Time Evolution of States for Open Quantum Systems. The quadratic case

Our main goal in this paper is to extend to any system of coupled quadratic Hamiltonians some properties known for systems of quantum harmonic oscillators related with the Brownian Quantum Motion model. In a first part we get a rather general formula for the purity (or the linear entropy) in a short time approximation. In a second part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for bilinearly coupled quadratic Hamiltonians. The Hamiltonians and the bilinear coupling can be time dependent. Moreover we give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time $t$ is connected with the reduced density at initial time $t_0$ for $t_0 \leq t <t_0 +t_c$ where $t_c\in ]0, \infty]$ is a critical time but reversibility is lost for $t \geq t_0 +t_c$.