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Physical Review A

Phase distribution and amplitude of a micromaser field in a semiclassical approximation

Jozef Škvarček, Mark S. Hillery

Published Jun 13, 2000• 4 citations
Publisher

Abstract

We present a semiclassical method for determining the phase distribution and the average photon number of a steady-state micromaser field pumped by a stream of resonant two-level atoms in a superposition of their upper and lower states. Assuming the field to have a photon number distribution with a single sharp peak, we find an equation for the amplitude and the phase of the steady-state field, which has two solutions. The stability analysis shows which one of the two is physically realizable. Using the very simple time evolution of particular atomic states, valid in the semiclassical limit, we find an expression for the phase distribution of the field that contains the mean photon number as a parameter. That can be determined from the stable solution.

DOI
10.1103/physreva.62.013807

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Journal of Physics B Atomic Molecular and Optical Physics•Sep 1, 1999•1 citation
We present a semiclassical method for determining the phase distribution and the average photon number of a steady-state micromaser field pumped by a stream of resonant two-level atoms in a superposition of their upper and lower states. Assuming the field to have a photon number distribution with a single sharp peak, we find an equation for the amplitude and the phase of the steady-state field, which has two solutions. The stability analysis shows which one of the two is physically realizable. Using the very simple time evolution of particular atomic states, valid in the semiclassical limit, we find an expression for the phase distribution of the field that contains the mean photon number as a parameter. That can be determined from the stable solution.
A detailed theoretical and experimental study of the atom-field interaction starting from first principles was made possible by the realization of the single-atom micromaser. The situation is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a superconducting cavity. In spite of the considerable amount of work devoted to the study of the micromaser, it still remains to be established how the system behaves under coherent pumping. We present a comprehensive study of the coherently pumped micromaser; we develop an analytic method to obtain the steady-state solution of the master equation governing the time evolution of the cavity field. We illustrate this method with a simple example and present results for the photon distribution and purity of the steady state of the coherently pumped micromaser.
A simple model of a maser or laser consisting of a single-mode field coupled to NNN identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.
Abstract In the photon number distribution p(θ,k) of the micromaser, distinct structures are formed by the ridges connecting the peaks in the θ-k space. We refer to these structures as phases. By a simple condition we can distinguish between "semiclassical" and "quantum" regimes of operation near and far above threshold, respectively. In the semiclassical regime, the equations of state of the phases θ= θ(k) are monotonous and coincide with the steady state solutions of the semiclassical theory. They reflect typical features of the micromaser dynamics. The transition jumps, e. g., between the phases decrease with increasing pumping parameter θ, signifying the onset of the Jaynes-Cummings collapse. On increasing 9 further, the phases first disintegrate and then restructure into new kinds of phases in the quantum regime. The equations of state are no longer monotonous. Large single peaks, the quantum island states (QIS), develop in the neighborhoods of minima. The system undergoes oscillations between two kinds of quantum island states, QIS- and QIS+ , as a function of θ. The disintegration and transformation of phases recur periodically as θ is varied, and the phases in the semiclassical region are followed by consecutive phase structures in the quantum regime. The subsequent collapses and revivals of phases are directly connected to the Jaynes-Cummings collapse and revival. The observation of these phase structures and the accompanying QIS is experimentally feasible.
The operation of one atom-maser is considered under the conditions, when the atoms enter the cavity being excited to the state, which is the mixture of Zeeman sublevels of both the upper and lower resonant atomic levels. The analytic expression for the steady state photon distribution in the cavity under such conditions is obtained. It is shown, that the micromaser parameters, such as the atomic flux and velocity of atoms, the portion of atoms at the lower maser level, the polarizations of laser excitation fields may be chosen in such a way as to obtain the sharp maximum of photon distribution function at a given number of photons, providing the possibility of creation of pure Fock states.
We formulate the theory of a micromaser operating simultaneously on a one- and a two-photon atomic transition. Departing from the complete microscopic Hamiltonian for this composite `atom plus field' system, we derive the equations governing its behaviour in the semiclassical approximation and also the fully quantum mechanical master equation. Using parameters corresponding to existing realizations of the one- and two-photon micromaser, we obtain illustrative examples of some novel aspects of this system in the steady state as well as in dynamical evolution.