Diffusion Approximation of Stochastic Master Equations with Jumps

TL;DR

This paper rigorously derives diffusion approximations for stochastic master equations with jumps in quantum systems, providing conditions and proofs for when such approximations are valid, especially in homodyne and heterodyne detection scenarios.

Contribution

It establishes a necessary condition for diffusion approximation of jump master equations and proves it rigorously using Markov process techniques.

Findings

Diffusion approximation is valid under specific conditions.

Rigorous proof using Markov generator convergence.

Application to homodyne and heterodyne detection cases.

Abstract

In the presence of quantum measurements with direct photon detection the evolution of open quantum systems is usually described by stochastic master equations with jumps. Heuristically, from these equations one can obtain diffusion models as approximation. A necessary condition for a general diffusion approximation for jump master equations is presented. This approximation is rigorously proved by using techniques for Markov process which are based upon the convergence of Markov generators and martingale problems. This result is illustrated by rigorously obtaining the diffusion approximation for homodyne and heterodyne detection.