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Physical Review A

Macroscopic and submacroscopic fields in micromasers with non-Poissonian pumping

Shi-Yao Zhu, Chang Su, Ma Ai-Qun

Published Mar 1, 1993• 4 citations
Publisher

Abstract

The differences between macroscopic (all time or entire ensemble average) and submacroscopic (partial time or partial ensemble average) fields of micromasers with non-Poissonian pumping are discussed. A formula to calculate the steady-state density matrix of the macroscopic field is derived. The photon statistics, average photon numbers, and photon-number variances of the macroscopic field, together with those of the submacroscopic fields, are obtained and compared. The photon statistics of a special submacroscopic field are almost the same as those of the macroscopic field.

DOI
10.1103/physreva.47.2319

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Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Jun 1, 1995•5 citations
The differences between macroscopic (all time or entire ensemble average) and submacroscopic (partial time or partial ensemble average) fields of micromasers with non-Poissonian pumping are discussed. A formula to calculate the steady-state density matrix of the macroscopic field is derived. The photon statistics, average photon numbers, and photon-number variances of the macroscopic field, together with those of the submacroscopic fields, are obtained and compared. The photon statistics of a special submacroscopic field are almost the same as those of the macroscopic field.
We present a theory of the micromaser including the effect of the arrival-time statistics of the active atoms in the theoretical treatments. Our starting point is an approach that explicitly incorporates the effect of pump statistics into the dynamical description of lasers and masers. We show that the application of standard laser theory to the micromaser is tantamount to the tacit assumption of Poissonian arrival-time statistics. Hence standard laser theory and the theory of the micromaser are, in general, not equivalent, but the latter contains the former, as a limiting case, for completely random arrival times. We show, as an application, that the steady-state value of the average photon number (first moment) is essentially unaffected by the pump statistics since it depends only on the flux of active atoms (average number of atoms entering the cavity per unit time) provided the flux is constant. The intensity fluctuations (second moment), however, exhibit a stronger dependence on the pump statistics. In particular, for sub-Poissonian pump statistics the sub-Poissonian character of the field is enhanced.
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 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.
It is proposed to use a result of indirect measurements—the average relative frequency of counts of a detector selective for the states of atoms escaping from a micromaser—for time selection of subensembles of a field mode. Analytical expressions for the reduced density matrix, average number of photons, and Mandel Q parameter for mode subensembles were obtained by the method of periodic paths. The generation of a random sequence of results of indirect quantum measurements was performed by the Monte Carlo method. The results of the analytical and numerical calculations are compared. It is found that the statistics of subensembles is sub-Poisson even when the calculation based on the matrix for the entire ensemble (in the absence of quantum measurements) gives super-Poisson statistics. It is noted that the time selection of subensembles is possible within a limited range of variation in the pumping parameter Θ, when the probability of quantum jumps is relatively low.
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.
The author presents a tutorial review of the quantum theory of the micromaser which allows for arbitrary (sub- as well as super-Poissonian) fluctuations of the pumping beam. In conventional reservoir theory the rate of change of the cavity field is a sum of the changes due to separate interaction with the individual reservoirs, i.e. the interactions are uncorrelated. In their approach, which is based on discrete mapping rather than a maser equation, corrections to reservoir theory arise due to correlations between these interactions. The magnitude of these terms is characterized by the quantity p/N ex . Here p, the parameter describing pump beam fluctuations, is the negative of the Mandel Q parameter of the pump beam so that p=1 corresponds to regular pumping, p=0 to Poissonian one and p<0 to super-Poissonian pump beam fluctuations. N ex is the number of atoms passing through the cavity during the lifetime of the intercavity field. The conventional reservoir limit (standard laser theory) is recovered if p=0 and/or N ex is large. In all other cases the interactions with the gain and loss reservoirs are correlated. The author presents analytical results to demonstrate the effect of pump regularity on steady-state problems (photon statistics, mean number of photons and photon number variance) as well as on transient phenomena (correlation function and spectrum). In particular, it is shown that the approach to equilibrium can have non-Markovian character.