Quantum dissipation for the kicked particle

The evolution of the density matrix is calculated in the Wigner representation for the periodically kicked particle [H=(1/2${\mathit{p}}^2$+1/2) ${\mathit{Kx}}^2$ ${\mathrm{\Sigma }}_{\mathit{n}}$δ(t-nT)] in the presence of dissipation. The dissipation is introduced via a linear coupling to a bath of harmonic oscillators. The behavior in various regions of the parameter space for the problem is analyzed in detail and compared with the Markovian approximation. The degree of agreement with the Markovian approximation is classified. In various regions in parameter space the model reduces to various examples that were studied in the past (damped particle, harmonic oscillator). Most regions of the parameter space were not explored in the past and are of potential interest. One of these regions exhibits a limit cycle.