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Ultracompact nonlinear optical devices utilizing two-dimensional (2D)materials and nanostructures are emerging as important elements of photoniccircuits. Integration of the nonlinear material into a subwavelength cavity orwaveguide leads to a strong Purcell enhancement of the nonlinear processes andcompensates for a small interaction volume. The generic feature of such deviceswhich makes them especially challenging for analysis is strong dissipation ofboth the nonlinear polarization and highly confined modes of a subwavelengthcavity. Here we solve a quantum-electrodynamic problem of the spontaneous andstimulated parametric down-conversion in a nonlinear quasi-2D waveguide orcavity. We develop a rigorous Heisenberg-Langevin approach which includesdissipation and fluctuations in the electron ensemble and in theelectromagnetic field of a cavity on equal footing. Within a relatively simplemodel, we take into account the nonlinear coupling of the quantized cavitymodes, their interaction with a dissipative reservoir and the outside world,amplification of thermal noise and zero-point fluctuations of theelectromagnetic field, and other relevant effects. We derive closed-formanalytic results for relevant quantities such as the spontaneous parametricsignal power and the threshold for parametric instability. We find a strongreduction in the parametric instability threshold for 2D nonlinear materials ina subwavelength cavity and provide a comparison with conventional nonlinearphotonic devices.

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