Quantum Brownian motion

We study the behavior of a subsystem (harmonic oscillator) in contact with a thermal reservoir (finite set of uncoupled harmonic oscillators). We exactly solve the eigenvalue problem and obtain the temporal evolution of the dynamical variables of interest. We show how the subsystem goes to equilibrium and give quantitative estimates of the Poincar\'e recurrence times. We study the behavior of the subsystem mean ocuppation number in the limit of a dense bath and compare it with the expected exponential decay law.