Gaussian Nonlinear Optics In Coupled Cavity Systems: Back-scattering in Micro-ring Resonators.

Systems of coupled cavities have the potential to provide bright quantum optical states of light in a highly versatile manner. Micro-ring resonators for instance are highly scalable candidates for photon sources thanks to CMOS fabrication techniques and their small footprint, however, surface roughness of the wave-guides, and defects in the coupler geometry routinely induce splitting of the cavity modes due to back-scattering and back-coupling. The parasitic back-propagating mode in the micro-ring leads to hybridisation of the modes, altering the linear and nonlinear properties of this coupled cavity system, and ultimately constraining the fidelity of quantum light sources that can be produced. In this paper, we derive a comprehensive general model for Gaussian non-linear processes in coupled cavity systems, based on an effective field Hamiltonian and a dispersive input-output model. The resulting dynamics of the equations of motion are evaluated in a Gaussian process formalism via the symplectic transformations on the optical modes. We numerically model and explore the problem of back-scattering in micro-ring resonators in physically relevant parameter regimes, involving the splitting of various resonances, we calculate the consequent impurity and heralding efficiency of various heralded photon schemes, we explore a perturbative explanation of the observations and assess the correspondence between spontaneous and stimulated processes in these systems.