5617:
4775:
5612:{\displaystyle {\begin{matrix}{\begin{aligned}{\hat {U}}(t)&=e^{-i{\hat {H}}_{\text{JC}}t/\hbar }\\&={\begin{pmatrix}e^{-i\omega _{c}t\left({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\right)}\left(\cos t{\sqrt {{\hat {\varphi }}+g^{2}}}-i\delta /2{\frac {\sin t{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\right)&-ige^{-i\omega _{c}t\left({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\right)}{\frac {\sin t{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\,{\hat {a}}\\-ige^{-i\omega _{c}t\left({\hat {a}}^{\dagger }{\hat {a}}-{\frac {1}{2}}\right)}{\frac {\sin t{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}}{\hat {a}}^{\dagger }&e^{-i\omega _{c}t\left({\hat {a}}^{\dagger }{\hat {a}}-{\frac {1}{2}}\right)}\left(\cos t{\sqrt {\hat {\varphi }}}+i\delta /2{\frac {\sin t{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}}\right)\end{pmatrix}}\end{aligned}}\end{matrix}}}
17290:
16540:
306:, & Klein were finally able to use a single-atom maser to demonstrate the revivals of probabilities predicted by the model. Before that time, research groups were unable to build experimental setups capable of enhancing the coupling of an atom with a single field mode, simultaneously suppressing other modes. Experimentally, the quality factor of the cavity must be high enough to consider the dynamics of the system as equivalent to the dynamics of a single mode field. This successful demonstration of dynamics that could only be explained by a quantum mechanical model of the field spurred further development of high quality cavities for use in this research.
12607:
17887:
17285:{\displaystyle {\begin{aligned}|\psi (t)\rangle &=e^{-i{\hat {H}}_{n}t/\hbar }|\psi (0)\rangle \\&=e^{-|\alpha |^{2}/2}|g,0\rangle +\sum _{n=1}^{\infty }e^{-|\alpha |^{2}/2}{\frac {\alpha ^{n}}{\sqrt {n!}}}e^{-in\omega _{c}t}\left(\cos {({\sqrt {n}}\Omega t/2)}{\hat {I}}^{(n)}-i\sin {({\sqrt {n}}\Omega t/2)}{\hat {\sigma }}_{x}^{(n)}\right)|g,n\rangle \\&=e^{-|\alpha |^{2}/2}|g,0\rangle +\sum _{n=1}^{\infty }e^{-|\alpha |^{2}/2}{\frac {\alpha ^{n}}{\sqrt {n!}}}e^{-in\omega _{c}t}\left(\cos {({\sqrt {n}}\Omega t/2)}|g,n\rangle -i\sin {({\sqrt {n}}\Omega t/2)}|e,n-1\rangle \right)\end{aligned}}}
12281:
9453:
14779:
14326:
8304:
11987:
17415:
13811:
8879:
14379:
13911:
7945:
11038:
12602:{\displaystyle {\hat {H}}_{JC}={\begin{bmatrix}H_{0}&0&0&0&\cdots &\cdots &\cdots \\0&{\hat {H}}_{1}&0&0&\ddots &\ddots &\ddots \\0&0&{\hat {H}}_{2}&0&\ddots &\ddots &\ddots \\\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\ddots \\\vdots &\ddots &\ddots &0&{\hat {H}}_{n}&0&\ddots \\\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\ddots \\\end{bmatrix}}}
2564:
1806:
17882:{\displaystyle {\begin{aligned}P_{e}(t)=\left|\langle e|\psi (t)\rangle \right|^{2}&=\sum _{n=1}^{\infty }{\frac {e^{-|\alpha |^{2}}}{n!}}|\alpha |^{2n}\sin ^{2}\left({\tfrac {1}{2}}{\sqrt {n}}\Omega t\right)\\&=\sum _{n=1}^{\infty }{\frac {e^{-\langle n\rangle }\langle n\rangle ^{n}}{n!}}\sin ^{2}\left({\tfrac {1}{2}}{\sqrt {n}}\Omega t\right)\\&=\sum _{n=1}^{\infty }{\frac {e^{-\langle n\rangle }\langle n\rangle ^{n}}{n!}}\sin ^{2}(\Omega _{n}t)\\{}\end{aligned}}}
13491:
7588:
6101:
19001:
15767:
9448:{\displaystyle {\hat {H}}_{AF}(t)=e^{i{\hat {H}}_{0}t/\hbar }{\hat {H}}_{AF}e^{-i{\hat {H}}_{0}t/\hbar }=\hbar \left(g_{c}{\hat {\sigma }}_{+}{\hat {a}}_{c}^{\dagger }e^{i(\omega _{c}+\omega _{eg})t}+g_{c}^{*}{\hat {\sigma }}_{-}{\hat {a}}_{c}e^{-i(\omega _{c}+\omega _{eg})t}-g_{c}^{*}{\hat {\sigma }}_{-}{\hat {a}}_{c}^{\dagger }e^{-i(\omega _{eg}-\omega _{c})t}-g_{c}{\hat {\sigma }}_{+}{\hat {a}}_{c}e^{i(\omega _{eg}-\omega _{c})t}\right)}
14774:{\displaystyle {\begin{aligned}E_{n,\pm }&=\left(n\hbar \omega _{c}-{\frac {1}{2}}\hbar \Delta \right)\pm {\frac {1}{2}}\hbar {\sqrt {\Delta ^{2}+n\Omega ^{2}}}\\|n,+\rangle &=\cos \left({\frac {\theta _{n}}{2}}\right)|e,n-1\rangle +\sin \left({\frac {\theta _{n}}{2}}\right)|g,n\rangle \\|n,-\rangle &=\cos \left({\frac {\theta _{n}}{2}}\right)|g,n\rangle -\sin \left({\frac {\theta _{n}}{2}}\right)|e,n-1\rangle \\\end{aligned}}}
2273:
1408:
14321:{\displaystyle {\begin{aligned}{\hat {I}}^{(n)}&=\left|e,n-1\right\rangle \left\langle e,n-1\right|+\left|g,n\right\rangle \left\langle g,n\right|\\{\hat {\sigma }}_{z}^{(n)}&=\left|e,n-1\right\rangle \left\langle e,n-1\right|-\left|g,n\right\rangle \left\langle g,n\right|\\{\hat {\sigma }}_{x}^{(n)}&=\left|e,n-1\right\rangle \left\langle g,n\right|+\left|g,n\right\rangle \left\langle e,n-1\right|.\\\,\end{aligned}}}
8299:{\displaystyle {\hat {H}}_{AF}=-\sum _{\mathbf {k} ,\lambda }\hbar \left(g_{\mathbf {k} ,\lambda }{\hat {\sigma }}_{+}{\hat {a}}_{\mathbf {k} ,\lambda }-g_{\mathbf {k} ,\lambda }^{*}{\hat {\sigma }}_{-}{\hat {a}}_{\mathbf {k} ,\lambda }^{\dagger }-g_{\mathbf {k} ,\lambda }{\hat {\sigma }}_{+}{\hat {a}}_{\mathbf {k} ,\lambda }^{\dagger }+g_{\mathbf {k} ,\lambda }^{*}{\hat {\sigma }}_{-}{\hat {a}}_{\mathbf {k} ,\lambda }\right),}
11982:{\displaystyle {\begin{aligned}\left&=\hbar g_{c}\left(\left+\left\right)\\&=\hbar g_{c}\left({\hat {a}}_{c}\left+\left{\hat {\sigma }}_{+}+{\hat {a}}_{c}^{\dagger }\left+\left{\hat {\sigma }}_{-}\right)\\&=\hbar g_{c}\left(-{\hat {a}}_{c}{\hat {\sigma }}_{+}+{\hat {a}}_{c}{\hat {\sigma }}_{+}+{\hat {a}}_{c}^{\dagger }{\hat {\sigma }}_{-}-{\hat {a}}_{c}^{\dagger }{\hat {\sigma }}_{-}\right)\\&=0\end{aligned}}}
768:
81:
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1952:, and it is valid since the fast oscillating term couples states of comparatively large energy difference: When the difference in energy is much larger than the coupling, the mixing of these states will be small, or put differently, the coupling is responsible for very little population transfer between the states. Transforming back into the Schrödinger picture the JCM Hamiltonian is thus written as
15485:
13806:{\displaystyle {\hat {H}}_{n}={\begin{bmatrix}n\hbar \omega _{c}-{\frac {1}{2}}\hbar \Delta &{\frac {{\sqrt {n}}\hbar \Omega }{2}}\\{\frac {{\sqrt {n}}\hbar \Omega }{2}}&n\hbar \omega _{c}+{\frac {1}{2}}\hbar \Delta \\\end{bmatrix}}=n\hbar \omega _{c}{\hat {I}}^{(n)}-{\frac {\hbar \Delta }{2}}{\hat {\sigma }}_{z}^{(n)}+{\frac {1}{2}}{\sqrt {n}}\hbar \Omega {\hat {\sigma }}_{x}^{(n)}}
2178:
25:
13245:
15319:
10923:
537:
12948:
3365:
2559:{\displaystyle {\begin{aligned}{\hat {H}}_{\text{I}}&=\hbar \omega _{c}\left({\hat {a}}^{\dagger }{\hat {a}}+{\frac {{\hat {\sigma }}_{z}}{2}}\right)\\{\hat {H}}_{\text{II}}&=\hbar \delta {\frac {{\hat {\sigma }}_{z}}{2}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right)\end{aligned}}}
1801:{\displaystyle {\hat {H}}_{\text{int}}(t)={\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{-}e^{-i(\omega _{c}+\omega _{a})t}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{+}e^{i(\omega _{c}+\omega _{a})t}+{\hat {a}}{\hat {\sigma }}_{+}e^{i(-\omega _{c}+\omega _{a})t}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}e^{-i(-\omega _{c}+\omega _{a})t}\right).}
10351:
18527:
10054:
10363:
13395:
6961:
1955:
7583:{\displaystyle {\hat {\mathbf {E} }}(\mathbf {R} )=i\sum _{\mathbf {k} ,\lambda }{\sqrt {\frac {2\pi \hbar \omega _{\mathbf {k} }}{V}}}\mathbf {u} _{\mathbf {k} ,\lambda }\left({\hat {a}}_{\mathbf {k} ,\lambda }e^{i\mathbf {k} \cdot \mathbf {R} }-{\hat {a}}_{\mathbf {k} ,\lambda }^{\dagger }e^{-i\mathbf {k} \cdot \mathbf {R} }\right)}
6096:{\displaystyle {\begin{aligned}{\frac {\sin t\,{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\;{\hat {a}}&={\hat {a}}\;{\frac {\sin t\,{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}},\\\cos t\,{\sqrt {{\hat {\varphi }}+g^{2}}}\;{\hat {a}}&={\hat {a}}\;\cos t{\sqrt {\hat {\varphi }}},\end{aligned}}}
7903:
317:
light frequencies, hour-glass-type optical modes can be helpful because of their large mode volume that eventually coincides with a strong field inside the cavity. A quantum dot inside a photonic crystal nano-cavity is also a promising system for observing collapse and revival of Rabi cycles in the visible light frequencies.
12955:
15067:
10679:
15762:{\displaystyle {\begin{aligned}|n,+\rangle &=\cos \left({\frac {\theta _{n}}{2}}\right)|e,n-1\rangle +\sin \left({\frac {\theta _{n}}{2}}\right)|g,n\rangle \\|n,-\rangle &=\cos \left({\frac {\theta _{n}}{2}}\right)|g,n\rangle -\sin \left({\frac {\theta _{n}}{2}}\right)|e,n-1\rangle \\\end{aligned}}}
7735:
18532:
12703:
9768:
of anti-resonant oscillation, the net effect of the quickly oscillating anti-resonant terms tends to average to 0 for the timescales over which we wish to analyze resonant behavior. We may thus neglect the anti-resonant terms altogether, since their value is negligible compared to that of the nearly
3183:
277:
so that the transitions between the states in the two-level system (typically two energy sub-levels in an atom) are coupled very strongly by the interaction of the atom with the field mode. This simultaneously suppresses any coupling between other sub-levels in atom and coupling to other modes of the
18207:
346:
Theoretical work has extended the original model to include the effects of dissipation and damping, typically via a phenomenological approach. Proposed extensions have also incorporated the inclusion of multiple modes of the quantum field, allowing for coupling to additional energy levels within the
201:
This approach is in contrast to the earlier semi-classical method, in which only the dynamics of the atom are treated quantum mechanically, while the field with which it interacts is assumed to behave according to classical electromagnetic theory. The quantum mechanical treatment of the field in the
316:
It was found that an hourglass geometry could be used to maximize the volume occupied by the mode, while simultaneously maintaining a high quality factor in order to maximize coupling strength, and thus better approximate the parameters of the model. To observe strong atom-field coupling in visible
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The Rabi oscillations can readily be seen in the sin and cos functions in the state vector. Different periods occur for different number states of photons. What is observed in experiment is the sum of many periodic functions that can be very widely oscillating and destructively sum to zero at some
18878:
The collapse can be easily understood as a consequence of destructive interference between the different frequency components as they de-phase and begin to destructively interfere over time. However, the fact that the frequencies have a discrete spectrum leads to another interesting result in the
10090:
330:
to the modes of a micro-cavity, potentially allowing it to be applied in a physical system of much smaller size. Other experiments have focused on demonstrating the non-linear nature of the Jaynes-Cummings ladder of energy levels by direct spectroscopic observation. These experiments have found
3942:
3799:
18212:
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13252:
763:{\displaystyle {\begin{aligned}{\hat {H}}_{\text{field}}&=\hbar \omega _{c}{\hat {a}}^{\dagger }{\hat {a}}\\{\hat {H}}_{\text{atom}}&=\hbar \omega _{a}{\frac {{\hat {\sigma }}_{z}}{2}}\\{\hat {H}}_{\text{int}}&={\frac {\hbar \Omega }{2}}{\hat {E}}{\hat {S}}.\end{aligned}}}
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385:. For example, it applies to a high quality resonant cavity containing multiple identical atoms with transitions near the cavity resonance, or a resonator coupled to multiple quantum dots on a superconducting circuit. It reduces to the Jaynes–Cummings model for the case
6812:
325:
Many recent experiments have focused on the application of the model to systems with potential applications in quantum information processing and coherent control. Various experiments have demonstrated the dynamics of the Jaynes–Cummings model in the coupling of a
278:
field, and thus makes any losses small enough to observe the dynamics predicted by the Jaynes–Cummings model. Because of the difficulty in realizing such an apparatus, the model remained a mathematical curiosity for quite some time. In 1985, several groups using
3536:
10625:{\displaystyle {\hat {H}}_{JC}=\hbar \omega _{c}\left({\hat {a}}_{c}^{\dagger }{\hat {a}}_{c}+{\frac {1}{2}}\right)+\hbar \omega _{eg}|e\rangle \langle e|+\hbar g_{c}\left({\hat {\sigma }}_{+}{\hat {a}}_{c}+{\hat {\sigma }}_{-}{\hat {a}}_{c}^{\dagger }\right)}
7218:
210:
between the states of the two-level system as it interacts with the quantum field. This was originally believed to be a purely quantum mechanical effect, although a semi-classical explanation for it was later provided in terms of linear dispersion and
2173:{\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega _{c}{\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega _{a}{\frac {{\hat {\sigma }}_{z}}{2}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right).}
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18101:
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4764:
moment of time, but will be non-zero again at later moments. Finiteness of this moment results just from discreteness of the periodicity arguments. If the field amplitude were continuous, the revival would have never happened at finite time.
16017:
343:. In the latter case, the presence or absence of a collective Rydberg excitation in the ensemble serves the role of the two level system, while the role of the bosonic field mode is played by the total number of spin flips that take place.
4359:
7327:
6775:
3803:
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18023:
3178:
13240:{\displaystyle \langle g,n|{\hat {H}}_{JC}|e,n-1\rangle =\hbar g_{c}\langle g,n|{\hat {a}}_{c}^{\dagger }{\hat {\sigma }}_{-}|e,n-1\rangle +\hbar g_{c}\langle g,n|{\hat {a}}_{c}{\hat {\sigma }}_{+}|e,n-1\rangle ={\sqrt {n}}\hbar g_{c}}
4051:
414:
The model provides the possibility to realize several exotic theoretical possibilities in an experimental setting. For example, it was realized that during the periods of collapsed Rabi oscillations, the atom-cavity system exists in a
15314:{\displaystyle {\begin{aligned}P_{e}(t)&=|\langle e,0|\psi (t)\rangle |^{2}=\cos ^{2}\left({\frac {\Omega t}{2}}\right)\\P_{g}(t)&=|\langle g,1|\psi (t)\rangle |^{2}=\sin ^{2}\left({\frac {\Omega t}{2}}\right)\\\end{aligned}}}
10918:{\displaystyle {\hat {H}}_{JC}=\hbar \omega _{c}{\hat {a}}_{c}^{\dagger }{\hat {a}}_{c}+\hbar \omega _{eg}|e\rangle \langle e|+\hbar g_{c}\left({\hat {\sigma }}_{+}{\hat {a}}_{c}+{\hat {\sigma }}_{-}{\hat {a}}_{c}^{\dagger }\right)}
7593:
6256:
11029:
883:
18946:
The revival of probability is due to the re-phasing of the various discrete frequencies. If the field were classical, the frequencies would have a continuous spectrum, and such re-phasing could never occur within a finite time.
9716:
5740:
2268:
16393:
16207:
6637:
14907:
1403:
3032:
15478:, as is the case for an atom initially in a definite state (ground or excited) entering a cavity containing a known number of photons, then the state of the atom-cavity system at later times becomes a superposition of the
19215:
Zhu, Yifu and
Gauthier, Daniel J. and Morin, S. E. and Wu, Qilin and Carmichael, H. J. and Mossberg, T. W. (1990). "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations".
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88:
in an optical cavity is shown as red dot on the top left. The energy levels of the atom that couple to the field mode within the cavity are shown in the circle on the bottom right. Transfer between the two states causes
15858:
4426:
17420:
12943:{\displaystyle {\hat {H}}_{n}={\begin{bmatrix}\hbar \omega _{c}(n-1)+\hbar \omega _{eg}&\langle e,n-1|{\hat {H}}_{JC}|g,n\rangle \\\langle g,n|{\hat {H}}_{JC}|e,n-1\rangle &n\hbar \omega _{c}\\\end{bmatrix}}}
3360:{\displaystyle H^{(n)}=\hbar {\begin{pmatrix}n\omega _{c}+{\frac {\omega _{a}}{2}}&{\frac {\Omega }{2}}{\sqrt {n+1}}\\{\frac {\Omega }{2}}{\sqrt {n+1}}&(n+1)\omega _{c}-{\frac {\omega _{a}}{2}}\end{pmatrix}}}
6528:
4138:
3617:
14864:
13916:
9535:
18832:
18640:
1068:
10346:{\displaystyle {\hat {H}}_{AF}=e^{-i{\hat {H}}_{0}t/\hbar }{\hat {H}}_{AF}(t)e^{i{\hat {H}}_{0}t/\hbar }=\hbar g_{c}\left({\hat {\sigma }}_{+}{\hat {a}}_{c}+{\hat {\sigma }}_{-}{\hat {a}}_{c}^{\dagger }\right)}
117:
field), with or without the presence of light (in the form of a bath of electromagnetic radiation that can cause spontaneous emission and absorption). It was originally developed to study the interaction of
6344:
18522:{\displaystyle P_{e}(t)\simeq {\frac {1}{2}}-{\frac {e^{-\langle n\rangle }}{4}}\cdot \left(e^{-i{\sqrt {\langle n\rangle }}\Omega t/2}\exp \left+e^{i{\sqrt {\langle n\rangle }}\Omega t/2}\exp \left\right)}
18950:
A plot of the probability to find an atom initially in the ground state to have transitioned to the excited state after interacting with a cavity prepared a in a coherent state vs. the unit-less parameter
17354:
10049:{\displaystyle {\hat {H}}_{AF}(t)=-\hbar g_{c}\left({\hat {\sigma }}_{+}{\hat {a}}_{c}e^{i(\omega _{eg}-\omega _{c})t}+{\hat {\sigma }}_{-}{\hat {a}}_{c}^{\dagger }e^{-i(\omega _{eg}-\omega _{c})t}\right)}
6423:
For ease of illustration, consider the interaction of two energy sub-levels of an atom with a quantized electromagnetic field. The behavior of any other two-state system coupled to a bosonic field will be
3425:
19133:
A more recent reference reviewing the
Physics of the Jaynes–Cummings model is Journal of Physics B, 2013, vol. 46, #22, containing numerous relevant articles, including two interesting editorials, one by
7052:
8874:
15394:
9643:
15915:
8426:
8365:
1190:
1129:
13390:{\displaystyle {\hat {H}}_{n}={\begin{bmatrix}n\hbar \omega _{c}-\hbar \Delta &{\frac {{\sqrt {n}}\hbar \Omega }{2}}\\{\frac {{\sqrt {n}}\hbar \Omega }{2}}&n\hbar \omega _{c}\\\end{bmatrix}}}
8480:
subspace of the atom. The application of the Jaynes–Cummings model allows suppression of this sum, and restrict the attention to a single mode of the field. Thus the atom-field
Hamiltonian becomes:
18874:
9766:
17392:
contribute to the dynamics beyond an overall phase, since they represent the zero-point energy. In this case, the probability to find the atom having flipped to the excited state at a later time
7940:
6956:{\displaystyle {\hat {H}}_{F}=\sum _{\mathbf {k} ,\lambda }\hbar \omega _{\mathbf {k} }\left({\hat {a}}_{\mathbf {k} ,\lambda }^{\dagger }{\hat {a}}_{\mathbf {k} ,\lambda }+{\frac {1}{2}}\right)}
16545:
15490:
15072:
14384:
11043:
7098:
are the photon creation and annihilation operators for each indexed mode of the field. The simplicity of the Jaynes–Cummings model comes from suppressing this general sum by considering only a
5834:
4784:
2866:
2278:
542:
15771:
This change in eigenstates due to the alteration of the
Hamiltonian caused by the atom-field interaction is sometimes called "dressing" the atom, and the new eigenstates are referred to as the
7096:
16058:
2802:
13902:
9584:
17936:
10670:
2610:
18590:. Note the initial collapse over short times, followed by revival at longer times. This behavior is attributable to the discrete spectrum of frequencies caused by quantization of the field.
15327:. In this case, there was only a single quantum in the atom-field system, carried in by the initially excited atom. In general, the Rabi oscillation associated with an atom-field system of
7898:{\displaystyle \hbar g_{\mathbf {k} ,\lambda }=i{\sqrt {\frac {2\pi \hbar \omega _{\mathbf {k} }}{V}}}\langle e|{\hat {\mathbf {d} }}|g\rangle \cdot \mathbf {u} _{\mathbf {k} ,\lambda },}
4060:
It is now possible to obtain the dynamics of a general state by expanding it on to the noted eigenstates. We consider a superposition of number states as the initial state for the field,
442:
13486:
8777:
5793:
2713:
18647:
347:
atom, or the presence of multiple atoms interacting with the same field. Some attempt has also been made to go beyond the so-called rotating-wave approximation that is usually employed
12167:
8478:
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1946:
1900:
1854:
16166:
15920:
4143:
7105:
3068:
2958:
19507:
J.A.C. Gallas; G. Leuchs; H. Walther; H. Figger (1985). "Rydberg atoms: high-resolution spectroscopy and radiation interaction-Rydberg molecules". In D. Bates; B. Bederson (eds.).
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6145:
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11998:
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7349:
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2190:
1307:
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236:
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3420:
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313:) with a single resonant mode of the electromagnetic field in a cavity from an experimental point of view, and study different aspects of the Jaynes–Cummings model.
18886:
17945:
12636:
10083:
9798:
15778:
3073:
6663:
409:
18556:
7730:{\displaystyle {\hat {\mathbf {d} }}={\hat {\sigma }}_{+}\langle e|{\hat {\mathbf {d} }}|g\rangle +{\hat {\sigma }}_{-}\langle g|{\hat {\mathbf {d} }}|e\rangle }
20038:
Rodríguez-Lara, B.; Moya-Cessa, H.; Klimov, A. (2005). "Combining Jaynes-Cummings and anti-Jaynes-Cummings dynamics in a trapped-ion system driven by a laser".
18096:
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790:
256:
18202:{\displaystyle \Omega _{n}\simeq {\frac {\Omega }{2}}{\sqrt {\langle n\rangle }}\left(1+{\frac {1}{2}}{\frac {n-\langle n\rangle }{\langle n\rangle }}\right)}
6435:
19525:
S.E. Morin; C.C. Yu; T.W. Mossberg (1994). "Strong Atom-Cavity
Coupling over Large Volumes and the Observation of Subnatural Intracavity Atomic Linewidths".
16192:
is known as the Jaynes-Cummings ladder. This non-linear splitting effect is purely quantum mechanical, and cannot be explained by any semi-classical model.
802:
15396:. As explained below, this discrete spectrum of frequencies is the underlying reason for the collapses and subsequent revivals probabilities in the model.
9648:
194:. In order to simplify the math and allow for a tractable calculation, Jaynes and Cummings restricted their attention to the interaction of an atom with a
9718:
that it takes for the resonant terms to complete one full oscillation, the anti-resonant terms will complete many full cycles. Since over each full cycle
6536:
3937:{\displaystyle |n,-\rangle =\sin \left({\frac {\alpha _{n}}{2}}\right)|\psi _{1n}\rangle -\cos \left({\frac {\alpha _{n}}{2}}\right)|\psi _{2n}\rangle }
3794:{\displaystyle |n,+\rangle =\cos \left({\frac {\alpha _{n}}{2}}\right)|\psi _{1n}\rangle +\sin \left({\frac {\alpha _{n}}{2}}\right)|\psi _{2n}\rangle }
1328:
6260:
2963:
1199:
20397:
19404:
J.A.C. Gallas; G. Leuchs; H. Walther; H. Figger (1985). "Rydberg Atoms: High-Resolution
Spectroscopy and Radiation Interaction-Rydberg Molecules".
19259:
6665:
are used to denote the excited and ground state respectively. Setting the zero of energy to the ground state energy of the atom simplifies this to
20073:
P. Kochanski; Z. Bialynicka-Birula; I. Bialynicki-Birula (2001). "Squeezing of electromagnetic field in a cavity by electrons in Trojan states".
258:
is the total number of quanta in the coupled system. This quantization of energies and non-linear scaling is purely quantum mechanical in nature.
15323:
Thus the probability amplitude to find the atom in either state oscillates. This is the quantum mechanical explanation for the phenomenon of
14811:
4063:
3541:
18837:
16528:{\displaystyle {\hat {H}}_{n}=\left(n+{\frac {1}{2}}\right){\hat {I}}^{(n)}+{\frac {\hbar {\sqrt {n}}\Omega }{2}}{\hat {\sigma }}_{x}^{(n)}}
9458:
16355:{\displaystyle |\psi (0)\rangle =|g,\alpha \rangle =\sum _{n=0}^{\infty }e^{-|\alpha |^{2}/2}{\frac {\alpha ^{n}}{\sqrt {n!}}}|g,n\rangle }
331:
direct evidence for the non-linear behavior predicted from the quantum nature of the field in both superconducting circuits containing an "
20179:
Hussin, V.; Nieto, L. M. (2005). "Ladder operators and coherent states for the Jaynes-Cummings model in the rotating-wave approximation".
19214:
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to be the mean photon number in a coherent state. If the mean photon number is large, then since the statistics of the coherent state are
15035:{\displaystyle |\psi (t)\rangle =\cos \left({\frac {\Omega t}{2}}\right)|e,0\rangle -i\sin \left({\frac {\Omega t}{2}}\right)|g,1\rangle }
1000:
290:
demonstrated the predicted Rabi oscillations. However, as noted before, this effect was later found to have a semi-classical explanation.
18791:
261:
The collapse and subsequent revivals of the probability to detect the two-level system in a given state when the field is initially in a
273:
To realize the dynamics predicted by the Jaynes–Cummings model experimentally requires a quantum mechanical resonator with a very high
49:
4754:{\displaystyle |\psi _{\text{tot}}(t)\rangle =e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|\psi _{\text{tot}}(0)\rangle =\sum _{n}C_{n}\left.}
19095:
19144:
E.T. Jaynes; F.W. Cummings (1963). "Comparison of quantum and semiclassical radiation theories with application to the beam maser".
18879:
longer time regime; in that case, the periodic nature of the slowly varying double exponential predicts that there should also be a
17295:
7008:
19324:
J.H. Eberly; N.B. Narozhny; J.J. Sanchez-Mondragon (1980). "Periodic spontaneous collapse and revival in a simple quantum model".
15860:
Of particular interest is the case where the cavity frequency is perfectly resonant with the transition frequency of the atom, so
8799:
19492:
S. Haroche; J.M. Raimond (1985). "Radiative properties of
Rydberg states in resonant cavities". In D. Bates; B. Bederson (eds.).
15350:
14878:. Moreover, one assumes that the angular frequency of the mode can be approximated to the atomic transition frequency, involving
9589:
5765:
15863:
8370:
8309:
1134:
1073:
20537:
20480:
20409:
20376:
20342:
19100:
9721:
534:
consists of the free field
Hamiltonian, the atomic excitation Hamiltonian, and the Jaynes–Cummings interaction Hamiltonian:
20126:
Kundu, A. (2005). "Quantum
Integrable Multiatom Matter-Radiation Models With and Without the Rotating-Wave Approximation".
19829:
J.M Fink; et al. (2008). "Climbing the Jaynes-Cummings Ladder and observing its nonlinearity in a cavity QED system".
11991:
Thus the number operator commutes with the atom-field
Hamiltonian. The eigenstates of the number operator are the basis of
9455:
We now assume that the resonance frequency of the cavity is near the transition frequency of the atom, that is, we assume
7908:
2807:
951:
20425:
A. A. Karatsuba; E. A. Karatsuba (2009). "A resummation formula for collapse and revival in the Jaynes–Cummings model".
7057:
16024:
3531:{\displaystyle E_{\pm }(n)=\hbar \omega _{c}\left(n+{\frac {1}{2}}\right)\pm {\frac {1}{2}}\hbar \Omega _{n}(\delta ),}
2749:
13858:
9540:
20556:
20518:
20499:
19433:
19388:
17892:
15044:
So the probabilities to find the system in the ground or excited states after interacting with the cavity for a time
10635:
4772:
It is possible in the Heisenberg notation to directly determine the unitary evolution operator from the Hamiltonian:
2569:
2187:
It is possible, and often very helpful, to write the Hamiltonian of the full system as a sum of two commuting parts:
154:
67:
19568:
Yoshie, T.; et al. (2004). "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity".
19678:
T. Yoshie; et al. (2004). "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity".
6106:
By the unitary evolution operator one can calculate the time evolution of the state of the system described by its
18642:, the inner exponential inside the double exponential in the last term can be expanded up second order to obtain:
10360:
Using the results gathered in the last two sections, we may now write down the full Jaynes-Cummings Hamiltonian:
13458:
8483:
6429:
2658:
7213:{\textstyle {\hat {H}}_{F}=\hbar \omega _{c}\left({\hat {a}}_{c}^{\dagger }{\hat {a}}_{c}+{\frac {1}{2}}\right)}
6963:
is the Hamiltonian of the quantized electromagnetic field. Note the infinite sum over all possible wave-vectors
19083:
12134:
8435:
7740:
1905:
1859:
1813:
162:
135:
16141:
214:
A ladder of quantized energy levels, called the Jaynes-Cummings ladder, that scales in energy non-linearly as
14365:
9773:, and it accords with the intuition that energy must be conserved. Then the interaction Hamiltonian (taking
3037:
2927:
19009:
18535:
A plot of the probability to find the system in the excited state as a function of the unit-less parameter
888:
795:
For deriving the JCM interaction Hamiltonian the quantized radiation field is taken to consist of a single
19995:
Gea-Banacloche, J. (1993). "Jaynes-Cummings model with quasiclassical fields: The effect of dissipation".
19449:
G. Rempe; H. Walther; N. Klein (1987). "Observation of quantum collapse and revival in a one-atom maser".
18561:
14874:
Consider an atom entering the cavity initially in its excited state, while the cavity is initially in the
19729:
J. Kasprzak; et al. (2010). "Up on the Jaynes-Cummings ladder of a quantum-dot/microcavity system".
12124:{\displaystyle \left\{|g,0\rangle ;|e,0\rangle ,|g,1\rangle ;\cdots ;|e,n-1\rangle ,|g,n\rangle \right\}}
9770:
4140:, and assume an atom in the excited state is injected into the field. The initial state of the system is
1949:
309:
With the advent of one-atom masers it was possible to study the interaction of a single atom (usually a
20249:
Ellinas, D.; Smyrnakis, I. (2005). "Asymptotics of a quantum random walk driven by an optical cavity".
18055:
7351:
is the position of the atom). Electric field operator of a quantized electromagnetic field is given by
6351:
6112:
5622:
2718:
14881:
6389:
16200:
Consider an atom initially in the ground state interacting with a field mode initially prepared in a
4366:
3626:
527:{\displaystyle {\hat {H}}={\hat {H}}_{\text{field}}+{\hat {H}}_{\text{atom}}+{\hat {H}}_{\text{int}}}
40:
17359:
16067:
15407:
13818:
12641:
2871:
2622:
18954:
18775:{\displaystyle P_{e}(t)\simeq {\frac {1}{2}}-{\frac {1}{2}}\cdot \cos \lefte^{-\Omega ^{2}t^{2}/8}}
16061:
1902:
oscillating components. To get a solvable model, the quickly oscillating "counter-rotating" terms,
265:. While the collapse has a simple classical explanation, the revivals can only be explained by the
16107:
15447:
7332:
6966:
6780:
20575:
19115:
18028:
16171:
16012:{\displaystyle |n,\pm \rangle ={\frac {1}{\sqrt {2}}}\left(|g,n\rangle \mp |e,n-1\rangle \right)}
15324:
14784:
13908:
between the frequencies of the cavity and atomic transition. We have also defined the operators:
3947:
1285:
957:
217:
19359:
S. Haroche; J.M. Raimond (1985). "Radiative properties of Rydberg states in resonant cavities".
18991:
is shown to the right. Note the initial collapse followed by the clear revival at longer times.
13430:
13402:
12252:
12196:
5800:
4354:{\displaystyle |\psi _{\text{tot}}(0)\rangle =\sum _{n}{C_{n}|n,e\rangle }=\sum _{n}C_{n}\left.}
924:
420:
16367:
14337:
12677:
8789:
7322:{\displaystyle {\hat {H}}_{AF}=-{\hat {\mathbf {d} }}\cdot {\hat {\mathbf {E} }}(\mathbf {R} )}
6770:{\displaystyle {\hat {H}}_{A}=E_{e}|e\rangle \langle e|=\hbar \omega _{eg}|e\rangle \langle e|}
981:
is the angular frequency of the mode. On the other hand, the two-level atom is equivalent to a
19006:
This plot of quantum oscillations of atomic inversion—for quadratic scaled detuning parameter
18018:{\displaystyle \langle (\Delta n)^{2}\rangle /\langle n\rangle ^{2}\simeq 1/\langle n\rangle }
6988:
1318:
19253:
19065:
3392:
3173:{\displaystyle {H}_{ij}^{(n)}:=\langle \psi _{in}|{\hat {H}}_{\text{JC}}|\psi _{jn}\rangle ,}
416:
191:
123:
4400:
4046:{\displaystyle \alpha _{n}:=\tan ^{-1}\left({\frac {\Omega {\sqrt {n+1}}}{\delta }}\right).}
354:
20434:
20268:
20223:
20188:
20145:
20092:
20047:
20004:
19961:
19918:
19848:
19795:
19738:
19687:
19644:
19579:
19534:
19458:
19413:
19368:
19333:
19285:
19225:
19184:
19110:
17939:
12614:
10061:
9776:
424:
127:
8:
20446:
19197:
19172:
15772:
8793:
6642:
1322:
428:
388:
150:
20438:
20272:
20227:
20192:
20149:
20096:
20051:
20008:
19965:
19922:
19852:
19799:
19742:
19691:
19648:
19583:
19538:
19462:
19417:
19372:
19337:
19289:
19229:
19188:
18538:
6251:{\displaystyle {\hat {\rho }}(t)={\hat {U}}^{\dagger }(t){\hat {\rho }}(0){\hat {U}}(t)}
20450:
20284:
20258:
20161:
20135:
20108:
20082:
19934:
19908:
19872:
19838:
19811:
19785:
19711:
19660:
19634:
19603:
19301:
18081:
17395:
15404:
As shown in the previous subsection, if the initial state of the atom-cavity system is
15330:
15047:
12229:
12176:
11024:{\displaystyle {\hat {N}}=|e\rangle \langle e|+{\hat {a}}_{c}^{\dagger }{\hat {a}}_{c}}
7223:
5745:
3372:
2907:
775:
241:
19952:
Kukliński, J.; Madajczyk, J. (1988). "Strong squeezing in the Jaynes-Cummings model".
19896:
19773:
19622:
19425:
19380:
10676:
of the field. It will not contribute to the dynamics, so it may be neglected, giving:
993:
spin, but rather a generic two-level quantum system whose Hilbert space is isomorphic
878:{\displaystyle {\hat {E}}=E_{\text{ZPF}}\left({\hat {a}}+{\hat {a}}^{\dagger }\right)}
20552:
20533:
20514:
20495:
20476:
20454:
20405:
20372:
20338:
20165:
20020:
19977:
19938:
19876:
19864:
19754:
19703:
19664:
19595:
19550:
19474:
19429:
19384:
19305:
19241:
10673:
9711:{\displaystyle \tau ={\frac {2\pi }{\Delta }},\Delta \equiv \omega _{eg}-\omega _{c}}
187:
35:
20288:
20280:
20112:
20072:
19815:
5735:{\displaystyle {\hat {\varphi }}=g^{2}{\hat {a}}^{\dagger }{\hat {a}}+\delta ^{2}/4}
2263:{\displaystyle {\hat {H}}_{\text{JC}}={\hat {H}}_{\text{I}}+{\hat {H}}_{\text{II}},}
20442:
20304:, "Quantum theory of electromagnetic fields interacting with atoms and molecules",
20276:
20231:
20196:
20153:
20100:
20055:
20012:
19969:
19926:
19856:
19803:
19746:
19715:
19695:
19652:
19607:
19587:
19570:
19542:
19466:
19421:
19376:
19341:
19293:
19233:
19192:
19153:
6147:, and from there the expectation value of any observable, given the initial state:
287:
158:
110:
6632:{\displaystyle {\hat {H}}_{A}=E_{g}|g\rangle \langle g|+E_{e}|e\rangle \langle e|}
8429:
1398:{\displaystyle {\hat {H}}_{0}={\hat {H}}_{\text{field}}+{\hat {H}}_{\text{atom}}}
1193:
997:
a spin-half.) The atom is coupled to the field through its polarization operator
332:
303:
19546:
19345:
19237:
19000:
4397:
are stationary states of the field-atom system, then the state vector for times
3027:{\displaystyle \operatorname {span} \{|\psi _{1n}\rangle ,|\psi _{2n}\rangle \}}
198:
of quantum electromagnetic field. (See below for further mathematical details.)
20104:
20059:
19930:
19807:
19656:
19470:
16201:
13905:
13852:
11992:
6107:
3620:
2613:
982:
340:
262:
146:
142:
106:
20235:
20157:
19323:
19297:
18937:{\displaystyle \tau _{r}={\frac {4\pi }{\Omega }}{\sqrt {\langle n\rangle }}.}
1275:{\displaystyle {\hat {\sigma }}_{z}=|e\rangle \langle e|-|g\rangle \langle g|}
20569:
20301:
20016:
19973:
19774:"Single-electron double quantum dot dipole-coupled to a single photonic mode"
15853:{\displaystyle \delta E=E_{+}-E_{-}=\hbar {\sqrt {\Delta ^{2}+n\Omega ^{2}}}}
14331:
6807:
is the resonance frequency of transitions between the sub-levels of the atom.
989:. (It should be understood that "two-level atom" here is not an actual atom
377:) two-state subsystems (equivalent to spins higher than 1/2) is known as the
335:" coupled to a very high quality oscillator in the form of a superconducting
310:
299:
279:
183:
12278:(total number states), the Hamiltonian takes on a block diagonal structure:
19868:
19758:
19707:
19599:
19554:
19478:
19276:
F.W. Cummings (1965). "Stimulated emission of radiation in a single mode".
19245:
19157:
19105:
18209:
Inserting this into the sum yields a complicated product of exponentials:
14361:
986:
423:", since it allows the exploration of the counter intuitive effects of how
207:
179:
20024:
19981:
18784:
This result shows that the probability of occupation of the excited state
419:
state on a macroscopic scale. Such a state is sometimes referred to as a "
20263:
20087:
19639:
19506:
19403:
19082:
is the detuning parameter—was built on the basis of formulas obtained by
14373:
14369:
13249:
We obtain the portion of the Hamiltonian that acts in the n subspace as:
378:
336:
327:
19897:"Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms"
19860:
19699:
19591:
19173:"Reminiscing about thesis work with E T Jaynes at Stanford in the 1950s"
6523:{\displaystyle {\hat {H}}={\hat {H}}_{A}+{\hat {H}}_{F}+{\hat {H}}_{AF}}
4133:{\textstyle |\psi _{\text{field}}(0)\rangle =\sum _{n}{C_{n}|n\rangle }}
3612:{\textstyle \Omega _{n}(\delta )={\sqrt {\delta ^{2}+\Omega ^{2}(n+1)}}}
80:
20214:
Shore, Bruce W.; Knight, Peter L. (1993). "The Jaynes-Cummings Model".
14859:{\displaystyle \tan \theta _{n}=-{\frac {{\sqrt {n}}\Omega }{\Delta }}}
12170:
11032:
7240:
indicates that we are considering only the resonant mode of the cavity.
6425:
20200:
10058:
With this approximation in hand (and absorbing the negative sign into
9586:
are nearly resonant, while the other exponential terms oscillating at
9530:{\displaystyle |\omega _{eg}-\omega _{c}|\ll \omega _{eg}+\omega _{c}}
20140:
19750:
18531:
202:
Jaynes–Cummings model reveals a number of novel features, including:
157:, both experimentally and theoretically. It also has applications in
18827:{\textstyle \Omega _{\text{eff}}={\sqrt {\langle n\rangle }}\Omega }
18635:{\displaystyle {\frac {\Omega t}{2}}\ll {\sqrt {\langle n\rangle }}}
14904:. Then the state of the atom-field system as a function of time is:
1063:{\displaystyle {\hat {S}}={\hat {\sigma }}_{+}+{\hat {\sigma }}_{-}}
19913:
427:
manifests in macroscopic systems. It can also be used to model how
348:
274:
19843:
19790:
45:
article was recently merged, duplicate information may be present.
18994:
16537:
Using this, the time evolution of the atom-field system will be:
8783:
2655:, being of tensor product form, are easily solved and denoted by
796:
131:
6339:{\displaystyle \langle {\hat {\Theta }}\rangle _{t}={\text{Tr}}}
93:
emission (absorption) by the atom into (out of) the cavity mode.
19620:
18834:. It also shows that it should decay over characteristic time:
14875:
14330:
to be the identity operator and Pauli x and z operators in the
8792:
into the so-called "co-rotating" frame. To do this, we use the
90:
20424:
20037:
19524:
17349:{\displaystyle {\frac {\hbar \omega _{c}}{2}}{\hat {I}}^{(n)}}
14360:
Hamiltonian is of the same form as what would be found in the
3214:
16060:. Thus the interaction of the atom with the field splits the
9537:. Under this condition, the exponential terms oscillating at
7047:{\displaystyle {\hat {a}}_{\mathbf {k} ,\lambda }^{\dagger }}
351:. The coupling of a single quantum field mode with multiple (
283:
119:
114:
113:
interacting with a quantized mode of an optical cavity (or a
14334:
of the n energy level of the atom-field system. This simple
8869:{\displaystyle {\hat {H}}_{0}={\hat {H}}_{A}+{\hat {H}}_{F}}
19623:"Spin-photon dynamics of quantum dots in two-mode cavities"
19177:
Journal of Physics B: Atomic, Molecular and Optical Physics
15389:{\displaystyle \Omega _{n}={\frac {{\sqrt {n}}\Omega }{2}}}
9638:{\displaystyle \omega _{eg}+\omega _{c}\simeq 2\omega _{c}}
85:
19448:
15910:{\displaystyle \omega _{eg}=\omega _{c}\implies \Delta =0}
269:
of the energy spectrum due to quantum nature of the field.
8421:{\displaystyle {\hat {\sigma }}_{-}=|g\rangle \langle e|}
8360:{\displaystyle {\hat {\sigma }}_{+}=|e\rangle \langle g|}
1185:{\displaystyle {\hat {\sigma }}_{-}=|g\rangle \langle e|}
1124:{\displaystyle {\hat {\sigma }}_{+}=|e\rangle \langle g|}
772:
Here, for convenience, the vacuum field energy is set to
339:, and in a collection of Rydberg atoms coupled via their
173:
19143:
16195:
16168:. This non-linear hierarchy of energy levels scaling as
2616:(frequency) between the field and the two-level system.
178:
The model was originally developed in a 1963 article by
15775:. The energy difference between the dressed states is:
7329:
is the dipole atom-field interaction Hamiltonian (here
985:
whose state can be described using a three-dimensional
190:
treatment to the behavior of atoms interacting with an
19511:. Vol. 20. New York: Academic Press. p. 414.
19496:. Vol. 20. New York: Academic Press. p. 350.
19491:
19358:
18869:{\displaystyle \tau _{c}={\frac {\sqrt {2}}{\Omega }}}
18794:
17728:
17600:
16390:), then the Hamiltonian for the n number subspace is:
13522:
13283:
12734:
12315:
9761:{\displaystyle {\frac {2\pi }{2\omega _{c}}}\ll \tau }
7108:
4873:
4780:
4066:
3623:
for the specific detuning parameter. The eigenstates
3544:
19170:
19068:
19012:
18957:
18889:
18840:
18650:
18600:
18564:
18541:
18215:
18104:
18084:
18058:
18031:
17948:
17895:
17418:
17398:
17362:
17298:
16543:
16396:
16370:
16210:
16174:
16144:
16110:
16070:
16027:
15923:
15866:
15781:
15488:
15450:
15410:
15353:
15333:
15070:
15050:
14910:
14884:
14814:
14787:
14382:
14340:
13914:
13861:
13821:
13494:
13461:
13433:
13405:
13255:
12958:
12706:
12680:
12644:
12617:
12284:
12255:
12232:
12199:
12179:
12137:
12001:
11041:
10937:
10682:
10638:
10366:
10093:
10085:), we may transform back to the Schrödinger picture:
10064:
9808:
9779:
9724:
9651:
9592:
9543:
9461:
8882:
8802:
8788:
Next, the analysis may be simplified by performing a
8486:
8438:
8373:
8312:
7948:
7911:
7773:
7743:
7596:
7357:
7335:
7248:
7226:
7060:
7011:
6991:
6969:
6815:
6783:
6671:
6645:
6539:
6438:
6392:
6354:
6263:
6153:
6115:
5832:
5803:
5768:
5748:
5654:
5625:
4778:
4429:
4403:
4369:
4146:
3977:
3950:
3806:
3663:
3629:
3428:
3395:
3375:
3186:
3076:
3040:
2966:
2930:
2910:
2874:
2810:
2752:
2721:
2661:
2625:
2572:
2276:
2193:
1958:
1908:
1862:
1816:
1411:
1331:
1288:
1202:
1137:
1076:
1003:
960:
927:
891:
805:
778:
540:
445:
391:
357:
244:
220:
16204:, so the initial state of the atom-field system is:
7935:{\displaystyle \mathbf {u} _{\mathbf {k} ,\lambda }}
3657:
associated with the energy eigenvalues are given by
2743:
denotes the number of radiation quanta in the mode.
20473:
Principles of Laser Spectroscopy and Quantum Optics
20335:
Principles of Laser Spectroscopy and Quantum Optics
9769:resonant terms. This approximation is known as the
2861:{\displaystyle |\psi _{2n}\rangle :=|n+1,g\rangle }
293:
20396:
19951:
19074:
19054:
18983:
18936:
18868:
18826:
18774:
18634:
18582:
18550:
18521:
18201:
18090:
18070:
18044:
18017:
17930:
17881:
17404:
17384:
17348:
17284:
16527:
16382:
16354:
16184:
16160:
16130:
16096:
16052:
16011:
15909:
15852:
15761:
15470:
15436:
15388:
15339:
15313:
15056:
15034:
14896:
14858:
14800:
14773:
14352:
14320:
13896:
13843:
13805:
13480:
13447:
13419:
13389:
13239:
12942:
12692:
12666:
12630:
12601:
12270:
12238:
12214:
12185:
12161:
12123:
11981:
11035:of this operator with the atom-field Hamiltonian:
11023:
10917:
10664:
10624:
10345:
10077:
10048:
9792:
9760:
9710:
9637:
9578:
9529:
9447:
8868:
8771:
8472:
8420:
8359:
8298:
7934:
7897:
7759:
7729:
7582:
7343:
7321:
7232:
7212:
7091:{\displaystyle {\hat {a}}_{\mathbf {k} ,\lambda }}
7090:
7046:
6997:
6977:
6955:
6799:
6769:
6657:
6639:is the Hamiltonian of the atom, where the letters
6631:
6522:
6407:
6378:
6338:
6250:
6139:
6095:
5818:
5787:
5754:
5734:
5640:
5611:
4753:
4415:
4389:
4353:
4132:
4045:
3963:
3936:
3793:
3649:
3611:
3530:
3414:
3381:
3359:
3172:
3062:
3026:
2952:
2916:
2896:
2860:
2796:
2735:
2707:
2647:
2604:
2558:
2262:
2172:
1940:
1894:
1848:
1800:
1397:
1301:
1274:
1184:
1123:
1062:
973:
942:
913:
877:
784:
762:
526:
403:
369:
250:
230:
141:The Jaynes–Cummings model is of great interest to
20333:Berman, Paul R.; Malinovsky, Vladimir S. (2011).
16053:{\displaystyle \delta E={\sqrt {n}}\hbar \Omega }
2797:{\displaystyle |\psi _{1n}\rangle :=|n,e\rangle }
20567:
20337:. Princeton University Press. pp. 360–370.
20332:
20248:
15917:. In the resonant case, the dressed states are:
13897:{\displaystyle \Delta =\omega _{c}-\omega _{eg}}
10355:
9579:{\displaystyle \omega _{eg}-\omega _{c}\simeq 0}
7942:s are the orthonormal field modes, we may write
6985:and two possible orthogonal polarization states
1312:
439:The Hamiltonian that describes the full system,
20470:
17931:{\displaystyle \langle n\rangle =|\alpha |^{2}}
10665:{\displaystyle {\frac {1}{2}}\hbar \omega _{c}}
4055:
2605:{\displaystyle \delta =\omega _{a}-\omega _{c}}
19994:
19945:
18995:Collapses and revivals of quantum oscillations
8784:Rotating frame and rotating-wave approximation
6530:Where we have made the following definitions:
6348:The initial state of the system is denoted by
4767:
1325:(a.k.a. rotating frame) defined by the choice
84:Illustration of the Jaynes-Cummings model. An
20508:
19275:
6418:
434:
20546:
19258:: CS1 maint: multiple names: authors list (
19224:(21). American Physical Society: 2499–2502.
18926:
18920:
18816:
18810:
18718:
18712:
18627:
18621:
18571:
18565:
18497:
18491:
18459:
18453:
18421:
18415:
18384:
18378:
18342:
18336:
18304:
18298:
18267:
18261:
18188:
18182:
18177:
18171:
18136:
18130:
18065:
18059:
18012:
18006:
17986:
17979:
17971:
17949:
17902:
17896:
17817:
17810:
17805:
17799:
17689:
17682:
17677:
17671:
17474:
17451:
17270:
17206:
17036:
16971:
16700:
16635:
16565:
16349:
16248:
16228:
16125:
16091:
16001:
15975:
15938:
15752:
15695:
15640:
15619:
15568:
15507:
15465:
15431:
15253:
15224:
15135:
15106:
15029:
14977:
14928:
14764:
14707:
14652:
14631:
14580:
14519:
13442:
13414:
13211:
13136:
13117:
13037:
13018:
12959:
12911:
12852:
12845:
12786:
12151:
12113:
12093:
12061:
12041:
12021:
11632:
11629:
11454:
11451:
11302:
11299:
11175:
11172:
10964:
10961:
10794:
10791:
10501:
10498:
8876:. Then the interaction Hamiltonian becomes:
8467:
8464:
8450:
8439:
8407:
8404:
8346:
8343:
7866:
7833:
7724:
7691:
7666:
7633:
6756:
6753:
6715:
6712:
6618:
6615:
6583:
6580:
6280:
6264:
4697:
4603:
4527:
4454:
4384:
4340:
4289:
4212:
4171:
4126:
4091:
3931:
3876:
3821:
3788:
3733:
3678:
3644:
3164:
3106:
3021:
3018:
2994:
2973:
2855:
2829:
2791:
2771:
2702:
2682:
1261:
1258:
1236:
1233:
1171:
1168:
1110:
1107:
20527:
20489:
20371:(3 ed.). Wiley-VCH. pp. 408–423.
20366:
20213:
20178:
19728:
17942:we have that the variance-to-mean ratio is
14869:
13481:{\displaystyle {\frac {1}{2}}\hbar \Delta }
8772:{\displaystyle {\hat {H}}_{AF}=\hbar \left}
5788:{\displaystyle g={\frac {\Omega }{\hbar }}}
2708:{\displaystyle |n+1,g\rangle ,|n,e\rangle }
186:to elucidate the effects of giving a fully
20404:(4 ed.). Springer. pp. 333–338.
20392:
20390:
20388:
19621:Florian Meier; David D. Awschalom (2004).
19271:
19269:
15897:
15893:
6062:
6030:
5941:
5909:
1948:, are ignored. This is referred to as the
20513:. Cambridge: Cambridge University Press.
20494:. Cambridge: Cambridge University Press.
20328:
20326:
20324:
20322:
20320:
20318:
20262:
20139:
20086:
19912:
19842:
19789:
19771:
19677:
19638:
19196:
15399:
14313:
12162:{\displaystyle \left\{|n\rangle \right\}}
8473:{\displaystyle \{|e\rangle ,|g\rangle \}}
7760:{\displaystyle \mathbf {R} =\mathbf {0} }
6000:
5954:
5849:
5254:
2729:
1941:{\displaystyle (\omega _{c}+\omega _{a})}
1895:{\displaystyle (\omega _{c}-\omega _{a})}
1849:{\displaystyle (\omega _{c}+\omega _{a})}
168:
126:in order to investigate the phenomena of
68:Learn how and when to remove this message
20362:
20360:
20358:
20356:
20354:
19828:
19509:Advances in Atomic and Molecular Physics
19494:Advances in Atomic and Molecular Physics
19406:Advances in Atomic and Molecular Physics
19361:Advances in Atomic and Molecular Physics
19319:
19317:
19315:
18530:
16364:For simplicity, take the resonant case (
16161:{\displaystyle {\sqrt {n}}\hbar \Omega }
7102:mode of the field, allowing us to write
6415:is an operator denoting the observable.
79:
20385:
19266:
15482:eigenstates of the atom-cavity system:
1810:This Hamiltonian contains both quickly
349:(see the mathematical derivation below)
320:
20568:
20509:Scully, M. O.; Zubairy, M. S. (1997).
20471:Berman, P.R.; Maliovsky, V.S. (2011).
20315:
19894:
19890:
19888:
19886:
19567:
9645:are nearly anti-resonant. In the time
3063:{\displaystyle {\hat {H}}_{\text{JC}}}
2953:{\displaystyle {\hat {H}}_{\text{JC}}}
1282:is the atomic inversion operator, and
174:1963: Edwin Jaynes & Fred Cummings
20547:Walls, D. F.; Milburn, G. J. (1995).
20351:
20125:
19520:
19518:
19312:
19210:
19208:
19055:{\displaystyle a=(\delta /2g)^{2}=40}
18558:for a system with mean photon number
17292:Note neither of the constant factors
16196:Collapse and revival of probabilities
6428:to these dynamics. In that case, the
914:{\displaystyle {\hat {a}}^{\dagger }}
20490:Gerry, C. C.; Knight, P. L. (2005).
20128:Theoretical and Mathematical Physics
18583:{\displaystyle \langle n\rangle =25}
12246:of quanta in the atom-field system.
1309:is the atomic transition frequency.
18:
19883:
952:creation and annihilation operators
431:is transferred in a quantum field.
13:
20464:
19515:
19205:
18967:
18913:
18861:
18821:
18796:
18743:
18723:
18604:
18478:
18426:
18365:
18309:
18120:
18106:
18033:
18025:. Using this result and expanding
17955:
17854:
17783:
17746:
17655:
17618:
17511:
17232:
17174:
17058:
16904:
16838:
16722:
16486:
16371:
16270:
16155:
16047:
15898:
15839:
15823:
15377:
15355:
15291:
15173:
14999:
14947:
14885:
14851:
14847:
14490:
14474:
14447:
13862:
13838:
13770:
13708:
13644:
13601:
13575:
13557:
13475:
13352:
13326:
13308:
9676:
9668:
6396:
6324:
6270:
5777:
4014:
3583:
3546:
3507:
3281:
3254:
2461:
2076:
1449:
720:
14:
20587:
18078:to lowest non-vanishing order in
18071:{\displaystyle \langle n\rangle }
16613:
16152:
16044:
15817:
14468:
14444:
14418:
13767:
13661:
13641:
13615:
13598:
13572:
13554:
13528:
13472:
13366:
13349:
13323:
13305:
13289:
13224:
13123:
13024:
12919:
12768:
12611:With the exception of the scalar
11765:
11379:
11100:
10808:
10767:
10708:
10649:
10515:
10474:
10392:
10236:
10228:
10157:
9846:
9025:
9017:
8952:
8512:
7995:
7811:
7417:
7131:
6856:
6729:
6379:{\displaystyle {\hat {\rho }}(0)}
6140:{\displaystyle {\hat {\rho }}(t)}
5780:
5641:{\displaystyle {\hat {\varphi }}}
4853:
4738:
4644:
4498:
3503:
3451:
3206:
2736:{\displaystyle n\in \mathbb {N} }
2420:
2307:
2028:
1981:
645:
571:
20367:Vogel, W.; Welsch, D-G. (2006).
18999:
14897:{\displaystyle \Delta \approx 0}
12249:In this basis of eigenstates of
12193:of photons. The number operator
8276:
8225:
8199:
8153:
8127:
8076:
8055:
8009:
7983:
7920:
7914:
7880:
7874:
7848:
7820:
7783:
7753:
7745:
7706:
7648:
7601:
7590:and dipole operator is given by
7569:
7561:
7532:
7508:
7500:
7479:
7447:
7441:
7426:
7395:
7376:
7362:
7337:
7312:
7298:
7281:
7076:
7027:
6971:
6923:
6891:
6865:
6844:
6408:{\displaystyle {\hat {\Theta }}}
6103:and their Hermitian conjugates.
5826:is guaranteed by the identities
294:1987: Rempe, Walther & Klein
23:
20528:Vogel, W.; Welsch, D-G (2006).
20418:
20295:
20242:
20207:
20181:Journal of Mathematical Physics
20172:
20119:
20066:
20031:
19988:
19822:
19772:J. Basset; et al. (2013).
19765:
19722:
19671:
19614:
19561:
19500:
19485:
9800:to be real for simplicity) is:
4390:{\displaystyle |n,\pm \rangle }
3650:{\displaystyle |n,\pm \rangle }
2868:are degenerate with respect to
109:. It describes the system of a
20475:. Princeton University Press.
20447:10.1088/1751-8113/42/19/195304
20400:; Sargent III, Murray (2007).
19442:
19397:
19352:
19198:10.1088/0953-4075/46/22/220202
19171:Frederick W. Cummings (2013).
19164:
19137:
19127:
19037:
19019:
18667:
18661:
18232:
18226:
17962:
17952:
17918:
17909:
17866:
17850:
17568:
17559:
17536:
17527:
17471:
17465:
17458:
17439:
17433:
17385:{\displaystyle {\hat {H}}_{0}}
17370:
17341:
17335:
17328:
17251:
17246:
17222:
17193:
17188:
17164:
17081:
17072:
17023:
17002:
16993:
16958:
16947:
16941:
16929:
16918:
16894:
16876:
16870:
16863:
16852:
16828:
16745:
16736:
16687:
16666:
16657:
16632:
16626:
16619:
16593:
16562:
16556:
16549:
16520:
16514:
16502:
16465:
16459:
16452:
16404:
16336:
16293:
16284:
16235:
16225:
16219:
16212:
16112:
16097:{\displaystyle |e,n-1\rangle }
16072:
15982:
15962:
15925:
15894:
15733:
15682:
15627:
15606:
15549:
15494:
15452:
15437:{\displaystyle |e,n-1\rangle }
15412:
15258:
15250:
15244:
15237:
15220:
15209:
15203:
15140:
15132:
15126:
15119:
15102:
15091:
15085:
15016:
14964:
14925:
14919:
14912:
14745:
14694:
14639:
14618:
14561:
14506:
14203:
14197:
14185:
14071:
14065:
14053:
13939:
13933:
13926:
13844:{\displaystyle 2g_{c}=\Omega }
13798:
13792:
13780:
13742:
13736:
13724:
13694:
13688:
13681:
13502:
13435:
13407:
13263:
13192:
13179:
13160:
13149:
13098:
13085:
13061:
13050:
12999:
12983:
12972:
12892:
12876:
12865:
12832:
12816:
12805:
12762:
12750:
12714:
12667:{\displaystyle {\hat {H}}_{n}}
12652:
12532:
12432:
12374:
12292:
12262:
12206:
12144:
12100:
12074:
12048:
12028:
12008:
11945:
11921:
11899:
11875:
11853:
11834:
11812:
11793:
11738:
11714:
11690:
11663:
11639:
11622:
11606:
11577:
11555:
11531:
11507:
11485:
11461:
11444:
11428:
11404:
11347:
11323:
11309:
11292:
11276:
11252:
11220:
11196:
11182:
11165:
11149:
11130:
11082:
11058:
11009:
10985:
10971:
10954:
10944:
10893:
10874:
10852:
10833:
10801:
10784:
10752:
10728:
10690:
10600:
10581:
10559:
10540:
10508:
10491:
10441:
10417:
10374:
10321:
10302:
10280:
10261:
10208:
10190:
10184:
10169:
10137:
10101:
10033:
10004:
9976:
9957:
9939:
9910:
9890:
9871:
9837:
9831:
9816:
9494:
9463:
9432:
9403:
9383:
9364:
9336:
9307:
9279:
9260:
9227:
9198:
9175:
9156:
9123:
9094:
9069:
9050:
8997:
8964:
8932:
8911:
8905:
8890:
8854:
8832:
8810:
8747:
8728:
8686:
8667:
8617:
8598:
8561:
8542:
8494:
8457:
8443:
8430:raising and lowering operators
8414:
8397:
8381:
8353:
8336:
8320:
8268:
8249:
8191:
8172:
8119:
8100:
8047:
8028:
7956:
7859:
7852:
7840:
7717:
7710:
7698:
7679:
7659:
7652:
7640:
7621:
7605:
7524:
7471:
7380:
7372:
7366:
7316:
7308:
7302:
7285:
7256:
7180:
7156:
7116:
7068:
7019:
6915:
6883:
6823:
6763:
6746:
6722:
6705:
6679:
6625:
6608:
6590:
6573:
6547:
6505:
6483:
6461:
6445:
6432:for the atom-field system is:
6399:
6373:
6367:
6361:
6333:
6327:
6318:
6312:
6306:
6297:
6273:
6245:
6239:
6233:
6224:
6218:
6212:
6203:
6197:
6185:
6172:
6166:
6160:
6134:
6128:
6122:
6079:
6056:
6037:
6009:
5976:
5962:
5935:
5916:
5887:
5858:
5810:
5705:
5687:
5661:
5632:
5580:
5566:
5523:
5476:
5458:
5408:
5393:
5379:
5334:
5316:
5261:
5232:
5203:
5157:
5139:
5065:
5036:
4978:
4930:
4912:
4833:
4805:
4799:
4793:
4727:
4721:
4684:
4633:
4627:
4590:
4524:
4518:
4504:
4478:
4451:
4445:
4431:
4371:
4327:
4276:
4199:
4168:
4162:
4148:
4119:
4088:
4082:
4068:
3914:
3859:
3808:
3771:
3716:
3665:
3631:
3604:
3592:
3561:
3555:
3522:
3516:
3445:
3439:
3407:
3401:
3316:
3304:
3198:
3192:
3147:
3134:
3123:
3098:
3092:
3048:
3001:
2977:
2938:
2924:, it is enough to diagonalize
2897:{\displaystyle {\hat {H}}_{I}}
2882:
2836:
2812:
2778:
2754:
2689:
2663:
2648:{\displaystyle {\hat {H}}_{I}}
2633:
2535:
2516:
2494:
2481:
2435:
2401:
2368:
2350:
2332:
2288:
2245:
2223:
2201:
2182:
2150:
2131:
2109:
2096:
2050:
2019:
2001:
1966:
1935:
1909:
1889:
1863:
1843:
1817:
1782:
1753:
1730:
1711:
1693:
1664:
1644:
1631:
1614:
1588:
1568:
1549:
1531:
1505:
1482:
1469:
1437:
1431:
1419:
1383:
1361:
1339:
1268:
1251:
1243:
1226:
1210:
1194:raising and lowering operators
1178:
1161:
1145:
1117:
1100:
1084:
1048:
1026:
1010:
934:
899:
858:
842:
812:
747:
735:
695:
667:
626:
609:
591:
552:
512:
490:
468:
452:
163:quantum information processing
1:
19426:10.1016/S0065-2199(08)60272-9
19381:10.1016/S0065-2199(08)60271-7
19121:
19101:Jaynes–Cummings–Hubbard model
18984:{\displaystyle gt=\Omega t/2}
10356:Jaynes-Cummings Hamiltonian 2
1313:Jaynes–Cummings Hamiltonian 1
799:mode with the field operator
19895:J. Lee; et al. (2017).
18594:For "small" times such that
16131:{\displaystyle |g,n\rangle }
15471:{\displaystyle |g,n\rangle }
13399:By shifting the energy from
12674:on the diagonal is itself a
7344:{\displaystyle \mathbf {R} }
6978:{\displaystyle \mathbf {k} }
6800:{\displaystyle \omega _{eg}}
4056:Schrödinger picture dynamics
3389:, the energy eigenvalues of
155:quantum information circuits
105:) is a theoretical model in
7:
20492:Introductory Quantum Optics
19547:10.1103/PhysRevLett.73.1489
19346:10.1103/PhysRevLett.44.1323
19238:10.1103/PhysRevLett.64.2499
19089:
18045:{\displaystyle \Omega _{n}}
16185:{\displaystyle {\sqrt {n}}}
15347:quanta will have frequency
14808:is defined by the relation
14801:{\displaystyle \theta _{n}}
10927:Next, define the so-called
9771:rotating wave approximation
4768:Heisenberg picture dynamics
3964:{\displaystyle \alpha _{n}}
1950:rotating wave approximation
1302:{\displaystyle \omega _{a}}
974:{\displaystyle \omega _{c}}
298:It was not until 1987 that
231:{\displaystyle {\sqrt {n}}}
43:. The specific problem is:
10:
20592:
20402:Elements of Quantum Optics
20105:10.1103/PhysRevA.63.013811
20060:10.1103/PhysRevA.71.023811
19931:10.1103/PhysRevA.95.041801
19808:10.1103/PhysRevB.88.125312
19657:10.1103/PhysRevB.70.205329
19471:10.1103/PhysRevLett.58.353
13448:{\displaystyle |g\rangle }
13420:{\displaystyle |e\rangle }
12271:{\displaystyle {\hat {N}}}
12215:{\displaystyle {\hat {N}}}
7767:and making the definition
6419:Mathematical formulation 2
5819:{\displaystyle {\hat {U}}}
3034:. The matrix elements of
1196:of the atom. The operator
943:{\displaystyle {\hat {a}}}
435:Mathematical formulation 1
20532:(3 ed.). Wiley-VCH.
20281:10.1088/1464-4266/7/7/004
20236:10.1080/09500349314551321
20158:10.1007/s11232-005-0125-7
19298:10.1103/PhysRev.140.A1051
18788:with effective frequency
17889:Where we have identified
16383:{\displaystyle \Delta =0}
14353:{\displaystyle 2\times 2}
13815:where we have identified
12952:Now, using the relation:
12693:{\displaystyle 2\times 2}
20216:Journal of Modern Optics
20017:10.1103/PhysRevA.47.2221
19974:10.1103/PhysRevA.37.3175
18883:of probability at time:
14870:Vacuum Rabi oscillations
6998:{\displaystyle \lambda }
20427:J. Phys. A: Math. Theor
19116:Vacuum Rabi oscillation
19075:{\displaystyle \delta }
16021:With energy difference
15325:vacuum Rabi oscillation
12173:with a definite number
3415:{\displaystyle H^{(n)}}
101:(sometimes abbreviated
16:Model in quantum optics
20081:(1): 013811–013811–8.
19158:10.1109/PROC.1963.1664
19096:Caldeira–Leggett model
19076:
19056:
18985:
18938:
18870:
18828:
18776:
18636:
18591:
18584:
18552:
18523:
18203:
18092:
18072:
18046:
18019:
17932:
17883:
17787:
17659:
17515:
17406:
17386:
17350:
17286:
17062:
16726:
16529:
16384:
16356:
16274:
16186:
16162:
16132:
16098:
16054:
16013:
15911:
15854:
15763:
15472:
15438:
15400:Jaynes-Cummings ladder
15390:
15341:
15315:
15058:
15036:
14898:
14860:
14802:
14775:
14354:
14322:
13898:
13845:
13807:
13482:
13449:
13421:
13391:
13241:
12944:
12694:
12668:
12632:
12603:
12272:
12240:
12216:
12187:
12163:
12125:
11983:
11025:
10919:
10666:
10626:
10347:
10079:
10050:
9794:
9762:
9712:
9639:
9580:
9531:
9449:
8870:
8790:passive transformation
8773:
8474:
8422:
8361:
8300:
7936:
7899:
7761:
7731:
7584:
7345:
7323:
7234:
7214:
7092:
7048:
6999:
6979:
6957:
6801:
6771:
6659:
6633:
6524:
6409:
6380:
6340:
6252:
6141:
6097:
5820:
5789:
5756:
5736:
5642:
5613:
4755:
4417:
4416:{\displaystyle t>0}
4391:
4355:
4134:
4047:
3965:
3938:
3795:
3651:
3613:
3532:
3416:
3383:
3361:
3174:
3064:
3028:
2954:
2918:
2898:
2862:
2798:
2737:
2709:
2649:
2606:
2560:
2264:
2174:
1942:
1896:
1850:
1802:
1399:
1303:
1276:
1186:
1125:
1064:
975:
944:
915:
885:, where the operators
879:
786:
764:
528:
405:
371:
370:{\displaystyle N>1}
252:
232:
169:Historical development
94:
19077:
19057:
18986:
18939:
18871:
18829:
18777:
18637:
18585:
18553:
18534:
18524:
18204:
18093:
18073:
18047:
18020:
17933:
17884:
17767:
17639:
17495:
17407:
17387:
17351:
17287:
17042:
16706:
16530:
16385:
16357:
16254:
16187:
16163:
16133:
16099:
16055:
16014:
15912:
15855:
15764:
15473:
15439:
15391:
15342:
15316:
15059:
15037:
14899:
14861:
14803:
14776:
14355:
14323:
13899:
13846:
13808:
13483:
13450:
13422:
13392:
13242:
12945:
12695:
12669:
12633:
12631:{\displaystyle H_{0}}
12604:
12273:
12241:
12217:
12188:
12164:
12126:
11984:
11026:
10920:
10667:
10627:
10348:
10080:
10078:{\displaystyle g_{c}}
10051:
9795:
9793:{\displaystyle g_{c}}
9763:
9713:
9640:
9581:
9532:
9450:
8871:
8774:
8475:
8423:
8362:
8301:
7937:
7900:
7762:
7732:
7585:
7346:
7324:
7235:
7215:
7093:
7049:
7000:
6980:
6958:
6802:
6772:
6660:
6634:
6525:
6410:
6381:
6341:
6253:
6142:
6098:
5821:
5790:
5757:
5737:
5643:
5614:
4756:
4418:
4392:
4356:
4135:
4048:
3966:
3939:
3796:
3652:
3614:
3533:
3417:
3384:
3362:
3175:
3065:
3029:
2955:
2919:
2899:
2863:
2799:
2738:
2710:
2650:
2607:
2561:
2265:
2175:
1943:
1897:
1851:
1803:
1400:
1304:
1277:
1187:
1126:
1065:
976:
945:
916:
880:
787:
765:
529:
417:quantum superposition
406:
372:
253:
233:
192:electromagnetic field
124:electromagnetic field
99:Jaynes–Cummings model
83:
19183:(22): 220202 (3pp).
19111:Spontaneous emission
19086:and E.A. Karatsuba.
19066:
19010:
18955:
18887:
18838:
18792:
18648:
18598:
18562:
18539:
18213:
18102:
18082:
18056:
18029:
17946:
17893:
17416:
17396:
17360:
17296:
16541:
16394:
16368:
16208:
16172:
16142:
16108:
16068:
16025:
15921:
15864:
15779:
15486:
15448:
15408:
15351:
15331:
15068:
15048:
14908:
14882:
14812:
14785:
14380:
14338:
13912:
13859:
13819:
13492:
13459:
13431:
13403:
13253:
12956:
12704:
12700:matrix of the form;
12678:
12642:
12615:
12282:
12253:
12230:
12197:
12177:
12135:
11999:
11039:
10935:
10680:
10636:
10364:
10091:
10062:
9806:
9777:
9722:
9649:
9590:
9541:
9459:
8880:
8800:
8484:
8436:
8371:
8310:
7946:
7909:
7771:
7741:
7594:
7355:
7333:
7246:
7224:
7220:where the subscript
7106:
7058:
7009:
6989:
6967:
6813:
6781:
6669:
6643:
6537:
6436:
6390:
6352:
6261:
6151:
6113:
5830:
5801:
5766:
5746:
5652:
5623:
4776:
4427:
4401:
4367:
4144:
4064:
3975:
3948:
3804:
3661:
3627:
3542:
3426:
3393:
3373:
3184:
3074:
3038:
2964:
2928:
2908:
2872:
2808:
2750:
2719:
2659:
2623:
2570:
2274:
2191:
1956:
1906:
1860:
1814:
1409:
1329:
1286:
1200:
1135:
1074:
1001:
958:
925:
889:
803:
776:
538:
443:
425:quantum entanglement
389:
383:Tavis–Cummings model
355:
321:Further developments
242:
218:
128:spontaneous emission
50:improve this article
39:to meet Knowledge's
20551:. Springer-Verlag.
20439:2009JPhA...42s5304K
20273:2005JOptB...7S.152E
20228:1993JMOp...40.1195S
20193:2005JMP....46l2102H
20150:2005TMP...144..975K
20097:2000PhRvA..63a3811K
20052:2005PhRvA..71b3811R
20009:1993PhRvA..47.2221G
19966:1988PhRvA..37.3175K
19923:2017PhRvA..95d1801L
19861:10.1038/nature07112
19853:2008Natur.454..315F
19800:2013PhRvB..88l5312B
19743:2010NatMa...9..304K
19700:10.1038/nature03119
19692:2004Natur.432..200Y
19649:2004PhRvB..70t5329M
19592:10.1038/nature03119
19584:2004Natur.432..200Y
19539:1994PhRvL..73.1489M
19463:1987PhRvL..58..353R
19418:1985AdAMP..20..413G
19373:1985AdAMP..20..347H
19338:1980PhRvL..44.1323E
19290:1965PhRv..140.1051C
19284:(4A): A1051–A1056.
19230:1990PhRvL..64.2499Z
19189:2013JPhB...46v0202C
16951:
16524:
14207:
14075:
13855:of the system, and
13802:
13746:
13455:with the amount of
13077:
11937:
11891:
11706:
11679:
11593:
11523:
11339:
11268:
11212:
11001:
10909:
10744:
10616:
10433:
10337:
9992:
9295:
9252:
9148:
9085:
8794:interaction picture
8720:
8702:
8633:
8590:
8241:
8215:
8143:
8092:
7548:
7172:
7043:
6907:
6658:{\displaystyle e,g}
5619:where the operator
3971:is defined through
3102:
2619:The eigenstates of
1323:interaction picture
1319:Schrödinger picture
429:quantum information
404:{\displaystyle N=1}
151:solid-state physics
122:with the quantized
20433:(19): 195304, 16.
20312:(1), 1–121 (1973).
19072:
19052:
18981:
18934:
18866:
18824:
18772:
18632:
18592:
18580:
18551:{\displaystyle gt}
18548:
18519:
18199:
18088:
18068:
18042:
18015:
17928:
17879:
17877:
17737:
17609:
17402:
17382:
17346:
17282:
17280:
16922:
16525:
16495:
16380:
16352:
16182:
16158:
16128:
16094:
16050:
16009:
15907:
15850:
15759:
15757:
15468:
15434:
15386:
15337:
15311:
15309:
15054:
15032:
14894:
14856:
14798:
14771:
14769:
14350:
14318:
14316:
14178:
14046:
13894:
13841:
13803:
13773:
13717:
13649:
13478:
13445:
13417:
13387:
13381:
13237:
13054:
12940:
12934:
12690:
12664:
12628:
12599:
12593:
12268:
12236:
12212:
12183:
12159:
12121:
11979:
11977:
11914:
11868:
11683:
11656:
11570:
11500:
11316:
11245:
11189:
11021:
10978:
10915:
10886:
10721:
10662:
10632:The constant term
10622:
10593:
10410:
10343:
10314:
10075:
10046:
9969:
9790:
9758:
9708:
9635:
9576:
9527:
9445:
9272:
9238:
9134:
9062:
8866:
8769:
8706:
8679:
8610:
8576:
8470:
8418:
8357:
8296:
8219:
8184:
8112:
8070:
7994:
7932:
7895:
7757:
7727:
7580:
7517:
7406:
7341:
7319:
7230:
7210:
7149:
7088:
7044:
7012:
6995:
6975:
6953:
6876:
6855:
6797:
6767:
6655:
6629:
6520:
6405:
6376:
6336:
6248:
6137:
6093:
6091:
5816:
5785:
5752:
5732:
5638:
5609:
5607:
5603:
5595:
4751:
4542:
4413:
4387:
4351:
4228:
4186:
4130:
4106:
4043:
3961:
3934:
3791:
3647:
3609:
3528:
3412:
3379:
3357:
3351:
3170:
3077:
3070:in this subspace,
3060:
3024:
2950:
2914:
2894:
2858:
2794:
2733:
2705:
2645:
2602:
2556:
2554:
2260:
2170:
1938:
1892:
1846:
1798:
1395:
1299:
1272:
1182:
1121:
1060:
971:
940:
911:
875:
782:
760:
758:
524:
401:
367:
248:
228:
188:quantum mechanical
130:and absorption of
95:
20539:978-3-527-40507-7
20482:978-0-691-14056-8
20411:978-3-540-74209-8
20378:978-3-527-40507-7
20344:978-0-691-14056-8
20201:10.1063/1.2137718
20040:Physical Review A
19997:Physical Review A
19954:Physical Review A
19837:(7202): 315–318.
19686:(7014): 200–203.
19578:(7014): 200–203.
19533:(11): 1489–1492.
19332:(20): 1323–1326.
18929:
18916:
18864:
18860:
18819:
18802:
18721:
18694:
18681:
18630:
18614:
18503:
18500:
18424:
18390:
18387:
18307:
18275:
18246:
18192:
18160:
18139:
18126:
18091:{\displaystyle n}
17835:
17744:
17736:
17707:
17616:
17608:
17556:
17405:{\displaystyle t}
17373:
17331:
17319:
17230:
17172:
17121:
17120:
16932:
16902:
16866:
16836:
16785:
16784:
16596:
16505:
16493:
16484:
16455:
16438:
16407:
16333:
16332:
16180:
16150:
16042:
15954:
15953:
15848:
15726:
15675:
15599:
15542:
15384:
15375:
15340:{\displaystyle n}
15301:
15183:
15057:{\displaystyle t}
15009:
14957:
14854:
14845:
14738:
14687:
14611:
14554:
14499:
14466:
14442:
14368:gives the energy
14188:
14056:
13929:
13904:is the so-called
13783:
13765:
13758:
13727:
13715:
13684:
13639:
13608:
13596:
13582:
13570:
13552:
13505:
13470:
13359:
13347:
13333:
13321:
13266:
13222:
13182:
13163:
13088:
13064:
12986:
12879:
12819:
12717:
12655:
12535:
12435:
12377:
12295:
12265:
12239:{\displaystyle n}
12209:
12186:{\displaystyle n}
12169:of the field are
12131:where the states
11948:
11924:
11902:
11878:
11856:
11837:
11815:
11796:
11741:
11717:
11693:
11666:
11609:
11580:
11558:
11534:
11510:
11488:
11431:
11407:
11350:
11326:
11279:
11255:
11223:
11199:
11152:
11133:
11085:
11061:
11012:
10988:
10947:
10896:
10877:
10855:
10836:
10755:
10731:
10693:
10674:zero-point energy
10647:
10603:
10584:
10562:
10543:
10464:
10444:
10420:
10377:
10324:
10305:
10283:
10264:
10211:
10172:
10140:
10104:
9979:
9960:
9893:
9874:
9819:
9750:
9671:
9386:
9367:
9282:
9263:
9178:
9159:
9072:
9053:
9000:
8967:
8935:
8893:
8857:
8835:
8813:
8750:
8731:
8689:
8670:
8620:
8601:
8564:
8545:
8497:
8384:
8323:
8271:
8252:
8194:
8175:
8122:
8103:
8050:
8031:
7977:
7959:
7855:
7831:
7830:
7713:
7682:
7655:
7624:
7608:
7527:
7474:
7437:
7436:
7389:
7369:
7305:
7288:
7259:
7233:{\displaystyle c}
7203:
7183:
7159:
7119:
7071:
7022:
6946:
6918:
6886:
6838:
6826:
6682:
6550:
6508:
6486:
6464:
6448:
6402:
6364:
6330:
6309:
6295:
6276:
6236:
6215:
6188:
6163:
6125:
6084:
6082:
6059:
6040:
6028:
6012:
5982:
5981:
5979:
5967:
5965:
5938:
5919:
5907:
5906:
5890:
5877:
5861:
5813:
5797:The unitarity of
5783:
5755:{\displaystyle g}
5708:
5690:
5664:
5635:
5586:
5585:
5583:
5571:
5569:
5528:
5526:
5493:
5479:
5461:
5411:
5399:
5398:
5396:
5384:
5382:
5351:
5337:
5319:
5264:
5252:
5251:
5235:
5222:
5206:
5174:
5160:
5142:
5085:
5084:
5068:
5055:
5039:
4997:
4981:
4947:
4933:
4915:
4842:
4836:
4796:
4677:
4583:
4533:
4515:
4487:
4481:
4442:
4423:is just given by
4320:
4269:
4219:
4177:
4159:
4097:
4079:
4034:
4028:
3907:
3852:
3764:
3709:
3607:
3501:
3483:
3382:{\displaystyle n}
3347:
3300:
3287:
3273:
3260:
3248:
3143:
3137:
3057:
3051:
2960:in the subspaces
2947:
2941:
2917:{\displaystyle n}
2885:
2636:
2538:
2519:
2497:
2484:
2468:
2450:
2438:
2410:
2404:
2383:
2371:
2353:
2335:
2297:
2291:
2254:
2248:
2232:
2226:
2210:
2204:
2153:
2134:
2112:
2099:
2083:
2065:
2053:
2022:
2004:
1975:
1969:
1733:
1714:
1647:
1634:
1571:
1552:
1485:
1472:
1456:
1428:
1422:
1392:
1386:
1370:
1364:
1342:
1213:
1148:
1087:
1070:. The operators
1051:
1029:
1013:
937:
902:
861:
845:
828:
815:
785:{\displaystyle 0}
750:
738:
727:
704:
698:
682:
670:
635:
629:
612:
594:
561:
555:
521:
515:
499:
493:
477:
471:
455:
251:{\displaystyle n}
226:
208:Rabi oscillations
206:The existence of
78:
77:
70:
41:quality standards
32:This article may
20583:
20562:
20543:
20524:
20505:
20486:
20459:
20458:
20422:
20416:
20415:
20394:
20383:
20382:
20364:
20349:
20348:
20330:
20313:
20299:
20293:
20292:
20266:
20264:quant-ph/0510112
20246:
20240:
20239:
20222:(7): 1195–1238.
20211:
20205:
20204:
20176:
20170:
20169:
20143:
20123:
20117:
20116:
20090:
20088:quant-ph/0007033
20070:
20064:
20063:
20035:
20029:
20028:
20003:(3): 2221–2234.
19992:
19986:
19985:
19960:(8): 3175–3178.
19949:
19943:
19942:
19916:
19892:
19881:
19880:
19846:
19826:
19820:
19819:
19793:
19769:
19763:
19762:
19751:10.1038/nmat2717
19731:Nature Materials
19726:
19720:
19719:
19675:
19669:
19668:
19642:
19640:cond-mat/0405342
19618:
19612:
19611:
19565:
19559:
19558:
19522:
19513:
19512:
19504:
19498:
19497:
19489:
19483:
19482:
19446:
19440:
19439:
19401:
19395:
19394:
19356:
19350:
19349:
19321:
19310:
19309:
19273:
19264:
19263:
19257:
19249:
19212:
19203:
19202:
19200:
19168:
19162:
19161:
19141:
19135:
19131:
19081:
19079:
19078:
19073:
19061:
19059:
19058:
19053:
19045:
19044:
19029:
19003:
18990:
18988:
18987:
18982:
18977:
18943:
18941:
18940:
18935:
18930:
18919:
18917:
18912:
18904:
18899:
18898:
18875:
18873:
18872:
18867:
18865:
18856:
18855:
18850:
18849:
18833:
18831:
18830:
18825:
18820:
18809:
18804:
18803:
18800:
18781:
18779:
18778:
18773:
18771:
18770:
18766:
18761:
18760:
18751:
18750:
18733:
18729:
18722:
18711:
18695:
18687:
18682:
18674:
18660:
18659:
18641:
18639:
18638:
18633:
18631:
18620:
18615:
18610:
18602:
18589:
18587:
18586:
18581:
18557:
18555:
18554:
18549:
18528:
18526:
18525:
18520:
18518:
18514:
18513:
18509:
18508:
18504:
18502:
18501:
18490:
18484:
18473:
18441:
18440:
18436:
18425:
18414:
18401:
18397:
18396:
18392:
18391:
18389:
18388:
18377:
18371:
18360:
18324:
18323:
18319:
18308:
18297:
18276:
18271:
18270:
18252:
18247:
18239:
18225:
18224:
18208:
18206:
18205:
18200:
18198:
18194:
18193:
18191:
18180:
18163:
18161:
18153:
18140:
18129:
18127:
18119:
18114:
18113:
18097:
18095:
18094:
18089:
18077:
18075:
18074:
18069:
18051:
18049:
18048:
18043:
18041:
18040:
18024:
18022:
18021:
18016:
18005:
17994:
17993:
17978:
17970:
17969:
17937:
17935:
17934:
17929:
17927:
17926:
17921:
17912:
17888:
17886:
17885:
17880:
17878:
17874:
17862:
17861:
17846:
17845:
17836:
17834:
17826:
17825:
17824:
17809:
17808:
17789:
17786:
17781:
17760:
17756:
17752:
17745:
17740:
17738:
17729:
17718:
17717:
17708:
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10974:
10957:
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10910:
10908:
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10884:
10879:
10878:
10870:
10863:
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10848:
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10829:
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10748:
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10704:
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10671:
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10663:
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10204:
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10133:
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10097:
10084:
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9895:
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9867:
9858:
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9830:
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9812:
9799:
9797:
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9767:
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9759:
9751:
9749:
9748:
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9726:
9717:
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9709:
9707:
9706:
9694:
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9672:
9667:
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9644:
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9636:
9634:
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9605:
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9577:
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9556:
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9536:
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9513:
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9466:
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9440:
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9418:
9417:
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9388:
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9379:
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9369:
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9360:
9356:
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9335:
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9294:
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9275:
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9264:
9256:
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9225:
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9084:
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9021:
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8978:
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8594:
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8460:
8446:
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8400:
8392:
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8364:
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8331:
8330:
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8316:
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8279:
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8264:
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8110:
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7993:
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7503:
7490:
7489:
7482:
7476:
7475:
7467:
7458:
7457:
7450:
7444:
7438:
7432:
7431:
7430:
7429:
7409:
7408:
7405:
7398:
7379:
7371:
7370:
7365:
7360:
7350:
7348:
7347:
7342:
7340:
7328:
7326:
7325:
7320:
7315:
7307:
7306:
7301:
7296:
7290:
7289:
7284:
7279:
7270:
7269:
7261:
7260:
7252:
7239:
7237:
7236:
7231:
7219:
7217:
7216:
7211:
7209:
7205:
7204:
7196:
7191:
7190:
7185:
7184:
7176:
7171:
7166:
7161:
7160:
7152:
7143:
7142:
7127:
7126:
7121:
7120:
7112:
7097:
7095:
7094:
7089:
7087:
7086:
7079:
7073:
7072:
7064:
7053:
7051:
7050:
7045:
7042:
7037:
7030:
7024:
7023:
7015:
7005:. The operators
7004:
7002:
7001:
6996:
6984:
6982:
6981:
6976:
6974:
6962:
6960:
6959:
6954:
6952:
6948:
6947:
6939:
6934:
6933:
6926:
6920:
6919:
6911:
6906:
6901:
6894:
6888:
6887:
6879:
6870:
6869:
6868:
6854:
6847:
6834:
6833:
6828:
6827:
6819:
6806:
6804:
6803:
6798:
6796:
6795:
6776:
6774:
6773:
6768:
6766:
6749:
6744:
6743:
6725:
6708:
6703:
6702:
6690:
6689:
6684:
6683:
6675:
6664:
6662:
6661:
6656:
6638:
6636:
6635:
6630:
6628:
6611:
6606:
6605:
6593:
6576:
6571:
6570:
6558:
6557:
6552:
6551:
6543:
6529:
6527:
6526:
6521:
6519:
6518:
6510:
6509:
6501:
6494:
6493:
6488:
6487:
6479:
6472:
6471:
6466:
6465:
6457:
6450:
6449:
6441:
6414:
6412:
6411:
6406:
6404:
6403:
6395:
6385:
6383:
6382:
6377:
6366:
6365:
6357:
6345:
6343:
6342:
6337:
6332:
6331:
6323:
6311:
6310:
6302:
6296:
6293:
6288:
6287:
6278:
6277:
6269:
6257:
6255:
6254:
6249:
6238:
6237:
6229:
6217:
6216:
6208:
6196:
6195:
6190:
6189:
6181:
6165:
6164:
6156:
6146:
6144:
6143:
6138:
6127:
6126:
6118:
6102:
6100:
6099:
6094:
6092:
6085:
6083:
6075:
6073:
6061:
6060:
6052:
6042:
6041:
6033:
6029:
6027:
6026:
6014:
6013:
6005:
6002:
5983:
5980:
5972:
5970:
5969:
5968:
5966:
5958:
5956:
5943:
5940:
5939:
5931:
5921:
5920:
5912:
5908:
5905:
5904:
5892:
5891:
5883:
5880:
5879:
5878:
5876:
5875:
5863:
5862:
5854:
5851:
5838:
5825:
5823:
5822:
5817:
5815:
5814:
5806:
5794:
5792:
5791:
5786:
5784:
5776:
5761:
5759:
5758:
5753:
5741:
5739:
5738:
5733:
5728:
5723:
5722:
5710:
5709:
5701:
5698:
5697:
5692:
5691:
5683:
5679:
5678:
5666:
5665:
5657:
5647:
5645:
5644:
5639:
5637:
5636:
5628:
5618:
5616:
5615:
5610:
5608:
5604:
5600:
5599:
5592:
5588:
5587:
5584:
5576:
5574:
5573:
5572:
5570:
5562:
5560:
5548:
5543:
5529:
5527:
5519:
5517:
5501:
5500:
5499:
5495:
5494:
5486:
5481:
5480:
5472:
5469:
5468:
5463:
5462:
5454:
5442:
5441:
5419:
5418:
5413:
5412:
5404:
5400:
5397:
5389:
5387:
5386:
5385:
5383:
5375:
5373:
5361:
5359:
5358:
5357:
5353:
5352:
5344:
5339:
5338:
5330:
5327:
5326:
5321:
5320:
5312:
5300:
5299:
5266:
5265:
5257:
5253:
5250:
5249:
5237:
5236:
5228:
5225:
5224:
5223:
5221:
5220:
5208:
5207:
5199:
5196:
5184:
5182:
5181:
5180:
5176:
5175:
5167:
5162:
5161:
5153:
5150:
5149:
5144:
5143:
5135:
5123:
5122:
5091:
5087:
5086:
5083:
5082:
5070:
5069:
5061:
5058:
5057:
5056:
5054:
5053:
5041:
5040:
5032:
5029:
5017:
5012:
4998:
4996:
4995:
4983:
4982:
4974:
4971:
4955:
4954:
4953:
4949:
4948:
4940:
4935:
4934:
4926:
4923:
4922:
4917:
4916:
4908:
4896:
4895:
4861:
4857:
4856:
4852:
4844:
4843:
4840:
4838:
4837:
4829:
4798:
4797:
4789:
4760:
4758:
4757:
4752:
4747:
4743:
4742:
4741:
4737:
4720:
4719:
4687:
4682:
4678:
4673:
4672:
4663:
4648:
4647:
4643:
4626:
4625:
4593:
4588:
4584:
4579:
4578:
4569:
4552:
4551:
4541:
4517:
4516:
4513:
4507:
4502:
4501:
4497:
4489:
4488:
4485:
4483:
4482:
4474:
4444:
4443:
4440:
4434:
4422:
4420:
4419:
4414:
4396:
4394:
4393:
4388:
4374:
4360:
4358:
4357:
4352:
4347:
4343:
4330:
4325:
4321:
4316:
4315:
4306:
4279:
4274:
4270:
4265:
4264:
4255:
4238:
4237:
4227:
4215:
4202:
4197:
4196:
4185:
4161:
4160:
4157:
4151:
4139:
4137:
4136:
4131:
4129:
4122:
4117:
4116:
4105:
4081:
4080:
4077:
4071:
4052:
4050:
4049:
4044:
4039:
4035:
4030:
4029:
4018:
4012:
4003:
4002:
3987:
3986:
3970:
3968:
3967:
3962:
3960:
3959:
3944:where the angle
3943:
3941:
3940:
3935:
3930:
3929:
3917:
3912:
3908:
3903:
3902:
3893:
3875:
3874:
3862:
3857:
3853:
3848:
3847:
3838:
3811:
3800:
3798:
3797:
3792:
3787:
3786:
3774:
3769:
3765:
3760:
3759:
3750:
3732:
3731:
3719:
3714:
3710:
3705:
3704:
3695:
3668:
3656:
3654:
3653:
3648:
3634:
3618:
3616:
3615:
3610:
3608:
3591:
3590:
3578:
3577:
3568:
3554:
3553:
3537:
3535:
3534:
3529:
3515:
3514:
3502:
3494:
3489:
3485:
3484:
3476:
3463:
3462:
3438:
3437:
3421:
3419:
3418:
3413:
3411:
3410:
3388:
3386:
3385:
3380:
3366:
3364:
3363:
3358:
3356:
3355:
3348:
3343:
3342:
3333:
3328:
3327:
3301:
3290:
3288:
3280:
3274:
3263:
3261:
3253:
3249:
3244:
3243:
3234:
3229:
3228:
3202:
3201:
3179:
3177:
3176:
3171:
3163:
3162:
3150:
3145:
3144:
3141:
3139:
3138:
3130:
3126:
3121:
3120:
3101:
3090:
3082:
3069:
3067:
3066:
3061:
3059:
3058:
3055:
3053:
3052:
3044:
3033:
3031:
3030:
3025:
3017:
3016:
3004:
2993:
2992:
2980:
2959:
2957:
2956:
2951:
2949:
2948:
2945:
2943:
2942:
2934:
2923:
2921:
2920:
2915:
2903:
2901:
2900:
2895:
2893:
2892:
2887:
2886:
2878:
2867:
2865:
2864:
2859:
2839:
2828:
2827:
2815:
2803:
2801:
2800:
2795:
2781:
2770:
2769:
2757:
2742:
2740:
2739:
2734:
2732:
2714:
2712:
2711:
2706:
2692:
2666:
2654:
2652:
2651:
2646:
2644:
2643:
2638:
2637:
2629:
2611:
2609:
2608:
2603:
2601:
2600:
2588:
2587:
2565:
2563:
2562:
2557:
2555:
2551:
2547:
2546:
2545:
2540:
2539:
2531:
2527:
2526:
2521:
2520:
2512:
2505:
2504:
2499:
2498:
2490:
2486:
2485:
2477:
2469:
2464:
2456:
2451:
2446:
2445:
2440:
2439:
2431:
2427:
2412:
2411:
2408:
2406:
2405:
2397:
2389:
2385:
2384:
2379:
2378:
2373:
2372:
2364:
2360:
2355:
2354:
2346:
2343:
2342:
2337:
2336:
2328:
2319:
2318:
2299:
2298:
2295:
2293:
2292:
2284:
2269:
2267:
2266:
2261:
2256:
2255:
2252:
2250:
2249:
2241:
2234:
2233:
2230:
2228:
2227:
2219:
2212:
2211:
2208:
2206:
2205:
2197:
2179:
2177:
2176:
2171:
2166:
2162:
2161:
2160:
2155:
2154:
2146:
2142:
2141:
2136:
2135:
2127:
2120:
2119:
2114:
2113:
2105:
2101:
2100:
2092:
2084:
2079:
2071:
2066:
2061:
2060:
2055:
2054:
2046:
2042:
2040:
2039:
2024:
2023:
2015:
2012:
2011:
2006:
2005:
1997:
1993:
1992:
1977:
1976:
1973:
1971:
1970:
1962:
1947:
1945:
1944:
1939:
1934:
1933:
1921:
1920:
1901:
1899:
1898:
1893:
1888:
1887:
1875:
1874:
1855:
1853:
1852:
1847:
1842:
1841:
1829:
1828:
1807:
1805:
1804:
1799:
1794:
1790:
1789:
1788:
1781:
1780:
1768:
1767:
1741:
1740:
1735:
1734:
1726:
1722:
1721:
1716:
1715:
1707:
1700:
1699:
1692:
1691:
1679:
1678:
1655:
1654:
1649:
1648:
1640:
1636:
1635:
1627:
1621:
1620:
1613:
1612:
1600:
1599:
1579:
1578:
1573:
1572:
1564:
1560:
1559:
1554:
1553:
1545:
1538:
1537:
1530:
1529:
1517:
1516:
1493:
1492:
1487:
1486:
1478:
1474:
1473:
1465:
1457:
1452:
1444:
1430:
1429:
1426:
1424:
1423:
1415:
1404:
1402:
1401:
1396:
1394:
1393:
1390:
1388:
1387:
1379:
1372:
1371:
1368:
1366:
1365:
1357:
1350:
1349:
1344:
1343:
1335:
1317:Moving from the
1308:
1306:
1305:
1300:
1298:
1297:
1281:
1279:
1278:
1273:
1271:
1254:
1246:
1229:
1221:
1220:
1215:
1214:
1206:
1191:
1189:
1188:
1183:
1181:
1164:
1156:
1155:
1150:
1149:
1141:
1130:
1128:
1127:
1122:
1120:
1103:
1095:
1094:
1089:
1088:
1080:
1069:
1067:
1066:
1061:
1059:
1058:
1053:
1052:
1044:
1037:
1036:
1031:
1030:
1022:
1015:
1014:
1006:
980:
978:
977:
972:
970:
969:
950:are the bosonic
949:
947:
946:
941:
939:
938:
930:
920:
918:
917:
912:
910:
909:
904:
903:
895:
884:
882:
881:
876:
874:
870:
869:
868:
863:
862:
854:
847:
846:
838:
830:
829:
826:
817:
816:
808:
791:
789:
788:
783:
769:
767:
766:
761:
759:
752:
751:
743:
740:
739:
731:
728:
723:
715:
706:
705:
702:
700:
699:
691:
683:
678:
677:
672:
671:
663:
659:
657:
656:
637:
636:
633:
631:
630:
622:
614:
613:
605:
602:
601:
596:
595:
587:
583:
582:
563:
562:
559:
557:
556:
548:
533:
531:
530:
525:
523:
522:
519:
517:
516:
508:
501:
500:
497:
495:
494:
486:
479:
478:
475:
473:
472:
464:
457:
456:
448:
410:
408:
407:
402:
376:
374:
373:
368:
288:microwave cavity
257:
255:
254:
249:
237:
235:
234:
229:
227:
222:
159:coherent control
73:
66:
62:
59:
53:
27:
26:
19:
20591:
20590:
20586:
20585:
20584:
20582:
20581:
20580:
20566:
20565:
20559:
20540:
20521:
20502:
20483:
20467:
20465:Further reading
20462:
20423:
20419:
20412:
20398:Meystre, Pierre
20395:
20386:
20379:
20365:
20352:
20345:
20331:
20316:
20306:Physics Reports
20300:
20296:
20247:
20243:
20212:
20208:
20177:
20173:
20124:
20120:
20071:
20067:
20036:
20032:
19993:
19989:
19950:
19946:
19893:
19884:
19827:
19823:
19770:
19766:
19727:
19723:
19676:
19672:
19619:
19615:
19566:
19562:
19527:Phys. Rev. Lett
19523:
19516:
19505:
19501:
19490:
19486:
19451:Phys. Rev. Lett
19447:
19443:
19436:
19402:
19398:
19391:
19357:
19353:
19326:Phys. Rev. Lett
19322:
19313:
19274:
19267:
19251:
19250:
19218:Phys. Rev. Lett
19213:
19206:
19169:
19165:
19142:
19138:
19132:
19128:
19124:
19092:
19067:
19064:
19063:
19040:
19036:
19025:
19011:
19008:
19007:
18997:
18973:
18956:
18953:
18952:
18918:
18905:
18903:
18894:
18890:
18888:
18885:
18884:
18854:
18845:
18841:
18839:
18836:
18835:
18808:
18799:
18795:
18793:
18790:
18789:
18762:
18756:
18752:
18746:
18742:
18738:
18734:
18710:
18709:
18705:
18686:
18673:
18655:
18651:
18649:
18646:
18645:
18619:
18603:
18601:
18599:
18596:
18595:
18563:
18560:
18559:
18540:
18537:
18536:
18489:
18485:
18474:
18472:
18468:
18452:
18448:
18432:
18413:
18409:
18405:
18376:
18372:
18361:
18359:
18355:
18351:
18335:
18331:
18315:
18296:
18289:
18285:
18284:
18280:
18257:
18253:
18251:
18238:
18220:
18216:
18214:
18211:
18210:
18181:
18164:
18162:
18152:
18145:
18141:
18128:
18118:
18109:
18105:
18103:
18100:
18099:
18083:
18080:
18079:
18057:
18054:
18053:
18036:
18032:
18030:
18027:
18026:
18001:
17989:
17985:
17974:
17965:
17961:
17947:
17944:
17943:
17922:
17917:
17916:
17908:
17894:
17891:
17890:
17876:
17875:
17873:
17870:
17869:
17857:
17853:
17841:
17837:
17827:
17820:
17816:
17795:
17791:
17790:
17788:
17782:
17771:
17758:
17757:
17739:
17727:
17726:
17722:
17713:
17709:
17699:
17692:
17688:
17667:
17663:
17662:
17660:
17654:
17643:
17630:
17629:
17611:
17599:
17598:
17594:
17585:
17581:
17572:
17567:
17566:
17558:
17548:
17540:
17535:
17534:
17526:
17522:
17518:
17516:
17510:
17499:
17488:
17482:
17457:
17450:
17446:
17445:
17427:
17423:
17419:
17417:
17414:
17413:
17397:
17394:
17393:
17376:
17365:
17364:
17363:
17361:
17358:
17357:
17334:
17323:
17322:
17321:
17309:
17305:
17301:
17299:
17297:
17294:
17293:
17279:
17278:
17250:
17238:
17225:
17221:
17192:
17180:
17167:
17163:
17156:
17152:
17141:
17137:
17127:
17123:
17107:
17103:
17101:
17091:
17085:
17080:
17079:
17071:
17067:
17063:
17057:
17046:
17022:
17012:
17006:
17001:
17000:
16992:
16988:
16984:
16975:
16974:
16957:
16940:
16935:
16924:
16923:
16910:
16897:
16893:
16869:
16858:
16857:
16856:
16844:
16831:
16827:
16820:
16816:
16805:
16801:
16791:
16787:
16771:
16767:
16765:
16755:
16749:
16744:
16743:
16735:
16731:
16727:
16721:
16710:
16686:
16676:
16670:
16665:
16664:
16656:
16652:
16648:
16639:
16638:
16618:
16608:
16599:
16588:
16587:
16586:
16579:
16575:
16568:
16548:
16544:
16542:
16539:
16538:
16513:
16508:
16497:
16496:
16479:
16475:
16473:
16458:
16447:
16446:
16445:
16430:
16423:
16419:
16410:
16399:
16398:
16397:
16395:
16392:
16391:
16369:
16366:
16365:
16335:
16319:
16315:
16313:
16303:
16297:
16292:
16291:
16283:
16279:
16275:
16269:
16258:
16234:
16211:
16209:
16206:
16205:
16198:
16175:
16173:
16170:
16169:
16145:
16143:
16140:
16139:
16111:
16109:
16106:
16105:
16071:
16069:
16066:
16065:
16037:
16026:
16023:
16022:
15981:
15961:
15960:
15956:
15944:
15924:
15922:
15919:
15918:
15887:
15883:
15871:
15867:
15865:
15862:
15861:
15842:
15838:
15826:
15822:
15820:
15808:
15804:
15795:
15791:
15780:
15777:
15776:
15756:
15755:
15732:
15717:
15713:
15711:
15707:
15681:
15666:
15662:
15660:
15656:
15643:
15626:
15623:
15622:
15605:
15590:
15586:
15584:
15580:
15548:
15533:
15529:
15527:
15523:
15510:
15493:
15489:
15487:
15484:
15483:
15451:
15449:
15446:
15445:
15411:
15409:
15406:
15405:
15402:
15370:
15369:
15367:
15358:
15354:
15352:
15349:
15348:
15332:
15329:
15328:
15308:
15307:
15290:
15288:
15284:
15275:
15271:
15262:
15257:
15256:
15236:
15219:
15212:
15197:
15193:
15190:
15189:
15172:
15170:
15166:
15157:
15153:
15144:
15139:
15138:
15118:
15101:
15094:
15079:
15075:
15071:
15069:
15066:
15065:
15049:
15046:
15045:
15015:
14998:
14996:
14992:
14963:
14946:
14944:
14940:
14911:
14909:
14906:
14905:
14883:
14880:
14879:
14872:
14840:
14839:
14837:
14825:
14821:
14813:
14810:
14809:
14792:
14788:
14786:
14783:
14782:
14768:
14767:
14744:
14729:
14725:
14723:
14719:
14693:
14678:
14674:
14672:
14668:
14655:
14638:
14635:
14634:
14617:
14602:
14598:
14596:
14592:
14560:
14545:
14541:
14539:
14535:
14522:
14505:
14502:
14501:
14493:
14489:
14477:
14473:
14471:
14458:
14434:
14425:
14421:
14414:
14410:
14403:
14391:
14387:
14383:
14381:
14378:
14377:
14366:Diagonalization
14339:
14336:
14335:
14315:
14314:
14310:
14309:
14285:
14281:
14266:
14262:
14244:
14240:
14219:
14215:
14208:
14196:
14191:
14180:
14179:
14175:
14174:
14159:
14155:
14140:
14136:
14112:
14108:
14087:
14083:
14076:
14064:
14059:
14048:
14047:
14043:
14042:
14027:
14023:
14008:
14004:
13980:
13976:
13955:
13951:
13944:
13932:
13921:
13920:
13919:
13915:
13913:
13910:
13909:
13885:
13881:
13872:
13868:
13860:
13857:
13856:
13829:
13825:
13820:
13817:
13816:
13791:
13786:
13775:
13774:
13760:
13750:
13735:
13730:
13719:
13718:
13704:
13702:
13687:
13676:
13675:
13674:
13668:
13664:
13648:
13647:
13631:
13622:
13618:
13610:
13591:
13590:
13588:
13585:
13584:
13565:
13564:
13562:
13560:
13544:
13535:
13531:
13518:
13517:
13508:
13497:
13496:
13495:
13493:
13490:
13489:
13462:
13460:
13457:
13456:
13434:
13432:
13429:
13428:
13406:
13404:
13401:
13400:
13380:
13379:
13373:
13369:
13361:
13342:
13341:
13339:
13336:
13335:
13316:
13315:
13313:
13311:
13296:
13292:
13279:
13278:
13269:
13258:
13257:
13256:
13254:
13251:
13250:
13231:
13227:
13217:
13191:
13185:
13174:
13173:
13172:
13166:
13155:
13154:
13153:
13148:
13130:
13126:
13097:
13091:
13080:
13079:
13078:
13072:
13067:
13056:
13055:
13049:
13031:
13027:
12998:
12989:
12978:
12977:
12976:
12971:
12957:
12954:
12953:
12933:
12932:
12926:
12922:
12914:
12891:
12882:
12871:
12870:
12869:
12864:
12849:
12848:
12831:
12822:
12811:
12810:
12809:
12804:
12784:
12775:
12771:
12744:
12740:
12730:
12729:
12720:
12709:
12708:
12707:
12705:
12702:
12701:
12679:
12676:
12675:
12658:
12647:
12646:
12645:
12643:
12640:
12639:
12622:
12618:
12616:
12613:
12612:
12592:
12591:
12586:
12581:
12576:
12571:
12566:
12561:
12555:
12554:
12549:
12544:
12538:
12527:
12526:
12525:
12523:
12518:
12513:
12508:
12502:
12501:
12496:
12491:
12486:
12481:
12476:
12471:
12465:
12464:
12459:
12454:
12449:
12444:
12438:
12427:
12426:
12425:
12423:
12418:
12412:
12411:
12406:
12401:
12396:
12391:
12386:
12380:
12369:
12368:
12367:
12365:
12359:
12358:
12353:
12348:
12343:
12338:
12333:
12328:
12322:
12318:
12311:
12310:
12298:
12287:
12286:
12285:
12283:
12280:
12279:
12257:
12256:
12254:
12251:
12250:
12231:
12228:
12227:
12201:
12200:
12198:
12195:
12194:
12178:
12175:
12174:
12143:
12142:
12138:
12136:
12133:
12132:
12099:
12073:
12047:
12027:
12007:
12006:
12002:
12000:
11997:
11996:
11976:
11975:
11963:
11962:
11951:
11940:
11939:
11938:
11932:
11927:
11916:
11915:
11905:
11894:
11893:
11892:
11886:
11881:
11870:
11869:
11859:
11848:
11847:
11846:
11840:
11829:
11828:
11827:
11818:
11807:
11806:
11805:
11799:
11788:
11787:
11786:
11782:
11778:
11772:
11768:
11756:
11755:
11744:
11733:
11732:
11731:
11720:
11709:
11708:
11707:
11701:
11696:
11685:
11684:
11674:
11669:
11658:
11657:
11655:
11651:
11638:
11621:
11612:
11601:
11600:
11599:
11598:
11594:
11588:
11583:
11572:
11571:
11561:
11550:
11549:
11548:
11537:
11526:
11525:
11524:
11518:
11513:
11502:
11501:
11491:
11480:
11479:
11478:
11477:
11473:
11460:
11443:
11434:
11423:
11422:
11421:
11420:
11416:
11410:
11399:
11398:
11397:
11396:
11392:
11386:
11382:
11370:
11369:
11353:
11342:
11341:
11340:
11334:
11329:
11318:
11317:
11308:
11291:
11282:
11271:
11270:
11269:
11263:
11258:
11247:
11246:
11244:
11240:
11226:
11215:
11214:
11213:
11207:
11202:
11191:
11190:
11181:
11164:
11155:
11144:
11143:
11142:
11136:
11125:
11124:
11123:
11122:
11118:
11117:
11113:
11107:
11103:
11093:
11077:
11076:
11064:
11053:
11052:
11051:
11050:
11046:
11042:
11040:
11037:
11036:
11031:. Consider the
11015:
11004:
11003:
11002:
10996:
10991:
10980:
10979:
10970:
10953:
10939:
10938:
10936:
10933:
10932:
10929:number operator
10904:
10899:
10888:
10887:
10880:
10869:
10868:
10867:
10858:
10847:
10846:
10845:
10839:
10828:
10827:
10826:
10825:
10821:
10815:
10811:
10800:
10783:
10774:
10770:
10758:
10747:
10746:
10745:
10739:
10734:
10723:
10722:
10715:
10711:
10696:
10685:
10684:
10683:
10681:
10678:
10677:
10672:represents the
10656:
10652:
10639:
10637:
10634:
10633:
10611:
10606:
10595:
10594:
10587:
10576:
10575:
10574:
10565:
10554:
10553:
10552:
10546:
10535:
10534:
10533:
10532:
10528:
10522:
10518:
10507:
10490:
10481:
10477:
10456:
10447:
10436:
10435:
10434:
10428:
10423:
10412:
10411:
10409:
10405:
10399:
10395:
10380:
10369:
10368:
10367:
10365:
10362:
10361:
10358:
10332:
10327:
10316:
10315:
10308:
10297:
10296:
10295:
10286:
10275:
10274:
10273:
10267:
10256:
10255:
10254:
10253:
10249:
10243:
10239:
10223:
10214:
10203:
10202:
10201:
10197:
10193:
10175:
10164:
10163:
10162:
10152:
10143:
10132:
10131:
10130:
10123:
10119:
10107:
10096:
10095:
10094:
10092:
10089:
10088:
10069:
10065:
10063:
10060:
10059:
10027:
10023:
10011:
10007:
9997:
9993:
9987:
9982:
9971:
9970:
9963:
9952:
9951:
9950:
9933:
9929:
9917:
9913:
9906:
9902:
9896:
9885:
9884:
9883:
9877:
9866:
9865:
9864:
9863:
9859:
9853:
9849:
9822:
9811:
9810:
9809:
9807:
9804:
9803:
9784:
9780:
9778:
9775:
9774:
9743:
9739:
9735:
9727:
9725:
9723:
9720:
9719:
9702:
9698:
9686:
9682:
9660:
9658:
9650:
9647:
9646:
9629:
9625:
9613:
9609:
9597:
9593:
9591:
9588:
9587:
9564:
9560:
9548:
9544:
9542:
9539:
9538:
9521:
9517:
9505:
9501:
9493:
9487:
9483:
9471:
9467:
9462:
9460:
9457:
9456:
9426:
9422:
9410:
9406:
9399:
9395:
9389:
9378:
9377:
9376:
9370:
9359:
9358:
9357:
9351:
9347:
9330:
9326:
9314:
9310:
9300:
9296:
9290:
9285:
9274:
9273:
9266:
9255:
9254:
9253:
9247:
9242:
9218:
9214:
9205:
9201:
9191:
9187:
9181:
9170:
9169:
9168:
9162:
9151:
9150:
9149:
9143:
9138:
9114:
9110:
9101:
9097:
9090:
9086:
9080:
9075:
9064:
9063:
9056:
9045:
9044:
9043:
9037:
9033:
9032:
9028:
9012:
9003:
8992:
8991:
8990:
8983:
8979:
8970:
8959:
8958:
8957:
8947:
8938:
8927:
8926:
8925:
8921:
8917:
8896:
8885:
8884:
8883:
8881:
8878:
8877:
8860:
8849:
8848:
8847:
8838:
8827:
8826:
8825:
8816:
8805:
8804:
8803:
8801:
8798:
8797:
8786:
8753:
8742:
8741:
8740:
8734:
8723:
8722:
8721:
8715:
8710:
8697:
8692:
8681:
8680:
8673:
8662:
8661:
8660:
8654:
8650:
8646:
8642:
8628:
8623:
8612:
8611:
8604:
8593:
8592:
8591:
8585:
8580:
8567:
8556:
8555:
8554:
8548:
8537:
8536:
8535:
8529:
8525:
8524:
8520:
8519:
8515:
8500:
8489:
8488:
8487:
8485:
8482:
8481:
8456:
8442:
8437:
8434:
8433:
8413:
8396:
8387:
8376:
8375:
8374:
8372:
8369:
8368:
8352:
8335:
8326:
8315:
8314:
8313:
8311:
8308:
8307:
8275:
8274:
8263:
8262:
8261:
8255:
8244:
8243:
8242:
8236:
8224:
8223:
8210:
8198:
8197:
8186:
8185:
8178:
8167:
8166:
8165:
8152:
8151:
8147:
8138:
8126:
8125:
8114:
8113:
8106:
8095:
8094:
8093:
8087:
8075:
8074:
8054:
8053:
8042:
8041:
8040:
8034:
8023:
8022:
8021:
8008:
8007:
8003:
8002:
7998:
7982:
7981:
7962:
7951:
7950:
7949:
7947:
7944:
7943:
7919:
7918:
7913:
7912:
7910:
7907:
7906:
7879:
7878:
7873:
7872:
7858:
7847:
7845:
7844:
7839:
7819:
7818:
7814:
7804:
7801:
7782:
7781:
7777:
7772:
7769:
7768:
7752:
7744:
7742:
7739:
7738:
7716:
7705:
7703:
7702:
7697:
7685:
7674:
7673:
7672:
7658:
7647:
7645:
7644:
7639:
7627:
7616:
7615:
7614:
7600:
7598:
7597:
7595:
7592:
7591:
7568:
7560:
7553:
7549:
7543:
7531:
7530:
7519:
7518:
7507:
7499:
7495:
7491:
7478:
7477:
7466:
7465:
7464:
7463:
7459:
7446:
7445:
7440:
7439:
7425:
7424:
7420:
7410:
7407:
7394:
7393:
7375:
7361:
7359:
7358:
7356:
7353:
7352:
7336:
7334:
7331:
7330:
7311:
7297:
7295:
7294:
7280:
7278:
7277:
7262:
7251:
7250:
7249:
7247:
7244:
7243:
7225:
7222:
7221:
7195:
7186:
7175:
7174:
7173:
7167:
7162:
7151:
7150:
7148:
7144:
7138:
7134:
7122:
7111:
7110:
7109:
7107:
7104:
7103:
7075:
7074:
7063:
7062:
7061:
7059:
7056:
7055:
7038:
7026:
7025:
7014:
7013:
7010:
7007:
7006:
6990:
6987:
6986:
6970:
6968:
6965:
6964:
6938:
6922:
6921:
6910:
6909:
6908:
6902:
6890:
6889:
6878:
6877:
6875:
6871:
6864:
6863:
6859:
6843:
6842:
6829:
6818:
6817:
6816:
6814:
6811:
6810:
6788:
6784:
6782:
6779:
6778:
6762:
6745:
6736:
6732:
6721:
6704:
6698:
6694:
6685:
6674:
6673:
6672:
6670:
6667:
6666:
6644:
6641:
6640:
6624:
6607:
6601:
6597:
6589:
6572:
6566:
6562:
6553:
6542:
6541:
6540:
6538:
6535:
6534:
6511:
6500:
6499:
6498:
6489:
6478:
6477:
6476:
6467:
6456:
6455:
6454:
6440:
6439:
6437:
6434:
6433:
6421:
6394:
6393:
6391:
6388:
6387:
6356:
6355:
6353:
6350:
6349:
6322:
6321:
6301:
6300:
6292:
6283:
6279:
6268:
6267:
6262:
6259:
6258:
6228:
6227:
6207:
6206:
6191:
6180:
6179:
6178:
6155:
6154:
6152:
6149:
6148:
6117:
6116:
6114:
6111:
6110:
6090:
6089:
6074:
6072:
6051:
6050:
6043:
6032:
6031:
6022:
6018:
6004:
6003:
6001:
5988:
5987:
5971:
5957:
5955:
5944:
5942:
5930:
5929:
5922:
5911:
5910:
5900:
5896:
5882:
5881:
5871:
5867:
5853:
5852:
5850:
5839:
5837:
5833:
5831:
5828:
5827:
5805:
5804:
5802:
5799:
5798:
5775:
5767:
5764:
5763:
5747:
5744:
5743:
5724:
5718:
5714:
5700:
5699:
5693:
5682:
5681:
5680:
5674:
5670:
5656:
5655:
5653:
5650:
5649:
5627:
5626:
5624:
5621:
5620:
5606:
5605:
5602:
5601:
5594:
5593:
5575:
5561:
5559:
5549:
5547:
5539:
5518:
5516:
5506:
5502:
5485:
5471:
5470:
5464:
5453:
5452:
5451:
5450:
5446:
5437:
5433:
5426:
5422:
5420:
5414:
5403:
5402:
5401:
5388:
5374:
5372:
5362:
5360:
5343:
5329:
5328:
5322:
5311:
5310:
5309:
5308:
5304:
5295:
5291:
5284:
5280:
5268:
5267:
5256:
5255:
5245:
5241:
5227:
5226:
5216:
5212:
5198:
5197:
5195:
5185:
5183:
5166:
5152:
5151:
5145:
5134:
5133:
5132:
5131:
5127:
5118:
5114:
5107:
5103:
5092:
5078:
5074:
5060:
5059:
5049:
5045:
5031:
5030:
5028:
5018:
5016:
5008:
4991:
4987:
4973:
4972:
4970:
4960:
4956:
4939:
4925:
4924:
4918:
4907:
4906:
4905:
4904:
4900:
4891:
4887:
4880:
4876:
4869:
4868:
4859:
4858:
4848:
4839:
4828:
4827:
4826:
4819:
4815:
4808:
4788:
4787:
4783:
4779:
4777:
4774:
4773:
4770:
4733:
4715:
4711:
4704:
4700:
4683:
4668:
4664:
4662:
4658:
4639:
4621:
4617:
4610:
4606:
4589:
4574:
4570:
4568:
4564:
4557:
4553:
4547:
4543:
4537:
4512:
4508:
4503:
4493:
4484:
4473:
4472:
4471:
4464:
4460:
4439:
4435:
4430:
4428:
4425:
4424:
4402:
4399:
4398:
4370:
4368:
4365:
4364:
4326:
4311:
4307:
4305:
4301:
4275:
4260:
4256:
4254:
4250:
4243:
4239:
4233:
4229:
4223:
4198:
4192:
4188:
4187:
4181:
4156:
4152:
4147:
4145:
4142:
4141:
4118:
4112:
4108:
4107:
4101:
4076:
4072:
4067:
4065:
4062:
4061:
4058:
4017:
4013:
4011:
4007:
3995:
3991:
3982:
3978:
3976:
3973:
3972:
3955:
3951:
3949:
3946:
3945:
3922:
3918:
3913:
3898:
3894:
3892:
3888:
3867:
3863:
3858:
3843:
3839:
3837:
3833:
3807:
3805:
3802:
3801:
3779:
3775:
3770:
3755:
3751:
3749:
3745:
3724:
3720:
3715:
3700:
3696:
3694:
3690:
3664:
3662:
3659:
3658:
3630:
3628:
3625:
3624:
3586:
3582:
3573:
3569:
3567:
3549:
3545:
3543:
3540:
3539:
3510:
3506:
3493:
3475:
3468:
3464:
3458:
3454:
3433:
3429:
3427:
3424:
3423:
3400:
3396:
3394:
3391:
3390:
3374:
3371:
3370:
3350:
3349:
3338:
3334:
3332:
3323:
3319:
3302:
3289:
3279:
3276:
3275:
3262:
3252:
3250:
3239:
3235:
3233:
3224:
3220:
3210:
3209:
3191:
3187:
3185:
3182:
3181:
3155:
3151:
3146:
3140:
3129:
3128:
3127:
3122:
3113:
3109:
3091:
3083:
3078:
3075:
3072:
3071:
3054:
3043:
3042:
3041:
3039:
3036:
3035:
3009:
3005:
3000:
2985:
2981:
2976:
2965:
2962:
2961:
2944:
2933:
2932:
2931:
2929:
2926:
2925:
2909:
2906:
2905:
2888:
2877:
2876:
2875:
2873:
2870:
2869:
2835:
2820:
2816:
2811:
2809:
2806:
2805:
2777:
2762:
2758:
2753:
2751:
2748:
2747:
2728:
2720:
2717:
2716:
2688:
2662:
2660:
2657:
2656:
2639:
2628:
2627:
2626:
2624:
2621:
2620:
2596:
2592:
2583:
2579:
2571:
2568:
2567:
2553:
2552:
2541:
2530:
2529:
2528:
2522:
2511:
2510:
2509:
2500:
2489:
2488:
2487:
2476:
2475:
2474:
2470:
2457:
2455:
2441:
2430:
2429:
2428:
2426:
2413:
2407:
2396:
2395:
2394:
2391:
2390:
2374:
2363:
2362:
2361:
2359:
2345:
2344:
2338:
2327:
2326:
2325:
2324:
2320:
2314:
2310:
2300:
2294:
2283:
2282:
2281:
2277:
2275:
2272:
2271:
2251:
2240:
2239:
2238:
2229:
2218:
2217:
2216:
2207:
2196:
2195:
2194:
2192:
2189:
2188:
2185:
2156:
2145:
2144:
2143:
2137:
2126:
2125:
2124:
2115:
2104:
2103:
2102:
2091:
2090:
2089:
2085:
2072:
2070:
2056:
2045:
2044:
2043:
2041:
2035:
2031:
2014:
2013:
2007:
1996:
1995:
1994:
1988:
1984:
1972:
1961:
1960:
1959:
1957:
1954:
1953:
1929:
1925:
1916:
1912:
1907:
1904:
1903:
1883:
1879:
1870:
1866:
1861:
1858:
1857:
1837:
1833:
1824:
1820:
1815:
1812:
1811:
1776:
1772:
1763:
1759:
1746:
1742:
1736:
1725:
1724:
1723:
1717:
1706:
1705:
1704:
1687:
1683:
1674:
1670:
1660:
1656:
1650:
1639:
1638:
1637:
1626:
1625:
1608:
1604:
1595:
1591:
1584:
1580:
1574:
1563:
1562:
1561:
1555:
1544:
1543:
1542:
1525:
1521:
1512:
1508:
1498:
1494:
1488:
1477:
1476:
1475:
1464:
1463:
1462:
1458:
1445:
1443:
1425:
1414:
1413:
1412:
1410:
1407:
1406:
1389:
1378:
1377:
1376:
1367:
1356:
1355:
1354:
1345:
1334:
1333:
1332:
1330:
1327:
1326:
1315:
1293:
1289:
1287:
1284:
1283:
1267:
1250:
1242:
1225:
1216:
1205:
1204:
1203:
1201:
1198:
1197:
1177:
1160:
1151:
1140:
1139:
1138:
1136:
1133:
1132:
1116:
1099:
1090:
1079:
1078:
1077:
1075:
1072:
1071:
1054:
1043:
1042:
1041:
1032:
1021:
1020:
1019:
1005:
1004:
1002:
999:
998:
965:
961:
959:
956:
955:
929:
928:
926:
923:
922:
905:
894:
893:
892:
890:
887:
886:
864:
853:
852:
851:
837:
836:
835:
831:
825:
821:
807:
806:
804:
801:
800:
777:
774:
773:
757:
756:
742:
741:
730:
729:
716:
714:
707:
701:
690:
689:
688:
685:
684:
673:
662:
661:
660:
658:
652:
648:
638:
632:
621:
620:
619:
616:
615:
604:
603:
597:
586:
585:
584:
578:
574:
564:
558:
547:
546:
545:
541:
539:
536:
535:
518:
507:
506:
505:
496:
485:
484:
483:
474:
463:
462:
461:
447:
446:
444:
441:
440:
437:
421:Schrödinger cat
390:
387:
386:
356:
353:
352:
333:artificial atom
323:
296:
243:
240:
239:
221:
219:
216:
215:
176:
171:
74:
63:
57:
54:
47:
28:
24:
17:
12:
11:
5:
20589:
20579:
20578:
20576:Quantum optics
20564:
20563:
20557:
20549:Quantum Optics
20544:
20538:
20530:Quantum Optics
20525:
20519:
20511:Quantum Optics
20506:
20500:
20487:
20481:
20466:
20463:
20461:
20460:
20417:
20410:
20384:
20377:
20369:Quantum Optics
20350:
20343:
20314:
20294:
20241:
20206:
20187:(12): 122102.
20171:
20134:(1): 975–984.
20118:
20065:
20030:
19987:
19944:
19882:
19821:
19784:(12): 125312.
19764:
19737:(4): 304–308.
19721:
19670:
19633:(20): 205329.
19613:
19560:
19514:
19499:
19484:
19457:(4): 353–356.
19441:
19434:
19396:
19389:
19351:
19311:
19265:
19204:
19163:
19136:
19125:
19123:
19120:
19119:
19118:
19113:
19108:
19103:
19098:
19091:
19088:
19084:A.A. Karatsuba
19071:
19051:
19048:
19043:
19039:
19035:
19032:
19028:
19024:
19021:
19018:
19015:
18996:
18993:
18980:
18976:
18972:
18969:
18966:
18963:
18960:
18933:
18928:
18925:
18922:
18915:
18911:
18908:
18902:
18897:
18893:
18863:
18859:
18853:
18848:
18844:
18823:
18818:
18815:
18812:
18807:
18798:
18769:
18765:
18759:
18755:
18749:
18745:
18741:
18737:
18732:
18728:
18725:
18720:
18717:
18714:
18708:
18704:
18701:
18698:
18693:
18690:
18685:
18680:
18677:
18672:
18669:
18666:
18663:
18658:
18654:
18629:
18626:
18623:
18618:
18613:
18609:
18606:
18579:
18576:
18573:
18570:
18567:
18547:
18544:
18517:
18512:
18507:
18499:
18496:
18493:
18488:
18483:
18480:
18477:
18471:
18467:
18464:
18461:
18458:
18455:
18451:
18447:
18444:
18439:
18435:
18431:
18428:
18423:
18420:
18417:
18412:
18408:
18404:
18400:
18395:
18386:
18383:
18380:
18375:
18370:
18367:
18364:
18358:
18354:
18350:
18347:
18344:
18341:
18338:
18334:
18330:
18327:
18322:
18318:
18314:
18311:
18306:
18303:
18300:
18295:
18292:
18288:
18283:
18279:
18274:
18269:
18266:
18263:
18260:
18256:
18250:
18245:
18242:
18237:
18234:
18231:
18228:
18223:
18219:
18197:
18190:
18187:
18184:
18179:
18176:
18173:
18170:
18167:
18159:
18156:
18151:
18148:
18144:
18138:
18135:
18132:
18125:
18122:
18117:
18112:
18108:
18087:
18067:
18064:
18061:
18039:
18035:
18014:
18011:
18008:
18004:
18000:
17997:
17992:
17988:
17984:
17981:
17977:
17973:
17968:
17964:
17960:
17957:
17954:
17951:
17925:
17920:
17915:
17911:
17907:
17904:
17901:
17898:
17872:
17871:
17868:
17865:
17860:
17856:
17852:
17849:
17844:
17840:
17833:
17830:
17823:
17819:
17815:
17812:
17807:
17804:
17801:
17798:
17794:
17785:
17780:
17777:
17774:
17770:
17766:
17763:
17761:
17759:
17755:
17751:
17748:
17743:
17735:
17732:
17725:
17721:
17716:
17712:
17705:
17702:
17695:
17691:
17687:
17684:
17679:
17676:
17673:
17670:
17666:
17657:
17652:
17649:
17646:
17642:
17638:
17635:
17633:
17631:
17627:
17623:
17620:
17615:
17607:
17604:
17597:
17593:
17588:
17584:
17578:
17575:
17570:
17565:
17561:
17554:
17551:
17543:
17538:
17533:
17529:
17525:
17521:
17513:
17508:
17505:
17502:
17498:
17494:
17491:
17489:
17485:
17480:
17476:
17473:
17470:
17467:
17464:
17460:
17456:
17453:
17449:
17444:
17441:
17438:
17435:
17430:
17426:
17422:
17421:
17401:
17379:
17372:
17369:
17343:
17340:
17337:
17330:
17327:
17318:
17312:
17308:
17304:
17276:
17272:
17269:
17266:
17263:
17260:
17257:
17253:
17248:
17245:
17241:
17237:
17234:
17229:
17224:
17220:
17217:
17214:
17211:
17208:
17205:
17202:
17199:
17195:
17190:
17187:
17183:
17179:
17176:
17171:
17166:
17162:
17159:
17155:
17149:
17144:
17140:
17136:
17133:
17130:
17126:
17119:
17116:
17110:
17106:
17098:
17094:
17088:
17083:
17078:
17074:
17070:
17066:
17060:
17055:
17052:
17049:
17045:
17041:
17038:
17035:
17032:
17029:
17025:
17019:
17015:
17009:
17004:
16999:
16995:
16991:
16987:
16983:
16980:
16978:
16976:
16973:
16970:
16967:
16964:
16960:
16955:
16949:
16946:
16943:
16938:
16931:
16928:
16920:
16917:
16913:
16909:
16906:
16901:
16896:
16892:
16889:
16886:
16883:
16878:
16875:
16872:
16865:
16862:
16854:
16851:
16847:
16843:
16840:
16835:
16830:
16826:
16823:
16819:
16813:
16808:
16804:
16800:
16797:
16794:
16790:
16783:
16780:
16774:
16770:
16762:
16758:
16752:
16747:
16742:
16738:
16734:
16730:
16724:
16719:
16716:
16713:
16709:
16705:
16702:
16699:
16696:
16693:
16689:
16683:
16679:
16673:
16668:
16663:
16659:
16655:
16651:
16647:
16644:
16642:
16640:
16637:
16634:
16631:
16628:
16625:
16621:
16615:
16611:
16607:
16602:
16595:
16592:
16585:
16582:
16578:
16574:
16571:
16569:
16567:
16564:
16561:
16558:
16555:
16551:
16547:
16546:
16522:
16519:
16516:
16511:
16504:
16501:
16492:
16488:
16483:
16478:
16472:
16467:
16464:
16461:
16454:
16451:
16443:
16437:
16434:
16429:
16426:
16422:
16418:
16413:
16406:
16403:
16379:
16376:
16373:
16351:
16348:
16345:
16342:
16338:
16331:
16328:
16322:
16318:
16310:
16306:
16300:
16295:
16290:
16286:
16282:
16278:
16272:
16267:
16264:
16261:
16257:
16253:
16250:
16247:
16244:
16241:
16237:
16233:
16230:
16227:
16224:
16221:
16218:
16214:
16202:coherent state
16197:
16194:
16179:
16157:
16154:
16149:
16127:
16124:
16121:
16118:
16114:
16093:
16090:
16087:
16084:
16081:
16078:
16074:
16064:of the states
16049:
16046:
16041:
16036:
16033:
16030:
16007:
16003:
16000:
15997:
15994:
15991:
15988:
15984:
15980:
15977:
15974:
15971:
15968:
15964:
15959:
15952:
15948:
15943:
15940:
15937:
15934:
15931:
15927:
15906:
15903:
15900:
15896:
15890:
15886:
15882:
15877:
15874:
15870:
15845:
15841:
15837:
15834:
15829:
15825:
15819:
15816:
15811:
15807:
15803:
15798:
15794:
15790:
15787:
15784:
15773:dressed states
15754:
15751:
15748:
15745:
15742:
15739:
15735:
15730:
15725:
15720:
15716:
15710:
15706:
15703:
15700:
15697:
15694:
15691:
15688:
15684:
15679:
15674:
15669:
15665:
15659:
15655:
15652:
15649:
15646:
15644:
15642:
15639:
15636:
15633:
15629:
15625:
15624:
15621:
15618:
15615:
15612:
15608:
15603:
15598:
15593:
15589:
15583:
15579:
15576:
15573:
15570:
15567:
15564:
15561:
15558:
15555:
15551:
15546:
15541:
15536:
15532:
15526:
15522:
15519:
15516:
15513:
15511:
15509:
15506:
15503:
15500:
15496:
15492:
15491:
15467:
15464:
15461:
15458:
15454:
15433:
15430:
15427:
15424:
15421:
15418:
15414:
15401:
15398:
15383:
15379:
15374:
15366:
15361:
15357:
15336:
15305:
15300:
15296:
15293:
15287:
15283:
15278:
15274:
15270:
15265:
15260:
15255:
15252:
15249:
15246:
15243:
15239:
15235:
15232:
15229:
15226:
15222:
15218:
15215:
15213:
15211:
15208:
15205:
15200:
15196:
15192:
15191:
15187:
15182:
15178:
15175:
15169:
15165:
15160:
15156:
15152:
15147:
15142:
15137:
15134:
15131:
15128:
15125:
15121:
15117:
15114:
15111:
15108:
15104:
15100:
15097:
15095:
15093:
15090:
15087:
15082:
15078:
15074:
15073:
15053:
15031:
15028:
15025:
15022:
15018:
15013:
15008:
15004:
15001:
14995:
14991:
14988:
14985:
14982:
14979:
14976:
14973:
14970:
14966:
14961:
14956:
14952:
14949:
14943:
14939:
14936:
14933:
14930:
14927:
14924:
14921:
14918:
14914:
14893:
14890:
14887:
14871:
14868:
14853:
14849:
14844:
14836:
14833:
14828:
14824:
14820:
14817:
14795:
14791:
14766:
14763:
14760:
14757:
14754:
14751:
14747:
14742:
14737:
14732:
14728:
14722:
14718:
14715:
14712:
14709:
14706:
14703:
14700:
14696:
14691:
14686:
14681:
14677:
14671:
14667:
14664:
14661:
14658:
14656:
14654:
14651:
14648:
14645:
14641:
14637:
14636:
14633:
14630:
14627:
14624:
14620:
14615:
14610:
14605:
14601:
14595:
14591:
14588:
14585:
14582:
14579:
14576:
14573:
14570:
14567:
14563:
14558:
14553:
14548:
14544:
14538:
14534:
14531:
14528:
14525:
14523:
14521:
14518:
14515:
14512:
14508:
14504:
14503:
14496:
14492:
14488:
14485:
14480:
14476:
14470:
14465:
14462:
14457:
14453:
14449:
14446:
14441:
14438:
14433:
14428:
14424:
14420:
14417:
14413:
14409:
14406:
14404:
14400:
14397:
14394:
14390:
14386:
14385:
14349:
14346:
14343:
14312:
14311:
14308:
14304:
14300:
14297:
14294:
14291:
14288:
14284:
14279:
14275:
14272:
14269:
14265:
14261:
14257:
14253:
14250:
14247:
14243:
14238:
14234:
14231:
14228:
14225:
14222:
14218:
14214:
14211:
14209:
14205:
14202:
14199:
14194:
14187:
14184:
14177:
14176:
14172:
14168:
14165:
14162:
14158:
14153:
14149:
14146:
14143:
14139:
14135:
14131:
14127:
14124:
14121:
14118:
14115:
14111:
14106:
14102:
14099:
14096:
14093:
14090:
14086:
14082:
14079:
14077:
14073:
14070:
14067:
14062:
14055:
14052:
14045:
14044:
14040:
14036:
14033:
14030:
14026:
14021:
14017:
14014:
14011:
14007:
14003:
13999:
13995:
13992:
13989:
13986:
13983:
13979:
13974:
13970:
13967:
13964:
13961:
13958:
13954:
13950:
13947:
13945:
13941:
13938:
13935:
13928:
13925:
13918:
13917:
13891:
13888:
13884:
13880:
13875:
13871:
13867:
13864:
13853:Rabi frequency
13840:
13837:
13832:
13828:
13824:
13800:
13797:
13794:
13789:
13782:
13779:
13772:
13769:
13764:
13757:
13754:
13749:
13744:
13741:
13738:
13733:
13726:
13723:
13714:
13710:
13707:
13701:
13696:
13693:
13690:
13683:
13680:
13671:
13667:
13663:
13660:
13657:
13652:
13646:
13643:
13638:
13635:
13630:
13625:
13621:
13617:
13614:
13611:
13607:
13603:
13600:
13595:
13587:
13586:
13581:
13577:
13574:
13569:
13561:
13559:
13556:
13551:
13548:
13543:
13538:
13534:
13530:
13527:
13524:
13523:
13521:
13516:
13511:
13504:
13501:
13477:
13474:
13469:
13466:
13444:
13441:
13437:
13416:
13413:
13409:
13384:
13376:
13372:
13368:
13365:
13362:
13358:
13354:
13351:
13346:
13338:
13337:
13332:
13328:
13325:
13320:
13312:
13310:
13307:
13304:
13299:
13295:
13291:
13288:
13285:
13284:
13282:
13277:
13272:
13265:
13262:
13234:
13230:
13226:
13221:
13216:
13213:
13210:
13207:
13204:
13201:
13198:
13194:
13188:
13181:
13178:
13169:
13162:
13159:
13151:
13147:
13144:
13141:
13138:
13133:
13129:
13125:
13122:
13119:
13116:
13113:
13110:
13107:
13104:
13100:
13094:
13087:
13084:
13075:
13070:
13063:
13060:
13052:
13048:
13045:
13042:
13039:
13034:
13030:
13026:
13023:
13020:
13017:
13014:
13011:
13008:
13005:
13001:
12995:
12992:
12985:
12982:
12974:
12970:
12967:
12964:
12961:
12937:
12929:
12925:
12921:
12918:
12915:
12913:
12910:
12907:
12904:
12901:
12898:
12894:
12888:
12885:
12878:
12875:
12867:
12863:
12860:
12857:
12854:
12851:
12850:
12847:
12844:
12841:
12838:
12834:
12828:
12825:
12818:
12815:
12807:
12803:
12800:
12797:
12794:
12791:
12788:
12785:
12781:
12778:
12774:
12770:
12767:
12764:
12761:
12758:
12755:
12752:
12747:
12743:
12739:
12736:
12735:
12733:
12728:
12723:
12716:
12713:
12689:
12686:
12683:
12661:
12654:
12651:
12625:
12621:
12596:
12590:
12587:
12585:
12582:
12580:
12577:
12575:
12572:
12570:
12567:
12565:
12562:
12560:
12557:
12556:
12553:
12550:
12548:
12545:
12541:
12534:
12531:
12524:
12522:
12519:
12517:
12514:
12512:
12509:
12507:
12504:
12503:
12500:
12497:
12495:
12492:
12490:
12487:
12485:
12482:
12480:
12477:
12475:
12472:
12470:
12467:
12466:
12463:
12460:
12458:
12455:
12453:
12450:
12448:
12445:
12441:
12434:
12431:
12424:
12422:
12419:
12417:
12414:
12413:
12410:
12407:
12405:
12402:
12400:
12397:
12395:
12392:
12390:
12387:
12383:
12376:
12373:
12366:
12364:
12361:
12360:
12357:
12354:
12352:
12349:
12347:
12344:
12342:
12339:
12337:
12334:
12332:
12329:
12325:
12321:
12317:
12316:
12314:
12309:
12304:
12301:
12294:
12291:
12264:
12261:
12235:
12208:
12205:
12182:
12157:
12153:
12150:
12146:
12141:
12119:
12115:
12112:
12109:
12106:
12102:
12098:
12095:
12092:
12089:
12086:
12083:
12080:
12076:
12072:
12069:
12066:
12063:
12060:
12057:
12054:
12050:
12046:
12043:
12040:
12037:
12034:
12030:
12026:
12023:
12020:
12017:
12014:
12010:
12005:
11993:tensor product
11974:
11971:
11968:
11966:
11964:
11960:
11954:
11947:
11944:
11935:
11930:
11923:
11920:
11913:
11908:
11901:
11898:
11889:
11884:
11877:
11874:
11867:
11862:
11855:
11852:
11843:
11836:
11833:
11826:
11821:
11814:
11811:
11802:
11795:
11792:
11785:
11781:
11775:
11771:
11767:
11764:
11761:
11759:
11757:
11753:
11747:
11740:
11737:
11729:
11723:
11716:
11713:
11704:
11699:
11692:
11689:
11682:
11677:
11672:
11665:
11662:
11654:
11650:
11646:
11641:
11637:
11634:
11631:
11628:
11624:
11620:
11615:
11608:
11605:
11597:
11591:
11586:
11579:
11576:
11569:
11564:
11557:
11554:
11546:
11540:
11533:
11530:
11521:
11516:
11509:
11506:
11499:
11494:
11487:
11484:
11476:
11472:
11468:
11463:
11459:
11456:
11453:
11450:
11446:
11442:
11437:
11430:
11427:
11419:
11413:
11406:
11403:
11395:
11389:
11385:
11381:
11378:
11375:
11373:
11371:
11367:
11362:
11356:
11349:
11346:
11337:
11332:
11325:
11322:
11315:
11311:
11307:
11304:
11301:
11298:
11294:
11290:
11285:
11278:
11275:
11266:
11261:
11254:
11251:
11243:
11239:
11235:
11229:
11222:
11219:
11210:
11205:
11198:
11195:
11188:
11184:
11180:
11177:
11174:
11171:
11167:
11163:
11158:
11151:
11148:
11139:
11132:
11129:
11121:
11116:
11110:
11106:
11102:
11099:
11096:
11094:
11091:
11084:
11081:
11075:
11070:
11067:
11060:
11057:
11049:
11045:
11044:
11018:
11011:
11008:
10999:
10994:
10987:
10984:
10977:
10973:
10969:
10966:
10963:
10960:
10956:
10952:
10946:
10943:
10913:
10907:
10902:
10895:
10892:
10883:
10876:
10873:
10866:
10861:
10854:
10851:
10842:
10835:
10832:
10824:
10818:
10814:
10810:
10807:
10803:
10799:
10796:
10793:
10790:
10786:
10780:
10777:
10773:
10769:
10766:
10761:
10754:
10751:
10742:
10737:
10730:
10727:
10718:
10714:
10710:
10707:
10702:
10699:
10692:
10689:
10659:
10655:
10651:
10646:
10643:
10620:
10614:
10609:
10602:
10599:
10590:
10583:
10580:
10573:
10568:
10561:
10558:
10549:
10542:
10539:
10531:
10525:
10521:
10517:
10514:
10510:
10506:
10503:
10500:
10497:
10493:
10487:
10484:
10480:
10476:
10473:
10469:
10463:
10460:
10455:
10450:
10443:
10440:
10431:
10426:
10419:
10416:
10408:
10402:
10398:
10394:
10391:
10386:
10383:
10376:
10373:
10357:
10354:
10341:
10335:
10330:
10323:
10320:
10311:
10304:
10301:
10294:
10289:
10282:
10279:
10270:
10263:
10260:
10252:
10246:
10242:
10238:
10235:
10230:
10226:
10222:
10217:
10210:
10207:
10200:
10196:
10192:
10189:
10186:
10181:
10178:
10171:
10168:
10159:
10155:
10151:
10146:
10139:
10136:
10129:
10126:
10122:
10118:
10113:
10110:
10103:
10100:
10072:
10068:
10044:
10038:
10035:
10030:
10026:
10022:
10017:
10014:
10010:
10006:
10003:
10000:
9996:
9990:
9985:
9978:
9975:
9966:
9959:
9956:
9949:
9944:
9941:
9936:
9932:
9928:
9923:
9920:
9916:
9912:
9909:
9905:
9899:
9892:
9889:
9880:
9873:
9870:
9862:
9856:
9852:
9848:
9845:
9842:
9839:
9836:
9833:
9828:
9825:
9818:
9815:
9787:
9783:
9757:
9754:
9746:
9742:
9738:
9733:
9730:
9705:
9701:
9697:
9692:
9689:
9685:
9681:
9678:
9675:
9670:
9666:
9663:
9657:
9654:
9632:
9628:
9624:
9621:
9616:
9612:
9608:
9603:
9600:
9596:
9575:
9572:
9567:
9563:
9559:
9554:
9551:
9547:
9524:
9520:
9516:
9511:
9508:
9504:
9500:
9496:
9490:
9486:
9482:
9477:
9474:
9470:
9465:
9443:
9437:
9434:
9429:
9425:
9421:
9416:
9413:
9409:
9405:
9402:
9398:
9392:
9385:
9382:
9373:
9366:
9363:
9354:
9350:
9346:
9341:
9338:
9333:
9329:
9325:
9320:
9317:
9313:
9309:
9306:
9303:
9299:
9293:
9288:
9281:
9278:
9269:
9262:
9259:
9250:
9245:
9241:
9237:
9232:
9229:
9224:
9221:
9217:
9213:
9208:
9204:
9200:
9197:
9194:
9190:
9184:
9177:
9174:
9165:
9158:
9155:
9146:
9141:
9137:
9133:
9128:
9125:
9120:
9117:
9113:
9109:
9104:
9100:
9096:
9093:
9089:
9083:
9078:
9071:
9068:
9059:
9052:
9049:
9040:
9036:
9031:
9027:
9024:
9019:
9015:
9011:
9006:
8999:
8996:
8989:
8986:
8982:
8976:
8973:
8966:
8963:
8954:
8950:
8946:
8941:
8934:
8931:
8924:
8920:
8916:
8913:
8910:
8907:
8902:
8899:
8892:
8889:
8863:
8856:
8853:
8846:
8841:
8834:
8831:
8824:
8819:
8812:
8809:
8785:
8782:
8781:
8780:
8767:
8762:
8756:
8749:
8746:
8737:
8730:
8727:
8718:
8713:
8709:
8705:
8700:
8695:
8688:
8685:
8676:
8669:
8666:
8657:
8653:
8649:
8645:
8641:
8637:
8631:
8626:
8619:
8616:
8607:
8600:
8597:
8588:
8583:
8579:
8575:
8570:
8563:
8560:
8551:
8544:
8541:
8532:
8528:
8523:
8518:
8514:
8511:
8506:
8503:
8496:
8493:
8469:
8466:
8463:
8459:
8455:
8452:
8449:
8445:
8441:
8432:acting in the
8416:
8412:
8409:
8406:
8403:
8399:
8395:
8390:
8383:
8380:
8355:
8351:
8348:
8345:
8342:
8338:
8334:
8329:
8322:
8319:
8295:
8291:
8285:
8282:
8278:
8270:
8267:
8258:
8251:
8248:
8239:
8234:
8231:
8227:
8222:
8218:
8213:
8208:
8205:
8201:
8193:
8190:
8181:
8174:
8171:
8162:
8159:
8155:
8150:
8146:
8141:
8136:
8133:
8129:
8121:
8118:
8109:
8102:
8099:
8090:
8085:
8082:
8078:
8073:
8069:
8064:
8061:
8057:
8049:
8046:
8037:
8030:
8027:
8018:
8015:
8011:
8006:
8001:
7997:
7992:
7989:
7985:
7980:
7976:
7973:
7968:
7965:
7958:
7955:
7929:
7926:
7922:
7916:
7894:
7889:
7886:
7882:
7876:
7871:
7868:
7865:
7861:
7854:
7850:
7842:
7838:
7835:
7829:
7822:
7817:
7813:
7810:
7807:
7800:
7797:
7792:
7789:
7785:
7780:
7776:
7755:
7751:
7747:
7726:
7723:
7719:
7712:
7708:
7700:
7696:
7693:
7688:
7681:
7678:
7671:
7668:
7665:
7661:
7654:
7650:
7642:
7638:
7635:
7630:
7623:
7620:
7613:
7607:
7603:
7578:
7571:
7567:
7563:
7559:
7556:
7552:
7546:
7541:
7538:
7534:
7526:
7523:
7516:
7510:
7506:
7502:
7498:
7494:
7488:
7485:
7481:
7473:
7470:
7462:
7456:
7453:
7449:
7443:
7435:
7428:
7423:
7419:
7416:
7413:
7404:
7401:
7397:
7392:
7388:
7385:
7382:
7378:
7374:
7368:
7364:
7339:
7318:
7314:
7310:
7304:
7300:
7293:
7287:
7283:
7276:
7273:
7268:
7265:
7258:
7255:
7241:
7229:
7208:
7202:
7199:
7194:
7189:
7182:
7179:
7170:
7165:
7158:
7155:
7147:
7141:
7137:
7133:
7130:
7125:
7118:
7115:
7085:
7082:
7078:
7070:
7067:
7041:
7036:
7033:
7029:
7021:
7018:
6994:
6973:
6951:
6945:
6942:
6937:
6932:
6929:
6925:
6917:
6914:
6905:
6900:
6897:
6893:
6885:
6882:
6874:
6867:
6862:
6858:
6853:
6850:
6846:
6841:
6837:
6832:
6825:
6822:
6808:
6794:
6791:
6787:
6765:
6761:
6758:
6755:
6752:
6748:
6742:
6739:
6735:
6731:
6728:
6724:
6720:
6717:
6714:
6711:
6707:
6701:
6697:
6693:
6688:
6681:
6678:
6654:
6651:
6648:
6627:
6623:
6620:
6617:
6614:
6610:
6604:
6600:
6596:
6592:
6588:
6585:
6582:
6579:
6575:
6569:
6565:
6561:
6556:
6549:
6546:
6517:
6514:
6507:
6504:
6497:
6492:
6485:
6482:
6475:
6470:
6463:
6460:
6453:
6447:
6444:
6420:
6417:
6401:
6398:
6375:
6372:
6369:
6363:
6360:
6335:
6329:
6326:
6320:
6317:
6314:
6308:
6305:
6299:
6291:
6286:
6282:
6275:
6272:
6266:
6247:
6244:
6241:
6235:
6232:
6226:
6223:
6220:
6214:
6211:
6205:
6202:
6199:
6194:
6187:
6184:
6177:
6174:
6171:
6168:
6162:
6159:
6136:
6133:
6130:
6124:
6121:
6108:density matrix
6088:
6081:
6078:
6071:
6068:
6065:
6058:
6055:
6049:
6046:
6044:
6039:
6036:
6025:
6021:
6017:
6011:
6008:
5999:
5996:
5993:
5990:
5989:
5986:
5978:
5975:
5964:
5961:
5953:
5950:
5947:
5937:
5934:
5928:
5925:
5923:
5918:
5915:
5903:
5899:
5895:
5889:
5886:
5874:
5870:
5866:
5860:
5857:
5848:
5845:
5842:
5836:
5835:
5812:
5809:
5782:
5779:
5774:
5771:
5751:
5731:
5727:
5721:
5717:
5713:
5707:
5704:
5696:
5689:
5686:
5677:
5673:
5669:
5663:
5660:
5648:is defined as
5634:
5631:
5598:
5591:
5582:
5579:
5568:
5565:
5558:
5555:
5552:
5546:
5542:
5538:
5535:
5532:
5525:
5522:
5515:
5512:
5509:
5505:
5498:
5492:
5489:
5484:
5478:
5475:
5467:
5460:
5457:
5449:
5445:
5440:
5436:
5432:
5429:
5425:
5421:
5417:
5410:
5407:
5395:
5392:
5381:
5378:
5371:
5368:
5365:
5356:
5350:
5347:
5342:
5336:
5333:
5325:
5318:
5315:
5307:
5303:
5298:
5294:
5290:
5287:
5283:
5279:
5276:
5273:
5270:
5269:
5263:
5260:
5248:
5244:
5240:
5234:
5231:
5219:
5215:
5211:
5205:
5202:
5194:
5191:
5188:
5179:
5173:
5170:
5165:
5159:
5156:
5148:
5141:
5138:
5130:
5126:
5121:
5117:
5113:
5110:
5106:
5102:
5099:
5096:
5093:
5090:
5081:
5077:
5073:
5067:
5064:
5052:
5048:
5044:
5038:
5035:
5027:
5024:
5021:
5015:
5011:
5007:
5004:
5001:
4994:
4990:
4986:
4980:
4977:
4969:
4966:
4963:
4959:
4952:
4946:
4943:
4938:
4932:
4929:
4921:
4914:
4911:
4903:
4899:
4894:
4890:
4886:
4883:
4879:
4875:
4874:
4872:
4867:
4864:
4862:
4860:
4855:
4851:
4847:
4835:
4832:
4825:
4822:
4818:
4814:
4811:
4809:
4807:
4804:
4801:
4795:
4792:
4786:
4785:
4782:
4781:
4769:
4766:
4750:
4746:
4740:
4736:
4732:
4729:
4726:
4723:
4718:
4714:
4710:
4707:
4703:
4699:
4696:
4693:
4690:
4686:
4681:
4676:
4671:
4667:
4661:
4657:
4654:
4651:
4646:
4642:
4638:
4635:
4632:
4629:
4624:
4620:
4616:
4613:
4609:
4605:
4602:
4599:
4596:
4592:
4587:
4582:
4577:
4573:
4567:
4563:
4560:
4556:
4550:
4546:
4540:
4536:
4532:
4529:
4526:
4523:
4520:
4511:
4506:
4500:
4496:
4492:
4480:
4477:
4470:
4467:
4463:
4459:
4456:
4453:
4450:
4447:
4438:
4433:
4412:
4409:
4406:
4386:
4383:
4380:
4377:
4373:
4350:
4346:
4342:
4339:
4336:
4333:
4329:
4324:
4319:
4314:
4310:
4304:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4278:
4273:
4268:
4263:
4259:
4253:
4249:
4246:
4242:
4236:
4232:
4226:
4222:
4218:
4214:
4211:
4208:
4205:
4201:
4195:
4191:
4184:
4180:
4176:
4173:
4170:
4167:
4164:
4155:
4150:
4128:
4125:
4121:
4115:
4111:
4104:
4100:
4096:
4093:
4090:
4087:
4084:
4075:
4070:
4057:
4054:
4042:
4038:
4033:
4027:
4024:
4021:
4016:
4010:
4006:
4001:
3998:
3994:
3990:
3985:
3981:
3958:
3954:
3933:
3928:
3925:
3921:
3916:
3911:
3906:
3901:
3897:
3891:
3887:
3884:
3881:
3878:
3873:
3870:
3866:
3861:
3856:
3851:
3846:
3842:
3836:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3810:
3790:
3785:
3782:
3778:
3773:
3768:
3763:
3758:
3754:
3748:
3744:
3741:
3738:
3735:
3730:
3727:
3723:
3718:
3713:
3708:
3703:
3699:
3693:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3667:
3646:
3643:
3640:
3637:
3633:
3621:Rabi frequency
3606:
3603:
3600:
3597:
3594:
3589:
3585:
3581:
3576:
3572:
3566:
3563:
3560:
3557:
3552:
3548:
3527:
3524:
3521:
3518:
3513:
3509:
3505:
3500:
3497:
3492:
3488:
3482:
3479:
3474:
3471:
3467:
3461:
3457:
3453:
3450:
3447:
3444:
3441:
3436:
3432:
3409:
3406:
3403:
3399:
3378:
3354:
3346:
3341:
3337:
3331:
3326:
3322:
3318:
3315:
3312:
3309:
3306:
3303:
3299:
3296:
3293:
3286:
3283:
3278:
3277:
3272:
3269:
3266:
3259:
3256:
3251:
3247:
3242:
3238:
3232:
3227:
3223:
3219:
3216:
3215:
3213:
3208:
3205:
3200:
3197:
3194:
3190:
3169:
3166:
3161:
3158:
3154:
3149:
3136:
3133:
3125:
3119:
3116:
3112:
3108:
3105:
3100:
3097:
3094:
3089:
3086:
3081:
3050:
3047:
3023:
3020:
3015:
3012:
3008:
3003:
2999:
2996:
2991:
2988:
2984:
2979:
2975:
2972:
2969:
2940:
2937:
2913:
2891:
2884:
2881:
2857:
2854:
2851:
2848:
2845:
2842:
2838:
2834:
2831:
2826:
2823:
2819:
2814:
2793:
2790:
2787:
2784:
2780:
2776:
2773:
2768:
2765:
2761:
2756:
2746:As the states
2731:
2727:
2724:
2704:
2701:
2698:
2695:
2691:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2665:
2642:
2635:
2632:
2599:
2595:
2591:
2586:
2582:
2578:
2575:
2550:
2544:
2537:
2534:
2525:
2518:
2515:
2508:
2503:
2496:
2493:
2483:
2480:
2473:
2467:
2463:
2460:
2454:
2449:
2444:
2437:
2434:
2425:
2422:
2419:
2416:
2414:
2403:
2400:
2393:
2392:
2388:
2382:
2377:
2370:
2367:
2358:
2352:
2349:
2341:
2334:
2331:
2323:
2317:
2313:
2309:
2306:
2303:
2301:
2290:
2287:
2280:
2279:
2259:
2247:
2244:
2237:
2225:
2222:
2215:
2203:
2200:
2184:
2181:
2169:
2165:
2159:
2152:
2149:
2140:
2133:
2130:
2123:
2118:
2111:
2108:
2098:
2095:
2088:
2082:
2078:
2075:
2069:
2064:
2059:
2052:
2049:
2038:
2034:
2030:
2027:
2021:
2018:
2010:
2003:
2000:
1991:
1987:
1983:
1980:
1968:
1965:
1937:
1932:
1928:
1924:
1919:
1915:
1911:
1891:
1886:
1882:
1878:
1873:
1869:
1865:
1845:
1840:
1836:
1832:
1827:
1823:
1819:
1797:
1793:
1787:
1784:
1779:
1775:
1771:
1766:
1762:
1758:
1755:
1752:
1749:
1745:
1739:
1732:
1729:
1720:
1713:
1710:
1703:
1698:
1695:
1690:
1686:
1682:
1677:
1673:
1669:
1666:
1663:
1659:
1653:
1646:
1643:
1633:
1630:
1624:
1619:
1616:
1611:
1607:
1603:
1598:
1594:
1590:
1587:
1583:
1577:
1570:
1567:
1558:
1551:
1548:
1541:
1536:
1533:
1528:
1524:
1520:
1515:
1511:
1507:
1504:
1501:
1497:
1491:
1484:
1481:
1471:
1468:
1461:
1455:
1451:
1448:
1442:
1439:
1436:
1433:
1421:
1418:
1385:
1382:
1375:
1363:
1360:
1353:
1348:
1341:
1338:
1314:
1311:
1296:
1292:
1270:
1266:
1263:
1260:
1257:
1253:
1249:
1245:
1241:
1238:
1235:
1232:
1228:
1224:
1219:
1212:
1209:
1180:
1176:
1173:
1170:
1167:
1163:
1159:
1154:
1147:
1144:
1119:
1115:
1112:
1109:
1106:
1102:
1098:
1093:
1086:
1083:
1057:
1050:
1047:
1040:
1035:
1028:
1025:
1018:
1012:
1009:
968:
964:
936:
933:
908:
901:
898:
873:
867:
860:
857:
850:
844:
841:
834:
824:
820:
814:
811:
781:
755:
749:
746:
737:
734:
726:
722:
719:
713:
710:
708:
697:
694:
687:
686:
681:
676:
669:
666:
655:
651:
647:
644:
641:
639:
628:
625:
618:
617:
611:
608:
600:
593:
590:
581:
577:
573:
570:
567:
565:
554:
551:
544:
543:
514:
511:
504:
492:
489:
482:
470:
467:
460:
454:
451:
436:
433:
400:
397:
394:
366:
363:
360:
322:
319:
295:
292:
275:quality factor
271:
270:
263:coherent state
259:
247:
225:
212:
175:
172:
170:
167:
147:quantum optics
143:atomic physics
111:two-level atom
107:quantum optics
76:
75:
31:
29:
22:
15:
9:
6:
4:
3:
2:
20588:
20577:
20574:
20573:
20571:
20560:
20558:3-540-58831-0
20554:
20550:
20545:
20541:
20535:
20531:
20526:
20522:
20520:0-521-43595-1
20516:
20512:
20507:
20503:
20501:0-521-52735-X
20497:
20493:
20488:
20484:
20478:
20474:
20469:
20468:
20456:
20452:
20448:
20444:
20440:
20436:
20432:
20428:
20421:
20413:
20407:
20403:
20399:
20393:
20391:
20389:
20380:
20374:
20370:
20363:
20361:
20359:
20357:
20355:
20346:
20340:
20336:
20329:
20327:
20325:
20323:
20321:
20319:
20311:
20307:
20303:
20298:
20290:
20286:
20282:
20278:
20274:
20270:
20265:
20260:
20256:
20252:
20245:
20237:
20233:
20229:
20225:
20221:
20217:
20210:
20202:
20198:
20194:
20190:
20186:
20182:
20175:
20167:
20163:
20159:
20155:
20151:
20147:
20142:
20137:
20133:
20129:
20122:
20114:
20110:
20106:
20102:
20098:
20094:
20089:
20084:
20080:
20076:
20069:
20061:
20057:
20053:
20049:
20046:(2): 023811.
20045:
20041:
20034:
20026:
20022:
20018:
20014:
20010:
20006:
20002:
19998:
19991:
19983:
19979:
19975:
19971:
19967:
19963:
19959:
19955:
19948:
19940:
19936:
19932:
19928:
19924:
19920:
19915:
19910:
19907:(4): 041801.
19906:
19902:
19898:
19891:
19889:
19887:
19878:
19874:
19870:
19866:
19862:
19858:
19854:
19850:
19845:
19840:
19836:
19832:
19825:
19817:
19813:
19809:
19805:
19801:
19797:
19792:
19787:
19783:
19779:
19775:
19768:
19760:
19756:
19752:
19748:
19744:
19740:
19736:
19732:
19725:
19717:
19713:
19709:
19705:
19701:
19697:
19693:
19689:
19685:
19681:
19674:
19666:
19662:
19658:
19654:
19650:
19646:
19641:
19636:
19632:
19628:
19624:
19617:
19609:
19605:
19601:
19597:
19593:
19589:
19585:
19581:
19577:
19573:
19572:
19564:
19556:
19552:
19548:
19544:
19540:
19536:
19532:
19528:
19521:
19519:
19510:
19503:
19495:
19488:
19480:
19476:
19472:
19468:
19464:
19460:
19456:
19452:
19445:
19437:
19435:9780120038206
19431:
19427:
19423:
19419:
19415:
19411:
19407:
19400:
19392:
19390:9780120038206
19386:
19382:
19378:
19374:
19370:
19366:
19362:
19355:
19347:
19343:
19339:
19335:
19331:
19327:
19320:
19318:
19316:
19307:
19303:
19299:
19295:
19291:
19287:
19283:
19279:
19272:
19270:
19261:
19255:
19247:
19243:
19239:
19235:
19231:
19227:
19223:
19219:
19211:
19209:
19199:
19194:
19190:
19186:
19182:
19178:
19174:
19167:
19159:
19155:
19152:(1): 89–109.
19151:
19147:
19140:
19130:
19126:
19117:
19114:
19112:
19109:
19107:
19104:
19102:
19099:
19097:
19094:
19093:
19087:
19085:
19069:
19049:
19046:
19041:
19033:
19030:
19026:
19022:
19016:
19013:
19004:
19002:
18992:
18978:
18974:
18970:
18964:
18961:
18958:
18948:
18944:
18931:
18923:
18909:
18906:
18900:
18895:
18891:
18882:
18876:
18857:
18851:
18846:
18842:
18813:
18805:
18787:
18782:
18767:
18763:
18757:
18753:
18747:
18739:
18735:
18730:
18726:
18715:
18706:
18702:
18699:
18696:
18691:
18688:
18683:
18678:
18675:
18670:
18664:
18656:
18652:
18643:
18624:
18616:
18611:
18607:
18577:
18574:
18568:
18545:
18542:
18533:
18529:
18515:
18510:
18505:
18494:
18486:
18481:
18475:
18469:
18465:
18462:
18456:
18449:
18445:
18442:
18437:
18433:
18429:
18418:
18410:
18406:
18402:
18398:
18393:
18381:
18373:
18368:
18362:
18356:
18352:
18348:
18345:
18339:
18332:
18328:
18325:
18320:
18316:
18312:
18301:
18293:
18290:
18286:
18281:
18277:
18272:
18264:
18258:
18254:
18248:
18243:
18240:
18235:
18229:
18221:
18217:
18195:
18185:
18174:
18168:
18165:
18157:
18154:
18149:
18146:
18142:
18133:
18123:
18115:
18110:
18085:
18062:
18037:
18009:
18002:
17998:
17995:
17990:
17982:
17975:
17966:
17958:
17941:
17923:
17913:
17905:
17899:
17863:
17858:
17847:
17842:
17838:
17831:
17828:
17821:
17813:
17802:
17796:
17792:
17778:
17775:
17772:
17768:
17764:
17762:
17753:
17749:
17741:
17733:
17730:
17723:
17719:
17714:
17710:
17703:
17700:
17693:
17685:
17674:
17668:
17664:
17650:
17647:
17644:
17640:
17636:
17634:
17625:
17621:
17613:
17605:
17602:
17595:
17591:
17586:
17582:
17576:
17573:
17563:
17552:
17549:
17541:
17531:
17523:
17519:
17506:
17503:
17500:
17496:
17492:
17490:
17483:
17478:
17468:
17462:
17454:
17447:
17442:
17436:
17428:
17424:
17399:
17377:
17367:
17338:
17325:
17316:
17310:
17306:
17302:
17274:
17267:
17264:
17261:
17258:
17255:
17243:
17239:
17235:
17227:
17218:
17215:
17212:
17209:
17203:
17200:
17197:
17185:
17181:
17177:
17169:
17160:
17157:
17153:
17147:
17142:
17138:
17134:
17131:
17128:
17124:
17117:
17114:
17108:
17104:
17096:
17092:
17086:
17076:
17068:
17064:
17053:
17050:
17047:
17043:
17039:
17033:
17030:
17027:
17017:
17013:
17007:
16997:
16989:
16985:
16981:
16979:
16968:
16965:
16962:
16953:
16944:
16936:
16926:
16915:
16911:
16907:
16899:
16890:
16887:
16884:
16881:
16873:
16860:
16849:
16845:
16841:
16833:
16824:
16821:
16817:
16811:
16806:
16802:
16798:
16795:
16792:
16788:
16781:
16778:
16772:
16768:
16760:
16756:
16750:
16740:
16732:
16728:
16717:
16714:
16711:
16707:
16703:
16697:
16694:
16691:
16681:
16677:
16671:
16661:
16653:
16649:
16645:
16643:
16629:
16623:
16609:
16605:
16600:
16590:
16583:
16580:
16576:
16572:
16570:
16559:
16553:
16535:
16517:
16509:
16499:
16490:
16481:
16476:
16470:
16462:
16449:
16441:
16435:
16432:
16427:
16424:
16420:
16416:
16411:
16401:
16377:
16374:
16362:
16346:
16343:
16340:
16329:
16326:
16320:
16316:
16308:
16304:
16298:
16288:
16280:
16276:
16265:
16262:
16259:
16255:
16251:
16245:
16242:
16239:
16231:
16222:
16216:
16203:
16193:
16177:
16147:
16122:
16119:
16116:
16088:
16085:
16082:
16079:
16076:
16063:
16039:
16034:
16031:
16028:
16019:
16005:
15998:
15995:
15992:
15989:
15986:
15978:
15972:
15969:
15966:
15957:
15950:
15946:
15941:
15935:
15932:
15929:
15904:
15901:
15888:
15884:
15880:
15875:
15872:
15868:
15843:
15835:
15832:
15827:
15814:
15809:
15805:
15801:
15796:
15792:
15788:
15785:
15782:
15774:
15769:
15749:
15746:
15743:
15740:
15737:
15728:
15723:
15718:
15714:
15708:
15704:
15701:
15698:
15692:
15689:
15686:
15677:
15672:
15667:
15663:
15657:
15653:
15650:
15647:
15645:
15637:
15634:
15631:
15616:
15613:
15610:
15601:
15596:
15591:
15587:
15581:
15577:
15574:
15571:
15565:
15562:
15559:
15556:
15553:
15544:
15539:
15534:
15530:
15524:
15520:
15517:
15514:
15512:
15504:
15501:
15498:
15481:
15462:
15459:
15456:
15428:
15425:
15422:
15419:
15416:
15397:
15381:
15372:
15364:
15359:
15334:
15326:
15321:
15303:
15298:
15294:
15285:
15281:
15276:
15272:
15268:
15263:
15247:
15241:
15233:
15230:
15227:
15216:
15214:
15206:
15198:
15194:
15185:
15180:
15176:
15167:
15163:
15158:
15154:
15150:
15145:
15129:
15123:
15115:
15112:
15109:
15098:
15096:
15088:
15080:
15076:
15051:
15042:
15026:
15023:
15020:
15011:
15006:
15002:
14993:
14989:
14986:
14983:
14980:
14974:
14971:
14968:
14959:
14954:
14950:
14941:
14937:
14934:
14931:
14922:
14916:
14891:
14888:
14877:
14867:
14842:
14834:
14831:
14826:
14822:
14818:
14815:
14793:
14789:
14761:
14758:
14755:
14752:
14749:
14740:
14735:
14730:
14726:
14720:
14716:
14713:
14710:
14704:
14701:
14698:
14689:
14684:
14679:
14675:
14669:
14665:
14662:
14659:
14657:
14649:
14646:
14643:
14628:
14625:
14622:
14613:
14608:
14603:
14599:
14593:
14589:
14586:
14583:
14577:
14574:
14571:
14568:
14565:
14556:
14551:
14546:
14542:
14536:
14532:
14529:
14526:
14524:
14516:
14513:
14510:
14494:
14486:
14483:
14478:
14463:
14460:
14455:
14451:
14439:
14436:
14431:
14426:
14422:
14415:
14411:
14407:
14405:
14398:
14395:
14392:
14388:
14375:
14371:
14367:
14363:
14347:
14344:
14341:
14333:
14332:Hilbert space
14328:
14306:
14302:
14298:
14295:
14292:
14289:
14286:
14282:
14277:
14273:
14270:
14267:
14263:
14259:
14255:
14251:
14248:
14245:
14241:
14236:
14232:
14229:
14226:
14223:
14220:
14216:
14212:
14210:
14200:
14192:
14182:
14170:
14166:
14163:
14160:
14156:
14151:
14147:
14144:
14141:
14137:
14133:
14129:
14125:
14122:
14119:
14116:
14113:
14109:
14104:
14100:
14097:
14094:
14091:
14088:
14084:
14080:
14078:
14068:
14060:
14050:
14038:
14034:
14031:
14028:
14024:
14019:
14015:
14012:
14009:
14005:
14001:
13997:
13993:
13990:
13987:
13984:
13981:
13977:
13972:
13968:
13965:
13962:
13959:
13956:
13952:
13948:
13946:
13936:
13923:
13907:
13889:
13886:
13882:
13878:
13873:
13869:
13865:
13854:
13835:
13830:
13826:
13822:
13813:
13795:
13787:
13777:
13762:
13755:
13752:
13747:
13739:
13731:
13721:
13712:
13705:
13699:
13691:
13678:
13669:
13665:
13658:
13655:
13650:
13636:
13633:
13628:
13623:
13619:
13612:
13605:
13593:
13579:
13567:
13549:
13546:
13541:
13536:
13532:
13525:
13519:
13514:
13509:
13499:
13488:, we can get
13467:
13464:
13439:
13411:
13397:
13382:
13374:
13370:
13363:
13356:
13344:
13330:
13318:
13302:
13297:
13293:
13286:
13280:
13275:
13270:
13260:
13247:
13232:
13228:
13219:
13214:
13208:
13205:
13202:
13199:
13196:
13186:
13176:
13167:
13157:
13145:
13142:
13139:
13131:
13127:
13120:
13114:
13111:
13108:
13105:
13102:
13092:
13082:
13073:
13068:
13058:
13046:
13043:
13040:
13032:
13028:
13021:
13015:
13012:
13009:
13006:
13003:
12993:
12990:
12980:
12968:
12965:
12962:
12950:
12935:
12927:
12923:
12916:
12908:
12905:
12902:
12899:
12896:
12886:
12883:
12873:
12861:
12858:
12855:
12842:
12839:
12836:
12826:
12823:
12813:
12801:
12798:
12795:
12792:
12789:
12779:
12776:
12772:
12765:
12759:
12756:
12753:
12745:
12741:
12737:
12731:
12726:
12721:
12711:
12687:
12684:
12681:
12659:
12649:
12623:
12619:
12609:
12594:
12588:
12583:
12578:
12573:
12568:
12563:
12558:
12551:
12546:
12539:
12529:
12520:
12515:
12510:
12505:
12498:
12493:
12488:
12483:
12478:
12473:
12468:
12461:
12456:
12451:
12446:
12439:
12429:
12420:
12415:
12408:
12403:
12398:
12393:
12388:
12381:
12371:
12362:
12355:
12350:
12345:
12340:
12335:
12330:
12323:
12319:
12312:
12307:
12302:
12299:
12289:
12259:
12247:
12233:
12225:
12203:
12180:
12172:
12155:
12148:
12139:
12117:
12110:
12107:
12104:
12096:
12090:
12087:
12084:
12081:
12078:
12070:
12067:
12064:
12058:
12055:
12052:
12044:
12038:
12035:
12032:
12024:
12018:
12015:
12012:
12003:
11994:
11989:
11972:
11969:
11967:
11958:
11952:
11942:
11933:
11928:
11918:
11911:
11906:
11896:
11887:
11882:
11872:
11865:
11860:
11850:
11841:
11831:
11824:
11819:
11809:
11800:
11790:
11783:
11779:
11773:
11769:
11762:
11760:
11751:
11745:
11735:
11727:
11721:
11711:
11702:
11697:
11687:
11680:
11675:
11670:
11660:
11652:
11648:
11644:
11635:
11626:
11618:
11613:
11603:
11595:
11589:
11584:
11574:
11567:
11562:
11552:
11544:
11538:
11528:
11519:
11514:
11504:
11497:
11492:
11482:
11474:
11470:
11466:
11457:
11448:
11440:
11435:
11425:
11417:
11411:
11401:
11393:
11387:
11383:
11376:
11374:
11365:
11360:
11354:
11344:
11335:
11330:
11320:
11313:
11305:
11296:
11288:
11283:
11273:
11264:
11259:
11249:
11241:
11237:
11233:
11227:
11217:
11208:
11203:
11193:
11186:
11178:
11169:
11161:
11156:
11146:
11137:
11127:
11119:
11114:
11108:
11104:
11097:
11095:
11089:
11079:
11073:
11068:
11065:
11055:
11047:
11034:
11016:
11006:
10997:
10992:
10982:
10975:
10967:
10958:
10950:
10941:
10930:
10925:
10911:
10905:
10900:
10890:
10881:
10871:
10864:
10859:
10849:
10840:
10830:
10822:
10816:
10812:
10805:
10797:
10788:
10778:
10775:
10771:
10764:
10759:
10749:
10740:
10735:
10725:
10716:
10712:
10705:
10700:
10697:
10687:
10675:
10657:
10653:
10644:
10641:
10618:
10612:
10607:
10597:
10588:
10578:
10571:
10566:
10556:
10547:
10537:
10529:
10523:
10519:
10512:
10504:
10495:
10485:
10482:
10478:
10471:
10467:
10461:
10458:
10453:
10448:
10438:
10429:
10424:
10414:
10406:
10400:
10396:
10389:
10384:
10381:
10371:
10353:
10339:
10333:
10328:
10318:
10309:
10299:
10292:
10287:
10277:
10268:
10258:
10250:
10244:
10240:
10233:
10224:
10220:
10215:
10205:
10198:
10194:
10187:
10179:
10176:
10166:
10153:
10149:
10144:
10134:
10127:
10124:
10120:
10116:
10111:
10108:
10098:
10086:
10070:
10066:
10056:
10042:
10036:
10028:
10024:
10020:
10015:
10012:
10008:
10001:
9998:
9994:
9988:
9983:
9973:
9964:
9954:
9947:
9942:
9934:
9930:
9926:
9921:
9918:
9914:
9907:
9903:
9897:
9887:
9878:
9868:
9860:
9854:
9850:
9843:
9840:
9834:
9826:
9823:
9813:
9801:
9785:
9781:
9772:
9755:
9752:
9744:
9740:
9736:
9731:
9728:
9703:
9699:
9695:
9690:
9687:
9683:
9679:
9673:
9664:
9661:
9655:
9652:
9630:
9626:
9622:
9619:
9614:
9610:
9606:
9601:
9598:
9594:
9573:
9570:
9565:
9561:
9557:
9552:
9549:
9545:
9522:
9518:
9514:
9509:
9506:
9502:
9498:
9488:
9484:
9480:
9475:
9472:
9468:
9441:
9435:
9427:
9423:
9419:
9414:
9411:
9407:
9400:
9396:
9390:
9380:
9371:
9361:
9352:
9348:
9344:
9339:
9331:
9327:
9323:
9318:
9315:
9311:
9304:
9301:
9297:
9291:
9286:
9276:
9267:
9257:
9248:
9243:
9239:
9235:
9230:
9222:
9219:
9215:
9211:
9206:
9202:
9195:
9192:
9188:
9182:
9172:
9163:
9153:
9144:
9139:
9135:
9131:
9126:
9118:
9115:
9111:
9107:
9102:
9098:
9091:
9087:
9081:
9076:
9066:
9057:
9047:
9038:
9034:
9029:
9022:
9013:
9009:
9004:
8994:
8987:
8984:
8980:
8974:
8971:
8961:
8948:
8944:
8939:
8929:
8922:
8918:
8914:
8908:
8900:
8897:
8887:
8861:
8851:
8844:
8839:
8829:
8822:
8817:
8807:
8795:
8791:
8765:
8760:
8754:
8744:
8735:
8725:
8716:
8711:
8707:
8703:
8698:
8693:
8683:
8674:
8664:
8655:
8651:
8647:
8643:
8639:
8635:
8629:
8624:
8614:
8605:
8595:
8586:
8581:
8577:
8573:
8568:
8558:
8549:
8539:
8530:
8526:
8521:
8516:
8509:
8504:
8501:
8491:
8461:
8453:
8447:
8431:
8410:
8401:
8393:
8388:
8378:
8349:
8340:
8332:
8327:
8317:
8293:
8289:
8283:
8280:
8265:
8256:
8246:
8237:
8232:
8229:
8220:
8216:
8211:
8206:
8203:
8188:
8179:
8169:
8160:
8157:
8148:
8144:
8139:
8134:
8131:
8116:
8107:
8097:
8088:
8083:
8080:
8071:
8067:
8062:
8059:
8044:
8035:
8025:
8016:
8013:
8004:
7999:
7990:
7987:
7978:
7974:
7971:
7966:
7963:
7953:
7927:
7924:
7892:
7887:
7884:
7869:
7863:
7836:
7827:
7815:
7808:
7805:
7798:
7795:
7790:
7787:
7778:
7774:
7749:
7721:
7694:
7686:
7676:
7669:
7663:
7636:
7628:
7618:
7611:
7576:
7565:
7557:
7554:
7550:
7544:
7539:
7536:
7521:
7514:
7504:
7496:
7492:
7486:
7483:
7468:
7460:
7454:
7451:
7433:
7421:
7414:
7411:
7402:
7399:
7390:
7386:
7383:
7291:
7274:
7271:
7266:
7263:
7253:
7242:
7227:
7206:
7200:
7197:
7192:
7187:
7177:
7168:
7163:
7153:
7145:
7139:
7135:
7128:
7123:
7113:
7101:
7083:
7080:
7065:
7039:
7034:
7031:
7016:
6992:
6949:
6943:
6940:
6935:
6930:
6927:
6912:
6903:
6898:
6895:
6880:
6872:
6860:
6851:
6848:
6839:
6835:
6830:
6820:
6809:
6792:
6789:
6785:
6759:
6750:
6740:
6737:
6733:
6726:
6718:
6709:
6699:
6695:
6691:
6686:
6676:
6652:
6649:
6646:
6621:
6612:
6602:
6598:
6594:
6586:
6577:
6567:
6563:
6559:
6554:
6544:
6533:
6532:
6531:
6515:
6512:
6502:
6495:
6490:
6480:
6473:
6468:
6458:
6451:
6442:
6431:
6427:
6416:
6370:
6358:
6346:
6315:
6303:
6289:
6284:
6242:
6230:
6221:
6209:
6200:
6192:
6182:
6175:
6169:
6157:
6131:
6119:
6109:
6104:
6086:
6076:
6069:
6066:
6063:
6053:
6047:
6045:
6034:
6023:
6019:
6015:
6006:
5997:
5994:
5991:
5984:
5973:
5959:
5951:
5948:
5945:
5932:
5926:
5924:
5913:
5901:
5897:
5893:
5884:
5872:
5868:
5864:
5855:
5846:
5843:
5840:
5807:
5795:
5772:
5769:
5749:
5729:
5725:
5719:
5715:
5711:
5702:
5694:
5684:
5675:
5671:
5667:
5658:
5629:
5596:
5589:
5577:
5563:
5556:
5553:
5550:
5544:
5540:
5536:
5533:
5530:
5520:
5513:
5510:
5507:
5503:
5496:
5490:
5487:
5482:
5473:
5465:
5455:
5447:
5443:
5438:
5434:
5430:
5427:
5423:
5415:
5405:
5390:
5376:
5369:
5366:
5363:
5354:
5348:
5345:
5340:
5331:
5323:
5313:
5305:
5301:
5296:
5292:
5288:
5285:
5281:
5277:
5274:
5271:
5258:
5246:
5242:
5238:
5229:
5217:
5213:
5209:
5200:
5192:
5189:
5186:
5177:
5171:
5168:
5163:
5154:
5146:
5136:
5128:
5124:
5119:
5115:
5111:
5108:
5104:
5100:
5097:
5094:
5088:
5079:
5075:
5071:
5062:
5050:
5046:
5042:
5033:
5025:
5022:
5019:
5013:
5009:
5005:
5002:
4999:
4992:
4988:
4984:
4975:
4967:
4964:
4961:
4957:
4950:
4944:
4941:
4936:
4927:
4919:
4909:
4901:
4897:
4892:
4888:
4884:
4881:
4877:
4870:
4865:
4863:
4849:
4845:
4830:
4823:
4820:
4816:
4812:
4810:
4802:
4790:
4765:
4761:
4748:
4744:
4734:
4730:
4724:
4716:
4712:
4708:
4705:
4701:
4694:
4691:
4688:
4679:
4674:
4669:
4665:
4659:
4655:
4652:
4649:
4640:
4636:
4630:
4622:
4618:
4614:
4611:
4607:
4600:
4597:
4594:
4585:
4580:
4575:
4571:
4565:
4561:
4558:
4554:
4548:
4544:
4538:
4534:
4530:
4521:
4509:
4494:
4490:
4475:
4468:
4465:
4461:
4457:
4448:
4436:
4410:
4407:
4404:
4381:
4378:
4375:
4361:
4348:
4344:
4337:
4334:
4331:
4322:
4317:
4312:
4308:
4302:
4298:
4295:
4292:
4286:
4283:
4280:
4271:
4266:
4261:
4257:
4251:
4247:
4244:
4240:
4234:
4230:
4224:
4220:
4216:
4209:
4206:
4203:
4193:
4189:
4182:
4178:
4174:
4165:
4153:
4123:
4113:
4109:
4102:
4098:
4094:
4085:
4073:
4053:
4040:
4036:
4031:
4025:
4022:
4019:
4008:
4004:
3999:
3996:
3992:
3988:
3983:
3979:
3956:
3952:
3926:
3923:
3919:
3909:
3904:
3899:
3895:
3889:
3885:
3882:
3879:
3871:
3868:
3864:
3854:
3849:
3844:
3840:
3834:
3830:
3827:
3824:
3818:
3815:
3812:
3783:
3780:
3776:
3766:
3761:
3756:
3752:
3746:
3742:
3739:
3736:
3728:
3725:
3721:
3711:
3706:
3701:
3697:
3691:
3687:
3684:
3681:
3675:
3672:
3669:
3641:
3638:
3635:
3622:
3601:
3598:
3595:
3587:
3579:
3574:
3570:
3564:
3558:
3550:
3525:
3519:
3511:
3498:
3495:
3490:
3486:
3480:
3477:
3472:
3469:
3465:
3459:
3455:
3448:
3442:
3434:
3430:
3404:
3397:
3376:
3367:
3352:
3344:
3339:
3335:
3329:
3324:
3320:
3313:
3310:
3307:
3297:
3294:
3291:
3284:
3270:
3267:
3264:
3257:
3245:
3240:
3236:
3230:
3225:
3221:
3217:
3211:
3203:
3195:
3188:
3167:
3159:
3156:
3152:
3131:
3117:
3114:
3110:
3103:
3095:
3087:
3084:
3079:
3045:
3013:
3010:
3006:
2997:
2989:
2986:
2982:
2970:
2967:
2935:
2911:
2889:
2879:
2852:
2849:
2846:
2843:
2840:
2832:
2824:
2821:
2817:
2788:
2785:
2782:
2774:
2766:
2763:
2759:
2744:
2725:
2722:
2699:
2696:
2693:
2685:
2679:
2676:
2673:
2670:
2667:
2640:
2630:
2617:
2615:
2597:
2593:
2589:
2584:
2580:
2576:
2573:
2548:
2542:
2532:
2523:
2513:
2506:
2501:
2491:
2478:
2471:
2465:
2458:
2452:
2447:
2442:
2432:
2423:
2417:
2415:
2398:
2386:
2380:
2375:
2365:
2356:
2347:
2339:
2329:
2321:
2315:
2311:
2304:
2302:
2285:
2257:
2242:
2235:
2220:
2213:
2198:
2180:
2167:
2163:
2157:
2147:
2138:
2128:
2121:
2116:
2106:
2093:
2086:
2080:
2073:
2067:
2062:
2057:
2047:
2036:
2032:
2025:
2016:
2008:
1998:
1989:
1985:
1978:
1963:
1951:
1930:
1926:
1922:
1917:
1913:
1884:
1880:
1876:
1871:
1867:
1838:
1834:
1830:
1825:
1821:
1808:
1795:
1791:
1785:
1777:
1773:
1769:
1764:
1760:
1756:
1750:
1747:
1743:
1737:
1727:
1718:
1708:
1701:
1696:
1688:
1684:
1680:
1675:
1671:
1667:
1661:
1657:
1651:
1641:
1628:
1622:
1617:
1609:
1605:
1601:
1596:
1592:
1585:
1581:
1575:
1565:
1556:
1546:
1539:
1534:
1526:
1522:
1518:
1513:
1509:
1502:
1499:
1495:
1489:
1479:
1466:
1459:
1453:
1446:
1440:
1434:
1416:
1380:
1373:
1358:
1351:
1346:
1336:
1324:
1320:
1310:
1294:
1290:
1264:
1255:
1247:
1239:
1230:
1222:
1217:
1207:
1195:
1174:
1165:
1157:
1152:
1142:
1113:
1104:
1096:
1091:
1081:
1055:
1045:
1038:
1033:
1023:
1016:
1007:
996:
992:
988:
984:
966:
962:
953:
931:
906:
896:
871:
865:
855:
848:
839:
832:
822:
818:
809:
798:
793:
779:
770:
753:
744:
732:
724:
717:
711:
709:
692:
679:
674:
664:
653:
649:
642:
640:
623:
606:
598:
588:
579:
575:
568:
566:
549:
509:
502:
487:
480:
465:
458:
449:
432:
430:
426:
422:
418:
412:
398:
395:
392:
384:
380:
364:
361:
358:
350:
344:
342:
338:
334:
329:
318:
314:
312:
307:
305:
301:
291:
289:
285:
282:along with a
281:
280:Rydberg atoms
276:
268:
264:
260:
245:
223:
213:
209:
205:
204:
203:
199:
197:
193:
189:
185:
184:Fred Cummings
181:
166:
164:
160:
156:
152:
148:
144:
139:
137:
133:
129:
125:
121:
116:
112:
108:
104:
100:
92:
87:
82:
72:
69:
61:
51:
46:
42:
38:
37:
30:
21:
20:
20548:
20529:
20510:
20491:
20472:
20430:
20426:
20420:
20401:
20368:
20334:
20309:
20305:
20297:
20254:
20250:
20244:
20219:
20215:
20209:
20184:
20180:
20174:
20141:nlin/0409032
20131:
20127:
20121:
20078:
20075:Phys. Rev. A
20074:
20068:
20043:
20039:
20033:
20000:
19996:
19990:
19957:
19953:
19947:
19904:
19901:Phys. Rev. A
19900:
19834:
19830:
19824:
19781:
19778:Phys. Rev. B
19777:
19767:
19734:
19730:
19724:
19683:
19679:
19673:
19630:
19627:Phys. Rev. B
19626:
19616:
19575:
19569:
19563:
19530:
19526:
19508:
19502:
19493:
19487:
19454:
19450:
19444:
19409:
19405:
19399:
19364:
19360:
19354:
19329:
19325:
19281:
19277:
19254:cite journal
19221:
19217:
19180:
19176:
19166:
19149:
19145:
19139:
19129:
19106:Rabi problem
19005:
18998:
18949:
18945:
18880:
18877:
18785:
18783:
18644:
18593:
16536:
16363:
16199:
16020:
15770:
15479:
15403:
15322:
15043:
14873:
14362:Rabi problem
14329:
13814:
13398:
13248:
12951:
12610:
12248:
12223:
11990:
10928:
10926:
10359:
10087:
10057:
9802:
8787:
7099:
6422:
6347:
6105:
5796:
5762:is given by
4771:
4762:
4362:
4059:
3369:For a given
3368:
2745:
2618:
2186:
1809:
1405:, we obtain
1316:
994:
990:
987:Bloch vector
794:
771:
438:
413:
382:
345:
324:
315:
311:Rydberg atom
308:
297:
272:
267:discreteness
266:
200:
195:
180:Edwin Jaynes
177:
140:
102:
98:
96:
64:
55:
48:Please help
44:
33:
20302:S. Stenholm
20257:(7): S152.
19412:: 413–466.
19367:: 347–411.
14374:eigenstates
14370:eigenvalues
12222:counts the
6430:Hamiltonian
2612:called the
2183:Eigenstates
1856:and slowly
379:Dicke model
337:RLC circuit
328:quantum dot
196:single mode
58:August 2018
52:if you can.
19914:1609.03940
19146:Proc. IEEE
19122:References
18786:oscillates
17940:Poissonian
16062:degeneracy
13906:"detuning"
11033:commutator
7905:where the
7737:. Setting
6426:isomorphic
4363:Since the
211:absorption
20455:120269208
20251:J. Opt. B
20166:119379204
19939:118360147
19877:205214018
19844:0902.1827
19791:1304.5141
19665:119454297
19306:117003321
19278:Phys. Rev
19134:Cummings.
19070:δ
19023:δ
18968:Ω
18927:⟩
18921:⟨
18914:Ω
18910:π
18892:τ
18862:Ω
18843:τ
18822:Ω
18817:⟩
18811:⟨
18797:Ω
18744:Ω
18740:−
18724:Ω
18719:⟩
18713:⟨
18703:
18697:⋅
18684:−
18671:≃
18628:⟩
18622:⟨
18617:≪
18605:Ω
18572:⟩
18566:⟨
18498:⟩
18492:⟨
18479:Ω
18466:
18460:⟩
18454:⟨
18446:
18427:Ω
18422:⟩
18416:⟨
18385:⟩
18379:⟨
18366:Ω
18357:−
18349:
18343:⟩
18337:⟨
18329:
18310:Ω
18305:⟩
18299:⟨
18291:−
18278:⋅
18268:⟩
18262:⟨
18259:−
18249:−
18236:≃
18189:⟩
18183:⟨
18178:⟩
18172:⟨
18169:−
18137:⟩
18131:⟨
18121:Ω
18116:≃
18107:Ω
18066:⟩
18060:⟨
18034:Ω
18013:⟩
18007:⟨
17996:≃
17987:⟩
17980:⟨
17972:⟩
17956:Δ
17950:⟨
17914:α
17903:⟩
17897:⟨
17855:Ω
17848:
17818:⟩
17811:⟨
17806:⟩
17800:⟨
17797:−
17784:∞
17769:∑
17747:Ω
17720:
17690:⟩
17683:⟨
17678:⟩
17672:⟨
17669:−
17656:∞
17641:∑
17619:Ω
17592:
17564:α
17532:α
17524:−
17512:∞
17497:∑
17475:⟩
17463:ψ
17452:⟨
17371:^
17329:^
17307:ω
17303:ℏ
17271:⟩
17265:−
17233:Ω
17219:
17210:−
17207:⟩
17175:Ω
17161:
17139:ω
17129:−
17105:α
17077:α
17069:−
17059:∞
17044:∑
17037:⟩
16998:α
16990:−
16972:⟩
16930:^
16927:σ
16905:Ω
16891:
16882:−
16864:^
16839:Ω
16825:
16803:ω
16793:−
16769:α
16741:α
16733:−
16723:∞
16708:∑
16701:⟩
16662:α
16654:−
16636:⟩
16624:ψ
16614:ℏ
16594:^
16581:−
16566:⟩
16554:ψ
16503:^
16500:σ
16487:Ω
16477:ℏ
16453:^
16405:^
16372:Δ
16350:⟩
16317:α
16289:α
16281:−
16271:∞
16256:∑
16249:⟩
16246:α
16229:⟩
16217:ψ
16156:Ω
16153:ℏ
16126:⟩
16092:⟩
16086:−
16048:Ω
16045:ℏ
16029:δ
16002:⟩
15996:−
15979:∓
15976:⟩
15939:⟩
15936:±
15899:Δ
15895:⟹
15885:ω
15869:ω
15840:Ω
15824:Δ
15818:ℏ
15810:−
15802:−
15783:δ
15753:⟩
15747:−
15715:θ
15705:
15699:−
15696:⟩
15664:θ
15654:
15641:⟩
15638:−
15620:⟩
15588:θ
15578:
15569:⟩
15563:−
15531:θ
15521:
15508:⟩
15466:⟩
15432:⟩
15426:−
15378:Ω
15356:Ω
15292:Ω
15282:
15254:⟩
15242:ψ
15225:⟨
15174:Ω
15164:
15136:⟩
15124:ψ
15107:⟨
15030:⟩
15000:Ω
14990:
14981:−
14978:⟩
14948:Ω
14938:
14929:⟩
14917:ψ
14889:≈
14886:Δ
14852:Δ
14848:Ω
14835:−
14823:θ
14819:
14790:θ
14765:⟩
14759:−
14727:θ
14717:
14711:−
14708:⟩
14676:θ
14666:
14653:⟩
14650:−
14632:⟩
14600:θ
14590:
14581:⟩
14575:−
14543:θ
14533:
14520:⟩
14491:Ω
14475:Δ
14469:ℏ
14456:±
14448:Δ
14445:ℏ
14432:−
14423:ω
14419:ℏ
14399:±
14345:×
14296:−
14230:−
14186:^
14183:σ
14134:−
14123:−
14098:−
14054:^
14051:σ
13991:−
13966:−
13927:^
13883:ω
13879:−
13870:ω
13863:Δ
13839:Ω
13781:^
13778:σ
13771:Ω
13768:ℏ
13725:^
13722:σ
13709:Δ
13706:ℏ
13700:−
13682:^
13666:ω
13662:ℏ
13645:Δ
13642:ℏ
13620:ω
13616:ℏ
13602:Ω
13599:ℏ
13576:Ω
13573:ℏ
13558:Δ
13555:ℏ
13542:−
13533:ω
13529:ℏ
13503:^
13476:Δ
13473:ℏ
13443:⟩
13415:⟩
13371:ω
13367:ℏ
13353:Ω
13350:ℏ
13327:Ω
13324:ℏ
13309:Δ
13306:ℏ
13303:−
13294:ω
13290:ℏ
13264:^
13225:ℏ
13212:⟩
13206:−
13180:^
13177:σ
13161:^
13137:⟨
13124:ℏ
13118:⟩
13112:−
13093:−
13086:^
13083:σ
13074:†
13062:^
13038:⟨
13025:ℏ
13019:⟩
13013:−
12984:^
12960:⟨
12924:ω
12920:ℏ
12912:⟩
12906:−
12877:^
12853:⟨
12846:⟩
12817:^
12799:−
12787:⟨
12773:ω
12769:ℏ
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12533:^
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12433:^
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12399:⋱
12375:^
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12094:⟩
12088:−
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12062:⟩
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12022:⟩
11953:−
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11835:^
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11715:^
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11532:^
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11429:^
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11284:−
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11253:^
11221:^
11209:†
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11083:^
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8933:^
8891:^
8855:^
8833:^
8811:^
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900:^
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8796:. Take
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12638:, each
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3619:is the
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304:Walther
132:photons
115:bosonic
36:cleanup
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19873:S2CID
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19812:S2CID
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19302:S2CID
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18476:i
18470:(
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17906:=
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17520:e
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17504:=
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17065:e
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16982:=
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16954:)
16948:)
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16737:|
16729:e
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16715:=
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16678:/
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16667:|
16658:|
16650:e
16646:=
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16630:0
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16610:/
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16584:i
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16573:=
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16521:)
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16375:=
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16330:!
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16277:e
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16263:=
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16232:=
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16220:(
16213:|
16178:n
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16089:1
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16040:n
16035:=
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16006:)
15999:1
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15987:e
15983:|
15973:n
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15942:=
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15815:=
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15789:=
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15750:1
15744:n
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15729:)
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15709:(
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15687:g
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15678:)
15673:2
15668:n
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15632:n
15628:|
15617:n
15614:,
15611:g
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15602:)
15597:2
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15582:(
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15566:1
15560:n
15557:,
15554:e
15550:|
15545:)
15540:2
15535:n
15525:(
15515:=
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15502:,
15499:n
15495:|
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15460:,
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15429:1
15423:n
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15413:|
15382:2
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15365:=
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15335:n
15304:)
15299:2
15295:t
15286:(
15277:2
15269:=
15264:2
15259:|
15251:)
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15234:1
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15228:g
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15217:=
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15151:=
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15141:|
15133:)
15130:t
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15116:0
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15110:e
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15099:=
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15077:P
15052:t
15027:1
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15021:g
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15012:)
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14994:(
14984:i
14975:0
14972:,
14969:e
14965:|
14960:)
14955:2
14951:t
14942:(
14932:=
14926:)
14923:t
14920:(
14913:|
14892:0
14843:n
14832:=
14827:n
14794:n
14762:1
14756:n
14753:,
14750:e
14746:|
14741:)
14736:2
14731:n
14721:(
14705:n
14702:,
14699:g
14695:|
14690:)
14685:2
14680:n
14670:(
14660:=
14647:,
14644:n
14640:|
14629:n
14626:,
14623:g
14619:|
14614:)
14609:2
14604:n
14594:(
14584:+
14578:1
14572:n
14569:,
14566:e
14562:|
14557:)
14552:2
14547:n
14537:(
14527:=
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14514:,
14511:n
14507:|
14495:2
14487:n
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14479:2
14464:2
14461:1
14452:)
14440:2
14437:1
14427:c
14416:n
14412:(
14408:=
14396:,
14393:n
14389:E
14348:2
14342:2
14307:.
14303:|
14299:1
14293:n
14290:,
14287:e
14274:n
14271:,
14268:g
14264:|
14260:+
14256:|
14252:n
14249:,
14246:g
14233:1
14227:n
14224:,
14221:e
14217:|
14213:=
14204:)
14201:n
14198:(
14193:x
14171:|
14167:n
14164:,
14161:g
14148:n
14145:,
14142:g
14138:|
14130:|
14126:1
14120:n
14117:,
14114:e
14101:1
14095:n
14092:,
14089:e
14085:|
14081:=
14072:)
14069:n
14066:(
14061:z
14039:|
14035:n
14032:,
14029:g
14016:n
14013:,
14010:g
14006:|
14002:+
13998:|
13994:1
13988:n
13985:,
13982:e
13969:1
13963:n
13960:,
13957:e
13953:|
13949:=
13940:)
13937:n
13934:(
13924:I
13890:g
13887:e
13874:c
13866:=
13836:=
13831:c
13827:g
13823:2
13799:)
13796:n
13793:(
13788:x
13763:n
13756:2
13753:1
13748:+
13743:)
13740:n
13737:(
13732:z
13713:2
13695:)
13692:n
13689:(
13679:I
13670:c
13659:n
13656:=
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13637:2
13634:1
13629:+
13624:c
13613:n
13606:2
13594:n
13580:2
13568:n
13550:2
13547:1
13537:c
13526:n
13520:[
13515:=
13510:n
13500:H
13468:2
13465:1
13440:g
13436:|
13412:e
13408:|
13383:]
13375:c
13364:n
13357:2
13345:n
13331:2
13319:n
13298:c
13287:n
13281:[
13276:=
13271:n
13261:H
13233:c
13229:g
13220:n
13215:=
13209:1
13203:n
13200:,
13197:e
13193:|
13187:+
13168:c
13158:a
13150:|
13146:n
13143:,
13140:g
13132:c
13128:g
13121:+
13115:1
13109:n
13106:,
13103:e
13099:|
13069:c
13059:a
13051:|
13047:n
13044:,
13041:g
13033:c
13029:g
13022:=
13016:1
13010:n
13007:,
13004:e
13000:|
12994:C
12991:J
12981:H
12973:|
12969:n
12966:,
12963:g
12936:]
12928:c
12917:n
12909:1
12903:n
12900:,
12897:e
12893:|
12887:C
12884:J
12874:H
12866:|
12862:n
12859:,
12856:g
12843:n
12840:,
12837:g
12833:|
12827:C
12824:J
12814:H
12806:|
12802:1
12796:n
12793:,
12790:e
12780:g
12777:e
12766:+
12763:)
12760:1
12754:n
12751:(
12746:c
12732:[
12727:=
12722:n
12712:H
12688:2
12682:2
12660:n
12650:H
12624:0
12620:H
12595:]
12547:0
12540:n
12530:H
12521:0
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12421:0
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12372:H
12363:0
12341:0
12336:0
12331:0
12324:0
12320:H
12313:[
12308:=
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12290:H
12260:N
12234:n
12204:N
12181:n
12156:}
12149:n
12145:|
12140:{
12118:}
12111:n
12108:,
12105:g
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12097:,
12091:1
12085:n
12082:,
12079:e
12075:|
12071:;
12065:;
12059:1
12056:,
12053:g
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12045:,
12039:0
12036:,
12033:e
12029:|
12025:;
12019:0
12016:,
12013:g
12009:|
12004:{
11973:0
11970:=
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11929:c
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11866:+
11861:+
11842:c
11832:a
11825:+
11820:+
11801:c
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11780:(
11774:c
11770:g
11763:=
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11728:]
11722:c
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11698:c
11688:a
11681:,
11671:c
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11653:[
11649:+
11645:]
11640:|
11636:e
11627:e
11623:|
11619:,
11596:[
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11545:]
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11475:[
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11467:]
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11445:|
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11436:+
11418:[
11412:c
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11377:=
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11361:]
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11314:+
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11297:e
11293:|
11289:,
11260:c
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11242:[
11238:+
11234:]
11228:c
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11204:c
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11162:,
11157:+
11138:c
11128:a
11120:[
11115:(
11109:c
11105:g
11098:=
11090:]
11080:N
11074:,
11069:F
11066:A
11056:H
11048:[
11017:c
11007:a
10993:c
10983:a
10976:+
10972:|
10968:e
10959:e
10955:|
10951:=
10942:N
10912:)
10901:c
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10865:+
10860:c
10850:a
10841:+
10823:(
10817:c
10813:g
10806:+
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10789:e
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10779:g
10776:e
10765:+
10760:c
10750:a
10736:c
10726:a
10717:c
10706:=
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10698:J
10688:H
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10567:c
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10548:+
10530:(
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10468:)
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10449:c
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10425:c
10415:a
10407:(
10401:c
10390:=
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10288:c
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10269:+
10251:(
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10234:=
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10216:0
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10185:(
10180:F
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10154:/
10150:t
10145:0
10135:H
10128:i
10121:e
10117:=
10112:F
10109:A
10099:H
10071:c
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10043:)
10037:t
10034:)
10029:c
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10013:e
10005:(
10002:i
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9911:(
9908:i
9904:e
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9879:+
9861:(
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9841:=
9838:)
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9827:F
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9814:H
9786:c
9782:g
9745:c
9737:2
9729:2
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9674:,
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9656:=
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9574:0
9566:c
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9550:e
9523:c
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9510:g
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9495:|
9489:c
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9464:|
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9095:(
9092:i
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9067:a
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9039:c
9035:g
9030:(
9023:=
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9010:t
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8995:H
8988:i
8981:e
8975:F
8972:A
8962:H
8949:/
8945:t
8940:0
8930:H
8923:i
8919:e
8915:=
8912:)
8909:t
8906:(
8901:F
8898:A
8888:H
8862:F
8852:H
8845:+
8840:A
8830:H
8823:=
8818:0
8808:H
8779:.
8766:]
8761:)
8755:c
8745:a
8712:c
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8704:+
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8640:+
8636:)
8625:c
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8559:a
8550:+
8531:c
8527:g
8522:(
8517:[
8510:=
8505:F
8502:A
8492:H
8468:}
8462:g
8458:|
8454:,
8448:e
8444:|
8440:{
8415:|
8411:e
8402:g
8398:|
8394:=
8354:|
8350:g
8341:e
8337:|
8333:=
8328:+
8294:,
8290:)
8281:,
8277:k
8266:a
8230:,
8226:k
8221:g
8217:+
8204:,
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8158:,
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8128:k
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8081:,
8077:k
8072:g
8060:,
8056:k
8045:a
8036:+
8014:,
8010:k
8005:g
8000:(
7988:,
7984:k
7972:=
7967:F
7964:A
7954:H
7925:,
7921:k
7915:u
7893:,
7885:,
7881:k
7875:u
7864:g
7860:|
7849:d
7841:|
7837:e
7828:V
7821:k
7806:2
7799:i
7796:=
7788:,
7784:k
7779:g
7754:0
7750:=
7746:R
7722:e
7718:|
7707:d
7699:|
7695:g
7670:+
7664:g
7660:|
7649:d
7641:|
7637:e
7629:+
7612:=
7602:d
7577:)
7570:R
7562:k
7558:i
7551:e
7537:,
7533:k
7522:a
7509:R
7501:k
7497:i
7493:e
7484:,
7480:k
7469:a
7461:(
7452:,
7448:k
7442:u
7434:V
7427:k
7412:2
7400:,
7396:k
7387:i
7384:=
7381:)
7377:R
7373:(
7363:E
7338:R
7317:)
7313:R
7309:(
7299:E
7282:d
7272:=
7267:F
7264:A
7254:H
7228:c
7207:)
7201:2
7198:1
7193:+
7188:c
7178:a
7164:c
7154:a
7146:(
7140:c
7129:=
7124:F
7114:H
7081:,
7077:k
7066:a
7032:,
7028:k
7017:a
6972:k
6950:)
6944:2
6941:1
6936:+
6928:,
6924:k
6913:a
6896:,
6892:k
6881:a
6873:(
6866:k
6849:,
6845:k
6836:=
6831:F
6821:H
6793:g
6790:e
6764:|
6760:e
6751:e
6747:|
6741:g
6738:e
6727:=
6723:|
6719:e
6710:e
6706:|
6700:e
6696:E
6692:=
6687:A
6677:H
6653:g
6650:,
6647:e
6626:|
6622:e
6613:e
6609:|
6603:e
6599:E
6595:+
6591:|
6587:g
6578:g
6574:|
6568:g
6564:E
6560:=
6555:A
6545:H
6516:F
6513:A
6503:H
6496:+
6491:F
6481:H
6474:+
6469:A
6459:H
6452:=
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6374:)
6371:0
6368:(
6334:]
6319:)
6316:t
6313:(
6298:[
6290:=
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6246:)
6243:t
6240:(
6231:U
6225:)
6222:0
6219:(
6204:)
6201:t
6198:(
6183:U
6176:=
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6170:t
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6135:)
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6087:,
6070:t
6054:a
6048:=
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6024:2
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5933:a
5927:=
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5865:+
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5773:=
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5730:4
5726:/
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5712:+
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5685:a
5676:2
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5668:=
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5590:)
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5534:i
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5497:)
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4951:)
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4942:1
4937:+
4928:a
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4871:(
4866:=
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4813:=
4806:)
4803:t
4800:(
4791:U
4749:.
4745:]
4735:/
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4725:n
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4713:E
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4623:+
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4581:2
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4566:(
4555:[
4549:n
4545:C
4539:n
4531:=
4525:)
4522:0
4519:(
4505:|
4495:/
4491:t
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4469:i
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4458:=
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4449:t
4446:(
4432:|
4411:0
4405:t
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4376:n
4372:|
4349:.
4345:]
4335:,
4332:n
4328:|
4323:)
4318:2
4313:n
4303:(
4293:+
4287:+
4284:,
4281:n
4277:|
4272:)
4267:2
4262:n
4252:(
4241:[
4235:n
4231:C
4225:n
4217:=
4210:e
4207:,
4204:n
4200:|
4194:n
4190:C
4183:n
4175:=
4169:)
4166:0
4163:(
4149:|
4124:n
4120:|
4114:n
4110:C
4103:n
4095:=
4089:)
4086:0
4083:(
4069:|
4041:.
4037:)
4026:1
4023:+
4020:n
4009:(
4000:1
3984:n
3957:n
3927:n
3924:2
3915:|
3910:)
3905:2
3900:n
3890:(
3872:n
3869:1
3860:|
3855:)
3850:2
3845:n
3835:(
3825:=
3816:,
3813:n
3809:|
3784:n
3781:2
3772:|
3767:)
3762:2
3757:n
3747:(
3737:+
3729:n
3726:1
3717:|
3712:)
3707:2
3702:n
3692:(
3682:=
3676:+
3673:,
3670:n
3666:|
3639:,
3636:n
3632:|
3605:)
3602:1
3599:+
3596:n
3593:(
3588:2
3580:+
3575:2
3565:=
3562:)
3556:(
3551:n
3526:,
3523:)
3517:(
3512:n
3499:2
3496:1
3487:)
3481:2
3478:1
3473:+
3470:n
3466:(
3460:c
3449:=
3446:)
3443:n
3440:(
3431:E
3408:)
3405:n
3402:(
3398:H
3377:n
3353:)
3345:2
3340:a
3325:c
3317:)
3314:1
3311:+
3308:n
3305:(
3298:1
3295:+
3292:n
3285:2
3271:1
3268:+
3265:n
3258:2
3246:2
3241:a
3231:+
3226:c
3218:n
3212:(
3204:=
3199:)
3196:n
3193:(
3189:H
3168:,
3160:n
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3148:|
3132:H
3124:|
3118:n
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3096:n
3093:(
3088:j
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3046:H
3022:}
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2998:,
2990:n
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2978:|
2974:{
2936:H
2912:n
2890:I
2880:H
2853:g
2850:,
2847:1
2844:+
2841:n
2837:|
2825:n
2822:2
2813:|
2789:e
2786:,
2783:n
2779:|
2767:n
2764:1
2755:|
2730:N
2723:n
2700:e
2697:,
2694:n
2690:|
2686:,
2680:g
2677:,
2674:1
2671:+
2668:n
2664:|
2641:I
2631:H
2598:c
2585:a
2577:=
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2514:a
2507:+
2502:+
2479:a
2472:(
2466:2
2453:+
2448:2
2443:z
2418:=
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2387:)
2381:2
2376:z
2357:+
2348:a
2330:a
2322:(
2316:c
2305:=
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2286:H
2258:,
2243:H
2236:+
2231:I
2221:H
2214:=
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2168:.
2164:)
2129:a
2122:+
2117:+
2094:a
2087:(
2081:2
2068:+
2063:2
2058:z
2037:a
2026:+
2017:a
1999:a
1990:c
1979:=
1964:H
1936:)
1931:a
1923:+
1918:c
1910:(
1890:)
1885:a
1872:c
1864:(
1844:)
1839:a
1831:+
1826:c
1818:(
1796:.
1792:)
1786:t
1783:)
1778:a
1770:+
1765:c
1754:(
1751:i
1744:e
1709:a
1702:+
1697:t
1694:)
1689:a
1681:+
1676:c
1665:(
1662:i
1658:e
1652:+
1629:a
1623:+
1618:t
1615:)
1610:a
1602:+
1597:c
1589:(
1586:i
1582:e
1576:+
1547:a
1540:+
1535:t
1532:)
1527:a
1519:+
1514:c
1506:(
1503:i
1496:e
1467:a
1460:(
1454:2
1441:=
1438:)
1435:t
1432:(
1417:H
1381:H
1374:+
1359:H
1352:=
1347:0
1337:H
1295:a
1269:|
1265:g
1256:g
1252:|
1244:|
1240:e
1231:e
1227:|
1223:=
1218:z
1179:|
1175:e
1166:g
1162:|
1158:=
1118:|
1114:g
1105:e
1101:|
1097:=
1092:+
1039:+
1034:+
1017:=
1008:S
967:c
932:a
897:a
872:)
856:a
849:+
840:a
833:(
823:E
819:=
810:E
780:0
754:.
745:S
733:E
725:2
712:=
693:H
680:2
675:z
654:a
643:=
624:H
607:a
589:a
580:c
569:=
550:H
510:H
503:+
488:H
481:+
466:H
459:=
450:H
399:1
396:=
393:N
365:1
359:N
246:n
224:n
71:)
65:(
60:)
56:(
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