\dot{\phi} - iqA^{0}\phi,\end{aligned}$$ we obtain the Hamiltonian for the field $$\begin{aligned}
H(t) = \int d^3 {\bf x} \Bigl[ \pi^{*} \pi + i qA^{0} (\pi \phi - \pi^{*} \phi) + (\partial_{k} + i qA^{*}_{k}) \phi^{*} (\partial_{k} - i qA_{k}) \phi
+ m^2 \phi^{*} \phi \Bigr].\end{aligned}$$ Then, the quantum dynamics is governed by the functional Schrödinger equation $$\begin{aligned}
i \frac{\partial}{\partial t} \Psi (t, \phi, \phi^{*}) = \hat{H} (t) \Psi (t, \phi, \phi^{*}). \label{sch eq}\end{aligned}$$ In this paper, we consider only the vector potential for a homogeneous, time-dependent electric and magnetic field along a fixed direction of the form $$\begin{aligned}
{\bf A}_{\parallel} (t) = - \int^t_{- \infty}  dt' {\bf E}_{\parallel} (t'),  \quad {\bf A}_{\perp} (t, {\bf x}_{\perp})  = \frac{1}{2} {\bf B} (t) \times {\bf x}. \label{vec pot}\end{aligned}$$ We further assume that ${\bf E}_{\parallel} (-\infty) = 0$ and ${\bf B} (- \infty) = {\bf B}_0$ so that the initial state is either the Minkowski vacuum or the Landau state, respectively.

Firstly, in the case of homogeneous, time-dependent electric fields, we decompose the fields by the Fourier modes as $$\begin{aligned}
\phi (t, {\bf x}) = \int \frac{d^3 {\bf k}}{(2 \pi)^{3}} \phi_{\bf k} (t)  e^{i {\bf k} \cdot {\bf x}}, \quad \phi^* (t, {\bf x}) = 