  The evolution of a quantum system undergoing repeated indirect measurements
naturally leads to a Markov chain on the set of states which is called a
quantum trajectory. In this paper we consider a specific model of such a
quantum trajectory associated to the one-atom maser model. It describes the
evolution of one mode of the quantized electromagnetic field in a cavity
interacting with two-level atoms. When the system is non-resonant we prove that
this Markov chain admits a unique invariant probability measure. We moreover
prove convergence in the Wasserstein metric towards this invariant measure.
These results rely on a purification theorem: almost surely the state of the
system approaches the set of pure states. Compared to similar results in the
literature, the system considered here is infinite dimensional. While existence
of an invariant measure is a consequence of the compactness of the set of
states in finite dimension, in infinite dimension existence of an invariant
measure is not free. Furthermore usual purification criterions in finite
dimension have no straightforward equivalent in infinite dimension.
