Explicit expressions are derived for calculating the two-time coincidence probability density, the Mandel Q parameter of the counting statistics, and the waiting-time distribution between detector clicks for the detection of de-excited atoms leaving a micromaser with Poissonian pumping. The procedure is illustrated by applying it to the simplest trapped micromaser state for which all results can be obtained analytically. Moreover, by making use of a symmetry relation which determines the properties of the time-dependent micromaser master equation solution, it is shown that in the stationary regime for negligible thermal photon number the normalized two-time intensity correlation function of the cavity field with respect to a given time difference is equal to the normalized two-time coincidence probability density of detecting two de-excited atoms with the same time difference at the exit of the micromaser.
Explicit expressions are derived for calculating the two-time coincidence probability density, the Mandel Q parameter of the counting statistics, and the waiting-time distribution between detector clicks for the detection of de-excited atoms leaving a micromaser with Poissonian pumping. The procedure is illustrated by applying it to the simplest trapped micromaser state for which all results can be obtained analytically. Moreover, by making use of a symmetry relation which determines the properties of the time-dependent micromaser master equation solution, it is shown that in the stationary regime for negligible thermal photon number the normalized two-time intensity correlation function of the cavity field with respect to a given time difference is equal to the normalized two-time coincidence probability density of detecting two de-excited atoms with the same time difference at the exit of the micromaser.
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.
We present a general method for the derivation of various statistical quantities describing the detection of a beam of atoms emerging from a micromaser. The use of non-normalized conditioned density operators and a linear master equation for the dynamics between detection events is discussed as are the counting statistics, sequence statistics, and waiting time statistics. In particular, we derive expressions for the mean number of successive detections of atoms in one of any two orthogonal states of the two-level atom. We also derive expressions for the mean waiting times between detections. We show that the mean waiting times between detections of atoms in like states are equivalent to the mean waiting times calculated from the uncorrelated steady state detection rates, although like atoms are indeed correlated. The mean waiting times between detections of atoms in unlike states exhibit correlations. We evaluate the expressions for various detector efficiencies using numerical integration, reporting results for the standard micromaser arrangement in which the cavity is pumped by excited atoms and the excitation levels of the emerging atoms are measured. In addition, the atomic inversion and the Fano-Mandel function for the detection of deexcited atoms are calculated for comparison to the experimental results of Weidinger et al. [Phys. Rev. Lett. 82, 3795 (1999)], who report the observation of trapping states.
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 investigate the effect of non-Poissonian pump fluctuations on the micromaser dynamics. Non-Poissonian micromaser pumping has been described by two, seemingly unrelated, theoretical approaches. The first employs a discrete pumping process where the pump atoms are allowed to arrive, with certain probability p, only at regularly spaced instants of time, whereas the second refers to continuously distributed arrival times of the pump atoms. Based on a generalization of the latter, we present a unified approach that can handle both situations on an equal footing. We find that for any kind of non-Poissonian pumping the resulting dynamics of the micromaser field is a non-Markovian one. For a micromaser with discrete non-Poissonian pumping, we show the equivalence of ensemble averaging and time averaging, providing a rare example where the validity of the ergodic hypothesis can be explicitly demonstrated. Moreover, we investigate the time-delayed field-field correlation function and a generalized k-photon spectrum of the cavity field, which for k=1 corresponds to the usual power spectrum. For the case that the micromaser is operated under the k-photon trapping condition, we derive exact analytical expressions for the k-photon spectrum and the corresponding correlation functions that result from the exact solution of a non-Markovian evolution problem. Provided that p>1/2, the spectrum is found to be split into several equidistant peaks for certain values of the interaction parameters. \textcopyright{} 1996 The American Physical Society.
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 Qe(Qg) for outcoming atoms detected in their excited states (ground states), for both N -> infinity and t -> infinity, as functions of Nex=RTc: R is the mean rate of entry for the incoming atoms and Tc is the cavity damping time. The behavior of these atomic Q-parameters is compared with that parameter for the cavity field.
The field in the micromaser cavity, the atoms which both pump and probe the micromaser, and the detectors that determine the outgoing states of the atoms are treated as a single system in order to derive a nonlinear master equation for the field in the micromaser cavity which takes the effect of undetected atoms into account. Analytic solutions for this master equation are found for both phase-insensitive and phase-sensitive detection. These solutions can be used to explicitly show the back-action effects that measurement has on the spectrum of the micromaser. In addition, these solutions are used to find analytic expressions for the mean number of successive detector clicks of the same kind as well as atom-atom and photon-photon correlation functions. In this way, the analytic solutions are linked to experimentally accessible data.