22:
119:
323:
969:
If the field had a classical stochastic process underlying it, say a positive definite probability distribution for photon number, the variance would have to be greater than or equal to the mean. This can be shown by an application of the Cauchy–Schwarz inequality to the definition of
1236:
454:
573:
168:
704:
1413:
826:
709:
This quantity is basically the probability of detecting two simultaneous photons, normalized by the probability of detecting two photons at once for a random photon source. Here and after we assume stationary counting statistics.
133:
generally refers to a light field with photons more equally spaced than a coherent laser field, a signature being a measured two-time correlation suppressed below that of a coherent laser field. More specifically, it can refer to
1497:
1091:
964:
334:
1309:
1012:. Sub-Poissonian fields violate this, and hence are nonclassical in the sense that there can be no underlying positive definite probability distribution for photon number (or intensity).
318:{\displaystyle V_{n}=\langle \Delta n^{2}\rangle =\langle n^{2}\rangle -\langle n\rangle ^{2}=\left\langle \left(a^{\dagger }a\right)^{2}\right\rangle -\langle a^{\dagger }a\rangle ^{2}.}
465:
588:
122:
Photon detections as a function of time for a) antibunching (e.g. light emitted from a single atom), b) random (e.g. a coherent state, laser beam), and c) bunching (chaotic light). Ď„
1535:
1320:
877:
1073:
1078:
A more general definition for photon antibunching concerns the slope of the correlation function away from zero time delay. It can also be shown by an application of the
1010:
1561:
138:
photon statistics, that is a photon number distribution for which the variance is less than the mean. A coherent state, as output by a laser far above threshold, has
903:
719:
1697:
H. J. Carmichael and D. F. Walls, A Quantum-Mechanical Master
Equation Treatment of the Dynamical Stark Effect, J. Phys. B: Atom. Mol. Phys. 9, 1199 (1976).
1075:. An experiment with more precision that did not require subtraction of a background count rate was done for a single atom in an ion trap by Walther et al.
1651:
1424:
1311:. Hence a rise in the second order intensity correlation function at early times is also nonclassical. This initial rise is photon antibunching.
1750:
Nogueira, W. A. T.; Walborn, S. P.; P\'adua, S.; Monken, C. H. (30 April 2001). "Experimental
Observation of Spatial Antibunching of Photons".
1231:{\displaystyle g^{(2)}(\tau )={{\langle a^{\dagger }(0)a^{\dagger }(\tau )a(\tau )a(0)\rangle } \over {\langle a^{\dagger }a\rangle ^{2}}}.}
915:
86:
449:{\displaystyle V_{n}=\langle {(a^{\dagger }})^{2}a^{2}\rangle +\langle a^{\dagger }a\rangle -\langle a^{\dagger }a\rangle ^{2}.}
58:
1572:
1241:
It can be shown that for a classical positive definite probability distribution to exist (i.e. for the field to be classical)
1811:
Nogueira, W. A. T.; Walborn, S. P.; P\'adua, S.; Monken, C. H. (30 January 2004). "Generation of a Two-Photon
Singlet Beam".
1631:
1584:
65:
39:
1652:
https://web.archive.org/web/20110615173635/http://www.ucd.ie/speclab/UCDSOPAMS/peoplehtml/quantumoptics2006/lecture5.pdf
105:
72:
1244:
568:{\displaystyle V_{n}-\langle n\rangle =\langle (a^{\dagger })^{2}a^{2}\rangle -\langle a^{\dagger }a\rangle ^{2}.}
1623:
699:{\displaystyle g^{(2)}(0)={{\langle (a^{\dagger })^{2}a^{2}\rangle } \over {\langle a^{\dagger }a\rangle ^{2}}}.}
54:
1627:
1604:
151:
43:
1599:
1314:
Another way of looking at this time dependent correlation function, inspired by quantum trajectory theory is
1079:
154:, the number of fluctuations is larger than a coherent state; for an antibunched source they are smaller.
1408:{\displaystyle g^{(2)}(\tau )={{\langle a^{\dagger }a\rangle _{C}} \over {\langle a^{\dagger }a\rangle }}}
1505:
1015:
Photon antibunching by this definition was first proposed by
Carmichael and Walls and first observed by
1874:
834:
831:
Then we see that sub-Poisson photon statistics, one definition of photon antibunching, is given by
1030:
1890:
1609:
79:
32:
973:
1024:
1540:
882:
821:{\displaystyle {{1} \over {(\langle n\rangle )^{2}}}(V_{n}-\langle n\rangle )=g^{(2)}(0)-1.}
1830:
1769:
1716:
1672:
143:
139:
8:
1589:
1083:
579:
1834:
1773:
1720:
1707:
Zou, X T; Mandel, L (1990). "Photon-antibunching and sub-Poissonian photon statistics".
1676:
1854:
1820:
1793:
1759:
906:
1846:
1785:
1732:
1858:
1797:
1838:
1777:
1724:
1680:
147:
1842:
1492:{\displaystyle \langle O\rangle _{C}\equiv \langle \Psi _{C}|O|\Psi _{C}\rangle .}
1781:
1684:
1020:
1016:
135:
1884:
1728:
1850:
1789:
126:
is the coherence time (the time scale of photon or intensity fluctuations).
1736:
1825:
1764:
1594:
1027:. A driven atom cannot emit two photons at once, and so in this case
21:
1571:
Spatial antibunching has been observed in photon pairs produced by
1537:
is the state conditioned on previous detection of a photon at time
118:
1810:
1749:
150:
statistics and yields bunched photon spacing. In the thermal
959:{\displaystyle Q\equiv {\frac {V_{n}}{\langle n\rangle }}-1.}
1543:
1508:
1427:
1323:
1247:
1094:
1033:
976:
918:
885:
837:
722:
591:
468:
337:
171:
328:
Using commutation relations, this can be written as
142:statistics yielding random photon spacing; while a
46:. Unsourced material may be challenged and removed.
1555:
1529:
1491:
1407:
1303:
1230:
1067:
1004:
958:
897:
871:
820:
698:
567:
448:
317:
162:The variance of the photon number distribution is
1882:
879:. We can equivalently express antibunching by
1304:{\displaystyle g^{(2)}(\tau )\leq g^{(2)}(0)}
1524:
1483:
1447:
1435:
1428:
1399:
1383:
1372:
1355:
1213:
1196:
1191:
1126:
944:
938:
778:
772:
740:
734:
681:
664:
659:
623:
553:
536:
530:
494:
488:
482:
434:
417:
411:
395:
389:
351:
303:
286:
233:
226:
220:
207:
201:
185:
1824:
1763:
1706:
106:Learn how and when to remove this message
117:
1663:Paul, H (1982). "Photon antibunching".
1883:
1573:spontaneous parametric down-conversion
1662:
1585:Correlation does not imply causation
582:(for zero delay time) is defined as
44:adding citations to reliable sources
15:
1877:(Becker & Hickl GmbH, web page)
13:
1530:{\displaystyle |\Psi _{C}\rangle }
1515:
1474:
1451:
188:
14:
1902:
1868:
1650:Anti-bunching and Entanglement -
1082:to the time dependent intensity
20:
872:{\displaystyle g^{(2)}(0)<1}
31:needs additional citations for
1804:
1743:
1700:
1691:
1656:
1644:
1628:GNU Free Documentation License
1605:Hanbury Brown and Twiss effect
1566:
1510:
1469:
1461:
1346:
1340:
1335:
1329:
1298:
1292:
1287:
1281:
1270:
1264:
1259:
1253:
1188:
1182:
1176:
1170:
1164:
1158:
1145:
1139:
1117:
1111:
1106:
1100:
1056:
1050:
1045:
1039:
999:
993:
988:
982:
860:
854:
849:
843:
809:
803:
798:
792:
781:
756:
744:
731:
640:
626:
614:
608:
603:
597:
511:
497:
370:
355:
157:
1:
1843:10.1103/PhysRevLett.92.043602
1638:
1068:{\displaystyle g^{(2)}(0)=0}
7:
1782:10.1103/PhysRevLett.86.4009
1622:Article based on text from
1578:
578:The second-order intensity
10:
1907:
1685:10.1103/RevModPhys.54.1061
1615:
1005:{\displaystyle g^{(2)}(0)}
1665:Reviews of Modern Physics
1080:Cauchy–Schwarz inequality
1729:10.1103/PhysRevA.41.475
1626:, reproduced under the
1610:Squeezed coherent state
1556:{\displaystyle \tau =0}
459:This can be written as
1557:
1531:
1493:
1409:
1305:
1232:
1069:
1025:resonance fluorescence
1006:
960:
899:
898:{\displaystyle Q<0}
873:
822:
700:
569:
450:
319:
127:
1600:Hong–Ou–Mandel effect
1558:
1532:
1494:
1410:
1306:
1233:
1070:
1007:
961:
900:
874:
823:
701:
570:
451:
320:
121:
55:"Photon antibunching"
1541:
1506:
1425:
1321:
1245:
1092:
1084:correlation function
1031:
974:
916:
883:
835:
720:
589:
580:correlation function
466:
335:
169:
40:improve this article
1875:Photon antibunching
1835:2004PhRvL..92d3602N
1774:2001PhRvL..86.4009N
1721:1990PhRvA..41..475Z
1677:1982RvMP...54.1061P
1632:Photon Antibunching
1590:Degree of coherence
131:Photon antibunching
1553:
1527:
1489:
1405:
1301:
1228:
1065:
1023:, and Dagenais in
1002:
956:
907:Mandel Q parameter
895:
869:
818:
696:
565:
446:
315:
128:
1758:(18): 4009–4012.
1634:
1403:
1223:
948:
754:
691:
116:
115:
108:
90:
1898:
1863:
1862:
1828:
1826:quant-ph/0503117
1808:
1802:
1801:
1767:
1765:quant-ph/0206039
1747:
1741:
1740:
1704:
1698:
1695:
1689:
1688:
1671:(4): 1061–1102.
1660:
1654:
1648:
1621:
1562:
1560:
1559:
1554:
1536:
1534:
1533:
1528:
1523:
1522:
1513:
1498:
1496:
1495:
1490:
1482:
1481:
1472:
1464:
1459:
1458:
1443:
1442:
1414:
1412:
1411:
1406:
1404:
1402:
1395:
1394:
1381:
1380:
1379:
1367:
1366:
1353:
1339:
1338:
1310:
1308:
1307:
1302:
1291:
1290:
1263:
1262:
1237:
1235:
1234:
1229:
1224:
1222:
1221:
1220:
1208:
1207:
1194:
1157:
1156:
1138:
1137:
1124:
1110:
1109:
1074:
1072:
1071:
1066:
1049:
1048:
1011:
1009:
1008:
1003:
992:
991:
965:
963:
962:
957:
949:
947:
936:
935:
926:
904:
902:
901:
896:
878:
876:
875:
870:
853:
852:
827:
825:
824:
819:
802:
801:
768:
767:
755:
753:
752:
751:
729:
724:
705:
703:
702:
697:
692:
690:
689:
688:
676:
675:
662:
658:
657:
648:
647:
638:
637:
621:
607:
606:
574:
572:
571:
566:
561:
560:
548:
547:
529:
528:
519:
518:
509:
508:
478:
477:
455:
453:
452:
447:
442:
441:
429:
428:
407:
406:
388:
387:
378:
377:
368:
367:
366:
347:
346:
324:
322:
321:
316:
311:
310:
298:
297:
282:
278:
277:
272:
268:
264:
263:
241:
240:
219:
218:
200:
199:
181:
180:
148:super-Poissonian
111:
104:
100:
97:
91:
89:
48:
24:
16:
1906:
1905:
1901:
1900:
1899:
1897:
1896:
1895:
1881:
1880:
1871:
1866:
1813:Phys. Rev. Lett
1809:
1805:
1752:Phys. Rev. Lett
1748:
1744:
1705:
1701:
1696:
1692:
1661:
1657:
1649:
1645:
1641:
1618:
1581:
1569:
1542:
1539:
1538:
1518:
1514:
1509:
1507:
1504:
1503:
1477:
1473:
1468:
1460:
1454:
1450:
1438:
1434:
1426:
1423:
1422:
1390:
1386:
1382:
1375:
1371:
1362:
1358:
1354:
1352:
1328:
1324:
1322:
1319:
1318:
1280:
1276:
1252:
1248:
1246:
1243:
1242:
1216:
1212:
1203:
1199:
1195:
1152:
1148:
1133:
1129:
1125:
1123:
1099:
1095:
1093:
1090:
1089:
1038:
1034:
1032:
1029:
1028:
981:
977:
975:
972:
971:
937:
931:
927:
925:
917:
914:
913:
884:
881:
880:
842:
838:
836:
833:
832:
791:
787:
763:
759:
747:
743:
730:
725:
723:
721:
718:
717:
684:
680:
671:
667:
663:
653:
649:
643:
639:
633:
629:
622:
620:
596:
592:
590:
587:
586:
556:
552:
543:
539:
524:
520:
514:
510:
504:
500:
473:
469:
467:
464:
463:
437:
433:
424:
420:
402:
398:
383:
379:
373:
369:
362:
358:
354:
342:
338:
336:
333:
332:
306:
302:
293:
289:
273:
259:
255:
254:
250:
249:
245:
236:
232:
214:
210:
195:
191:
176:
172:
170:
167:
166:
160:
125:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
1904:
1894:
1893:
1891:Quantum optics
1879:
1878:
1870:
1869:External links
1867:
1865:
1864:
1803:
1742:
1715:(1): 475–476.
1699:
1690:
1655:
1642:
1640:
1637:
1636:
1635:
1617:
1614:
1613:
1612:
1607:
1602:
1597:
1592:
1587:
1580:
1577:
1568:
1565:
1552:
1549:
1546:
1526:
1521:
1517:
1512:
1500:
1499:
1488:
1485:
1480:
1476:
1471:
1467:
1463:
1457:
1453:
1449:
1446:
1441:
1437:
1433:
1430:
1416:
1415:
1401:
1398:
1393:
1389:
1385:
1378:
1374:
1370:
1365:
1361:
1357:
1351:
1348:
1345:
1342:
1337:
1334:
1331:
1327:
1300:
1297:
1294:
1289:
1286:
1283:
1279:
1275:
1272:
1269:
1266:
1261:
1258:
1255:
1251:
1239:
1238:
1227:
1219:
1215:
1211:
1206:
1202:
1198:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1155:
1151:
1147:
1144:
1141:
1136:
1132:
1128:
1122:
1119:
1116:
1113:
1108:
1105:
1102:
1098:
1064:
1061:
1058:
1055:
1052:
1047:
1044:
1041:
1037:
1001:
998:
995:
990:
987:
984:
980:
967:
966:
955:
952:
946:
943:
940:
934:
930:
924:
921:
909:is defined as
894:
891:
888:
868:
865:
862:
859:
856:
851:
848:
845:
841:
829:
828:
817:
814:
811:
808:
805:
800:
797:
794:
790:
786:
783:
780:
777:
774:
771:
766:
762:
758:
750:
746:
742:
739:
736:
733:
728:
707:
706:
695:
687:
683:
679:
674:
670:
666:
661:
656:
652:
646:
642:
636:
632:
628:
625:
619:
616:
613:
610:
605:
602:
599:
595:
576:
575:
564:
559:
555:
551:
546:
542:
538:
535:
532:
527:
523:
517:
513:
507:
503:
499:
496:
493:
490:
487:
484:
481:
476:
472:
457:
456:
445:
440:
436:
432:
427:
423:
419:
416:
413:
410:
405:
401:
397:
394:
391:
386:
382:
376:
372:
365:
361:
357:
353:
350:
345:
341:
326:
325:
314:
309:
305:
301:
296:
292:
288:
285:
281:
276:
271:
267:
262:
258:
253:
248:
244:
239:
235:
231:
228:
225:
222:
217:
213:
209:
206:
203:
198:
194:
190:
187:
184:
179:
175:
159:
156:
152:(bunched) case
136:sub-Poissonian
123:
114:
113:
28:
26:
19:
9:
6:
4:
3:
2:
1903:
1892:
1889:
1888:
1886:
1876:
1873:
1872:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1827:
1822:
1819:(4): 043602.
1818:
1814:
1807:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1771:
1766:
1761:
1757:
1753:
1746:
1738:
1734:
1730:
1726:
1722:
1718:
1714:
1710:
1703:
1694:
1686:
1682:
1678:
1674:
1670:
1666:
1659:
1653:
1647:
1643:
1633:
1629:
1625:
1620:
1619:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1582:
1576:
1574:
1564:
1550:
1547:
1544:
1519:
1486:
1478:
1465:
1455:
1444:
1439:
1431:
1421:
1420:
1419:
1396:
1391:
1387:
1376:
1368:
1363:
1359:
1349:
1343:
1332:
1325:
1317:
1316:
1315:
1312:
1295:
1284:
1277:
1273:
1267:
1256:
1249:
1225:
1217:
1209:
1204:
1200:
1185:
1179:
1173:
1167:
1161:
1153:
1149:
1142:
1134:
1130:
1120:
1114:
1103:
1096:
1088:
1087:
1086:
1085:
1081:
1076:
1062:
1059:
1053:
1042:
1035:
1026:
1022:
1018:
1013:
996:
985:
978:
953:
950:
941:
932:
928:
922:
919:
912:
911:
910:
908:
892:
889:
886:
866:
863:
857:
846:
839:
815:
812:
806:
795:
788:
784:
775:
769:
764:
760:
748:
737:
726:
716:
715:
714:
713:Then we have
711:
693:
685:
677:
672:
668:
654:
650:
644:
634:
630:
617:
611:
600:
593:
585:
584:
583:
581:
562:
557:
549:
544:
540:
533:
525:
521:
515:
505:
501:
491:
485:
479:
474:
470:
462:
461:
460:
443:
438:
430:
425:
421:
414:
408:
403:
399:
392:
384:
380:
374:
363:
359:
348:
343:
339:
331:
330:
329:
312:
307:
299:
294:
290:
283:
279:
274:
269:
265:
260:
256:
251:
246:
242:
237:
229:
223:
215:
211:
204:
196:
192:
182:
177:
173:
165:
164:
163:
155:
153:
149:
145:
144:thermal light
141:
137:
132:
120:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
1816:
1812:
1806:
1755:
1751:
1745:
1712:
1709:Phys. Rev. A
1708:
1702:
1693:
1668:
1664:
1658:
1646:
1570:
1501:
1417:
1313:
1240:
1077:
1014:
968:
830:
712:
708:
577:
458:
327:
161:
130:
129:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
1567:Experiments
158:Explanation
1639:References
1595:Fock state
905:where the
146:field has
140:Poissonian
66:newspapers
1545:τ
1525:⟩
1516:Ψ
1484:⟩
1475:Ψ
1452:Ψ
1448:⟨
1445:≡
1436:⟩
1429:⟨
1400:⟩
1392:†
1384:⟨
1373:⟩
1364:†
1356:⟨
1344:τ
1274:≤
1268:τ
1214:⟩
1205:†
1197:⟨
1192:⟩
1174:τ
1162:τ
1154:†
1135:†
1127:⟨
1115:τ
951:−
945:⟩
939:⟨
923:≡
813:−
779:⟩
773:⟨
770:−
741:⟩
735:⟨
682:⟩
673:†
665:⟨
660:⟩
635:†
624:⟨
554:⟩
545:†
537:⟨
534:−
531:⟩
506:†
495:⟨
489:⟩
483:⟨
480:−
435:⟩
426:†
418:⟨
415:−
412:⟩
404:†
396:⟨
390:⟩
364:†
352:⟨
304:⟩
295:†
287:⟨
284:−
261:†
234:⟩
227:⟨
224:−
221:⟩
208:⟨
202:⟩
189:Δ
186:⟨
1885:Category
1859:25022990
1851:14995372
1798:25655506
1790:11328082
1579:See also
280:⟩
247:⟨
96:May 2008
1831:Bibcode
1770:Bibcode
1737:9902890
1717:Bibcode
1673:Bibcode
1616:Sources
80:scholar
1857:
1849:
1796:
1788:
1735:
1630:: see
1418:where
1021:Mandel
1017:Kimble
82:
75:
68:
61:
53:
1855:S2CID
1821:arXiv
1794:S2CID
1760:arXiv
1624:Qwiki
1502:with
87:JSTOR
73:books
1847:PMID
1786:PMID
1733:PMID
890:<
864:<
59:news
1839:doi
1778:doi
1725:doi
1681:doi
1575:.
42:by
1887::
1853:.
1845:.
1837:.
1829:.
1817:92
1815:.
1792:.
1784:.
1776:.
1768:.
1756:86
1754:.
1731:.
1723:.
1713:41
1711:.
1679:.
1669:54
1667:.
1563:.
1019:,
954:1.
816:1.
1861:.
1841::
1833::
1823::
1800:.
1780::
1772::
1762::
1739:.
1727::
1719::
1687:.
1683::
1675::
1551:0
1548:=
1520:C
1511:|
1487:.
1479:C
1470:|
1466:O
1462:|
1456:C
1440:C
1432:O
1397:a
1388:a
1377:C
1369:a
1360:a
1350:=
1347:)
1341:(
1336:)
1333:2
1330:(
1326:g
1299:)
1296:0
1293:(
1288:)
1285:2
1282:(
1278:g
1271:)
1265:(
1260:)
1257:2
1254:(
1250:g
1226:.
1218:2
1210:a
1201:a
1189:)
1186:0
1183:(
1180:a
1177:)
1171:(
1168:a
1165:)
1159:(
1150:a
1146:)
1143:0
1140:(
1131:a
1121:=
1118:)
1112:(
1107:)
1104:2
1101:(
1097:g
1063:0
1060:=
1057:)
1054:0
1051:(
1046:)
1043:2
1040:(
1036:g
1000:)
997:0
994:(
989:)
986:2
983:(
979:g
942:n
933:n
929:V
920:Q
893:0
887:Q
867:1
861:)
858:0
855:(
850:)
847:2
844:(
840:g
810:)
807:0
804:(
799:)
796:2
793:(
789:g
785:=
782:)
776:n
765:n
761:V
757:(
749:2
745:)
738:n
732:(
727:1
694:.
686:2
678:a
669:a
655:2
651:a
645:2
641:)
631:a
627:(
618:=
615:)
612:0
609:(
604:)
601:2
598:(
594:g
563:.
558:2
550:a
541:a
526:2
522:a
516:2
512:)
502:a
498:(
492:=
486:n
475:n
471:V
444:.
439:2
431:a
422:a
409:a
400:a
393:+
385:2
381:a
375:2
371:)
360:a
356:(
349:=
344:n
340:V
313:.
308:2
300:a
291:a
275:2
270:)
266:a
257:a
252:(
243:=
238:2
230:n
216:2
212:n
205:=
197:2
193:n
183:=
178:n
174:V
124:c
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
