Basic properties of a mean field laser equation

TL;DR

This paper analyzes a mean field laser equation using quantum master equations, establishing existence, uniqueness, and deriving Maxwell-Bloch equations, thus providing a rigorous mathematical foundation for laser dynamics.

Contribution

It introduces a rigorous mathematical framework for the mean field laser equation, including existence, uniqueness, and derivation of Maxwell-Bloch equations from quantum master equations.

Findings

Proved existence and uniqueness of solutions to the non-linear quantum master equation.

Provided a probabilistic representation via a mean field stochastic Schrödinger equation.

Derived Maxwell-Bloch equations rigorously from the mean field laser model.

Abstract

We study the non-linear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the non-linear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schr\"ondiger equation. To this end, we find a regular solution for the non-autonomous linear quantum master equation in Gorini-Kossakowski-Sudarshan-Lindblad form, and we prove the uniqueness of the solution to the non-autonomous linear adjoint quantum master equation in Gorini-Kossakowski-Sudarshan-Lindblad form. Moreover, we obtain rigorously the Maxwell-Bloch equations from the mean field laser equation.