傑恩斯-卡明斯模型

通過把薛丁格繪景轉換為相互作用繪景(又名旋轉框架(rotating frame)) ，使得${\displaystyle {\begin{smallmatrix}H_{0}={\hat {H}}_{\text{field}}+{\hat {H}}_{\text{atom}}\end{smallmatrix}}}$，可以得到：

${\displaystyle {\hat {H}}_{\text{int}}(t)={\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{-}e^{-i(\omega _{c}+\omega _{a})t}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{+}e^{i(\omega _{c}+\omega _{a})t}+{\hat {a}}{\hat {\sigma }}_{+}e^{i(-\omega _{c}+\omega _{a})t}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}e^{-i(-\omega _{c}+\omega _{a})t}\right).}$

這個哈密頓量同時包含了兩個部分：

• ${\displaystyle {\begin{smallmatrix}(\omega _{c}+\omega _{a})\end{smallmatrix}}}$ 是快速震蕩，
• ${\displaystyle {\begin{smallmatrix}(\omega _{c}-\omega _{a})\end{smallmatrix}}}$ 是慢速震蕩。

為了求解這個方程，簡化模型是再所難免的。注意到，當 ${\displaystyle {\begin{smallmatrix}|\omega _{c}-\omega _{a}|\ll \omega _{c}+\omega _{a}\end{smallmatrix}}}$ 的時候，快速振盪的 “反向旋轉”項（也就是快速震蕩項）可被忽略，這被稱為旋波近似。再將之轉換回薛丁格繪景，JCM哈密頓量就變成了：

${\displaystyle {\hat {H}}_{\text{JC}}=\hbar \omega _{c}{\hat {a}}^{\dagger }{\hat {a}}+\hbar \omega _{a}{\frac {{\hat {\sigma }}_{z}}{2}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right).}$

其中，

• ${\displaystyle {\begin{smallmatrix}\hbar \Omega /2=d(\omega _{a}/\hbar V\epsilon _{0})^{1/2}\end{smallmatrix}}}$是原子場的耦合常數，
• ${\displaystyle {\begin{smallmatrix}d\end{smallmatrix}}}$是原子躍遷時刻，
• ${\displaystyle {\begin{smallmatrix}V\end{smallmatrix}}}$是腔模的體積。

一般情況下，將哈密頓量拆分為2部分有助於對其進行求解:

${\displaystyle {\hat {H}}_{\text{JC}}={\hat {H}}_{I}+{\hat {H}}_{II},}$

其中，

${\displaystyle {\begin{array}{lcl}{\hat {H}}_{I}&=&\hbar \omega _{c}\left({\hat {a}}^{\dagger }{\hat {a}}+{\frac {{\hat {\sigma }}_{z}}{2}}\right)\\{\hat {H}}_{II}&=&\hbar \delta {\frac {{\hat {\sigma }}_{z}}{2}}+{\frac {\hbar \Omega }{2}}\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right)\end{array}}}$

${\displaystyle {\begin{smallmatrix}\delta =\omega _{a}-\omega _{c}\end{smallmatrix}}}$ 稱之為場與雙態系統的失諧量（頻率）。

為了更好地求解哈密頓量，把${\displaystyle {\begin{smallmatrix}{\begin{smallmatrix}{\hat {H}}_{I}\end{smallmatrix}}\end{smallmatrix}}}$的本徵態轉換成張量積 ${\displaystyle {\begin{smallmatrix}|n,g\rangle ,|n,e\rangle \end{smallmatrix}}}$（${\displaystyle {\begin{smallmatrix}n\in \mathbb {N} \end{smallmatrix}}}$，表示模型中輻射量子的數量。）

對位任意正整數n，狀態${\displaystyle {\begin{smallmatrix}|\psi _{1n}\rangle$ :=|n,e\rangle \end{smallmatrix}}} 與狀態${\displaystyle {\begin{smallmatrix}|\psi _{2n}\rangle$ :=|n+1,g\rangle \end{smallmatrix}}} 會退化為${\displaystyle {\begin{smallmatrix}{\hat {H}}_{I}\end{smallmatrix}}}$ ，${\displaystyle {\begin{smallmatrix}{\hat {H}}_{\text{JC}}\end{smallmatrix}}}$ 足以在子空間${\displaystyle {\begin{smallmatrix}{\text{span}}\{|\psi _{1n}\rangle ,|\psi _{2n}\rangle \}\end{smallmatrix}}}$對角化。 ${\displaystyle {\begin{smallmatrix}{\hat {H}}_{\text{JC}}\end{smallmatrix}}}$的元素屬於${\displaystyle {\begin{smallmatrix}{H}_{ij}^{(n)}:=\langle \psi _{in}|{\hat {H}}_{\text{JC}}|\psi _{jn}\rangle \end{smallmatrix}}}$的子空間，表示為：

${\displaystyle H^{(n)}=\hbar {\begin{pmatrix}n\omega _{c}+{\frac {\omega _{a}}{2}}&{\frac {\Omega }{2}}{\sqrt {n+1}}\\[8pt]{\frac {\Omega }{2}}{\sqrt {n+1}}&(n+1)\omega _{c}-{\frac {\omega _{a}}{2}}\end{pmatrix}}}$

對於任意正整數n，能量本徵態${\textstyle {\begin{smallmatrix}H^{(n)}\end{smallmatrix}}}$為：

${\displaystyle E_{\pm }(n)=\hbar \omega _{c}\left(n+{\frac {1}{2}}\right)\pm {\frac {1}{2}}\hbar \Omega _{n}(\delta ),}$

其中，${\displaystyle {\begin{smallmatrix}\Omega _{n}(\delta )={\sqrt {\delta ^{2}+\Omega ^{2}(n+1)}}\end{smallmatrix}}}$ 是拉比頻率特殊的失諧參數。

含能量本徵態 ${\displaystyle {\begin{smallmatrix}|n,\pm \rangle ~\end{smallmatrix}}}$的特徵值是：

${\displaystyle |n,+\rangle =\cos \left({\frac {\alpha _{n}}{2}}\right)|\psi _{1n}\rangle +\sin \left({\frac {\alpha _{n}}{2}}\right)|\psi _{2n}\rangle }$

${\displaystyle |n,-\rangle =-\sin \left({\frac {\alpha _{n}}{2}}\right)|\psi _{1n}\rangle +\cos \left({\frac {\alpha _{n}}{2}}\right)|\psi _{2n}\rangle }$

其中，${\displaystyle {\begin{smallmatrix}\angle \alpha _{n}=\tan ^{-1}\left({\frac {\Omega {\sqrt {n+1}}}{\delta }}\right)\end{smallmatrix}}}$

為了得到動量的一般情況。 首先考慮一個場疊加態的初態 ${\displaystyle {\begin{smallmatrix}~|\psi _{\text{field}}(0)\rangle =\sum _{n}{C_{n}|n\rangle }~\end{smallmatrix}}}$，若置一激發態原子于場內，則系統初態為：

${\displaystyle |\psi _{\text{tot}}(0)\rangle =\sum _{n}C_{n}\left[\cos \left({\frac {\alpha _{n}}{2}}\right)|n,+\rangle -\sin \left({\frac {\alpha _{n}}{2}}\right)|n,-\rangle \right].}$

其中 ${\displaystyle {\begin{smallmatrix}~|n,\pm \rangle ~\end{smallmatrix}}}$ 是該系統的定態, 含時狀態向量是：

${\displaystyle |\psi _{\text{tot}}(t)\rangle =e^{-i{\hat {H}}_{\text{JC}}t/\hbar }|\psi _{\text{tot}}(0)\rangle =\sum _{n}C_{n}\left[\cos \left({\frac {\alpha _{n}}{2}}\right)|n,+\rangle e^{-iE_{+}(n)t/\hbar }-\sin \left({\frac {\alpha _{n}}{2}}\right)|n,-\rangle e^{-iE_{-}(n)t/\hbar }\right],t>0}$

可以直接通過海森堡記法（Heisenberg notation）來確定么正演化算符（unitary evolution operator） :[1]

${\displaystyle {\begin{matrix}{\begin{aligned}{\hat {U}}(t)&=e^{-i{\hat {H}}_{\text{JC}}t/\hbar }\\&={\begin{pmatrix}e^{-i\omega _{c}t({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}})}\left(\cos t{\sqrt {{\hat {\varphi }}+g^{2}}}-i\delta /2{\frac {\sin t{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\right)&-ige^{-i\omega _{c}t({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}})}{\frac {\sin t{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\,{\hat {a}}\\-ige^{-i\omega _{c}t({\hat {a}}^{\dagger }{\hat {a}}-{\frac {1}{2}})}{\frac {\sin t{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}}{\hat {a}}^{\dagger }&e^{-i\omega _{c}t({\hat {a}}^{\dagger }{\hat {a}}-{\frac {1}{2}})}\left(\cos t{\sqrt {\hat {\varphi }}}+i\delta /2{\frac {\sin t{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}}\right)\end{pmatrix}}\end{aligned}}\end{matrix}}}$

其中，定義算符${\displaystyle ~{\hat {\varphi }}~}$為

${\displaystyle {\hat {\varphi }}=g^{2}{\hat {a}}^{\dagger }{\hat {a}}+\delta ^{2}/4}$

${\displaystyle ~{\hat {U}}~}$的么正(unitary )被恆等定義：

${\displaystyle {\frac {\sin t\,{\sqrt {{\hat {\varphi }}+g^{2}}}}{\sqrt {{\hat {\varphi }}+g^{2}}}}\;{\hat {a}}={\hat {a}}\;{\frac {\sin t\,{\sqrt {\hat {\varphi }}}}{\sqrt {\hat {\varphi }}}},}$

${\displaystyle \cos t\,{\sqrt {{\hat {\varphi }}+g^{2}}}\;{\hat {a}}={\hat {a}}\;\cos t{\sqrt {\hat {\varphi }}},}$

么正算符可以計算被密度矩陣${\displaystyle ~{\hat {\rho }}(t)~}$所描述的含時系統狀態的演變，么正算符包含了所有可觀測量。給定初態${\displaystyle ~{\hat {\rho }}(0)~}$，則有：

${\displaystyle {\hat {\rho }}(t)={\hat {U}}^{\dagger }(t){\hat {\rho }}(0){\hat {U}}(t)}$，

${\displaystyle \langle {\hat {\Theta }}\rangle _{t}={\text{Tr}}[{\hat {\rho }}(t){\hat {\Theta }}]}$ ，

其中，${\displaystyle ~{\hat {\Theta }}~}$ 是表示可觀測量的算符。