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First-Hitting-Time Probabilities for Regime-Switching Diffusions in Nonlinear Photonic Cavities
Moussa Kounta and Maria Woodside-Oriakhi
Nonlinear optical cavities with structured refractive index profiles can exhibit resonant field enhancement, optical bistability, and noise-driven switching between metastable photonic configurations. Motivated by biologically realized photonic crystals, we develop a stochastic model for nonlinear cavity dynamics. Starting from a driven Kerr-type stochastic differential equation for the complex cavity field amplitude, we derive amplitude–phase equations and perform a phase reduction near steady operation. The resulting dynamics form a regime-switching diffusion in which the phase drift and diffusion depend on a discrete photonic configuration represented by a continuous-time Markov chain. The probability that the phase first reaches a prescribed boundary is characterized by an associated boundary value problem, which we evaluate explicitly. A discrete approximation scheme is introduced and shown to converge to the continuous diffusion limit. Finally, the stochastic dynamics admit a quantum-state representation, establishing a connection between nonlinear cavity fluctuations and quantum master-equation frameworks.
Keywords: Markov chains, regime switching, first-hitting time, stochastic differential equations, nonlinear optical cavities