Directed Transport in classical and quantum chaotic billiards

Abstract

We construct an autonomous chaotic Hamiltonian ratchet as a channel billiard subdivided by equidistant walls attached perpendicularly to one side of the channel, leaving an opening on the opposite side. A static homogeneous magnetic field penetrating the billiard breaks time-reversal invariance and renders the classical motion partially chaotic. We show that the classical dynamics exhibits directed transport, owing to the asymmetric distribution of regular regions in phase space. The billiard is quantized by a numerical method based on a finite-element algorithm combined with the Landau gauge and the Bloch formalism for periodic potentials. We discuss features of the billiard eigenstates such as node lines and vortices in the probability flow. Evidence for directed quantum transport, inherited from the corresponding features of the classical dynamics, is presented in terms of level-velocity statistics.