does not couple Rindler modes with different frequencies, the input-output formalism is substantially simplified. The coefficients $\alpha^{mn}_{k \omega}$ and $\beta^{mn}_{k \omega}$ are now proportional to $\delta(k - \omega)$ so Eq. (\[generalcoupling\]) becomes $$\label{nomixingfrequency}
\hat{b}_{m\omega}^{\prime R} = \alpha^{m1}_{\omega \omega} \hat{b}_{1\omega}^R 
+ \beta^{m1}_{\omega \omega} \hat{b}_{1\omega}^{R \dagger} 
+ \alpha^{m2}_{\omega \omega} \hat{b}_{2 \omega}^R 
+ \beta^{m2}_{\omega \omega} \hat{b}_{2\omega}^{R \dagger}.$$ Since modes with different frequencies are independent, we can propose a quantum circuit model for each single frequency. The quantum circuit is shown in Fig. \[circuit:Fig\]. A pair of left-moving Unruh modes $(\hat{c}_{1\omega}, \hat{d}_{1\omega})$ and a pair of right-moving Unruh modes $(\hat{c}_{2\omega}, \hat{d}_{2\omega})$ pass through the two-mode squeezers $S_{\omega}$, from which emerge left-moving Rindler modes $(\hat{b}_{1\omega}^R, \hat{b}_{1\omega}^L)$ and right-moving Rindler modes $(\hat{b}_{2\omega}^R, \hat{b}_{2\omega}^L)$, respectively. $\hat{b}_{1\omega}^R$ and $\hat{b}_{2\omega}^R$ interact with each other when passing through the object (symbolized by the black dot in Fig. \[circuit:Fig\]) and emerge as $\hat{b}_{1\omega}^{\prime R}$ and $\hat{b}_{2\omega}^{\prime R}$, which can be described by a unitary transformation $U_{\omega}$ according to Eq. (\[nomixingfrequency\]). After that, the Rindler modes are combined by two-mode antisqueezers $S^{-1}_{\omega}$, ending up with Unruh modes again.

![(color online). Unruh modes pass through the squeezers and then become Rindler modes. The Rindler modes in the right Rindler wedge interact with the 