as $${\hat{\Phi}}(x)=\int\frac{{\mathrm{d}}^3{\bm p}}{(2\pi)^3}\Bigl(\phi_{\bm p}^{(+)}(x)\hat{b}_{\bm p}(t)+\phi_{\bm p}^{(-)}(x)\hat{d}_{-{\bm p}}^\dagger(t)\Bigr),
\label{bf18}$$ where the new functions $$\begin{aligned}
\phi_{\bm p}^{(+)}(x)&=\eta_{\bm p}^*(t)\Phi_{\bm p}^{(+)}(x)-\xi_{\bm p}^*(t)\Phi_{\bm p}^{(-)}(x),\label{bf19}\\
\phi_{\bm p}^{(-)}(x)&=\eta_{\bm p}(t)\Phi_{\bm p}^{(-)}(x)-\xi_{\bm p}(t)\Phi_{\bm p}^{(+)}(x),\label{bf20}\end{aligned}$$ have been introduced in compliance with the Bogolyubov transformation \[Eqs.  and \]. Moreover, we assume that $$\phi_{\bm p}^{(\beta)}(x)={\mathrm{e}}^{{\mathrm{i}}{\bm p}\cdot {\bm x}-{\mathrm{i}}\beta\int^t{\mathrm{d}}t'\omega_{\bm p}(t')}\tilde{\phi}_{\bm p}^{(\beta)}(t).
\label{bf21}$$ Thus, it follows from Eqs. , , , and  that $$\begin{aligned}
\Phi_{\bm p}^{(+)}(t)=\eta_{\bm p}(t)&{\mathrm{e}}^{-{\mathrm{i}}\int^t{\mathrm{d}}t'\omega_{\bm p}(t')}\tilde{\phi}_{\bm p}^{(+)}(t)\nonumber\\
&+\xi_{\bm p}^*(t){\mathrm{e}}^{{\mathrm{i}}\int^t{\mathrm{d}}t'\omega_{\bm p}(t')}\tilde{\phi}_{\bm p}^{(-)}(t),\label{bf22}\\
\Phi_{\bm p}^{(-)}(t)=\eta_{\bm p}^*(t)&{\mathrm{e}}^{{\mathrm{i}}\int^t{\mathrm{d}}t'\omega_{\bm p}(t')}\tilde{\phi}_{\bm p}^{(-)}(t)\nonumber\\
&+\xi_{\bm p}(t){\mathrm{e}}^{-{\mathrm{i}}\int^t{\mathrm{d}}t'\omega_{\bm p}(t')}\tilde{\phi}_{\bm p}^{(+)}(t).\label{bf23}\end{aligned}$$ One can show that these functions solve Eq.  provided that $$\tilde{\phi}_{\bm p}^{(\beta)}(t)=\sqrt{\frac{mc^2}{\omega_{\bm p}(t)}},
\label{bf24}$$ with the coefficients $\eta_{\bm p}(t)$ and $\xi_{\bm p}(t)$ coupled through the equations, $$\begin{aligned}
\dot{\eta}_{\bm p}(t)&=\frac{\dot{\omega}_{\bm p}(t)}{2\omega_{\bm p}(t)}\,\xi_{\bm p}^*(t){\mathrm{e}}^{2{\mathrm{i}}\int^t{\mathrm{d}}t'\omega_{\bm p}(t')},\label{bf25}\\
\dot{\xi}_{\bm p}^*(t)&=\frac{\dot{\omega}_{\bm p}(t)}{2\omega_{\bm p}(t)}\,\eta_{\bm p}(t){\mathrm{e}}^{-2{\mathrm{i}}\int^t{\mathrm{d}}t'\omega_{\bm p}(t')}.\label{bf26}\end{aligned}$$ This system of equations has to be solved with the initial conditions such that at time $t_0$, which is before the pulsed electric field starts to act, $\eta_{\bm p}(t_0)=1$ and $\xi_{\bm p}(t_0)=0$. It is useful to introduce new coefficients, $$\begin{aligned}
c_{\bm p}^{(1)}(t)&=\eta_{\bm p}(t){\mathrm{e}}^{-{\mathrm{i}}\int^t{\mathrm{d}}t'\omega_{\bm p}(t')},\label{bf27}\\
c_{\bm p}^{(2)}(t)&=\xi_{\bm p}^*(t){\mathrm{e}}^{{\mathrm{i}}\int^t{\mathrm{d}}t'\omega_{\bm p}(t')},\label{bf28}\end{aligned}$$ as it allows to remove the rapidly oscillating in time phase factors in  and . In this case, Eqs.  and  become $${\mathrm{i}}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\begin{bmatrix}
                     c_{\bm p}^{(1)}(t)\\
                     c_{\bm p}^{(2)}(t)
                    \end{bmatrix}
                    =
                    \begin{pmatrix} \omega_{\bm p}(t) & -{\mathrm{i}}\Omega_{\bm p}(t) \cr 