where we have approximated $\tilde{J}\simeq J$, since $J\gg \hbar\Delta\omega, U_{\mathrm{nl}}, \hbar\gamma$. Eqs. (\[eq:lowF1\])-(\[eq:lowF5\]) can be simplified by considering the condition $C_{00} \simeq 1$ and $C_{00} \gg C_{10}, C_{01} \gg C_{20}, C_{11}, C_{02}$. The second-order correlation for light emitted from cavity 1 can be analytically approximated as $g^{(2)}(0)\simeq 2|C_{20}|^2 / |C_{10}|^4$. Hence, minimizing $|C_{20}|^2 $ with respect to $\Delta\omega_1= \Delta\omega_2 = \Delta\omega$ (keeping fixed $\gamma_{+}+\gamma_{-}$, $J$, $U_{\mathrm{nl}}$), which corresponds to minimizing the complex function $|2\tilde{\Delta}^3 + J^2 U_{\mathrm{nl}}|^2$, we get the optimal conditions for the antibunching behavior at fixed $\Delta\omega$ as $$[\gamma_{+}+\gamma_{-} ]_{\mathrm{opt}} \simeq 4\sqrt{3} \Delta\omega \, .$$ This result quantitatively explains the dependence of the optimal antibunching (minimum in $g^{(2)}(0)$) in figure \[fig:gamma\_pm\]. At small pumping rate, the value of $g^{(2)}(0)$ grows monotonously with $F$ [@savona10prl]. In this sense, the smaller the pumping rate, the better the single-photon behavior of the output. However, in a practical device this would also lead to a vanishingly small number photons in the systems, and hence of single-photons emitted from the cavity 1. In our conditions, the average number of photons $\langle \hat{a}^{\dagger}_1\hat{a}_1 \rangle$ is always much smaller than 1, but on the order of $10^{-5} - 10^{-4}$, depending on the pumping strength, which 