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

TL;DR

This paper analyzes the decay and survival of a localized quantum state under harmonic forcing, providing exact results that reveal conditions for decay, non-decay, and the influence of resonances.

Contribution

It offers exact analytical results on the survival probability of a localized quantum state under time-dependent harmonic potentials, including conditions for decay and non-decay scenarios.

Findings

Survival probability tends to zero for almost all parameters as time approaches infinity.

Decay is exponential initially, then follows a power law unless near resonances or with large forcing.

Certain parameter sets lead to non-decaying states due to bound states in the Floquet operator.

Abstract

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