i \varphi_{\omega}}\sin \theta_{\omega} \cosh(r_{\omega}) \sinh(r_{\omega}) \delta(\omega - \omega'),\end{aligned}$$  $$\begin{aligned} \label{B0} \langle 0_M | \hat{c}'_{0 \omega}  \hat{d}'_{0 \omega'} |0_M \rangle &=& \langle 0_M | \hat{d}'_{0 \omega}  \hat{c}'_{0 \omega'} |0_M \rangle = \langle 0_M | \hat{c}^{\prime \dag}_{0 \omega}  \hat{d}^{\prime \dag}_{0 \omega'} |0_M \rangle ^* = \langle 0_M | \hat{d}^{\prime \dag}_{0 \omega}  \hat{c}^{\prime \dag}_{0 \omega'} |0_M \rangle^* \nonumber \\ \nonumber \\ &=& 