object ($U_{\omega}$) and then combine with the Rindler modes from the left Rindler wedge in the antisqueezers, going back to Unruh modes again. []{data-label="circuit:Fig"}](f2.png){width="8.0cm"}

For computational purposes, we introduce operator vectors $\hat{\bf c}_{\omega}$, $\hat{\bf d}_{\omega}$, $\hat{\bf b}_{\omega}^R$ and $\hat{\bf b}_{\omega}^L$, which are defined as $$\begin{aligned}
\hat{\bf c}_{\omega} = {\hat{c}_{\omega} \choose \hat{c}_{\omega}^{\dag}}, \,\,\,\, \hat{\bf d}_{\omega} = {\hat{d}_{\omega} \choose \hat{d}_{\omega}^{\dag}}, \,\,\,\,
\hat{\bf b}_{\omega}^R = {\hat{b}_{\omega}^R \choose \hat{b}_{\omega}^{R\dag}},\,\,\,\, \hat{\bf b}_{\omega}^L = {\hat{b}_{\omega}^L \choose \hat{b}_{\omega}^{L\dag}}.\end{aligned}$$ Then Eqs. (\[UnruhRindler\]) and (\[UnruhRindler:inverse\]) can be rewritten as $$\begin{aligned}
{\hat{\bf c}_{m\omega} \choose \hat{\bf d}_{m\omega}} = S^{-1}_{\omega} {\hat{\bf b}_{m\omega}^R \choose \hat{\bf b}_{m\omega}^L}, \,\,\,\,\,\,
{\hat{\bf b}_{m\omega}^R \choose \hat{\bf b}_{m\omega}^L} =S_{\omega}{\hat{\bf c}_{m\omega} \choose \hat{\bf d}_{m\omega}},\end{aligned}$$ with $$\begin{aligned}
S_{\omega} \equiv \left({I \text{cosh}(r_{\omega}) \atop \sigma_x \text{sinh}(r_{\omega})}{ \sigma_x \text{sinh}(r_{\omega}) 
\atop I \text{cosh}(r_{\omega})}\right)\end{aligned}$$ where $I = \left({1 \atop 0}{0 \atop 1}  \right)$ is the identity matrix and $\sigma_x = \left({0 \atop 1}{1 \atop 0} \right)$ is one of the Pauli matrices. The transformation between the input Unruh modes $(\hat{\bf c}_{1\omega}, \hat{\bf d}_{1\omega}, \hat{\bf c}_{2\omega}, \hat{\bf d}_{2\omega})^T$ and the output Unruh modes $(\hat{\bf c}^{\prime}_{1\omega} \hat{\bf d}^{\prime}_{1\omega}, \hat{\bf c}^{\prime}_{2\omega}, \hat{\bf d}^{\prime}_{2\omega})^T$ can be represented as $$\begin{aligned}
\label{input-output}
\begin{pmatrix}
\hat{\bf c}^{\prime}_{1\omega} \\ \hat{\bf d}^{\prime}_{1\omega} \\ \hat{\bf c}^{\prime}_{2\omega}  \\ \hat{\bf d}^{\prime}_{2\omega}
\end{pmatrix}
=\mathcal{S}^{-1}_{\omega} \mathcal{U}_{\omega} \mathcal{S}_{\omega}
\begin{pmatrix}
\hat{\bf c}_{1\omega} \\ 