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Exact results for ionization of a model atom

A. Rokhlenko, Ovidiu Costin, Joel L. Lebowitz

Published Nov 14, 2002• 0 citations
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Abstract

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.

DOI
10.1109/ppps.2001.1002009

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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 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 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.
We investigate the time evolution of the decay (or ionization) probability of a D -dimensional model atom ( D = 1, 2, 3) in the presence of a uniform (i.e., static and homogeneous) background field. The model atom consists in a non-relativistic point particle in the presence of a point-like attractive well. It is shown that the model exhibits infinitely many resonances leading to possible deviations from the naive exponential decay law of the non-decay (or survival) probability of the initial atomic quantum state. Almost stable states exist due to the presence of the attractive interaction, no matter how weak it is. Analytic estimates as well as numerical evaluation of the decay rates are explicitly given and discussed.
We study the time dependence of the ionization probability of Rydberg atoms driven by a microwave field, both in classical and in quantum mechanics. The quantum survival probability follows the classical one up to the Heisenberg time and then decays algebraically as P(t) approximately 1/t. This decay law derives from the exponentially long times required to escape from some region of the phase space, due to tunneling and localization effects. We also provide parameter values which should allow one to observe such decay in laboratory experiments.
In the paper, a simple model of alpha decay with Dirac delta potential is studied. The model leads to breakdown of the exponential decay and to power law behavior at asymptotic times. Time dependence of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of Green's function representation and numerical solution of the time-dependent Schrödinger equation. In particular, finite depth potential wells and behavior between the exponential and power law regimes, which are situations that could not be described in detail analytically, are studied. The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime. Moreover, the constructive and destructive interference is observed in the intermediate stage of the process. The simple alpha decay model is compared to the results of Rothe- Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.