We study the dynamics of the solution of a non-linear quantum master equation describing a simple laser under the mean field 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 is born at the regular stationary state as a relevant parameter, which is denoted by $C_\mathfrak{b} $, 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 $ C_\mathfrak{b} < 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 $C_\mathfrak{b} =1 $.
Abstract We derive from first principles the experimentally observed effective dynamics of a spinor Bose gas initially prepared as a Bose–Einstein condensate and then left free to expand ballistically. In spinor condensates, which represent one of the recent frontiers in the manipulation of ultra-cold atoms, particles interact with a two-body spatial interaction and a spin–spin interaction. The effective dynamics is well-known to be governed by a system of coupled semi-linear Schrödinger equations: we recover this system, in the sense of marginals in the limit of infinitely many particles, with a mean-field re-scaling of the many-body Hamiltonian. When the resulting control of the dynamical persistence of condensation is quantified with the parameters of modern observations, we obtain a bound that remains quite accurate for the whole typical duration of the experiment.