Stroboscopic theory of atomic statistics in the micromaser

We study the statistics of the atoms emerging from the cavity of a micromaser in a dynamical, discrete-time `stroboscopic' description which takes into account the measurements made, in general, with imperfect efficiencies, on the states of the outcoming atoms. Inverted atoms enter stochastically, in general, with a binomial distribution in discrete time; but we also consider the continuous-time limit of this input statistics which is Poissonian. We envisage two alternative experimental procedures: one of these is to consider a fixed number N of atoms pumped into the cavity and subsequently leaving it to undergo state detection; the other is to consider input of the excited atoms and their subsequent detection and collection in a fixed time t. We consider, in particular, the steady state behaviors achieved in the two limits, N -> infinity and t -> infinity, as well as the approaches to these two limits. Although these limits are the same for the state of the cavity field, they are not the same, in general, for the observable outcoming atom statistics. We evaluate, in particular, Mandel's Q-parameters $Q_{e}$ $(Q_{g})$ for outcoming atoms detected in their excited states (ground states), for both N -> infinity and t -> infinity, as functions of $N_{ex} = RT_{c}$: R is the mean rate of entry for the incoming atoms and $T_c$ is the cavity damping time. The behavior of these atomic Q-parameters is compared with that parameter for the cavity field.