Quantum chaos of a particle in a square well: Competing length scales and dynamical localization

The classical and quantum dynamics of a particle trapped in a one-dimensional infinite square well with a time-periodic pulsed field is investigated. This is a two-parameter non-KAM (Kolmogorov-Arnold-Moser) generalization of the kicked rotor, which can be seen as the standard map of particles subjected to both smooth and hard potentials. The virtue of the generalization lies in the introduction of an extra parameter R, which is the ratio of two length scales, namely, the well width and the field wavelength. If R is a noninteger the dynamics is discontinuous and non-KAM. We have explored the role of R in controlling the localization properties of the eigenstates. In particular, the connection between classical diffusion and localization is found to generalize reasonably well. In unbounded chaotic systems such as these, while the nearest neighbor spacing distribution of the eigenvalues is less sensitive to the nature of the classical dynamics, the distribution of participation ratios of the eigenstates proves to be a sensitive measure; in the chaotic regimes the latter is log-normal. We find that the tails of the well converged localized states are exponentially localized despite the discontinuous dynamics while the bulk part shows fluctuations that tend to be closer to random matrix theory predictions. Time evolving states show considerable R dependence, and tuning R to enhance classical diffusion can lead to significantly larger quantum diffusion for the same field strengths, an effect that is potentially observable in present day experiments.