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

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.

References in corpus (5)

• Nonautonomous Hamiltonians
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• On the absence of bound-state stabilization through short ultra-intense fields
• Ionization Probabilities through ultra-intense Fields in the extreme Limit
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Cited by in corpus (8)

• Effective Hamiltonians for periodically driven systems
• Time independent description of rapidly oscillating potentials
• Time asymptotics of the Schroedinger wave function in time-periodic potentials
• Ionization for Three Dimensional Time-dependent Point Interactions
• Nonlinear Schroedinger equation with two symmetric point interactions in one dimension
• Decay of a Bound State under a Time-Periodic Perturbation: a Toy Case
• Dispersive estimates for Schroedinger operators: A survey
• Exact theory and numeric results for short pulse ionization of simple model atom in one dimension