of the nonlinear susceptibility and relative dielectric permittivity, $\overline{\chi}^{(3)}$ and $\overline{\varepsilon}_r$ respectively, to have realistic order of magnitude estimates on materials of current technological interest [@nota2]. The hamiltonian can be expressed in a rotating frame with respect to the pump frequency, $\Delta\omega_{i}=\omega_{i}-\omega_{L}$, $$\begin{aligned}
\label{eq:model}
\hat{H}=\sum_{i=1,2}[\hbar\Delta\omega_{i} \hat{a}_{i}^{\dag}\hat{a}_{i}+
U_{\mathrm{nl}}\hat{a}_{i}^{\dag}\hat{a}_{i}^{\dag}\hat{a}_{i}\hat{a}_{i}]
&+& J(\hat{a}_{1}^{\dag}\hat{a}_{2}+\hat a_{2}^{\dag}\hat{a}_{1}) \nonumber \\
&+&F\hat{a}_{1}^{\dagger}+F^{\ast}\hat{a}_{1} \, .\end{aligned}$$

The driven/dissipative character of this model is taken into account by a master equation treatment for the system density matrix in Markov approximation $$\label{rho2res}
\dot{\rho}=\frac{1}{i \hbar}[\rho,\hat{H}] + \mathcal{L}^{(1,2)} + \mathcal{L}^{deph}\, ,$$ where $$\mathcal{L}^{(1,2)}= \sum_{i=1,2} \frac{\gamma_{i}}{2}  [2\hat{a}_{i}\rho
\hat{a}_{i}^{\dag} -
\hat{a}_{i}^{\dag}\hat{a}_{i}\rho-\rho\hat{a}_{i}^{\dag}\hat{a}_{i}] \, ,$$ is the Liouvillian operator in the usual Lindblad form for the two resonators modes, and $$\mathcal{L}^{deph}= 
              \sum_{i=1,2} \frac{\gamma_i^{\ast}}{2}  [2\hat{a}_{i}^{\dag}\hat{a}_{i}\rho\hat{a}_{i}^{\dag}\hat{a}_{i}
            - \hat{a}_{i}^{\dag}\hat{a}_{i} \hat{a}_{i}^{\dag}\hat{a}_{i} \rho
            - \rho\hat{a}_{i}^{\dag}\hat{a}_{i} \hat{a}_{i}^{\dag}\hat{a}_{i} ] \, ,
\label{eq:deph12}$$ models the pure dephasing for the two resonators modes.

In this work, we generalize the model to account for possible unbalanced losses between the two normal modes of the system. In the following, we will only consider the case $\Delta\omega_{1}=\Delta\omega_{2}$ (with straightforward generalization), under which conditions one has the two symmetric and antisymmetric combinations, respectively given by $$\hat{a}_{+}=\frac{1}{\sqrt{2}}(\hat{a}_{1}+\hat{a}_{2})\, ; \, \, \, \,
\hat{a}_{-}=\frac{1}{\sqrt{2}}(\hat{a}_{1}-\hat{a}_{2})\, .$$ In this case, the losses can be described through 