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Physical Review Letters

Spontaneous and Induced Emission of Soft Bosons: Exact Non-Markovian Solution

Benjamin Fain

Published Nov 7, 1988• 13 citations
Publisher

Abstract

An exact solution of the time-dependent problem has been obtained for a degenerate two-level system interacting with a harmonic boson field. Unlike the Weisskopf-Wigner solution this solution takes into account various many-boson processes and is non-Markovian. There is no exponential decay and the asymptotic behavior explicitly depends on the initial conditions. Both spontaneous and stimulated processes have been considered.

DOI
10.1103/physrevlett.61.2197

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Journal of Mathematical Physics•Apr 1, 1974•18 citations
An exact solution of the time-dependent problem has been obtained for a degenerate two-level system interacting with a harmonic boson field. Unlike the Weisskopf-Wigner solution this solution takes into account various many-boson processes and is non-Markovian. There is no exponential decay and the asymptotic behavior explicitly depends on the initial conditions. Both spontaneous and stimulated processes have been considered.
An isomorphism is established between the configurational interaction of one discrete level with a continuum, and a relaxation via spontaneous emission of bosons in one-boson approximation. Conditions of one-boson approximation are derived. This approach enables one to overcome limitations of the Markovian or Weisskopf-Wigner approximations. The results qualitatively differ from those obtained in the master-equation approach. In a general case, the probability to remain in an excited state has asymptotically oscillatory behavior, unlike in the Weisskopf-Wigner approximation in which it tends to zero.
In this paper, a model is studied in which it was possible to obtain an expression for the exact solution of the master equation for the problem of spontaneous emission in a finite system. The model chosen for discussion is that of the Wigner-Weisskopf atom in interaction with a massless boson field, here 1-dimensional. The solution takes the form of a constant term plus a time-dependent one, expressed as the sum of residues at a series of poles along the real axis of a Laplace transform variable. Numerical calculations were performed on various aspects of the solution, and although these were too delicate to be quite certain around sensitive values of the time, the general picture is clear: After an initial decay to a value near zero, the series gave rise at fairly regular intervals to rapid and large fluctuations, the size of which never quite attains the initial value, but may nonetheless be large even after very long times. This result seems to be in agreement with the observation made by Zwanzig, namely, that for finite systems the master equation might demonstrate properties associated with the finite size of the system which would become important over certain long-time scales and would be ignored by the use of the customary ``thermodynamic limit'' of statistical mechanics. The relationship between this work and that of Montroll and Mazur and of Rubin is also discussed.
The temporal evolution of a degenerate two-level system (spin) coupled to a dissipative bath (bosons) is studied in this Rapid Communication. Closed-form expressions for the occupancy probabilities and correlation functions are found by functional methods. The non-Markovian characteristics such as explicit dependence on the initial condition and a possible long-time tail behavior are shown to be present. We also suggest a reasonable procedure for the construction of the Green's function of the nondegenerate system.
A full study, both analytic and numerical, is made of an exact solution, given in a previous paper of the series, for the problem of the spontaneous emission of a Wigner-Weisskopf atom in a one-dimensional radiation field, when the system is considered to be finite in extent. The solution is obtained directly from the Schrödinger equation of the problem. The numerical solution and results are compared extensively with two separate weak-coupling approximations, treated earlier in the series, which were derived respectively from the Prigogine-Résibois master equation and from the solution of the Schrödinger equation. It is found that the latter corresponds much better, except for exceedingly small systems, with the exact results, and that it accordingly takes better account of the effects due to the nonanalyticity of the solution when the coupling tends to zero. Some proposals are made for the general exploitation in nonequilibrium statistical mechanics of this feature, and also for a possible application to the study of radiationless transitions in molecules.
An investigation is made of the effect on the dynamics of spontaneous emission of the Wigner-Weisskopf atom in an infinite-system limit of various choices of the coupling function or form factor describing the atom's interaction with the spectrum of the radiation field. This is carried out both for the exact solution to the problem of spontaneous emission, obtained in the earlier papers in the series, and for some approximate solutions, also previously considered, in particular one based on the Schrödinger equation of the problem and one based on the weak-coupling Prigogine-Résibois master equation. The details of the form factor are found, by numerical computation of the solutions, to be critical in determining the nonexponential parts of the solutions, and these parts are seen to be capable in some cases of dominating the exponential parts, which are given only by the values of the form factor near the resonance energy. The approximate solutions discussed are found to vary widely in their worth, and one, which yields the exact solution for the Wigner-Weisskopf problem, is singled out as being of probable use in the statistical-mechanical description of more complicated systems.