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Journal of Modern Optics

Exact Equations of Motion for Density-matrix Elements and Numerical Results for Many-atom Spontaneous Emission in a Damped Cavity

JOSIP SEKE, Frank Rattay

Published May 1, 1987• 3 citations
Publisher

Abstract

An exact closed set of first-order linear differential equations for density-matrix elements, expressible in terms of a simple recurrence equation, is derived. This recurrence equation makes possible numerical calculations for a larger number N of atoms and arbitrary cavity damping in the case of spontaneous emission. For the first time, numerical results are presented for N = 3 10 and 18 in the case of high-Q cavities. A comparison with the approximate calculations of Bonifacio et al. is also given for low-Q cavities.

DOI
10.1080/09500348714550671

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An exact closed set of first-order linear differential equations for density-matrix elements, expressible in terms of a simple recurrence equation, is derived. This recurrence equation makes possible numerical calculations for a larger number N of atoms and arbitrary cavity damping in the case of spontaneous emission. For the first time, numerical results are presented for N = 3 10 and 18 in the case of high-Q cavities. A comparison with the approximate calculations of Bonifacio et al. is also given for low-Q cavities.
An exact closed system of linear differential equations of the first order, which can be expressed by a simple recurrence equation, is derived for matrix elements of the statistical density operator. This exact recurrence equation opens the way for numerical solutions for collective many-atom spontaneous emission in cavities with arbitrary damping and detuning. For the first time to the authors’ knowledge, numerical results are presented for N = 10 and N = 15. Moreover, the influence of the cavity damping and detuning on the collective spontaneous emission is discussed. New effects, such as the suppression of the superfluorescent effects in strongly detuned cavities, are also pointed out.
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In this paper the equations of motion for open systems, derived recently by one of the present authors, are applied to the case of spontaneous emission from a system of NNN identical two-level atoms. Three different observation levels of the atomic system are considered. First, exact closed integrodifferential equations for the expectation values of each observation level are obtained. Second, making the standard approximations---Born, Markov, and long-time approximations, we find that these equations reduce to closed differential equations for the expectation values. The quality of the Born approximation for different observation levels is also discussed. Moreover, in the obtained equations no kind of rotating-wave approximation was made.