Introduction

Dynamical localization is a variant of Anderson localization appearing in quantum systems whose classical limit is chaotic. It is a novel quantum coherence effect—perhaps the most important new physical effect appearing in the field of quantum chaos. Like the familiar Anderson localization it is based on destructive interference of waves in random systems. What is new in dynamical localization is the fact that the randomness is not externally imposed, e.g., by a random medium, but is produced dynamically by a simple and completely deterministic system. Here the parallel to chaos (i.e. stochasticity) in deterministic classical dynamical systems with few degrees of freedom is apparent.

Dynamical localization has been reviewed in depth in the lectures of Fishman, Shepelyansky and Izrailev. Another useful review has been given in [1]. In order to set the stage it is, therefore, enough to recall a few basic facts and to mention some physical examples where this effect appears.

The discussion will be restricted to Hamiltonian systems which are either autonomous with two degrees of freedom or externally driven periodically in time with one degree of freedom. In fact, extending phase space the latter case can be viewed just as a special case of the former [2]. Classically, under conditions of chaos, the two action variables, describing the system together with the canonically conjugate angles, will undergo a diffusion process. We shall always assume that the chaotic part of phase space is sufficiently large to neglect boundary effects on the diffusion.