The effects of classical resonance islands in time-dependent quantum mechanics

Abstract

We investigate the effect of the main classical resonance island on the quantal ionization probabilities for an excited one-dimensional hydrogen atom perturbed by a near resonant, periodic electric field. The classical and quantal ionization probabilities produced by a periodic electric field near the (1:1)-resonance show similarities and also remarkable differences; specifically both probabilities increase sharply at a particular frequency, just below resonance, but at a slightly higher frequency the quantal ionization probability is very sensitive to the field frequency whilst the classical probability is not. This sensitivity decreases as the resonant frequency is approached and very close to resonance classical and quantal probabilities are similar. We describe the origin of this behaviour and show how the classical dynamics profoundly affects the quantum mechanics even when the observable behaviour seems quite different. In order to compare classical and quantal dynamics near the (1:1)-resonance we develop a new approximation to the continuum. We also obtain a necessary condition that long-period orbits, producing islands within islands, affect the quantum dynamics; our conclusions differ from the interpretations of recently published results and we offer an alternative explanation of these results.