Fields {#sec4}
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In the case of time-dependent electric fields, the Hamiltonian (\[E-ham\]) is a system of decoupled, time-dependent oscillators, for which we may use a pair of invariants for each oscillator as the time-dependent annihilation and the creation operators, and construct the Fock space of number states [@MMT70; @kim-kim99; @kim-lee00; @kim-page01; @kim-page13]. From Eq. (\[q-inv\]), we introduce the first class invariant for each $\alpha$ $$\begin{aligned}
\hat{A}_{\alpha} (t) = \frac{i}{\sqrt{2}} \Bigl[ \varphi^*_{\alpha} (\hat{\pi}^*_{\alpha} + \hat{\pi}_{\alpha}) - \dot{\varphi}^*_{\alpha} (\hat{\phi}^*_{\alpha} + \hat{\phi}_{\alpha}) \Bigr], \label{E-in1}\end{aligned}$$ and the second class invariant $$\begin{aligned}
\hat{A}^{\dagger}_{\alpha} (t) = - \frac{i}{\sqrt{2}} \Bigl[ \varphi_{\alpha} (\hat{\pi}^*_{\alpha} + \hat{\pi}_{\alpha}) - \dot{\varphi}_{\alpha} (\hat{\phi}^*_{\alpha} + \hat{\phi}_{\alpha}) \Bigr]. \label{E-in2}\end{aligned}$$ Here, $\varphi_{\alpha} (t)$ is a complex solution to the mode equation (\[E-eq\]) and satisfies the Wronskian condition from the quantization rule $$\begin{aligned}
\varphi_{\alpha} (t)  \dot{\varphi}^*_{\alpha} (t) - \varphi^*_{\alpha} (t) \dot{\varphi}_{\alpha} (t) = i. \label{wr con}\end{aligned}$$ Each complex solution $\varphi_{\alpha} (t)$ determines a pair of the invariants (\[E-in1\]) and (\[E-in2\]), which is analogous to a Fock basis in the standard Bogoliubov transformation. However, under the assumption of ${\bf E}_{\parallel} (-\infty) = 0$ and ${\bf B} (- \infty) = {\bf B}_0$, the complex solution can be uniquely chosen such that in the electric field it coincides with 