3809:
3285:
3804:{\displaystyle {\begin{bmatrix}u\\v\\w\end{bmatrix}}={\begin{bmatrix}{\frac {(\kappa E_{0})^{2}+\delta ^{2}\cos \Omega t}{\Omega ^{2}}}&-{\frac {\delta }{\Omega }}\sin {\Omega t}&-{\frac {\delta \kappa E_{0}}{\Omega ^{2}}}(1-\cos \Omega t)\\{\frac {\delta }{\Omega }}\sin \Omega t&\cos \Omega t&{\frac {\kappa E_{0}}{\Omega }}\sin \Omega t\\{\frac {\delta \kappa E_{0}}{\Omega ^{2}}}(1-\cos \Omega t)&-{\frac {\kappa E_{0}}{\Omega }}\sin {\Omega t}&{\frac {\delta ^{2}+(\kappa E_{0})^{2}\cos \Omega t}{\Omega ^{2}}}\end{bmatrix}}{\begin{bmatrix}u_{0}\\v_{0}\\w_{0}\end{bmatrix}}.}
2071:, which can be interpreted as the population difference between the excited and ground state (varying from −1 to represent completely in the ground state to +1, completely in the excited state). Keep in mind that for the classical case, there was a continuous energy spectrum that the atomic oscillator could occupy, while for the quantum case (as we've assumed) there are only two possible (eigen)states of the problem.
2921:
2694:
2254:
2705:
2080:
4002:
1813:
2508:
209:
1653:
1428:
5454:
4904:
2322:
5504:
An exact solution of spin-1/2 system in an oscillating magnetic field is solved by Rabi (1937). From their work, it is clear that the Rabi oscillation frequency is proportional to the magnitude of oscillation magnetic field.
5150:
2916:{\displaystyle {\begin{bmatrix}u'\\v'\\w'\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&\cos \Omega t&\sin \Omega t\\0&-\sin \Omega t&\cos \Omega t\end{bmatrix}}{\begin{bmatrix}u''\\v''\\w''\end{bmatrix}},}
1195:
376:
3060:
4278:
2249:{\displaystyle {\frac {d}{dt}}{\begin{bmatrix}u\\v\\w\\\end{bmatrix}}={\begin{bmatrix}0&-\delta &0\\\delta &0&\kappa E\\0&-\kappa E&0\end{bmatrix}}{\begin{bmatrix}u\\v\\w\\\end{bmatrix}}.}
5220:
628:
2066:
There is a clear analogy here between these equations and those that defined the evolution of the in-phase and out-of-phase components of oscillation in the classical case. Now, however, there is a third term
3845:
1041:
1668:
4696:
2689:{\displaystyle {\begin{bmatrix}u\\v\\w\end{bmatrix}}={\begin{bmatrix}\cos \chi &0&\sin \chi \\0&1&0\\-\sin \chi &0&\cos \chi \end{bmatrix}}{\begin{bmatrix}u'\\v'\\w'\end{bmatrix}}}
2981:
2423:
4534:
68:
2500:
970:
907:
848:
4763:
1982:
789:
737:
4471:
501:
1125:
3222:
685:
4012:
In the quantum approach, the periodic driving force can be considered as periodic perturbation and, therefore, the problem can be solved using time-dependent perturbation theory, with
2371:
2030:
1864:
approximation, in which one only treats two energy levels of the atom in question. No atom with only two energy levels exists in reality, but a transition between, for example, two
5238:
1928:
4999:
4080:
4771:
4367:
3092:
5499:
3171:
250:
4963:
4935:
4309:
3137:
1434:
1209:
1071:
277:
4403:
4146:
2061:
4110:
3274:
3250:
4166:
2269:
1880:
280:
5007:
1133:
289:
1841:
The classical Rabi problem gives some basic results and a simple to understand picture of the issue, but in order to understand phenomena such as
2987:
4174:
5158:
3997:{\displaystyle w(t;\delta )=-1+{\frac {2(\kappa E_{0})^{2}}{(\kappa E_{0})^{2}+\delta ^{2}}}\sin ^{2}\left({\frac {\Omega t}{2}}\right).}
1868:
in an atom can be treated, to first approximation, as if only those two levels existed, assuming the drive is not too far off resonance.
537:
1808:{\displaystyle x_{a}(t)={\frac {e}{m}}E_{0}\left({\frac {e^{i\omega t}}{\omega _{a}^{2}-\omega ^{2}+2i\omega /T}}+{\text{c.c.}}\right),}
976:
4542:
2932:
2039:
in throwing out terms with high angular velocity (and thus small effect on the total spin dynamics over long time periods) and
2376:
204:{\displaystyle {\ddot {x}}_{a}+{\frac {2}{\tau _{0}}}{\dot {x}}_{a}+\omega _{a}^{2}x_{a}={\frac {e}{m}}E(t,\mathbf {r} _{a}),}
5564:
4476:
5513:
In Bloch's approach, the field is not quantized, and neither the resulting coherence nor the resonance is well explained.
2442:
918:
2502:. In this case, the solution can be found by applying two successive rotations to the matrix equation above, of the form
853:
794:
4704:
55:
1934:
742:
690:
4408:
398:
1091:
3180:
912:
With these assumptions, the
Lorentz force equations for the in-phase and out-of-phase parts can be rewritten as
657:
5449:{\displaystyle P_{m}=d_{m}(t)^{*}d_{m}(t)=d_{m,-}^{2}(0)+d_{m,+}^{2}(0)+2d_{m,-}(0)d_{m,+}(0)\cos(2\Omega t).}
2330:
1988:
1871:
The convenience of the two-level atom is that any two-level system evolves in essentially the same way as a
1889:
4968:
4899:{\displaystyle i\hbar {\dot {d}}_{m}=\sum _{n}\langle m|H^{1}|n\rangle e^{i(\omega _{mn}-\omega )t}d_{n}.}
4018:
1127:, which serves equally well to distinguish atoms of different resonant frequencies. Finally, the constant
2036:
4314:
3173:. If this (on-resonance) pulse is shone on a collection of atoms originally all in their ground state (
3095:
3068:
5520:
3109:
axis (given by the first coordinate transformation above). As an example, if the electric field (or
42:. It provides a simple and generally solvable example of light–atom interactions and is named after
5462:
1861:
3146:
1648:{\displaystyle v(t;\delta )=e^{-t/T}-\kappa E_{0}\int _{0}^{t}dt'\cos \delta (t-t')e^{-(t-t')/T}.}
1423:{\displaystyle u(t;\delta )=e^{-t/T}+\kappa E_{0}\int _{0}^{t}dt'\sin \delta (t-t')e^{-(t-t')/T},}
5630:
5542:
1850:
1836:
228:
4940:
4912:
4286:
3116:
1049:
255:
4372:
4115:
2046:
5635:
5593:
4088:
1846:
1842:
1081:(which could include other interactions such as collisions) and have dropped the subscript
3259:
3235:
2317:{\displaystyle {\frac {d\mathbf {\rho } }{dt}}=\mathbf {\Omega } \times \mathbf {\rho } ,}
8:
2040:
1865:
687:), then these values will be slowly varying in time, and we can make the assumption that
5597:
214:
where it has been assumed that the atom can be treated as a charged particle (of charge
5001:, it becomes a normal-mode problem of a two-level system, and it is easy to find that
4151:
3818:
can be greatly simplified if the atom is assumed to be initially in its ground state (
54:
In the classical approach, the Rabi problem can be represented by the solution to the
5609:
5570:
5560:
1854:
1819:
43:
39:
5601:
4311:
represents the eigenstates of the unperturbed states. For an unperturbed system,
1876:
1659:
5145:{\displaystyle d_{m,n}(t)=d_{m,n,+}(0)e^{i\Omega t}+d_{m,n,-}(0)e^{-i\Omega t},}
1190:{\displaystyle \kappa \ {\stackrel {\text{def}}{=}}\ {\frac {e}{m\omega x_{0}}}}
654:
are allowed to vary in time. However, if the system is very close to resonance (
371:{\displaystyle {\frac {2}{\tau _{0}}}={\frac {2e^{2}\omega _{a}^{2}}{3mc^{3}}},}
5537:
1086:
31:
5624:
5613:
5584:
Rabi, I. I. (1937-04-15). "Space
Quantization in a Gyrating Magnetic Field".
4112:
is the time-independent
Hamiltonian that gives the original eigenstates, and
59:
5574:
5605:
3055:{\displaystyle \Omega (\delta )={\sqrt {\delta ^{2}+(\kappa E_{0})^{2}}}.}
2428:
As before, the Rabi problem is solved by assuming that the electric field
218:) oscillating about its equilibrium position around a neutral atom. Here
28:
4273:{\displaystyle \phi (t)=\sum _{n}d_{n}(t)e^{-iE_{n}t/\hbar }|n\rangle ,}
5532:
3099:
2260:
388:
To apply this to the Rabi problem, one assumes that the electric field
4007:
35:
5215:{\displaystyle \Omega ={\frac {\langle m|H^{1}|n\rangle }{\hbar }}.}
1872:
623:{\displaystyle x_{a}=x_{0}(u_{a}\cos \omega t+v_{a}\sin \omega t).}
4909:
When the excitation frequency is at resonance between two states
2259:
It is noteworthy that these equations can be written as a vector
1662:
have died away, the steady-state solution takes the simple form
1036:{\displaystyle {\dot {v}}=\delta u-{\frac {v}{T}}+\kappa E_{0},}
382:
4691:{\displaystyle 0=\sum _{n}e^{-iE_{n}^{0}t/\hbar }|n\rangle ,}
3110:
3224:, then after the pulse, the atoms will now all be in their
2976:{\displaystyle \tan \chi ={\frac {\delta }{\kappa E_{0}}},}
24:
385:
oscillator's energy loss from electromagnetic radiation.
2418:{\displaystyle \mathbf {\Omega } =(-\kappa E,0,\delta )}
4529:{\displaystyle i\hbar \partial /\partial t-H^{0}-H^{1}}
3749:
3330:
3294:
3139:), then the pseudo-spin vector will precess about the
2867:
2765:
2714:
2643:
2553:
2517:
2215:
2140:
2104:
5465:
5241:
5161:
5010:
4971:
4943:
4915:
4774:
4707:
4545:
4479:
4411:
4375:
4317:
4289:
4177:
4154:
4118:
4091:
4021:
3848:
3288:
3262:
3238:
3183:
3149:
3119:
3071:
2990:
2935:
2708:
2511:
2445:
2379:
2333:
2272:
2083:
2049:
1991:
1937:
1892:
1671:
1437:
1212:
1136:
1094:
1052:
979:
921:
856:
797:
745:
693:
660:
540:
401:
292:
258:
231:
71:
2495:{\displaystyle E=E_{0}(e^{i\omega t}+{\text{c.c.}})}
965:{\displaystyle {\dot {u}}=-\delta v-{\frac {u}{T}},}
531:
that is out of phase (corresponding to absorption):
4536:on both sides of the previous equation, we can get
4148:is the time-dependent perturbation. Assume at time
3276:-pulse and has the result of a complete inversion.
2074:These equations can also be stated in matrix form:
902:{\displaystyle {\ddot {v}}_{a}\ll \omega ^{2}v_{a}}
843:{\displaystyle {\ddot {u}}_{a}\ll \omega ^{2}u_{a}}
5493:
5448:
5214:
5144:
4993:
4957:
4929:
4898:
4757:
4690:
4528:
4465:
4397:
4361:
4303:
4272:
4160:
4140:
4104:
4074:
4008:Rabi Problem in time-dependent perturbation theory
3996:
3803:
3268:
3244:
3216:
3165:
3131:
3086:
3054:
2975:
2915:
2688:
2494:
2417:
2365:
2316:
2248:
2055:
2043:into a set of coordinates rotating at a frequency
2024:
1976:
1922:
1807:
1647:
1422:
1189:
1119:
1065:
1035:
964:
901:
842:
783:
731:
679:
622:
495:
370:
271:
244:
203:
5501:depends on the initial condition of the system.
5622:
5508:
4758:{\displaystyle \langle m|e^{iE_{m}^{0}t/\hbar }}
4701:and then multiply both sides of the equation by
3102:of the pseudo-spin vector about the transformed
5592:(8). American Physical Society (APS): 652–654.
524:field (corresponding to dispersion) and a part
225:is its instantaneous magnitude of oscillation,
1977:{\displaystyle {\dot {v}}=\delta u+\kappa Ew,}
784:{\displaystyle {\dot {v}}_{a}\ll \omega v_{a}}
732:{\displaystyle {\dot {u}}_{a}\ll \omega u_{a}}
392:is oscillatory in time and constant in space:
5200:
5171:
4952:
4924:
4842:
4813:
4708:
4682:
4466:{\displaystyle H^{1}(t)=H^{1}e^{-i\omega t}}
4298:
4264:
1046:where we have replaced the natural lifetime
496:{\displaystyle E=E_{0}=2E_{0}\cos \omega t,}
5554:
1120:{\displaystyle \delta =\omega -\omega _{a}}
3217:{\displaystyle \Delta t=\pi /\kappa E_{0}}
1203:These equations can be solved as follows:
680:{\displaystyle \omega \approx \omega _{a}}
1830:
49:
2366:{\displaystyle \mathbf {\rho } =(u,v,w)}
2432:is oscillatory with constant magnitude
381:which has been calculated based on the
252:its natural oscillation frequency, and
5623:
5224:The possibility of being in the state
2025:{\displaystyle {\dot {w}}=-\kappa Ev,}
5557:Optical resonance and two-level atoms
3113:) is exactly on resonance (such that
1923:{\displaystyle {\dot {u}}=-\delta v,}
1860:The simplest approach is through the
5583:
4994:{\displaystyle \omega =\omega _{mn}}
4369:is a constant. Now, let's calculate
4075:{\displaystyle H(t)=H^{0}+H^{1}(t),}
1879:, which define the dynamics of the
13:
5434:
5162:
5131:
5078:
4494:
4486:
3975:
3725:
3717:
3660:
3648:
3617:
3591:
3560:
3549:
3524:
3510:
3499:
3481:
3455:
3422:
3410:
3389:
3381:
3184:
3072:
2991:
2848:
2834:
2810:
2796:
520:that is in-phase with the driving
14:
5647:
5204:
4778:
4750:
4669:
4568:
4483:
4362:{\displaystyle d_{n}(t)=d_{n}(0)}
4251:
3814:The expression for the inversion
1825:
56:driven damped harmonic oscillator
5555:Allen, L; Eberly, J. H. (1987).
3087:{\displaystyle \Omega (\delta )}
2381:
2299:
185:
3279:The general result is given by
2373:is the pseudo-spin vector, and
640:is assumed to be constant, but
5488:
5482:
5440:
5428:
5419:
5413:
5394:
5388:
5363:
5357:
5330:
5324:
5297:
5291:
5272:
5265:
5193:
5178:
5117:
5111:
5067:
5061:
5033:
5027:
4945:
4917:
4875:
4853:
4835:
4820:
4715:
4675:
4632:
4562:
4428:
4422:
4405:under a periodic perturbation
4392:
4386:
4356:
4350:
4334:
4328:
4291:
4257:
4219:
4213:
4187:
4181:
4135:
4129:
4066:
4060:
4031:
4025:
3930:
3913:
3902:
3885:
3864:
3852:
3702:
3685:
3623:
3602:
3487:
3466:
3353:
3336:
3081:
3075:
3038:
3021:
3000:
2994:
2489:
2462:
2412:
2388:
2360:
2342:
1688:
1682:
1629:
1612:
1601:
1584:
1509:
1459:
1453:
1441:
1404:
1387:
1376:
1359:
1284:
1234:
1228:
1216:
1085:in favor of the newly defined
614:
564:
459:
418:
195:
174:
58:with the electric part of the
1:
5548:
5519:for the QFT approach, mainly
5509:Quantum field theory approach
5494:{\displaystyle d_{m,\pm }(0)}
4168:, we can expand the state as
3252:(or 180°) rotation about the
2425:acts as an effective torque.
1875:system, in accordance to the
3166:{\displaystyle \kappa E_{0}}
1818:where "c.c." stands for the
23:concerns the response of an
7:
5526:
2037:rotating wave approximation
245:{\displaystyle \omega _{a}}
10:
5652:
4958:{\displaystyle |n\rangle }
4930:{\displaystyle |m\rangle }
4304:{\displaystyle |n\rangle }
3098:, which gives the rate of
3096:generalized Rabi frequency
1834:
513:is decomposed into a part
3256:axis. This is known as a
3132:{\displaystyle \delta =0}
1066:{\displaystyle \tau _{0}}
272:{\displaystyle \tau _{0}}
38:very close to the atom's
16:Problem in quantum optics
4398:{\displaystyle d_{n}(t)}
4141:{\displaystyle H^{1}(t)}
1857:treatment is necessary.
5543:Vacuum Rabi oscillation
2056:{\displaystyle \omega }
2035:where we have made the
1837:optical Bloch equations
5606:10.1103/physrev.51.652
5495:
5450:
5216:
5146:
4995:
4959:
4931:
4900:
4759:
4692:
4530:
4467:
4399:
4363:
4305:
4274:
4162:
4142:
4106:
4076:
3998:
3805:
3270:
3246:
3218:
3167:
3133:
3088:
3056:
2977:
2917:
2690:
2496:
2419:
2367:
2318:
2250:
2057:
2026:
1978:
1924:
1883:in an electric field:
1831:Semiclassical approach
1822:of the opposing term.
1809:
1649:
1424:
1191:
1121:
1067:
1037:
966:
903:
844:
785:
733:
681:
624:
497:
372:
273:
246:
205:
50:Classical Rabi problem
5521:Jaynes–Cummings model
5496:
5451:
5217:
5147:
4996:
4960:
4932:
4901:
4760:
4693:
4531:
4468:
4400:
4364:
4306:
4275:
4163:
4143:
4107:
4105:{\displaystyle H^{0}}
4077:
3999:
3806:
3271:
3247:
3232:= +1) because of the
3219:
3168:
3134:
3089:
3057:
2978:
2918:
2691:
2497:
2420:
2368:
2319:
2251:
2058:
2027:
1979:
1925:
1810:
1650:
1425:
1192:
1122:
1068:
1038:
967:
904:
845:
786:
734:
682:
625:
498:
373:
274:
247:
206:
62:as the driving term:
5463:
5239:
5159:
5008:
4969:
4941:
4913:
4772:
4705:
4543:
4477:
4473:. Applying operator
4409:
4373:
4315:
4287:
4175:
4152:
4116:
4089:
4019:
3846:
3286:
3269:{\displaystyle \pi }
3260:
3245:{\displaystyle \pi }
3236:
3181:
3147:
3117:
3069:
2988:
2933:
2706:
2509:
2443:
2377:
2331:
2270:
2081:
2047:
1989:
1935:
1890:
1847:spontaneous emission
1669:
1435:
1210:
1134:
1092:
1073:with a more general
1050:
977:
919:
854:
795:
743:
691:
658:
538:
399:
290:
256:
229:
69:
5598:1937PhRv...51..652R
5559:. New York: Dover.
5356:
5323:
4741:
4660:
3839:= 0, in which case
3065:Here the frequency
1851:Bloch–Siegert shift
1752:
1563:
1338:
343:
147:
5491:
5446:
5336:
5303:
5212:
5142:
4991:
4955:
4927:
4896:
4812:
4755:
4727:
4688:
4646:
4561:
4526:
4463:
4395:
4359:
4301:
4270:
4202:
4158:
4138:
4102:
4072:
3994:
3801:
3792:
3738:
3316:
3266:
3242:
3214:
3163:
3143:axis at a rate of
3129:
3084:
3052:
2973:
2913:
2904:
2856:
2751:
2686:
2680:
2632:
2539:
2492:
2415:
2363:
2314:
2246:
2237:
2204:
2126:
2053:
2022:
1974:
1920:
1881:pseudo-spin vector
1855:quantum-mechanical
1805:
1738:
1645:
1549:
1420:
1324:
1200:has been defined.
1187:
1117:
1063:
1033:
962:
899:
840:
781:
729:
677:
620:
493:
368:
329:
269:
242:
201:
133:
34:, with an applied
5566:978-0-486-65533-8
5207:
4803:
4791:
4581:
4552:
4193:
4161:{\displaystyle t}
3985:
3953:
3734:
3651:
3600:
3552:
3502:
3464:
3413:
3398:
3177:= −1) for a time
3047:
2968:
2487:
2293:
2097:
2001:
1947:
1902:
1820:complex conjugate
1795:
1787:
1702:
1185:
1161:
1156:
1154:
1142:
1012:
989:
957:
931:
867:
808:
756:
704:
363:
308:
169:
121:
109:
82:
44:Isidor Isaac Rabi
40:natural frequency
5643:
5617:
5578:
5518:
5500:
5498:
5497:
5492:
5481:
5480:
5455:
5453:
5452:
5447:
5412:
5411:
5387:
5386:
5355:
5350:
5322:
5317:
5290:
5289:
5280:
5279:
5264:
5263:
5251:
5250:
5221:
5219:
5218:
5213:
5208:
5203:
5196:
5191:
5190:
5181:
5169:
5151:
5149:
5148:
5143:
5138:
5137:
5110:
5109:
5085:
5084:
5060:
5059:
5026:
5025:
5000:
4998:
4997:
4992:
4990:
4989:
4964:
4962:
4961:
4956:
4948:
4936:
4934:
4933:
4928:
4920:
4905:
4903:
4902:
4897:
4892:
4891:
4882:
4881:
4868:
4867:
4838:
4833:
4832:
4823:
4811:
4799:
4798:
4793:
4792:
4784:
4764:
4762:
4761:
4756:
4754:
4753:
4749:
4740:
4735:
4718:
4697:
4695:
4694:
4689:
4678:
4673:
4672:
4668:
4659:
4654:
4631:
4630:
4621:
4620:
4602:
4601:
4589:
4588:
4583:
4582:
4574:
4560:
4535:
4533:
4532:
4527:
4525:
4524:
4512:
4511:
4493:
4472:
4470:
4469:
4464:
4462:
4461:
4443:
4442:
4421:
4420:
4404:
4402:
4401:
4396:
4385:
4384:
4368:
4366:
4365:
4360:
4349:
4348:
4327:
4326:
4310:
4308:
4307:
4302:
4294:
4279:
4277:
4276:
4271:
4260:
4255:
4254:
4250:
4242:
4241:
4212:
4211:
4201:
4167:
4165:
4164:
4159:
4147:
4145:
4144:
4139:
4128:
4127:
4111:
4109:
4108:
4103:
4101:
4100:
4081:
4079:
4078:
4073:
4059:
4058:
4046:
4045:
4003:
4001:
4000:
3995:
3990:
3986:
3981:
3973:
3964:
3963:
3954:
3952:
3951:
3950:
3938:
3937:
3928:
3927:
3911:
3910:
3909:
3900:
3899:
3880:
3810:
3808:
3807:
3802:
3797:
3796:
3789:
3788:
3775:
3774:
3761:
3760:
3743:
3742:
3735:
3733:
3732:
3723:
3710:
3709:
3700:
3699:
3681:
3680:
3670:
3666:
3652:
3647:
3646:
3645:
3632:
3601:
3599:
3598:
3589:
3588:
3587:
3571:
3553:
3548:
3547:
3546:
3533:
3503:
3495:
3465:
3463:
3462:
3453:
3452:
3451:
3435:
3428:
3414:
3406:
3399:
3397:
3396:
3387:
3374:
3373:
3361:
3360:
3351:
3350:
3334:
3321:
3320:
3275:
3273:
3272:
3267:
3251:
3249:
3248:
3243:
3223:
3221:
3220:
3215:
3213:
3212:
3200:
3172:
3170:
3169:
3164:
3162:
3161:
3138:
3136:
3135:
3130:
3107:
3094:is known as the
3093:
3091:
3090:
3085:
3061:
3059:
3058:
3053:
3048:
3046:
3045:
3036:
3035:
3017:
3016:
3007:
2982:
2980:
2979:
2974:
2969:
2967:
2966:
2965:
2949:
2922:
2920:
2919:
2914:
2909:
2908:
2901:
2889:
2877:
2861:
2860:
2756:
2755:
2748:
2736:
2724:
2695:
2693:
2692:
2687:
2685:
2684:
2677:
2665:
2653:
2637:
2636:
2544:
2543:
2501:
2499:
2498:
2493:
2488:
2485:
2480:
2479:
2461:
2460:
2424:
2422:
2421:
2416:
2384:
2372:
2370:
2369:
2364:
2338:
2323:
2321:
2320:
2315:
2310:
2302:
2294:
2292:
2284:
2283:
2274:
2255:
2253:
2252:
2247:
2242:
2241:
2209:
2208:
2131:
2130:
2098:
2096:
2085:
2062:
2060:
2059:
2054:
2031:
2029:
2028:
2023:
2003:
2002:
1994:
1983:
1981:
1980:
1975:
1949:
1948:
1940:
1929:
1927:
1926:
1921:
1904:
1903:
1895:
1866:hyperfine states
1814:
1812:
1811:
1806:
1801:
1797:
1796:
1793:
1788:
1786:
1782:
1765:
1764:
1751:
1746:
1736:
1735:
1720:
1713:
1712:
1703:
1695:
1681:
1680:
1654:
1652:
1651:
1646:
1641:
1640:
1636:
1628:
1600:
1574:
1562:
1557:
1548:
1547:
1532:
1531:
1527:
1496:
1495:
1471:
1470:
1429:
1427:
1426:
1421:
1416:
1415:
1411:
1403:
1375:
1349:
1337:
1332:
1323:
1322:
1307:
1306:
1302:
1271:
1270:
1246:
1245:
1196:
1194:
1193:
1188:
1186:
1184:
1183:
1182:
1163:
1159:
1158:
1157:
1155:
1152:
1150:
1145:
1140:
1126:
1124:
1123:
1118:
1116:
1115:
1072:
1070:
1069:
1064:
1062:
1061:
1042:
1040:
1039:
1034:
1029:
1028:
1013:
1005:
991:
990:
982:
971:
969:
968:
963:
958:
950:
933:
932:
924:
908:
906:
905:
900:
898:
897:
888:
887:
875:
874:
869:
868:
860:
849:
847:
846:
841:
839:
838:
829:
828:
816:
815:
810:
809:
801:
790:
788:
787:
782:
780:
779:
764:
763:
758:
757:
749:
738:
736:
735:
730:
728:
727:
712:
711:
706:
705:
697:
686:
684:
683:
678:
676:
675:
629:
627:
626:
621:
601:
600:
576:
575:
563:
562:
550:
549:
502:
500:
499:
494:
477:
476:
458:
457:
436:
435:
417:
416:
377:
375:
374:
369:
364:
362:
361:
360:
344:
342:
337:
328:
327:
314:
309:
307:
306:
294:
281:natural lifetime
278:
276:
275:
270:
268:
267:
251:
249:
248:
243:
241:
240:
210:
208:
207:
202:
194:
193:
188:
170:
162:
157:
156:
146:
141:
129:
128:
123:
122:
114:
110:
108:
107:
95:
90:
89:
84:
83:
75:
5651:
5650:
5646:
5645:
5644:
5642:
5641:
5640:
5621:
5620:
5586:Physical Review
5567:
5551:
5529:
5516:
5511:
5470:
5466:
5464:
5461:
5460:
5401:
5397:
5376:
5372:
5351:
5340:
5318:
5307:
5285:
5281:
5275:
5271:
5259:
5255:
5246:
5242:
5240:
5237:
5236:
5192:
5186:
5182:
5177:
5170:
5168:
5160:
5157:
5156:
5124:
5120:
5093:
5089:
5074:
5070:
5043:
5039:
5015:
5011:
5009:
5006:
5005:
4982:
4978:
4970:
4967:
4966:
4944:
4942:
4939:
4938:
4916:
4914:
4911:
4910:
4887:
4883:
4860:
4856:
4849:
4845:
4834:
4828:
4824:
4819:
4807:
4794:
4783:
4782:
4781:
4773:
4770:
4769:
4745:
4736:
4731:
4723:
4719:
4714:
4706:
4703:
4702:
4674:
4664:
4655:
4650:
4639:
4635:
4626:
4622:
4607:
4603:
4597:
4593:
4584:
4573:
4572:
4571:
4556:
4544:
4541:
4540:
4520:
4516:
4507:
4503:
4489:
4478:
4475:
4474:
4448:
4444:
4438:
4434:
4416:
4412:
4410:
4407:
4406:
4380:
4376:
4374:
4371:
4370:
4344:
4340:
4322:
4318:
4316:
4313:
4312:
4290:
4288:
4285:
4284:
4256:
4246:
4237:
4233:
4226:
4222:
4207:
4203:
4197:
4176:
4173:
4172:
4153:
4150:
4149:
4123:
4119:
4117:
4114:
4113:
4096:
4092:
4090:
4087:
4086:
4054:
4050:
4041:
4037:
4020:
4017:
4016:
4010:
3974:
3972:
3968:
3959:
3955:
3946:
3942:
3933:
3929:
3923:
3919:
3912:
3905:
3901:
3895:
3891:
3881:
3879:
3847:
3844:
3843:
3838:
3831:
3824:
3791:
3790:
3784:
3780:
3777:
3776:
3770:
3766:
3763:
3762:
3756:
3752:
3745:
3744:
3737:
3736:
3728:
3724:
3705:
3701:
3695:
3691:
3676:
3672:
3671:
3669:
3667:
3659:
3641:
3637:
3633:
3631:
3626:
3594:
3590:
3583:
3579:
3572:
3570:
3567:
3566:
3542:
3538:
3534:
3532:
3530:
3516:
3494:
3491:
3490:
3458:
3454:
3447:
3443:
3436:
3434:
3429:
3421:
3405:
3400:
3392:
3388:
3369:
3365:
3356:
3352:
3346:
3342:
3335:
3333:
3326:
3325:
3315:
3314:
3308:
3307:
3301:
3300:
3290:
3289:
3287:
3284:
3283:
3261:
3258:
3257:
3237:
3234:
3233:
3208:
3204:
3196:
3182:
3179:
3178:
3157:
3153:
3148:
3145:
3144:
3118:
3115:
3114:
3105:
3070:
3067:
3066:
3041:
3037:
3031:
3027:
3012:
3008:
3006:
2989:
2986:
2985:
2961:
2957:
2953:
2948:
2934:
2931:
2930:
2903:
2902:
2894:
2891:
2890:
2882:
2879:
2878:
2870:
2863:
2862:
2855:
2854:
2840:
2823:
2817:
2816:
2802:
2788:
2782:
2781:
2776:
2771:
2761:
2760:
2750:
2749:
2741:
2738:
2737:
2729:
2726:
2725:
2717:
2710:
2709:
2707:
2704:
2703:
2679:
2678:
2670:
2667:
2666:
2658:
2655:
2654:
2646:
2639:
2638:
2631:
2630:
2619:
2614:
2599:
2598:
2593:
2588:
2582:
2581:
2570:
2565:
2549:
2548:
2538:
2537:
2531:
2530:
2524:
2523:
2513:
2512:
2510:
2507:
2506:
2484:
2469:
2465:
2456:
2452:
2444:
2441:
2440:
2437:
2380:
2378:
2375:
2374:
2334:
2332:
2329:
2328:
2306:
2298:
2285:
2279:
2275:
2273:
2271:
2268:
2267:
2236:
2235:
2229:
2228:
2222:
2221:
2211:
2210:
2203:
2202:
2197:
2186:
2180:
2179:
2171:
2166:
2160:
2159:
2154:
2146:
2136:
2135:
2125:
2124:
2118:
2117:
2111:
2110:
2100:
2099:
2089:
2084:
2082:
2079:
2078:
2048:
2045:
2044:
1993:
1992:
1990:
1987:
1986:
1939:
1938:
1936:
1933:
1932:
1894:
1893:
1891:
1888:
1887:
1877:Bloch equations
1839:
1833:
1828:
1792:
1778:
1760:
1756:
1747:
1742:
1737:
1725:
1721:
1719:
1718:
1714:
1708:
1704:
1694:
1676:
1672:
1670:
1667:
1666:
1632:
1621:
1608:
1604:
1593:
1567:
1558:
1553:
1543:
1539:
1523:
1516:
1512:
1491:
1487:
1466:
1462:
1436:
1433:
1432:
1407:
1396:
1383:
1379:
1368:
1342:
1333:
1328:
1318:
1314:
1298:
1291:
1287:
1266:
1262:
1241:
1237:
1211:
1208:
1207:
1178:
1174:
1167:
1162:
1151:
1146:
1144:
1143:
1135:
1132:
1131:
1111:
1107:
1093:
1090:
1089:
1057:
1053:
1051:
1048:
1047:
1024:
1020:
1004:
981:
980:
978:
975:
974:
949:
923:
922:
920:
917:
916:
893:
889:
883:
879:
870:
859:
858:
857:
855:
852:
851:
834:
830:
824:
820:
811:
800:
799:
798:
796:
793:
792:
775:
771:
759:
748:
747:
746:
744:
741:
740:
723:
719:
707:
696:
695:
694:
692:
689:
688:
671:
667:
659:
656:
655:
652:
645:
638:
596:
592:
571:
567:
558:
554:
545:
541:
539:
536:
535:
529:
518:
511:
472:
468:
444:
440:
425:
421:
412:
408:
400:
397:
396:
356:
352:
345:
338:
333:
323:
319:
315:
313:
302:
298:
293:
291:
288:
287:
263:
259:
257:
254:
253:
236:
232:
230:
227:
226:
223:
189:
184:
183:
161:
152:
148:
142:
137:
124:
113:
112:
111:
103:
99:
94:
85:
74:
73:
72:
70:
67:
66:
52:
17:
12:
11:
5:
5649:
5639:
5638:
5633:
5631:Atomic physics
5619:
5618:
5580:
5579:
5565:
5550:
5547:
5546:
5545:
5540:
5538:Rabi frequency
5535:
5528:
5525:
5510:
5507:
5490:
5487:
5484:
5479:
5476:
5473:
5469:
5457:
5456:
5445:
5442:
5439:
5436:
5433:
5430:
5427:
5424:
5421:
5418:
5415:
5410:
5407:
5404:
5400:
5396:
5393:
5390:
5385:
5382:
5379:
5375:
5371:
5368:
5365:
5362:
5359:
5354:
5349:
5346:
5343:
5339:
5335:
5332:
5329:
5326:
5321:
5316:
5313:
5310:
5306:
5302:
5299:
5296:
5293:
5288:
5284:
5278:
5274:
5270:
5267:
5262:
5258:
5254:
5249:
5245:
5211:
5206:
5202:
5199:
5195:
5189:
5185:
5180:
5176:
5173:
5167:
5164:
5153:
5152:
5141:
5136:
5133:
5130:
5127:
5123:
5119:
5116:
5113:
5108:
5105:
5102:
5099:
5096:
5092:
5088:
5083:
5080:
5077:
5073:
5069:
5066:
5063:
5058:
5055:
5052:
5049:
5046:
5042:
5038:
5035:
5032:
5029:
5024:
5021:
5018:
5014:
4988:
4985:
4981:
4977:
4974:
4954:
4951:
4947:
4926:
4923:
4919:
4907:
4906:
4895:
4890:
4886:
4880:
4877:
4874:
4871:
4866:
4863:
4859:
4855:
4852:
4848:
4844:
4841:
4837:
4831:
4827:
4822:
4818:
4815:
4810:
4806:
4802:
4797:
4790:
4787:
4780:
4777:
4752:
4748:
4744:
4739:
4734:
4730:
4726:
4722:
4717:
4713:
4710:
4699:
4698:
4687:
4684:
4681:
4677:
4671:
4667:
4663:
4658:
4653:
4649:
4645:
4642:
4638:
4634:
4629:
4625:
4619:
4616:
4613:
4610:
4606:
4600:
4596:
4592:
4587:
4580:
4577:
4570:
4567:
4564:
4559:
4555:
4551:
4548:
4523:
4519:
4515:
4510:
4506:
4502:
4499:
4496:
4492:
4488:
4485:
4482:
4460:
4457:
4454:
4451:
4447:
4441:
4437:
4433:
4430:
4427:
4424:
4419:
4415:
4394:
4391:
4388:
4383:
4379:
4358:
4355:
4352:
4347:
4343:
4339:
4336:
4333:
4330:
4325:
4321:
4300:
4297:
4293:
4281:
4280:
4269:
4266:
4263:
4259:
4253:
4249:
4245:
4240:
4236:
4232:
4229:
4225:
4221:
4218:
4215:
4210:
4206:
4200:
4196:
4192:
4189:
4186:
4183:
4180:
4157:
4137:
4134:
4131:
4126:
4122:
4099:
4095:
4083:
4082:
4071:
4068:
4065:
4062:
4057:
4053:
4049:
4044:
4040:
4036:
4033:
4030:
4027:
4024:
4009:
4006:
4005:
4004:
3993:
3989:
3984:
3980:
3977:
3971:
3967:
3962:
3958:
3949:
3945:
3941:
3936:
3932:
3926:
3922:
3918:
3915:
3908:
3904:
3898:
3894:
3890:
3887:
3884:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3836:
3829:
3822:
3812:
3811:
3800:
3795:
3787:
3783:
3779:
3778:
3773:
3769:
3765:
3764:
3759:
3755:
3751:
3750:
3748:
3741:
3731:
3727:
3722:
3719:
3716:
3713:
3708:
3704:
3698:
3694:
3690:
3687:
3684:
3679:
3675:
3668:
3665:
3662:
3658:
3655:
3650:
3644:
3640:
3636:
3630:
3627:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3597:
3593:
3586:
3582:
3578:
3575:
3569:
3568:
3565:
3562:
3559:
3556:
3551:
3545:
3541:
3537:
3531:
3529:
3526:
3523:
3520:
3517:
3515:
3512:
3509:
3506:
3501:
3498:
3493:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3461:
3457:
3450:
3446:
3442:
3439:
3433:
3430:
3427:
3424:
3420:
3417:
3412:
3409:
3404:
3401:
3395:
3391:
3386:
3383:
3380:
3377:
3372:
3368:
3364:
3359:
3355:
3349:
3345:
3341:
3338:
3332:
3331:
3329:
3324:
3319:
3313:
3310:
3309:
3306:
3303:
3302:
3299:
3296:
3295:
3293:
3265:
3241:
3211:
3207:
3203:
3199:
3195:
3192:
3189:
3186:
3160:
3156:
3152:
3128:
3125:
3122:
3083:
3080:
3077:
3074:
3063:
3062:
3051:
3044:
3040:
3034:
3030:
3026:
3023:
3020:
3015:
3011:
3005:
3002:
2999:
2996:
2993:
2983:
2972:
2964:
2960:
2956:
2952:
2947:
2944:
2941:
2938:
2924:
2923:
2912:
2907:
2900:
2897:
2893:
2892:
2888:
2885:
2881:
2880:
2876:
2873:
2869:
2868:
2866:
2859:
2853:
2850:
2847:
2844:
2841:
2839:
2836:
2833:
2830:
2827:
2824:
2822:
2819:
2818:
2815:
2812:
2809:
2806:
2803:
2801:
2798:
2795:
2792:
2789:
2787:
2784:
2783:
2780:
2777:
2775:
2772:
2770:
2767:
2766:
2764:
2759:
2754:
2747:
2744:
2740:
2739:
2735:
2732:
2728:
2727:
2723:
2720:
2716:
2715:
2713:
2697:
2696:
2683:
2676:
2673:
2669:
2668:
2664:
2661:
2657:
2656:
2652:
2649:
2645:
2644:
2642:
2635:
2629:
2626:
2623:
2620:
2618:
2615:
2613:
2610:
2607:
2604:
2601:
2600:
2597:
2594:
2592:
2589:
2587:
2584:
2583:
2580:
2577:
2574:
2571:
2569:
2566:
2564:
2561:
2558:
2555:
2554:
2552:
2547:
2542:
2536:
2533:
2532:
2529:
2526:
2525:
2522:
2519:
2518:
2516:
2491:
2483:
2478:
2475:
2472:
2468:
2464:
2459:
2455:
2451:
2448:
2435:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2383:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2337:
2325:
2324:
2313:
2309:
2305:
2301:
2297:
2291:
2288:
2282:
2278:
2257:
2256:
2245:
2240:
2234:
2231:
2230:
2227:
2224:
2223:
2220:
2217:
2216:
2214:
2207:
2201:
2198:
2196:
2193:
2190:
2187:
2185:
2182:
2181:
2178:
2175:
2172:
2170:
2167:
2165:
2162:
2161:
2158:
2155:
2153:
2150:
2147:
2145:
2142:
2141:
2139:
2134:
2129:
2123:
2120:
2119:
2116:
2113:
2112:
2109:
2106:
2105:
2103:
2095:
2092:
2088:
2052:
2033:
2032:
2021:
2018:
2015:
2012:
2009:
2006:
2000:
1997:
1984:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1946:
1943:
1930:
1919:
1916:
1913:
1910:
1907:
1901:
1898:
1862:two-level atom
1832:
1829:
1827:
1826:Two-level atom
1824:
1816:
1815:
1804:
1800:
1791:
1785:
1781:
1777:
1774:
1771:
1768:
1763:
1759:
1755:
1750:
1745:
1741:
1734:
1731:
1728:
1724:
1717:
1711:
1707:
1701:
1698:
1693:
1690:
1687:
1684:
1679:
1675:
1656:
1655:
1644:
1639:
1635:
1631:
1627:
1624:
1620:
1617:
1614:
1611:
1607:
1603:
1599:
1596:
1592:
1589:
1586:
1583:
1580:
1577:
1573:
1570:
1566:
1561:
1556:
1552:
1546:
1542:
1538:
1535:
1530:
1526:
1522:
1519:
1515:
1511:
1508:
1505:
1502:
1499:
1494:
1490:
1486:
1483:
1480:
1477:
1474:
1469:
1465:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1430:
1419:
1414:
1410:
1406:
1402:
1399:
1395:
1392:
1389:
1386:
1382:
1378:
1374:
1371:
1367:
1364:
1361:
1358:
1355:
1352:
1348:
1345:
1341:
1336:
1331:
1327:
1321:
1317:
1313:
1310:
1305:
1301:
1297:
1294:
1290:
1286:
1283:
1280:
1277:
1274:
1269:
1265:
1261:
1258:
1255:
1252:
1249:
1244:
1240:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1198:
1197:
1181:
1177:
1173:
1170:
1166:
1149:
1139:
1114:
1110:
1106:
1103:
1100:
1097:
1060:
1056:
1044:
1043:
1032:
1027:
1023:
1019:
1016:
1011:
1008:
1003:
1000:
997:
994:
988:
985:
972:
961:
956:
953:
948:
945:
942:
939:
936:
930:
927:
896:
892:
886:
882:
878:
873:
866:
863:
837:
833:
827:
823:
819:
814:
807:
804:
778:
774:
770:
767:
762:
755:
752:
726:
722:
718:
715:
710:
703:
700:
674:
670:
666:
663:
650:
643:
636:
631:
630:
619:
616:
613:
610:
607:
604:
599:
595:
591:
588:
585:
582:
579:
574:
570:
566:
561:
557:
553:
548:
544:
527:
516:
509:
504:
503:
492:
489:
486:
483:
480:
475:
471:
467:
464:
461:
456:
453:
450:
447:
443:
439:
434:
431:
428:
424:
420:
415:
411:
407:
404:
379:
378:
367:
359:
355:
351:
348:
341:
336:
332:
326:
322:
318:
312:
305:
301:
297:
266:
262:
239:
235:
221:
212:
211:
200:
197:
192:
187:
182:
179:
176:
173:
168:
165:
160:
155:
151:
145:
140:
136:
132:
127:
120:
117:
106:
102:
98:
93:
88:
81:
78:
51:
48:
32:electric field
27:to an applied
15:
9:
6:
4:
3:
2:
5648:
5637:
5634:
5632:
5629:
5628:
5626:
5615:
5611:
5607:
5603:
5599:
5595:
5591:
5587:
5582:
5581:
5576:
5572:
5568:
5562:
5558:
5553:
5552:
5544:
5541:
5539:
5536:
5534:
5531:
5530:
5524:
5522:
5514:
5506:
5502:
5485:
5477:
5474:
5471:
5467:
5459:The value of
5443:
5437:
5431:
5425:
5422:
5416:
5408:
5405:
5402:
5398:
5391:
5383:
5380:
5377:
5373:
5369:
5366:
5360:
5352:
5347:
5344:
5341:
5337:
5333:
5327:
5319:
5314:
5311:
5308:
5304:
5300:
5294:
5286:
5282:
5276:
5268:
5260:
5256:
5252:
5247:
5243:
5235:
5234:
5233:
5231:
5227:
5222:
5209:
5197:
5187:
5183:
5174:
5165:
5139:
5134:
5128:
5125:
5121:
5114:
5106:
5103:
5100:
5097:
5094:
5090:
5086:
5081:
5075:
5071:
5064:
5056:
5053:
5050:
5047:
5044:
5040:
5036:
5030:
5022:
5019:
5016:
5012:
5004:
5003:
5002:
4986:
4983:
4979:
4975:
4972:
4949:
4921:
4893:
4888:
4884:
4878:
4872:
4869:
4864:
4861:
4857:
4850:
4846:
4839:
4829:
4825:
4816:
4808:
4804:
4800:
4795:
4788:
4785:
4775:
4768:
4767:
4766:
4746:
4742:
4737:
4732:
4728:
4724:
4720:
4711:
4685:
4679:
4665:
4661:
4656:
4651:
4647:
4643:
4640:
4636:
4627:
4623:
4617:
4614:
4611:
4608:
4604:
4598:
4594:
4590:
4585:
4578:
4575:
4565:
4557:
4553:
4549:
4546:
4539:
4538:
4537:
4521:
4517:
4513:
4508:
4504:
4500:
4497:
4490:
4480:
4458:
4455:
4452:
4449:
4445:
4439:
4435:
4431:
4425:
4417:
4413:
4389:
4381:
4377:
4353:
4345:
4341:
4337:
4331:
4323:
4319:
4295:
4267:
4261:
4247:
4243:
4238:
4234:
4230:
4227:
4223:
4216:
4208:
4204:
4198:
4194:
4190:
4184:
4178:
4171:
4170:
4169:
4155:
4132:
4124:
4120:
4097:
4093:
4069:
4063:
4055:
4051:
4047:
4042:
4038:
4034:
4028:
4022:
4015:
4014:
4013:
3991:
3987:
3982:
3978:
3969:
3965:
3960:
3956:
3947:
3943:
3939:
3934:
3924:
3920:
3916:
3906:
3896:
3892:
3888:
3882:
3876:
3873:
3870:
3867:
3861:
3858:
3855:
3849:
3842:
3841:
3840:
3835:
3828:
3821:
3817:
3798:
3793:
3785:
3781:
3771:
3767:
3757:
3753:
3746:
3739:
3729:
3720:
3714:
3711:
3706:
3696:
3692:
3688:
3682:
3677:
3673:
3663:
3656:
3653:
3642:
3638:
3634:
3628:
3620:
3614:
3611:
3608:
3605:
3595:
3584:
3580:
3576:
3573:
3563:
3557:
3554:
3543:
3539:
3535:
3527:
3521:
3518:
3513:
3507:
3504:
3496:
3484:
3478:
3475:
3472:
3469:
3459:
3448:
3444:
3440:
3437:
3431:
3425:
3418:
3415:
3407:
3402:
3393:
3384:
3378:
3375:
3370:
3366:
3362:
3357:
3347:
3343:
3339:
3327:
3322:
3317:
3311:
3304:
3297:
3291:
3282:
3281:
3280:
3277:
3263:
3255:
3239:
3231:
3227:
3209:
3205:
3201:
3197:
3193:
3190:
3187:
3176:
3158:
3154:
3150:
3142:
3126:
3123:
3120:
3112:
3108:
3101:
3097:
3078:
3049:
3042:
3032:
3028:
3024:
3018:
3013:
3009:
3003:
2997:
2984:
2970:
2962:
2958:
2954:
2950:
2945:
2942:
2939:
2936:
2929:
2928:
2927:
2910:
2905:
2898:
2895:
2886:
2883:
2874:
2871:
2864:
2857:
2851:
2845:
2842:
2837:
2831:
2828:
2825:
2820:
2813:
2807:
2804:
2799:
2793:
2790:
2785:
2778:
2773:
2768:
2762:
2757:
2752:
2745:
2742:
2733:
2730:
2721:
2718:
2711:
2702:
2701:
2700:
2681:
2674:
2671:
2662:
2659:
2650:
2647:
2640:
2633:
2627:
2624:
2621:
2616:
2611:
2608:
2605:
2602:
2595:
2590:
2585:
2578:
2575:
2572:
2567:
2562:
2559:
2556:
2550:
2545:
2540:
2534:
2527:
2520:
2514:
2505:
2504:
2503:
2481:
2476:
2473:
2470:
2466:
2457:
2453:
2449:
2446:
2438:
2431:
2426:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2385:
2357:
2354:
2351:
2348:
2345:
2339:
2335:
2311:
2307:
2303:
2295:
2289:
2286:
2280:
2276:
2266:
2265:
2264:
2262:
2243:
2238:
2232:
2225:
2218:
2212:
2205:
2199:
2194:
2191:
2188:
2183:
2176:
2173:
2168:
2163:
2156:
2151:
2148:
2143:
2137:
2132:
2127:
2121:
2114:
2107:
2101:
2093:
2090:
2086:
2077:
2076:
2075:
2072:
2070:
2064:
2050:
2042:
2038:
2019:
2016:
2013:
2010:
2007:
2004:
1998:
1995:
1985:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1944:
1941:
1931:
1917:
1914:
1911:
1908:
1905:
1899:
1896:
1886:
1885:
1884:
1882:
1878:
1874:
1869:
1867:
1863:
1858:
1856:
1852:
1848:
1844:
1838:
1823:
1821:
1802:
1798:
1789:
1783:
1779:
1775:
1772:
1769:
1766:
1761:
1757:
1753:
1748:
1743:
1739:
1732:
1729:
1726:
1722:
1715:
1709:
1705:
1699:
1696:
1691:
1685:
1677:
1673:
1665:
1664:
1663:
1661:
1642:
1637:
1633:
1625:
1622:
1618:
1615:
1609:
1605:
1597:
1594:
1590:
1587:
1581:
1578:
1575:
1571:
1568:
1564:
1559:
1554:
1550:
1544:
1540:
1536:
1533:
1528:
1524:
1520:
1517:
1513:
1506:
1503:
1500:
1497:
1492:
1488:
1484:
1481:
1478:
1475:
1472:
1467:
1463:
1456:
1450:
1447:
1444:
1438:
1431:
1417:
1412:
1408:
1400:
1397:
1393:
1390:
1384:
1380:
1372:
1369:
1365:
1362:
1356:
1353:
1350:
1346:
1343:
1339:
1334:
1329:
1325:
1319:
1315:
1311:
1308:
1303:
1299:
1295:
1292:
1288:
1281:
1278:
1275:
1272:
1267:
1263:
1259:
1256:
1253:
1250:
1247:
1242:
1238:
1231:
1225:
1222:
1219:
1213:
1206:
1205:
1204:
1201:
1179:
1175:
1171:
1168:
1164:
1147:
1137:
1130:
1129:
1128:
1112:
1108:
1104:
1101:
1098:
1095:
1088:
1084:
1080:
1076:
1058:
1054:
1030:
1025:
1021:
1017:
1014:
1009:
1006:
1001:
998:
995:
992:
986:
983:
973:
959:
954:
951:
946:
943:
940:
937:
934:
928:
925:
915:
914:
913:
910:
894:
890:
884:
880:
876:
871:
864:
861:
835:
831:
825:
821:
817:
812:
805:
802:
776:
772:
768:
765:
760:
753:
750:
724:
720:
716:
713:
708:
701:
698:
672:
668:
664:
661:
653:
646:
639:
617:
611:
608:
605:
602:
597:
593:
589:
586:
583:
580:
577:
572:
568:
559:
555:
551:
546:
542:
534:
533:
532:
530:
523:
519:
512:
490:
487:
484:
481:
478:
473:
469:
465:
462:
454:
451:
448:
445:
441:
437:
432:
429:
426:
422:
413:
409:
405:
402:
395:
394:
393:
391:
386:
384:
365:
357:
353:
349:
346:
339:
334:
330:
324:
320:
316:
310:
303:
299:
295:
286:
285:
284:
282:
264:
260:
237:
233:
224:
217:
198:
190:
180:
177:
171:
166:
163:
158:
153:
149:
143:
138:
134:
130:
125:
118:
115:
104:
100:
96:
91:
86:
79:
76:
65:
64:
63:
61:
60:Lorentz force
57:
47:
45:
41:
37:
33:
30:
26:
22:
5589:
5585:
5556:
5515:
5512:
5503:
5458:
5229:
5225:
5223:
5154:
4908:
4700:
4282:
4084:
4011:
3833:
3826:
3819:
3815:
3813:
3278:
3253:
3229:
3225:
3174:
3140:
3103:
3064:
2925:
2698:
2433:
2429:
2427:
2326:
2258:
2073:
2068:
2065:
2034:
1870:
1859:
1840:
1817:
1657:
1202:
1199:
1082:
1078:
1074:
1045:
911:
648:
641:
634:
632:
525:
521:
514:
507:
505:
389:
387:
380:
219:
215:
213:
53:
21:Rabi problem
20:
18:
5636:Spintronics
3825:= −1) with
2041:transformed
5625:Categories
5549:References
5533:Rabi cycle
3100:precession
2263:equation:
2261:precession
1853:, a fully
1849:, and the
1835:See also:
1660:transients
1658:After all
5614:0031-899X
5517:Need work
5478:±
5435:Ω
5426:
5384:−
5315:−
5277:∗
5205:ℏ
5201:⟩
5172:⟨
5163:Ω
5132:Ω
5126:−
5107:−
5079:Ω
4980:ω
4973:ω
4953:⟩
4925:⟩
4873:ω
4870:−
4858:ω
4843:⟩
4814:⟨
4805:∑
4789:˙
4779:ℏ
4751:ℏ
4709:⟨
4683:⟩
4670:ℏ
4641:−
4615:ω
4609:−
4591:−
4579:˙
4569:ℏ
4554:∑
4514:−
4501:−
4495:∂
4487:∂
4484:ℏ
4456:ω
4450:−
4299:⟩
4265:⟩
4252:ℏ
4228:−
4195:∑
4179:ϕ
3976:Ω
3966:
3944:δ
3917:κ
3889:κ
3871:−
3862:δ
3726:Ω
3718:Ω
3715:
3689:κ
3674:δ
3661:Ω
3657:
3649:Ω
3635:κ
3629:−
3618:Ω
3615:
3609:−
3592:Ω
3577:κ
3574:δ
3561:Ω
3558:
3550:Ω
3536:κ
3525:Ω
3522:
3511:Ω
3508:
3500:Ω
3497:δ
3482:Ω
3479:
3473:−
3456:Ω
3441:κ
3438:δ
3432:−
3423:Ω
3419:
3411:Ω
3408:δ
3403:−
3390:Ω
3382:Ω
3379:
3367:δ
3340:κ
3264:π
3240:π
3202:κ
3194:π
3185:Δ
3151:κ
3121:δ
3079:δ
3073:Ω
3025:κ
3010:δ
2998:δ
2992:Ω
2955:κ
2951:δ
2943:χ
2940:
2849:Ω
2846:
2835:Ω
2832:
2826:−
2811:Ω
2808:
2797:Ω
2794:
2628:χ
2625:
2612:χ
2609:
2603:−
2579:χ
2576:
2563:χ
2560:
2474:ω
2410:δ
2395:κ
2392:−
2382:Ω
2336:ρ
2308:ρ
2304:×
2300:Ω
2281:ρ
2192:κ
2189:−
2174:κ
2164:δ
2152:δ
2149:−
2051:ω
2011:κ
2008:−
1999:˙
1963:κ
1954:δ
1945:˙
1912:δ
1909:−
1900:˙
1843:inversion
1776:ω
1758:ω
1754:−
1740:ω
1730:ω
1619:−
1610:−
1591:−
1582:δ
1579:
1551:∫
1537:κ
1534:−
1518:−
1504:δ
1501:
1479:δ
1476:
1451:δ
1394:−
1385:−
1366:−
1357:δ
1354:
1326:∫
1312:κ
1293:−
1279:δ
1276:
1260:−
1254:δ
1251:
1226:δ
1172:ω
1138:κ
1109:ω
1105:−
1102:ω
1096:δ
1077:lifetime
1075:effective
1055:τ
1018:κ
1002:−
996:δ
987:˙
947:−
941:δ
938:−
929:˙
881:ω
877:≪
865:¨
822:ω
818:≪
806:¨
769:ω
766:≪
754:˙
717:ω
714:≪
702:˙
669:ω
665:≈
662:ω
609:ω
606:
584:ω
581:
485:ω
482:
452:ω
446:−
430:ω
331:ω
300:τ
261:τ
234:ω
135:ω
119:˙
101:τ
80:¨
36:frequency
5575:17233252
5527:See also
5228:at time
2899:″
2887:″
2875:″
2746:′
2734:′
2722:′
2675:′
2663:′
2651:′
1873:spin-1/2
1626:′
1598:′
1572:′
1401:′
1373:′
1347:′
1087:detuning
29:harmonic
5594:Bibcode
4965:, i.e.
3228:state (
3226:excited
5612:
5573:
5563:
5155:where
4283:where
4085:where
2926:where
2327:where
1160:
1141:
383:dipole
3111:laser
3106:'
633:Here
5610:ISSN
5571:OCLC
5561:ISBN
4937:and
2699:and
2486:c.c.
1794:c.c.
791:and
647:and
506:and
279:its
25:atom
19:The
5602:doi
5423:cos
5232:is
3957:sin
3712:cos
3654:sin
3612:cos
3555:sin
3519:cos
3505:sin
3476:cos
3416:sin
3376:cos
2937:tan
2843:cos
2829:sin
2805:sin
2791:cos
2622:cos
2606:sin
2573:sin
2557:cos
1576:cos
1498:sin
1473:cos
1351:sin
1273:sin
1248:cos
1153:def
603:sin
578:cos
479:cos
5627::
5608:.
5600:.
5590:51
5588:.
5569:.
5523:.
4765::
3832:=
2439::
2063:.
1845:,
909:.
850:,
739:,
283::
46:.
5616:.
5604::
5596::
5577:.
5489:)
5486:0
5483:(
5475:,
5472:m
5468:d
5444:.
5441:)
5438:t
5432:2
5429:(
5420:)
5417:0
5414:(
5409:+
5406:,
5403:m
5399:d
5395:)
5392:0
5389:(
5381:,
5378:m
5374:d
5370:2
5367:+
5364:)
5361:0
5358:(
5353:2
5348:+
5345:,
5342:m
5338:d
5334:+
5331:)
5328:0
5325:(
5320:2
5312:,
5309:m
5305:d
5301:=
5298:)
5295:t
5292:(
5287:m
5283:d
5273:)
5269:t
5266:(
5261:m
5257:d
5253:=
5248:m
5244:P
5230:t
5226:m
5210:.
5198:n
5194:|
5188:1
5184:H
5179:|
5175:m
5166:=
5140:,
5135:t
5129:i
5122:e
5118:)
5115:0
5112:(
5104:,
5101:n
5098:,
5095:m
5091:d
5087:+
5082:t
5076:i
5072:e
5068:)
5065:0
5062:(
5057:+
5054:,
5051:n
5048:,
5045:m
5041:d
5037:=
5034:)
5031:t
5028:(
5023:n
5020:,
5017:m
5013:d
4987:n
4984:m
4976:=
4950:n
4946:|
4922:m
4918:|
4894:.
4889:n
4885:d
4879:t
4876:)
4865:n
4862:m
4854:(
4851:i
4847:e
4840:n
4836:|
4830:1
4826:H
4821:|
4817:m
4809:n
4801:=
4796:m
4786:d
4776:i
4747:/
4743:t
4738:0
4733:m
4729:E
4725:i
4721:e
4716:|
4712:m
4686:,
4680:n
4676:|
4666:/
4662:t
4657:0
4652:n
4648:E
4644:i
4637:e
4633:]
4628:n
4624:d
4618:t
4612:i
4605:e
4599:1
4595:H
4586:n
4576:d
4566:i
4563:[
4558:n
4550:=
4547:0
4522:1
4518:H
4509:0
4505:H
4498:t
4491:/
4481:i
4459:t
4453:i
4446:e
4440:1
4436:H
4432:=
4429:)
4426:t
4423:(
4418:1
4414:H
4393:)
4390:t
4387:(
4382:n
4378:d
4357:)
4354:0
4351:(
4346:n
4342:d
4338:=
4335:)
4332:t
4329:(
4324:n
4320:d
4296:n
4292:|
4268:,
4262:n
4258:|
4248:/
4244:t
4239:n
4235:E
4231:i
4224:e
4220:)
4217:t
4214:(
4209:n
4205:d
4199:n
4191:=
4188:)
4185:t
4182:(
4156:t
4136:)
4133:t
4130:(
4125:1
4121:H
4098:0
4094:H
4070:,
4067:)
4064:t
4061:(
4056:1
4052:H
4048:+
4043:0
4039:H
4035:=
4032:)
4029:t
4026:(
4023:H
3992:.
3988:)
3983:2
3979:t
3970:(
3961:2
3948:2
3940:+
3935:2
3931:)
3925:0
3921:E
3914:(
3907:2
3903:)
3897:0
3893:E
3886:(
3883:2
3877:+
3874:1
3868:=
3865:)
3859:;
3856:t
3853:(
3850:w
3837:0
3834:v
3830:0
3827:u
3823:0
3820:w
3816:w
3799:.
3794:]
3786:0
3782:w
3772:0
3768:v
3758:0
3754:u
3747:[
3740:]
3730:2
3721:t
3707:2
3703:)
3697:0
3693:E
3686:(
3683:+
3678:2
3664:t
3643:0
3639:E
3624:)
3621:t
3606:1
3603:(
3596:2
3585:0
3581:E
3564:t
3544:0
3540:E
3528:t
3514:t
3488:)
3485:t
3470:1
3467:(
3460:2
3449:0
3445:E
3426:t
3394:2
3385:t
3371:2
3363:+
3358:2
3354:)
3348:0
3344:E
3337:(
3328:[
3323:=
3318:]
3312:w
3305:v
3298:u
3292:[
3254:u
3230:w
3210:0
3206:E
3198:/
3191:=
3188:t
3175:w
3159:0
3155:E
3141:u
3127:0
3124:=
3104:u
3082:)
3076:(
3050:.
3043:2
3039:)
3033:0
3029:E
3022:(
3019:+
3014:2
3004:=
3001:)
2995:(
2971:,
2963:0
2959:E
2946:=
2911:,
2906:]
2896:w
2884:v
2872:u
2865:[
2858:]
2852:t
2838:t
2821:0
2814:t
2800:t
2786:0
2779:0
2774:0
2769:1
2763:[
2758:=
2753:]
2743:w
2731:v
2719:u
2712:[
2682:]
2672:w
2660:v
2648:u
2641:[
2634:]
2617:0
2596:0
2591:1
2586:0
2568:0
2551:[
2546:=
2541:]
2535:w
2528:v
2521:u
2515:[
2490:)
2482:+
2477:t
2471:i
2467:e
2463:(
2458:0
2454:E
2450:=
2447:E
2436:0
2434:E
2430:E
2413:)
2407:,
2404:0
2401:,
2398:E
2389:(
2386:=
2361:)
2358:w
2355:,
2352:v
2349:,
2346:u
2343:(
2340:=
2312:,
2296:=
2290:t
2287:d
2277:d
2244:.
2239:]
2233:w
2226:v
2219:u
2213:[
2206:]
2200:0
2195:E
2184:0
2177:E
2169:0
2157:0
2144:0
2138:[
2133:=
2128:]
2122:w
2115:v
2108:u
2102:[
2094:t
2091:d
2087:d
2069:w
2020:,
2017:v
2014:E
2005:=
1996:w
1972:,
1969:w
1966:E
1960:+
1957:u
1951:=
1942:v
1918:,
1915:v
1906:=
1897:u
1803:,
1799:)
1790:+
1784:T
1780:/
1773:i
1770:2
1767:+
1762:2
1749:2
1744:a
1733:t
1727:i
1723:e
1716:(
1710:0
1706:E
1700:m
1697:e
1692:=
1689:)
1686:t
1683:(
1678:a
1674:x
1643:.
1638:T
1634:/
1630:)
1623:t
1616:t
1613:(
1606:e
1602:)
1595:t
1588:t
1585:(
1569:t
1565:d
1560:t
1555:0
1545:0
1541:E
1529:T
1525:/
1521:t
1514:e
1510:]
1507:t
1493:0
1489:v
1485:+
1482:t
1468:0
1464:u
1460:[
1457:=
1454:)
1448:;
1445:t
1442:(
1439:v
1418:,
1413:T
1409:/
1405:)
1398:t
1391:t
1388:(
1381:e
1377:)
1370:t
1363:t
1360:(
1344:t
1340:d
1335:t
1330:0
1320:0
1316:E
1309:+
1304:T
1300:/
1296:t
1289:e
1285:]
1282:t
1268:0
1264:v
1257:t
1243:0
1239:u
1235:[
1232:=
1229:)
1223:;
1220:t
1217:(
1214:u
1180:0
1176:x
1169:m
1165:e
1148:=
1113:a
1099:=
1083:a
1079:T
1059:0
1031:,
1026:0
1022:E
1015:+
1010:T
1007:v
999:u
993:=
984:v
960:,
955:T
952:u
944:v
935:=
926:u
895:a
891:v
885:2
872:a
862:v
836:a
832:u
826:2
813:a
803:u
777:a
773:v
761:a
751:v
725:a
721:u
709:a
699:u
673:a
651:a
649:v
644:a
642:u
637:0
635:x
618:.
615:)
612:t
598:a
594:v
590:+
587:t
573:a
569:u
565:(
560:0
556:x
552:=
547:a
543:x
528:a
526:v
522:E
517:a
515:u
510:a
508:x
491:,
488:t
474:0
470:E
466:2
463:=
460:]
455:t
449:i
442:e
438:+
433:t
427:i
423:e
419:[
414:0
410:E
406:=
403:E
390:E
366:,
358:3
354:c
350:m
347:3
340:2
335:a
325:2
321:e
317:2
311:=
304:0
296:2
265:0
238:a
222:a
220:x
216:e
199:,
196:)
191:a
186:r
181:,
178:t
175:(
172:E
167:m
164:e
159:=
154:a
150:x
144:2
139:a
131:+
126:a
116:x
105:0
97:2
92:+
87:a
77:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.