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Physical Review A

Deviations from exponential decay in the case of spontaneous emission from a two-level atom

JOSIP SEKE, Wolfgang N. Herfort

Published Jul 1, 1988• 49 citations
Publisher

Abstract

A mathematically rigorous treatment of the Weisskopf-Wigner model of atomic spontaneous emission is presented. For the first time to the authors' knowledge, it is shown that in the correct asymptotic treatment with a cutoff frequency at which the dipole approximation breaks down, the main contribution to the long-time deviation from the exponential decay is of the order of 1/(t ${\mathrm{ln}}^{2}$t). This contradicts the results of previous authors who have obtained a long-time behavior of the order of 1/${t}^{2}$ by nonrigorous mathematical treatment of the same model in the dipole approximation. However, we will show that the result 1/${t}^{2}$ can still be obtained if the retardation effects are taken into account, i.e., if no dipole approximation is made.

DOI
10.1103/physreva.38.833

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A mathematically rigorous treatment of the Weisskopf-Wigner model of atomic spontaneous emission is presented. For the first time to the authors' knowledge, it is shown that in the correct asymptotic treatment with a cutoff frequency at which the dipole approximation breaks down, the main contribution to the long-time deviation from the exponential decay is of the order of 1/(t ln2{\mathrm{ln}}^{2}ln2t). This contradicts the results of previous authors who have obtained a long-time behavior of the order of 1/t2{t}^{2}t2 by nonrigorous mathematical treatment of the same model in the dipole approximation. However, we will show that the result 1/t2{t}^{2}t2 can still be obtained if the retardation effects are taken into account, i.e., if no dipole approximation is made.
For the first time to the authors' knowledge a mathematically rigorous method is used for treating finite-time deviations from the exponential decay in the case of spontaneous Lyman-\ensuremath{\alpha} transitions in a two-level hydrogenic atom. First, in the so-called Weisskopf-Wigner model (where the rotating-wave approximation is implied) finite-time deviations with a rigorous validity range, based on accurate error estimations, are derived. Second, in order to obtain the frequency shift (Lamb shift), counter-rotating terms are taken into account by using a projection-operator method developed a decade ago by one of the present authors [J. Seke, Phys. Rev. A 21, 2156 (1980)]. By means of this method, equations of motion for dipole-moment expectation values are derived without employing the usual Born approximation. These equations are solved by using the method of the Laplace transform and its inverse. Again finite-time deviations from the exponential decay of the dipole-moment expectation values with a definite validity range are obtained. Finally, it should be emphasized that all the results presented in this paper contain accurate error estimates which are absolutely necessary in a rigorous mathematical method.
The exact determination of the spontaneous emission coefficient for an excited atom is an extremely complex problem so various approximations are typically used. One of the most popular ones is the use of the dipole approximation of a two-level atom followed by rotating wave approximation (RWA). However, such an approach applied to the entire frequency spectrum results in the appearance of divergent integrals in the derivations, which are not treated rigorously in typical student textbooks. It is known from the literature that the introduction of cutoff for frequencies, justified by the finite size of the atom, may solve this problem. For didactic purposes, in this paper, we introduce a mathematically simple cutoff, which allows for a straightforward yet mathematically consistent rederivation of the Weisskopf-Wigner spontaneous emission rate (up to the small correction) within RWA. Importantly, this cutoff is not a mathematical trick to make calculations easier but is related to a real feature of the physical system, the neglect of which leads to inconsistency. More precise analysis demand going beyond RWA and dipole approximation.
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.
The time-dependent electric energy density surrounding a two-level atom fixed at r = 0 is studied, the atom being taken in its excited state at t = 0 and the field being initially in the vacuum state. The atom-field coupling includes both rotating and counter-rotating terms. The energy density of the spontaneously emitted field in the rotating wave approximation is shown to behave non-causally, while in the presence of the complete coupling it is shown to vanish outside a sphere of radius r = ct centred on the atom. The deviations of atomic dynamics from the exponential Wigner-Weisskopf behaviour during spontaneous decay are shown to be deeply influenced by the counter-rotating terms. It is concluded that the virtual photons induced by the counter-rotating terms in the atom-field coupling are essential in order to ensure causality and cannot be neglected in any accurate treatment of spontaneous emission.
We consider a quantum-mechanical analysis of spontaneous emission in terms of an effective two-level system with a vacuum decay rate ${\ensuremath{\Gamma}}_{0}$ and transition angular frequency ${\ensuremath{\omega}}_{A}$. Our analysis is in principle exact, even though presented as a numerical solution of the time evolution including memory effects. The results so obtained are confronted with previous discussions in the literature. In terms of the dimensionless lifetime $\ensuremath{\tau}=t{\ensuremath{\Gamma}}_{0}$ of spontaneous emission, we obtain deviations from exponential decay of the form $O(1/\ensuremath{\tau})$ for the decay amplitude as well as the previously obtained asymptotic behaviors of the form $O(1/{\ensuremath{\tau}}^{2})$ or $O(1/\ensuremath{\tau}{\mathrm{ln}}^{2}\ensuremath{\tau})$ for $\ensuremath{\tau}\ensuremath{\gg}1$. The actual asymptotic behavior depends on the adopted regularization procedure as well as on the physical parameters at hand. We show that for any reasonable range of $\ensuremath{\tau}$ and for a sufficiently large value of the required angular frequency cutoff ${\ensuremath{\omega}}_{c}$ of the electromagnetic fluctuations, i.e., ${\ensuremath{\omega}}_{c}\ensuremath{\gg}{\ensuremath{\omega}}_{A}$, one obtains either a $O(1/\ensuremath{\tau})$ or a $O(1/{\ensuremath{\tau}}^{2})$ dependence. In the presence of physical boundaries, which can change the decay rate with many orders of magnitude, the conclusions remains the same after a suitable rescaling of parameters.