efficiency as SPS.

Model parameters 
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We consider passive materials where the non-resonant nonlinearity is determined by the bulk contribution to the third-order material susceptibility, where the real part of the $\chi^{(3)}$ elements is responsible for the nonlinear frequency shift, while the imaginary part gives rise to an additional contribution to losses, such as two-photon absorption (TPA). As shown in [@ferretti2012], TPA is essentially negligible for the low power intensities that would be required for single-photon nonlinear operation in such devices. For this reason, we will neglect its effect here. Typical semiconductor or insulator materials employed in the optoelectronic industry possess a real part of third-order susceptibility of the order of $10^{-19} - 10^{-18}$ in Si units (m$^2/$V$^2$) [@boyd_book; @ferretti2012]. We focus here on photonic crystal type confinement, where remarkable figures of merit (such as, e.g., ultra-high quality factors) have been already shown both with silicon (for a review, see [@notomi_review]) and GaAs materials [@derossi2008]. Moreover, strong enhancement of second and third order nonlinearities has been also reported [@derossi2008; @galli2010; @graphene2012]. In such nanostructures, a realistic cavity mode confinement can be on the order of $V_{\mathrm{eff}}=(\lambda / n_r)^3\simeq 0.1$ $\mu$m$^3$ (for near infrared operation, i.e. $\lambda\sim 1$ $\mu$m and $n_r=3$ 