in figure \[sistema\]. Each resonator is assumed to be single-mode, where $\hat{a}_i$ ($\hat{a}_i^{\dagger}$) define the destruction (creation) operator of a single photon ($i=1,2$ labels the two resonators). As in [@savona10prl; @bamba], we consider the nonlinear second-quantized hamiltonian $$\begin{aligned}
\hat{H}=\sum_{i=1,2}[\hbar\omega_{i} \hat{a}_{i}^{\dag}\hat{a}_{i}+
U_{\mathrm{nl}}\hat{a}_{i}^{\dag}\hat{a}_{i}^{\dag}\hat{a}_{i}\hat{a}_{i}] + \nonumber \\
+ J(\hat{a}_{1}^{\dag}\hat{a}_{2}+\hat a_{2}^{\dag}\hat{a}_{1})+F
e^{-i\omega_L t} \hat{a}_{1}^{\dagger}+F^{\ast} e^{i\omega_L
t}\hat{a}_{1} \, ,\end{aligned}$$ where the linear part describes two harmonic oscillators, neglecting the zero point energy, $J/ \hbar$ is the tunnel coupling rate between the two resonators [@nota1], $F/ \hbar$ is the coherent pump rate at the continuous wave (cw) laser frequency $\omega_L$. Following the scheme introduced in [@savona10prl], only one of the two resonators is driven, and light will be assumed to be collected only from the same. For the case of passive nonlinear resonator, the nonlinear energy shift for each mode can be approximated as [@drummond80; @ferretti2012] $$\label{shift_kerr}
U_{\mathrm{nl}}\simeq \frac{3(\hbar\omega_0)^2 } {4\varepsilon_0}
\frac{\overline{\chi}^{(3)}}{\overline{\varepsilon}_r^2}
\int  |\vec{\alpha}_i(\mathbf{r})|^4 \mathrm{d}\mathbf{r} 
=  \frac{3(\hbar\omega_0)^2 } {4\varepsilon_0 V_{\mathrm{eff}}}
\frac{\overline{\chi}^{(3)}}{\overline{\varepsilon}_r^2} \, ,$$ where the effective mode volume is defined as $V^{-1}_{i,\mathrm{eff}}=V^{-1}_{\mathrm{eff}}=\int  |\vec{\alpha}_i(\mathbf{r})|^4 \mathrm{d}\mathbf{r}$, from the normalized field profile in each cavity, $\vec{\alpha}_i$, for the specific nanocavity implementation (see [@vahala_review] for a review). To this purpose, we will assume constant values for the average real part 