  Both in atomic physics and in mesoscopic physics it is sometimes interesting
to consider the energy time-dependence of a parametrically-driven chaotic
system. We assume an Hamiltonian ${\cal H}(Q,P;x(t))$ where $x(t)=Vt$. The
velocity $V$ is slow in the classical sense but not necessarily in the
quantum-mechanical sense. The crossover (in time) from ballistic to diffusive
energy-spreading is studied. The associated irreversible growth of the average
energy has the meaning of dissipation. It is found that a dimensionless
velocity $v_{PR}$ determines the nature of the dynamics, and controls the route
towards quantal-classical correspondence (QCC). A perturbative regime and a
non-perturbative semiclassical regime are distinguished.
