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Physical Review A

Calculation of the micromaser spectrum. II. Eigenvalue approach

Karl Vogel, W. P. Schleich, Marlan O. Scully, HERBERT WALTHER

Published Jul 1, 1993• 38 citations
Publisher

Abstract

We calculate the micromaser spectrum via the eigenvalues and eigenvectors of the master equation. We show that (i) it is not always the lowest eigenvalue which governs the linewidth and (ii) the oscillations in the linewidth originate from an interplay between two eigenvalues.
DOI
10.1103/physreva.48.813

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We first calculate the ``exact'' micromaser spectrum by solving numerically for the Green's function of the approximate master equation for the one-atom (micro-) maser. We then proceed to calculate approximate analytical expressions for the maser linewidth using (i) a phase-operator approach, (ii) an ansatz similar to the standard quantum theory of the laser, and (iii) a linear expansion of the correlation function. Novel features, quite distinct from the familiar Schawlow-Townes linewidth, are found, e.g., the linewidth decreases as the number of thermal photons increases.
In earlier papers, the linewidth of a micromaser was approximated by a phase diffusion rate that was obtained using a single decay rate assumption to model the decay of the electric field in the micromaser cavity. These methods gave analytic but approximate answers. Exact methods calculated the spectrum via the two-time correlation function, but required extensive numerical calculations. In this paper, we present an approximate analytic derivation of the micromaser spectrum via the two-time correlation function that is more accurate than other approximate analytic methods yet requires much less numerical calculation than the exact methods. We also review the other methods, both analytical and exact, and present numerical comparisons.
We examine the approach to equilibrium of the micromaser.Analytic methods are first used to show that for large times (i.e.many atoms) the convergence is governed by the next to leading eigenvalue of the corresponding discrete evolution matrix.The model is then studied numerically.The numerical results confirm the phase structure expected from analytic approximation methods and agree for large times with the analysis of Elmfors et al in terms of the "continuous master equation".For short times, however, we see evidence for interesting new structure not previously reported in the literature.
By calculating the two-time correlation function of a one-photon micromaser, we study the line shapes and linewidth of the micromaser spectrum. We find that the half-width at half maximum of the spectrum can be less than the smallest decay rate of the off-diagonal density-matrix elements ${\mathrm{\ensuremath{\rho}}}_{\mathit{n},}$n+1{\mathit{n}}_{+1}n+1​, that a hole can appear in the spectrum, and that the spectrum can be split by shifted spectral components.
We examine the approach to equilibrium of the micromaser. Analytic methods are first used to show that for large times (i.e. many atoms) the convergence is governed by the next to leading eigenvalue of the corresponding discrete evolution matrix. The model is then studied numerically. The numerical results confirm the phase structure expected from analytic approximation methods and agree for large times with the analysis of Elmfors et al in terms of the continuous master equation. For short times, however, we see evidence for interesting new structure not previously reported in the literature.