On the relaxation to quantum-statistical equilibrium of the Wigner–Weisskopf atom in a one-dimensional radiation field. VII. Emission in a finite system in the presence of an extra photon
In this paper, we present the exact solution to a problem previously unsolved in radiation theory: the emission of a two-level atom in a (one-dimensional) radiation field in the presence of an extra photon. The solution is obtained directly from the Schrödinger equation of the problem using techniques suggested by the work of Muskhelishvili on singular integral equations. The solution corresponding to a finite system as well as the one corresponding to a system infinite in extent are given, although our primary concern in this paper is the finite-system problem. For a particular choice of initial condition, the probability at time t that the two-level atom is in the excited state is found, and the effects of system size and choice of coupling function are studied numerically for a given coupling constant. Our results are compared with those obtained in an earlier paper of the series, wherein the spontaneous emission of a Wigner-Weisskopf atom in a (one-dimensional) field of radiation for a system finite in extent was studied. The physical effects calculated and the conclusions drawn from our comparative studies are all in accord with simple intuition regarding the problem. The paper concludes with some brief remarks on problems in radiation theory which are accessible to study given the methods laid down in this paper.
In this paper, we present the exact solution to a problem previously unsolved in radiation theory: the emission of a two-level atom in a (one-dimensional) radiation field in the presence of an extra photon. The solution is obtained directly from the Schrödinger equation of the problem using techniques suggested by the work of Muskhelishvili on singular integral equations. The solution corresponding to a finite system as well as the one corresponding to a system infinite in extent are given, although our primary concern in this paper is the finite-system problem. For a particular choice of initial condition, the probability at time t that the two-level atom is in the excited state is found, and the effects of system size and choice of coupling function are studied numerically for a given coupling constant. Our results are compared with those obtained in an earlier paper of the series, wherein the spontaneous emission of a Wigner-Weisskopf atom in a (one-dimensional) field of radiation for a system finite in extent was studied. The physical effects calculated and the conclusions drawn from our comparative studies are all in accord with simple intuition regarding the problem. The paper concludes with some brief remarks on problems in radiation theory which are accessible to study given the methods laid down in this paper.
In this paper we study the emission of a two-level atom in a radiation field in the case where one mode of the field is assumed to be excited initially, and where the system is assumed to be of infinite extent. (The restriction to a one-dimensional field, which has been made throughout this series, is not essential: It is made chiefly for ease of presentation of the mathematical methods.) An exact expression is obtained for the probability ρ (t) that the two-level quantum system is in the excited state at time t. This problem, previously unsolved in radiation theory, is tackled by reformulating the expression found in VII [J. Math. Phys. 16, 1013 (1975)] of this series for the time evolution of ρ (t) in a finite system in the presence of an extra photon, and then constructing the infinite-system limit. A quantitative assessment of the role of the extra photon and of the coupling constant in influencing the dynamics is obtained by studying numerically the expression derived for ρ (t) for a particular choice of initial condition. The study presented here casts light on the problem of time-reversal invariance and clarifies the sense in which exponential decay is universal; in particular, we find that: (1) It is the infinite-system limit which converts the time-reversible solutions of VII into the irreversible solution obtained here, and (2) it is the weak-coupling limit that imposes exponential form on the time dependence of the evolution of the system. The anticipated generalization of our methods to more complicated radiation-matter problems is discussed, and finally, several problems in radiation chemistry and physics, already accessible to exact analysis given the approach introduced here, are cited.
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.
In pursuit of the study begun in the preceding paper of the spontaneous emission of a Wigner-Weisskopf atom in a one-dimensional radiation field, the exact solution of this problem in the thermodynamic limit of a system of infinite extent is examined. The analysis reveals the existence in general of three contributions to the probability of the atom's being excited at any moment: a constant contribution and two sorts of time-dependent ones, exponential and nonexponential. For a specific choice of coupling function, this probability is written down in closed form. The link between the constant contribution and the appearance of ghost states and ergodicity is investigated, and criteria given to determine when this contribution does not arise. A numerical study is made of the expressions obtained, and the nonexponential part of the solution compared with approximations for it which have been previously obtained by various means. It is concluded that the predictions of the weak-coupling Prigogine-Résibois master equation are inadequate except when the model is ergodic in a certain sense, and some suggestions are made concerning the relevance of these studies to nonequilibrium statistical mechanics.
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.
A study is undertaken to cast light on difficulties, which arose in the first two papers of this series, pertaining to the occurrence of negative probabilities in the weak-coupling solution of the generalized Prigogine-Résibois master equation for the model of the Wigner-Weisskopf atom in a one-dimensional radiation field. The Schrödinger equation is solved exactly for the model with the initial condition for spontaneous emission, and then the weak-coupling approximations to the solution, both for an infinite and for a finite system, are derived as inverse Laplace transform integrals. An extensive analysis, theoretical and numerical, of these is undertaken, and comparison is made with the corresponding results based on the master equation. In particular, quantitative estimates of the Poincaré recurrence times for finite systems are made. It is found that considerable differences exist between the statistical-mechanical and quantum-mechanical results, but that both manifest nonanalyticity in the coupling parameter as it tends to zero. Suggestions are given for further work toward the resolution of these discrepancies and a better understanding of the weak-coupling limit.
A theoretical study of the phenomenon of spontaneous emission has been carried out, using as a model the Wigner-Weisskopf atom in a one-dimensional radiation field. The calculation is performed within the framework of the Prigogine theory of nonequilibrium statistical mechanics. When the model is solved exactly to first order in the coupling parameter α and the evolution in time of the diagonal elements of the density matrix ρ is studied, it is found that 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. An approximate theory is developed alongside the exact one, and corresponding terms in each treatment are compared numerically. It is found that, for small values of the coupling parameter α (α ≤ 0.1) and for sufficiently large values of τ, defined as τ = αEt where E is frequency and t is time, the approximate theory gives a satisfactory representation of the exact solution to first order in α. The importance and relevance of the model introduced by Van Hove and coworkers, in which nonexponential behavior was also observed, will be noted but not stressed in this paper, as this relationship will be developed in considerable detail in a subsequent contribution. Finally, the possible relevance of the theory to a problem of interest in magnetic resonance is mentioned.