Semiclassics for a Dissipative Quantum Map

Abstract

We present a semiclassical analysis for a dissipative quantum map with an area-nonpreserving classical limit. We show that in the limit of Planck's constant to 0 the trace of an arbitrary natural power of the propagator is dominated by contributions from periodic orbits of the corresponding classical dissipative motion. We derive trace formulae of the Gutzwiller type for such quantum maps. In comparison to Tabor's formula for area-preserving maps, both classical action and stability prefactor are modified by the dissipation. We evaluate the traces explicitly in the case of a dissipative kicked top with integrable classical motion and find good agreement with numerical results.