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Journal of the Optical Society of America B

Regularly injected one-atom maser in the high-flux regime: a numerical study

Giacomo Mauro D’Ariano, R. D. Seno, Nicoletta Sterpi

Published Feb 1, 1995• 1 citation
Publisher

Abstract

We present a numerical study of the regularly injected one-atom maser in the high-flux regime, namely, when the time spacing between two consecutive injected atoms is comparable with the atom–field interaction time. Gain and losses are treated simultaneously in a general master equation that takes into account atomic incoherent decay. At stroboscopic times the dynamics of photon-number probability distribution is given by a suitably reduced Green operator, which has the form of a Markoff matrix. We perform a spectral analysis of the Green operator, showing the influence of photon traps on the eigenvalues. A comparison with the opposite case of Poissonian injection and low flux is given for a wide range of the pumping parameter θ. Regular injection leads to larger gain than Poissonian, but for high values of θ the opposite result can be found. Anomalous behaviors occur in which the normalized field fluctuations are increased by regularization of pumping and decreased by atomic decay: these features confirm similar anomalies found by other authors and are ascribed to the occurrence of nonclassical multiple-peak photon distributions and to different responses of the peaks to dissipation and gain. Atomic elastic collisions destroy any signature of trapping states on the stationary field. A comparison with a previously studied semiclassical model is given.

DOI
10.1364/josab.12.000348

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We present a numerical study of the regularly injected one-atom maser in the high-flux regime, namely, when the time spacing between two consecutive injected atoms is comparable with the atom–field interaction time. Gain and losses are treated simultaneously in a general master equation that takes into account atomic incoherent decay. At stroboscopic times the dynamics of photon-number probability distribution is given by a suitably reduced Green operator, which has the form of a Markoff matrix. We perform a spectral analysis of the Green operator, showing the influence of photon traps on the eigenvalues. A comparison with the opposite case of Poissonian injection and low flux is given for a wide range of the pumping parameter θ. Regular injection leads to larger gain than Poissonian, but for high values of θ the opposite result can be found. Anomalous behaviors occur in which the normalized field fluctuations are increased by regularization of pumping and decreased by atomic decay: these features confirm similar anomalies found by other authors and are ascribed to the occurrence of nonclassical multiple-peak photon distributions and to different responses of the peaks to dissipation and gain. Atomic elastic collisions destroy any signature of trapping states on the stationary field. A comparison with a previously studied semiclassical model is given.
We study the one-atom maser with a periodic-pump scheme for the injected atoms. The dynamics of the system is solved also in a regime where the losses of the field cannot be neglected while the atom is inside the cavity, as is the situation in an optical maser. The time development of the maser field is described by a general evolution operator, whose matrix representation in the damping basis is known analytically for zero temperature. The numerical evaluation of this matrix yields the stationary as well as the transient properties of the maser. Noise-producing mechanisms of the pump are included explicitly in the evolution operator for the field. These are a finite efficiency of the pump laser, a velocity spread in the atomic beam, and spontaneous transitions between the atomic levels and to the ground state. We find that features such as trapped states and sub-Poissonian photon statistics also occur in the high-loss regime like in an optical resonator. The observation of such effects will, as in the ordinary micromaser experiments, depend very sensitively on the reduction of the pump noise. For highly regular excitation, we find a pronounced oscillation of the field-correlation function for a wide range of the transit time, which leads to a distinct line splitting in the spectrum of the field. The amount of the splitting is determined by the periodicity of the pump. For less regular excitation, and in particular in the Poissonian limit, these oscillations disappear.
We derive a master equation for the dynamics of a maser or laser with periodic injection of excited atoms. The model includes regular pumping with single atoms, but it also covers the case of periodic injection of groups of atoms, whose number may fluctuate. To obtain a quasicontinuous master equation for such a periodically kicked dynamics, we consider a macroscopic state of the field that is obtained from the true microscopic state by a time average over many periods. The master equation for that time-averaged state correctly describes both the stationary and the transient properties of the maser on a coarse, macroscopic scale. Numerical and analytical results based on this equation are compared with a microscopic, step-by-step resolution of the field's evolution along the periods. Deviations from the microscopic results only occur when frequencies of the order of the (mechanical) injection frequency of the pump become relevant. The recently predicted regularity-induced line splitting in the emission spectrum of a maser is an example for such a situation.
The statistics of the beam of excited atoms that pumps a micromaser influence the photon number distribution of the cavity field. Of the cases studied so far, regular injection statistics minimize the number fluctuations and Poisson statistics maximize them. Here we extend these studies in two directions. Previous master equations describing sub-Poissonian injection statistics were derived by using standard reservoir theory. Recently, corrections to these equations have been found that involve the commutator of the gain and loss operators. Describing the system by a discrete map, we can avoid the approximations inherent in reservoir theory. The results are compared to those of the master equation. Differences are found to be small for large photon number but increasingly important for decreasing photon number and pump beam intensity, indicating the breakdown of reservoir theory in this regime. We give an approximate expression for the photon number distribution for weak damping and discuss its connection with semiclassical theory. Finally, we generalize our treatment for super-Poissonian pump fluctuations.
We present the theory of a truly microscopic maser consisting of a single-mode high-Q resonator in which a monoenergetic beam of excited two-level atoms is injected at such a low flux that at most one atom at a time is present inside the cavity. Both a microscopic theory and a heuristic Fokker-Planck approach are presented. We show that the micromaser exhibits a number of novel features that are averaged out in usual masers and lasers. First, the field is in general sub-Poissonian, which reflects the quantization of both the field and its sources. Second, the onset of maser oscillations may be followed by a succession of abrupt transitions in the state of the field. Finally, as the atomic flux through the resonator is increased, the maser threshold acquires characteristics of a continuous phase transition, whereas the subsequent changes in the field distribution become analogous to first-order phase transitions.
A method is presented for theoretically investigating the properties of a one-atom micromaser with stationary non-Poissonian pumping. The method is based on treating the statistics of the arrival times of the individual pump atoms with the help of the theory of stochastic point processes. Considerable simplification is achieved by assuming the pump statistics to be described by a stationary renewal process. Thus the influence of super- as well as sub-Poissonian pumping with different strengths of correlation between the pump atoms and different correlation decay times may be studied quantitatively. The level-selective statistics of the atoms leaving the cavity is investigated as well as the photon statistics of the cavity field. Moreover, the relation to the other models used in the literature for describing the micromaser pump statistics is discussed. It is found that for sub- (super-)Poissonian pumping, the stationary expectation value of the cavity photon number (which corresponds to its time-averaged value) is larger (smaller) than the conditioned mean photon number that would result from averaging over the number of photons present in the cavity at the time instants immediately before the injection of the individual pump atoms. For Poissonian pumping, both quantities are shown to be equal. The relative standard deviation of the stationary cavity photon number is decreased (increased) for sub- (super-) Poissonian pumping, in comparison to the corresponding values that would result from Poissonian pumping. Moreover, it turns out that for sub- (super-)Poissonian pumping, the normalized coincidence probability density for the detection of the outgoing deexcited atoms is smaller (larger) than the normalized cavity field intensity correlation function at zero time delay. The difference decreases with increasing lifetime of the pump-atom correlations. Finally, the level-selective delayed coincidence probability densities of the outgoing atoms and their waiting-time distributions are found to be affected greatly by the correlation strength as well as the correlation decay time of the incoming pump atoms.
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.