  We perform numerical studies of the wave packet propagation through open
quantum billiards whose classical counterparts exhibit regular and chaotic
dynamics. We show that for t less or similar to tau (tau being the Heisenberg
time), the features in the transmitted and reflected currents are directly
related to specific classical trajectories connecting the billiard leads. In
contrast, the long-time asymptotics of the wave packet dynamics is
qualitatively different for classical and quantum billiards. In particularly,
the decay of the quantum system obeys a power law that depends on the number of
decay channels, and is not sensitive to the nature of classical dynamics
(chaotic or regular).
