随机薛定谔方程

约化密度矩阵的重构

对环境部分，采用相干态表象，满足完备性关系

$$\begin{equation} \int d^2\alpha e^{-|\alpha|^2}\ket{\alpha}\bra{\alpha}=I \end{equation}$$

对于系统与环境的初态，满足

$$\begin{equation} \begin{aligned} \ket{\Phi_{tot}(t)}&\\ &=\int d^2\alpha\ket{\alpha}\bra{\alpha}\ket{\phi_{tot}(t)} \\ &=\int d^2\alpha e^{-|\alpha|^2}\ket{\varphi_{a^*}(t)}\otimes\ket{\alpha} \end{aligned} \end{equation}$$

构造约化密度矩阵

$$\begin{equation}\begin{aligned} \hat{\rho}_{sys}(t)&=\int d^2\alpha e^{-|\alpha|^2}\ket{\varphi_{a^*}(t)}\otimes\ket{\alpha}\int d^2\alpha e^{-|\alpha|^2}\bra{\alpha}\otimes\bra{\varphi_{a^*}} \\&=\int d^2\alpha e^{-|\alpha|^2}\ket{\varphi_{a^*}(t)}\bra{\varphi_{a^*}(t)}\otimes\int d^2 \alpha\ket{\alpha}\bra{\alpha} \\&=\int d^2\alpha e^{-|\alpha|^2}\ket{\varphi_{a^*}(t)}\bra{\varphi_{a^*}(t)} \\&=M_z[\ket{\varphi_{a^*}(t)}\bra{\varphi_{a^*}(t)}]\end{aligned} \end{equation}$$

对于$\int d^2\alpha e^{-|\alpha|^2}$的计算，可假设$\alpha$满足高斯分布，由积分运算换为求和运算。

系统波函数的演化方程

对波函数进行表象变换$\ket{\phi}=e^{i\omega a^{\dagger}at}\ket{\Phi_{tot}(t)}$

$$\begin{equation} \begin{aligned} \frac{d\ket{\phi}}{dt}&=\frac{de^{i\omega a^{\dagger}at}}{dt}\ket{\Phi_{tot}(t)}+e^{i\omega a^{\dagger}at}\frac{d\ket{\Phi_{tot}(t)}}{dt} \\&=\omega a^{\dagger}a\ket{\Phi_{tot}(t)}+e^{i\omega a^{\dagger}at}\frac{H_{sys}+V+H_{env}}{i\hbar}\ket{\Phi_{tot}(t)} \\&=[\omega a^{\dagger}a-\omega a^{\dagger}a+-\frac{i}{\hbar}e^{i\omega a^{\dagger}at}(H_{sys}+V)]\ket{\Phi_{tot}(t)} \\&=[-\frac{i}{\hbar}e^{i\omega a^{\dagger}at}(H_{sys}+V)]e^{-i\omega a^{\dagger}at}e^{i\omega a^{\dagger}at}\ket{\Phi_{tot}(t)} \\&=-\frac{i}{\hbar}(H_{sys}-e^{i\omega a^{\dagger}at}Ve^{-i\omega a^{\dagger}at})\ket{\phi} \end{aligned} \end{equation}$$

对于相干态 $\ket{\alpha(t)}=e^{-i\omega_ia^{\dagger}_ia_it}\ket{\alpha_0}$，满足如下方程

$$\begin{equation} \begin{aligned} \frac{d}{dt}e^{i\omega a^{\dagger}at}&\int d^2\alpha\ket{\varphi_{a^*}(t)}\otimes\ket{\alpha(t)} \\ &=\frac{d}{dt}de^{i\omega a^{\dagger}at}\int d^2\alpha\ket{\varphi_{a^*}(t)}\otimes e^{-i\omega_ia^{\dagger}_ia_it}\ket{\alpha_0} \\ &=\frac{d\int d^2\alpha\ket{\varphi_{a^*}(t)}\otimes \ket{\alpha_0}}{dt} \end{aligned} \end{equation}$$

整理得到

$$\begin{equation} \begin{aligned} \frac{d}{dt}&\int d^2\alpha\ket{\varphi_{a^*}}\otimes \ket{\alpha_0} \\ &=-\frac{i}{\hbar}(H_{sys}-e^{i\omega a^{\dagger}at}Ve^{-i\omega a^{\dagger}at})\int d^2\alpha\ket{\varphi_{a^*}}\otimes \ket{\alpha_0} \end{aligned} \end{equation}$$

对系统波函数部分

$$\begin{equation} \frac{d}{dt}\ket{\varphi_{a^*}(t)}=-\frac{i}{\hbar}(H_{sys}-e^{i\omega a^{\dagger}at}Ve^{-i\omega a^{\dagger}at})\ket{\varphi_{a^*}(t)} \end{equation}$$

假设相互作用具有$V=(a^\dagger+a)$，则

$$\begin{equation} \begin{aligned} \frac{d}{dt}\ket{\varphi_{a^*}(t)}&=-\frac{i}{\hbar}(H_{sys}+e^{i\omega a^{\dagger}at}Ve^{-i\omega a^{\dagger}at})\ket{\varphi_{a^*}(t)}\\&=-\frac{i}{\hbar}H_{sys}+\frac{iq}{\hbar}\sum\limits_i\chi_i(e^{i\omega t}a^{\dagger}+e^{-i\omega t}a)\ket{\varphi_{a^*}(t)} \label{NMSDM0} \end{aligned} \end{equation}$$

在相干态表象下运用下列关系

$$\begin{equation} a^{\dagger}\rightarrow a^{*},a\rightarrow \frac{\partial}{\partial \alpha^*} \end{equation}$$

式$\ref{NMSDM0}$进一步化简为

$$\begin{equation} \begin{aligned} \frac{d}{dt}\ket{\varphi_{a^*}(t)}=-\frac{i}{\hbar}H_{sys}+\frac{iq}{\hbar}\sum\limits_i\chi_i(e^{i\omega_i t}a^{*}+e^{-i\omega_i t}\frac{\partial}{\partial a^*})\ket{\varphi_{a^*}(t)} \end{aligned} \end{equation}$$

定义随机过程

$$\begin{equation} \begin{aligned} Z_a(t)=\sum\limits_i\chi_ia^*e^{i\omega t} \end{aligned} \end{equation}$$

通过链式规则

$$\begin{equation} \begin{aligned} \sum\limits_{i}\chi _ie^{-i\omega_it}\frac{\partial}{\partial a_{i}^*} &=\int ds\sum\limits_{i}\chi _ie^{-i\omega_it}\frac{\delta}{\delta Z_a(s)}\frac{\delta Z_a(s)}{\delta a_i^*} \\&=\int ds\sum\limits_{i}\chi _ie^{-i\omega_it}\chi_ie^{-i\omega_is}\frac{\delta}{\delta Z_a(s)} \\&=\int ds\sum\limits_{i}\chi_i^2e^{-i\omega_i(t-s)}\frac{\delta}{\delta Z_a(s)} \end{aligned} \end{equation}$$

方程进一步化简为

$$\begin{equation} \begin{aligned} \frac{d}{dt}&\ket{\varphi_{a^*}(t)}=\\ &[-\frac{i}{\hbar}H_{sys}+\frac{iq}{\hbar}Z_a(t)+\frac{i}{\hbar}q^{\dagger}\int ds\sum\limits_{i}\chi_i^2e^{-i\omega_i(t-s)}\frac{\delta}{\delta Z_a(s)}]\ket{\varphi_{a^*}(t)} \end{aligned} \end{equation}$$