Dissipation, diffusion, and the quantum-classical limit in phase space

We discuss the role of dissipation and diffusion in the classical limit of quantum mechanics, for a kicked harmonic oscillator, in terms of phase-space distributions. A simple physical derivation is presented for the separation time when dissipation without diffusion is present, which would be the case for a zero-temperature reservoir. This expression displays a logarithmic dependence of this time with respect to the classicality parameter, which in this case coincides with the Lamb-Dicke parameter. We show that, for sufficiently large diffusion, this dependence becomes algebraic, so that classical and quantum dynamics stay together for much longer times, the separation between the classical and the quantum distributions remaining bounded throughout the evolution of the system.