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Optics and Spectroscopy

Phase probability density in subensembles of a quantum mode of a one-atom maser

George P. Miroshnichenko

Published Sep 1, 2004• 2 citations
Publisher

Abstract

The method of periodic trajectories is applied to the analysis of the phase states of a one-atom maser mode, information on which can be obtained from a series of consequent indirect phase-sensitive quantum measurements of atoms leaving the cavity. Such information allows one to study in detail the evolution of a maser mode in a stationary state. The evolution pattern is represented as a random sequence of subensembles in which the mode exists during different time intervals. An approximate stochastic recurrence relation is obtained, which allows us, using the Monte Carlo method, to generate a sequence of relative frequencies of detection of the states of a chosen basis in escaping atoms. Formulas for the phase probability density for subensembles of the mode are derived. These formulas are obtained using as initial data the average relative frequencies measured by an experimenter in a region of a stable trajectory.

DOI
10.1134/1.1803645

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Physical Review A•Apr 1, 1994•87 citations
The method of periodic trajectories is applied to the analysis of the phase states of a one-atom maser mode, information on which can be obtained from a series of consequent indirect phase-sensitive quantum measurements of atoms leaving the cavity. Such information allows one to study in detail the evolution of a maser mode in a stationary state. The evolution pattern is represented as a random sequence of subensembles in which the mode exists during different time intervals. An approximate stochastic recurrence relation is obtained, which allows us, using the Monte Carlo method, to generate a sequence of relative frequencies of detection of the states of a chosen basis in escaping atoms. Formulas for the phase probability density for subensembles of the mode are derived. These formulas are obtained using as initial data the average relative frequencies measured by an experimenter in a region of a stable trajectory.
The method of stochastic recurrent relation is used for simulating a random sequence (trajectory) of the counts of a detector of atomic states at the output of a single-atom micromaser. A random sequence of the relative frequencies of counts of a detector during a fixed time interval is calculated. The frequencies of counts vary randomly near the average time-independent level. It is assumed that these average levels are reproducible observables for each random trajectory. A micromaser can jumpwise transfer from one average level of the relative frequencies of counts to another. It is assumed that a certain subensemble of the states of a field mode corresponds to each measured average level of the frequencies of counts. A method is proposed for calculating possible average relative frequencies of counts and corresponding (mean) reduced density matrices of the mode ρ st . The matrix ρ st characterizes a micromaser during its development along a specified periodic trajectory. It is found by solving the eigenvalue problem for the evolution operator on the period. An analytic method for solving this problem is developed. The matrix ρ st is the solution to the inverse problem of the reconstruction of the statistics of a field mode from the statistics of a random trajectory. The procedure of selecting the parameters of the evolution operator on the period is discussed through a numerical example.
The evolution of a quantum system undergoing repeated indirect measurements naturally leads to a Markov chain on the set of states which is called a quantum trajectory. In this paper we consider a specific model of such a quantum trajectory associated to the one-atom maser model. It describes the evolution of one mode of the quantized electromagnetic field in a cavity interacting with two-level atoms. When the system is non-resonant we prove that this Markov chain admits a unique invariant probability measure. We moreover prove convergence in the Wasserstein metric towards this invariant measure. These results rely on a purification theorem: almost surely the state of the system approaches the set of pure states. Compared to similar results in the literature, the system considered here is infinite dimensional. While existence of an invariant measure is a consequence of the compactness of the set of states in finite dimension, in infinite dimension existence of an invariant measure is not free. Furthermore usual purification criterions in finite dimension have no straightforward equivalent in infinite dimension.
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.
A random sequence (path) of counts of an ideal detector, selective for energy states of an atom, at the output of a one-atom maser is simulated numerically by the Monte Carlo method. Fragments of a path for the cases in which a micromaser undergoes quantum jumps with an increase (jump up) or a decrease in the average number of photons in a mode are obtained. An idealized dynamic model of a quantum jump is formulated on the basis of the concept of a critical fluctuation, when several tens of atoms in the same state are detected successively. Approximate formulas for the dependence of the average number of photons and the rate of its change on the number of atoms passing through a cavity along a jump path are obtained. The probability of a quantum jump in the idealized model is estimated. It is shown that accounting for the nonideality of a jump increases its probability by several orders of magnitude. The high rate of a quantum jump up (the superpump effect) is attributed to the possibility of redistribution of photons between subensembles of the mode states. The probability of a nonideal quantum jump and the most likely number of interruptions (opposite escapes) in an ideal jump path are estimated.
Trapping states in the one-atom maser or micromaser are predicted by maser theory and are a direct consequence of the quantization of the electromagnetic field. They have been found in our experiment when the inversion of atoms leaving the cavity is observed while the interaction time is scanned. They correspond to quantum states of the atom-cavity system and under certain conditions represent Fock states. The trapping states occur when the atom-field dynamics complete a full evolution (or multiples thereof). Owing to feedback through the maser dynamics, a stabilization of the photon number in the steady state field of the maser cavity is achieved.
We present a general theory for the calculation of various characteristic properties of the beam of atoms emerging from a resonator in one-atom-maser experiments. The beam is described in terms of the statistics of the detector clicks. The evolution of the state of the maser photons between clicks is governed by a nonlinear master equation. The nonlinearity originates in the necessity to account for the atoms that escape detection. Despite the permanent reductions of the photon state, resulting from the detections, the steady state of the conventional linear master equation determines the statistics of the detector clicks. The whole process is ergodic, in the sense that a single run of the experiment contains all reproducible statistical data, provided the duration of the run is much longer than all relevant correlation times. The nonlinear master equation is used to calculate the distribution of waiting times between detector clicks. Other statistical properties of the clicks that are derived include correlation functions and variances of the counting statistics. The formalism is applied to standard one-atom-maser experiments and to parity measurements on both unpumped and pumped cavities. We find that, for the standard one-atom-maser operation, none of the said statistical properties is a simple immediate indicator for a sub-Poissonian variance of the photon number. For example, the detector clicks may be antibunched although the photon distribution has a super-Poissonian variance.