Purcell enhancement of the parametric down-conversion in two-dimensional nonlinear materials

Ultracompact nonlinear optical devices utilizing two-dimensional (2D) materials and nanostructures are emerging as important elements of photonic circuits. Integration of the nonlinear material into a subwavelength cavity or waveguide leads to a strong Purcell enhancement of the nonlinear processes and compensates for a small interaction volume. The generic feature of such devices which makes them especially challenging for analysis is strong dissipation of both the nonlinear polarization and highly confined modes of a subwavelength cavity. Here we solve a quantum-electrodynamic problem of the spontaneous and stimulated parametric down-conversion in a nonlinear quasi-2D waveguide or cavity. We develop a rigorous Heisenberg-Langevin approach which includes dissipation and fluctuations in the electron ensemble and in the electromagnetic field of a cavity on equal footing. Within a relatively simple model, we take into account the nonlinear coupling of the quantized cavity modes, their interaction with a dissipative reservoir and the outside world, amplification of thermal noise and zero-point fluctuations of the electromagnetic field, and other relevant effects. We derive closed-form analytic results for relevant quantities such as the spontaneous parametric signal power and the threshold for parametric instability. We find a strong reduction in the parametric instability threshold for 2D nonlinear materials in a subwavelength cavity and provide a comparison with conventional nonlinear photonic devices.