Didier Robert
Time Evolution of States for Open Quantum Systems
(618K, pdf)
ABSTRACT. 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.
For this formula the quadratic assumption is not necessary, more general Hamiltonians can be considered.\
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 written as a convolution integral for the reduced
density at initial time $t_0=0$, with a Gaussian kernel, for $0 \leq t < t_c$ where $t_c\in ]0, \infty]$ is a critical time. Reversibility is lost for $t \geq t_c$.