many-body systems and quantum-optical correlations, which has made it very suitable approach for
1161:
737:
480:
2414:
847:
while the explicit momentum indices are suppressed for the sake of briefness. These quantities are normally ordered, which means that all creation operators are on the left-hand side while all annihilation operators are on the right-hand side in the expectation value. It is straight forward to show
2458:
However, as a major difference to a direct expectation-value approach, both many-body and quantum-optical interactions generate correlations sequentially. In several relevant problems, one indeed has a situation where only the lowest-order clusters are initially nonvanishing while the higher-order
2778:
This completely mathematical problem has a direct physical application. One can apply the cluster-expansion transformation to robustly project classical measurement into a quantum-optical measurement. This property is largely based on CET's ability to describe any distribution in the form where a
1164:
Schematic representation of the cluster-expansion-based classification. The full correlation is composed of singlets, doublets, triplets, and higher-order correlations, all uniquely defined by the cluster-expansion approach. Each blue sphere corresponds to one particle operator and yellow
2454:
among clusters. Obviously, introducing clusters cannot remove the hierarchy problem of the direct approach because the hierarchical contributions remains in the dynamics. This property and the appearance of the nonlinear terms seem to suggest complications for the applicability of the
1169:
The hierarchy problem can be systematically truncated after identifying correlated clusters. The simplest definitions follow after one identifies the clusters recursively. At the lowest level, one finds the class of single-particle expectation values (singlets) that are symbolized by
732:{\displaystyle \langle {\hat {N}}\rangle \equiv \langle {\hat {B}}_{1}^{\dagger }\cdots {\hat {B}}_{K}^{\dagger }\ {\hat {a}}_{1}^{\dagger }\cdots {\hat {a}}_{N_{\hat {a}}}^{\dagger }{\hat {a}}_{N_{\hat {a}}}\cdots {\hat {a}}_{1}\ {\hat {B}}_{J}\cdots {\hat {B}}_{1}\rangle }
1070:
1661:
1577:
2775:, defined by the singlet–doublet contributions, multiplied by a polynomial, defined by the higher-order clusters. It turns out that this formulation provides extreme convergence in representation-to-representation transformations.
2174:
1276:
2589:
Besides describing quantum dynamics, one can naturally apply the cluster-expansion approach to represent the quantum distributions. One possibility is to represent the quantum fluctuations of a quantized light mode
1423:
350:
2079:
where each product term represents one factorization symbolically and implicitly includes a sum over all factorizations within the class of terms identified. The purely correlated part is denoted by
936:
2169:
As this identification is applied recursively, one may directly identify which correlations appear in the hierarchy problem. One then determines the quantum dynamics of the correlations, yielding
206:
1673:
2765:
2691:
394:
1513:
1337:
2581:
is typically much smaller than the overall particle number, the cluster-expansion approach yields a pragmatic and systematic solution scheme for many-body and quantum-optics investigations.
799:
244:
2164:
2135:
2106:
2519:
2452:
419:
269:
1478:
1452:
1302:
1220:
1194:
845:
1150:
2617:
439:
1599:
1152:. Since all levels of expectation values can be nonzero, up to the actual particle number, this equation cannot be directly truncated without further considerations.
137:
that mainly operates with many-body wavefunctions. The coupled-clusters approach is one of the most successful methods to solve quantum states of complex molecules.
925:
1604:
1520:
2579:
2559:
2539:
1357:
1095:
893:
873:
293:
2409:{\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}\Delta \langle {\hat {N}}\rangle =\mathrm {T} \left+\mathrm {NL} \left+\mathrm {Hi} \left\,,}
2619:
in terms of clusters, yielding the cluster-expansion representation. Alternatively, one can express them in terms of the expectation-value representation
1225:
2811:
52:
163:
3066:
3047:
2965:
2945:
2868:
2848:
1378:
31:
problem that arises when quantum dynamics of interacting systems is solved. This method is well suited for producing a closed set of
306:
1065:{\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}\langle {\hat {N}}\rangle =\mathrm {T} \left+\mathrm {Hi} \left}
3014:
Kira, M.; Koch, S. W.; Smith, R. P.; Hunter, A. E.; Cundiff, S. T. (2011). "Quantum spectroscopy with Schrödinger-cat states".
2767:
to the density matrix is unique but can result in a numerically diverging series. This problem can be solved by introducing a
117:
The idea of cumulants was converted into quantum physics by Fritz
Coester and Hermann Kümmel with the intention of studying
2806:
168:
48:
2801:
2780:
2696:
2622:
355:
2768:
1483:
1307:
1097:
symbolizes contributions without hierarchy problem and the functional for hierarchical (Hi) coupling is symbolized by
299:
nature of the quantity. When the many-body state consists of electronic excitations of matter, it is fully defined by
3085:
145:
44:
848:
that this expectation value vanishes if the amount of
Fermion creation and annihilation operators are not equal.
744:
450:
133:
in order to describe many-body phenomena in complex atoms and molecules. This work introduced the basis for the
1075:
211:
2541:-particle clusters. As a result, the equations become closed and one only needs to compute the dynamics up to
2140:
2111:
2082:
1364:
1278:
that contains a formal sum over all possible products of single-particle expectation values. More generally,
79:
63:
1165:
circles/ellipses to correlations. The number of spheres within a correlation identifies the cluster number.
2779:
Gaussian is multiplied by a polynomial factor. This technique is already being used to access and derive
2462:
2421:
399:
249:
1457:
1431:
1281:
1199:
1173:
804:
2906:
2887:
1372:
1100:
852:
156:
465:
When the many-body system is studied together with its quantum-optical properties, all measurable
1425:. The singlet factorization constitutes the first level of the cluster-expansion representation.
296:
114:, and so on, that identify the distribution with increasing accuracy as more cumulants are used.
87:
2976:
Mootz, M.; Kira, M.; Koch, S. W. (2012). "Sequential build-up of quantum-optical correlations".
2593:
122:
3003:
2925:
931:. More mathematically, all particles interact with each other leading to an equation structure
3062:
3043:
2961:
2941:
2864:
2844:
875:-particle operator. However, the many-body as well as quantum-optical interactions couple the
466:
442:
424:
152:
130:
24:
2783:
from a set of classical spectroscopy measurements, which can be performed using high-quality
1584:
94:
that describe probabilistic distributions with as few quantities as possible; he called them
1160:
1656:{\displaystyle \Delta \langle 2\rangle =\langle 2\rangle -\langle 2\rangle _{\mathrm {S} }}
1572:{\displaystyle \langle 2\rangle =\langle 2\rangle _{\mathrm {S} }+\Delta \langle 2\rangle }
898:
2796:
134:
118:
28:
2984:
82:. Conceptually, there is always, at least formally, probability distribution behind each
2561:-particle correlations in order to explain the relevant properties of the system. Since
2564:
2544:
2524:
1342:
1080:
878:
858:
446:
278:
159:
to describe the physics involved. For example, a light field is then described through
75:
40:
2898:
Coester, F.; Kümmel, H. (1960). "Short-range correlations in nuclear wave functions".
86:
that is measured. Already in 1889, a long time before quantum physics was formulated,
3079:
2459:
clusters build up slowly. In this situation, one can omit the hierarchical coupling,
67:
1271:{\displaystyle \langle 2\rangle _{\mathrm {S} }=\langle 1\rangle \langle 1\rangle }
126:
71:
2995:
Kira, M.; Koch, S. (2008). "Cluster-expansion representation in quantum optics".
32:
83:
3022:
2917:
Kira, M.; Koch, S. (2006). "Quantum-optical spectroscopy of semiconductors".
91:
36:
3059:
Many-Body
Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory
2958:
Many-Body
Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory
1375:
where Boson operators are formally replaced by a coherent amplitude, i.e.,
2772:
1360:
1359:-particle expectation value. Physically, the singlet factorization among
111:
107:
103:
300:
35:
computable equations that can be applied to analyze a great variety of
2784:
1368:
272:
141:
1663:. The next levels of identifications follow recursively by applying
1159:
160:
1418:{\displaystyle {\hat {B}}\rightarrow \langle {\hat {B}}\rangle }
99:
2879:
Coester, F. (1958). "Bound states of a many-particle system".
929:
Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy problem
345:{\displaystyle {\hat {a}}_{\lambda ,\mathbf {k} }^{\dagger }}
2137:
determine doublets while the three-particle correlations
2938:
Statistische Physik: Gleichgewichtstheorie und
Kinetik
2418:
where the factorizations produce a nonlinear coupling
851:
155:
or quantum optics, it is most convenient to apply the
98:. The cumulants form a sequence of quantities such as
66:
essentially replaces classically accurate values by a
2771:(CET) that represents the distribution in terms of a
2699:
2625:
2596:
2567:
2547:
2527:
2465:
2424:
2177:
2143:
2114:
2085:
1671:
1607:
1587:
1523:
1486:
1460:
1434:
1381:
1345:
1310:
1284:
1228:
1202:
1176:
1103:
1083:
939:
901:
881:
861:
807:
747:
483:
427:
402:
358:
309:
281:
252:
214:
171:
927:-particle expectation values, which is known as the
201:{\displaystyle {\hat {B}}_{\mathbf {q} }^{\dagger }}
70:distribution that can be formulated using, e.g., a
2760:{\displaystyle \langle ^{J}{\hat {B}}^{K}\rangle }
2759:
2686:{\displaystyle \langle ^{J}{\hat {B}}^{K}\rangle }
2685:
2611:
2573:
2553:
2533:
2513:
2446:
2408:
2158:
2129:
2100:
2069:
1655:
1593:
1571:
1507:
1472:
1446:
1417:
1351:
1331:
1296:
1270:
1214:
1188:
1144:
1089:
1064:
919:
887:
867:
839:
793:
731:
433:
413:
389:{\displaystyle {\hat {a}}_{\lambda ,\mathbf {k} }}
388:
344:
287:
263:
238:
200:
1601:contribution denotes the correlated part, i.e.,
1508:{\displaystyle \langle 2\rangle _{\mathrm {S} }}
1332:{\displaystyle \langle N\rangle _{\mathrm {S} }}
43:problems. For example, it is widely applied in
855:of motion to generate the dynamics of a given
8:
2754:
2700:
2680:
2626:
2503:
2482:
2394:
2373:
2346:
2331:
2316:
2301:
2292:
2277:
2253:
2238:
2219:
2204:
2153:
2147:
2124:
2118:
2108:. From these, the two-particle correlations
2095:
2089:
2056:
2050:
2027:
2021:
2012:
2006:
1989:
1976:
1962:
1956:
1939:
1926:
1906:
1900:
1891:
1885:
1868:
1855:
1841:
1835:
1818:
1805:
1783:
1776:
1766:
1760:
1749:
1743:
1734:
1728:
1719:
1713:
1699:
1692:
1682:
1676:
1642:
1635:
1629:
1623:
1617:
1611:
1566:
1560:
1543:
1536:
1530:
1524:
1494:
1487:
1467:
1461:
1441:
1435:
1412:
1397:
1318:
1311:
1291:
1285:
1265:
1259:
1256:
1250:
1236:
1229:
1209:
1203:
1183:
1177:
1136:
1115:
1054:
1033:
1009:
994:
978:
963:
726:
505:
499:
484:
2978:Journal of the Optical Society of America B
794:{\displaystyle N=N_{\hat {B}}+N_{\hat {a}}}
2835:
2833:
2831:
2829:
2827:
2748:
2737:
2736:
2729:
2719:
2708:
2707:
2698:
2674:
2663:
2662:
2655:
2645:
2634:
2633:
2624:
2598:
2597:
2595:
2566:
2546:
2526:
2486:
2485:
2466:
2464:
2425:
2423:
2402:
2377:
2376:
2357:
2335:
2334:
2305:
2304:
2281:
2280:
2264:
2242:
2241:
2225:
2208:
2207:
2186:
2178:
2176:
2142:
2113:
2084:
2059:
1993:
1992:
1943:
1942:
1872:
1871:
1822:
1821:
1787:
1786:
1752:
1703:
1702:
1672:
1670:
1646:
1645:
1606:
1586:
1547:
1546:
1522:
1498:
1497:
1485:
1459:
1433:
1401:
1400:
1383:
1382:
1380:
1344:
1322:
1321:
1309:
1283:
1240:
1239:
1227:
1201:
1175:
1119:
1118:
1104:
1102:
1082:
1037:
1036:
1020:
998:
997:
984:
967:
966:
948:
940:
938:
900:
880:
860:
813:
812:
806:
779:
778:
759:
758:
746:
720:
709:
708:
698:
687:
686:
676:
665:
664:
646:
645:
640:
629:
628:
621:
608:
607:
602:
591:
590:
580:
575:
564:
563:
553:
548:
537:
536:
526:
521:
510:
509:
488:
487:
482:
426:
406:
401:
379:
372:
361:
360:
357:
336:
330:
323:
312:
311:
308:
280:
256:
251:
239:{\displaystyle {\hat {B}}_{\mathbf {q} }}
229:
228:
217:
216:
213:
192:
186:
185:
174:
173:
170:
2159:{\displaystyle \Delta \langle 3\rangle }
2130:{\displaystyle \Delta \langle 2\rangle }
2101:{\displaystyle \Delta \langle N\rangle }
47:and it can be applied to generalize the
16:Quantum mechanical calculation technique
2823:
2183:
945:
403:
253:
3057:Shavitt, I.; Bartlett, R. J. (2009).
1454:is then the difference of the actual
1222:can be approximated by factorization
1196:. Any two-particle expectation value
303:creation and annihilation operators
7:
2812:Semiconductor luminescence equations
2693:. In this case, the connection from
421:refers to particle's momentum while
53:semiconductor luminescence equations
1339:is the singlet factorization of an
469:can be expressed in the form of an
164:creation and annihilation operators
2514:{\displaystyle \mathrm {Hi} \left}
2479:
2470:
2467:
2447:{\displaystyle \mathrm {NL} \left}
2429:
2426:
2370:
2361:
2358:
2328:
2298:
2268:
2265:
2235:
2226:
2201:
2192:
2188:
2179:
2144:
2115:
2086:
2047:
2018:
2003:
1994:
1953:
1944:
1897:
1882:
1873:
1832:
1823:
1788:
1740:
1725:
1704:
1647:
1608:
1588:
1557:
1548:
1499:
1323:
1241:
1108:
1105:
1024:
1021:
985:
954:
950:
941:
414:{\displaystyle \hbar \mathbf {k} }
264:{\displaystyle \hbar \mathbf {q} }
27:that systematically truncates the
14:
1515:. More mathematically, one finds
1473:{\displaystyle \langle 2\rangle }
1447:{\displaystyle \langle 2\rangle }
1297:{\displaystyle \langle 1\rangle }
1215:{\displaystyle \langle 2\rangle }
1189:{\displaystyle \langle 1\rangle }
2907:doi:10.1016/0029-5582(60)90140-1
2888:doi:10.1016/0029-5582(58)90280-3
2769:cluster-expansion transformation
1156:Recursive definition of clusters
840:{\displaystyle N_{\hat {B}}=J+K}
407:
380:
331:
257:
230:
187:
2043:
1972:
1922:
1851:
1801:
3061:. Cambridge University Press.
3042:. Cambridge University Press.
3038:Kira, M.; Koch, S. W. (2011).
3004:doi:10.1103/PhysRevA.78.022102
2960:. Cambridge University Press.
2926:doi:10.1103/PhysRevA.73.013813
2843:. Cambridge University Press.
2839:Kira, M.; Koch, S. W. (2011).
2742:
2726:
2713:
2703:
2668:
2652:
2639:
2629:
2603:
2491:
2382:
2340:
2310:
2286:
2247:
2213:
1480:and the singlet factorization
1406:
1394:
1388:
1139:
1124:
1112:
1042:
1003:
972:
914:
902:
818:
784:
764:
714:
692:
670:
651:
634:
613:
596:
569:
542:
515:
493:
366:
317:
222:
179:
1:
2861:Thiele: Pioneer in Statistics
2807:Semiconductor Bloch equations
1145:{\displaystyle \mathrm {Hi} }
157:second-quantization formalism
49:semiconductor Bloch equations
3040:Semiconductor Quantum Optics
2841:Semiconductor Quantum Optics
2802:Quantum-optical spectroscopy
2781:quantum-optical spectroscopy
2455:cluster-expansion approach.
146:semiconductor quantum optics
121:many-body phenomena. Later,
45:semiconductor quantum optics
2985:doi:10.1364/JOSAB.29.000A17
474:-particle expectation value
3102:
2612:{\displaystyle {\hat {B}}}
1365:Hartree–Fock approximation
271:defines the momentum of a
129:extended the approach for
21:cluster-expansion approach
2859:Lauritzen, S. L. (2002).
2521:, at the level exceeding
1304:defines the singlets and
2956:Bartlett, R. J. (2009).
895:-particle quantities to
434:{\displaystyle \lambda }
135:coupled-cluster approach
80:phase-space distribution
1594:{\displaystyle \Delta }
1428:The correlated part of
1373:classical approximation
461:-particle contributions
2863:. Oxford Univ. Press.
2761:
2687:
2613:
2575:
2555:
2535:
2515:
2448:
2410:
2160:
2131:
2102:
2071:
1657:
1595:
1573:
1509:
1474:
1448:
1419:
1353:
1333:
1298:
1272:
1216:
1190:
1166:
1146:
1091:
1066:
921:
889:
869:
841:
795:
733:
435:
415:
396:, respectively, where
390:
346:
289:
265:
246:, respectively, where
240:
202:
151:Like almost always in
3023:doi:10.1038/nphys2091
2762:
2688:
2614:
2576:
2556:
2536:
2516:
2449:
2411:
2166:are called triplets.
2161:
2132:
2103:
2072:
1658:
1596:
1574:
1510:
1475:
1449:
1420:
1354:
1334:
1299:
1273:
1217:
1191:
1163:
1147:
1092:
1067:
922:
920:{\displaystyle (N+1)}
890:
870:
842:
796:
734:
436:
416:
391:
347:
290:
266:
241:
203:
2697:
2623:
2594:
2565:
2545:
2525:
2463:
2422:
2175:
2141:
2112:
2083:
1669:
1605:
1585:
1521:
1484:
1458:
1432:
1379:
1343:
1308:
1282:
1226:
1200:
1174:
1101:
1081:
937:
899:
879:
859:
805:
745:
481:
425:
400:
356:
307:
279:
250:
212:
169:
853:Heisenberg equation
626:
585:
558:
531:
341:
197:
2757:
2683:
2609:
2571:
2551:
2531:
2511:
2444:
2406:
2156:
2127:
2098:
2067:
2065:
1653:
1591:
1569:
1505:
1470:
1444:
1415:
1349:
1329:
1294:
1268:
1212:
1186:
1167:
1142:
1087:
1062:
917:
885:
865:
837:
791:
729:
589:
562:
535:
508:
467:expectation values
457:Classification of
431:
411:
386:
342:
310:
285:
261:
236:
198:
172:
88:Thorvald N. Thiele
23:is a technique in
3086:Quantum mechanics
2997:Physical Review A
2936:Haug, H. (2006).
2919:Physical Review A
2745:
2716:
2671:
2642:
2606:
2574:{\displaystyle C}
2554:{\displaystyle C}
2534:{\displaystyle C}
2494:
2385:
2343:
2313:
2289:
2250:
2216:
2199:
2017:
2002:
1952:
1896:
1881:
1831:
1724:
1409:
1391:
1352:{\displaystyle N}
1127:
1090:{\displaystyle T}
1045:
1006:
975:
961:
888:{\displaystyle N}
868:{\displaystyle N}
821:
787:
767:
717:
695:
684:
673:
654:
637:
616:
599:
572:
561:
545:
518:
496:
443:degree of freedom
441:is some internal
369:
320:
288:{\displaystyle B}
275:. The "hat" over
225:
182:
153:many-body physics
131:quantum chemistry
25:quantum mechanics
3093:
3072:
3053:
3025:
3012:
3006:
2993:
2987:
2974:
2968:
2954:
2948:
2934:
2928:
2915:
2909:
2896:
2890:
2877:
2871:
2857:
2851:
2837:
2766:
2764:
2763:
2758:
2753:
2752:
2747:
2746:
2738:
2734:
2733:
2724:
2723:
2718:
2717:
2709:
2692:
2690:
2689:
2684:
2679:
2678:
2673:
2672:
2664:
2660:
2659:
2650:
2649:
2644:
2643:
2635:
2618:
2616:
2615:
2610:
2608:
2607:
2599:
2580:
2578:
2577:
2572:
2560:
2558:
2557:
2552:
2540:
2538:
2537:
2532:
2520:
2518:
2517:
2512:
2510:
2506:
2496:
2495:
2487:
2473:
2453:
2451:
2450:
2445:
2443:
2432:
2415:
2413:
2412:
2407:
2401:
2397:
2387:
2386:
2378:
2364:
2353:
2349:
2345:
2344:
2336:
2315:
2314:
2306:
2291:
2290:
2282:
2271:
2260:
2256:
2252:
2251:
2243:
2229:
2218:
2217:
2209:
2200:
2198:
2187:
2182:
2165:
2163:
2162:
2157:
2136:
2134:
2133:
2128:
2107:
2105:
2104:
2099:
2076:
2074:
2073:
2068:
2066:
2039:
2015:
2000:
1999:
1998:
1997:
1968:
1950:
1949:
1948:
1947:
1918:
1894:
1879:
1878:
1877:
1876:
1847:
1829:
1828:
1827:
1826:
1797:
1793:
1792:
1791:
1722:
1709:
1708:
1707:
1662:
1660:
1659:
1654:
1652:
1651:
1650:
1600:
1598:
1597:
1592:
1578:
1576:
1575:
1570:
1553:
1552:
1551:
1514:
1512:
1511:
1506:
1504:
1503:
1502:
1479:
1477:
1476:
1471:
1453:
1451:
1450:
1445:
1424:
1422:
1421:
1416:
1411:
1410:
1402:
1393:
1392:
1384:
1358:
1356:
1355:
1350:
1338:
1336:
1335:
1330:
1328:
1327:
1326:
1303:
1301:
1300:
1295:
1277:
1275:
1274:
1269:
1246:
1245:
1244:
1221:
1219:
1218:
1213:
1195:
1193:
1192:
1187:
1151:
1149:
1148:
1143:
1129:
1128:
1120:
1111:
1096:
1094:
1093:
1088:
1071:
1069:
1068:
1063:
1061:
1057:
1047:
1046:
1038:
1027:
1016:
1012:
1008:
1007:
999:
988:
977:
976:
968:
962:
960:
949:
944:
926:
924:
923:
918:
894:
892:
891:
886:
874:
872:
871:
866:
846:
844:
843:
838:
824:
823:
822:
814:
800:
798:
797:
792:
790:
789:
788:
780:
770:
769:
768:
760:
738:
736:
735:
730:
725:
724:
719:
718:
710:
703:
702:
697:
696:
688:
682:
681:
680:
675:
674:
666:
659:
658:
657:
656:
655:
647:
639:
638:
630:
625:
620:
619:
618:
617:
609:
601:
600:
592:
584:
579:
574:
573:
565:
559:
557:
552:
547:
546:
538:
530:
525:
520:
519:
511:
498:
497:
489:
440:
438:
437:
432:
420:
418:
417:
412:
410:
395:
393:
392:
387:
385:
384:
383:
371:
370:
362:
351:
349:
348:
343:
340:
335:
334:
322:
321:
313:
294:
292:
291:
286:
270:
268:
267:
262:
260:
245:
243:
242:
237:
235:
234:
233:
227:
226:
218:
207:
205:
204:
199:
196:
191:
190:
184:
183:
175:
3101:
3100:
3096:
3095:
3094:
3092:
3091:
3090:
3076:
3075:
3069:
3056:
3050:
3037:
3034:
3032:Further reading
3029:
3028:
3021:(10): 799–804.
3013:
3009:
2994:
2990:
2975:
2971:
2955:
2951:
2935:
2931:
2916:
2912:
2900:Nuclear Physics
2897:
2893:
2881:Nuclear Physics
2878:
2874:
2858:
2854:
2838:
2825:
2820:
2797:BBGKY hierarchy
2793:
2735:
2725:
2706:
2695:
2694:
2661:
2651:
2632:
2621:
2620:
2592:
2591:
2587:
2563:
2562:
2543:
2542:
2523:
2522:
2478:
2474:
2461:
2460:
2433:
2420:
2419:
2369:
2365:
2276:
2272:
2234:
2230:
2191:
2173:
2172:
2139:
2138:
2110:
2109:
2081:
2080:
2064:
2063:
2037:
2036:
1988:
1966:
1965:
1938:
1916:
1915:
1867:
1845:
1844:
1817:
1795:
1794:
1782:
1769:
1757:
1756:
1698:
1685:
1667:
1666:
1641:
1603:
1602:
1583:
1582:
1542:
1519:
1518:
1493:
1482:
1481:
1456:
1455:
1430:
1429:
1377:
1376:
1341:
1340:
1317:
1306:
1305:
1280:
1279:
1235:
1224:
1223:
1198:
1197:
1172:
1171:
1158:
1099:
1098:
1079:
1078:
1032:
1028:
993:
989:
953:
935:
934:
897:
896:
877:
876:
857:
856:
808:
803:
802:
774:
754:
743:
742:
707:
685:
663:
641:
627:
603:
479:
478:
463:
423:
422:
398:
397:
359:
354:
353:
305:
304:
277:
276:
248:
247:
215:
210:
209:
167:
166:
96:half-invariants
61:
41:quantum-optical
29:BBGKY hierarchy
17:
12:
11:
5:
3099:
3097:
3089:
3088:
3078:
3077:
3074:
3073:
3068:978-0521818322
3067:
3054:
3049:978-0521875097
3048:
3033:
3030:
3027:
3026:
3016:Nature Physics
3007:
2988:
2969:
2966:978-0521818322
2949:
2946:978-3540256298
2929:
2910:
2891:
2872:
2869:978-0198509721
2852:
2849:978-0521875097
2822:
2821:
2819:
2816:
2815:
2814:
2809:
2804:
2799:
2792:
2789:
2756:
2751:
2744:
2741:
2732:
2728:
2722:
2715:
2712:
2705:
2702:
2682:
2677:
2670:
2667:
2658:
2654:
2648:
2641:
2638:
2631:
2628:
2605:
2602:
2586:
2583:
2570:
2550:
2530:
2509:
2505:
2502:
2499:
2493:
2490:
2484:
2481:
2477:
2472:
2469:
2442:
2439:
2436:
2431:
2428:
2405:
2400:
2396:
2393:
2390:
2384:
2381:
2375:
2372:
2368:
2363:
2360:
2356:
2352:
2348:
2342:
2339:
2333:
2330:
2327:
2324:
2321:
2318:
2312:
2309:
2303:
2300:
2297:
2294:
2288:
2285:
2279:
2275:
2270:
2267:
2263:
2259:
2255:
2249:
2246:
2240:
2237:
2233:
2228:
2224:
2221:
2215:
2212:
2206:
2203:
2197:
2194:
2190:
2185:
2181:
2155:
2152:
2149:
2146:
2126:
2123:
2120:
2117:
2097:
2094:
2091:
2088:
2062:
2058:
2055:
2052:
2049:
2046:
2042:
2040:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2014:
2011:
2008:
2005:
1996:
1991:
1987:
1984:
1981:
1978:
1975:
1971:
1969:
1967:
1964:
1961:
1958:
1955:
1946:
1941:
1937:
1934:
1931:
1928:
1925:
1921:
1919:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1893:
1890:
1887:
1884:
1875:
1870:
1866:
1863:
1860:
1857:
1854:
1850:
1848:
1846:
1843:
1840:
1837:
1834:
1825:
1820:
1816:
1813:
1810:
1807:
1804:
1800:
1798:
1796:
1790:
1785:
1781:
1778:
1775:
1772:
1770:
1768:
1765:
1762:
1759:
1758:
1755:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1721:
1718:
1715:
1712:
1706:
1701:
1697:
1694:
1691:
1688:
1686:
1684:
1681:
1678:
1675:
1674:
1649:
1644:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1590:
1568:
1565:
1562:
1559:
1556:
1550:
1545:
1541:
1538:
1535:
1532:
1529:
1526:
1501:
1496:
1492:
1489:
1469:
1466:
1463:
1443:
1440:
1437:
1414:
1408:
1405:
1399:
1396:
1390:
1387:
1371:it yields the
1348:
1325:
1320:
1316:
1313:
1293:
1290:
1287:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1243:
1238:
1234:
1231:
1211:
1208:
1205:
1185:
1182:
1179:
1157:
1154:
1141:
1138:
1135:
1132:
1126:
1123:
1117:
1114:
1110:
1107:
1086:
1060:
1056:
1053:
1050:
1044:
1041:
1035:
1031:
1026:
1023:
1019:
1015:
1011:
1005:
1002:
996:
992:
987:
983:
980:
974:
971:
965:
959:
956:
952:
947:
943:
916:
913:
910:
907:
904:
884:
864:
836:
833:
830:
827:
820:
817:
811:
786:
783:
777:
773:
766:
763:
757:
753:
750:
728:
723:
716:
713:
706:
701:
694:
691:
679:
672:
669:
662:
653:
650:
644:
636:
633:
624:
615:
612:
606:
598:
595:
588:
583:
578:
571:
568:
556:
551:
544:
541:
534:
529:
524:
517:
514:
507:
504:
501:
495:
492:
486:
462:
455:
430:
409:
405:
382:
378:
375:
368:
365:
339:
333:
329:
326:
319:
316:
295:signifies the
284:
259:
255:
232:
224:
221:
195:
189:
181:
178:
76:density matrix
64:Quantum theory
60:
57:
15:
13:
10:
9:
6:
4:
3:
2:
3098:
3087:
3084:
3083:
3081:
3070:
3064:
3060:
3055:
3051:
3045:
3041:
3036:
3035:
3031:
3024:
3020:
3017:
3011:
3008:
3005:
3001:
2998:
2992:
2989:
2986:
2982:
2979:
2973:
2970:
2967:
2963:
2959:
2953:
2950:
2947:
2943:
2939:
2933:
2930:
2927:
2923:
2920:
2914:
2911:
2908:
2904:
2901:
2895:
2892:
2889:
2885:
2882:
2876:
2873:
2870:
2866:
2862:
2856:
2853:
2850:
2846:
2842:
2836:
2834:
2832:
2830:
2828:
2824:
2817:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2794:
2790:
2788:
2786:
2782:
2776:
2774:
2770:
2749:
2739:
2730:
2720:
2710:
2675:
2665:
2656:
2646:
2636:
2600:
2584:
2582:
2568:
2548:
2528:
2507:
2500:
2497:
2488:
2475:
2456:
2440:
2437:
2434:
2416:
2403:
2398:
2391:
2388:
2379:
2366:
2354:
2350:
2337:
2325:
2322:
2319:
2307:
2295:
2283:
2273:
2261:
2257:
2244:
2231:
2222:
2210:
2195:
2170:
2167:
2150:
2121:
2092:
2077:
2060:
2053:
2044:
2041:
2033:
2030:
2024:
2009:
1985:
1982:
1979:
1973:
1970:
1959:
1935:
1932:
1929:
1923:
1920:
1912:
1909:
1903:
1888:
1864:
1861:
1858:
1852:
1849:
1838:
1814:
1811:
1808:
1802:
1799:
1779:
1773:
1771:
1763:
1753:
1746:
1737:
1731:
1716:
1710:
1695:
1689:
1687:
1679:
1664:
1638:
1632:
1626:
1620:
1614:
1579:
1563:
1554:
1539:
1533:
1527:
1516:
1490:
1464:
1438:
1426:
1403:
1385:
1374:
1370:
1366:
1363:produces the
1362:
1346:
1314:
1288:
1262:
1253:
1247:
1232:
1206:
1180:
1162:
1155:
1153:
1133:
1130:
1121:
1084:
1077:
1072:
1058:
1051:
1048:
1039:
1029:
1017:
1013:
1000:
990:
981:
969:
957:
932:
930:
911:
908:
905:
882:
862:
854:
849:
834:
831:
828:
825:
815:
809:
781:
775:
771:
761:
755:
751:
748:
739:
721:
711:
704:
699:
689:
677:
667:
660:
648:
642:
631:
622:
610:
604:
593:
586:
581:
576:
566:
554:
549:
539:
532:
527:
522:
512:
502:
490:
476:
475:
473:
468:
460:
456:
454:
452:
448:
444:
428:
376:
373:
363:
337:
327:
324:
314:
302:
298:
282:
274:
219:
193:
176:
165:
162:
158:
154:
149:
147:
143:
138:
136:
132:
128:
124:
120:
115:
113:
109:
105:
101:
97:
93:
90:proposed the
89:
85:
81:
77:
73:
69:
68:probabilistic
65:
58:
56:
54:
50:
46:
42:
38:
34:
30:
26:
22:
3058:
3039:
3018:
3015:
3010:
2999:
2996:
2991:
2980:
2977:
2972:
2957:
2952:
2940:. Springer.
2937:
2932:
2921:
2918:
2913:
2902:
2899:
2894:
2883:
2880:
2875:
2860:
2855:
2840:
2777:
2588:
2457:
2417:
2171:
2168:
2078:
1665:
1580:
1517:
1427:
1168:
1073:
933:
928:
850:
740:
477:
471:
470:
464:
458:
150:
139:
127:Josef Paldus
116:
95:
72:wavefunction
62:
20:
18:
2905:: 477–485.
2886:: 421–424.
33:numerically
2983:(2): A17.
2818:References
2585:Extensions
1581:where the
1367:while for
1076:functional
451:band index
445:, such as
123:Jiři Čížek
84:observable
59:Background
2755:⟩
2743:^
2721:†
2714:^
2701:⟨
2681:⟩
2669:^
2647:†
2640:^
2627:⟨
2604:^
2504:⟩
2492:^
2483:⟨
2480:Δ
2438:⋯
2395:⟩
2383:^
2374:⟨
2371:Δ
2347:⟩
2341:^
2332:⟨
2329:Δ
2323:⋯
2317:⟩
2311:^
2302:⟨
2299:Δ
2293:⟩
2287:^
2278:⟨
2254:⟩
2248:^
2239:⟨
2236:Δ
2220:⟩
2214:^
2205:⟨
2202:Δ
2193:∂
2189:∂
2184:ℏ
2154:⟩
2148:⟨
2145:Δ
2125:⟩
2119:⟨
2116:Δ
2096:⟩
2090:⟨
2087:Δ
2057:⟩
2051:⟨
2048:Δ
2034:…
2028:⟩
2022:⟨
2019:Δ
2013:⟩
2007:⟨
2004:Δ
1990:⟩
1983:−
1977:⟨
1963:⟩
1957:⟨
1954:Δ
1940:⟩
1933:−
1927:⟨
1913:…
1907:⟩
1901:⟨
1898:Δ
1892:⟩
1886:⟨
1883:Δ
1869:⟩
1862:−
1856:⟨
1842:⟩
1836:⟨
1833:Δ
1819:⟩
1812:−
1806:⟨
1784:⟩
1777:⟨
1767:⟩
1761:⟨
1750:⟩
1744:⟨
1741:Δ
1735:⟩
1729:⟨
1726:Δ
1720:⟩
1714:⟨
1700:⟩
1693:⟨
1683:⟩
1677:⟨
1643:⟩
1636:⟨
1633:−
1630:⟩
1624:⟨
1618:⟩
1612:⟨
1609:Δ
1589:Δ
1567:⟩
1561:⟨
1558:Δ
1544:⟩
1537:⟨
1531:⟩
1525:⟨
1495:⟩
1488:⟨
1468:⟩
1462:⟨
1442:⟩
1436:⟨
1413:⟩
1407:^
1398:⟨
1395:→
1389:^
1319:⟩
1312:⟨
1292:⟩
1286:⟨
1266:⟩
1260:⟨
1257:⟩
1251:⟨
1237:⟩
1230:⟨
1210:⟩
1204:⟨
1184:⟩
1178:⟨
1137:⟩
1125:^
1116:⟨
1055:⟩
1043:^
1034:⟨
1010:⟩
1004:^
995:⟨
979:⟩
973:^
964:⟨
955:∂
951:∂
946:ℏ
819:^
785:^
765:^
727:⟩
715:^
705:⋯
693:^
671:^
661:⋯
652:^
635:^
623:†
614:^
597:^
587:⋯
582:†
570:^
555:†
543:^
533:⋯
528:†
516:^
506:⟨
503:≡
500:⟩
494:^
485:⟨
429:λ
404:ℏ
374:λ
367:^
338:†
325:λ
318:^
254:ℏ
223:^
194:†
180:^
92:cumulants
37:many-body
3080:Category
2791:See also
2773:Gaussian
1361:Fermions
297:operator
112:kurtosis
108:skewness
104:variance
301:Fermion
119:nuclear
78:, or a
39:and/or
3065:
3046:
2964:
2944:
2867:
2847:
2785:lasers
2016:
2001:
1951:
1895:
1880:
1830:
1723:
1369:Bosons
1074:where
741:where
683:
560:
273:photon
142:solids
3002:(2).
2924:(1).
161:Boson
3063:ISBN
3044:ISBN
2962:ISBN
2942:ISBN
2865:ISBN
2845:ISBN
801:and
447:spin
352:and
208:and
125:and
100:mean
74:, a
51:and
19:The
449:or
140:In
3082::
3000:78
2981:29
2922:73
2903:17
2826:^
2787:.
453:.
148:.
110:,
106:,
102:,
55:.
3071:.
3052:.
3019:7
2884:7
2750:K
2740:B
2731:J
2727:]
2711:B
2704:[
2676:K
2666:B
2657:J
2653:]
2637:B
2630:[
2601:B
2569:C
2549:C
2529:C
2508:]
2501:1
2498:+
2489:C
2476:[
2471:i
2468:H
2441:]
2435:[
2430:L
2427:N
2404:,
2399:]
2392:1
2389:+
2380:N
2367:[
2362:i
2359:H
2355:+
2351:]
2338:N
2326:,
2320:,
2308:2
2296:,
2284:1
2274:[
2269:L
2266:N
2262:+
2258:]
2245:N
2232:[
2227:T
2223:=
2211:N
2196:t
2180:i
2151:3
2122:2
2093:N
2061:,
2054:N
2045:+
2031:+
2025:2
2010:3
1995:S
1986:5
1980:N
1974:+
1960:3
1945:S
1936:3
1930:N
1924:+
1910:+
1904:2
1889:2
1874:S
1865:4
1859:N
1853:+
1839:2
1824:S
1815:2
1809:N
1803:+
1789:S
1780:N
1774:=
1764:N
1754:,
1747:3
1738:+
1732:2
1717:1
1711:+
1705:S
1696:3
1690:=
1680:3
1648:S
1639:2
1627:2
1621:=
1615:2
1564:2
1555:+
1549:S
1540:2
1534:=
1528:2
1500:S
1491:2
1465:2
1439:2
1404:B
1386:B
1347:N
1324:S
1315:N
1289:1
1263:1
1254:1
1248:=
1242:S
1233:2
1207:2
1181:1
1140:]
1134:1
1131:+
1122:N
1113:[
1109:i
1106:H
1085:T
1059:]
1052:1
1049:+
1040:N
1030:[
1025:i
1022:H
1018:+
1014:]
1001:N
991:[
986:T
982:=
970:N
958:t
942:i
915:)
912:1
909:+
906:N
903:(
883:N
863:N
835:K
832:+
829:J
826:=
816:B
810:N
782:a
776:N
772:+
762:B
756:N
752:=
749:N
722:1
712:B
700:J
690:B
678:1
668:a
649:a
643:N
632:a
611:a
605:N
594:a
577:1
567:a
550:K
540:B
523:1
513:B
491:N
472:N
459:N
408:k
381:k
377:,
364:a
332:k
328:,
315:a
283:B
258:q
231:q
220:B
188:q
177:B
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.