the time-dependent coefficients $\gamma_{\bm p}^{(\beta\beta')}(t)$ are defined as $$\gamma_{\bm p}^{(\beta\beta')}(t)=\left\{ 
\begin{array}{ll}
\displaystyle\frac{1}{2mc^2}\Bigl(|\dot{\Phi}_{\bm p}^{(\beta)}(t)|^2+\omega_{\bm p}^2(t)|\Phi_{\bm p}^{(\beta)}(t)|^2\Bigr) &\quad{\mbox{if}}\,\,\beta=\beta',\\
\displaystyle\frac{1}{2mc^2}\Bigl([\dot{\Phi}_{\bm p}^{(\beta)}(t)]^*\dot{\Phi}_{\bm p}^{(\beta')}(t)+\omega_{\bm p}^2(t)
[\Phi_{\bm p}^{(\beta)}(t)]^*\Phi_{\bm p}^{(\beta')}(t)\Bigr)&\quad{\mbox{if}}\,\,\beta\neq\beta'.
\end{array}
\right.
\label{bf5}$$ One can show using the asymptotic condition  that $$\lim_{t\rightarrow-\infty}\gamma_{\bm p}^{(\beta\beta')}(t)=\omega_{\bm p}\delta_{\beta\beta'}.
\label{bf7}$$ Thus, in the remote past the Hamiltonian  describes a free bosonic field. It is the interaction with the pulsed electric field which leads to the appearance of nondiagonal terms, ${\hat{b}}_{\bm p}^\dagger{\hat{d}}_{-{\bm p}}^\dagger$ and ${\hat{d}}_{-{\bm p}}{\hat{b}}_{\bm p}$, in . These terms, however, can be removed by means of the Bogolyubov transformation [@Bogolyubov].

In order to diagonalize the Hamiltonian , we introduce new annihilation operators [@Bogolyubov], $$\begin{aligned}
{\hat{b}}_{\bm p}(t)&=\eta_{\bm p}(t){\hat{b}}_{\bm p}+\xi_{\bm p}(t){\hat{d}}^\dagger_{-{\bm p}},\label{bf8}\\
{\hat{d}}_{\bm p}(t)&=\eta_{-{\bm p}}(t){\hat{d}}_{\bm p}+\xi_{-{\bm p}}(t){\hat{b}}_{-{\bm p}}^\dagger,
\label{bf9}\end{aligned}$$ and the corresponding creation operators as well. They are defined through unknown time-dependent functions $\eta_{\bm p}(t)$ and $\xi_{\bm p}(t)$. As the instantaneous operators, ${\hat{b}}_{\bm p}(t)$ and ${\hat{d}}_{\bm p}(t)$, should evolve from the corresponding in-operators, ${\hat{b}}_{\bm p}$ and ${\hat{d}}_{\bm p}$, we infer that $$\lim_{t\rightarrow -\infty}\eta_{\bm p}(t)=1,\quad\lim_{t\rightarrow-\infty}\xi_{\bm p}(t)=0.
\label{bf10}$$ In addition, after imposing the bosonic commutation relations on the new set of annihilation and creation operators, we obtain that at all times $$|\eta_{\bm p}(t)|^2-|\xi_{\bm p}(t)|^2=1.
\label{bf11}$$ Keeping this in mind, we rewrite the bosonic field operator  