with $\Omega = 0$, may be used, but the transverse Hamiltonian still has off-diagonal terms, $\langle m, t_0 \vert \hat{H}_{\perp} (t) \vert n, t_0 \rangle \neq 0$ for $m = n \pm 2$. Further note that in the second quantization, we quantize not the Landau states (\[lan ham\]) but the field modes $\phi_{n} (t)$ and $\pi_{n} (t)$ and their complex conjugates. In fact, the Landau states (\[lan ham\]) are used as a mathematical tool to expand the field (\[lan exp\]). The Heisenberg equation $$\begin{aligned}
\ddot{\phi}_n - 2 \Omega_{nm} \dot{\phi}_m + \Bigl( \omega_n^2 (t) \delta_{nm} + (\Omega^2)_{nm} - \dot{\Omega}_{nm} \Bigr) \phi_m = 0, \label{B-eq}\end{aligned}$$ is the vector component of the KG equation [@kim14a].

Time-Dependent Schrödinger Equation {#sec3}
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The Hamiltonian (\[E-ham\]) or (\[B-ham\]) is an infinite sum of time-dependent oscillators. Using a compact notation, in which $\alpha, \beta$ stand for ${\bf k}, {\bf k}'$ for electric fields or $(m, k_z), (n, k'_z)$ for magnetic fields such that $\sum_{\alpha} = \int d^3 {\bf k}/(2\pi)^{3}$ or $ \sum_{\alpha} =  \sum_{n} \int d k_z/(2 \pi)$, and introducing the extended phase space variables $$\begin{aligned}
Z_{\alpha} = \begin{pmatrix} \pi_{\alpha}\\ \phi^*_{\alpha} \end{pmatrix}, \quad Z^*_{\alpha} = \begin{pmatrix} \pi^*_{\alpha}\\ \phi_{\alpha} \end{pmatrix}, \label{phase}\end{aligned}$$ the Hamiltonian can be rewritten in the 