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Journal of Physics A Mathematical and General

Estimate for regeneration up to the golden rule time

Ioannis E. Antoniou, Jacob Levitan, Lawrence Paul Horwitz

Published Jul 7, 1993• 2 citations
Publisher

Abstract

The authors study the regeneration contribution to the decay law in Wigner-Weisskopf theory for times less than and up to the golden rule time. A power series expansion for the regeneration term and the part of the product of the amplitudes which has the semigroup property is carried out in second-order perturbation theory, the same order to which the Wigner-Weisskopf calculation is carried out in their estimate of the line widths in atomic decay. They show that the regeneration contribution as a smaller leading behaviour in t than the amplitudes at times of the order of the golden rule time, thus accounting for an approximate semigroup behaviour, on this scale, within the framework of the Wigner-Weisskopf theory. For very short times, the estimates of Misra and Sinha (1977) are obtained.

DOI
10.1088/0305-4470/26/13/026

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The authors study the regeneration contribution to the decay law in Wigner-Weisskopf theory for times less than and up to the golden rule time. A power series expansion for the regeneration term and the part of the product of the amplitudes which has the semigroup property is carried out in second-order perturbation theory, the same order to which the Wigner-Weisskopf calculation is carried out in their estimate of the line widths in atomic decay. They show that the regeneration contribution as a smaller leading behaviour in t than the amplitudes at times of the order of the golden rule time, thus accounting for an approximate semigroup behaviour, on this scale, within the framework of the Wigner-Weisskopf theory. For very short times, the estimates of Misra and Sinha (1977) are obtained.
There is given a generalization of the theorem of Misra and Sinha [Helv. Phys. Acta 50, 99 (1977)], which shows that in the decay process the regeneration of the undecayed unstable states from the decay products cannot proceed too slowly at arbitrarily short time and, correspondingly, the deviation from exponential decay at arbitrarily short time cannot be too small either. The generalization presented here is shown to be valid in the region where the theorem of Misra and Sinha cannot be applied.
BEYOND THE WEISSKOPF–WIGNER APPROXIMATION IN THE K0−Kˉ0K^0-\bar K^0K0−Kˉ0 SYSTEM
We reanalyze the time evolution of the [Formula: see text] system in the language of certain spectral function whose Fourier transforms give the time-dependent survival and transition amplitudes. Approximating the spectral function by a one-pole ansatz we obtain insight into the limitation of the validity of the one-pole approximation. It will be shown that the same validity restrictions apply to the known formulae of the Weisskopf–Wigner approximation as well. The present analysis can also be applied to the effect of so-called vacuum regeneration of KL and KS. As a result of this possibility new contributions to the well-known oscillatory terms will enter the time-dependent transition probabilities. It will be shown that the order of magnitude of this new effect is very small and, in principle, its exact determination lies outside the scope of the one-pole ansatz.
We reconsider the problem of the spontaneous emission of light by an excited atomic state. We scrutinize the survival probability of this excited state for very short times, in the so-called Zeno regime, for which we show that the dynamics is dictated by a coherent, in-phase, response of the on-shell and off-shell vacuum modes. We also develop a perturbative approach in order to interpolate between different temporal regimes: the Zeno, golden rule (linear), and Wigner-Weisskopf (exponential) regimes. We compare results obtained with the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{E}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{x}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$ interaction Hamiltonians, using successively the dipole approximation and the exact coupling.
A novel and readily understandable derivation of the Golden Rule of time dependent perturbation theory is presented. The derivation is based on adiabatic turning on of the perturbation as used, for instance, in some formal developments of scattering theory. Energy conservation is expressed in terms of an intuitively and physically appealing Lorentzian line shape rather than the artificial, oscillatory sin(x)=x type line shape that appears in conventional derivations. The conditions for the derivation's validity are compactly and conveniently expressed in the frequency/energy domain rather than in the usual time domain. The derivation also highlights how, along with energy conservation, the transition rate faithfully and instantaneously follows the variations in the square of the perturbing potential as one may expect in the adiabatic limit. In the first instance, the adiabatic turning on is achieved, as usual, by a single exponential time variation. But we demonstrate that the instantaneous following of the square of the perturbing potential by the transition rate is more general and that one can derive the Golden Rule for a general slowly varying time dependent perturbation. This allows one to derive generalisations of the simple decay law, originally derived in the classic paper by Weisskopf and Wigner; a tutorial exposition of the essence of this classic work is provided. The Oppenheimer method for applying the Golden rule to problems, such as the electric field ionisation of atoms, in which the perturbing potential can also create the final states, is reviewed. No use of an energy gap condition is needed to derive our results on adiabatic behaviour in contrast to the origianl derivations of the adiabatic theorem in quantum mechanics.
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.
We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation, the evolution of a total system subspace is not an exact semigroup for multichannel decay unless the projectors into eigenstates of the reduced evolution generator W(z)W(z)W(z) are orthogonal. With multichannel decay, the projectors must be evaluated at different pole locations ${z}_{\ensuremath{\alpha}}\ensuremath{\ne}{z}_{\ensuremath{\beta}}$, and since the orthogonality relation does not generally hold at different values of zzz, the semigroup evolution is a poor approximation for the multichannel decay, even for very weak coupling. Nevertheless, if the theory is generalized to take into account interactions with an environment, one can ensure orthogonality of the W(z)W(z)W(z) projectors regardless of the number of poles. Such a possibility occurs when W(z)W(z)W(z), and hence its eigenvectors, is independent of zzz, which corresponds to the Markovian limit of the coupling to the continuum spectrum.