468:
2778:
3258:
992:
282:
that determine which transitions are allowed and which are not. Often these selection rules can be boiled down to conservation of angular momentum (the photon has angular momentum). In most cases we only consider an atom interacting with the electric dipole field of the photon. Then some transitions
213:
of the atom no longer interacts with the laser simply because no photons of the right frequency are present to induce a transition to a different level. In practice, the term dark state is often used for a state that is not accessible by the specific laser in use even though transitions from this
3263:
This means that when the atoms are in this state, they will stay in this state indefinitely. This is a dark state, because it can not absorb or emit any photons from the applied fields. It is, therefore, effectively transparent to the probe laser, even when the laser is exactly resonant with the
1411:
806:
454:
state can not decay to the ground state by emitting a single photon. It can only decay by collisions with other atoms or by emitting multiple photons. Since these events are rare, the atom can remain in this excited state for a very long time, such an excited state is called a
3292:
can result in an atom being in this dark state or another coherent state, known as a bright state. Therefore, in a collection of atoms, over time, decay into the dark state will inevitably result in the system being "trapped" coherently in that state, a phenomenon known as
2567:
2219:
63:
superposition of two states, both of which are coupled by lasers at the right frequency to a third state. With the system in a particular superposition of the two states, the system can be made dark to both lasers as the probability of absorbing a photon goes to 0.
2978:
2575:
3117:
812:
1868:
51:
light to induce transitions between energy levels, when atoms can spontaneously decay into a state that is not coupled to any other level by the laser light, preventing the atom from absorbing or emitting light from that state.
1724:
3106:
1199:
2841:
667:
1477:
1622:
1552:
599:
557:
515:
2227:
1879:
2849:
2773:{\displaystyle c_{3}(t)=-ic_{1}(0){\frac {\Omega _{p}}{\Omega }}\sin {\frac {\Omega t}{2}}-ic_{2}(0){\frac {\Omega _{c}}{\Omega }}\sin {\frac {\Omega t}{2}}+c_{3}(0)\cos {\frac {\Omega t}{2}}}
171:
1187:
1147:
3253:{\displaystyle \cos \theta ={\frac {\Omega _{c}}{\sqrt {\Omega _{p}^{2}+\Omega _{c}^{2}}}},\qquad \sin \theta ={\frac {\Omega _{p}}{\sqrt {\Omega _{p}^{2}+\Omega _{c}^{2}}}}.}
47:. However, not all transitions between arbitrary states are allowed. A state that cannot absorb an incident photon is called a dark state. This can occur in experiments using
211:
656:
432:
379:
326:
1107:
1080:
1049:
1022:
3290:
3009:
987:{\displaystyle H_{1}=-{\frac {\hbar }{2}}\left(\Omega _{p}e^{+i\omega _{p}t}|1\rangle \langle 3|+\Omega _{c}e^{+i\omega _{c}t}|2\rangle \langle 3|\right)+{\mbox{H.c.}},}
276:
248:
3029:
95:
3350:
122:
1734:
283:
are not allowed at all, others are only allowed for photons of a certain polarization. Consider for example the hydrogen atom. The transition from the state
1628:
3306:
1406:{\displaystyle |\psi (t)\rangle =c_{1}(t)e^{-i\omega _{1}t}|1\rangle +c_{2}(t)e^{-i\omega _{2}t}|2\rangle +c_{3}(t)e^{-i\omega _{3}t}|3\rangle .}
3037:
173:. However, the atom can still decay spontaneously into a third state by emitting a photon of a different frequency. The new state with energy
278:
is allowed often depends on how detailed the model is that we use for the atom-light interaction. From a particular model follow a set of
801:{\displaystyle H_{0}=\hbar \omega _{1}|1\rangle \langle 1|+\hbar \omega _{2}|2\rangle \langle 2|+\hbar \omega _{3}|3\rangle \langle 3|,}
2786:
1422:
1558:
1488:
606:
2562:{\displaystyle c_{2}(t)=c_{1}(0)\left+c_{2}(0)\left\quad -ic_{3}(0){\frac {\Omega _{c}}{\Omega }}\sin {\frac {\Omega t}{2}}}
2214:{\displaystyle c_{1}(t)=c_{1}(0)\left+c_{2}(0)\left\quad -ic_{3}(0){\frac {\Omega _{p}}{\Omega }}\sin {\frac {\Omega t}{2}}}
2973:{\displaystyle c_{1}(0)={\frac {\Omega _{c}}{\Omega }},\qquad c_{2}(0)=-{\frac {\Omega _{p}}{\Omega }},\qquad c_{3}(0)=0,}
562:
520:
478:
602:
127:
1152:
1112:
97:(meaning the photons have a specific energy), so they only couple one set of states with a particular energy
2983:
which gives a time-independent solution to these equations with no probability of the system being in state
1417:
3311:
389:
is only allowed for light with polarization along the z axis (quantization axis) of the atom. The state
56:
176:
1190:
615:
392:
339:
286:
3355:
1085:
1058:
1027:
1000:
3267:
2986:
253:
225:
60:
17:
3014:
80:
456:
1863:{\displaystyle |\psi (0)\rangle =c_{1}(0)|1\rangle +c_{2}(0)|2\rangle +c_{3}(0)|3\rangle ,}
100:
39:; different states can have different energies and a system can make a transition from one
279:
1052:
24:
3344:
36:
40:
467:
77:
Experiments in atomic physics are often done with a laser of a specific frequency
1719:{\displaystyle {\dot {c}}_{3}={\frac {i}{2}}(\Omega _{p}c_{1}+\Omega _{c}c_{2}).}
35:
that cannot absorb (or emit) photons. All atoms and molecules are described by
442:
therefore appears dark for light of other polarizations. Transitions from the
32:
3101:{\displaystyle |D\rangle =\cos \theta |1\rangle -\sin \theta |2\rangle }
44:
1193:
of the entire expression. We will write the atomic wave function as
466:
48:
2836:{\displaystyle \Omega ={\sqrt {\Omega _{c}^{2}+\Omega _{p}^{2}}}}
3011:. This state can also be expressed in terms of a mixing angle
1472:{\displaystyle i\hbar |{\dot {\psi }}\rangle =H|\psi \rangle }
1617:{\displaystyle {\dot {c}}_{2}={\frac {i}{2}}\Omega _{c}c_{3}}
1547:{\displaystyle {\dot {c}}_{1}={\frac {i}{2}}\Omega _{p}c_{3}}
2843:. We observe that we can choose the initial conditions
3332:
Fundamentals of
Quantum Optics and Quantum Information
975:
3270:
3120:
3040:
3017:
2989:
2852:
2789:
2578:
2230:
1882:
1737:
1631:
1561:
1491:
1425:
1202:
1155:
1115:
1088:
1061:
1030:
1003:
815:
670:
618:
565:
523:
481:
395:
342:
289:
256:
228:
179:
130:
103:
83:
594:{\displaystyle |1\rangle \leftrightarrow |2\rangle }
552:{\displaystyle |2\rangle \leftrightarrow |3\rangle }
510:{\displaystyle |1\rangle \leftrightarrow |3\rangle }
222:Whether or not we say a transition between a state
3284:
3252:
3100:
3023:
3003:
2972:
2835:
2772:
2561:
2213:
1862:
1718:
1616:
1546:
1471:
1405:
1181:
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1101:
1074:
1043:
1016:
986:
800:
650:
593:
551:
509:
426:
373:
320:
270:
242:
205:
165:
116:
89:
475:We start with a three-state Λ-type system, where
43:to another by emitting or absorbing one or more
1109:) in resonance with the transition frequencies
59:in a three-level system, when an atom is in a
8:
3279:
3095:
3072:
3049:
2998:
1854:
1821:
1788:
1755:
1466:
1449:
1397:
1338:
1279:
1220:
955:
952:
894:
891:
784:
781:
746:
743:
708:
705:
588:
574:
546:
532:
504:
490:
265:
237:
3330:P. Lambropoulos & D. Petrosyan (2007).
3271:
3269:
3238:
3233:
3220:
3215:
3204:
3198:
3173:
3168:
3155:
3150:
3139:
3133:
3119:
3087:
3064:
3041:
3039:
3016:
2990:
2988:
2946:
2927:
2921:
2900:
2881:
2875:
2857:
2851:
2825:
2820:
2807:
2802:
2796:
2788:
2755:
2734:
2712:
2695:
2689:
2674:
2649:
2632:
2626:
2611:
2583:
2577:
2544:
2527:
2521:
2506:
2475:
2461:
2451:
2446:
2440:
2429:
2419:
2414:
2408:
2388:
2361:
2347:
2336:
2326:
2319:
2308:
2297:
2287:
2280:
2257:
2235:
2229:
2196:
2179:
2173:
2158:
2127:
2113:
2102:
2092:
2085:
2074:
2063:
2053:
2046:
2023:
1996:
1982:
1972:
1967:
1961:
1950:
1940:
1935:
1929:
1909:
1887:
1881:
1846:
1831:
1813:
1798:
1780:
1765:
1738:
1736:
1704:
1694:
1681:
1671:
1654:
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1634:
1633:
1630:
1608:
1598:
1584:
1575:
1564:
1563:
1560:
1538:
1528:
1514:
1505:
1494:
1493:
1490:
1458:
1438:
1437:
1432:
1424:
1389:
1378:
1367:
1348:
1330:
1319:
1308:
1289:
1271:
1260:
1249:
1230:
1203:
1201:
1173:
1160:
1154:
1133:
1120:
1114:
1093:
1087:
1066:
1060:
1035:
1029:
1008:
1002:
974:
961:
944:
933:
922:
912:
900:
883:
872:
861:
851:
832:
820:
814:
790:
773:
767:
752:
735:
729:
714:
697:
691:
675:
669:
642:
629:
617:
580:
566:
564:
538:
524:
522:
496:
482:
480:
414:
410:
400:
394:
361:
357:
347:
341:
308:
304:
294:
288:
257:
255:
229:
227:
197:
184:
178:
166:{\displaystyle E_{2}=E_{1}+\hbar \omega }
148:
135:
129:
108:
102:
82:
3307:Electromagnetically induced transparency
1189:, respectively, and H.c. stands for the
124:to another set of states with an energy
3322:
1873:we can solve these equations to obtain
1429:
1182:{\displaystyle \omega _{2}-\omega _{3}}
1142:{\displaystyle \omega _{1}-\omega _{3}}
1082:) and the coupling field (of frequency
834:
760:
722:
684:
157:
55:A dark state can also be the result of
3351:Atomic, molecular, and optical physics
3264:transition. Spontaneous emission from
7:
559:are dipole-allowed transitions and
3230:
3212:
3201:
3165:
3147:
3136:
2933:
2924:
2887:
2878:
2817:
2799:
2790:
2758:
2715:
2701:
2692:
2652:
2638:
2629:
2547:
2533:
2524:
2478:
2458:
2443:
2426:
2411:
2364:
2344:
2333:
2323:
2305:
2294:
2284:
2199:
2185:
2176:
2130:
2110:
2099:
2089:
2071:
2060:
2050:
1999:
1979:
1964:
1947:
1932:
1691:
1668:
1595:
1525:
1032:
1005:
909:
848:
450:level are not allowed at all. The
14:
1055:of the probe field (of frequency
214:state are in principle allowed.
31:refers to a state of an atom or
3185:
2941:
2895:
2495:
2147:
3272:
3088:
3065:
3042:
2991:
2958:
2952:
2912:
2906:
2869:
2863:
2746:
2740:
2686:
2680:
2623:
2617:
2595:
2589:
2518:
2512:
2400:
2394:
2269:
2263:
2247:
2241:
2170:
2164:
2035:
2029:
1921:
1915:
1899:
1893:
1847:
1843:
1837:
1814:
1810:
1804:
1781:
1777:
1771:
1752:
1746:
1739:
1710:
1664:
1459:
1433:
1390:
1360:
1354:
1331:
1301:
1295:
1272:
1242:
1236:
1217:
1211:
1204:
962:
945:
901:
884:
791:
774:
753:
736:
715:
698:
581:
577:
567:
539:
535:
525:
497:
493:
483:
258:
230:
206:{\displaystyle E_{3}<E_{2}}
1:
3334:. Berlin; New York: Springer.
651:{\displaystyle H=H_{0}+H_{1}}
3295:coherent population trapping
1728:Using the initial condition
427:{\displaystyle 2^{2}P_{3/2}}
374:{\displaystyle 2^{2}P_{3/2}}
321:{\displaystyle 1^{2}S_{1/2}}
1102:{\displaystyle \omega _{c}}
1075:{\displaystyle \omega _{p}}
1044:{\displaystyle \Omega _{c}}
1017:{\displaystyle \Omega _{p}}
603:rotating wave approximation
471:A three-state Λ-type system
3372:
3285:{\displaystyle |3\rangle }
3004:{\displaystyle |3\rangle }
1479:, we obtain the solutions
271:{\displaystyle |2\rangle }
243:{\displaystyle |1\rangle }
15:
3024:{\displaystyle \theta }
90:{\displaystyle \omega }
16:For the 2021 film, see
3312:List of laser articles
3286:
3254:
3102:
3025:
3005:
2974:
2837:
2774:
2563:
2215:
1864:
1720:
1618:
1548:
1473:
1407:
1183:
1143:
1103:
1076:
1045:
1018:
988:
802:
652:
595:
553:
511:
472:
428:
375:
322:
272:
244:
207:
167:
118:
91:
3287:
3255:
3103:
3026:
3006:
2975:
2838:
2775:
2564:
2216:
1865:
1721:
1619:
1549:
1474:
1408:
1184:
1144:
1104:
1077:
1046:
1019:
989:
803:
653:
605:, the semi-classical
601:is forbidden. In the
596:
554:
512:
470:
429:
376:
323:
273:
245:
208:
168:
119:
117:{\displaystyle E_{1}}
92:
3268:
3118:
3038:
3015:
2987:
2850:
2787:
2576:
2228:
1880:
1735:
1629:
1559:
1489:
1423:
1418:Schrödinger equation
1200:
1153:
1113:
1086:
1059:
1028:
1001:
813:
668:
616:
563:
521:
479:
393:
340:
287:
254:
226:
177:
128:
101:
81:
57:quantum interference
3243:
3225:
3178:
3160:
2830:
2812:
2456:
2424:
1977:
1945:
1191:Hermitian conjugate
463:Three-level systems
3282:
3250:
3229:
3211:
3164:
3146:
3098:
3021:
3001:
2970:
2833:
2816:
2798:
2770:
2559:
2442:
2410:
2211:
1963:
1931:
1860:
1716:
1614:
1544:
1469:
1403:
1179:
1139:
1099:
1072:
1041:
1014:
984:
979:
798:
648:
591:
549:
507:
473:
424:
371:
318:
268:
240:
203:
163:
114:
87:
3245:
3244:
3180:
3179:
2936:
2890:
2831:
2768:
2725:
2704:
2662:
2641:
2557:
2536:
2488:
2467:
2435:
2374:
2353:
2314:
2209:
2188:
2140:
2119:
2080:
2009:
1988:
1956:
1662:
1642:
1592:
1572:
1522:
1502:
1446:
978:
840:
68:Two-level systems
18:Dark State (film)
3363:
3336:
3335:
3327:
3291:
3289:
3288:
3283:
3275:
3259:
3257:
3256:
3251:
3246:
3242:
3237:
3224:
3219:
3210:
3209:
3208:
3199:
3181:
3177:
3172:
3159:
3154:
3145:
3144:
3143:
3134:
3107:
3105:
3104:
3099:
3091:
3068:
3045:
3030:
3028:
3027:
3022:
3010:
3008:
3007:
3002:
2994:
2979:
2977:
2976:
2971:
2951:
2950:
2937:
2932:
2931:
2922:
2905:
2904:
2891:
2886:
2885:
2876:
2862:
2861:
2842:
2840:
2839:
2834:
2832:
2829:
2824:
2811:
2806:
2797:
2779:
2777:
2776:
2771:
2769:
2764:
2756:
2739:
2738:
2726:
2721:
2713:
2705:
2700:
2699:
2690:
2679:
2678:
2663:
2658:
2650:
2642:
2637:
2636:
2627:
2616:
2615:
2588:
2587:
2568:
2566:
2565:
2560:
2558:
2553:
2545:
2537:
2532:
2531:
2522:
2511:
2510:
2494:
2490:
2489:
2484:
2476:
2468:
2466:
2465:
2455:
2450:
2441:
2436:
2434:
2433:
2423:
2418:
2409:
2393:
2392:
2380:
2376:
2375:
2370:
2362:
2354:
2352:
2351:
2342:
2341:
2340:
2331:
2330:
2320:
2315:
2313:
2312:
2303:
2302:
2301:
2292:
2291:
2281:
2262:
2261:
2240:
2239:
2220:
2218:
2217:
2212:
2210:
2205:
2197:
2189:
2184:
2183:
2174:
2163:
2162:
2146:
2142:
2141:
2136:
2128:
2120:
2118:
2117:
2108:
2107:
2106:
2097:
2096:
2086:
2081:
2079:
2078:
2069:
2068:
2067:
2058:
2057:
2047:
2028:
2027:
2015:
2011:
2010:
2005:
1997:
1989:
1987:
1986:
1976:
1971:
1962:
1957:
1955:
1954:
1944:
1939:
1930:
1914:
1913:
1892:
1891:
1869:
1867:
1866:
1861:
1850:
1836:
1835:
1817:
1803:
1802:
1784:
1770:
1769:
1742:
1725:
1723:
1722:
1717:
1709:
1708:
1699:
1698:
1686:
1685:
1676:
1675:
1663:
1655:
1650:
1649:
1644:
1643:
1635:
1623:
1621:
1620:
1615:
1613:
1612:
1603:
1602:
1593:
1585:
1580:
1579:
1574:
1573:
1565:
1553:
1551:
1550:
1545:
1543:
1542:
1533:
1532:
1523:
1515:
1510:
1509:
1504:
1503:
1495:
1478:
1476:
1475:
1470:
1462:
1448:
1447:
1439:
1436:
1412:
1410:
1409:
1404:
1393:
1388:
1387:
1383:
1382:
1353:
1352:
1334:
1329:
1328:
1324:
1323:
1294:
1293:
1275:
1270:
1269:
1265:
1264:
1235:
1234:
1207:
1188:
1186:
1185:
1180:
1178:
1177:
1165:
1164:
1148:
1146:
1145:
1140:
1138:
1137:
1125:
1124:
1108:
1106:
1105:
1100:
1098:
1097:
1081:
1079:
1078:
1073:
1071:
1070:
1053:Rabi frequencies
1050:
1048:
1047:
1042:
1040:
1039:
1023:
1021:
1020:
1015:
1013:
1012:
993:
991:
990:
985:
980:
976:
970:
966:
965:
948:
943:
942:
938:
937:
917:
916:
904:
887:
882:
881:
877:
876:
856:
855:
841:
833:
825:
824:
807:
805:
804:
799:
794:
777:
772:
771:
756:
739:
734:
733:
718:
701:
696:
695:
680:
679:
657:
655:
654:
649:
647:
646:
634:
633:
600:
598:
597:
592:
584:
570:
558:
556:
555:
550:
542:
528:
516:
514:
513:
508:
500:
486:
457:metastable state
433:
431:
430:
425:
423:
422:
418:
405:
404:
380:
378:
377:
372:
370:
369:
365:
352:
351:
327:
325:
324:
319:
317:
316:
312:
299:
298:
277:
275:
274:
269:
261:
249:
247:
246:
241:
233:
212:
210:
209:
204:
202:
201:
189:
188:
172:
170:
169:
164:
153:
152:
140:
139:
123:
121:
120:
115:
113:
112:
96:
94:
93:
88:
3371:
3370:
3366:
3365:
3364:
3362:
3361:
3360:
3341:
3340:
3339:
3329:
3328:
3324:
3320:
3303:
3266:
3265:
3200:
3135:
3116:
3115:
3036:
3035:
3013:
3012:
2985:
2984:
2942:
2923:
2896:
2877:
2853:
2848:
2847:
2785:
2784:
2757:
2730:
2714:
2691:
2670:
2651:
2628:
2607:
2579:
2574:
2573:
2546:
2523:
2502:
2477:
2457:
2425:
2407:
2403:
2384:
2363:
2343:
2332:
2322:
2321:
2304:
2293:
2283:
2282:
2276:
2272:
2253:
2231:
2226:
2225:
2198:
2175:
2154:
2129:
2109:
2098:
2088:
2087:
2070:
2059:
2049:
2048:
2042:
2038:
2019:
1998:
1978:
1946:
1928:
1924:
1905:
1883:
1878:
1877:
1827:
1794:
1761:
1733:
1732:
1700:
1690:
1677:
1667:
1632:
1627:
1626:
1604:
1594:
1562:
1557:
1556:
1534:
1524:
1492:
1487:
1486:
1421:
1420:
1374:
1363:
1344:
1315:
1304:
1285:
1256:
1245:
1226:
1198:
1197:
1169:
1156:
1151:
1150:
1129:
1116:
1111:
1110:
1089:
1084:
1083:
1062:
1057:
1056:
1031:
1026:
1025:
1004:
999:
998:
929:
918:
908:
868:
857:
847:
846:
842:
816:
811:
810:
763:
725:
687:
671:
666:
665:
638:
625:
614:
613:
561:
560:
519:
518:
477:
476:
465:
439:
406:
396:
391:
390:
386:
353:
343:
338:
337:
333:
300:
290:
285:
284:
280:selection rules
252:
251:
224:
223:
220:
193:
180:
175:
174:
144:
131:
126:
125:
104:
99:
98:
79:
78:
75:
70:
21:
12:
11:
5:
3369:
3367:
3359:
3358:
3353:
3343:
3342:
3338:
3337:
3321:
3319:
3316:
3315:
3314:
3309:
3302:
3299:
3281:
3278:
3274:
3261:
3260:
3249:
3241:
3236:
3232:
3228:
3223:
3218:
3214:
3207:
3203:
3197:
3194:
3191:
3188:
3184:
3176:
3171:
3167:
3163:
3158:
3153:
3149:
3142:
3138:
3132:
3129:
3126:
3123:
3109:
3108:
3097:
3094:
3090:
3086:
3083:
3080:
3077:
3074:
3071:
3067:
3063:
3060:
3057:
3054:
3051:
3048:
3044:
3020:
3000:
2997:
2993:
2981:
2980:
2969:
2966:
2963:
2960:
2957:
2954:
2949:
2945:
2940:
2935:
2930:
2926:
2920:
2917:
2914:
2911:
2908:
2903:
2899:
2894:
2889:
2884:
2880:
2874:
2871:
2868:
2865:
2860:
2856:
2828:
2823:
2819:
2815:
2810:
2805:
2801:
2795:
2792:
2781:
2780:
2767:
2763:
2760:
2754:
2751:
2748:
2745:
2742:
2737:
2733:
2729:
2724:
2720:
2717:
2711:
2708:
2703:
2698:
2694:
2688:
2685:
2682:
2677:
2673:
2669:
2666:
2661:
2657:
2654:
2648:
2645:
2640:
2635:
2631:
2625:
2622:
2619:
2614:
2610:
2606:
2603:
2600:
2597:
2594:
2591:
2586:
2582:
2570:
2569:
2556:
2552:
2549:
2543:
2540:
2535:
2530:
2526:
2520:
2517:
2514:
2509:
2505:
2501:
2498:
2493:
2487:
2483:
2480:
2474:
2471:
2464:
2460:
2454:
2449:
2445:
2439:
2432:
2428:
2422:
2417:
2413:
2406:
2402:
2399:
2396:
2391:
2387:
2383:
2379:
2373:
2369:
2366:
2360:
2357:
2350:
2346:
2339:
2335:
2329:
2325:
2318:
2311:
2307:
2300:
2296:
2290:
2286:
2279:
2275:
2271:
2268:
2265:
2260:
2256:
2252:
2249:
2246:
2243:
2238:
2234:
2222:
2221:
2208:
2204:
2201:
2195:
2192:
2187:
2182:
2178:
2172:
2169:
2166:
2161:
2157:
2153:
2150:
2145:
2139:
2135:
2132:
2126:
2123:
2116:
2112:
2105:
2101:
2095:
2091:
2084:
2077:
2073:
2066:
2062:
2056:
2052:
2045:
2041:
2037:
2034:
2031:
2026:
2022:
2018:
2014:
2008:
2004:
2001:
1995:
1992:
1985:
1981:
1975:
1970:
1966:
1960:
1953:
1949:
1943:
1938:
1934:
1927:
1923:
1920:
1917:
1912:
1908:
1904:
1901:
1898:
1895:
1890:
1886:
1871:
1870:
1859:
1856:
1853:
1849:
1845:
1842:
1839:
1834:
1830:
1826:
1823:
1820:
1816:
1812:
1809:
1806:
1801:
1797:
1793:
1790:
1787:
1783:
1779:
1776:
1773:
1768:
1764:
1760:
1757:
1754:
1751:
1748:
1745:
1741:
1715:
1712:
1707:
1703:
1697:
1693:
1689:
1684:
1680:
1674:
1670:
1666:
1661:
1658:
1653:
1648:
1641:
1638:
1611:
1607:
1601:
1597:
1591:
1588:
1583:
1578:
1571:
1568:
1541:
1537:
1531:
1527:
1521:
1518:
1513:
1508:
1501:
1498:
1484:
1483:
1468:
1465:
1461:
1457:
1454:
1451:
1445:
1442:
1435:
1431:
1428:
1414:
1413:
1402:
1399:
1396:
1392:
1386:
1381:
1377:
1373:
1370:
1366:
1362:
1359:
1356:
1351:
1347:
1343:
1340:
1337:
1333:
1327:
1322:
1318:
1314:
1311:
1307:
1303:
1300:
1297:
1292:
1288:
1284:
1281:
1278:
1274:
1268:
1263:
1259:
1255:
1252:
1248:
1244:
1241:
1238:
1233:
1229:
1225:
1222:
1219:
1216:
1213:
1210:
1206:
1176:
1172:
1168:
1163:
1159:
1136:
1132:
1128:
1123:
1119:
1096:
1092:
1069:
1065:
1038:
1034:
1011:
1007:
995:
994:
983:
973:
969:
964:
960:
957:
954:
951:
947:
941:
936:
932:
928:
925:
921:
915:
911:
907:
903:
899:
896:
893:
890:
886:
880:
875:
871:
867:
864:
860:
854:
850:
845:
839:
836:
831:
828:
823:
819:
808:
797:
793:
789:
786:
783:
780:
776:
770:
766:
762:
759:
755:
751:
748:
745:
742:
738:
732:
728:
724:
721:
717:
713:
710:
707:
704:
700:
694:
690:
686:
683:
678:
674:
659:
658:
645:
641:
637:
632:
628:
624:
621:
590:
587:
583:
579:
576:
573:
569:
548:
545:
541:
537:
534:
531:
527:
506:
503:
499:
495:
492:
489:
485:
464:
461:
437:
421:
417:
413:
409:
403:
399:
384:
368:
364:
360:
356:
350:
346:
331:
315:
311:
307:
303:
297:
293:
267:
264:
260:
239:
236:
232:
219:
216:
200:
196:
192:
187:
183:
162:
159:
156:
151:
147:
143:
138:
134:
111:
107:
86:
74:
71:
69:
66:
37:quantum states
25:atomic physics
13:
10:
9:
6:
4:
3:
2:
3368:
3357:
3354:
3352:
3349:
3348:
3346:
3333:
3326:
3323:
3317:
3313:
3310:
3308:
3305:
3304:
3300:
3298:
3296:
3276:
3247:
3239:
3234:
3226:
3221:
3216:
3205:
3195:
3192:
3189:
3186:
3182:
3174:
3169:
3161:
3156:
3151:
3140:
3130:
3127:
3124:
3121:
3114:
3113:
3112:
3092:
3084:
3081:
3078:
3075:
3069:
3061:
3058:
3055:
3052:
3046:
3034:
3033:
3032:
3018:
2995:
2967:
2964:
2961:
2955:
2947:
2943:
2938:
2928:
2918:
2915:
2909:
2901:
2897:
2892:
2882:
2872:
2866:
2858:
2854:
2846:
2845:
2844:
2826:
2821:
2813:
2808:
2803:
2793:
2765:
2761:
2752:
2749:
2743:
2735:
2731:
2727:
2722:
2718:
2709:
2706:
2696:
2683:
2675:
2671:
2667:
2664:
2659:
2655:
2646:
2643:
2633:
2620:
2612:
2608:
2604:
2601:
2598:
2592:
2584:
2580:
2572:
2571:
2554:
2550:
2541:
2538:
2528:
2515:
2507:
2503:
2499:
2496:
2491:
2485:
2481:
2472:
2469:
2462:
2452:
2447:
2437:
2430:
2420:
2415:
2404:
2397:
2389:
2385:
2381:
2377:
2371:
2367:
2358:
2355:
2348:
2337:
2327:
2316:
2309:
2298:
2288:
2277:
2273:
2266:
2258:
2254:
2250:
2244:
2236:
2232:
2224:
2223:
2206:
2202:
2193:
2190:
2180:
2167:
2159:
2155:
2151:
2148:
2143:
2137:
2133:
2124:
2121:
2114:
2103:
2093:
2082:
2075:
2064:
2054:
2043:
2039:
2032:
2024:
2020:
2016:
2012:
2006:
2002:
1993:
1990:
1983:
1973:
1968:
1958:
1951:
1941:
1936:
1925:
1918:
1910:
1906:
1902:
1896:
1888:
1884:
1876:
1875:
1874:
1857:
1851:
1840:
1832:
1828:
1824:
1818:
1807:
1799:
1795:
1791:
1785:
1774:
1766:
1762:
1758:
1749:
1743:
1731:
1730:
1729:
1726:
1713:
1705:
1701:
1695:
1687:
1682:
1678:
1672:
1659:
1656:
1651:
1646:
1639:
1636:
1624:
1609:
1605:
1599:
1589:
1586:
1581:
1576:
1569:
1566:
1554:
1539:
1535:
1529:
1519:
1516:
1511:
1506:
1499:
1496:
1482:
1481:
1480:
1463:
1455:
1452:
1443:
1440:
1426:
1419:
1400:
1394:
1384:
1379:
1375:
1371:
1368:
1364:
1357:
1349:
1345:
1341:
1335:
1325:
1320:
1316:
1312:
1309:
1305:
1298:
1290:
1286:
1282:
1276:
1266:
1261:
1257:
1253:
1250:
1246:
1239:
1231:
1227:
1223:
1214:
1208:
1196:
1195:
1194:
1192:
1174:
1170:
1166:
1161:
1157:
1134:
1130:
1126:
1121:
1117:
1094:
1090:
1067:
1063:
1054:
1036:
1009:
981:
971:
967:
958:
949:
939:
934:
930:
926:
923:
919:
913:
905:
897:
888:
878:
873:
869:
865:
862:
858:
852:
843:
837:
829:
826:
821:
817:
809:
795:
787:
778:
768:
764:
757:
749:
740:
730:
726:
719:
711:
702:
692:
688:
681:
676:
672:
664:
663:
662:
643:
639:
635:
630:
626:
622:
619:
612:
611:
610:
608:
604:
585:
571:
543:
529:
501:
487:
469:
462:
460:
458:
453:
449:
446:level to the
445:
441:
419:
415:
411:
407:
401:
397:
388:
366:
362:
358:
354:
348:
344:
336:to the state
335:
313:
309:
305:
301:
295:
291:
281:
262:
234:
217:
215:
198:
194:
190:
185:
181:
160:
154:
149:
145:
141:
136:
132:
109:
105:
84:
72:
67:
65:
62:
58:
53:
50:
46:
42:
38:
34:
30:
26:
19:
3331:
3325:
3294:
3262:
3110:
2982:
2782:
1872:
1727:
1625:
1555:
1485:
1416:Solving the
1415:
996:
660:
609:is given by
474:
451:
447:
443:
435:
382:
329:
250:and a state
221:
76:
54:
41:energy level
28:
22:
607:Hamiltonian
73:In practice
3345:Categories
3318:References
29:dark state
3356:Photonics
3280:⟩
3231:Ω
3213:Ω
3202:Ω
3193:θ
3190:
3166:Ω
3148:Ω
3137:Ω
3128:θ
3125:
3096:⟩
3085:θ
3082:
3076:−
3073:⟩
3062:θ
3059:
3050:⟩
3019:θ
2999:⟩
2934:Ω
2925:Ω
2919:−
2888:Ω
2879:Ω
2818:Ω
2800:Ω
2791:Ω
2759:Ω
2753:
2716:Ω
2710:
2702:Ω
2693:Ω
2665:−
2653:Ω
2647:
2639:Ω
2630:Ω
2602:−
2548:Ω
2542:
2534:Ω
2525:Ω
2497:−
2479:Ω
2473:
2459:Ω
2444:Ω
2427:Ω
2412:Ω
2365:Ω
2359:
2345:Ω
2334:Ω
2324:Ω
2306:Ω
2295:Ω
2285:Ω
2278:−
2200:Ω
2194:
2186:Ω
2177:Ω
2149:−
2131:Ω
2125:
2111:Ω
2100:Ω
2090:Ω
2072:Ω
2061:Ω
2051:Ω
2044:−
2000:Ω
1994:
1980:Ω
1965:Ω
1948:Ω
1933:Ω
1855:⟩
1822:⟩
1789:⟩
1756:⟩
1744:ψ
1692:Ω
1669:Ω
1640:˙
1596:Ω
1570:˙
1526:Ω
1500:˙
1467:⟩
1464:ψ
1450:⟩
1444:˙
1441:ψ
1430:ℏ
1398:⟩
1376:ω
1369:−
1339:⟩
1317:ω
1310:−
1280:⟩
1258:ω
1251:−
1221:⟩
1209:ψ
1171:ω
1167:−
1158:ω
1131:ω
1127:−
1118:ω
1091:ω
1064:ω
1033:Ω
1006:Ω
956:⟨
953:⟩
931:ω
910:Ω
895:⟨
892:⟩
870:ω
849:Ω
835:ℏ
830:−
785:⟨
782:⟩
765:ω
761:ℏ
747:⟨
744:⟩
727:ω
723:ℏ
709:⟨
706:⟩
689:ω
685:ℏ
589:⟩
578:↔
575:⟩
547:⟩
536:↔
533:⟩
505:⟩
494:↔
491:⟩
266:⟩
238:⟩
218:In theory
161:ω
158:ℏ
85:ω
3301:See also
1051:are the
61:coherent
33:molecule
45:photons
997:where
3111:with
2783:with
661:with
440:=-1/2
434:with
387:=-1/2
381:with
334:=-1/2
328:with
49:laser
1149:and
1024:and
977:H.c.
517:and
191:<
27:, a
3187:sin
3122:cos
3079:sin
3056:cos
3031:as
2750:cos
2707:sin
2644:sin
2539:sin
2470:cos
2356:cos
2191:sin
2122:cos
1991:cos
23:In
3347::
3297:.
459:.
452:2S
448:1S
444:2S
3277:3
3273:|
3248:.
3240:2
3235:c
3227:+
3222:2
3217:p
3206:p
3196:=
3183:,
3175:2
3170:c
3162:+
3157:2
3152:p
3141:c
3131:=
3093:2
3089:|
3070:1
3066:|
3053:=
3047:D
3043:|
2996:3
2992:|
2968:,
2965:0
2962:=
2959:)
2956:0
2953:(
2948:3
2944:c
2939:,
2929:p
2916:=
2913:)
2910:0
2907:(
2902:2
2898:c
2893:,
2883:c
2873:=
2870:)
2867:0
2864:(
2859:1
2855:c
2827:2
2822:p
2814:+
2809:2
2804:c
2794:=
2766:2
2762:t
2747:)
2744:0
2741:(
2736:3
2732:c
2728:+
2723:2
2719:t
2697:c
2687:)
2684:0
2681:(
2676:2
2672:c
2668:i
2660:2
2656:t
2634:p
2624:)
2621:0
2618:(
2613:1
2609:c
2605:i
2599:=
2596:)
2593:t
2590:(
2585:3
2581:c
2555:2
2551:t
2529:c
2519:)
2516:0
2513:(
2508:3
2504:c
2500:i
2492:]
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2416:p
2405:[
2401:)
2398:0
2395:(
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2386:c
2382:+
2378:]
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2299:c
2289:p
2274:[
2270:)
2267:0
2264:(
2259:1
2255:c
2251:=
2248:)
2245:t
2242:(
2237:2
2233:c
2207:2
2203:t
2181:p
2171:)
2168:0
2165:(
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2144:]
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2094:p
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2076:2
2065:c
2055:p
2040:[
2036:)
2033:0
2030:(
2025:2
2021:c
2017:+
2013:]
2007:2
2003:t
1984:2
1974:2
1969:p
1959:+
1952:2
1942:2
1937:c
1926:[
1922:)
1919:0
1916:(
1911:1
1907:c
1903:=
1900:)
1897:t
1894:(
1889:1
1885:c
1858:,
1852:3
1848:|
1844:)
1841:0
1838:(
1833:3
1829:c
1825:+
1819:2
1815:|
1811:)
1808:0
1805:(
1800:2
1796:c
1792:+
1786:1
1782:|
1778:)
1775:0
1772:(
1767:1
1763:c
1759:=
1753:)
1750:0
1747:(
1740:|
1714:.
1711:)
1706:2
1702:c
1696:c
1688:+
1683:1
1679:c
1673:p
1665:(
1660:2
1657:i
1652:=
1647:3
1637:c
1610:3
1606:c
1600:c
1590:2
1587:i
1582:=
1577:2
1567:c
1540:3
1536:c
1530:p
1520:2
1517:i
1512:=
1507:1
1497:c
1460:|
1456:H
1453:=
1434:|
1427:i
1401:.
1395:3
1391:|
1385:t
1380:3
1372:i
1365:e
1361:)
1358:t
1355:(
1350:3
1346:c
1342:+
1336:2
1332:|
1326:t
1321:2
1313:i
1306:e
1302:)
1299:t
1296:(
1291:2
1287:c
1283:+
1277:1
1273:|
1267:t
1262:1
1254:i
1247:e
1243:)
1240:t
1237:(
1232:1
1228:c
1224:=
1218:)
1215:t
1212:(
1205:|
1175:3
1162:2
1135:3
1122:1
1095:c
1068:p
1037:c
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982:,
972:+
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963:|
959:3
950:2
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940:t
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924:+
920:e
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902:|
898:3
889:1
885:|
879:t
874:p
866:i
863:+
859:e
853:p
844:(
838:2
827:=
822:1
818:H
796:,
792:|
788:3
779:3
775:|
769:3
758:+
754:|
750:2
741:2
737:|
731:2
720:+
716:|
712:1
703:1
699:|
693:1
682:=
677:0
673:H
644:1
640:H
636:+
631:0
627:H
623:=
620:H
586:2
582:|
572:1
568:|
544:3
540:|
530:2
526:|
502:3
498:|
488:1
484:|
438:j
436:m
420:2
416:/
412:3
408:P
402:2
398:2
385:j
383:m
367:2
363:/
359:3
355:P
349:2
345:2
332:j
330:m
314:2
310:/
306:1
302:S
296:2
292:1
263:2
259:|
235:1
231:|
199:2
195:E
186:3
182:E
155:+
150:1
146:E
142:=
137:2
133:E
110:1
106:E
20:.
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