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Journal of the Optical Society of America B

Stroboscopic dynamics of the micromaser

Ravinder Rupchand Puri, Fritz Haake, Dieter Forster

Published Dec 1, 1996• 6 citations
Publisher

Abstract

The successive passage of excited atoms through a micromaser suggests a stroboscopic description. Surprisingly, such a treatment is simpler than its various, more familiar continuous-time approximations. An especially interesting result arises for the longest time scale of approach to equilibrium, which in continuous-time dynamics would appear as being due to barrier hopping and/or tunneling.

DOI
10.1364/josab.13.002689

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The successive passage of excited atoms through a micromaser suggests a stroboscopic description. Surprisingly, such a treatment is simpler than its various, more familiar continuous-time approximations. An especially interesting result arises for the longest time scale of approach to equilibrium, which in continuous-time dynamics would appear as being due to barrier hopping and/or tunneling.
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 QeQ_{e}Qe​ (Qg)(Q_{g})(Qg​) for outcoming atoms detected in their excited states (ground states), for both N -> infinity and t -> infinity, as functions of Nex=RTcN_{ex} = RT_{c}Nex​=RTc​: R is the mean rate of entry for the incoming atoms and TcT_cTc​ is the cavity damping time. The behavior of these atomic Q-parameters is compared with that parameter for the cavity field.
We examine the approach to equilibrium of the micromaser. Analytic methods are first used to show that for large times (i.e. many atoms) the convergence is governed by the next to leading eigenvalue of the corresponding discrete evolution matrix. The model is then studied numerically. The numerical results confirm the phase structure expected from analytic approximation methods and agree for large times with the analysis of Elmfors et al in terms of the continuous master equation. For short times, however, we see evidence for interesting new structure not previously reported in the literature.
We examine the approach to equilibrium of the micromaser.Analytic methods are first used to show that for large times (i.e.many atoms) the convergence is governed by the next to leading eigenvalue of the corresponding discrete evolution matrix.The model is then studied numerically.The numerical results confirm the phase structure expected from analytic approximation methods and agree for large times with the analysis of Elmfors et al in terms of the "continuous master equation".For short times, however, we see evidence for interesting new structure not previously reported in the literature.
We examine the approach to equilibrium of the micromaser. Analytic methods are first used to show that for large times (i.e., many atoms) the convergence is governed by the next-to-leading eigenvalue of the corresponding discrete evolution matrix. The model is then studied numerically. The numerical results confirm the phase structure expected from analytic approximation methods and agree for large times with the analysis of Elmfors et al. in terms of the ``continuous master equation.'' For short times, however, we see evidence for interesting new structure not previously reported in the literature. PACS No.: 42.55Sa
The micromaser possesses a variety of dynamical phase transitions parametrized by the flux of atoms and the time of flight of the atom within the cavity. We discuss how these phases may be revealed to an observer outside the cavity using the long-time correlation length in the atomic beam. Some of the phase transitions are not reflected in the average excitation level of the outgoing atom, which is the commonly used observable. The correlation length is directly related to the leading eigenvalue of the time evolution operator, which we study in order to elucidate the phase structure. We find that as a function of the time of flight the transition from the thermal to the maser phase is characterized by a sharp peak in the correlation length. For longer times of flight there is a transition to a phase where the correlation length grows exponentially with the flux. We present a detailed numerical and analytical treatment of the different phases and discuss the physics behind them. \textcopyright{} 1996 The American Physical Society.
The realization of the micromaser prompted theoretical interest in the fundamental problems of laser physics. In particular, it has been shown that the usual quantum theory of the laser can be recovered from the micromaser theory if a Poissonian average with respect to the time intervals between subsequent injections is taken. The problem of general injection statistics has not been fully resolved but, instead, has been circumvented in the following ways. Instead of the time period between successive injections, the number of injected atoms during a fixed time interval was considered as a stochastic variable and was averaged with respect to a general sub-Poissonian distribution. This led to a closed-form master equa tion, but correction terms arise when the effect of cavity damping is significant. In another approach a stroboscopic theory has been developed: The system is observed when a full cycle of pumping, firing, and decay is completed. This assumes observation of the system on a nonuniform time scale. Here we carry out the averaging over the time interval between subsequent injections with sub-Poissonian distribution, suggest a new compact master equation, and exploit some of its consequences. We show that this approach reproduces the previous ones if the cavity damping is negligible. The surprising result of our approach is that altering the pump statistics affects both the gain and the loss part of the laser dynamics.