Chaos in dissipative quantum maps: a Wigner function description

We present a semiclassical analysis of dissipative quantum maps. We show th at in the presence of a small amount of dissipation the propagator of suffi ciently smooth Wigner functions reduces to the classical Frobenius-Perron p ropagator of the phase space density for the corresponding dissipative clas sical map. As a consequence, the invariant Wigner function is a smeared out classical attractor. Time-dependent expectation values and correlation fun ctions of observables are given by quantum-classical hybrid formulae, where the quantum mechanical character enters only through the initial Wigner fu nction. If this function is classical or if the map is iterated many times, classical periodic orbit theories can be applied (L. Graham, T. Tel, Z. Ph ys. B 60 (1985) 127). (C) 2001 Elsevier Science B.V. All rights reserved.