Chaos in dissipative quantum maps: a Wigner function description

We present a semiclassical analysis of dissipative quantum maps. We show that in the presence of a small amount of dissipation the propagator of sufficiently smooth Wigner functions reduces to the classical Frobenius–Perron propagator of the phase space density for the corresponding dissipative classical map. As a consequence, the invariant Wigner function is a smeared out classical attractor. Time-dependent expectation values and correlation functions of observables are given by quantum–classical hybrid formulae, where the quantum mechanical character enters only through the initial Wigner function. If this function is classical or if the map is iterated many times, classical periodic orbit theories can be applied (L. Graham, T. Tél, Z. Phys. B 60 (1985) 127).