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Journal of Physics A Mathematical and General

Limited quantum decay

Bernard Gaveau, Lawrence S. Schulman

Published Dec 21, 1995• 45 citations
Publisher

Abstract

An isolated level coupled to a continuum of levels need not decay fully into them. Ordinarily one expects exponential decay, module finite-size effects, transients and long-time power-law tails. In the phenomenon we describe the amplitude for remaining in the initial state at first drops, but it then levels off and remains O(1) indefinitely. Because of the available continuum, this is different from certain quantum localization phenomena where there is an absence of on-shell levels. The origin of the effect we describe is the existence of thresholds and band edges. A condition relating the proximity to threshold with the strength of the coupling determines whether the limited decay occurs.

DOI
10.1088/0305-4470/28/24/029

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An isolated level coupled to a continuum of levels need not decay fully into them. Ordinarily one expects exponential decay, module finite-size effects, transients and long-time power-law tails. In the phenomenon we describe the amplitude for remaining in the initial state at first drops, but it then levels off and remains O(1) indefinitely. Because of the available continuum, this is different from certain quantum localization phenomena where there is an absence of on-shell levels. The origin of the effect we describe is the existence of thresholds and band edges. A condition relating the proximity to threshold with the strength of the coupling determines whether the limited decay occurs.
An isolated level coupled to a continuum of levels need not decay fully into them Ordinarily one expects exponential decay. modulo finite-sire effects. transients and long-time power-law tails. In the phenomenon we describe the amplitude for remaining in the initial state at first drops, but then levels off and remains O(1) indefinitely. Because-of the available continuum, this is different from certain quantum localization phenomena where there is an absence of on-shell levels. The origin of the effect we describe is, the existence af thresiiolds and band edges. A condition relating the proximity to threshold with the strength ofthe coupling determines whether the limited decay occurs.
We study the decay of a prepared state into non-flat continuum. We find that the survival probability P ( t ) might exhibit either stretched-exponential or power-law decay, depending on non-universal features of the model. Still there is a universal characteristic time t 0 that does not depend on the functional form. It is only for a flat continuum that we get a robust exponential decay that is insensitive to the nature of the intra-continuum couplings. The analysis highlights the co-existence of perturbative and non-perturbative features in the local density of states, and the nonlinear dependence of 1/ t 0 on the strength of the coupling.
The dynamics of a single quantum state embedded in one continuum or several (quasi)continua is one of the most studied phenomena in quantum mechanics. In this paper we investigate its discrete analog and consider short- and long-time dynamics based on numerical and analytical solutions of the Schr\"odinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Johansson, Phys. Rev. A 95, 053821 (2017)], describing a qubit coupled to a phonon reservoir with energy dependent coupling parameters, is identical to a qubit interacting with a finite number of parallel regularly spaced bands of states via constant couplings. As a consequence, the characteristic near periodic initial-state revivals can be viewed as a transition of probability between different continua via the reviving initial state. Furthermore, polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.
The quantum Zeno effect--suppression of decay by frequent measurements--was believed to occur only when the response of the detector is so quick that the initial tiny deviation from the exponential decay law is detectable. However, we show that it can occur even for exactly exponentially decaying systems, for which this condition is never satisfied, by considering a realistic case where the detector has a finite energy band of detection. The conventional theories correspond to the limit of an infinite bandwidth. This implies that the Zeno effect occurs more widely than expected thus far.
Controlling the decay dynamics of an unstable state coupled to a continuum is a longstanding problem of major interest in different areas of quantum and classical physics. Suppression of quan- tum mechanical decay and slow-down of the escape dynamics in macroscopic quantum and optical tunneling via dynamical decoupling methods have been proposed and experimentally demonstrated in different physical systems over the last two decades. So far most of dynamical decoupling schemes have been focused to Hermitian dynamics. In this work we consider the decay of an unstable state coupled to a continuum via a time-dependent non-Hermitian coupling with finite (albeit arbitrarily long) duration, and show rather generally that for a wide class of coupling temporal shapes, in which the real and imaginary parts of the coupling are related each other by a Hilbert transform, the metastable state returns to its initial condition after the interaction with the continuum. Re- markably, the absence of decay holds even beyond the weak-coupling (Markovian) limit and for arbitrarily slow (adiabatic) interactions. The method is illustrated by considering a paradigmatic example of decay dynamics of a metastable state coupled to a tight-binding continuum described by a Fano-Anderson model and realized by a semi-infinite tight-binding lattice. Non-Hermitian coupling can be implemented by fast modulation of the complex site energy of the edge site in the lattice.
Abstract It is known that quantum systems yield non‐exponential (power law) decay on long time scales, associated with continuum threshold effects contributing to the survival probability for a prepared initial state. For an open quantum system consisting of a discrete state coupled to continuum, we study the case in which a discrete bound state of the full Hamiltonian approaches the energy continuum as the system parameters are varied. We find in this case that at least two regions exist yielding qualitatively different power law decay behaviors; we term these the long time ‘near zone’ and long time ‘far zone’. In the near zone the survival probability falls off according to a t‐1 power law, and in the far zone i t falls off as t‐3. We show that the timescale TQ separating these two regions is inversely related to the gap between the discrete bound state energy and the continuum threshold. In the case that the bound state is absorbed into the continuum and vanishes, then the time scale TQ diverges and the survival probability follows the t‐1 power law even on asymptotic scales. Conversely, one could study the case of an anti‐bound state approaching the threshold before being ejected from the continuum to form a bound state. Again the t‐1 power law dominates precisely at the point of ejection.