Jaynes–Cummings model

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: 7354: 5829: 10630: 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

4763:

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: 9805: 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}}} 16533: 16360: 15040: 4759: 7770: 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).} 12129: 18101: 532: 18780: 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: 3660: 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".

18942: 1280: 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: 7765: 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.).

19060: 919: 18588: 3974: 18076: 6384: 6145: 5646: 2741: 14902: 11998: 6413: 6150: 4395: 3655: 17390: 16102: 15442: 13849: 12672: 10934: 2902: 2653: 18989: 16136: 15476: 7349: 6983: 6805: 18050: 16190: 14806: 5651: 3969: 2190: 1307: 979: 236: 13453: 13425: 12276: 12220: 5824: 948: 16388: 14358: 12698: 7003: 19080: 3420: 7245: 6668: 4421: 375: 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: 17410: 15345: 15062: 12244: 12191: 7238: 5760: 3387: 2922: 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".

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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".

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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

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P. Kochanski; Z. Bialynicka-Birula; I. Bialynicki-Birula (2001). "Squeezing of electromagnetic field in a cavity by electrons in Trojan states".

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is the total number of quanta in the coupled system. This quantization of energies and non-linear scaling is purely quantum mechanical in nature.

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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 "

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Hussin, V.; Nieto, L. M. (2005). "Ladder operators and coherent states for the Jaynes-Cummings model in the rotating-wave approximation".

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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.

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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

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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".

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longer time regime; in that case, the periodic nature of the slowly varying double exponential predicts that there should also be a

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J.H. Eberly; N.B. Narozhny; J.J. Sanchez-Mondragon (1980). "Periodic spontaneous collapse and revival in a simple quantum model".

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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

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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: 17706: 17698: 17697: 17696: 17681: 17680: 17661: 17658: 17653: 17632: 17628: 17624: 17617: 17612: 17610: 17601: 17590: 17589: 17580: 17579: 17571: 17562: 17557: 17555: 17547: 17546: 17545: 17544: 17539: 17530: 17517: 17514: 17509: 17487: 17486: 17481: 17477: 17461: 17432: 17431: 17411: 17409: 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16457: 16456: 16448: 16444: 16440: 16439: 16431: 16415: 16414: 16409: 16408: 16400: 16389: 16387: 16386: 16381: 16361: 16359: 16358: 16353: 16339: 16334: 16325: 16324: 16323: 16314: 16312: 16311: 16307: 16302: 16301: 16296: 16287: 16273: 16268: 16238: 16215: 16191: 16189: 16188: 16183: 16181: 16176: 16167: 16165: 16164: 16159: 16151: 16146: 16137: 16135: 16134: 16129: 16115: 16103: 16101: 16100: 16095: 16075: 16059: 16057: 16056: 16051: 16043: 16038: 16018: 16016: 16015: 16010: 16008: 16004: 15985: 15965: 15955: 15949: 15945: 15928: 15916: 15914: 15913: 15908: 15892: 15891: 15879: 15878: 15859: 15857: 15856: 15851: 15849: 15847: 15846: 15831: 15830: 15821: 15813: 15812: 15800: 15799: 15768: 15766: 15765: 15760: 15758: 15736: 15731: 15727: 15722: 15721: 15712: 15685: 15680: 15676: 15671: 15670: 15661: 15630: 15609: 15604: 15600: 15595: 15594: 15585: 15552: 15547: 15543: 15538: 15537: 15528: 15497: 15477: 15475: 15474: 15469: 15455: 15443: 15441: 15440: 15435: 15415: 15395: 15393: 15392: 15387: 15385: 15380: 15376: 15371: 15368: 15363: 15362: 15346: 15344: 15343: 15338: 15320: 15318: 15317: 15312: 15310: 15306: 15302: 15297: 15289: 15280: 15279: 15267: 15266: 15261: 15240: 15223: 15202: 15201: 15188: 15184: 15179: 15171: 15162: 15161: 15149: 15148: 15143: 15122: 15105: 15084: 15083: 15063: 15061: 15060: 15055: 15041: 15039: 15038: 15033: 15019: 15014: 15010: 15005: 14997: 14967: 14962: 14958: 14953: 14945: 14915: 14903: 14901: 14900: 14895: 14865: 14863: 14862: 14857: 14855: 14850: 14846: 14841: 14838: 14830: 14829: 14807: 14805: 14804: 14799: 14797: 14796: 14781:Where the angle 14780: 14778: 14777: 14772: 14770: 14748: 14743: 14739: 14734: 14733: 14724: 14697: 14692: 14688: 14683: 14682: 14673: 14642: 14621: 14616: 14612: 14607: 14606: 14597: 14564: 14559: 14555: 14550: 14549: 14540: 14509: 14500: 14498: 14497: 14482: 14481: 14472: 14467: 14459: 14454: 14450: 14443: 14435: 14430: 14429: 14402: 14401: 14359: 14357: 14356: 14351: 14327: 14325: 14324: 14319: 14317: 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13273: 13268: 13267: 13259: 13246: 13244: 13243: 13238: 13236: 13235: 13223: 13218: 13195: 13190: 13189: 13184: 13183: 13175: 13171: 13170: 13165: 13164: 13156: 13152: 13135: 13134: 13101: 13096: 13095: 13090: 13089: 13081: 13076: 13071: 13066: 13065: 13057: 13053: 13036: 13035: 13002: 12997: 12996: 12988: 12987: 12979: 12975: 12949: 12947: 12946: 12941: 12939: 12938: 12931: 12930: 12895: 12890: 12889: 12881: 12880: 12872: 12868: 12835: 12830: 12829: 12821: 12820: 12812: 12808: 12783: 12782: 12749: 12748: 12725: 12724: 12719: 12718: 12710: 12699: 12697: 12696: 12691: 12673: 12671: 12670: 12665: 12663: 12662: 12657: 12656: 12648: 12637: 12635: 12634: 12629: 12627: 12626: 12608: 12606: 12605: 12600: 12598: 12597: 12543: 12542: 12537: 12536: 12528: 12443: 12442: 12437: 12436: 12428: 12385: 12384: 12379: 12378: 12370: 12327: 12326: 12306: 12305: 12297: 12296: 12288: 12277: 12275: 12274: 12269: 12267: 12266: 12258: 12245: 12243: 12242: 12237: 12221: 12219: 12218: 12213: 12211: 12210: 12202: 12192: 12190: 12189: 12184: 12168: 12166: 12165: 12160: 12158: 12154: 12147: 12130: 12128: 12127: 12122: 12120: 12116: 12103: 12077: 12051: 12031: 12011: 11988: 11986: 11985: 11980: 11978: 11965: 11961: 11957: 11956: 11955: 11950: 11949: 11941: 11936: 11931: 11926: 11925: 11917: 11910: 11909: 11904: 11903: 11895: 11890: 11885: 11880: 11879: 11871: 11864: 11863: 11858: 11857: 11849: 11845: 11844: 11839: 11838: 11830: 11823: 11822: 11817: 11816: 11808: 11804: 11803: 11798: 11797: 11789: 11777: 11776: 11758: 11754: 11750: 11749: 11748: 11743: 11742: 11734: 11730: 11726: 11725: 11724: 11719: 11718: 11710: 11705: 11700: 11695: 11694: 11686: 11678: 11673: 11668: 11667: 11659: 11647: 11643: 11642: 11625: 11617: 11616: 11611: 11610: 11602: 11592: 11587: 11582: 11581: 11573: 11566: 11565: 11560: 11559: 11551: 11547: 11543: 11542: 11541: 11536: 11535: 11527: 11522: 11517: 11512: 11511: 11503: 11496: 11495: 11490: 11489: 11481: 11469: 11465: 11464: 11447: 11439: 11438: 11433: 11432: 11424: 11415: 11414: 11409: 11408: 11400: 11391: 11390: 11372: 11368: 11364: 11363: 11359: 11358: 11357: 11352: 11351: 11343: 11338: 11333: 11328: 11327: 11319: 11312: 11295: 11287: 11286: 11281: 11280: 11272: 11267: 11262: 11257: 11256: 11248: 11236: 11232: 11231: 11230: 11225: 11224: 11216: 11211: 11206: 11201: 11200: 11192: 11185: 11168: 11160: 11159: 11154: 11153: 11145: 11141: 11140: 11135: 11134: 11126: 11112: 11111: 11092: 11088: 11087: 11086: 11078: 11072: 11071: 11063: 11062: 11054: 11030: 11028: 11027: 11022: 11020: 11019: 11014: 11013: 11005: 11000: 10995: 10990: 10989: 10981: 10974: 10957: 10949: 10948: 10940: 10924: 10922: 10921: 10916: 10914: 10910: 10908: 10903: 10898: 10897: 10889: 10885: 10884: 10879: 10878: 10870: 10863: 10862: 10857: 10856: 10848: 10844: 10843: 10838: 10837: 10829: 10820: 10819: 10804: 10787: 10782: 10781: 10763: 10762: 10757: 10756: 10748: 10743: 10738: 10733: 10732: 10724: 10720: 10719: 10704: 10703: 10695: 10694: 10686: 10671: 10669: 10668: 10663: 10661: 10660: 10648: 10640: 10631: 10629: 10628: 10623: 10621: 10617: 10615: 10610: 10605: 10604: 10596: 10592: 10591: 10586: 10585: 10577: 10570: 10569: 10564: 10563: 10555: 10551: 10550: 10545: 10544: 10536: 10527: 10526: 10511: 10494: 10489: 10488: 10470: 10466: 10465: 10457: 10452: 10451: 10446: 10445: 10437: 10432: 10427: 10422: 10421: 10413: 10404: 10403: 10388: 10387: 10379: 10378: 10370: 10352: 10350: 10349: 10344: 10342: 10338: 10336: 10331: 10326: 10325: 10317: 10313: 10312: 10307: 10306: 10298: 10291: 10290: 10285: 10284: 10276: 10272: 10271: 10266: 10265: 10257: 10248: 10247: 10232: 10231: 10227: 10219: 10218: 10213: 10212: 10204: 10183: 10182: 10174: 10173: 10165: 10161: 10160: 10156: 10148: 10147: 10142: 10141: 10133: 10115: 10114: 10106: 10105: 10097: 10084: 10082: 10081: 10076: 10074: 10073: 10055: 10053: 10052: 10047: 10045: 10041: 10040: 10039: 10032: 10031: 10019: 10018: 9991: 9986: 9981: 9980: 9972: 9968: 9967: 9962: 9961: 9953: 9946: 9945: 9938: 9937: 9925: 9924: 9901: 9900: 9895: 9894: 9886: 9882: 9881: 9876: 9875: 9867: 9858: 9857: 9830: 9829: 9821: 9820: 9812: 9799: 9797: 9796: 9791: 9789: 9788: 9767: 9765: 9764: 9759: 9751: 9749: 9748: 9747: 9734: 9726: 9717: 9715: 9714: 9709: 9707: 9706: 9694: 9693: 9672: 9667: 9659: 9644: 9642: 9641: 9636: 9634: 9633: 9618: 9617: 9605: 9604: 9585: 9583: 9582: 9577: 9569: 9568: 9556: 9555: 9536: 9534: 9533: 9528: 9526: 9525: 9513: 9512: 9497: 9492: 9491: 9479: 9478: 9466: 9454: 9452: 9451: 9446: 9444: 9440: 9439: 9438: 9431: 9430: 9418: 9417: 9394: 9393: 9388: 9387: 9379: 9375: 9374: 9369: 9368: 9360: 9356: 9355: 9343: 9342: 9335: 9334: 9322: 9321: 9294: 9289: 9284: 9283: 9275: 9271: 9270: 9265: 9264: 9256: 9251: 9246: 9234: 9233: 9226: 9225: 9210: 9209: 9186: 9185: 9180: 9179: 9171: 9167: 9166: 9161: 9160: 9152: 9147: 9142: 9130: 9129: 9122: 9121: 9106: 9105: 9084: 9079: 9074: 9073: 9065: 9061: 9060: 9055: 9054: 9046: 9042: 9041: 9021: 9020: 9016: 9008: 9007: 9002: 9001: 8993: 8978: 8977: 8969: 8968: 8960: 8956: 8955: 8951: 8943: 8942: 8937: 8936: 8928: 8904: 8903: 8895: 8894: 8886: 8875: 8873: 8872: 8867: 8865: 8864: 8859: 8858: 8850: 8843: 8842: 8837: 8836: 8828: 8821: 8820: 8815: 8814: 8806: 8778: 8776: 8775: 8770: 8768: 8764: 8763: 8759: 8758: 8757: 8752: 8751: 8743: 8739: 8738: 8733: 8732: 8724: 8719: 8714: 8701: 8696: 8691: 8690: 8682: 8678: 8677: 8672: 8671: 8663: 8659: 8658: 8638: 8634: 8632: 8627: 8622: 8621: 8613: 8609: 8608: 8603: 8602: 8594: 8589: 8584: 8572: 8571: 8566: 8565: 8557: 8553: 8552: 8547: 8546: 8538: 8534: 8533: 8508: 8507: 8499: 8498: 8490: 8479: 8477: 8476: 8471: 8460: 8446: 8427: 8425: 8424: 8419: 8417: 8400: 8392: 8391: 8386: 8385: 8377: 8366: 8364: 8363: 8358: 8356: 8339: 8331: 8330: 8325: 8324: 8316: 8305: 8303: 8302: 8297: 8292: 8288: 8287: 8286: 8279: 8273: 8272: 8264: 8260: 8259: 8254: 8253: 8245: 8240: 8235: 8228: 8214: 8209: 8202: 8196: 8195: 8187: 8183: 8182: 8177: 8176: 8168: 8164: 8163: 8156: 8142: 8137: 8130: 8124: 8123: 8115: 8111: 8110: 8105: 8104: 8096: 8091: 8086: 8079: 8066: 8065: 8058: 8052: 8051: 8043: 8039: 8038: 8033: 8032: 8024: 8020: 8019: 8012: 7993: 7986: 7970: 7969: 7961: 7960: 7952: 7941: 7939: 7938: 7933: 7931: 7930: 7923: 7917: 7904: 7902: 7901: 7896: 7891: 7890: 7883: 7877: 7862: 7857: 7856: 7851: 7846: 7843: 7832: 7826: 7825: 7824: 7823: 7803: 7802: 7794: 7793: 7786: 7766: 7764: 7763: 7758: 7756: 7748: 7736: 7734: 7733: 7728: 7720: 7715: 7714: 7709: 7704: 7701: 7690: 7689: 7684: 7683: 7675: 7662: 7657: 7656: 7651: 7646: 7643: 7632: 7631: 7626: 7625: 7617: 7610: 7609: 7604: 7599: 7589: 7587: 7586: 7581: 7579: 7575: 7574: 7573: 7572: 7564: 7547: 7542: 7535: 7529: 7528: 7520: 7513: 7512: 7511: 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 12757:− 12742:ω 12738:ℏ 12715:^ 12685:× 12653:^ 12589:⋱ 12584:⋱ 12579:⋱ 12574:⋱ 12569:⋱ 12564:⋱ 12559:⋮ 12552:⋱ 12533:^ 12516:⋱ 12511:⋱ 12506:⋮ 12499:⋱ 12494:⋱ 12489:⋱ 12484:⋱ 12479:⋱ 12474:⋱ 12469:⋮ 12462:⋱ 12457:⋱ 12452:⋱ 12433:^ 12409:⋱ 12404:⋱ 12399:⋱ 12375:^ 12356:⋯ 12351:⋯ 12346:⋯ 12293:^ 12263:^ 12207:^ 12152:⟩ 12114:⟩ 12094:⟩ 12088:− 12068:⋯ 12062:⟩ 12042:⟩ 12022:⟩ 11953:− 11946:^ 11943:σ 11934:† 11922:^ 11912:− 11907:− 11900:^ 11897:σ 11888:† 11876:^ 11854:^ 11851:σ 11835:^ 11813:^ 11810:σ 11794:^ 11784:− 11766:ℏ 11746:− 11739:^ 11736:σ 11715:^ 11703:† 11691:^ 11676:† 11664:^ 11633:⟨ 11630:⟩ 11614:− 11607:^ 11604:σ 11590:† 11578:^ 11556:^ 11553:σ 11532:^ 11520:† 11508:^ 11486:^ 11455:⟨ 11452:⟩ 11429:^ 11426:σ 11405:^ 11380:ℏ 11348:^ 11336:† 11324:^ 11303:⟨ 11300:⟩ 11284:− 11277:^ 11274:σ 11265:† 11253:^ 11221:^ 11209:† 11197:^ 11176:⟨ 11173:⟩ 11150:^ 11147:σ 11131:^ 11101:ℏ 11083:^ 11059:^ 11010:^ 10998:† 10986:^ 10965:⟨ 10962:⟩ 10945:^ 10906:† 10894:^ 10882:− 10875:^ 10872:σ 10853:^ 10834:^ 10831:σ 10809:ℏ 10795:⟨ 10792:⟩ 10772:ω 10768:ℏ 10753:^ 10741:† 10729:^ 10713:ω 10709:ℏ 10691:^ 10654:ω 10650:ℏ 10613:† 10601:^ 10589:− 10582:^ 10579:σ 10560:^ 10541:^ 10538:σ 10516:ℏ 10502:⟨ 10499:⟩ 10479:ω 10475:ℏ 10442:^ 10430:† 10418:^ 10397:ω 10393:ℏ 10375:^ 10334:† 10322:^ 10310:− 10303:^ 10300:σ 10281:^ 10262:^ 10259:σ 10237:ℏ 10229:ℏ 10209:^ 10170:^ 10158:ℏ 10138:^ 10125:− 10102:^ 10025:ω 10021:− 10009:ω 9999:− 9989:† 9977:^ 9965:− 9958:^ 9955:σ 9931:ω 9927:− 9915:ω 9891:^ 9872:^ 9869:σ 9847:ℏ 9844:− 9817:^ 9756:τ 9753:≪ 9741:ω 9732:π 9700:ω 9696:− 9684:ω 9680:≡ 9677:Δ 9669:Δ 9665:π 9653:τ 9627:ω 9620:≃ 9611:ω 9595:ω 9571:≃ 9562:ω 9558:− 9546:ω 9519:ω 9503:ω 9499:≪ 9485:ω 9481:− 9469:ω 9424:ω 9420:− 9408:ω 9384:^ 9365:^ 9362:σ 9345:− 9328:ω 9324:− 9312:ω 9302:− 9292:† 9280:^ 9268:− 9261:^ 9258:σ 9249:∗ 9236:− 9216:ω 9203:ω 9193:− 9176:^ 9164:− 9157:^ 9154:σ 9145:∗ 9112:ω 9099:ω 9082:† 9070:^ 9051:^ 9048:σ 9026:ℏ 9018:ℏ 8998:^ 8985:− 8965:^ 8953:ℏ 8933:^ 8891:^ 8855:^ 8833:^ 8811:^ 8748:^ 8736:− 8729:^ 8726:σ 8717:∗ 8699:† 8687:^ 8668:^ 8665:σ 8648:− 8630:† 8618:^ 8606:− 8599:^ 8596:σ 8587:∗ 8574:− 8562:^ 8543:^ 8540:σ 8513:ℏ 8495:^ 8465:⟩ 8451:⟩ 8408:⟨ 8405:⟩ 8389:− 8382:^ 8379:σ 8347:⟨ 8344:⟩ 8321:^ 8318:σ 8284:λ 8269:^ 8257:− 8250:^ 8247:σ 8238:∗ 8233:λ 8212:† 8207:λ 8192:^ 8173:^ 8170:σ 8161:λ 8145:− 8140:† 8135:λ 8120:^ 8108:− 8101:^ 8098:σ 8089:∗ 8084:λ 8068:− 8063:λ 8048:^ 8029:^ 8026:σ 8017:λ 7996:ℏ 7991:λ 7979:∑ 7975:− 7957:^ 7928:λ 7888:λ 7870:⋅ 7867:⟩ 7853:^ 7834:⟨ 7816:ω 7812:ℏ 7809:π 7791:λ 7775:ℏ 7725:⟩ 7711:^ 7692:⟨ 7687:− 7680:^ 7677:σ 7667:⟩ 7653:^ 7634:⟨ 7622:^ 7619:σ 7606:^ 7566:⋅ 7555:− 7545:† 7540:λ 7525:^ 7515:− 7505:⋅ 7487:λ 7472:^ 7455:λ 7422:ω 7418:ℏ 7415:π 7403:λ 7391:∑ 7367:^ 7303:^ 7292:⋅ 7286:^ 7275:− 7257:^ 7181:^ 7169:† 7157:^ 7136:ω 7132:ℏ 7117:^ 7084:λ 7069:^ 7040:† 7035:λ 7020:^ 6993:λ 6931:λ 6916:^ 6904:† 6899:λ 6884:^ 6861:ω 6857:ℏ 6852:λ 6840:∑ 6824:^ 6786:ω 6757:⟨ 6754:⟩ 6734:ω 6730:ℏ 6716:⟨ 6713:⟩ 6680:^ 6619:⟨ 6616:⟩ 6584:⟨ 6581:⟩ 6548:^ 6506:^ 6484:^ 6462:^ 6446:^ 6400:^ 6397:Θ 6362:^ 6359:ρ 6328:^ 6325:Θ 6307:^ 6304:ρ 6281:⟩ 6274:^ 6271:Θ 6265:⟨ 6234:^ 6213:^ 6210:ρ 6193:† 6186:^ 6161:^ 6158:ρ 6123:^ 6120:ρ 6080:^ 6077:φ 6067:⁡ 6057:^ 6038:^ 6010:^ 6007:φ 5995:⁡ 5977:^ 5974:φ 5963:^ 5960:φ 5949:⁡ 5936:^ 5917:^ 5888:^ 5885:φ 5859:^ 5856:φ 5844:⁡ 5811:^ 5781:ℏ 5778:Ω 5716:δ 5706:^ 5695:† 5688:^ 5662:^ 5659:φ 5633:^ 5630:φ 5581:^ 5578:φ 5567:^ 5564:φ 5554:⁡ 5537:δ 5524:^ 5521:φ 5511:⁡ 5483:− 5477:^ 5466:† 5459:^ 5435:ω 5428:− 5416:† 5409:^ 5394:^ 5391:φ 5380:^ 5377:φ 5367:⁡ 5341:− 5335:^ 5324:† 5317:^ 5293:ω 5286:− 5272:− 5262:^ 5233:^ 5230:φ 5204:^ 5201:φ 5190:⁡ 5158:^ 5147:† 5140:^ 5116:ω 5109:− 5095:− 5066:^ 5063:φ 5037:^ 5034:φ 5023:⁡ 5006:δ 5000:− 4979:^ 4976:φ 4965:⁡ 4931:^ 4920:† 4913:^ 4889:ω 4882:− 4854:ℏ 4834:^ 4821:− 4794:^ 4739:ℏ 4717:− 4706:− 4698:⟩ 4695:− 4666:α 4656:⁡ 4645:ℏ 4612:− 4604:⟩ 4572:α 4562:⁡ 4535:∑ 4528:⟩ 4510:ψ 4499:ℏ 4479:^ 4466:− 4455:⟩ 4437:ψ 4385:⟩ 4382:± 4341:⟩ 4338:− 4309:α 4299:⁡ 4290:⟩ 4258:α 4248:⁡ 4221:∑ 4213:⟩ 4179:∑ 4172:⟩ 4154:ψ 4127:⟩ 4099:∑ 4092:⟩ 4074:ψ 4032:δ 4015:Ω 4005:⁡ 3997:− 3980:α 3953:α 3932:⟩ 3920:ψ 3896:α 3886:⁡ 3880:− 3877:⟩ 3865:ψ 3841:α 3831:⁡ 3822:⟩ 3819:− 3789:⟩ 3777:ψ 3753:α 3743:⁡ 3734:⟩ 3722:ψ 3698:α 3688:⁡ 3679:⟩ 3645:⟩ 3642:± 3584:Ω 3571:δ 3559:δ 3547:Ω 3520:δ 3508:Ω 3504:ℏ 3491:± 3456:ω 3452:ℏ 3435:± 3336:ω 3330:− 3321:ω 3282:Ω 3255:Ω 3237:ω 3222:ω 3207:ℏ 3165:⟩ 3153:ψ 3135:^ 3111:ψ 3107:⟨ 3049:^ 3019:⟩ 3007:ψ 2995:⟩ 2983:ψ 2971:⁡ 2939:^ 2883:^ 2856:⟩ 2830:⟩ 2818:ψ 2792:⟩ 2772:⟩ 2760:ψ 2726:∈ 2703:⟩ 2683:⟩ 2634:^ 2594:ω 2590:− 2581:ω 2574:δ 2543:− 2536:^ 2533:σ 2524:† 2517:^ 2495:^ 2492:σ 2482:^ 2462:Ω 2459:ℏ 2436:^ 2433:σ 2424:δ 2421:ℏ 2402:^ 2369:^ 2366:σ 2351:^ 2340:† 2333:^ 2312:ω 2308:ℏ 2289:^ 2246:^ 2224:^ 2202:^ 2158:− 2151:^ 2148:σ 2139:† 2132:^ 2110:^ 2107:σ 2097:^ 2077:Ω 2074:ℏ 2051:^ 2048:σ 2033:ω 2029:ℏ 2020:^ 2009:† 2002:^ 1986:ω 1982:ℏ 1967:^ 1927:ω 1914:ω 1881:ω 1877:− 1868:ω 1835:ω 1822:ω 1774:ω 1761:ω 1757:− 1748:− 1738:− 1731:^ 1728:σ 1719:† 1712:^ 1685:ω 1672:ω 1668:− 1645:^ 1642:σ 1632:^ 1606:ω 1593:ω 1569:^ 1566:σ 1557:† 1550:^ 1523:ω 1510:ω 1500:− 1490:− 1483:^ 1480:σ 1470:^ 1450:Ω 1447:ℏ 1420:^ 1384:^ 1362:^ 1340:^ 1321:into the 1291:ω 1262:⟨ 1259:⟩ 1248:− 1237:⟨ 1234:⟩ 1211:^ 1208:σ 1172:⟨ 1169:⟩ 1153:− 1146:^ 1143:σ 1111:⟨ 1108:⟩ 1085:^ 1082:σ 1056:− 1049:^ 1046:σ 1027:^ 1024:σ 1011:^ 983:spin-half 963:ω 935:^ 907:† 900:^ 866:† 859:^ 843:^ 813:^ 748:^ 736:^ 721:Ω 718:ℏ 696:^ 668:^ 665:σ 650:ω 646:ℏ 627:^ 610:^ 599:† 592:^ 576:ω 572:ℏ 553:^ 513:^ 491:^ 469:^ 453:^ 20570:Category 20289:15489395 20113:36895794 19869:18633413 19816:56422948 19759:20208523 19708:15538363 19600:15538363 19555:10056806 19479:10034912 19246:10041728 19090:See also 19062:, where 14283:⟨ 14278:⟩ 14242:⟨ 14237:⟩ 14157:⟨ 14152:⟩ 14110:⟨ 14105:⟩ 14025:⟨ 14020:⟩ 13978:⟨ 13973:⟩ 8796:. Take 8428:are the 2904:for all 2715:, where 2614:detuning 1192:are the 34:require 20435:Bibcode 20269:Bibcode 20224:Bibcode 20189:Bibcode 20146:Bibcode 20093:Bibcode 20048:Bibcode 20025:9909174 20005:Bibcode 19982:9900058 19962:Bibcode 19919:Bibcode 19849:Bibcode 19796:Bibcode 19739:Bibcode 19716:4391322 19688:Bibcode 19645:Bibcode 19608:4391322 19580:Bibcode 19535:Bibcode 19459:Bibcode 19414:Bibcode 19369:Bibcode 19334:Bibcode 19286:Bibcode 19226:Bibcode 19185:Bibcode 18881:revival 18098:gives: 18052:around 14376:to be: 13851:as the 12638:, each 12226:number 11995:states 3619:is the 797:bosonic 381:or the 304:Walther 132:photons 115:bosonic 36:cleanup 20555: 20536: 20517: 20498: 20479: 20453: 20408: 20375: 20341: 20287: 20164: 20111: 20023: 19980: 19937: 19875: 19867: 19831:Nature 19814: 19757: 19714: 19706: 19680:Nature 19663: 19606: 19598: 19571:Nature 19553: 19477: 19432: 19387: 19304: 19244: 14876:vacuum 8306:where 7100:single 6777:where 3538:where 2270:where 238:where 136:cavity 91:photon 20451:S2CID 20285:S2CID 20259:arXiv 20162:S2CID 20136:arXiv 20109:S2CID 20083:arXiv 19935:S2CID 19909:arXiv 19873:S2CID 19839:arXiv 19812:S2CID 19786:arXiv 19712:S2CID 19661:S2CID 19635:arXiv 19604:S2CID 19302:S2CID 15064:are: 12224:total 12171:those 4078:field 3180:read 2566:with 1369:field 560:field 476:field 341:spins 300:Rempe 286:in a 284:maser 134:in a 120:atoms 20553:ISBN 20534:ISBN 20515:ISBN 20496:ISBN 20477:ISBN 20406:ISBN 20373:ISBN 20339:ISBN 20021:PMID 19978:PMID 19865:PMID 19755:PMID 19704:PMID 19596:PMID 19551:PMID 19475:PMID 19430:ISBN 19385:ISBN 19260:link 19242:PMID 17412:is: 17356:nor 16138:by 16104:and 14372:and 10931:by: 8367:and 7054:and 6386:and 5742:and 4408:> 3422:are 2968:span 2804:and 1391:atom 1131:and 991:with 954:and 921:and 634:atom 498:atom 362:> 182:and 161:and 153:and 97:The 86:atom 20443:doi 20277:doi 20232:doi 20197:doi 20154:doi 20132:144 20101:doi 20056:doi 20013:doi 19970:doi 19927:doi 19857:doi 19835:454 19804:doi 19747:doi 19696:doi 19684:432 19653:doi 19588:doi 19576:432 19543:doi 19467:doi 19422:doi 19377:doi 19342:doi 19294:doi 19282:140 19234:doi 19193:doi 19154:doi 18801:eff 18700:cos 18463:exp 18443:exp 18346:exp 18326:exp 17839:sin 17711:sin 17583:sin 17216:sin 17158:cos 16888:sin 16822:cos 15702:sin 15651:cos 15575:sin 15518:cos 15480:new 15444:or 15273:sin 15155:cos 14987:sin 14935:cos 14816:tan 14714:sin 14663:cos 14587:sin 14530:cos 13427:to 6064:cos 5992:cos 5946:sin 5841:sin 5551:sin 5508:cos 5364:sin 5187:sin 5020:sin 4962:cos 4653:sin 4559:cos 4514:tot 4441:tot 4296:sin 4245:cos 4158:tot 3993:tan 3883:cos 3828:sin 3740:sin 3685:cos 1427:int 827:ZPF 703:int 520:int 103:JCM 20572:: 20449:. 20441:. 20431:42 20429:. 20387:^ 20353:^ 20317:^ 20308:, 20283:. 20275:. 20267:. 20253:. 20230:. 20220:40 20218:. 20195:. 20185:46 20183:. 20160:. 20152:. 20144:. 20130:. 20107:. 20099:. 20091:. 20079:63 20077:. 20054:. 20044:71 20042:. 20019:. 20011:. 20001:47 19999:. 19976:. 19968:. 19958:37 19956:. 19933:. 19925:. 19917:. 19905:95 19903:. 19899:. 19885:^ 19871:. 19863:. 19855:. 19847:. 19833:. 19810:. 19802:. 19794:. 19782:88 19780:. 19776:. 19753:. 19745:. 19733:. 19710:. 19702:. 19694:. 19682:. 19659:. 19651:. 19643:. 19631:70 19629:. 19625:. 19602:. 19594:. 19586:. 19574:. 19549:. 19541:. 19531:73 19529:. 19517:^ 19473:. 19465:. 19455:58 19453:. 19428:. 19420:. 19410:20 19408:. 19383:. 19375:. 19365:20 19363:. 19340:. 19330:44 19328:. 19314:^ 19300:. 19292:. 19280:. 19268:^ 19256:}} 19252:{{ 19240:. 19232:. 19222:64 19220:. 19207:^ 19191:. 19181:46 19179:. 19175:. 19150:51 19148:. 19050:40 18578:25 14866:. 14364:. 6294:Tr 4841:JC 4486:JC 3989::= 3142:JC 3104::= 3056:JC 2946:JC 2833::= 2775::= 2409:II 2253:II 2209:JC 1974:JC 995:to 792:. 411:. 302:, 165:. 149:, 145:, 138:. 20561:. 20542:. 20523:. 20504:. 20485:. 20457:. 20445:: 20437:: 20414:. 20381:. 20347:. 20310:6 20291:. 20279:: 20271:: 20261:: 20255:7 20238:. 20234:: 20226:: 20203:. 20199:: 20191:: 20168:. 20156:: 20148:: 20138:: 20115:. 20103:: 20095:: 20085:: 20062:. 20058:: 20050:: 20027:. 20015:: 20007:: 19984:. 19972:: 19964:: 19941:. 19929:: 19921:: 19911:: 19879:. 19859:: 19851:: 19841:: 19818:. 19806:: 19798:: 19788:: 19761:. 19749:: 19741:: 19735:9 19718:. 19698:: 19690:: 19667:. 19655:: 19647:: 19637:: 19610:. 19590:: 19582:: 19557:. 19545:: 19537:: 19481:. 19469:: 19461:: 19438:. 19424:: 19416:: 19393:. 19379:: 19371:: 19348:. 19344:: 19336:: 19308:. 19296:: 19288:: 19262:) 19248:. 19236:: 19228:: 19201:. 19195:: 19187:: 19160:. 19156:: 19047:= 19042:2 19038:) 19034:g 19031:2 19027:/ 19020:( 19017:= 19014:a 18979:2 18975:/ 18971:t 18965:= 18962:t 18959:g 18932:. 18924:n 18907:4 18901:= 18896:r 18858:2 18852:= 18847:c 18814:n 18806:= 18768:8 18764:/ 18758:2 18754:t 18748:2 18736:e 18731:] 18727:t 18716:n 18707:[ 18692:2 18689:1 18679:2 18676:1 18668:) 18665:t 18662:( 18657:e 18653:P 18625:n 18612:2 18608:t 18575:= 18569:n 18546:t 18543:g 18516:) 18511:] 18506:) 18495:n 18487:2 18482:t 18476:i 18470:( 18457:n 18450:[ 18438:2 18434:/ 18430:t 18419:n 18411:i 18407:e 18403:+ 18399:] 18394:) 18382:n 18374:2 18369:t 18363:i 18353:( 18340:n 18333:[ 18321:2 18317:/ 18313:t 18302:n 18294:i 18287:e 18282:( 18273:4 18265:n 18255:e 18244:2 18241:1 18233:) 18230:t 18227:( 18222:e 18218:P 18196:) 18186:n 18175:n 18166:n 18158:2 18155:1 18150:+ 18147:1 18143:( 18134:n 18124:2 18111:n 18086:n 18063:n 18038:n 18010:n 18003:/ 17999:1 17991:2 17983:n 17976:/ 17967:2 17963:) 17959:n 17953:( 17924:2 17919:| 17910:| 17906:= 17900:n 17867:) 17864:t 17859:n 17851:( 17843:2 17832:! 17829:n 17822:n 17814:n 17803:n 17793:e 17779:1 17776:= 17773:n 17765:= 17754:) 17750:t 17742:n 17734:2 17731:1 17724:( 17715:2 17704:! 17701:n 17694:n 17686:n 17675:n 17665:e 17651:1 17648:= 17645:n 17637:= 17626:) 17622:t 17614:n 17606:2 17603:1 17596:( 17587:2 17577:n 17574:2 17569:| 17560:| 17553:! 17550:n 17542:2 17537:| 17528:| 17520:e 17507:1 17504:= 17501:n 17493:= 17484:2 17479:| 17472:) 17469:t 17466:( 17459:| 17455:e 17448:| 17443:= 17440:) 17437:t 17434:( 17429:e 17425:P 17400:t 17378:0 17368:H 17342:) 17339:n 17336:( 17326:I 17317:2 17311:c 17275:) 17268:1 17262:n 17259:, 17256:e 17252:| 17247:) 17244:2 17240:/ 17236:t 17228:n 17223:( 17213:i 17204:n 17201:, 17198:g 17194:| 17189:) 17186:2 17182:/ 17178:t 17170:n 17165:( 17154:( 17148:t 17143:c 17135:n 17132:i 17125:e 17118:! 17115:n 17109:n 17097:2 17093:/ 17087:2 17082:| 17073:| 17065:e 17054:1 17051:= 17048:n 17040:+ 17034:0 17031:, 17028:g 17024:| 17018:2 17014:/ 17008:2 17003:| 16994:| 16986:e 16982:= 16969:n 16966:, 16963:g 16959:| 16954:) 16948:) 16945:n 16942:( 16937:x 16919:) 16916:2 16912:/ 16908:t 16900:n 16895:( 16885:i 16877:) 16874:n 16871:( 16861:I 16853:) 16850:2 16846:/ 16842:t 16834:n 16829:( 16818:( 16812:t 16807:c 16799:n 16796:i 16789:e 16782:! 16779:n 16773:n 16761:2 16757:/ 16751:2 16746:| 16737:| 16729:e 16718:1 16715:= 16712:n 16704:+ 16698:0 16695:, 16692:g 16688:| 16682:2 16678:/ 16672:2 16667:| 16658:| 16650:e 16646:= 16633:) 16630:0 16627:( 16620:| 16610:/ 16606:t 16601:n 16591:H 16584:i 16577:e 16573:= 16563:) 16560:t 16557:( 16550:| 16521:) 16518:n 16515:( 16510:x 16491:2 16482:n 16471:+ 16466:) 16463:n 16460:( 16450:I 16442:) 16436:2 16433:1 16428:+ 16425:n 16421:( 16417:= 16412:n 16402:H 16378:0 16375:= 16347:n 16344:, 16341:g 16337:| 16330:! 16327:n 16321:n 16309:2 16305:/ 16299:2 16294:| 16285:| 16277:e 16266:0 16263:= 16260:n 16252:= 16243:, 16240:g 16236:| 16232:= 16226:) 16223:0 16220:( 16213:| 16178:n 16148:n 16123:n 16120:, 16117:g 16113:| 16089:1 16083:n 16080:, 16077:e 16073:| 16040:n 16035:= 16032:E 16006:) 15999:1 15993:n 15990:, 15987:e 15983:| 15973:n 15970:, 15967:g 15963:| 15958:( 15951:2 15947:1 15942:= 15933:, 15930:n 15926:| 15905:0 15902:= 15889:c 15881:= 15876:g 15873:e 15844:2 15836:n 15833:+ 15828:2 15815:= 15806:E 15797:+ 15793:E 15789:= 15786:E 15750:1 15744:n 15741:, 15738:e 15734:| 15729:) 15724:2 15719:n 15709:( 15693:n 15690:, 15687:g 15683:| 15678:) 15673:2 15668:n 15658:( 15648:= 15635:, 15632:n 15628:| 15617:n 15614:, 15611:g 15607:| 15602:) 15597:2 15592:n 15582:( 15572:+ 15566:1 15560:n 15557:, 15554:e 15550:| 15545:) 15540:2 15535:n 15525:( 15515:= 15505:+ 15502:, 15499:n 15495:| 15463:n 15460:, 15457:g 15453:| 15429:1 15423:n 15420:, 15417:e 15413:| 15382:2 15373:n 15365:= 15360:n 15335:n 15304:) 15299:2 15295:t 15286:( 15277:2 15269:= 15264:2 15259:| 15251:) 15248:t 15245:( 15238:| 15234:1 15231:, 15228:g 15221:| 15217:= 15210:) 15207:t 15204:( 15199:g 15195:P 15186:) 15181:2 15177:t 15168:( 15159:2 15151:= 15146:2 15141:| 15133:) 15130:t 15127:( 15120:| 15116:0 15113:, 15110:e 15103:| 15099:= 15092:) 15089:t 15086:( 15081:e 15077:P 15052:t 15027:1 15024:, 15021:g 15017:| 15012:) 15007:2 15003:t 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:= 14517:+ 14514:, 14511:n 14507:| 14495:2 14487:n 14484:+ 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:= 13651:] 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 12447:0 12440:2 12430:H 12421:0 12416:0 12394:0 12389:0 12382:1 12372:H 12363:0 12341:0 12336:0 12331:0 12324:0 12320:H 12313:[ 12308:= 12303:C 12300:J 12290:H 12260:N 12234:n 12204:N 12181:n 12156:} 12149:n 12145:| 12140:{ 12118:} 12111:n 12108:, 12105:g 12101:| 12097:, 12091:1 12085:n 12082:, 12079:e 12075:| 12071:; 12065:; 12059:1 12056:, 12053:g 12049:| 12045:, 12039:0 12036:, 12033:e 12029:| 12025:; 12019:0 12016:, 12013:g 12009:| 12004:{ 11973:0 11970:= 11959:) 11929:c 11919:a 11883:c 11873:a 11866:+ 11861:+ 11842:c 11832:a 11825:+ 11820:+ 11801:c 11791:a 11780:( 11774:c 11770:g 11763:= 11752:) 11728:] 11722:c 11712:a 11698:c 11688:a 11681:, 11671:c 11661:a 11653:[ 11649:+ 11645:] 11640:| 11636:e 11627:e 11623:| 11619:, 11596:[ 11585:c 11575:a 11568:+ 11563:+ 11545:] 11539:c 11529:a 11515:c 11505:a 11498:, 11493:c 11483:a 11475:[ 11471:+ 11467:] 11462:| 11458:e 11449:e 11445:| 11441:, 11436:+ 11418:[ 11412:c 11402:a 11394:( 11388:c 11384:g 11377:= 11366:) 11361:] 11355:c 11345:a 11331:c 11321:a 11314:+ 11310:| 11306:e 11297:e 11293:| 11289:, 11260:c 11250:a 11242:[ 11238:+ 11234:] 11228:c 11218:a 11204:c 11194:a 11187:+ 11183:| 11179:e 11170:e 11166:| 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 10891:a 10865:+ 10860:c 10850:a 10841:+ 10823:( 10817:c 10813:g 10806:+ 10802:| 10798:e 10789:e 10785:| 10779:g 10776:e 10765:+ 10760:c 10750:a 10736:c 10726:a 10717:c 10706:= 10701:C 10698:J 10688:H 10658:c 10645:2 10642:1 10619:) 10608:c 10598:a 10572:+ 10567:c 10557:a 10548:+ 10530:( 10524:c 10520:g 10513:+ 10509:| 10505:e 10496:e 10492:| 10486:g 10483:e 10472:+ 10468:) 10462:2 10459:1 10454:+ 10449:c 10439:a 10425:c 10415:a 10407:( 10401:c 10390:= 10385:C 10382:J 10372:H 10340:) 10329:c 10319:a 10293:+ 10288:c 10278:a 10269:+ 10251:( 10245:c 10241:g 10234:= 10225:/ 10221:t 10216:0 10206:H 10199:i 10195:e 10191:) 10188:t 10185:( 10180:F 10177:A 10167:H 10154:/ 10150:t 10145:0 10135:H 10128:i 10121:e 10117:= 10112:F 10109:A 10099:H 10071:c 10067:g 10043:) 10037:t 10034:) 10029:c 10016:g 10013:e 10005:( 10002:i 9995:e 9984:c 9974:a 9948:+ 9943:t 9940:) 9935:c 9922:g 9919:e 9911:( 9908:i 9904:e 9898:c 9888:a 9879:+ 9861:( 9855:c 9851:g 9841:= 9838:) 9835:t 9832:( 9827:F 9824:A 9814:H 9786:c 9782:g 9745:c 9737:2 9729:2 9704:c 9691:g 9688:e 9674:, 9662:2 9656:= 9631:c 9623:2 9615:c 9607:+ 9602:g 9599:e 9574:0 9566:c 9553:g 9550:e 9523:c 9515:+ 9510:g 9507:e 9495:| 9489:c 9476:g 9473:e 9464:| 9442:) 9436:t 9433:) 9428:c 9415:g 9412:e 9404:( 9401:i 9397:e 9391:c 9381:a 9372:+ 9353:c 9349:g 9340:t 9337:) 9332:c 9319:g 9316:e 9308:( 9305:i 9298:e 9287:c 9277:a 9244:c 9240:g 9231:t 9228:) 9223:g 9220:e 9212:+ 9207:c 9199:( 9196:i 9189:e 9183:c 9173:a 9140:c 9136:g 9132:+ 9127:t 9124:) 9119:g 9116:e 9108:+ 9103:c 9095:( 9092:i 9088:e 9077:c 9067:a 9058:+ 9039:c 9035:g 9030:( 9023:= 9014:/ 9010:t 9005:0 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 8708:g 8704:+ 8694:c 8684:a 8675:+ 8656:c 8652:g 8644:( 8640:+ 8636:) 8625:c 8615:a 8582:c 8578:g 8569:c 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:, 8200:k 8189:a 8180:+ 8158:, 8154:k 8149:g 8132:, 8128:k 8117:a 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:= 6443:H 6374:) 6371:0 6368:( 6334:] 6319:) 6316:t 6313:( 6298:[ 6290:= 6285:t 6246:) 6243:t 6240:( 6231:U 6225:) 6222:0 6219:( 6204:) 6201:t 6198:( 6183:U 6176:= 6173:) 6170:t 6167:( 6135:) 6132:t 6129:( 6087:, 6070:t 6054:a 6048:= 6035:a 6024:2 6020:g 6016:+ 5998:t 5985:, 5952:t 5933:a 5927:= 5914:a 5902:2 5898:g 5894:+ 5873:2 5869:g 5865:+ 5847:t 5808:U 5773:= 5770:g 5750:g 5730:4 5726:/ 5720:2 5712:+ 5703:a 5685:a 5676:2 5672:g 5668:= 5597:) 5590:) 5557:t 5545:2 5541:/ 5534:i 5531:+ 5514:t 5504:( 5497:) 5491:2 5488:1 5474:a 5456:a 5448:( 5444:t 5439:c 5431:i 5424:e 5406:a 5370:t 5355:) 5349:2 5346:1 5332:a 5314:a 5306:( 5302:t 5297:c 5289:i 5282:e 5278:g 5275:i 5259:a 5247:2 5243:g 5239:+ 5218:2 5214:g 5210:+ 5193:t 5178:) 5172:2 5169:1 5164:+ 5155:a 5137:a 5129:( 5125:t 5120:c 5112:i 5105:e 5101:g 5098:i 5089:) 5080:2 5076:g 5072:+ 5051:2 5047:g 5043:+ 5026:t 5014:2 5010:/ 5003:i 4993:2 4989:g 4985:+ 4968:t 4958:( 4951:) 4945:2 4942:1 4937:+ 4928:a 4910:a 4902:( 4898:t 4893:c 4885:i 4878:e 4871:( 4866:= 4850:/ 4846:t 4831:H 4824:i 4817:e 4813:= 4806:) 4803:t 4800:( 4791:U 4749:. 4745:] 4735:/ 4731:t 4728:) 4725:n 4722:( 4713:E 4709:i 4702:e 4692:, 4689:n 4685:| 4680:) 4675:2 4670:n 4660:( 4650:+ 4641:/ 4637:t 4634:) 4631:n 4628:( 4623:+ 4619:E 4615:i 4608:e 4601:+ 4598:, 4595:n 4591:| 4586:) 4581:2 4576:n 4566:( 4555:[ 4549:n 4545:C 4539:n 4531:= 4525:) 4522:0 4519:( 4505:| 4495:/ 4491:t 4476:H 4469:i 4462:e 4458:= 4452:) 4449:t 4446:( 4432:| 4411:0 4405:t 4379:, 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 3157:j 3148:| 3132:H 3124:| 3118:n 3115:i 3099:) 3096:n 3093:( 3088:j 3085:i 3080:H 3046:H 3022:} 3014:n 3011:2 3002:| 2998:, 2990:n 2987:1 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:= 2549:) 2514:a 2507:+ 2502:+ 2479:a 2472:( 2466:2 2453:+ 2448:2 2443:z 2418:= 2399:H 2387:) 2381:2 2376:z 2357:+ 2348:a 2330:a 2322:( 2316:c 2305:= 2296:I 2286:H 2258:, 2243:H 2236:+ 2231:I 2221:H 2214:= 2199:H 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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Index