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\end{equation}.    Taking $\Gamma_i\in\mathbb{R}$ and remembering $\sigma_x^{\dagger}=\sigma_x$ one has $\hat{\gamma}^{\dagger}=\hat{\gamma}$.  The Assuming the qubits totally-degenerate ($\omega_1=0$ and $\omega_2=0$), the  eigenstates of Hamiltonian (\ref{eqn:H2R}) are: \begin{equation}\label{eqn:basis}  |N_{nm}nm>=\ddag{\omega_0^{-1}\gamma_{nm}}|N>|n>|m>  \end{equation}