系统哈密顿量
H
^
=
H
^
field
+
H
^
atom
+
H
^
int
{\displaystyle {\hat {H}}={\hat {H}}_{\text{field}}+{\hat {H}}_{\text{atom}}+{\hat {H}}_{\text{int}}}
由自由场哈密顿量，原子激发态哈密顿量，JCM哈密顿量组成：
H
^
field
=
ℏ
ω
c
a
^
†
a
^
H
^
atom
=
ℏ
ω
a
σ
^
z
2
H
^
int
=
ℏ
Ω
2
E
^
S
^
.
{\displaystyle {\begin{array}{lcl}{\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{array}}}
为方便起见，设真空场能量为
0
{\displaystyle 0}
其中：
E
^
=
a
^
+
a
^
†
{\displaystyle {\begin{smallmatrix}{\hat {E}}={\hat {a}}+{\hat {a}}^{\dagger }\end{smallmatrix}}}
场运算符 ，目的是把量化辐射场 转化为玻色子 的模型，另外双态原子 是能被三维布洛赫球面 所描述的半自旋 粒子
a
^
†
{\displaystyle {\begin{smallmatrix}{\hat {a}}^{\dagger }\end{smallmatrix}}}
创生算符
a
^
{\displaystyle {\begin{smallmatrix}{\hat {a}}\end{smallmatrix}}}
湮灭算符
S
^
=
σ
^
+
+
σ
^
−
{\displaystyle {\begin{smallmatrix}{\hat {S}}={\hat {\sigma }}_{+}+{\hat {\sigma }}_{-}\end{smallmatrix}}}
偏振 运算符
σ
^
+
=
|
e
⟩
⟨
g
|
{\displaystyle {\begin{smallmatrix}{\hat {\sigma }}_{+}=|e\rangle \langle g|\end{smallmatrix}}}
σ
^
−
=
|
g
⟩
⟨
e
|
{\displaystyle {\begin{smallmatrix}{\hat {\sigma }}_{-}=|g\rangle \langle e|\end{smallmatrix}}}
阶梯算符
σ
^
z
=
|
e
⟩
⟨
e
|
−
|
g
⟩
⟨
g
|
{\displaystyle {\begin{smallmatrix}{\hat {\sigma }}_{z}=|e\rangle \langle e|-|g\rangle \langle g|\end{smallmatrix}}}
反转运算 符
ω
a
{\displaystyle {\begin{smallmatrix}\omega _{a}\end{smallmatrix}}}
ω
c
{\displaystyle {\begin{smallmatrix}\omega _{c}\end{smallmatrix}}}
JCM哈密顿量
编辑
通过把薛定谔绘景 转换为相互作用绘景 (又名旋转框架(rotating frame)) ，使得
H
0
=
H
^
field
+
H
^
atom
{\displaystyle {\begin{smallmatrix}H_{0}={\hat {H}}_{\text{field}}+{\hat {H}}_{\text{atom}}\end{smallmatrix}}}
H
^
int
(
t
)
=
ℏ
Ω
2
(
a
^
σ
^
−
e
−
i
(
ω
c
+
ω
a
)
t
+
a
^
†
σ
^
+
e
i
(
ω
c
+
ω
a
)
t
+
a
^
σ
^
+
e
i
(
−
ω
c
+
ω
a
)
t
+
a
^
†
σ
^
−
e
−
i
(
−
ω
c
+
ω
a
)
t
)
.
{\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).}
这个哈密顿量同时包含了两个部分：
(
ω
c
+
ω
a
)
{\displaystyle {\begin{smallmatrix}(\omega _{c}+\omega _{a})\end{smallmatrix}}}
(
ω
c
−
ω
a
)
{\displaystyle {\begin{smallmatrix}(\omega _{c}-\omega _{a})\end{smallmatrix}}}
为了求解这个方程，简化模型是再所难免的。注意到，当
|
ω
c
−
ω
a
|
≪
ω
c
+
ω
a
{\displaystyle {\begin{smallmatrix}|\omega _{c}-\omega _{a}|\ll \omega _{c}+\omega _{a}\end{smallmatrix}}}
旋波近似 。再将之转换回薛定谔绘景，JCM哈密顿量就变成了：
H
^
JC
=
ℏ
ω
c
a
^
†
a
^
+
ℏ
ω
a
σ
^
z
2
+
ℏ
Ω
2
(
a
^
σ
^
+
+
a
^
†
σ
^
−
)
.
{\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，
ℏ
Ω
/
2
=
d
(
ω
a
/
ℏ
V
ϵ
0
)
1
/
2
{\displaystyle {\begin{smallmatrix}\hbar \Omega /2=d(\omega _{a}/\hbar V\epsilon _{0})^{1/2}\end{smallmatrix}}}
d
{\displaystyle {\begin{smallmatrix}d\end{smallmatrix}}}
V
{\displaystyle {\begin{smallmatrix}V\end{smallmatrix}}}
一般情况下，将哈密顿量拆分为2部分有助于对其进行求解:
H
^
JC
=
H
^
I
+
H
^
I
I
,
{\displaystyle {\hat {H}}_{\text{JC}}={\hat {H}}_{I}+{\hat {H}}_{II},}
其中，
H
^
I
=
ℏ
ω
c
(
a
^
†
a
^
+
σ
^
z
2
)
H
^
I
I
=
ℏ
δ
σ
^
z
2
+
ℏ
Ω
2
(
a
^
σ
^
+
+
a
^
†
σ
^
−
)
{\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}}}
δ
=
ω
a
−
ω
c
{\displaystyle {\begin{smallmatrix}\delta =\omega _{a}-\omega _{c}\end{smallmatrix}}}
为了更好地求解哈密顿量，把
H
^
I
{\displaystyle {\begin{smallmatrix}{\begin{smallmatrix}{\hat {H}}_{I}\end{smallmatrix}}\end{smallmatrix}}}
本征态 转换成张量积
|
n
,
g
⟩
,
|
n
,
e
⟩
{\displaystyle {\begin{smallmatrix}|n,g\rangle ,|n,e\rangle \end{smallmatrix}}}
n
∈
N
{\displaystyle {\begin{smallmatrix}n\in \mathbb {N} \end{smallmatrix}}}
对位任意正整数n，状态
|
ψ
1
n
⟩
:=
|
n
,
e
⟩
{\displaystyle {\begin{smallmatrix}|\psi _{1n}\rangle :=|n,e\rangle \end{smallmatrix}}}
|
ψ
2
n
⟩
:=
|
n
+
1
,
g
⟩
{\displaystyle {\begin{smallmatrix}|\psi _{2n}\rangle :=|n+1,g\rangle \end{smallmatrix}}}
H
^
I
{\displaystyle {\begin{smallmatrix}{\hat {H}}_{I}\end{smallmatrix}}}
H
^
JC
{\displaystyle {\begin{smallmatrix}{\hat {H}}_{\text{JC}}\end{smallmatrix}}}
span
{
|
ψ
1
n
⟩
,
|
ψ
2
n
⟩
}
{\displaystyle {\begin{smallmatrix}{\text{span}}\{|\psi _{1n}\rangle ,|\psi _{2n}\rangle \}\end{smallmatrix}}}
H
^
JC
{\displaystyle {\begin{smallmatrix}{\hat {H}}_{\text{JC}}\end{smallmatrix}}}
H
i
j
(
n
)
:=
⟨
ψ
i
n
|
H
^
JC
|
ψ
j
n
⟩
{\displaystyle {\begin{smallmatrix}{H}_{ij}^{(n)}:=\langle \psi _{in}|{\hat {H}}_{\text{JC}}|\psi _{jn}\rangle \end{smallmatrix}}}
H
(
n
)
=
ℏ
(
n
ω
c
+
ω
a
2
Ω
2
n
+
1
Ω
2
n
+
1
(
n
+
1
)
ω
c
−
ω
a
2
)
{\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}}}
对于任意正整数n，能量本征态
H
(
n
)
{\textstyle {\begin{smallmatrix}H^{(n)}\end{smallmatrix}}}
E
±
(
n
)
=
ℏ
ω
c
(
n
+
1
2
)
±
1
2
ℏ
Ω
n
(
δ
)
,
{\displaystyle E_{\pm }(n)=\hbar \omega _{c}\left(n+{\frac {1}{2}}\right)\pm {\frac {1}{2}}\hbar \Omega _{n}(\delta ),}
其中，
Ω
n
(
δ
)
=
δ
2
+
Ω
2
(
n
+
1
)
{\displaystyle {\begin{smallmatrix}\Omega _{n}(\delta )={\sqrt {\delta ^{2}+\Omega ^{2}(n+1)}}\end{smallmatrix}}}
拉比频率 特殊的失谐 参数。
含能量本征态
|
n
,
±
⟩
{\displaystyle {\begin{smallmatrix}|n,\pm \rangle ~\end{smallmatrix}}}
特征值 是：
|
n
,
+
⟩
=
cos
(
α
n
2
)
|
ψ
1
n
⟩
+
sin
(
α
n
2
)
|
ψ
2
n
⟩
{\displaystyle |n,+\rangle =\cos \left({\frac {\alpha _{n}}{2}}\right)|\psi _{1n}\rangle +\sin \left({\frac {\alpha _{n}}{2}}\right)|\psi _{2n}\rangle }
|
n
,
−
⟩
=
−
sin
(
α
n
2
)
|
ψ
1
n
⟩
+
cos
(
α
n
2
)
|
ψ
2
n
⟩
{\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，
∠
α
n
=
tan
−
1
(
Ω
n
+
1
δ
)
{\displaystyle {\begin{smallmatrix}\angle \alpha _{n}=\tan ^{-1}\left({\frac {\Omega {\sqrt {n+1}}}{\delta }}\right)\end{smallmatrix}}}
薛定谔绘景动量
编辑
为了得到动量的一般情况。 首先考虑一个场叠加态的初态
|
ψ
field
(
0
)
⟩
=
∑
n
C
n
|
n
⟩
{\displaystyle {\begin{smallmatrix}~|\psi _{\text{field}}(0)\rangle =\sum _{n}{C_{n}|n\rangle }~\end{smallmatrix}}}
|
ψ
tot
(
0
)
⟩
=
∑
n
C
n
[
cos
(
α
n
2
)
|
n
,
+
⟩
−
sin
(
α
n
2
)
|
n
,
−
⟩
]
.
{\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
|
n
,
±
⟩
{\displaystyle {\begin{smallmatrix}~|n,\pm \rangle ~\end{smallmatrix}}}
|
ψ
tot
(
t
)
⟩
=
e
−
i
H
^
JC
t
/
ℏ
|
ψ
tot
(
0
)
⟩
=
∑
n
C
n
[
cos
(
α
n
2
)
|
n
,
+
⟩
e
−
i
E
+
(
n
)
t
/
ℏ
−
sin
(
α
n
2
)
|
n
,
−
⟩
e
−
i
E
−
(
n
)
t
/
ℏ
]
,
t
>
0
{\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}
相互作用绘景动量
编辑
可以直接通过海森堡 记法（Heisenberg notation）来确定幺正演化算符（unitary evolution operator） :[1]
U
^
(
t
)
=
e
−
i
H
^
JC
t
/
ℏ
=
(
e
−
i
ω
c
t
(
a
^
†
a
^
+
1
2
)
(
cos
t
φ
^
+
g
2
−
i
δ
/
2
sin
t
φ
^
+
g
2
φ
^
+
g
2
)
−
i
g
e
−
i
ω
c
t
(
a
^
†
a
^
+
1
2
)
sin
t
φ
^
+
g
2
φ
^
+
g
2
a
^
−
i
g
e
−
i
ω
c
t
(
a
^
†
a
^
−
1
2
)
sin
t
φ
^
φ
^
a
^
†
e
−
i
ω
c
t
(
a
^
†
a
^
−
1
2
)
(
cos
t
φ
^
+
i
δ
/
2
sin
t
φ
^
φ
^
)
)
{\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，定义算符
φ
^
{\displaystyle ~{\hat {\varphi }}~}
φ
^
=
g
2
a
^
†
a
^
+
δ
2
/
4
{\displaystyle {\hat {\varphi }}=g^{2}{\hat {a}}^{\dagger }{\hat {a}}+\delta ^{2}/4}
U
^
{\displaystyle ~{\hat {U}}~}
sin
t
φ
^
+
g
2
φ
^
+
g
2
a
^
=
a
^
sin
t
φ
^
φ
^
,
{\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 }}}},}
cos
t
φ
^
+
g
2
a
^
=
a
^
cos
t
φ
^
,
{\displaystyle \cos t\,{\sqrt {{\hat {\varphi }}+g^{2}}}\;{\hat {a}}={\hat {a}}\;\cos t{\sqrt {\hat {\varphi }}},}
幺正算符 可以计算被密度矩阵
ρ
^
(
t
)
{\displaystyle ~{\hat {\rho }}(t)~}
幺正算符 包含了所有可观测量。给定初态
ρ
^
(
0
)
{\displaystyle ~{\hat {\rho }}(0)~}
ρ
^
(
t
)
=
U
^
†
(
t
)
ρ
^
(
0
)
U
^
(
t
)
{\displaystyle {\hat {\rho }}(t)={\hat {U}}^{\dagger }(t){\hat {\rho }}(0){\hat {U}}(t)}
⟨
Θ
^
⟩
t
=
Tr
[
ρ
^
(
t
)
Θ
^
]
{\displaystyle \langle {\hat {\Theta }}\rangle _{t}={\text{Tr}}[{\hat {\rho }}(t){\hat {\Theta }}]}
其中，
Θ
^
{\displaystyle ~{\hat {\Theta }}~}
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0980c40fba0505d45a5d6288cccfa28d9848196f)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c1ec60da813df4da9f2162e63250307d6b3b6)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d9c02dbbd2a85adfcd56e46c21e40242225c48)
![{\displaystyle \langle {\hat {\Theta }}\rangle _{t}={\text{Tr}}[{\hat {\rho }}(t){\hat {\Theta }}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b150730d27198760537729ff5e322dbd51f5af6c)
