  We investigate entangled photon pair generation in a lossy microring
resonator using an input-output formalism based on the work of Raymer and
McKinstrie (Phys. Rev. A 88, 043819 (2013)) and 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.
