Quantum chaos in the configurational quantum cat map

The motion of a classical or quantum-mechanical charged particle in the unit square (with periodic boundary conditions) is investigated under the influence of periodic electromagnetic fields. It is shown that the external fields can be chosen in such a way that the configuration space of the particle is mapped periodically to itself according to Arnold’s cat map. The time evolution of the quantum system shows the same degree of irregularity as does the classical time evolution which is completely dominated by the properties of the hyperbolic map. In particular, the eigenfunctions of the Floquet operator are determined analytically, and, as an immediate consequence, the spectrum of quasienergies in this system is seen to be absolutely continuous. Furthermore, spatial correlations decay exponentially. The observed features are in striking similarity to properties of classically chaotic systems; for example, long-time predictions of the future behavior of the system turn out to be extremely sensitive to the specification of the initial state. In other words, the time evolution of the quantum system is algorithmically complex. These phenomena, based on the formation of arbitrarily fine structures in the two-dimensional configuration space, require that the system absorb energy (provided by the external kicks) at an exponential rate.