{"title":"热浴谐振子的弛豫:基于量子群论形式的研究","authors":"Yan Gu , Jiao Wang","doi":"10.1016/j.physa.2025.131004","DOIUrl":null,"url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":20152,"journal":{"name":"Physica A: Statistical Mechanics and its Applications","volume":"679 ","pages":"Article 131004"},"PeriodicalIF":3.1000,"publicationDate":"2025-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Relaxation of a thermally bathed harmonic oscillator: A study based on the quantum group-theoretical formalism\",\"authors\":\"Yan Gu , Jiao Wang\",\"doi\":\"10.1016/j.physa.2025.131004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>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.</div></div>\",\"PeriodicalId\":20152,\"journal\":{\"name\":\"Physica A: Statistical Mechanics and its Applications\",\"volume\":\"679 \",\"pages\":\"Article 131004\"},\"PeriodicalIF\":3.1000,\"publicationDate\":\"2025-09-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Physica A: Statistical Mechanics and its Applications\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0378437125006569\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica A: Statistical Mechanics and its Applications","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0378437125006569","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Relaxation of a thermally bathed harmonic oscillator: A study based on the quantum group-theoretical formalism
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.
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.