to the evolution operator and therefrom the pair-production rate [@KLR12]. Furthermore, the one-loop effective action is obtained from the scattering matrix between the out-vacuum and the in-vacuum under the action of the electric field modulo the scattering matrix in the absence of the field [@KLY08]: $$\begin{aligned}
e^{i \int  d^4x {\cal L}_{\rm eff} (E)} = \prod_{\alpha, T = \infty} \frac{\langle 0_{\alpha}, T/2 \vert 0_{\alpha}, -T/2 \rangle (E) }{\langle 0_{\alpha}, T/2 \vert 0_{\alpha}, -T/2 \rangle (E=0)}.\end{aligned}$$

Invariant Operators in Time-Dependent Magnetic Fields {#sec5}
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The charged scalar field in a time-dependent magnetic field along the fixed direction has the Hamiltonian (\[B-ham\]) or (\[q-ham\]), which may be written in the form $$\begin{aligned}
H_{\epsilon} (t) = H_{\rm D} (t) + \epsilon H_{\rm O} (t). \label{B-tham}\end{aligned}$$ Here, $\epsilon$ is an expansion parameter for the perturbation theory and will be set $\epsilon =1$ in the end of calculations, and $$\begin{aligned}
H_{\rm D} = \sum_{\alpha} \pi^*_{\alpha} \pi_{\alpha} + \omega^2_{\alpha} (t) \phi^*_{\alpha} \phi_{\alpha} \label{diag}\end{aligned}$$ is the diagonal Hamiltonian, while $$\begin{aligned}
H_{\rm O} = \sum_{\alpha \beta} \pi^*_{\alpha} \Omega_{\alpha \beta} \phi^*_{\beta} + \pi_{\alpha} \Omega_{\alpha \beta} \phi_{\beta} \label{off}\end{aligned}$$ is the off-diagonal Hamiltonian. The Hamiltonian (\[B-tham\]) is an infinite system of coupled oscillators due to the off-diagonal Hamiltonian (\[off\]), which induces continuous 