Generation of photon pairs by spontaneous four-wave mixing in linearly uncoupled resonators

We present a detailed study of the generation of photon pairs by spontaneous four-wave mixing in a structure composed of two linearly uncoupled resonators, where energy can be transferred from one resonator to another only through a nonlinear interaction. Specifically, we consider the case of two racetrack-shaped resonators connected by a coupler designed to guarantee that the resonance comb of each resonator can be tuned independently, and to allow the nonlinear interaction between modes that belong to different combs. We show that such a coupler can be realized in at least two ways: a directional coupler or a Mach-Zehnder interferometer. For these two scenarios, we derive analytic expressions for the pair generation rate via single-pump spontaneous four-wave mixing, and compare these results with that achievable in a single ring resonator.