electric field pulses.

The monodromy matrix in the fermionic case has been introduced in [@KTK] (see also, Appendix \[appendix1\]). It was shown there that it can be parametrized using four real parameters such that $0\leqslant \vartheta_0,\vartheta,\beta < 2\pi$ and $0\leqslant \gamma \leqslant\pi$, $${\hat{U}}(T+t_0,t_0)={\mathrm{e}}^{-{\mathrm{i}}\vartheta_0}
\begin{pmatrix}
\cos\vartheta+{\mathrm{i}}\sin\vartheta\cos\gamma & {\mathrm{i}}{\mathrm{e}}^{-{\mathrm{i}}\beta}\sin\vartheta\sin\gamma
\cr
{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\beta}\sin\vartheta\sin\gamma & \cos\vartheta-{\mathrm{i}}\sin\vartheta\cos\gamma
\end{pmatrix}.
\label{monodromyFF}$$ One can check that its eigenvalues are $$\begin{aligned}
\lambda_1&={\mathrm{e}}^{-{\mathrm{i}}\vartheta_1}\quad{\rm with}\quad\vartheta_1=\vartheta_0-\vartheta,\nonumber\\
\lambda_2&={\mathrm{e}}^{-{\mathrm{i}}\vartheta_2}\quad{\rm with}\quad\vartheta_2=\vartheta_0+\vartheta,
\label{eigenvaluesFF}\end{aligned}$$ meaning that $|\lambda_{1,2}|=1$. As it was presented in Ref. [@KTK], the respective phases $\vartheta_1,\vartheta_2\in\mathbb{R}$ play a significant role in interpreting interference patterns in the momentum distributions of created fermions. The same is true for bosons. The difference, however, is that for bosons the phases $\vartheta_1$ and $\vartheta_2$ are not necessarily real.

As shown in Appendix \[SU(1,1)\], the pseudo-unitary monodromy matrix for bosons can be either parametrized as $$\begin{aligned}
{\hat{U}}(T+t_0,t_0)={\mathrm{e}}^{-{\mathrm{i}}\vartheta_0}
\begin{pmatrix}
\cos\vartheta+{\mathrm{i}}\sin\vartheta\cosh\gamma  &-{\mathrm{i}}{\mathrm{e}}^{-{\mathrm{i}}\beta}\sin\vartheta\sinh\gamma
\cr
{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\beta}\sin\vartheta\sinh\gamma  &\cos\vartheta-{\mathrm{i}}\sin\vartheta\cosh\gamma
\end{pmatrix} ,
\label{monodromyBB}\end{aligned}$$ with $0\leqslant \vartheta_0,\vartheta,\beta < 2\pi$ and $\gamma\geqslant 0$, or as $$\begin{aligned}
{\hat{U}}(T+t_0,t_0)={\mathrm{e}}^{-{\mathrm{i}}\vartheta_0}
\begin{pmatrix}
\cosh\vartheta+{\mathrm{i}}\sinh\vartheta\sinh\gamma  &-{\mathrm{i}}{\mathrm{e}}^{-{\mathrm{i}}\beta}\sinh\vartheta\cosh\gamma
\cr
{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\beta}\sinh\vartheta\cosh\gamma  &\cosh\vartheta-{\mathrm{i}}\sinh\vartheta\sinh\gamma
\end{pmatrix} ,
\label{monodromyBBpseudo}\end{aligned}$$ with $0\leqslant \vartheta_0,\beta < 2\pi$ and $\vartheta,\gamma\geqslant 0$. Both these matrices satisfy the condition  but their eigenvalues have different character. While in the first case the eigenvalues are given by , in the second case 