0_R \rangle = \hat{b}_{m\omega}^L | 0_R \rangle =0$.

It proves useful to introduce Unruh modes (instead of Minkowski modes) that cover the whole Minkowski spacetime for two reasons: 1) the Unruh and Minkowski modes share the same vacuum; 2) the transformation between Rindler modes and Unruh modes is a two-mode squeezing transformation. The Unruh modes are defined as $$\begin{aligned}
\label{UnruhRindler}
\hat{c}_{m\omega} &=& \text{cosh}(r_{\omega}) \hat{b}_{m\omega}^R - \text{sinh}(r_{\omega}) \hat{b}_{m\omega}^{L\dag}, \nonumber \\
\hat{d}_{m\omega} &=& \text{cosh}(r_{\omega}) \hat{b}_{m\omega}^L - \text{sinh}(r_{\omega}) \hat{b}_{m\omega}^{R\dag},\end{aligned}$$ where $r_{\omega}$ satisfies $\text{tanh}(r_{\omega}) = e^{-\pi \omega/a}$. It is easy to find the inverse transformation, $$\begin{aligned}
\label{UnruhRindler:inverse}
\hat{b}_{m\omega}^R &=& \text{cosh}(r_{\omega}) \hat{c}_{m\omega} + \text{sinh}(r_{\omega}) \hat{d}_{m\omega}^{\dag}, \nonumber\\
\hat{b}_{m\omega}^L &=& \text{cosh}(r_{\omega}) \hat{d}_{m\omega} + \text{sinh}(r_{\omega}) \hat{c}_{m\omega}^{\dag}. \end{aligned}$$ We can see that the Rindler modes $(\hat{b}_{m\omega}^R, \hat{b}_{m\omega}^L)$ and Unruh modes $(\hat{c}_{m\omega}, \hat{d}_{m\omega} )$ are related by a two-mode squeezing operator with a frequency dependent squeezing parameter $r_{\omega}$. In terms of Unruh modes, the scalar field $\hat{\Phi}$ can be expressed as $$\begin{aligned}
\label{UnruhMode}
\hat{\Phi} &=& \int d \omega (\hat{c}_{1\omega} G_{1\omega}   + \hat{d}_{1\omega} \bar{G}_{1\omega}    \nonumber \\&&\qquad\qquad + \hat{c}_{2\omega} G_{2\omega}  + \hat{d}_{2\omega} \bar{G}_{2\omega} + \text{h.c.}   )\end{aligned}$$ where $$\begin{aligned}
\label{Unruhmodes}
G_{1\omega}(V) &=& F(\omega, a) (aV)^{-i \omega/a},  \nonumber\\
\bar{G}_{1\omega}(V) &=& F(\omega, a) (-aV)^{i \omega/a}, \nonumber\\
G_{2\omega}(U) &=& F(\omega, a) (-aU)^{i \omega/a}, \nonumber\\
\bar{G}_{2\omega}(U) &=& 