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.
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.
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.
We study the stochastic dynamics of the electromagnetic field in a lossless cavity interacting with a beam of two-level atoms, given that the atomic states are measured after they have crossed the cavity. The atoms first interact at the exit of the cavity with a classical laser field ℰ and then enter into a detector which measures their states. Each measurement disentangles the field and the atoms and changes in a random way the state |ψ(t)〉 of the cavity field. For weak atom-field coupling, the evolution of |ψ(t)〉 when many atoms cross the cavity and the detector is characterized by a succession of quantum jumps occurring at random times, separated by quasi-Hamiltonian evolutions, both of which depend on the laser field ℰ. For ℰ=0, the dynamics is the same as in the Monte Carlo wave function model of Dalibard et al. [Phys. Rev. Lett. 68, 580 (1992)] and Carmichael, An Open System Approach to Quantum Optics, Lecture Notes in Physics Vol. 18 (Springer, Berlin, 1991)]. The density matrix of the quantum field, obtained by averaging the projector |ψ(t)〉〈ψ(t)| over all results of the measurements, is independent of ℰ and follows the master equation of the damped harmonic oscillator at finite temperature. We provide numerical evidence showing that for large ℰ, an arbitrary initial field state |ψ(0)〉 evolves under the monitoring of the atoms and the measurements toward squeezed states |α,re2iφ〉, moving in the α-complex plane but with almost constant squeezing parameters r and φ. The values of r and φ are determined analytically. On the other hand, for ℰ=0, the dynamics transforms the initial state into Fock states |n〉 with fluctuating numbers of photons n, as shown in Kist et al. [J. Opt. B: Quantum Semiclassical Opt. 1, 251 (1999)]. In the last part, we derive the quantum jump dynamics from the linear quantum jump model proposed in Spehner and Bellissard [J. Stat. Phys. 104, 525 (2001)], for arbitrary open quantum systems having a Lindblad-type evolution. A careful derivation of the infinite jump rates limit, where the dynamics can be approximated by a diffusion process of the quantum state, is also presented.
We study the dynamics of the single-atom maser, or micromaser, by means of the recently introduced method of thermodynamics of quantum jump trajectories. We find that the dynamics of the micromaser displays multiple space-time phase transitions, i.e., phase transitions in ensembles of quantum jump trajectories. This rich dynamical phase structure becomes apparent when trajectories are classified by dynamical observables that quantify dynamical activity, such as the number of atoms that have changed state while traversing the cavity. The space-time transitions can be either first order or continuous, and are controlled not just by standard parameters of the micromaser but also by nonequilibrium "counting" fields. We discuss how the dynamical phase behavior relates to the better known stationary-state properties of the micromaser.
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 field of the one-atom maser, or micromaser, shows strong variations in certain parameter regions usually identified as phase transitions. In this paper the time development of the field in the vicinity of those regions is investigated. Depending on the parameters, spontaneous jumps or slow transitions between two metastable field values can be observed. Hystersis behavior also occurs. The results are explained by means of a quantum Monte Carlo simulation of the maser field as well as a Fokker-Planck description used earlier in micromaser theory.
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.