GENERALIZED UNCERTAINTY RELATIONS AND LONG-TIME LIMITS FOR QUANTUM BROWNIAN-MOTION MODELS
GENERALIZED UNCERTAINTY RELATIONS AND LONG-TIME LIMITS FOR QUANTUM BROWNIAN-MOTION MODELS
复制标题
DOI：
10.1103/physrevd.51.6870
复制
发表时间：
1995-06-15
影响因子：
5
通讯作者：
HALLIWELL, JJ
中科院分区：
文献类型：
--
作者：
ANASTOPOULOS, C;HALLIWELL, JJ
研究了一类量子布朗运动模型的约化密度算符的时间演化，该模型由在势V（x）中运动的粒子和热态的谐振子组成.我们的主要工具是维格纳函数的约化密度算子和线性系统，我们推导出一个明确的表达的维格纳函数传播。我们用它来推导两个广义不确定关系。第一个由在环境存在下演化为时间t后的不确定性函数U=（Δp）2（Δq）2的急剧下界组成。第二个结果是一个更强更简单的结果，它包含了在时间t上量Δ p A2 = U-Cpq 2的一个下界，其中Cpq = 1/2< Δp^ Δq^+ Δq^ Δp^>。（$Σ A-实质上是维格纳函数的1-σ轮廓所包围的面积。在这两种情况下，最小化的初始状态是一个相关的相干态（一个非最小高斯纯态），在第一种情况下，下限只是一个信封。这些广义的不确定性关系提供了量子涨落和热涨落的相对大小的度量。我们证明了两个简单的不等式，将不确定性与冯诺依曼熵-Tr（ρ lnρ）和冯诺依曼熵与线性熵1-Tr ρ 2联系起来。我们还证明了任意初态的维格纳函数的长时间极限的一些结果。对于谐振子，所有初始状态的维格纳函数在很大程度上变成高斯（通常但不总是热态）。我们推导出了自由粒子（一般不属于高斯分布）的长时间极限的显式形式，以及在高温近似下更一般的势的显式形式。我们讨论了与胡和张以及帕兹和祖雷克以前的工作的联系。
We study the time evolution of the reduced density operator for a class of quantum Brownian motion models consisting of a particle moving in a potential V (x) and coupled to an environment of harmonic oscillators in a thermal state. Our principal tool is the Wigner function of the reduced density operator and for linear systems we derive an explicit expression for the Wigner function propagator. We use it to derive two generalized uncertainty relations. The first consists of a sharp lower bound on the uncertainty function U=(Δp) 2 (Δq) 2 after evolution for time t in the presence of an environment. The second, a stronger and simpler result, consists of a lower bound at time t on the quantity scrA 2= U-C p q 2, where C p q= 1/2< Δp^ Δq^+ Δq^ Δp^>.($ scrA—is essentially the area enclosed by the 1-σ contour of the Wigner function.) In both cases the minimizing initial state is a correlated coherent state (a nonminimal Gaussian pure state), and in the first case the lower bound is only an envelope. These generalized uncertainty relations supply a measure of the comparative size of quantum and thermal fluctuations. We prove two simple inequalities, relating uncertainty to von Neumann entropy,-Tr (ρ lnρ), and the von Neumann entropy to linear entropy, 1-Tr ρ 2. We also prove some results on the long-time limit of the Wigner function for arbitrary initial states. For the harmonic oscillator the Wigner function for all initial states becomes a Gaussian at large times (often, but not always, a thermal state). We derive the explicit forms of the long-time limit for the free particle (which does not in general go to a Gaussian), and also for more general potentials in the approximation of high temperature. We discuss connections with previous work by Hu and Zhang and by Paz and Zurek.