Influence of finite wavelength on the quantum kicked rotator in the semiclassical regime

Abstract

The quantum kicked rotator, the classical limit of which is described by the standard map, is considered. Particular attention is devoted to a study of the effect of finite quantum wavelength on the detailed structure of phase space which appears in the classical limit. In the classical case, for large values of the nonlinearity parameter, most of the trajectories are ergodic. However, in addition to these ergodic trajectories, there can be small integrable regions of phase space, known as accelerator modes, which dominate the long-time evolution of the expected value of the particle energy. In this paper it is shown that this behavior is modified in the quantum case for small but finite values of the wavelength (i.e., Planck's constant). A simple model is presented to explain this modification. Based on our results, it is speculated that certain problems in the application of statistical concepts to intrinsically stochastic problems of classical mechanics may, in some cases, be mitigated by quantum effects.