标题： 开放量子系统的态的时间演化。 二次情形 显示英文标题

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

摘要： 本文的主要目标是将一些已知于布朗量子运动模型相关的量子谐振子系统性质推广到任意耦合二次哈密顿量的系统。在第一部分，我们得到了一个在短时间近似下的纯度（或线性熵）的一般公式。在第二部分，我们为双线性耦合二次哈密顿量的约化矩阵密度的时间演化建立了主方程（或福克-普朗克型方程）。哈密顿量和双线性耦合可以是时间相关的。此外，我们给出了此主方程解的一个显式公式，使得在时间 $t$ 的约化密度的时间演化与初始时间为 $t_0$ 时的约化密度相关联，在 $t_0 \leq t <t_0 +t_c$ 处存在一个临界时间 $t_c\in ]0, \infty]$，但当时间达到 $t \geq t_0 +t_c$ 时可逆性丧失。 显示英文摘要

摘要： 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 $t \geq t_0 +t_c$.