DOCSLIB.ORG

Energy-Temperature Uncertainty Relation in Quantum Thermodynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Energy-Temperature Uncertainty Relation in Quantum Thermodynamics ARTICLE DOI: 10.1038/s41467-018-04536-7 OPEN Energy-temperature uncertainty relation in quantum thermodynamics H.J.D. Miller1 & J. Anders1 It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we 1234567890():,; include such interactions and, using quantum estimation theory, derive a generalised ther- modynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system’s state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase- Dyson skew information. We demonstrate that the temperature’s signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers. 1 Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, UK. Correspondence and requests for materials should be addressed to H.J.D.M. (email: [email protected]) or to J.A. (email: [email protected]) NATURE COMMUNICATIONS | (2018) 9:2203 | DOI: 10.1038/s41467-018-04536-7 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 ohr suggested that there should exist a form of com- estimate, and we illustrate our bound with an example of a Bplementarity between temperature and energy in thermo- damped harmonic oscillator. dynamics similar to that of position and momentum in quantum theory1. His reasoning was that in order to assign a Results definite temperature T to a system it must be brought in contact The Wigner-Yanase-Dyson skew information. Our analysis with a thermal reservoir, in which case the energy U of the system throughout the paper will rely on distinguishing between classical fluctuates due to exchanges with the reservoir. On the other hand, and non-classical fluctuations of observables in quantum to assign a sharp energy to the system it must be isolated from the mechanics, and we first present a framework for quantifying these reservoir, rendering the system’s temperature T uncertain. Based different forms of statistical uncertainty for arbitrary mixed states. on this heuristic argument Bohr conjectured the thermodynamic Let us consider a quantum state ^ρ and an observable A^. Wigner uncertainty relation: and Yanase considered the problem of quantifying the quantum ^ ^ρ 41 1 uncertainty in observable A for the case where is mixed . Δβ ; ð1Þ However, they observed that the standard measure of uncertainty, ΔU namely the variance Var½^ρ; A^ := tr½^ρδA^2 with δA^ = A^ À A^ , − with β = (k T) 1 the inverse temperature. While Eq. (1) has since contains classical contributions due to mixing, and thus fails to B – ^ been derived in various settings2 9, it was Mandelbrot who first fully quantify the non-classical fluctuations in the observable A. based the concept of fluctuating temperature on the theory of This problem can be resolved by finding a quantum measure of ^ ^ statistical inference. Concretely, for a thermal system in canonical uncertainty Q½^ρ; A and classical measure K½^ρ; A such that the equilibrium, Δβ can be interpreted as the standard deviation variance can be partitioned according to associated with estimates of the parameter β. Mandelbrot proved Var½^ρ; A^¼Q½^ρ; A^þK½^ρ; A^: ð2Þ that Eq. (1) sets the ultimate limit on simultaneous estimates of energy and temperature in classical statistical physics2. fl The notion of uctuating temperature has proved to be fun- 47 fi Following the framework introduced by Luo , these functions damental in the emerging eld of quantum thermometry, where fi advances in nanotechnology now allow temperature sensing at are required to ful l three conditions: (i) both terms should be – ½^ρ; ^ ½^ρ; ^ sub-micron scales10 22. Using the tools of quantum metrology23, non-negative, Q A 0 and K A 0, so that they can be interpreted as forms of statistical uncertainty, (ii) if the state ^ρ is the relation Eq. (1) can also be derived for weakly coupled ½^ρ; ^ = ½^ρ; ^ ½^ρ; ^¼ quantum systems11,12,14, where the equilibrium state is best pure, then Q A Var A while K A 0 as all uncertainty should be associated to quantum fluctuations alone, (iii) Q½^ρ; A^ described by the canonical ensemble. Within the grand-canonical ^ρ ensemble the impact of the indistinguishability of quantum par- must be convex with respect to , so that it decreases under classical mixing. Correspondingly, K½^ρ; A^ must be concave with ticles on the estimation of temperature and the chemical potential ^ρ has also been explored24. Relation Eq. (1) informs us that when respect to . The following function, known as the WYD skew informa- designing an accurate quantum thermometer one should search 41 for systems with Hamiltonians that produce a large energy tion was shown to be a valid measure of quantum uncertainty: 14 1 À variance . Q ½^ρ; A^ :¼À tr½½A^; ^ρa½A^; ^ρ1 a; a 2ð0; 1Þ; ð3Þ Recently there has been an emerging interest into the effects of a 2 strong coupling on temperature estimation13,15,25. At the nanoscale the strength of interactions between the system and the with the complementary classical uncertainty given by À reservoir may become non-negligible, and the local equilibrium K ½^ρ; A^ :¼ tr½^ρa δA^ ^ρ1 aδA^; a 2ð0; 1Þ: ð4Þ state of the system will not be of Gibbs form26,27. In this regime a thermodynamics needs to be adapted as the equilibrium prop- While conditions (i) and (ii) are easily verified, the convexity/ erties of the system must now depend on the interaction ½^ρ; ^ ½^ρ; ^ 28–40 fl concavity of Qa A and Ka A respectively can be proven energy . We will see that the internal energy U and its uc- using Lieb’s concavity theorem. tuations ΔU are determined by a modified internal energy ^à The presence of the parameter a demonstrates that there is no operator, denoted by ES, that differs from the bare Hamiltonian unique way of separating the quantum and classical contributions 35,39 of the system . This modification brings into question the to the variance. We here follow the suggestion made in refs. 43,44 validity of Eq. (1) for general classical and quantum systems, and and average over the interval a∈ (0, 1) to define two new the aim of this paper is to investigate the impact of strong cou- quantities: Z pling on the thermodynamic uncertainty relation. 1 Taking into account quantum properties of the effective ½^ρ; ^ :¼ ½^ρ; ^; ð Þ Q A daQa A 5 internal energy operator and its temperature dependence, we here 0 derive the general thermodynamic uncertainty principle valid at Z all coupling strengths. Formally this result follows from a general 1 ½^ρ; ^ :¼ ½^ρ; ^: ð Þ upper bound on the quantum Fisher information (QFI) for K A daKa A 6 exponential states. We prove that quantum fluctuations arising 0 from coherences between energy states of the system lead to fl increased uctuations in the underlying temperature. Most It is not only the Q ½^ρ; A^ and K ½^ρ; A^ that separate the interestingly, the non-classical modifications to Eq. (1) are a a ^ fi quantum and classical fluctuations of a quantum observable A in quanti ed by the average Wigner-Yanase-Dyson (WYD) skew ^ 41–44 a state ^ρ according to Eq. (2), but also the averaged Q½^ρ; A and information , which is a quantity closely linked to measures ^ of coherence, asymmetry and quantum speed limits45,46. We then K½^ρ; A. This follows from the linearity of the integrals in Eqs. (5) demonstrate that the skew information is also linked to the heat and (6) which also preserve the conditions (i)–(iii). Throughout capacity of the system through a modified fluctuation-dissipation the remainder of the paper we will consider Q½^ρ; A^ and K½^ρ; A^ as relation (FDR). This result is used to find an upper bound on the the relevant measures of quantum and classical uncertainty, achievable signal-to-noise ratio of an unbiased temperature respectively. While this may appear to be an arbitrary choice, we 2 NATURE COMMUNICATIONS | (2018) 9:2203 | DOI: 10.1038/s41467-018-04536-7 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 ARTICLE will subsequently prove that the average skew information is Generalised thermodynamic uncertainty relation. We will now intimately connected to thermodynamics. use the results of the previous section to derive an uncertainty relation between energy and temperature for a quantum system Bound on quantum Fisher information for exponential states. strongly interacting with a reservoir. To achieve this we will first We now prove that the average skew information is linked to the discuss the appropriate energy operator for such a system, and quality of a parameter estimate for a quantum exponential state. then proceed to generalise Eq. (1). À^ ^ρ = Aθ = A quantum system S that interacts with a reservoir R is A quantum^ exponential state is of the form θ e Zθ where ÀAθ ^ Zθ ¼ tr½e and Aθ is a hermitian operator that is here assumed described by a Hamiltonian to depend analytically on a smooth parameter θ. For any state of ^ ^ ^ ^ ^ ^ ^ HS ∪ R :¼ HS IR þ IS HR þ VS ∪ R; ð9Þ full rank, an operator Aθ can be found such that the state can be expressed in this form, i.e.
Load more

Recommended publications
  • Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations
    Article Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations Jen-Tsung Hsiang 1,*,† and Bei-Lok Hu 2,† ID 1 Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China 2 Maryland Center for Fundamental Physics and Joint Quantum Institute, University of Maryland, College Park, MD 20742-4111, USA; [email protected] * Correspondence: [email protected]; Tel.: +86-21-3124-3754 † These authors contributed equally to this work. Received: 26 April 2018; Accepted: 30 May 2018; Published: 31 May 2018 Abstract: Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir.
    [Show full text]
  • The Quantum Thermodynamics Revolution
    I N F O R M A T I O N T H E O R Y The Quantum Thermodynamics Revolution By N A T A L I E W O L C H O V E R May 2, 2017 As physicists extend the 19th­century laws of thermodynamics to the quantum realm, they’re rewriting the relationships among energy, entropy and information. 48 Ricardo Bessa for Quanta Magazine In his 1824 book, Reflections on the Motive Power of Fire, the 28- year-old French engineer Sadi Carnot worked out a formula for how efficiently steam engines can convert heat — now known to be a random, diffuse kind of energy — into work, an orderly kind of energy that might push a piston or turn a wheel. To Carnot’s surprise, he discovered that a perfect engine’s efficiency depends only on the difference in temperature between the engine’s heat source (typically a fire) and its heat sink (typically the outside air). Work is a byproduct, Carnot realized, of heat naturally passing to a colder body from a warmer one. Carnot died of cholera eight years later, before he could see his efficiency formula develop over the 19th century into the theory of thermodynamics: a set of universal laws dictating the interplay among temperature, heat, work, energy and entropy — a measure of energy’s incessant spreading from more- to less-energetic bodies. The laws of thermodynamics apply not only to steam engines but also to everything else: the sun, black holes, living beings and the entire universe. The theory is so simple and general that Albert Einstein deemed it likely to “never be overthrown.” Yet since the beginning, thermodynamics has held a singularly strange status among the theories of nature.
    [Show full text]
  • Quantum Thermodynamics and Quantum Coherence Engines
    Turkish Journal of Physics Turk J Phys (2020) 44: 404 – 436 http://journals.tubitak.gov.tr/physics/ © TÜBİTAK Review Article doi:10.3906/fiz-2009-12 Quantum thermodynamics and quantum coherence engines Aslı TUNCER∗, Özgür E. MÜSTECAPLIOĞLU Department of Physics, College of Sciences, Koç University, İstanbul, Turkey Received: 17.09.2020 • Accepted/Published Online: 20.10.2020 • Final Version: 30.10.2020 Abstract: The advantages of quantum effects in several technologies, such as computation and communication, have already been well appreciated. Some devices, such as quantum computers and communication links, exhibiting superiority to their classical counterparts, have been demonstrated. The close relationship between information and energy motivates us to explore if similar quantum benefits can be found in energy technologies. Investigation of performance limits fora broader class of information-energy machines is the subject of the rapidly emerging field of quantum thermodynamics. Extension of classical thermodynamical laws to the quantum realm is far from trivial. This short review presents some of the recent efforts in this fundamental direction. It focuses on quantum heat engines and their efficiency boundswhen harnessing energy from nonthermal resources, specifically those containing quantum coherence and correlations. Key words: Quantum thermodynamics, quantum coherence, quantum information, quantum heat engines 1. Introduction Nowadays, we enjoy an unprecedented degree of control on physical systems at the quantum level. Quantum technologies promise exciting developments of machines, such as quantum computers, which can be supreme to their classical counterparts [1]. Learning from the history of industrial revolutions (IRs) associated with game- changing inventions such as steam engines, lasers, and computers, the natural question is the fundamental limits of those promised machines of quantum future.
    [Show full text]
  • Quantum Thermodynamics of Single Particle Systems Md
    www.nature.com/scientificreports OPEN Quantum thermodynamics of single particle systems Md. Manirul Ali, Wei‑Ming Huang & Wei‑Min Zhang* Thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through the dynamics that follows the principle of quantum mechanics. In this paper, we develop a theory of quantum thermodynamics that is applicable for arbitrary small systems, even for single particle systems coupled with a reservoir. We generalize the concept of temperature beyond equilibrium that depends on the detailed dynamics of quantum states. We apply the theory to a cavity system and a two‑level system interacting with a reservoir, respectively. The results unravels (1) the emergence of thermodynamics naturally from the exact quantum dynamics in the weak system‑reservoir coupling regime without introducing the hypothesis of equilibrium between the system and the reservoir from the beginning; (2) the emergence of thermodynamics in the intermediate system‑reservoir coupling regime where the Born‑Markovian approximation is broken down; (3) the breakdown of thermodynamics due to the long-time non- Markovian memory efect arisen from the occurrence of localized bound states; (4) the existence of dynamical quantum phase transition characterized by infationary dynamics associated with negative dynamical temperature. The corresponding dynamical criticality provides a border separating classical and quantum worlds. The infationary dynamics may also relate to the origin of big bang and universe infation. And the third law of thermodynamics, allocated in the deep quantum realm, is naturally proved. In the past decade, many eforts have been devoted to understand how, starting from an isolated quantum system evolving under Hamiltonian dynamics, equilibration and efective thermodynamics emerge at long times1–5.
    [Show full text]
  • What Is Quantum Thermodynamics?
    What is Quantum Thermodynamics? Gian Paolo Beretta Universit`a di Brescia, via Branze 38, 25123 Brescia, Italy What is the physical significance of entropy? What is the physical origin of irreversibility? Do entropy and irreversibility exist only for complex and macroscopic systems? For everyday laboratory physics, the mathematical formalism of Statistical Mechanics (canonical and grand-canonical, Boltzmann, Bose-Einstein and Fermi-Dirac distributions) allows a successful description of the thermodynamic equilibrium properties of matter, including entropy values. How- ever, as already recognized by Schr¨odinger in 1936, Statistical Mechanics is impaired by conceptual ambiguities and logical inconsistencies, both in its explanation of the meaning of entropy and in its implications on the concept of state of a system. An alternative theory has been developed by Gyftopoulos, Hatsopoulos and the present author to eliminate these stumbling conceptual blocks while maintaining the mathematical formalism of ordinary quantum theory, so successful in applications. To resolve both the problem of the meaning of entropy and that of the origin of irreversibility, we have built entropy and irreversibility into the laws of microscopic physics. The result is a theory that has all the necessary features to combine Mechanics and Thermodynamics uniting all the successful results of both theories, eliminating the logical inconsistencies of Statistical Mechanics and the paradoxes on irreversibility, and providing an entirely new perspective on the microscopic origin of irreversibility, nonlinearity (therefore including chaotic behavior) and maximal-entropy-generation non-equilibrium dynamics. In this long introductory paper we discuss the background and formalism of Quantum Thermo- dynamics including its nonlinear equation of motion and the main general results regarding the nonequilibrium irreversible dynamics it entails.
    [Show full text]
  • Similarity Between Quantum Mechanics and Thermodynamics: Entropy, Temperature, and Carnot Cycle
    Similarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle Sumiyoshi Abe1,2,3 and Shinji Okuyama1 1 Department of Physical Engineering, Mie University, Mie 514-8507, Japan 2 Institut Supérieur des Matériaux et Mécaniques Avancés, 44 F. A. Bartholdi, 72000 Le Mans, France 3 Inspire Institute Inc., Alexandria, Virginia 22303, USA Abstract Similarity between quantum mechanics and thermodynamics is discussed. It is found that if the Clausius equality is imposed on the Shannon entropy and the analogue of the quantity of heat, then the value of the Shannon entropy comes to formally coincide with that of the von Neumann entropy of the canonical density matrix, and pure-state quantum mechanics apparently transmutes into quantum thermodynamics. The corresponding quantum Carnot cycle of a simple two-state model of a particle confined in a one-dimensional infinite potential well is studied, and its efficiency is shown to be identical to the classical one. PACS number(s): 03.65.-w, 05.70.-a 1 In their work [1], Bender, Brody, and Meister have developed an interesting discussion about a quantum-mechanical analogue of the Carnot engine. They have treated a two-state model of a single particle confined in a one-dimensional potential well with width L and have considered reversible cycle. Using the “internal” energy, E (L) = ψ H ψ , they define the pressure (i.e., the force, because of the single dimensionality) as f = − d E(L) / d L , where H is the system Hamiltonian and ψ is a quantum state. Then, the analogue of “isothermal” process is defined to be f L = const.
    [Show full text]
  • A Lindblad Model of Quantum Brownian Motion
    A Lindblad model of quantum Brownian motion Aniello Lampo,1, ∗ Soon Hoe Lim,2 Jan Wehr,2 Pietro Massignan,1 and Maciej Lewenstein1, 3 1ICFO { Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 2Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, USA 3ICREA { Instituci´oCatalana de Recerca i Estudis Avan¸cats, Lluis Companys 23, E-08010 Barcelona, Spain (Dated: December 16, 2016) The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment. PACS numbers: 05.40.-a,03.65.Yz,72.70.+m,03.75.Gg I. INTRODUCTION default choice for evaluating decoherence and dissipation processes, and in general it provides a way to treat quan- A physical system interacting with the environment is titatively the effects experienced by an open system due referred to as open [1]. In reality, such an interaction to the interaction with the environment [27{30]. Here- is unavoidable, so every physical system is affected by after, we will refer to the central particle studied in the the presence of its surroundings, leading to dissipation, model as the Brownian particle.
    [Show full text]
  • Quantum Dissipation and Decoherence of Collective Telefon 08 21/598-32 34 Prof
    N§ MAI %NRAISONDU -OBILITÏ DÏMÏNAGEMENT E DU3ERVICEDELA RENCONTREDÏTUDIANTS !UPROGRAMMERENCONTREDÏTU TAIRE ÌL5,0!JOUTERÌ COMMUNICATION DIANTSETDENSEIGNANTS CHERCHEURS LACOMPÏTENCEDISCIPLINAIREETLIN ETDENSEIGNANTS LEPROCHAIN CHERCHEURSDANSLE VISITEDUCAMPUS INFORMATIONSSUR GUISTIQUE UNE EXPÏRIENCE BINATIO LES ÏCHANGES %RASMUS3OCRATES NALEETBICULTURELLE TELESTLOBJECTIF BULLETIN DOMAINEDESSCIENCESDE ENTRELES&ACULTÏSDEBIOLOGIEDES DECETTECOLLABORATIONNOVATRICEEN DINFORMATIONS LAVIE5,0 5NIVERSITËT DEUX UNIVERSITÏS PRÏSENTATION DE BIOLOGIEBIOCHIMIE PARAÔTRA DES3AARLANDES PROJETS DE RECHERCHE EN PHARMA LEMAI COLOGIE VIROLOGIE IMMUNOLOGIE ET #ONTACTS 5NE PREMIÒRE RENCONTRE AVAIT EU GÏNÏTIQUEHUMAINEDU:(-" &ACULTÏDESSCIENCESDELAVIE LIEU Ì 3AARBRàCKEN EN MAI $R*OERN0àTZ #ETTERENCONTRE ORGANISÏEÌLINI JPUETZ IBMCU STRASBGFR SUIVIEENFÏVRIERPARLAVENUE TIATIVEDU$R*0àTZETDU#ENTRE #ENTREDELANGUES)," Ì 3TRASBOURG DUN GROUPE DÏTU DELANGUESDEL)NSTITUT,E"ELCÙTÏ #ATHERINE#HOUISSA DIANTS ET DENSEIGNANTS DE LUNI STRASBOURGEOIS DU 0R - 3CHMITT BUREAU( VERSITÏ DE LA 3ARRE ,A E ÏDITION CÙTÏ SARROIS SINSCRIT DANS LE CATHERINECHOUISSA LABLANG ULPU STRASBGFR LE MAI PERMETTRA CETTE CADREDESÏCHANGESENTRELESDEUX FOISAUXSTRASBOURGEOISDEDÏCOU FACULTÏS VISANT Ì LA MISE SUR PIED ;SOMMAIRE VRIR LES LABORATOIRES DU :ENTRUM DUN CURSUS FRANCO ALLEMAND EN FàR(UMANUND-OLEKULARBIOLOGIE 3CIENCES DE LA VIE 5NE ÏTUDIANTE !CTUALITÏS :(-" DEL5NIVERSITÏDELA3ARRE DE L5,0 TERMINE ACTUELLEMENT SA IMPLANTÏS SUR LE SITE DU #(5 DE LICENCE, ÌL5D3 DEUXÏTUDIANTS /RGANISATION 3AARBRàCKEN
    [Show full text]
  • Frontiers of Quantum and Mesoscopic Thermodynamics 14 - 20 July 2019, Prague, Czech Republic
    Frontiers of Quantum and Mesoscopic Thermodynamics 14 - 20 July 2019, Prague, Czech Republic Under the auspicies of Ing. Miloš Zeman President of the Czech Republic Jaroslav Kubera President of the Senate of the Parliament of the Czech Republic Milan Štˇech Vice-President of the Senate of the Parliament of the Czech Republic Prof. RNDr. Eva Zažímalová, CSc. President of the Czech Academy of Sciences Dominik Cardinal Duka OP Archbishop of Prague Supported by • Committee on Education, Science, Culture, Human Rights and Petitions of the Senate of the Parliament of the Czech Republic • Institute of Physics, the Czech Academy of Sciences • Department of Physics, Texas A&M University, USA • Institute for Theoretical Physics, University of Amsterdam, The Netherlands • College of Engineering and Science, University of Detroit Mercy, USA • Quantum Optics Lab at the BRIC, Baylor University, USA • Institut de Physique Théorique, CEA/CNRS Saclay, France Topics • Non-equilibrium quantum phenomena • Foundations of quantum physics • Quantum measurement, entanglement and coherence • Dissipation, dephasing, noise and decoherence • Many body physics, quantum field theory • Quantum statistical physics and thermodynamics • Quantum optics • Quantum simulations • Physics of quantum information and computing • Topological states of quantum matter, quantum phase transitions • Macroscopic quantum behavior • Cold atoms and molecules, Bose-Einstein condensates • Mesoscopic, nano-electromechanical and nano-optical systems • Biological systems, molecular motors and
    [Show full text]
  • Quantum Thermodynamics: a Dynamical Viewpoint
    Entropy 2013, 15, 2100-2128; doi:10.3390/e15062100 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Review Quantum Thermodynamics: A Dynamical Viewpoint Ronnie Kosloff Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel; E-Mail: [email protected]; Tel.: + 972-26585485; Fax: + 972-26513742 Received: 26 March 2013; in revised form: 21 May 2013 / Accepted: 23 May 2013 / Published: 29 May 2013 Abstract: Quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. The viewpoint advocated is based on the intimate connection of quan- tum 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 consistency between the two theories, which address the same subject from different foundations. We claim that inconsistency is the result of faulty analysis, pointing to flaws in approximations. Keywords: thermodynamics; open quantum systems; heat engines; refrigerators; quantum devices; finite time thermodynamics 1. Introduction Quantum thermodynamics is the study of thermodynamic processes within the context of quantum dynamics. Thermodynamics preceded quantum mechanics; consistency with thermodynamics led to Planck’s law, the dawn of quantum theory. Following the ideas of Planck on black body radiation, Einstein (1905) quantized the electromagnetic field [1]. Quantum thermodynamics is devoted to unraveling the intimate connection between the laws of thermodynamics and their quantum origin, requiring consistency. For many decades, the two theories developed separately. An exception is the study of Scovil et al.[2,3], which showed the equivalence of the Carnot engine [4] with the three level Maser, setting the stage for new developments.
    [Show full text]
  • Arxiv:Quant-Ph/0208026V1 5 Aug 2002
    Path Integrals and Their Application to Dissipative Quantum Systems Gert-Ludwig Ingold Institut f¨ur Physik, Universit¨at Augsburg, D-86135 Augsburg arXiv:quant-ph/0208026v1 5 Aug 2002 to be published in “Coherent Evolution in Noisy Environments”, Lecture Notes in Physics, http://link.springer.de/series/lnpp/ c Springer Verlag, Berlin-Heidelberg-New York Path Integrals and Their Application to Dissipative Quantum Systems Gert-Ludwig Ingold Institut f¨ur Physik, Universit¨at Augsburg, D-86135 Augsburg, Germany 1 Introduction The coupling of a system to its environment is a recurrent subject in this collec- tion of lecture notes. The consequences of such a coupling are threefold. First of all, energy may irreversibly be transferred from the system to the environment thereby giving rise to the phenomenon of dissipation. In addition, the fluctuating force exerted by the environment on the system causes fluctuations of the system degree of freedom which manifest itself for example as Brownian motion. While these two effects occur both for classical as well as quantum systems, there exists a third phenomenon which is specific to the quantum world. As a consequence of the entanglement between system and environmental degrees of freedom a coherent superposition of quantum states may be destroyed in a process referred to as decoherence. This effect is of major concern if one wants to implement a quantum computer. Therefore, decoherence is discussed in detail in Chap. 5. Quantum computation, however, is by no means the only topic where the coupling to an environment is relevant. In fact, virtually no real system can be considered as completely isolated from its surroundings.
    [Show full text]
  • The Second Laws of Quantum Thermodynamics
    The second laws of quantum thermodynamics Fernando Brandãoa,1, Michał Horodeckib, Nelly Ngc, Jonathan Oppenheimc,d,2, and Stephanie Wehnerc,e aDepartment of Computer Science, University College London, London WC1E 6BT, United Kingdom; bInstytut Fizyki Teoretycznej i Astrofizyki, University of Gdansk, 80-952 Gdansk, Poland; cCentre for Quantum Technologies, National University of Singapore, 117543 Singapore; dDepartment of Physics & Astronomy, University College London, London WC1E 6BT, United Kingdom; and eSchool of Computing, National University of Singapore, 117417 Singapore Edited by Peter W. Shor, Massachusetts Institute of Technology, Cambridge, MA, and approved January 12, 2015 (received for review June 26, 2014) The second law of thermodynamics places constraints on state Here, we show that this is not the case in the microscopic regime, transformations. It applies to systems composed of many particles, and we therefore needs to talk about “how cyclic” a process is when however, we are seeing that one can formulate laws of thermo- stating the second law. We also derive in this work, a zeroth law of dynamics when only a small number of particles are interacting thermodynamics, which is stronger than the ordinary zeroth law. with a heat bath. Is there a second law of thermodynamics in this For thermodynamics at the macroscopic scale, a system in state regime? Here, we find that for processes which are approximately ρ can be transformed into state ρ′ provided that the free energy cyclic, the second law for microscopic systems takes on a different goes down, where the free energy for a state ρ is form compared to the macroscopic scale, imposing not just one constraint on state transformations, but an entire family of FðρÞ = hEðρÞi − kTSðρÞ; [1] constraints.
    [Show full text]