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Physical Review Letters

Resonant Spectra and the Time Evolution of the Survival and Nonescape Probabilities

Gastón García‐Calderón, J. L. Mateos, M. Moshińsky

Published Jan 16, 1995• 87 citations
Publisher

Abstract

The time evolution of quantum decay of an arbitrary state initially located within an interaction region of finite range is investigated. We compare the survival $S(t)$ and nonescape $P(t)$ probabilities. Our approach considers the full resonant spectra of the system using a novel representation of the time-dependent Green function. It is shown that, for an initial state near a resonance, $S(t)$ and $P(t)$ exhibit at long times a different behavior; not only the onset to a power law decay occurs at a different time, but also instead of the well known $S(t)\ensuremath{\sim}{t}^{\ensuremath{-}3}$ we obtain $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}$.

DOI
10.1103/physrevlett.74.337

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In [G. Garcia-Calderon, J. L. Mateos, and M. Moshinsky, Phys. Rev. Lett. 74, 337 (1995)], the time evolution of the quantum decay of a state initially located within an interaction region of finite range was investigated. In particular, it was shown that the survival and nonescape probabilities behave differently at very large times. The purpose of this Comment is to show that they have the same asymptotic behavior.

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An analytical solution for the time evolution of decay of two identical noninteracting quantum particles seated initially within a potential of finite range is derived using the formalism of resonant states. It is shown that the wave function, and hence also the survival and nonescape probabilities, for factorized symmetric and entangled symmetric or antisymmetric initial states evolve in a distinctive form along the exponentially decaying and nonexponential regimes. Our findings show the influence of the Pauli exclusion principle on decay. We exemplify our results by solving exactly the $s$-wave $\ensuremath{\delta}$ shell potential model.

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The time evolution of quantum decay of an arbitrary state initially located within an interaction region of finite range is investigated. We compare the survival S(t)S(t)S(t) and nonescape P(t)P(t)P(t) probabilities. Our approach considers the full resonant spectra of the system using a novel representation of the time-dependent Green function. It is shown that, for an initial state near a resonance, S(t)S(t)S(t) and P(t)P(t)P(t) exhibit at long times a different behavior; not only the onset to a power law decay occurs at a different time, but also instead of the well known S(t)\ensuremath∼t\ensuremath−3S(t)\ensuremath{\sim}{t}^{\ensuremath{-}3}S(t)\ensuremath∼t\ensuremath−3 we obtain P(t)\ensuremath∼t\ensuremath−1P(t)\ensuremath{\sim}{t}^{\ensuremath{-}1}P(t)\ensuremath∼t\ensuremath−1.
The short-time behavior of quantum decay of an unstable state initially located within an interaction region of finite range is investigated using a resonant expansion of the survival amplitude. It is shown that in general the short-time behavior of the survival probability S(t)S(t)S(t) has a dependence on the initial state and may behave as either $S(t)=1\ensuremath{-}O({t}^{3/2})$ or $1\ensuremath{-}O({t}^{2})$. These cases are illustrated by solvable models. The experiment reported by Wilkinson et al. [Nature (London) 387, 575 (1997)] does not distinguish between the above short-time behaviors.
ABSTRACT. In this work we compare the notions of survival 5(t) and nonescape P(t) probabilities for quantum decay. \\Ve consider the solution of the time dependent Schriidinger equation for a delta-potential as an initial value problem and obtain an exact solution far both probabilities as an infinite sum that involves the set oC resonant states associated to the complex energies oí the system. \\Ve discuss sorne numerical results that show that 5(t) and P(t) only coincide in a restricted time interva1. RESUMEN.En este trabajo se comparan las probabilidades de sobrevivencia 5(t) y no escape P(t) en el decaimiento cuántico. Se considera la solución de la ecuación de Schriidinger dependiente del tiempo para un potencial delta y se obtiene la solución exacta para ambas probabilidades como una suma infinita que involucra al conjunto de estados resonantes asociados a las energías complejas del sistema. Se discuten algunos resultados numéricos que muestran que 5(t) y P(t) coinciden en un intervalo restringido del tiempo. PACS: 03.65.Nk; 21.10.Tg 1.
An analytical solution for the time evolution of decay of two identical noninteracting quantum particles seated initially within a potential of finite range is derived using the formalism of resonant states. It is shown that the wave function, and hence also the survival and nonescape probabilities, for factorized symmetric and entangled symmetric or antisymmetric initial states evolve in a distinctive form along the exponentially decaying and nonexponential regimes. Our findings show the influence of the Pauli exclusion principle on decay. We exemplify our results by solving exactly the sss-wave $\ensuremath{\delta}$ shell potential model.
The behavior of both the survival S(t)S(t)S(t) and the nonescape P(t)P(t)P(t) probabilities at long times for the one-dimensional free-particle system is shown to be closely connected to that of the initial wave packet at small momentum. We prove that both S(t)S(t)S(t) and P(t)P(t)P(t) asymptotically exhibit the same power-law decrease at long times, when the initial wave packet in momentum representation behaves as km{k}^{m}km with m=0m=0m=0 or 111 at small momentum. On the other hand, if the integer m becomes greater than 1,1,1, S(t)S(t)S(t) and P(t)P(t)P(t) decrease in different power laws at long times.
An analytical solution to the time evolution of decay of one and two identical noninteracting particles is presented using the formalism of resonant states. It is shown that the time-dependent wave function and hence the survival and nonescape probabilities for the initial state of a single particle and entangled symmetric and antisymmetric initial states of two identical particles evolve in a distinctive form along the exponential and long-time nonexponential decaying regimes. In particular, for the last regime, they exhibit different inverse power of time behaviors.