-{\mathrm{i}}\Omega_{\bm p}(t) & -\omega_{\bm p}(t) \end{pmatrix}
                    \begin{bmatrix}
                     c_{\bm p}^{(1)}(t)\\
                     c_{\bm p}^{(2)}(t)
                    \end{bmatrix},
\label{bf29}$$ where $$\Omega_{\bm p}(t)=-\frac{\dot{\omega}_{\bm p}(t)}{2\omega_{\bm p}(t)}=-ce{\cal E}(t)\frac{c(p_{\|}-eA(t))}{2\omega_{\bm p}^2(t)},
\label{bf30}$$ and we impose the initial conditions such that $c_{\bm p}^{(1)}(t_0)=1$ and $c_{\bm p}^{(2)}(t_0)=0$.

In closing this section, let us rewrite the instantaneous Hamiltonian  using the time-dependent operators. Namely, $${\hat{H}}(t)=\int\frac{{\mathrm{d}}^3{\bm p}}{(2\pi)^3}\,\omega_{\bm p}(t)\Bigl({\hat{b}}_{\bm p}^\dagger(t){\hat{b}}_{\bm p}(t)+{\hat{d}}_{-{\bm p}}^\dagger(t){\hat{d}}_{-{\bm p}}(t)\Bigr),
\label{bf17}$$ where we have removed an infinite constant by normal ordering the operators ${\hat{d}}_{-{\bm p}}(t)$ and ${\hat{d}}_{-{\bm p}}^\dagger(t)$. As one can see, at each time $t$, Eq.  represents a collection of harmonic oscillators with energy $\omega_{\bm p}(t)$. Interestingly, the functions which define the Bogolyubov transformation and, thus, allow to diagonalize the Hamiltonian, are obtained from solutions of . Hence, for a single eigenmode of the bosonic field, the problem is equivalent to solving a two-level system  whose dynamics is not determined by a unitary matrix. The latter is an element of the $SU(1,1)$ group, the properties of which are analyzed in Appendix \[SU(1,1)\]. Further, we will investigate physical consequences of a nonunitary character of time evolution of the bosonic field as compared to the fermionic case.

### Normalized charge distribution of created boson pairs {#mombos}

Before we define the quantity that will be analyzed 