3095:
2719:
2710:. As a result, the D1 line has a saturation intensity 3 times larger than the D2 line. However, when the detuning from these two lines is much larger than the fine-structure splitting, the overall saturation intensity takes the value of the D2 line. In the case where the light's detuning is comparable to the fine-structure splitting but still much larger than the hyperfine splitting, the D2 line contributes twice as much dipole potential as the D1 line, as shown in Equation (19) of.
3927:, although the distinction may be difficult to determine experimentally. While both Autler-Townes splitting and EIT can produce a transparent window in an absorption band, EIT refers to a window that maintains transparency in a weak pump field, and thus requires Fano interference. Because Autler-Townes splitting will wash out Fano interference at stronger fields, a smooth transition between the two effects is evident in materials exhibiting EIT.
3090:{\displaystyle {\begin{aligned}U_{\text{dipole}}(\mathbf {r} )&=-{\frac {\hbar \Omega ^{2}}{4}}\left({\frac {1}{\omega _{0}-\omega }}+{\frac {1}{\omega _{0}+\omega }}\right)\\&=-{\frac {\hbar \Gamma ^{2}}{8I_{sat}}}\left({\frac {1}{\omega _{0}-\omega }}+{\frac {1}{\omega _{0}+\omega }}\right)I(\mathbf {r} )\\&=-{\frac {{\text{Re}}}{2\varepsilon _{0}c}}I(\mathbf {r} )=-{\frac {1}{2}}\langle \mathbf {p\cdot E} \rangle .\end{aligned}}}
148:. Before the availability of lasers, the AC Stark effect was observed with radio frequency sources. Autler and Townes' original observation of the effect used a radio frequency source tuned to 12.78 and 38.28 MHz, corresponding to the separation between two doublet microwave absorption lines of
3672:
In applications that utilize the optical dipole force, it is common practice to use a far-off-resonance light frequency. This is because a smaller detuning would increase the photon-atom scattering rate much faster than it increases the dipole potential energy, leading to undesirable heating of the
2697:
1592:
General expressions for AC Stark shifts must usually be calculated numerically and tend to provide little insight. However, there are important individual examples of the effect that are informative. Analytical solutions in these specific cases are usually obtained assuming the
2006:
155:
The notion of quasi-energy in treating the general AC Stark effect was later developed by
Nikishov and Ritis in 1964 and onward. This more general method of approaching the problem developed into the "dressed atom" model describing the interaction between lasers and atoms.
2412:
Another consequence of the AC Stark splitting here is the appearance of Mollow triplets, a triple peaked fluorescence profile. Historically an important confirmation of Rabi flopping, they were first predicted by Mollow in 1969 and confirmed in the 1970s experimentally.
159:
Prior to the 1970s there were various conflicting predictions concerning the fluorescence spectra of atoms due to the AC Stark effect at optical frequencies. In 1974 the observation of Mollow triplets verified the form of the AC Stark effect using visible light.
2172:
754:
3203:
1327:
1475:
1579:
is useful in a general case of atoms with infinite bound states, obtaining even a limited spectrum of shifted energies for the AC Stark effect is difficult in all but simple models, although calculations for systems such as the
1826:
1739:) can be approximated as a two level quantum system since the off resonance states have low occupation probability. The Hamiltonian can be divided into the bare atom term plus a term for the interaction with the field as:
1089:
337:
3342:
can be omitted; However, in some cases, the ODT light is so far detuned that counter-rotating term must be included in calculations, as well as contributions from adjacent atomic transitions with appreciable linewidth
3747:
1006:
886:
3529:
2583:
2230:
3667:
4671:
Anisimov, Petr M.; Dowling, Jonathan P.; Sanders, Barry C. (2011). "Autler-Townes
Splitting vs. Electromagnetically Induced Transparency: Objective Criterion to Discern Between Them in any Experiment".
1874:
2448:, and the light frequency is often far detuned from the atomic transition to avoid heating the atoms from the photon-atom scattering; in turn, the intensity of the light field (i.e. AC electric field)
85:
While generally referring to atomic spectral shifts due to AC fields at any (single) frequency, the effect is more pronounced when the field frequency is close to that of a natural atomic or molecular
2421:
For ultracold atoms experiments utilizing the optical dipole force from AC Stark shift, the light is usually linearly polarized to avoid the splitting of different magnetic substates with different
581:
3249:
of the atom, with its imaginary counterpart representing the dissipative optical scattering force. The factor of 1/2 takes into account that the dipole moment is an induced, not a permanent one.
2724:
2548:
1639:
3923:
Electromagnetically induced transparency (EIT), which gives some materials a small transparent area within an absorption line, can be thought of as a combination of Autler-Townes splitting and
3293:
1737:
1169:
3340:
2075:
1558:
2087:
1129:
663:
400:
370:
1516:
641:
2477:
2375:
3437:
3123:
524:
82:
which splits the spectral lines of atoms and molecules in a constant electric field. Compared to its DC counterpart, the AC Stark effect is computationally more complex.
3589:
3243:
1694:
4137:
Nikishov, A. I., and V. I. Ritus. Quantum
Processes in the Field of a Plane Electromagnetic Wave and in a Constant Field. PART I. Lebedev Inst. of Physics, Moscow, 1964.
3895:
3867:
3839:
3811:
3783:
2352:
2324:
2293:
2265:
1866:
1181:
3384:
3361:
3118:
2571:
2395:
2029:
1667:
937:
456:
428:
3897:
through large detuning and raising intensity of the driving field. Adiabatic elimination has been used to create comparatively stable effective two level systems in
1338:
3411:
2446:
916:
777:
498:
477:
3552:
89:. In this case, the alternating field has the effect of splitting the two bare transition states into doublets or "dressed states" that are separated by the
3413:. This is the principle of operation for Optical Dipole Trap (ODT, also known as Far Off Resonance Trap, FORT), in which case the light is red-detuned
1745:
1017:
2377:. Evidence of this shift is apparent in the atom's absorption spectrum, which shows two peaks around the bare transition frequency, separated by
182:
3669:, are crucial parameters in understanding the dynamics of atoms in light fields, especially in the context of atom optics and momentum transfer.
3679:
2702:
Note the above expression for saturation intensity does not apply to all cases. For example, the above applies for the D2 line transition of
945:
789:
3452:
106:
2692:{\displaystyle I_{\text{sat}}={\frac {\pi }{3}}{hc\Gamma \over \lambda _{0}^{3}}={\frac {\pi }{3}}{hc \over {\lambda _{0}^{3}\tau }}\,.}
124:. Vacuum Rabi oscillations have also been described as a manifestation of the AC Stark effect from atomic coupling to the vacuum field.
4180:
Zel'Dovich, Ya B. "Scattering and emission of a quantum system in a strong electromagnetic wave." Physics-Uspekhi 16.3 (1973): 427-433.
2184:
2001:{\displaystyle {\hat {H}}=-\hbar \Delta |e\rangle \langle e|+{\frac {\hbar \Omega }{2}}(|e\rangle \langle g|+|g\rangle \langle e|).}
3946:
3918:
3594:
117:
1576:
172:
model where the electromagnetic field is treated classically, a system of charges in a monochromatic electromagnetic field has a
3869:, the dissipative influence of the spontaneous decay can be eliminated. This is achieved by increasing the AC Stark shift on
4423:
3446:, which is the kinetic energy imparted in an atom initially at rest by "recoil" during the spontaneous emission of a photon:
1832:
145:
532:
2298:
The result of the AC field on the atom is thus to shift the strongly coupled bare atom energy eigenstates into two states
2482:
1599:
4288:
Grimm, Rudolf; Weidemüller, Matthias; Ovchinnikov, Yurii B. (2000-01-01), Bederson, Benjamin; Walther, Herbert (eds.),
4357:
4086:
Barnett, Stephen, and Paul M. Radmore. Methods in theoretical quantum optics. Vol. 15. Oxford
University Press, 2002.
654:
4318:
Grimm, Rudolf; Weidemüller, Matthias; Ovchinnikov, Yurii B. (1999-02-24). "Optical dipole traps for neutral atoms".
4025:
Schuda, F; Stroud, C R; Hercher, M (1974-05-11). "Observation of the resonant Stark effect at optical frequencies".
1521:
The solution for the joint particle-field system is, therefore, a linear combination of stationary states of energy
48:
173:
102:
3255:
2574:
1699:
1137:
3298:
2167:{\displaystyle E_{\pm }={\frac {-\hbar \Delta }{2}}\pm {\frac {\hbar {\sqrt {\Omega ^{2}+\Delta ^{2}}}}{2}}}
749:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (\mathbf {\xi } ,t)=H(t)\psi (\mathbf {\xi } ,t),}
2078:
2038:
651:
1524:
1836:
1097:
375:
345:
2398:
1483:
86:
4559:
Radmore, P M; Knight, P L (1982-02-28). "Population trapping and dispersion in a three-level system".
4289:
593:
3198:{\displaystyle s\equiv {\frac {I(\mathbf {r} )}{I_{\text{sat}}}}={\frac {2\Omega ^{2}}{\Gamma ^{2}}}}
2451:
4074:
Fox, Mark. Quantum Optics: An
Introduction: An Introduction. Vol. 15. Oxford university press, 2006.
3391:
2357:
1322:{\displaystyle \psi (\mathbf {\xi } ,t)=\exp\sum _{k=-\infty }^{\infty }C_{k}(\mathbf {\xi } )\exp}
3416:
507:
4762:
4757:
4189:
Crance, Michèle. "Nonperturbative ac Stark shifts in hydrogen atoms." JOSA B 7.4 (1990): 449-455.
3561:
3208:
1672:
63:
spectra of that spectral line. The AC Stark effect was discovered in 1955 by
American physicists
4602:
Linskens, A. F.; Holleman, I.; Dam, N.; Reuss, J. (1996-12-01). "Two-photon Rabi oscillations".
3872:
3844:
3816:
3788:
3760:
2329:
2301:
2270:
2242:
1842:
647:
4387:
Ytterbium and
Lithium Quantum Gases: Heteronuclear Molecules and Bose-Fermi Superfluid Mixtures
1470:{\displaystyle \psi (\mathbf {\xi } ,t)=\sum _{k=-\infty }^{\infty }C_{k}(\mathbf {\xi } )\exp}
141:
4498:
Brion, E; Pedersen, L H; Mølmer, K (2007-01-17). "Adiabatic elimination in a lambda system".
3369:
3346:
3103:
2556:
2380:
2014:
1652:
921:
433:
405:
169:
4385:
4691:
4611:
4568:
4517:
4462:
4394:
4337:
4251:
4208:
4160:
4112:
4034:
3396:
2424:
894:
780:
2409:
laser sweeps for a second transition between a third atomic state and the dressed states.
762:
8:
4100:
3979:
Delone, N B; Krainov, Vladimir P (1999-07-31). "AC Stark shift of atomic energy levels".
3295:, the rotating wave approximation applies, and the counter-rotating term proportional to
2707:
2236:
482:
461:
75:
56:
4695:
4615:
4572:
4521:
4466:
4398:
4341:
4255:
4212:
4164:
4148:
4116:
4038:
4715:
4681:
4653:
4541:
4507:
4452:
4363:
4327:
4199:
Cardoso, G.C.; Tabosa, J.W.R. (2000). "Four-wave mixing in dressed cold cesium atoms".
4004:
3941:
3537:
3443:
4349:
4242:
Mollow, B. R. (1969-12-25). "Power
Spectrum of Light Scattered by Two-Level Systems".
4220:
4707:
4645:
4637:
4584:
4580:
4533:
4529:
4480:
4419:
4353:
4267:
4224:
4149:"Shift and splitting of atomic energy levels by the field of an electromagnetic wave"
4050:
4008:
3996:
3951:
3924:
3906:
3387:
2551:
60:
4719:
4657:
4367:
4046:
3992:
4703:
4699:
4627:
4619:
4576:
4545:
4525:
4470:
4345:
4259:
4216:
4120:
4042:
3988:
501:
149:
121:
94:
3757:
In quantum system with three (or more) states, where a transition from one level,
2397:(Autler-Townes splitting). The modified absorption spectrum can be obtained by a
4413:
1821:{\displaystyle {\hat {H}}={\hat {H}}_{\mathrm {A} }+{\hat {H}}_{\mathrm {int} }.}
4475:
4440:
4096:
3439:. When blue-detuned, the light beam provides a potential bump/barrier instead.
3246:
2032:
1594:
1172:
1084:{\displaystyle \phi (\mathbf {\xi } ,t+2\pi /\omega )=\phi (\mathbf {\xi } ,t)}
137:
133:
113:
90:
68:
64:
36:
4751:
4641:
4623:
4588:
4537:
4484:
4271:
4263:
4228:
4054:
4000:
3898:
1581:
52:
4124:
783:
guarantees that the solutions to an equation of this form can be written as
332:{\displaystyle H=\sum _{i}{\frac {1}{2m_{i}}}\left^{2}+V(\mathbf {r} _{i}),}
4711:
3936:
2706:, but not the D1 line, which obeys a different sum rule in calculating the
2479:
is typically high to compensate for the large detuning. Typically, we have
1572:
79:
32:
20:
4649:
3742:{\displaystyle R_{sc}={\frac {U(\mathbf {r} )}{\hbar \Delta /\Gamma }}\,.}
4512:
3555:
1641:
is small compared to a characteristic frequency of the radiating system.
98:
4332:
3956:
1001:{\displaystyle \phi (\mathbf {\xi } ,t+\tau )=\phi (\mathbf {\xi } ,t)}
918:
is the "bare" energy for no coupling to the electromagnetic field, and
4632:
3591:
when detuned). The recoil energy, along with related recoil frequency
881:{\displaystyle \psi (\mathbf {\xi } ,t)=\exp\phi (\mathbf {\xi } ,t).}
44:
3524:{\displaystyle \varepsilon _{recoil}={\frac {\hbar ^{2}k^{2}}{2m}},}
4686:
4457:
132:
The AC Stark effect was discovered in 1955 by
American physicists
1134:
Because of its periodicity, it is often further useful to expand
458:
are respectively the position, momentum, mass, and charge of the
3442:
The optical dipole potential is often expressed in terms of the
2416:
2225:{\displaystyle E_{\pm }\approx \pm {\frac {\hbar \Omega }{2}}.}
3902:
3662:{\displaystyle \omega _{recoil}=\varepsilon _{recoil}/\hbar }
40:
4317:
4287:
3912:
2703:
112:
The AC Stark splitting is integral to several phenomena in
4441:"Laser-driven Sisyphus cooling in an optical dipole trap"
4095:
4740:, Quantum Mechanics, Vol 2, p 1358, trans. S. R. Hemley
4601:
3673:
atoms. Quantitatively, the scattering rate is given by:
3987:(7). Uspekhi Fizicheskikh Nauk (UFN) Journal: 669–687.
3120:
is related to the (dimensionless) saturation parameter
2077:
are the strongly coupled bare atom states. The energy
4670:
4439:
Ivanov, Vladyslav V.; Gupta, Subhadeep (2011-12-20).
3875:
3847:
3819:
3791:
3763:
3682:
3597:
3564:
3540:
3455:
3419:
3399:
3372:
3349:
3301:
3258:
3211:
3126:
3106:
2722:
2586:
2559:
2485:
2454:
2427:
2383:
2360:
2332:
2304:
2273:
2245:
2187:
2090:
2041:
2017:
1877:
1845:
1748:
1702:
1675:
1655:
1602:
1527:
1486:
1341:
1184:
1140:
1100:
1020:
948:
924:
897:
792:
765:
666:
596:
576:{\displaystyle \mathbf {A} (t+\tau )=\mathbf {A} (t)}
535:
510:
485:
464:
436:
408:
378:
348:
185:
105:. The resulting fluorescence spectrum is known as a
4497:
4024:
2543:{\displaystyle |\Delta |/\Gamma \gg I/I_{sat}\gg 1}
1649:An atom driven by an electric field with frequency
1634:{\displaystyle \Delta \equiv (\omega -\omega _{0})}
4561:Journal of Physics B: Atomic and Molecular Physics
4500:Journal of Physics A: Mathematical and Theoretical
4294:Advances In Atomic, Molecular, and Optical Physics
4103:(1955). "Stark Effect in Rapidly Varying Fields".
4027:Journal of Physics B: Atomic and Molecular Physics
3889:
3861:
3833:
3805:
3777:
3741:
3661:
3583:
3546:
3523:
3431:
3405:
3378:
3355:
3334:
3287:
3237:
3197:
3112:
3089:
2691:
2565:
2542:
2471:
2440:
2389:
2369:
2346:
2318:
2287:
2259:
2224:
2166:
2069:
2023:
2000:
1860:
1820:
1731:
1688:
1661:
1633:
1552:
1510:
1469:
1321:
1163:
1123:
1083:
1000:
939:has the same time-periodicity as the Hamiltonian,
931:
910:
880:
771:
748:
635:
575:
518:
492:
471:
450:
422:
394:
364:
331:
4250:(5). American Physical Society (APS): 1969–1975.
1564:state and the new set of energies are called the
163:
4749:
4320:Advances in Atomic Molecular and Optical Physics
2405:laser drives the bare transition while a weaker
55:, and resulting in a change of the shape of the
4296:, vol. 42, Academic Press, pp. 95–170
4153:Journal of Experimental and Theoretical Physics
35:corresponding to the case when an oscillating
4558:
4198:
3978:
3884:
3856:
3828:
3800:
3772:
3077:
3063:
2417:Optical Dipole Trap (Far-Off-Resonance Trap)
2341:
2313:
2282:
2254:
2235:The eigenstates of the atom-field system or
2064:
2050:
1981:
1978:
1956:
1953:
1913:
1910:
97:between the bare states which are no longer
93:. Alternatively, this can be described as a
4438:
2713:The optical dipole potential is therefore:
1644:
4685:
4631:
4511:
4474:
4456:
4331:
4111:(2). American Physical Society: 703–722.
4082:
4080:
4020:
4018:
3735:
2685:
1507:
928:
489:
468:
447:
419:
391:
361:
4418:. Oxford University Press. p. 199.
4290:"Optical Dipole Traps for Neutral Atoms"
3974:
3972:
3947:Electromagnetically induced transparency
3919:Electromagnetically induced transparency
3913:Electromagnetically induced transparency
3752:
3288:{\displaystyle |\Delta |\ll \omega _{0}}
1732:{\displaystyle |\Delta |\ll \omega _{0}}
1669:close to an atomic transition frequency
1164:{\displaystyle \phi (\mathbf {\xi } ,t)}
587:The Hamiltonian is thus also periodic:
118:electromagnetically induced transparency
3335:{\displaystyle 1/(\omega _{0}+\omega )}
4750:
4241:
4077:
4015:
4379:
4377:
4313:
4311:
4309:
4283:
4281:
4176:
4174:
4146:
4070:
4068:
4066:
4064:
3969:
2070:{\displaystyle |g\rangle ,|e\rangle }
1518:is the frequency of the laser field.
146:Massachusetts Institute of Technology
4411:
4089:
2550:, where the atomic transition has a
1553:{\displaystyle E_{b}+k\hbar \omega }
1131:the angular frequency of the field.
650:, under a periodic Hamiltonian is a
4383:
4131:
1124:{\displaystyle \omega =2\pi /\tau }
13:
4730:
4374:
4306:
4278:
4171:
4140:
4061:
3813:can be driven by an AC field, but
3729:
3721:
3420:
3373:
3350:
3264:
3184:
3173:
3107:
2863:
2765:
2619:
2560:
2504:
2491:
2384:
2364:
2210:
2147:
2134:
2113:
2018:
1933:
1899:
1809:
1806:
1803:
1779:
1708:
1603:
1384:
1379:
1266:
1261:
779:here represents all coordinates.
679:
675:
395:{\displaystyle \mathbf {p} _{i}\,}
365:{\displaystyle \mathbf {r} _{i}\,}
14:
4774:
3841:only decays to states other than
3656:
2110:
1896:
1544:
1511:{\displaystyle \omega =2\pi /T\,}
1461:
1444:
1241:
849:
670:
655:homogeneous differential equation
140:while at Columbia University and
4033:(7). IOP Publishing: L198–L202.
3709:
3366:Note that the natural linewidth
3245:is the real part of the complex
3143:
3073:
3067:
3040:
2966:
2741:
2462:
636:{\displaystyle H(t+\tau )=H(t).}
560:
537:
512:
381:
351:
313:
272:
263:
232:
4664:
4595:
4552:
4491:
4432:
4405:
4235:
4192:
3993:10.1070/pu1999v042n07abeh000557
2472:{\displaystyle I(\mathbf {r} )}
4704:10.1103/PhysRevLett.107.163604
4183:
3877:
3849:
3821:
3793:
3765:
3713:
3705:
3329:
3310:
3268:
3260:
3232:
3229:
3223:
3217:
3147:
3139:
3044:
3036:
3009:
3006:
3000:
2994:
2970:
2962:
2745:
2737:
2495:
2487:
2466:
2458:
2334:
2306:
2275:
2247:
2057:
2043:
1992:
1988:
1971:
1963:
1946:
1942:
1920:
1903:
1884:
1852:
1795:
1771:
1755:
1712:
1704:
1628:
1609:
1464:
1450:
1425:
1416:
1407:
1399:
1359:
1345:
1316:
1298:
1289:
1281:
1244:
1214:
1202:
1188:
1158:
1144:
1078:
1064:
1055:
1024:
995:
981:
972:
952:
872:
858:
852:
822:
810:
796:
740:
726:
720:
714:
705:
691:
627:
621:
612:
600:
570:
564:
553:
541:
323:
308:
288:
267:
164:General semiclassical approach
1:
4350:10.1016/S1049-250X(08)60186-X
4221:10.1016/s0030-4018(00)01033-6
4207:(4–6). Elsevier BV: 353–358.
3962:
2370:{\displaystyle \hbar \Omega }
3901:, which are of interest for
3432:{\displaystyle \Delta <0}
657:with periodic coefficients,
519:{\displaystyle \mathbf {A} }
500:is the speed of light. The
7:
3930:
3584:{\displaystyle k\neq k_{0}}
3238:{\displaystyle {\text{Re}}}
2354:which are now separated by
1837:rotating wave approximation
1689:{\displaystyle \omega _{0}}
1587:
1566:spectrum of quasi-harmonics
10:
4779:
4581:10.1088/0022-3700/15/4/009
4530:10.1088/1751-8113/40/5/011
4476:10.1103/PhysRevA.84.063417
3916:
3890:{\displaystyle |g\rangle }
3862:{\displaystyle |g\rangle }
3834:{\displaystyle |e\rangle }
3806:{\displaystyle |e\rangle }
3778:{\displaystyle |g\rangle }
2347:{\displaystyle |-\rangle }
2319:{\displaystyle |+\rangle }
2288:{\displaystyle |-\rangle }
2260:{\displaystyle |+\rangle }
1861:{\displaystyle {\hat {H}}}
127:
4047:10.1088/0022-3700/7/7/002
3100:Here, the Rabi frequency
78:equivalent of the static
4624:10.1103/PhysRevA.54.4854
4384:Roy, Richard J. (2017).
4264:10.1103/physrev.188.1969
3390:, and is the inverse of
2178:and for small detuning,
176:that can be written as:
4674:Physical Review Letters
4125:10.1103/PhysRev.100.703
3379:{\displaystyle \Gamma }
3356:{\displaystyle \Gamma }
3113:{\displaystyle \Omega }
2566:{\displaystyle \Gamma }
2390:{\displaystyle \Omega }
2024:{\displaystyle \Omega }
1662:{\displaystyle \omega }
1645:Two level atom dressing
932:{\displaystyle \phi \,}
451:{\displaystyle q_{i}\,}
423:{\displaystyle m_{i}\,}
3891:
3863:
3835:
3807:
3779:
3743:
3663:
3585:
3548:
3525:
3433:
3407:
3380:
3357:
3336:
3289:
3239:
3199:
3114:
3091:
2693:
2567:
2544:
2473:
2442:
2391:
2371:
2348:
2320:
2289:
2261:
2226:
2168:
2071:
2025:
2002:
1862:
1822:
1733:
1690:
1663:
1635:
1560:, which is known as a
1554:
1512:
1471:
1388:
1323:
1270:
1165:
1125:
1085:
1002:
933:
912:
882:
773:
750:
637:
577:
520:
494:
473:
452:
424:
396:
366:
333:
16:Dynamical Stark effect
4744:, Hermann, Paris 1977
4201:Optics Communications
4147:Ritus, V. I. (1967).
3892:
3864:
3836:
3808:
3780:
3753:Adiabatic elimination
3744:
3664:
3586:
3549:
3526:
3434:
3408:
3406:{\displaystyle \tau }
3381:
3358:
3337:
3290:
3240:
3200:
3115:
3092:
2694:
2568:
2545:
2474:
2443:
2441:{\displaystyle m_{F}}
2399:pump-probe experiment
2392:
2372:
2349:
2321:
2290:
2262:
2227:
2169:
2072:
2026:
2003:
1863:
1823:
1734:
1691:
1664:
1636:
1555:
1513:
1472:
1365:
1324:
1247:
1166:
1126:
1086:
1003:
934:
913:
911:{\displaystyle E_{b}}
883:
774:
751:
638:
578:
521:
495:
474:
453:
425:
397:
367:
334:
4412:Foot, C. J. (2005).
3873:
3845:
3817:
3789:
3761:
3680:
3595:
3562:
3538:
3453:
3417:
3397:
3370:
3347:
3299:
3256:
3209:
3124:
3104:
2720:
2584:
2575:saturation intensity
2557:
2483:
2452:
2425:
2381:
2358:
2330:
2302:
2271:
2243:
2185:
2088:
2039:
2015:
1875:
1843:
1746:
1700:
1673:
1653:
1600:
1525:
1484:
1339:
1182:
1138:
1098:
1018:
946:
922:
895:
790:
772:{\displaystyle \xi }
763:
664:
648:Schrödinger equation
594:
533:
508:
483:
462:
434:
406:
376:
346:
183:
49:transition frequency
25:Autler–Townes effect
4696:2011PhRvL.107p3604A
4616:1996PhRvA..54.4854L
4573:1982JPhB...15..561R
4522:2007JPhA...40.1033B
4467:2011PhRvA..84f3417I
4399:2017PhDT........64R
4342:2000AAMOP..42...95G
4256:1969PhRv..188.1969M
4213:2000OptCo.185..353C
4165:1967JETP...24.1041R
4117:1955PhRv..100..703A
4101:Charles Hard Townes
4039:1974JPhB....7L.198S
2708:oscillator strength
2678:
2637:
2401:, wherein a strong
1577:perturbation theory
493:{\displaystyle c\,}
472:{\displaystyle i\,}
3942:Stark spectroscopy
3887:
3859:
3831:
3803:
3775:
3739:
3659:
3581:
3558:of the ODT light (
3544:
3521:
3429:
3403:
3388:radians per second
3376:
3353:
3332:
3285:
3235:
3195:
3110:
3087:
3085:
2689:
2664:
2623:
2563:
2540:
2469:
2438:
2387:
2367:
2344:
2316:
2285:
2257:
2222:
2164:
2067:
2021:
1998:
1858:
1831:In an appropriate
1818:
1729:
1686:
1659:
1631:
1550:
1508:
1467:
1319:
1161:
1121:
1081:
998:
929:
908:
878:
769:
746:
633:
573:
516:
490:
479:-th particle, and
469:
448:
420:
392:
362:
329:
201:
101:of the atom–field
47:(or close) to the
31:), is a dynamical
4604:Physical Review A
4445:Physical Review A
4425:978-0-19-850695-9
3952:Fano interference
3925:Fano interference
3907:quantum computing
3905:manipulations in
3733:
3547:{\displaystyle k}
3516:
3215:
3193:
3161:
3158:
3061:
3031:
2992:
2952:
2924:
2894:
2836:
2808:
2778:
2734:
2683:
2651:
2638:
2608:
2594:
2552:natural linewidth
2217:
2162:
2156:
2120:
1940:
1887:
1855:
1835:, and making the
1798:
1774:
1758:
781:Floquet's theorem
686:
260:
222:
192:
87:dipole transition
39:(e.g., that of a
4770:
4736:Cohen-Tannoudji
4724:
4723:
4689:
4668:
4662:
4661:
4635:
4610:(6): 4854–4862.
4599:
4593:
4592:
4556:
4550:
4549:
4515:
4513:quant-ph/0610056
4506:(5): 1033–1043.
4495:
4489:
4488:
4478:
4460:
4436:
4430:
4429:
4409:
4403:
4402:
4392:
4381:
4372:
4371:
4335:
4315:
4304:
4303:
4302:
4301:
4285:
4276:
4275:
4239:
4233:
4232:
4196:
4190:
4187:
4181:
4178:
4169:
4168:
4159:(5): 1041–1044.
4144:
4138:
4135:
4129:
4128:
4093:
4087:
4084:
4075:
4072:
4059:
4058:
4022:
4013:
4012:
3976:
3896:
3894:
3893:
3888:
3880:
3868:
3866:
3865:
3860:
3852:
3840:
3838:
3837:
3832:
3824:
3812:
3810:
3809:
3804:
3796:
3784:
3782:
3781:
3776:
3768:
3748:
3746:
3745:
3740:
3734:
3732:
3728:
3716:
3712:
3700:
3695:
3694:
3668:
3666:
3665:
3660:
3655:
3650:
3649:
3622:
3621:
3590:
3588:
3587:
3582:
3580:
3579:
3553:
3551:
3550:
3545:
3530:
3528:
3527:
3522:
3517:
3515:
3507:
3506:
3505:
3496:
3495:
3485:
3480:
3479:
3438:
3436:
3435:
3430:
3412:
3410:
3409:
3404:
3385:
3383:
3382:
3377:
3362:
3360:
3359:
3354:
3341:
3339:
3338:
3333:
3322:
3321:
3309:
3294:
3292:
3291:
3286:
3284:
3283:
3271:
3263:
3244:
3242:
3241:
3236:
3216:
3213:
3204:
3202:
3201:
3196:
3194:
3192:
3191:
3182:
3181:
3180:
3167:
3162:
3160:
3159:
3156:
3150:
3146:
3134:
3119:
3117:
3116:
3111:
3096:
3094:
3093:
3088:
3086:
3076:
3062:
3054:
3043:
3032:
3030:
3026:
3025:
3012:
2993:
2990:
2987:
2976:
2969:
2958:
2954:
2953:
2951:
2944:
2943:
2930:
2925:
2923:
2916:
2915:
2902:
2895:
2893:
2892:
2891:
2872:
2871:
2870:
2857:
2846:
2842:
2838:
2837:
2835:
2828:
2827:
2814:
2809:
2807:
2800:
2799:
2786:
2779:
2774:
2773:
2772:
2759:
2744:
2736:
2735:
2732:
2698:
2696:
2695:
2690:
2684:
2682:
2677:
2672:
2662:
2654:
2652:
2644:
2639:
2636:
2631:
2622:
2611:
2609:
2601:
2596:
2595:
2592:
2572:
2570:
2569:
2564:
2549:
2547:
2546:
2541:
2533:
2532:
2517:
2503:
2498:
2490:
2478:
2476:
2475:
2470:
2465:
2447:
2445:
2444:
2439:
2437:
2436:
2396:
2394:
2393:
2388:
2376:
2374:
2373:
2368:
2353:
2351:
2350:
2345:
2337:
2325:
2323:
2322:
2317:
2309:
2294:
2292:
2291:
2286:
2278:
2266:
2264:
2263:
2258:
2250:
2231:
2229:
2228:
2223:
2218:
2213:
2205:
2197:
2196:
2173:
2171:
2170:
2165:
2163:
2158:
2157:
2155:
2154:
2142:
2141:
2132:
2126:
2121:
2116:
2105:
2100:
2099:
2076:
2074:
2073:
2068:
2060:
2046:
2030:
2028:
2027:
2022:
2007:
2005:
2004:
1999:
1991:
1974:
1966:
1949:
1941:
1936:
1928:
1923:
1906:
1889:
1888:
1880:
1867:
1865:
1864:
1859:
1857:
1856:
1848:
1827:
1825:
1824:
1819:
1814:
1813:
1812:
1800:
1799:
1791:
1784:
1783:
1782:
1776:
1775:
1767:
1760:
1759:
1751:
1738:
1736:
1735:
1730:
1728:
1727:
1715:
1707:
1695:
1693:
1692:
1687:
1685:
1684:
1668:
1666:
1665:
1660:
1640:
1638:
1637:
1632:
1627:
1626:
1584:have been done.
1559:
1557:
1556:
1551:
1537:
1536:
1517:
1515:
1514:
1509:
1503:
1476:
1474:
1473:
1468:
1460:
1437:
1436:
1406:
1398:
1397:
1387:
1382:
1352:
1328:
1326:
1325:
1320:
1288:
1280:
1279:
1269:
1264:
1240:
1232:
1231:
1195:
1170:
1168:
1167:
1162:
1151:
1130:
1128:
1127:
1122:
1117:
1090:
1088:
1087:
1082:
1071:
1051:
1031:
1007:
1005:
1004:
999:
988:
959:
938:
936:
935:
930:
917:
915:
914:
909:
907:
906:
887:
885:
884:
879:
865:
848:
840:
839:
803:
778:
776:
775:
770:
755:
753:
752:
747:
733:
698:
687:
685:
674:
642:
640:
639:
634:
582:
580:
579:
574:
563:
540:
525:
523:
522:
517:
515:
502:vector potential
499:
497:
496:
491:
478:
476:
475:
470:
457:
455:
454:
449:
446:
445:
429:
427:
426:
421:
418:
417:
401:
399:
398:
393:
390:
389:
384:
371:
369:
368:
363:
360:
359:
354:
338:
336:
335:
330:
322:
321:
316:
301:
300:
295:
291:
281:
280:
275:
266:
261:
256:
255:
246:
241:
240:
235:
223:
221:
220:
219:
203:
200:
122:Sisyphus cooling
95:Rabi oscillation
4778:
4777:
4773:
4772:
4771:
4769:
4768:
4767:
4748:
4747:
4733:
4731:Further reading
4728:
4727:
4669:
4665:
4600:
4596:
4557:
4553:
4496:
4492:
4437:
4433:
4426:
4410:
4406:
4390:
4382:
4375:
4360:
4333:physics/9902072
4316:
4307:
4299:
4297:
4286:
4279:
4244:Physical Review
4240:
4236:
4197:
4193:
4188:
4184:
4179:
4172:
4145:
4141:
4136:
4132:
4105:Physical Review
4094:
4090:
4085:
4078:
4073:
4062:
4023:
4016:
3981:Physics-Uspekhi
3977:
3970:
3965:
3933:
3921:
3915:
3876:
3874:
3871:
3870:
3848:
3846:
3843:
3842:
3820:
3818:
3815:
3814:
3792:
3790:
3787:
3786:
3764:
3762:
3759:
3758:
3755:
3724:
3717:
3708:
3701:
3699:
3687:
3683:
3681:
3678:
3677:
3651:
3630:
3626:
3602:
3598:
3596:
3593:
3592:
3575:
3571:
3563:
3560:
3559:
3539:
3536:
3535:
3508:
3501:
3497:
3491:
3487:
3486:
3484:
3460:
3456:
3454:
3451:
3450:
3418:
3415:
3414:
3398:
3395:
3394:
3371:
3368:
3367:
3348:
3345:
3344:
3317:
3313:
3305:
3300:
3297:
3296:
3279:
3275:
3267:
3259:
3257:
3254:
3253:
3212:
3210:
3207:
3206:
3187:
3183:
3176:
3172:
3168:
3166:
3155:
3151:
3142:
3135:
3133:
3125:
3122:
3121:
3105:
3102:
3101:
3084:
3083:
3066:
3053:
3039:
3021:
3017:
3013:
2989:
2988:
2986:
2974:
2973:
2965:
2939:
2935:
2934:
2929:
2911:
2907:
2906:
2901:
2900:
2896:
2881:
2877:
2873:
2866:
2862:
2858:
2856:
2844:
2843:
2823:
2819:
2818:
2813:
2795:
2791:
2790:
2785:
2784:
2780:
2768:
2764:
2760:
2758:
2748:
2740:
2731:
2727:
2723:
2721:
2718:
2717:
2673:
2668:
2663:
2655:
2653:
2643:
2632:
2627:
2612:
2610:
2600:
2591:
2587:
2585:
2582:
2581:
2558:
2555:
2554:
2522:
2518:
2513:
2499:
2494:
2486:
2484:
2481:
2480:
2461:
2453:
2450:
2449:
2432:
2428:
2426:
2423:
2422:
2419:
2382:
2379:
2378:
2359:
2356:
2355:
2333:
2331:
2328:
2327:
2305:
2303:
2300:
2299:
2274:
2272:
2269:
2268:
2246:
2244:
2241:
2240:
2206:
2204:
2192:
2188:
2186:
2183:
2182:
2150:
2146:
2137:
2133:
2131:
2127:
2125:
2106:
2104:
2095:
2091:
2089:
2086:
2085:
2056:
2042:
2040:
2037:
2036:
2016:
2013:
2012:
1987:
1970:
1962:
1945:
1929:
1927:
1919:
1902:
1879:
1878:
1876:
1873:
1872:
1847:
1846:
1844:
1841:
1840:
1802:
1801:
1790:
1789:
1788:
1778:
1777:
1766:
1765:
1764:
1750:
1749:
1747:
1744:
1743:
1723:
1719:
1711:
1703:
1701:
1698:
1697:
1696:(that is, when
1680:
1676:
1674:
1671:
1670:
1654:
1651:
1650:
1647:
1622:
1618:
1601:
1598:
1597:
1590:
1532:
1528:
1526:
1523:
1522:
1499:
1485:
1482:
1481:
1456:
1432:
1428:
1402:
1393:
1389:
1383:
1369:
1348:
1340:
1337:
1336:
1284:
1275:
1271:
1265:
1251:
1236:
1227:
1223:
1191:
1183:
1180:
1179:
1147:
1139:
1136:
1135:
1113:
1099:
1096:
1095:
1067:
1047:
1027:
1019:
1016:
1015:
984:
955:
947:
944:
943:
923:
920:
919:
902:
898:
896:
893:
892:
861:
844:
835:
831:
799:
791:
788:
787:
764:
761:
760:
729:
694:
678:
673:
665:
662:
661:
595:
592:
591:
559:
536:
534:
531:
530:
511:
509:
506:
505:
484:
481:
480:
463:
460:
459:
441:
437:
435:
432:
431:
413:
409:
407:
404:
403:
385:
380:
379:
377:
374:
373:
355:
350:
349:
347:
344:
343:
317:
312:
311:
296:
276:
271:
270:
262:
251:
247:
245:
236:
231:
230:
229:
225:
224:
215:
211:
207:
202:
196:
184:
181:
180:
166:
130:
29:AC Stark effect
27:(also known as
17:
12:
11:
5:
4776:
4766:
4765:
4763:Quantum optics
4760:
4758:Atomic physics
4746:
4745:
4732:
4729:
4726:
4725:
4680:(16): 163604.
4663:
4594:
4567:(4): 561–573.
4551:
4490:
4431:
4424:
4415:Atomic physics
4404:
4393:. p. 10.
4373:
4358:
4305:
4277:
4234:
4191:
4182:
4170:
4139:
4130:
4088:
4076:
4060:
4014:
3967:
3966:
3964:
3961:
3960:
3959:
3954:
3949:
3944:
3939:
3932:
3929:
3917:Main article:
3914:
3911:
3886:
3883:
3879:
3858:
3855:
3851:
3830:
3827:
3823:
3802:
3799:
3795:
3774:
3771:
3767:
3754:
3751:
3750:
3749:
3738:
3731:
3727:
3723:
3720:
3715:
3711:
3707:
3704:
3698:
3693:
3690:
3686:
3658:
3654:
3648:
3645:
3642:
3639:
3636:
3633:
3629:
3625:
3620:
3617:
3614:
3611:
3608:
3605:
3601:
3578:
3574:
3570:
3567:
3543:
3532:
3531:
3520:
3514:
3511:
3504:
3500:
3494:
3490:
3483:
3478:
3475:
3472:
3469:
3466:
3463:
3459:
3428:
3425:
3422:
3402:
3375:
3352:
3331:
3328:
3325:
3320:
3316:
3312:
3308:
3304:
3282:
3278:
3274:
3270:
3266:
3262:
3247:polarizability
3234:
3231:
3228:
3225:
3222:
3219:
3190:
3186:
3179:
3175:
3171:
3165:
3154:
3149:
3145:
3141:
3138:
3132:
3129:
3109:
3098:
3097:
3082:
3079:
3075:
3072:
3069:
3065:
3060:
3057:
3052:
3049:
3046:
3042:
3038:
3035:
3029:
3024:
3020:
3016:
3011:
3008:
3005:
3002:
2999:
2996:
2985:
2982:
2979:
2977:
2975:
2972:
2968:
2964:
2961:
2957:
2950:
2947:
2942:
2938:
2933:
2928:
2922:
2919:
2914:
2910:
2905:
2899:
2890:
2887:
2884:
2880:
2876:
2869:
2865:
2861:
2855:
2852:
2849:
2847:
2845:
2841:
2834:
2831:
2826:
2822:
2817:
2812:
2806:
2803:
2798:
2794:
2789:
2783:
2777:
2771:
2767:
2763:
2757:
2754:
2751:
2749:
2747:
2743:
2739:
2730:
2726:
2725:
2700:
2699:
2688:
2681:
2676:
2671:
2667:
2661:
2658:
2650:
2647:
2642:
2635:
2630:
2626:
2621:
2618:
2615:
2607:
2604:
2599:
2590:
2562:
2539:
2536:
2531:
2528:
2525:
2521:
2516:
2512:
2509:
2506:
2502:
2497:
2493:
2489:
2468:
2464:
2460:
2457:
2435:
2431:
2418:
2415:
2386:
2366:
2363:
2343:
2340:
2336:
2315:
2312:
2308:
2284:
2281:
2277:
2256:
2253:
2249:
2237:dressed states
2233:
2232:
2221:
2216:
2212:
2209:
2203:
2200:
2195:
2191:
2176:
2175:
2161:
2153:
2149:
2145:
2140:
2136:
2130:
2124:
2119:
2115:
2112:
2109:
2103:
2098:
2094:
2066:
2063:
2059:
2055:
2052:
2049:
2045:
2033:Rabi frequency
2020:
2009:
2008:
1997:
1994:
1990:
1986:
1983:
1980:
1977:
1973:
1969:
1965:
1961:
1958:
1955:
1952:
1948:
1944:
1939:
1935:
1932:
1926:
1922:
1918:
1915:
1912:
1909:
1905:
1901:
1898:
1895:
1892:
1886:
1883:
1854:
1851:
1833:rotating frame
1829:
1828:
1817:
1811:
1808:
1805:
1797:
1794:
1787:
1781:
1773:
1770:
1763:
1757:
1754:
1726:
1722:
1718:
1714:
1710:
1706:
1683:
1679:
1658:
1646:
1643:
1630:
1625:
1621:
1617:
1614:
1611:
1608:
1605:
1589:
1586:
1571:Unlike the DC
1549:
1546:
1543:
1540:
1535:
1531:
1506:
1502:
1498:
1495:
1492:
1489:
1478:
1477:
1466:
1463:
1459:
1455:
1452:
1449:
1446:
1443:
1440:
1435:
1431:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1405:
1401:
1396:
1392:
1386:
1381:
1378:
1375:
1372:
1368:
1364:
1361:
1358:
1355:
1351:
1347:
1344:
1330:
1329:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1287:
1283:
1278:
1274:
1268:
1263:
1260:
1257:
1254:
1250:
1246:
1243:
1239:
1235:
1230:
1226:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1194:
1190:
1187:
1173:Fourier series
1160:
1157:
1154:
1150:
1146:
1143:
1120:
1116:
1112:
1109:
1106:
1103:
1092:
1091:
1080:
1077:
1074:
1070:
1066:
1063:
1060:
1057:
1054:
1050:
1046:
1043:
1040:
1037:
1034:
1030:
1026:
1023:
1009:
1008:
997:
994:
991:
987:
983:
980:
977:
974:
971:
968:
965:
962:
958:
954:
951:
927:
905:
901:
889:
888:
877:
874:
871:
868:
864:
860:
857:
854:
851:
847:
843:
838:
834:
830:
827:
824:
821:
818:
815:
812:
809:
806:
802:
798:
795:
768:
757:
756:
745:
742:
739:
736:
732:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
697:
693:
690:
684:
681:
677:
672:
669:
644:
643:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
585:
584:
572:
569:
566:
562:
558:
555:
552:
549:
546:
543:
539:
514:
504:of the field,
488:
467:
444:
440:
416:
412:
388:
383:
358:
353:
340:
339:
328:
325:
320:
315:
310:
307:
304:
299:
294:
290:
287:
284:
279:
274:
269:
265:
259:
254:
250:
244:
239:
234:
228:
218:
214:
210:
206:
199:
195:
191:
188:
165:
162:
138:Charles Townes
134:Stanley Autler
129:
126:
114:quantum optics
107:Mollow triplet
91:Rabi frequency
69:Charles Townes
65:Stanley Autler
43:) is tuned in
37:electric field
15:
9:
6:
4:
3:
2:
4775:
4764:
4761:
4759:
4756:
4755:
4753:
4743:
4739:
4735:
4734:
4721:
4717:
4713:
4709:
4705:
4701:
4697:
4693:
4688:
4683:
4679:
4675:
4667:
4659:
4655:
4651:
4647:
4643:
4639:
4634:
4629:
4625:
4621:
4617:
4613:
4609:
4605:
4598:
4590:
4586:
4582:
4578:
4574:
4570:
4566:
4562:
4555:
4547:
4543:
4539:
4535:
4531:
4527:
4523:
4519:
4514:
4509:
4505:
4501:
4494:
4486:
4482:
4477:
4472:
4468:
4464:
4459:
4454:
4451:(6): 063417.
4450:
4446:
4442:
4435:
4427:
4421:
4417:
4416:
4408:
4400:
4396:
4389:
4388:
4380:
4378:
4369:
4365:
4361:
4359:9780120038428
4355:
4351:
4347:
4343:
4339:
4334:
4329:
4325:
4321:
4314:
4312:
4310:
4295:
4291:
4284:
4282:
4273:
4269:
4265:
4261:
4257:
4253:
4249:
4245:
4238:
4230:
4226:
4222:
4218:
4214:
4210:
4206:
4202:
4195:
4186:
4177:
4175:
4166:
4162:
4158:
4154:
4150:
4143:
4134:
4126:
4122:
4118:
4114:
4110:
4106:
4102:
4098:
4092:
4083:
4081:
4071:
4069:
4067:
4065:
4056:
4052:
4048:
4044:
4040:
4036:
4032:
4028:
4021:
4019:
4010:
4006:
4002:
3998:
3994:
3990:
3986:
3982:
3975:
3973:
3968:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3940:
3938:
3935:
3934:
3928:
3926:
3920:
3910:
3908:
3904:
3900:
3899:Rydberg atoms
3881:
3853:
3825:
3797:
3769:
3736:
3725:
3718:
3702:
3696:
3691:
3688:
3684:
3676:
3675:
3674:
3670:
3652:
3646:
3643:
3640:
3637:
3634:
3631:
3627:
3623:
3618:
3615:
3612:
3609:
3606:
3603:
3599:
3576:
3572:
3568:
3565:
3557:
3541:
3518:
3512:
3509:
3502:
3498:
3492:
3488:
3481:
3476:
3473:
3470:
3467:
3464:
3461:
3457:
3449:
3448:
3447:
3445:
3444:recoil energy
3440:
3426:
3423:
3400:
3393:
3389:
3364:
3326:
3323:
3318:
3314:
3306:
3302:
3280:
3276:
3272:
3250:
3248:
3226:
3220:
3188:
3177:
3169:
3163:
3152:
3136:
3130:
3127:
3080:
3070:
3058:
3055:
3050:
3047:
3033:
3027:
3022:
3018:
3014:
3003:
2997:
2983:
2980:
2978:
2959:
2955:
2948:
2945:
2940:
2936:
2931:
2926:
2920:
2917:
2912:
2908:
2903:
2897:
2888:
2885:
2882:
2878:
2874:
2867:
2859:
2853:
2850:
2848:
2839:
2832:
2829:
2824:
2820:
2815:
2810:
2804:
2801:
2796:
2792:
2787:
2781:
2775:
2769:
2761:
2755:
2752:
2750:
2728:
2716:
2715:
2714:
2711:
2709:
2705:
2686:
2679:
2674:
2669:
2665:
2659:
2656:
2648:
2645:
2640:
2633:
2628:
2624:
2616:
2613:
2605:
2602:
2597:
2588:
2580:
2579:
2578:
2576:
2553:
2537:
2534:
2529:
2526:
2523:
2519:
2514:
2510:
2507:
2500:
2455:
2433:
2429:
2414:
2410:
2408:
2404:
2400:
2361:
2338:
2310:
2296:
2279:
2251:
2238:
2219:
2214:
2207:
2201:
2198:
2193:
2189:
2181:
2180:
2179:
2159:
2151:
2143:
2138:
2128:
2122:
2117:
2107:
2101:
2096:
2092:
2084:
2083:
2082:
2080:
2061:
2053:
2047:
2034:
1995:
1984:
1975:
1967:
1959:
1950:
1937:
1930:
1924:
1916:
1907:
1893:
1890:
1881:
1871:
1870:
1869:
1849:
1838:
1834:
1815:
1792:
1785:
1768:
1761:
1752:
1742:
1741:
1740:
1724:
1720:
1716:
1681:
1677:
1656:
1642:
1623:
1619:
1615:
1612:
1606:
1596:
1585:
1583:
1582:hydrogen atom
1578:
1574:
1569:
1567:
1563:
1547:
1541:
1538:
1533:
1529:
1519:
1504:
1500:
1496:
1493:
1490:
1487:
1457:
1453:
1447:
1441:
1438:
1433:
1429:
1422:
1419:
1413:
1410:
1403:
1394:
1390:
1376:
1373:
1370:
1366:
1362:
1356:
1353:
1349:
1342:
1335:
1334:
1333:
1313:
1310:
1307:
1304:
1301:
1295:
1292:
1285:
1276:
1272:
1258:
1255:
1252:
1248:
1237:
1233:
1228:
1224:
1220:
1217:
1211:
1208:
1205:
1199:
1196:
1192:
1185:
1178:
1177:
1176:
1174:
1155:
1152:
1148:
1141:
1132:
1118:
1114:
1110:
1107:
1104:
1101:
1075:
1072:
1068:
1061:
1058:
1052:
1048:
1044:
1041:
1038:
1035:
1032:
1028:
1021:
1014:
1013:
1012:
992:
989:
985:
978:
975:
969:
966:
963:
960:
956:
949:
942:
941:
940:
925:
903:
899:
875:
869:
866:
862:
855:
845:
841:
836:
832:
828:
825:
819:
816:
813:
807:
804:
800:
793:
786:
785:
784:
782:
766:
743:
737:
734:
730:
723:
717:
711:
708:
702:
699:
695:
688:
682:
667:
660:
659:
658:
656:
653:
649:
630:
624:
618:
615:
609:
606:
603:
597:
590:
589:
588:
567:
556:
550:
547:
544:
529:
528:
527:
503:
486:
465:
442:
438:
414:
410:
386:
356:
326:
318:
305:
302:
297:
292:
285:
282:
277:
257:
252:
248:
242:
237:
226:
216:
212:
208:
204:
197:
193:
189:
186:
179:
178:
177:
175:
171:
170:semiclassical
161:
157:
153:
151:
147:
143:
139:
135:
125:
123:
119:
115:
110:
108:
104:
100:
96:
92:
88:
83:
81:
77:
72:
70:
66:
62:
58:
54:
53:spectral line
50:
46:
42:
38:
34:
30:
26:
22:
4741:
4737:
4677:
4673:
4666:
4607:
4603:
4597:
4564:
4560:
4554:
4503:
4499:
4493:
4448:
4444:
4434:
4414:
4407:
4386:
4323:
4319:
4298:, retrieved
4293:
4247:
4243:
4237:
4204:
4200:
4194:
4185:
4156:
4152:
4142:
4133:
4108:
4104:
4097:Autler, S. H
4091:
4030:
4026:
3984:
3980:
3937:Stark effect
3922:
3756:
3671:
3533:
3441:
3365:
3251:
3099:
2712:
2701:
2420:
2411:
2406:
2402:
2297:
2234:
2177:
2010:
1830:
1648:
1591:
1573:Stark effect
1570:
1565:
1562:quasi-energy
1561:
1520:
1479:
1331:
1175:, obtaining
1133:
1093:
1010:
890:
758:
645:
586:
526:, satisfies
341:
167:
158:
154:
142:Lincoln Labs
131:
111:
84:
80:Stark effect
73:
33:Stark effect
28:
24:
21:spectroscopy
18:
3785:to another
3386:here is in
2239:are dubbed
2079:eigenvalues
1868:reduces to
174:Hamiltonian
103:Hamiltonian
99:eigenstates
51:of a given
4752:Categories
4633:2066/27687
4300:2023-07-26
3963:References
3957:Rabi cycle
3556:wavevector
116:, such as
74:It is the
57:absorption
4687:1102.0546
4642:1050-2947
4589:0022-3700
4538:1751-8113
4485:1050-2947
4458:1110.3439
4272:0031-899X
4229:0030-4018
4055:0022-3700
4009:202602476
4001:1063-7869
3885:⟩
3857:⟩
3829:⟩
3801:⟩
3773:⟩
3730:Γ
3722:Δ
3719:ℏ
3657:ℏ
3628:ε
3600:ω
3569:≠
3489:ℏ
3458:ε
3421:Δ
3401:τ
3374:Γ
3351:Γ
3327:ω
3315:ω
3277:ω
3273:≪
3265:Δ
3227:ω
3221:α
3185:Γ
3174:Ω
3131:≡
3108:Ω
3078:⟩
3071:⋅
3064:⟨
3051:−
3019:ε
3004:ω
2998:α
2984:−
2949:ω
2937:ω
2921:ω
2918:−
2909:ω
2864:Γ
2860:ℏ
2854:−
2833:ω
2821:ω
2805:ω
2802:−
2793:ω
2766:Ω
2762:ℏ
2756:−
2680:τ
2666:λ
2646:π
2625:λ
2620:Γ
2603:π
2561:Γ
2535:≫
2508:≫
2505:Γ
2492:Δ
2385:Ω
2365:Ω
2362:ℏ
2342:⟩
2339:−
2314:⟩
2283:⟩
2280:−
2255:⟩
2211:Ω
2208:ℏ
2202:±
2199:≈
2194:±
2148:Δ
2135:Ω
2129:ℏ
2123:±
2114:Δ
2111:ℏ
2108:−
2097:±
2065:⟩
2051:⟩
2019:Ω
1982:⟨
1979:⟩
1957:⟨
1954:⟩
1934:Ω
1931:ℏ
1914:⟨
1911:⟩
1900:Δ
1897:ℏ
1894:−
1885:^
1853:^
1796:^
1772:^
1756:^
1721:ω
1717:≪
1709:Δ
1678:ω
1657:ω
1620:ω
1616:−
1613:ω
1607:≡
1604:Δ
1548:ω
1545:ℏ
1497:π
1488:ω
1462:ℏ
1448:ω
1445:ℏ
1420:−
1414:
1404:ξ
1385:∞
1380:∞
1377:−
1367:∑
1350:ξ
1343:ψ
1311:ω
1302:−
1296:
1286:ξ
1267:∞
1262:∞
1259:−
1249:∑
1242:ℏ
1218:−
1212:
1193:ξ
1186:ψ
1149:ξ
1142:ϕ
1119:τ
1111:π
1102:ω
1069:ξ
1062:ϕ
1053:ω
1045:π
1029:ξ
1022:ϕ
986:ξ
979:ϕ
970:τ
957:ξ
950:ϕ
926:ϕ
863:ξ
856:ϕ
850:ℏ
826:−
820:
801:ξ
794:ψ
767:ξ
731:ξ
724:ψ
696:ξ
689:ψ
680:∂
676:∂
671:ℏ
646:Now, the
610:τ
551:τ
243:−
194:∑
45:resonance
4720:15372792
4712:22107383
4658:40169348
4368:16499267
3931:See also
3392:lifetime
1595:detuning
1588:Examples
1575:, where
61:emission
4692:Bibcode
4650:9914052
4612:Bibcode
4569:Bibcode
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