Relaxation of a thermally bathed harmonic oscillator: A study based on the quantum group-theoretical formalism

IF 3.1 3区 物理与天体物理 Q2 PHYSICS, MULTIDISCIPLINARY
Yan Gu , Jiao Wang
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Abstract

The quantum dynamics of a damped harmonic oscillator has been extensively studied since the 1960s of the last century. Here, with a distinct tool termed the “group-theoretical characteristic function (GCF)”, we investigate analytically how a harmonic oscillator immersed in a thermal environment would relax to its equilibrium state. We assume that the oscillator is at a pure state initially and its evolution is governed by a well-known quantum-optical master equation. Taking advantage of the GCF, the master equation can be transformed into a first-order linear partial differential equation, allowing us to write down its solution explicitly. Based on the solution, it is found that, in clear contrast with the monotonic relaxation process of its classical counterpart, the quantum oscillator may demonstrate some intriguing non-monotonic relaxation characteristics. In particular, when the initial state is a Gaussian state (i.e., a squeezed coherent state), there is a critical value of the environmental temperature below which the entropy will first increase to reach its maximum value, then turn down and converge to its equilibrium value from above. Conversely, when the temperature exceeds the critical value, the entropy converges monotonically to its equilibrium value from below. In contrast, for an initial Fock state, there are two critical temperatures instead and, in between, a new additional phase emerges, where the time curve of entropy features two extreme points. Namely, the entropy will increase to reach its maximum first, then turn down to reach its minimum, from where it begins to increase and converges to the equilibrium value eventually. Other related issues are discussed as well.
热浴谐振子的弛豫:基于量子群论形式的研究
自上世纪60年代以来,阻尼谐振子的量子动力学得到了广泛的研究。在这里,我们用一种独特的工具称为“群论特征函数(GCF)”,分析研究了沉浸在热环境中的谐振子如何松弛到其平衡状态。我们假设振荡器最初处于纯态,它的演化由一个众所周知的量子光学主方程控制。利用GCF,可以将主方程转化为一阶线性偏微分方程,使我们能够明确地写出它的解。基于该解,我们发现,与经典对应物的单调弛豫过程明显相反,量子振荡器可能表现出一些有趣的非单调弛豫特性。特别是,当初始态为高斯态(即压缩相干态)时,存在一个环境温度临界值,在该临界值以下,熵首先增加达到最大值,然后下降并从上方收敛到其平衡值。反之,当温度超过临界值时,熵从以下单调收敛到其平衡值。相反,对于初始Fock状态,存在两个临界温度,并且在两者之间出现一个新的附加相,其中熵的时间曲线具有两个极值点。也就是说,熵首先增加到最大值,然后下降到最小值,从这里开始增加并最终收敛到平衡值。其他相关问题也进行了讨论。
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来源期刊
CiteScore
7.20
自引率
9.10%
发文量
852
审稿时长
6.6 months
期刊介绍: Physica A: Statistical Mechanics and its Applications Recognized by the European Physical Society Physica A publishes research in the field of statistical mechanics and its applications. Statistical mechanics sets out to explain the behaviour of macroscopic systems by studying the statistical properties of their microscopic constituents. Applications of the techniques of statistical mechanics are widespread, and include: applications to physical systems such as solids, liquids and gases; applications to chemical and biological systems (colloids, interfaces, complex fluids, polymers and biopolymers, cell physics); and other interdisciplinary applications to for instance biological, economical and sociological systems.
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