in the model hamiltonian (\[eq:model\]) that are consistent with state-of-the art semiconductor-based photonic technology: $U_{\mathrm{nl}} \sim 5\times10^{-5}$, $J\sim 50$ (in units of $\hbar\gamma$).

![Numerical solution for the zero-time-delay second-order correlation represented in color scale plot as a function of $\Delta\omega_{1}\neq\Delta\omega_{2}$, with parameters (units of $\hbar \gamma$): $\hbar \gamma=25$ $\mu$eV, $U_{\mathrm{nl}}=4\times10^{-5} $ and $J=56$, $F=10$. []{data-label="fig:omega12"}](figure3.pdf){width="50.00000%"}

Optimal antibunching
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Given the basic numbers, we now explore the efficiency of such SPS spanning the parameters space for the corresponding master equation, which is solved in its steady state by numerically determining a quantum average on $\rho_{ss}$, which is the density matrix corresponding to the eigenvalue $\lambda_{ss}=0$ in the linear eigenvalue problem $\cal{L}\rho = \lambda \rho$.

First, in order to numerically infer the optimal resonance-pump detunings, we show in figure \[fig:omega12\] the calculated $g^{(2)}(0)$ as a function of $\Delta\omega_{1}$ and $\Delta\omega_{2}$, respectively. We truncate the Hilbert space to consider a maximum number of photons $N_{\mathrm{max}}=12$ in the basis set of Fock states, which we checked to be sufficient for convergence. Realistic model parameters are assumed, as described in the previous section and reported in the figure caption. Again, we point out that all the energies are intended in units of $\hbar\gamma = 25$ 