光场量子化及光的量子效应

麦克斯韦方程组：

\[\begin{equation} \begin{aligned} \nabla\times\mathbf{H}&={\frac{\partial\mathbf{D}}{\partial t}}\\\\ \nabla\times\mathbf{E}&=-{\frac{\partial\mathbf{B}}{\partial t}} \\\\\nabla\cdot\mathbf{B}& = 0 \\\\\nabla\cdot\mathbf{D}&=0. \end{aligned} \end{equation}\]

电磁场的量子化

一维空腔中的电场 $x $ 分量做傅里叶展开：

\[ E_{x}(z,t)=\sum_{j}A_{j}q_{j}(t)\sin(k_{j}z) \]

其中\(k_{j} = j\pi/L, j =1,2,3..\)，定义系数\(A_{j}=\left(\frac{2\nu_{j}^{2}m_{j}}{V\epsilon_{0}}\right)^{1/2}\)，\(\nu_{j}=j\pi c/L\)，由\(\nabla\times\mathbf{H}=\frac{\partial\mathbf{D}}{\partial t}\)得到：

\[ H_{y}=\sum_{j}A_{j}({\frac{\dot{q}_{j}\epsilon_{0}}{k_{j}}})\cos(k_{j}z) \]

The classical Hamiltonian for the field is

\[\begin{equation} \begin{aligned} \mathcal H&=\frac{1}{2}\int_{V}d\tau(\epsilon_{0}E_{x}^{2}+\mu_{0}H_{y}^{2}) \\&=\frac{1}{2}\sum_{j}(m_{j}\nu_{j}^{2}q_{j}^{2}+m_{j}\dot{q}_{j}^{2}) \\&={\frac{1}{2}}\sum_{j}\left(m_{j}v_{j}^{2}q_{j}^{2}+{\frac{p_{j}^{2}}{m_{j}}}\right) \end{aligned} \end{equation}\]

其中升降算符满足：

\[\begin{equation} \begin{aligned} &[a_j,a_{j'}] = \delta_{jj'} \\ &[a_{j},a_{j^{\prime}}]=[a_{j}^{\dagger},a_{j^{\prime}}]=0 \end{aligned} \end{equation}\]

则电场和磁场算符写为：

\[\begin{equation} \begin{aligned} &E_{x}(z,t)=\sum_{j}\mathscr{E}_{j}(a_{j}e^{-i\nu_{j}t}+a_{j}^{\dagger}e^{i\nu_{j}t})\sin k_{j}z \\ &H_{y}(z,t)=-i\epsilon_{0}c\sum_{j}\mathscr{E}_{j}(a_{j}e^{-i\nu_{j}t}-a_{j}^{\dagger}e^{i\nu_{j}t})\cos k_{j}z \end{aligned} \end{equation}\]

where the quantity \(\mathscr{E}_{j}=\left(\frac{\hbar\nu_{j}}{\epsilon_{0}V}\right)^{1/2}\)，自由空间，\(k\)可以任意，且可以视作是空腔的箱极限：

\[\begin{equation} \begin{aligned} &E(r,t)= \sum_{k}{ \hat{\epsilon_{k}} \mathscr{E}_{k} \alpha_k e^{-i\nu_kt+i k\cdot r} }+ c.c. \\&H(r,t)=\frac{1}{\mu_0}\sum_{ k}\frac{ k\times\hat{\epsilon_{k}}}{v_{k}}\mathscr{E}_{k} {\alpha_{k}}e^{-i v_{k}t+i{k\cdot r}}+c.c. \end{aligned} \end{equation}\]

其中：\(\mathscr{E}_{k}=\left(\frac{\hbar\nu_{k}}{2\epsilon_{0}V}\right)^{1/2},k\cdot\hat{\epsilon}_{k}=0\)，易得模密度的表达式：

\[ D(\nu)d\nu=\frac{L^{3}\nu^{2}}{\pi^{2}c^{3}}d\nu, \]

电场与磁场的对易关系，将电场磁场写为：

\[ E(r,t)=\sum_{ k,\lambda}\hat{\epsilon}_{ k}^(\lambda)E_k a_k,\lambda e^{-i\nu_kt+i k\cdot r}+H.c., \]