Research on Topological operator on degenerate or non-degenerate modes

non-degenerate case

Hamiltonian

$$\begin{aligned}
H=&\omega_c a^\dagger a+\sum_{k=1}^2\Omega_kc_k^\dagger c_k+\sum_{k=1}^2g_k(c_k+c_k^\dagger)a^\dagger a\\
&+i\sqrt{\kappa_{in}}a_{in,1}(a^\dagger e^{-i\omega_1t}+ae^{i\omega_1t})\\
&+i\sqrt{\kappa_{in}}a_{in,2}(a^\dagger e^{-i\omega_2t}+ae^{i\omega_2t})
\end{aligned}$$

Equations of motion

$$\begin{aligned}
&\dot{c_1}=-(i\Omega_1+\gamma_1/2)c_1-ig_1a^\dagger a\\
&\dot{c_2}=-(i\Omega_2+\gamma_2/2)c_2-ig_2a^\dagger a\\
&\dot{a}=-(i\omega_c+\kappa/2)a-i\sum_{k=1}^2g_k(c_k+c_k^\dagger)a+\sqrt{\kappa_{in}}(a_{in,1}e^{-i\omega_1t}+a_{in,2}e^{-i\omega_2t})
\end{aligned}$$

Linearizing

Suppose $a=(\alpha+d)e^{-i\omega_ct}$