基于几何量子力学与经典力学之间的相似性，我们探索了量子热力学的另一种基础，该基础利用底层状态空间的微分几何。我们发展了微正则系综和正则系综，引入连续混合态作为量子态流形上的分布。我们指出了其对 qudit（量子 d 态系统）气体的实验影响。我们以内在方式定义了量子热和功，包括单轨迹功，并以一种与经典熵、量子熵和信息论熵相一致的方式重新表述热力学熵。我们给出了热力学第一定律、第二定律以及 Jarzynski 涨落定理。所得结果比通常可得的物理图景更加透明，在其中，经典动力学与量子动力学背后的数学结构和物理直觉被视为紧密一致。
英文摘要：
Building on parallels between geometric quantum mechanics and classical mechanics, we explore an alternative basis for quantum thermodynamics that exploits the differential geometry of the underlying state space. We develop both microcanonical and canonical ensembles, introducing continuous mixed states as distributions on the manifold of quantum states. We call out the experimental consequences for a gas of qudits. We define quantum heat and work in an intrinsic way, including single-trajectory work, and reformulate thermodynamic entropy in a way that accords with classical, quantum, and information-theoretic entropies. We give both the First and Second Laws of Thermodynamics and Jarzynki's Fluctuation Theorem. The result is a more transparent physics, than conventionally available, in which the mathematical structure and physical intuitions underlying classical and quantum dynamics are seen to be closely aligned.
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理中心引力物理与量子信息）；Department of Physics, Kyoto University（京都大学物理学系）
Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We review and connect different frameworks where it emerges in the quantum regime: adiabatically driven closed systems, time-dependent Lindblad master equations, and discrete processes. A geometric lower bound on entropy production in finitetime is then presented, which represents a quantum generalisation of the original classical bound. Following this, we review a…
Geometric quantum thermodynamics: A fibre bundle approach
Theoretical and Experimental Physics Department, Federal University of Rio Grande do Norte（北里奥格兰德联邦大学理论与实验物理系）；QPequi Group, Institute of Physics, Federal University of Goiás（戈亚斯联邦大学物理研究所 QPequi 小组）
Force-current structure in Markovian open quantum systems and its applications: geometric housekeeping-excess decomposition and thermodynamic trade-off relations
The problem of extracting the work from a quantum-thermodynamic system driven by slowly varying external parameters is discussed. It is shown that there naturally emerges a gauge-theoretic structure. The field strength identically vanishes if the system is in an equilibrium state, i.e., the nonvanishing field strength implies that the system is in a nonequilibrium quasi-stationary state. The work done through a cyclic process in the parameter space is given in terms of the flux of the field. This general formalism is applied to an example of a single spin in a varying magnetic field, and the m…
Geometric foundations of thermodynamics in the quantum regime
Universidad de Granada（格拉纳达大学）；Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada（格拉纳达大学卡洛斯一世理论与计算物理研究所）；Universidad de Córdoba（科尔多瓦大学）
A general investigation is made into the intrinsic Riemannian geometry for complex systems, from the perspective of statistical mechanics. The entropic formulation of statistical mechanics is the ingredient which enables a connection between statistical mechanics and the corresponding Riemannian geometry. The form of the entropy used commonly is the Shannon entropy. However, for modelling complex systems, it is often useful to make use of entropies such as the Rényi and Tsallis entropies. We consider, here, Shannon, Rényi, Tsallis, Abe and structural entropies, for our analysis. We focus on on…
The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior are new challenges for quantum thermodynamics, as traditional thermodynamics is built on large systems vanishingly weakly coupled to a non-dynamical reservoir. Important also are the quantum attributes, as in quantum coherence, correlations, entanglement and fluctuations. All told, one needs to reexamine the meaning of the thermodynamic functions, the viability of the thermodynamic relations and the validity of the thermodynamic laws anew. In one popular approac…
Zhejiang Key Laboratory of Quantum State Control and Optical Field Manipulation, Department of Physics, Zhejiang Sci-Tech University（浙江理工大学物理系量子态控制与光场调控浙江省重点实验室）；School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学院）；Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University（北京师范大学多尺度自旋物理教育部重点实验室）
Quantum decoherence is seen as an undesired source of irreversibility that destroys quantum resources. Quantum coherences seem to be a property that vanishes at thermodynamic equilibrium. Away from equilibrium, quantum coherences challenge the classical notions of a thermodynamic bath in a Carnot engines, affect the efficiency of quantum transport, lead to violations of Fourier's law, and can be used to dynamically control the temperature of a state. However, the role of quantum coherence in thermodynamics is not fully understood. Here we show that the relative entropy of a state with quantum …
This article sets up a new formalism to investigate stochastic thermodynamics of out-of-equilibrium quantum systems, where stochasticity primarily comes from quantum measurement. In the absence of any bath, we define a purely quantum component to heat exchange, that corresponds to energy fluctuations induced by measurement back-action. By providing a central position to quantum measurement, our formalism bridges the gap between quantum optics and quantum thermodynamics: Thermal signatures of measurement induced irreversibility are identified in a simple implementation of Jarzynski equality, wh…
The thermodynamic framework of repeated interactions is generalized to an arbitrary open quantum system in contact with a heat bath. Based on these findings the theory is then extended to arbitrary measurements performed on the system. This constitutes a direct experimentally testable framework in strong coupling quantum thermodynamics. By construction, it provides many quantum stochastic processes and quantum causal models with a consistent thermodynamic interpretation. The setting can be further used, for instance, to rigorously investigate the interplay between non-Markovianity and nonequil…
We develop a geometric framework to describe the thermodynamics of microscopic heat engines driven by slow periodic temperature variations and modulations of a mechanical control parameter. Covering both the classical and the quantum regime, our approach reveals a universal trade-off relation between efficiency and power that follows solely from geometric arguments and holds for any thermodynamically consistent microdynamics. Focusing on Lindblad dynamics, we derive a second bound showing that coherence as a genuine quantum effect inevitably reduces the performance of slow engine cycles regard…
Irreversible entropy production (IEP) plays an important role in quantum thermodynamic processes. Here we investigate the geometrical bounds of IEP in nonequilibrium thermodynamics by exemplifying a system coupled to a squeezed thermal bath subject to dissipation and dephasing, respectively. We find that the geometrical bounds of the IEP always shift in contrary way under dissipation and dephasing, where the lower and upper bounds turning to be tighter occurs in the situation of dephasing and dissipation, respectively. However, either under dissipation or under dephasing, we may reduce both th…
In this paper the geometric entanglement (GE) of systems in one spatial dimension (1D) and in the thermodynamic limit is analyzed focusing on two aspects. First, we reexamine the calculation of the GE for translation-invariant matrix product states (MPSs) in the limit of infinite system size. We obtain a lower bound to the GE which collapses to an equality under certain sufficient conditions that are fulfilled by many physical systems, such as those having unbroken space (P) or space-time (PT) inversion symmetry. Our analysis justifies the validity of several derivations carried out in previou…
A short introduction on quantum thermodynamics is given and three new topics are discussed: 1) Maximal work extraction from a finite quantum system. The thermodynamic prediction fails and a new, general result is derived, the ``ergotropy''. 2) In work extraction from two-temperature setups, the presence of correlations can push the effective efficiency beyond the Carnot bound. 3) In the presence of level crossing, non-slow changes may be more optimal than slow ones.
Thermodynamic and quantum thermodynamic analyses of Brownian movement of a solvent and a colloid passing through neutral thermodynamic equilibrium states only. It is shown that Brownian motors and E. coli do not represent Brownian movement.
This book provides an introduction to the emerging field of quantum thermodynamics, with particular focus on its relation to quantum information and its implications for quantum computers and next generation quantum technologies. The text, aimed at graduate level physics students with a working knowledge of quantum mechanics and statistical physics, provides a brief overview of the development of classical thermodynamics and its quantum formulation in Chapter 1. Chapter 2 then explores typical thermodynamic settings, such as cycles and work extraction protocols, when the working material is ge…
In this work we explore the use of thermodynamic length to improve the performance of experimental protocols. In particular, we implement Landauer erasure on a driven electron level in a semiconductor quantum dot, and compare the standard protocol in which the energy is increased linearly in time with the one coming from geometric optimisation. The latter is obtained by choosing a suitable metric structure, whose geodesics correspond to optimal finite-time thermodynamic protocols in the slow driving regime. We show experimentally that geodesic drivings minimise dissipation for slow protocols, …
The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The so-called thermodynamic arrow of time introduces an ordering that divides the system's state space into past, future and incomparable regions. In this work, we analyse the structure of the resulting thermal cones, i.e., sets of states that a given state can thermodynamically evolve to (the future thermal cone) or evolve from (the past thermal cone). Specifically, for a $d$-dimensional classical state of a system interacting with a heat bath, we find explicit construction of the past thermal cone and the…
External driving of bath temperatures with a phase difference of a nonequilibrium quantum engine leads to the emergence of geometric effects on the thermodynamics. In this work, we modulate the amplitude of the external driving protocols by introducing envelope functions and study the role of geometric effects on the flux, noise and efficiency of a four-level driven quantum heat engine coupled with two thermal baths and a unimodal cavity. We observe that having a finite width of the modulation envelope introduces an additional control knob for studying the thermodynamics in the adiabatic limit…
Quantum thermodynamics addresses the emergence of thermodynamical laws from quantum mechanics. The link is based on the intimate connection of quantum thermodynamics with the theory of open quantum systems. Quantum mechanics inserts dynamics into thermodynamics giving a sound foundation to finite-time-thermodynamics. The emergence of the 0-law I-law II-law and III-law of thermodynamics from quantum considerations is presented. The emphasis is on consistence between the two theories which address the same subject from different foundations. We claim that inconsistency is the result of faulty an…
This approach to the incorporation of stochastic thermodynamics into quantum theory is based on the conception of consistent inclusion of the holistic stochastic environmental influence described by wave functions of the arbitrary vacuum, which was proposed in our paper previously. In this study, we implement the possibility of explicitly incorporating the zeroth law of stochastic thermodynamics in the form of the saturated Schrödinger uncertainty relation into quantum theory. This allows comparatively analyzing the sets of states of arbitrary vacuums, namely, squeezed coherent states (SCSs) a…
In this work we will show that there exists a fundamental difference between microscopic quantum thermodynamics and macroscopic classical thermodynamics. It will be proved that the entropy production in quantum thermodynamics always vanishes for both closed and open quantum thermodynamic systems. This novel and very surprising result is derived based on the genuine reasoning Clausius used to establish the science of thermodynamics in the first place. This result will interestingly lead to define the generalized temperature for any non-equilibrium quantum system.
Geometric Phase Effect in Thermodynamic Properties and in the Imaginary-Time Multi-Electronic-State Path Integral Formulation
Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University（北京大学化学与分子工程学院理论计算化学研究所北京分子科学国家实验室）
Thermodynamics is a highly successful macroscopic theory widely used across the natural sciences and for the construction of everyday devices, from car engines and fridges to power plants and solar cells. With thermodynamics predating quantum theory, research now aims to uncover the thermodynamic laws that govern finite size systems which may in addition host quantum effects. Here we identify information processing tasks, the so-called "projections", that can only be formulated within the framework of quantum mechanics. We show that the physical realisation of such projections can come with a …
The physical variables of classical thermodynamics occur in conjugate pairs such as pressure/volume, entropy/temperature, chemical potential/particle number. Nevertheless, and unlike in classical mechanics, there are an odd number of such thermodynamic co-ordinates. We review the formulation of thermodynamics and geometrical optics in terms of contact geometry. The Lagrange bracket provides a generalization of canonical commutation relations. Then we explore the quantization of this algebra by analogy to the quantization of mechanics. The quantum contact algebra is associative, but the constan…
The quantum master equation obtained by generalizing the geometric formulation of nonequilibrium thermodynamics to dissipative quantum systems is seriously nonlinear. We argue that nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature of the nonlinear contribution. The thermodynamic nonlinearity leads to proper equilibrium solutions and improves the dynamic behavior of dissipative quantum systems. We also discuss the Markovian character of the thermodynamic quantum master equation. The general ideas are illustrated for two-level systems.
This article elaborates on entanglement entropy and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. A set of basic geometric and quantum mechanics and probability concepts together with methods of computation are developed in general covariant form for curved phase spaces modelled as cotangent Lorentz bundles. The constructions are based on ideas relating the Grigory Perelman's entropy for geometric flows and associated statistical thermodynamic systems to the quantum von Neumann entropy, classical and quantum relative and conditional entr…
