  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.
