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Physical Review A

Nonexponential decay of a stochastic one-channel system

Frank-Michael Dittes, Hanns Ludwig Harney, A. Müller

Published Jan 1, 1992• 43 citations
Publisher

Abstract

A general formula is presented that expresses the temporal evolution of compound systems. It connects the time dependence of the density matrix with the energy dependence of the scattering matrix. Using results of random matrix theory, we then study the decay behavior of stochastic compound systems with arbitrary coupling between bound states and decay channels. As an example we consider the case of one open channel and prove a nonexponential decay law for all times that asymptotically is of the form ${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$.

DOI
10.1103/physreva.45.701

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A simple solvable model of two-channel inelastic scattering is constructed. The contact-type interaction is realized mathematically by a self-adjoint extension; if the interchannel interaction is switched off it reduces to one point interaction in each channel. The scattering matrix is calculated and the bound states and resonances are found in the weak-coupling case. The decay of the resonance state is also considered; it is shown that the asymptotic behaviour of the non-exponential corrections to the decay law depends substantially on the spectrum of the unperturbed system
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The most unstable quantum states and elementary particles possess more than a single decay channel. At the same time, it is well known that typically the decay law is not simply exponential. Therefore, it is natural to ask how to spot the nonexponential decay when (at least) two decay channels are opened. In this work, we study the tunneling phenomenon of an initially localized particle in two spatially opposite directions through two different barriers, mimicking two decay channels. In this framework, through specific quantum mechanical examples which can be accurately solved, we study the general properties of a two-channel decay that apply for various unstable quantum states (including unstable particles). Apart from small deviations at early times, the survival probability and the partial tunneling probability along the chosen direction are very well described by the exponential-decay model. In contrast, the ratios of the decay probabilities and probability currents are evidently not a simple constant (as they would be in the exponential limit), but display time-persisting oscillations. Hence, these ratios are optimal witnesses of deviations from the exponential-decay law.