Comparative Study of Spontaneous Emission and Spin-Lattice Relaxation atT=0

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Abstract
In this paper we present a comparative study of spontaneous emission and spin-lattice relaxation at zero temperature. In particular, we study the time evolution of the density matrix for two simple models as determined from an analysis of the Prigogine-Résibois master equation. The first model treated is that of the Wigner-Weisskopf atom in a three-dimensional radiation field; the second model is that of a single, effective spin in interaction with the phonon modes of a three-dimensional lattice. The divergence which arises in the solution of the master equation for the first model is avoided using a frequency cutoff. A frequency cutoff in the second model is imposed by the upper bound of the spectrum of modes in the crystal, and this fact manifests itself when one integrates over the first Brillouin zone only. From a detailed numerical study of the analytic results obtained in solving the master equation, we find that for both models the relaxation to equilibrium is characterized, in part, by a sequence of slowly damped oscillations. This result seems to be in agreement with the observation made by Zwanzig, namely, that exponential decay in time seems not to be universal, and may, in fact, be hidden behind some other kind of time dependence. The numerical study also reveals, however, that the nonexponential modes of decay can be quantitatively different in magnitude and qualitatively different in structure for atomic versus spin systems. Finally, based on the solution obtained for the spin problem, an estimate is made of the relaxation time for cerium ethyl sulfate, and this estimate is found to be consistent with experiment.