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Taking $\Gamma_i\in\mathbb{R}$ and remembering $\sigma_x^{\dagger}=\sigma_x$ one has $\hat{\gamma}^{\dagger}=\hat{\gamma}$.
The eigenstates of Hamiltonian
\ref{eqn:H2R} (\ref{eqn:H2R}) are the displaced Fock states:
\begin{equation}
|N_{nm}nm>=\ddag{\omega^{-1}\gamma_{nm}}|N>|n>|m>
\end{equation}