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arXiv:math-ph/0608031 · Journal of Physics A Mathematical and General

Decay versus survival of a localized state subjected to harmonic forcing: exact results

A. Rokhlenko, Ovidiu Costin, Joel L. Lebowitz

Published Oct 8, 2002• 18 citations
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Abstract

We investigate the survival probability of a localized 1D quantum particle subjected to a time-dependent potential of the form rU ( x ) sin ω t with U ( x ) = 2δ( x − a ) or U ( x ) = 2δ( x − a ) − 2δ( x + a ). The particle is initially in a bound state produced by the binding potential −2δ( x ). We prove that this probability goes to zero as t → ∞ for almost all values of r , ω and a . The decay is initially exponential followed by a t −3 law if ω is not close to resonances and r is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters r , ω and a the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behaviour even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.

DOI
10.1088/0305-4470/35/42/305

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We investigate the survival probability of a localized 1D quantum particle subjected to a time-dependent potential of the form rU ( x ) sin ω t with U ( x ) = 2δ( x − a ) or U ( x ) = 2δ( x − a ) − 2δ( x + a ). The particle is initially in a bound state produced by the binding potential −2δ( x ). We prove that this probability goes to zero as t → ∞ for almost all values of r , ω and a . The decay is initially exponential followed by a t −3 law if ω is not close to resonances and r is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters r , ω and a the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behaviour even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.
We investigate the ionization of a 1-D quantum system with a zero-range attractive potential at x = 0 when it is subjected to a time dependent harmonic zero range potential forcing of frequency /spl omega/, strength r, at position x = a. We show that the ionization probability goes to 1 as time t /spl rarr/ /spl infin/ for almost all values of r, /spl omega/, and a. The initially exponential decay of the probability to stay in the bound state is followed at later time by a t/sup -3/ law if r is small and /spl omega/ is not close to resonances. When this is not true the separation between the exponential and power law decay regimes disappears and Fermi's golden rule fails. There exist exceptional sets of parameters where the survival probability never decays to zero and complete ionization is impossible.
We study the time evolution of the wave function of a particle bound by an attractive δ-function potential when it is subjected to time-dependent variations of the binding strength (parametric excitation). The simplicity of this model permits certain nonperturbative calculations to be carried out analytically both in one and three dimensions. Thus the survival probability of bound state |θ(t)|2, following a pulse of strength r and duration t, behaves as |θ(t)|2−|θ(∞)|2∼t−α, with both θ(∞) and α depending on r. On the other hand, a sequence of short pulses produces an exponential decay over an intermediate time scale.
Total Absorption in Finite Time in an iδiδiδ Potential
We consider the evolution of Green's function of the one-dimensional Schrödinger equation in the presence of the complex potential −ikδ(x)-ikδ(x)−ikδ(x). Our result is the construction of an explicit time-dependent solution which we use to calculate the time-dependent survival probability of a quantum particle. The survival probability decays to zero in finite time, which means that the complex delta potential well is a total absorber for quantum particles. This potential can be interpreted as a killing measure with infinite killing rate concentrated at the origin.
For a model 1d asymmetric double-well potential we calculated so-called survival probability (i.e. the probability for a particle initially localised in one well to remain there). We use a semiclassical (WKB) solution of Schroedinger equation. It is shown that behaviour essentially depends on transition probability, and on dimensionless parameter which is a ratio of characteristic frequencies for low energy non-linear in-well oscillations and inter wells tunnelling. For the potential describing a finite motion (double-well) one has always a regular behaviour. For the small value of the parameter there is well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However for the large value of the parameter no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case one may not restrict oneself to only resonance pair levels. The number of perturbed by tunnelling levels grows proportionally to the value of this parameter (by other words instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of the parameter one has a crossover between both limiting cases, namely the exponential decay with subsequent long period recurrent behaviour.
The long time behaviour of the survival probability of the initial state and its dependence on the initial states are considered, for the one-dimensional free quantum particle. We derive the asymptotic expansion of the time evolution operator at long times, in terms of the integral operators. This enables us to obtain the asymptotic formula for the survival probability of the initial state ψ( x ), which is assumed to decrease sufficiently rapidly at large | x |. We then show that the behaviour of the survival probability at long times is determined by that of the initial state ψ at zero momentum k = 0. Indeed, it is proved that the survival probability can exhibit the various power decays such as t −2 m −1 for an arbitrary non-negative integer m as t → ∞, corresponding to the initial states with the condition ( k ) = O ( k m ) as k → 0.
We investigate the exact quantum dynamics of a free particle damped through its interaction with an harmonic bath, when the effective coupling strength behaves as ωδ at low frequency. We find that various regimes can occur depending on the value of δ and of the temperature. At large times, the mean square displacement is shown either to diverge as tv, the exponent v being never greater than 2 or to tend towards a finite value, indicating a confinement in the broad sense. In addition, for δ < 1, an oscillation (absent in the uncoupled system) is found : due to the friction on the dominant low-frequency modes and to the existence of retardation effects, the particle is forced to relax in the mean towards its initial location, the net dynamical effect of the bath being similar to that of an applied potential. In the limit δ → 0, the particle is frozen.