  We deal with the dynamical system properties of a
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation with mean-field
Hamiltonian that models a simple laser by applying a mean field approximation
to a quantum system describing a single-mode optical cavity and a set of two
level atoms, each coupled to a reservoir. We prove that the mean field quantum
master equation has a unique regular stationary solution. In case a relevant
parameter $C_\mathfrak{b} $, i.e., the cavity cooperative parameter, is less
than $1$, we prove that any regular solution converges exponentially fast to
the equilibrium, and so the regular stationary state is a globally
asymptotically stable equilibrium solution. We obtain that a locally
exponential stable limit cycle is born at the regular stationary state as
$C_\mathfrak{b} $ passes through the critical value $1$. Then, the mean-field
laser equation has a Poincar\'e-Andronov-Hopf bifurcation at $C_\mathfrak{b} =1
$ of supercritical-like type. Namely, we derive rigorously, at the level of
density matrices --for the first time--, the transition from a global attractor
quantum state, where the light is not emitted, to a locally stable set of
coherent quantum states producing coherent light. Moreover, we establish the
local exponential stability of the limit cycle in case a relevant parameter is
between the first and second laser thresholds appearing in the semiclassical
laser theory. Thus, we get that the coherent laser light persists over time
under this condition. In order to prove the exponential convergence of the
quantum state, we develop a new technique for proving the exponential
convergence in open quantum systems that is based in a new variation of
constant formula. Applying our main results we find the long-time behavior of
the von Neumann entropy, the photon-number statistics, and the quantum variance
of the quadratures.
