the corresponding Liouvillian for the two coupled modes in the Lindblad form $$\begin{aligned}
\mathcal{L}^{(\pm)}= &&\frac{\gamma_{+}}{2}[2\hat{a}_{+}\rho
\hat{a}_{+}^{\dag} -
\hat{a}_{+}^{\dag}\hat{a}_{+}\rho-\rho\hat{a}_{+}^{\dag}\hat{a}_{+}]  \nonumber \\
&&   + \frac{\gamma_{-}}{2}[2\hat{a}_{-}\rho \hat{a}_{-}^{\dag} -
\hat{a}_{-}^{\dag}\hat{a}_{-}\rho-\rho\hat{a}_{-}^{\dag}\hat{a}_{-}]\, \, .\end{aligned}$$ One can easily show that, when $\gamma_{1}=\gamma_{2}=\gamma$, and hence $\gamma_{\pm}=\gamma$, we get $\mathcal{L}^{(\pm)}=\mathcal{L}^{(1,2)}$, as expected. For the latter model, pure dephasing is taken into account by the Lindblad term as in Eq. (\[eq:deph12\]), but for the operators $\hat{a}_{+}$ and $\hat{a}_{-}$.

![ Sketch of a possible realization of a SPS based on the scheme shown in figure \[sistema\], with a photonic crystal integrated circuit fabricated with a nonlinear material as the underlying platform.[]{data-label="schema"}](figure2.pdf){width="50.00000%"}

Towards passive single-photon sources {#sec:results}
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We hereby analyze the device represented in the scheme of figure \[sistema\] as a potential SPS: a photonic molecule is driven by a low power input laser impinging only on cavity 1, and output light is collected from the same cavity (see next section for a possible realization). The device could realize an efficient SPS if the light emitted from cavity 1 is strongly antibunched [@savona10prl]. In the following, we assume the normalized degree of second order coherence from the first resonator, $g^{(2)}(0) = \langle\hat{a}_1^{\dag2}\hat{a}_1^{2}\rangle/ \langle\hat{a}_1^{\dag}\hat{a}_1\rangle^2$, as the figure of merit quantifying the device 