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

We investigate the survival probability of a localized 1D 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 → ∞ 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.