(\[nomixingfrequency\]) we have $$\begin{aligned}
\alpha^{11}_{\omega \omega} &=& \alpha^{22}_{\omega \omega} = \cos \theta_{\omega},  \nonumber \\
\alpha^{12}_{\omega \omega} &=& - \alpha^{21*}_{\omega \omega} = -i e^{i \phi_{\omega}} \sin \theta_{\omega},\end{aligned}$$ and all $\beta^{mn}_{\omega \omega}$ are zero. We can therefore express the action of the beamsplitter as $$\begin{aligned}
\label{mirrortransformation}
\mathcal{U}_{\omega} = 
\begin{pmatrix}
I \text{cos}\,\theta_{\omega}& 0 & Z \text{sin}\,\theta_{\omega}  & 0 \\ 
0 & I  & 0 & 0 \\ 
-Z^* \text{sin}\,\theta_{\omega} & 0 & I \text{cos}\,\theta_{\omega}  & 0\\ 
0 & 0 & 0 & I 
\end{pmatrix},\end{aligned}$$ where $I$ is the $2\times 2$ identity matrix and $$Z = \left({-i e^{i \phi_{\omega}} \atop 0}{0 \atop i e^{-i \phi_{\omega}}} \right).
$$

The explicit expressions for the transformation Eq. (\[input-output\]) can be calculated straightforwardly and are summarized in Appendix \[appendixA\]. With these transformations, it is easy to calculate the expectation value of the particle number of the output $\hat{c}^{\prime}_{1\omega}$, $$\begin{aligned}
\label{ParticleNumber:1}
&&\langle 0_M | \hat{c}_{1\omega}^{\prime \dag} \hat{c}_{1\omega^{\prime}}^{\prime} | 0_M \rangle \nonumber \\
&=& 2(1-\text{cos} \,\theta_{\omega})\text{cosh}^2(r_{\omega}) \text{sinh}^2(r_{\omega}) \delta(\omega-\omega^{\prime}) \nonumber \\
&=& 2(1-\text{cos} \,\theta_{\omega})\frac{e^{2\pi \omega/a}}{(e^{2\pi \omega/a}-1)^2} \delta(\omega-\omega^{\prime}) \nonumber \\
&\equiv& n(\omega) \delta(\omega-\omega^{\prime}).\end{aligned}$$ The corresponding expectation values for the other three outputs is the same as Eq. (\[ParticleNumber:1\]). Hence the number of Unruh particles in every output is generally not zero. The 