## Preprint 2018-10

## Franco Fagnola, Carlos M. Mora:

### Bifurcation analysis of a mean field laser equation

### Abstract:

We study the dynamics of the solution of a non-linear quantum master equation describing a simple laser under the mean eld approximation. The quantum system is formed by a single mode optical cavity and a set of two level atoms that are coupled to two reservoirs. First, we establish the existence of a unique regular stationary state for the non-linear evolution equation under consideration. Second, we examine the free interaction solutions, i.e., the solutions to the non-linear quantum master equation that coincide with unitary evolutions generated by the Hamiltonian resulting from neglecting the interactions between the laser mode, atoms and the bath. We obtain that a family of non-constant free interaction solutions borns at the regular stationary state as a relevant parameter, which is denoted by Cb, passes through the critical value 1. These free interaction solutions yield the periodic solutions of the Maxwell Bloch equations modeling our physical system in the framework of the semiclassical laser theory. Third, in case Cb < 1 we deduce that the system converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. Thus, the quantum system has a Hopf bifurcation at Cb = 1.