one finds that $$\begin{aligned}
\lambda_1&={\mathrm{e}}^{-{\mathrm{i}}\vartheta_1}\quad{\rm with}\quad\vartheta_1=\vartheta_0-{\mathrm{i}}\vartheta,\nonumber\\
\lambda_2&={\mathrm{e}}^{-{\mathrm{i}}\vartheta_2}\quad{\rm with}\quad\vartheta_2=\vartheta_0+{\mathrm{i}}\vartheta,
\label{eigenvaluesBB}\end{aligned}$$ where the phases $\vartheta_1,\vartheta_2\in\mathbb{C}$. In either case we have $\vartheta_1+\vartheta_2=2\vartheta_0$ and $|\lambda_1\lambda_2|=1$.

In closing, we note that all of the aforementioned parametrized matrices share the same property. Namely, if we denote ${\hat{U}}(T+t_0,t_0)\equiv{\hat{U}}(\vartheta_0,\vartheta;\beta,\gamma)$ then it follows from explicit derivations that $${\hat{U}}(\vartheta_0^{(1)}+\vartheta_0^{(2)},\vartheta^{(1)}+\vartheta^{(2)};\beta,\gamma)={\hat{U}}(\vartheta_0^{(1)},\vartheta^{(1)};\beta,\gamma){\hat{U}}(\vartheta_0^{(2)},\vartheta^{(2)};\beta,\gamma).
\label{sram1}$$ This becomes important in light of Eq. . Based on this property, one can show that (for details, see Appendices \[SU(1,1)\] and \[appendix1\]) $${\hat{U}}(N_{\rm rep}T+t_0,t_0)=[{\hat{U}}(\vartheta_0,\vartheta;\beta,\gamma)]^{N_{\rm rep}}={\hat{U}}(N_{\rm rep}\vartheta_0,N_{\rm rep}\vartheta;\beta,\gamma).
\label{sram2}$$ This has been already proven in Ref. [@KTK] for the fermionic case. We have also realized there that the parameters $\beta$ and $\gamma$ do not play a role in interpreting the interference patterns in momentum distributions of created pairs. Actually, the parameter $\vartheta_0$ does not either, as it only enters the formulas through the global phase factor. This means that, up to an irrelevant value $\vartheta_0$, the phases of the eigenvalues of the monodromy matrices considered in this paper can be chosen either real \[Eq. \] or purely complex \[Eq. \]. We will use this convention when presenting our numerical results. For completeness, let us note that $\vartheta_0$ depends on arbitrarily chosen phases of amplitudes $c_{\bm p}^{(1)}(t)$ and $c_{\bm p}^{(2)}(t)$ in 