A Mean-Field Laser Quantum Master Equation

We discuss the properties of a master equation on density matrices with nonlinear mean-field Hamiltonian that models a simple laser. We show the existence of a unique regular family (ρ_t )t≥0 of density matrices which is a stationary solution. In case a relevant parameter Cb is less than 1, we prove that any regular solution converges exponentially fast to the equilibrium. A locally exponential stable limit cycle arises at the regular stationary state as Cb crosses the critical value 1 so that the nonlinear GKLS equation has a Poincaré-Andronov-Hopf bifurcation at Cb = 1 of supercritical-like type. Moreover, the limit cycle is locally exponentially stable if another function of relevant parameters is between the first and second laser thresholds appearing in the semiclassical laser theory. Furthermore, applying our main results we find the long-time behaviour of the von Neumann entropy, the photon-number statistics, and the quantum variance of the quadratures