作者姓名全海涛.doc

作者姓名：全海涛 论文题目：量子信息启发的量子热力学和量子相变问题作者简介：全海涛，男， 1980年8月出生，2002年9月师从于中科院理论物理研究所孙昌璞教授，于2007年7月获博士学位。中 文 摘 要热力学是描述热现象的宏观唯象理论。人们在无数的经验中总结出了热力学三大定律,这三个定律在描述宏观系统时具有高度的可靠性和普适性。但是我们知道，任何热力学系统，比如气体或固体，本质上服从的是比牛顿力学更基本的量子力学。随着量子信息科学和纳米技术的发展，现在人们可以在纳米尺度的范围内研究物理系统的性质。由于系统尺度偏离热力学极限，传统的基于宏观系统的热力学理论便不再适用；另外在纳米尺度上系统的量子效应会很明显。基于上述原因，现在人们开始探讨把热力学直接而且是完全建立在量子力学基础上的可能性，这个学科就是量子热力学。这个新兴领域人们主要关注的是如何从第一原理，例如量子力学，去描述理解热力学现象。量子热力学的发展对物理学，特别是统计物理和量子力学的一些基本问题的理解有重要意义。比如量子纠缠，熵与黑洞的信息丢失之谜，麦克斯韦妖佯谬与热力学第二定律的普适性问题，量子热力学的研究为这些物理学的基本问题提供了一个全新的视角，并带来新的启示。除此之外，量子热力学的研究还具有非常重要的应用价值。比如，量子热力学循环在纳米机械振子的冷却中的应用，使得用纳米机械振子作为量子计算的数据总线和纳米传感器成为可能。另外一个例子是Landauer原理（计算过程中每擦除1比特的信息至少要消耗Kln2的能量）。它帮助我们认识到计算的物理极限的存在，并为克服这些限制提供了新的思路。在本文前半部分，我们将就这些与量子信息密切相关的量子热力学问题进行深入地研究。量子信息启发的另一个热点方向是量子计算集成中的多体问题与相变。量子信息和理论凝聚态物理的交叉领域日渐成为一个蓬勃发展的新领域。在这个新交叉领域中物理学家普遍关注的一个问题是量子多体的相变问题。一方面，量子计算的方案在固体物理系统的实现面临着多量子比特的集成和量子退相干等量子多体问题。为了在物理上实现有实际用途的量子计算机，我们需要把普适的量子逻辑门有机地集成起来，保持量子比特间的量子纠缠，并能对它们加以操控。随着集成量子比特数目N的增多，量子相干性损失或称量子退相干会变得越来越严重，有时甚至呈e指数增长关系。一般认为量子系统与环境的耦合导致了其相干性的损失，也就是退相干。由于量子计算的实验都是在极低温下进行（这也是保证较长时间量子相干性的需要），人们会首先考虑发生在零温时环境的量子相变对量子比特的相干性的影响。另一方面，近年来人们也在试图利用量子信息中发展起来的一些新的理论工具和方法，去研究量子多体系统的一些性质。比如对于一个拓扑量子相变，它可能不伴随着对称性破缺，也找不到一个合适的局域序参量来描述，因而传统的Landau-Ginzuburg-Wilson理论无法刻画。而借用量子信息中量子纠缠的各种不同的度量，比如并发度，块纠缠以及负值性很好地刻画。在本文的后半部分我们将就量子相变与量子信息交叉领域的问题展开详细讨论。除去第一章引言，本文可以分为三部分。第一部分为第二章到第四章。这一部分我们介绍了量子热力学的一些基本概念，讨论了平衡态和非平衡态的量子热机模型。并以实际量子力学系统为例讨论了量子热机的性质，并与经典热机作比较。本部分内容是第二部分内容的基础。第二章中我们首先介绍了量子热力学的一些基本概念，比如对热力学的一些基本物理量，如做功，热传递，温度，压强，熵等进行了量子力学推广，并且在量子力学框架内表述热力学第一定律。这些推广把热力学中的物理量和量子力学中的能级变化及能级间的跃迁联系起来了。然后我们把一些基本热力学过程，如等温过程，等容过程，推广到量子力学的框架中。通过对一些基本的热力学过程的量子力学推广，我们可以直接从量子力学去理解热力学。后面我们介绍一些量子热力学的最新进展。这些新进展包括非平衡态热力学中的Jarzynski等式，这个等式把非平衡过程的跟平衡态过程联系起来。另外还有用量子纠缠代替“等概率假设”的尝试。这种尝试可能会让我们对统计力学的一些基本问题有更深刻的认识。第三章中我们研究了两种常见的热机循环-卡诺热机和奥托热机-的量子力学对应。在上一章有关量子热力学知识的基础上，我们给出了平衡态量子热机循环的普适的定义，以及构造这些量子热机循环的条件，并证明了量子热机的效率与经典热机效率的等价性。我们的研究澄清了现有文献中有关量子热机定义的十分混乱，甚至互相矛盾的状况。同时我们的研究也表明，由于工作物质的量子属性，量子卡诺热机和量子奥托热机表现出与经典卡诺热机和经典奥托热机不同的性质。我们还讨论了用一些真实的量子力学系统，如二能级系统，量子谐振子，一维无限深势阱系统来实现我们的热机模型。第四章的内容是在第三章中关于平衡态（未考虑工作物质的量子相干性）量子热机模型上的基础上的进一步推广。具体地说，在本章中我们讨论了热机的工作物质的量子相干性对热机的影响。由于量子相干性是量子力学系统的特有属性，没有经典对应，有人从理论上预言量子热机可能会有超越经典热机的一些奇异性质，比如一种理想的基于腔量子电动力学系统的量子热机（光子热机）可能会如超越经典热机的效率极限。考虑到物理系统的实际情况，如腔的漏损（腔壁并不能完全反射光子），和原子在穿过腔时和环境相互耦合会不可避免地导致它自身的退相干，我们改进了上述的光子热机模型，也就是在上述理想光子热机模型中引入了光场耗散和原子退相干。通过仔细计算和实验参数估计我们发现，由于腔的漏损和原子的退相干上述超越经典的结论无法通过现有的实验条件中观测到。我们的研究也表明由于工作物质的退相干导致了热机从量子到经典的转变。本文的第二部分是第五章。在这一部分我们借用量子信息中的一些概念和方法来理解热力学和统计物理学的一些基本问题。确切地说，从量子热力学的角度研究了由来已久并富有挑战性的麦克斯韦妖佯谬。这个问题和热力学第二定律的普适性密切相关。我们在前面量子热机研究的基础上，用两个二能级系统来模拟有麦克斯韦妖参与的复合量子热机。其中利用量子信息中的两个控制非门来模拟麦克斯韦妖对工作物质的信息测量，以及对工作物质的反馈控制并让它对外做功这两个功能，用热库冷却其中的一个量子比特来模拟对麦克斯韦妖的信息擦除过程。我们发现当麦克斯韦妖的信息擦除未被包含到热力学循环中来的时候，就出现了所谓的永动机；如果我们在每次循环完成后用热库冷却麦克斯韦妖，也就是把麦克斯韦妖的信息擦除被包含到热力学循环中，所谓的永动机就自动消失了。这样我们在更广泛的范围内证明了热力学第二定律的普适性。最后我们还讨论了如何在超导量子电路上演示我们的结论。本文的第三部分是第六章和第七章。在这一部分我们讨论了量子相变对量子比特的退相干的影响，以及如何用量子信息中的一些概念和方法去刻画量子相变。第六章中我们先简单介绍了自旋系统的量子相变，并且介绍了探测量子相变点的一些基本方法-寻找能级简并点或者量子纠缠的奇异性。然后，我们研究了一些与量子相变有关的量子开系统的临界动力学行为。通过一个典型的量子测量模型，我们研究了一个二能级系统的退相干与和他耦合的环境的量子相变之间的关系。解析分析和数值计算结果都揭示了一个导致二能级系统退相干（或环境的克斯密特回波衰减）的新机制-当环境处在量子临界点附近时，它的量子相变会加剧系统的退相干。在我们的模型中，系统环境之间达到最大纠缠，即中心系统从一个纯态转化为一个混态时，伴随着环境的一个演化分支上发生了量子相变。这个结果揭示了纠缠，洛克斯密特回波，退相干以及临界性之间的内在联系。另外，我们研究了一个与各向同性Lipkin-Meshkov-Glick(LMG)环境耦合的单比特的动力学性质。我们考虑了单比特和环境之间的两种耦合。在任一种情况中，单比特的动力学行为很好地揭示了环境的量子相变。我们的研究不但建立了LMG模型和反Jaynes-Cummings模型间的联系，而且也揭示了环境的量子相变会如何影响与它耦合的单比特的量子相干性，并提供了一种保持单比特的量子相干性的可能的新方案。第七章中我们把上一章环境在量子临界点附近的“动力学”敏感性推广到“静态”敏感性。在这一章我们考查的是量子相变系统的基态在相变参数临界点的奇异行为。我们用两个控制参数稍有不同的基态的内积定义了一个新物理量-量子保真度。量子保真度在临界点的敏感性可以用来探测系统的量子相变。而且我们将上述思想进一步推广到有限温度，研究在有限温度的时候量子相变的痕迹。对于一类重要的准自由的费米子系统，我们给出了它们在有限温度时的保真度和洛克斯密特回波的精确表达式。由这些解析结果得出的数值结果清楚地展示，在一定的温度范围内，保真度和洛克斯密特回波都可以标出零温时量子相变的相图。本章的研究结果还提示了用混态的量子保真度来研究热力学相变的可能。关键词：量子信息，量子热力学，量子热机，量子相变 Theoretical Studies of Quantum Thermodynamics and Quantum Phase Transition in the Light of Quantum InformationQuan Haitao ABSTRACTThermodynamics is a phenomenon theory of heat. Three laws of thermodynamics, which comes from the empirical life of people, are highly universal and acute in describing macroscopic systems. Nevertheless, it is well known now that any thermodynamic systems, such as gases or solids, actually obey quantum mechanics, which is regarded to be more fundamental than Newtonian mechanics. With the development of quantum information science and nanotechonology, we are now able to study system on nanometer scale. Since such tiny systems deviate from thermodynamic limit, traditional thermodynamics is no longer applicable. Meanwhile, in nanometer scale systems, quantum property will play important role. Based on the above facts, physicists begin to consider the possibility of base thermodynamics on quantum mechanics. This emerging field is called quantum thermodynamics. In this emerging field, physicists are trying to understand and describe thermodynamic phenomena from the first principle, quantum mechanics. It is expected that the development of quantum thermodynamics will shed new light on some fundamental problems of physics, especially of statistical mechanics and quantum mechanics. For example, the study on quantum thermodynamics is expected to offer a new angle and bring important insights to information loss paradox of Black Holes, Maxwells demon and the universality of the second law. Besides, studies on quantum thermodynamics also promise important applications. For example, the application of quantum thermodynamic cycle in cooling of nano-mechanical oscillator, which enables the nano-mechanical resonator become a possible candidate of quantum data bus and nano-sensor. Another good example in case is the Landauers principle (erasing of every bit of information leads to dissipation of kln2 energy), which predicts the existence of physical limit of computation, and provides us ideas on how to overcome the problem. In the first part of this thesis, we will thoroughly study some problems about quantum thermodynamics, which is closely related to the quantum information.Another important research topic illuminated by quantum information science is the many-body problems in integrating units of quantum computing and phase transitions. The disciplinary area of quantum information science and condensed matter theory is an emerging field, in which phase transitions of many-body system is a focus topic. On the one hand, many solid-state-system-based scheme of quantum computing are facing the problems of integrating of qubits and quantum decoherence. In order to put quantum computing into practice, it is required that we integrating those universal logic gates, and keep on the entanglements between them, and control these qubits properly. With the increase of the numbers of qubits that are integrated, the loss of quantum coherence, or decoherence will become serious, and even increase exponentially. Generally speaking, it is the coupling of the quantum system with its environment that results in the loss of quantum coherence of the system. Due to the fact that experiments about quantum computation are usually done under extremely low temperature (so that quantum coherence can be kept for a longer time), we can intuitively consider the influence of quantum phase transitions of environment on decoherence of qubits. On the other hand, physicists are trying to borrow new tools and methods developed in quantum information to study the properties of quantum many-body system. For example, a topological quantum phase transition without symmetry breaking, there is no local order parameter to describe it, hence traditional Landau-Ginzuburg-Wilson fails. However, this phenomenon can be described by different kinds of measurements of quantum entanglement from quantum information, for example, concurrence, entangled entropy, or negativity. In the latter part of our thesis, we will study the interplay between quantum information and quantum phase transitions.Except the fist chapter, introduction, the thesis can be divided into three parts. From the second chapter through the fourth chapter is the first part. In this part, we introduce some basic concepts of quantum thermodynamics, and study both equilibrium and nonequilibrium quantum heat engine models. We also study the properties of quantum heat engines based on realistic quantum mechanical systems, and compare them with classical heat engines. This part is the foundations for discussion of next part.In the second paragraph, we first introduced some basics of quantum thermodynamics, for example, the quantum mechanical generalization of some observables, including work, heat, temperature, pressure, entropy. The first law of thermodynamics is also rewritten in the frame of quantum mechanics. These generalizations connect observables in thermodynamics with the changes of eigenenergies and occupation probabilities in quantum mechanics. And then, we make quantum mechanical generalization of some typical thermodynamic processes, such as isothermal process, isochoric process. These generalizations enable us to understand thermodynamics from quantum mechanics directly. In the following, we introduce some recent development in quantum thermodynamics. These recent development including Jarzynski equality, which connects the nonequilibirum processes with equilibrium processes, and an attempt to replace “equal probability” postulate with quantum entanglement, which may shed new lights on a fundamental problem of statistical mechanics.In the third chapter, we study quantum mechanical analogue of two typical thermodynamic cyclesCarnot cycle and Otto cycle. Based on the fundamentals of quantum thermodynamics in last chapter, we give universal definitions of equilibrium quantum heat cycles, and the conditions to construct these cycles. We also prove the equivalence of the efficiencies of quantum heat engines with their classical counterpart. Our studies clarify the ambiguities and even discrepancies in the definition of quantum heat engines. In addition, we indicate that quantum Carnot engine and quantum Otto engine have different properties from their classical counterpart due to the quantum nature of the working substance. Finally, we consider implementing our quantum heat engines with some realistic quantum mechanical system, such as two-level system, harmonic oscillator, one-dimensional infinite square potential system. The fourth chapter is the further work on the equilibrium quantum heat engine model of chapter three. Specifically, we consider the influence of the quantum coherence of the working substance on the heat engine. As the quantum coherence is the special property of quantum mechanical system, physicists predict some novel property of quantum heat engine beyond that of classical cases. For example, a group proposes an ideal quantum heat engine based on cavity quantum electrodynamics system, and proclaim that its efficiency can surpass the limit of classical heat engines. Considering the practice of physical systems， for example, the loss of cavity wall and the decoherence of the atoms duet their coupling with the environment, we improve the quantum photon engine mentioned above by introducing photon dissipation and atom decoherence. Through detailed calculation and experimental parameter estimation, we find that the novel effect beyond classical thermodynamics cannot yet be observed due to the limitation of current experiment capability. Our study also demonstrates the quantum-classical transition of heat engines due to decoherence of the working substance.The fifth chapter is the second part of the thesis. In this chapter, we borrow some concepts and tools from quantum information to study some fundamental problems in thermodynamics and statistical mechanics. Specifically, we consider the long-standing and challenging Maxwells demon paradox from a new angle quantum thermodynamics. This problem is closely associated with the universality of the second law of thermodynamics. Based on our preceding work on quantum heat engines, we use two two-level systems to simulate a Maxwells demon-assisted composite quantum heat engine. Therein two CNOTs are used to simulate the Maxwells demons functions of monitoring the information of the system and feedback control of the system, the thermalization of one qubit with low-temperature heat bath is used to simulate the erasure of the information in the demon. It is illustrated that when the erasure of the demon is not included into the thermodynamic cycle, the apparent violation of the second law occurs; however, when the erasure of the demons information is included into the cycle, the apparent perpetual machine of the second kind disappears. We thus validate the universality of the second law in a more general domain. Finally, we discuss how to illustrate our main idea in superconducting circuits.The third part of the thesis consists of chapter 6 and chapter 7. In this part, we discuss the influence of quantum phase transitions on decoherence of qubit and how to describe quantum phase transitions by borrowing some tools from quantum information. In Chapter 6, we first briefly introduce quantum phase transitions in spin systems, and some methods to characterize the quantum critical point - the energy degenerate point and singularity of quantum entanglement. And then, we study dynamics of quantum open system concerning quantum phase transitions. Through a typical quantum measurement model, we study the relation between decoherence of a central two-level system and quantum phase transition in its environment. Both analytical and numerical results indicate a new mechanism of decoherence: when the environment is near to the critical point, quantum phase transition will enhance decoherence of the system. In our model, the maximum entanglement between the system and environment, or the transition of the system from a pure state to a mixed one, accompanies occurrence of quantum phase transition in one of the two evolution branches. This result indicates the intrinsic relation between entanglement, Loschmidt echo, decoherence and quantum criticality. Moreover, we study the dynamics of a central spin coupled to a homogeneous environment described by Lipkin-Meshkov-Glick (LMG) model. We consider two kinds of couplings of the system to the environment. In either case, the dynamics