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Physical Review A

Comment on “Theory of detection in the micromaser”

Ulrike Herzog

Published Mar 2, 2000• 2 citations
Publisher

Abstract

In a recent paper [R. R. McGowan and W. C. Schieve, Phys. Rev. A 59, 778 (1999)], the authors claim that the detection of the outcoming atoms has a back-action effect on the spectrum of the micromaser field. We point out that this assertion is wrong. In particular, we discuss the connection between atom statistics and field statistics, which has been stated to be the ultimate goal of the cited paper.

DOI
10.1103/physreva.61.047801

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In the preceding Comment [Phys. Rev. A 67, 027801 (2003)] Johnson and Schieve raise doubts on the validity of a result obtained in our treatment of collective atomic effects in micromaser dynamics [Phys. Rev. A 60, 1582 (1999)]. We fully confirm the validity of our results, showing that the conclusions derived from their analysis are not relevant.

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arXiv:quant-ph/0008082•Physical Review A•Feb 9, 2001•7 citations
In a recent paper [R. R. McGowan and W. C. Schieve, Phys. Rev. A 59, 778 (1999)], the authors claim that the detection of the outcoming atoms has a back-action effect on the spectrum of the micromaser field. We point out that this assertion is wrong. In particular, we discuss the connection between atom statistics and field statistics, which has been stated to be the ultimate goal of the cited paper.
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.
The lifetimes of Rydberg atoms have been considered in micromaser theory. The difference from the usual micromaser theory [P. Filipowicz, J. Javanainen, and P. Meystre, Phys. Rev. A 34, 3077 (1986)] without level decay rates is discussed. It is shown that when the atom-field interaction time is large, the quantum collapse and revival are significantly different from the prediction of the usual micromaser theory. The relation between the photon fluctuation and the fluctuation of the population of the lower level after interacting with a maser field is also discussed. We found that the consistency of the sub-Poisson statistics of the lower-level population and photon number is violated.
A quantum trajectory analysis of the steady-state photon statistics of a micromaser by Casagrande and collaborators [F. Casagrande et al., Phys. Rev. A 60, 1582 (1999)] shows a significant suppression of the trapping behavior due to the inclusion of two-atom events. We use numerical integration in our investigation and obtain results that disagree with their analysis and show less suppression of the trapping behavior. A comparison with two extreme cases places our results between the two extremes and gives us confidence that our calculations are correct. The results of Casagrande and collaborators do not satisfy this condition.
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 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.