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

O. Costin·J. L. Lebowitz·A. Rokhlenko
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Abstract

We investigate the survival probability of a localized 1-d quantum particle subjected to a time dependent potential of the form $rU(x)\sin{ωt}$ 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 $t\to\infty$ 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 behavior even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential.

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O. Costin, J. L. Lebowitz, A. Rokhlenko. 2006-08-13. Decay versus survival of a localized state subjected to harmonic forcing: exact results. https://doi.org/10.1088/0305-4470%2F35%2F42%2F305

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