Lyapunov spreading of semiclassical wave packets for the Lorentz gas: theory and applications

We consider the quantum-mechanical propagator for a particle moving in a d -dimensional Lorentz gas, with fixed, hard-sphere scatterers. To evaluate this propagator in the semiclassical region, and for times less than the Ehrenfest time, we express its effect on an initial Gaussian wave packet in terms of quantities analogous to those used to describe the exponential separation of trajectories in the classical version of this system. This result relates the spread of the wave packet to the rate of separation of classical trajectories, characterized by positive Lyapunov exponents. We consider applications of these results, first to illustrate the behavior of the wave-packet autocorrelation functions for wave packets on periodic orbits. The autocorrelation function can be related to the fidelity, or Loschmidt echo, for the special case that the perturbation is a small change in the mass of the particle. An exact expression for the fidelity, appropriate for this perturbation, leads to an analytical result valid over very long time intervals, inversely proportional to the size of the mass perturbation. For such perturbations, we then calculate the long-time echo for semiclassical wave packets on periodic orbits.