Photon-pair generation in a lossy microring resonator. I. Theory

We investigate entangled photon-pair generation in a lossy microring resonator using an input-output formalism based on the work of M. G. Raymer and C. J. McKinstrie [Phys. Rev. A 88, 043819 (2013)] and P. M. Alsing et al. [Phys. Rev. A 95, 053828 (2017)] that incorporates circulation factors that account for the multiple round-trips of the fields within the cavity. We consider the nonlinear processes of spontaneous parametric down-conversion and spontaneous four-wave mixing, and we compute the generated biphoton signal-idler state from a single-bus microring resonator, along with the generation, coincidence-to-accidental, and heralding efficiency rates. We compare these generalized results to those obtained by previous works employing the standard Langevin input-output formalism.