REVIEWS OF MODERN PHYSICS, VOLUME 83, JULY–SEPTEMBER 2011 Colloquium: Quantum fluctuation relations: Foundations and applications
Michele Campisi, Peter Ha¨nggi, and Peter Talkner Institute of Physics, University of Augsburg, Universita¨tsstrasse 1, D-86135 Augsburg, Germany (Received 10 December 2010; published 6 July 2011)
Two fundamental ingredients play a decisive role in the foundation of fluctuation relations: the principle of microreversibility and the fact that thermal equilibrium is described by the Gibbs canonical ensemble. Building on these two pillars the reader is guided through a self-contained exposition of the theory and applications of quantum fluctuation relations. These are exact results that constitute the fulcrum of the recent development of nonequilibrium thermodynamics beyond the linear response regime. The material is organized in a way that emphasizes the historical connection between quantum fluctuation relations and (non)linear response theory. A number of fundamental issues are clarified which were not completely settled in the prior literature. The main focus is on (i) work fluctuation relations for transiently driven closed or open quantum systems, and (ii) on fluctuation relations for heat and matter exchange in quantum transport settings. Recently performed and proposed experimental applications are presented and discussed.
DOI: 10.1103/RevModPhys.83.771 PACS numbers: 05.30. d, 05.40. a, 05.60.Gg, 05.70.Ln
CONTENTS I. INTRODUCTION I. Introduction 771 II. Nonlinear Response Theory and Classical Fluctuation This Colloquium focuses on fluctuation relations and, in Relations 773 particular, on their quantum versions. These relations con- A. Microreversibility of nonautonomous classical stitute a research topic that recently has attracted a great deal systems 773 of attention. At the microscopic level, matter is in a perma- B. Bochkov-Kuzovlev approach 774 nent state of agitation; consequently, many physical quanti- C. Jarzynski approach 775 ties of interest continuously undergo random fluctuations. III. Fundamental Issues 776 The purpose of statistical mechanics is the characterization A. Inclusive, exclusive, and dissipated work 776 of the statistical properties of those fluctuating quantities B. The problem of gauge freedom 777 from the known laws of classical and quantum physics that C. Work is not a quantum observable 777 govern the dynamics of the constituents of matter. A para- IV. Quantum Work Fluctuation Relations 778 digmatic example is the Maxwell distribution of velocities in A. Microreversibility of nonautonomous quantum a rarefied gas at equilibrium, which follows from the sole systems 778 assumptions that the microdynamics are Hamiltonian, and B. The work probability density function 778 that the very many system constituents interact via negligible, C. The characteristic function of work 779 short range forces (Khinchin, 1949). Besides the fluctuation D. Quantum generating functional 780 of velocity (or energy) at equilibrium, one might be interested in the properties of other fluctuating quantities, e.g., heat E. Microreversibility, conditional probabilities, and entropy 780 and work, characterizing nonequilibrium transformations. F. Weak-coupling case 781 Imposed by the reversibility of microscopic dynamical laws, the fluctuation relations put severe restrictions on the G. Strong-coupling case 782 V. Quantum Exchange Fluctuation Relations 783 form that the probability density function (PDF) of the con- VI. Experiments 784 sidered nonequilibrium fluctuating quantities may assume. A. Work fluctuation relations 784 Fluctuation relations are typically expressed in the form 1. Proposal for an experiment employing trapped exp cold ions 784 pFðxÞ¼pBð xÞ ½aðx bÞ ; (1) 2. Proposal for an experiment employing circuit quantum electrodynamics 785 where pFðxÞ is the PDF of the fluctuating quantity x during a B. Exchange fluctuation relations 785 nonequilibrium thermodynamic transformation, referred to 1. An electron counting statistics experiment 785 for simplicity as the forward (F) transformation, and pBðxÞ 2. Nonlinear response relations in a quantum is the PDF of x during the reversed (backward B) trans- coherent conductor 786 formation. The precise meaning of these expressions will VII. Outlook 787 be clarified below. The real-valued constants a and b contain Appendix A: Derivation of the Bochkov-Kuzvlev Relation 788 information about the equilibrium starting points of the B and Appendix B: Quantum Microreversibility 788 F transformations. Figure 1 shows a probability distribution Appendix C: Tasaki-Crooks Relation for the Characteristic satisfying the fluctuation relation, as measured in a recent Function 788 experiment of electron transport through a nanojunction
0034-6861= 2011=83(3)=771(21) 771 Ó 2011 American Physical Society 772 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ...
action of an external, in general, time-dependent force t that couples to an observable Q of the system. The dynamics of the system then is governed by the modified, time-dependent Hamiltonian
H ð tÞ¼H 0 tQ: (3)
FIG. 1. Example of statistics obeying the fluctuation relation, The approach of Callen and Welton (1951) was further Eq. (1). Left panel: Probability distribution pðqÞ of number q of systematized by Green (1952, 1954) and, in particular, by electrons, transported through a nanojunction subject to an electrical Kubo (1957) who proved that the linear response is deter- potential difference. Right panel: The linear behavior of mined by a response function BQðtÞ, which gives the de- ln½pðqÞ=pð qÞ evidences that pðqÞ obeys the fluctuation relation, viation h BðtÞi of the expectation value of an observable B to Eq. (1). In this example forward and backward protocols coincide its unperturbed value as 0 yielding pB ¼ pF p, and consequently b ¼ in Eq. (1). From Z Utsumi et al., 2010. t h BðtÞi ¼ BQðt sÞ sds: (4) 1
Kubo (1957) showed that the response function can be ex- (Utsumi et al., 2010). We analyze this experiment in detail in pressed in terms of the commutator of the observables Q and H H Sec. VI. B ðtÞ as BQðsÞ¼h½Q; B ðsÞ i=iℏ (the superscript H de- As often happens in science, the historical development of notes the Heisenberg picture with respect to the unperturbed theories is quite tortuous. Fluctuation relations are no excep- dynamics.) Moreover, Kubo derived the general relation tion in this respect. Without any intention of providing a thorough and complete historical account, we mention below hQBHðtÞi ¼ hBHðt iℏ ÞQi (5) a few milestones that, in our view, mark crucial steps in the historical development of quantum fluctuation relations. The between differently ordered thermal correlation functions beginning of the story might be traced back to the early years and deduced from it the celebrated quantum fluctuation- of the last century, with the work of Sutherland (1902, 1905) dissipation theorem (Callen and Welton, 1951), reading and Einstein (1905, 1906a, 1906b) first, and of Johnson (1928) and Nyquist (1928) later, when it was found that the ^ ^ linear response of a system in thermal equilibrium, as it is BQð!Þ¼ðℏ=2iÞ cothð ℏ!=2Þ BQð!Þ; (6) driven out of equilibrium by an external force, is determined R by the fluctuation properties of the system in the initial ^ 1 i!s where BQð!Þ¼ 1 e BQðsÞds denotes the Fourier equilibrium state. Specifically, Sutherland (1902, 1905) and transform of the symmetrized, stationary equilibrium corre- Einstein (1905, 1906a, 1906b) found a relation between the QBH BH Q 2 lation functionR BQðsÞ¼h ðsÞþ ðsÞ i= , and mobility of a Brownian particle (encoding information about ^ 1 i!s BQð!Þ¼ e BQðsÞds denotes the Fourier trans- its response to an externally applied force) and its diffusion 1 BQðsÞ constant (encoding information about its equilibrium fluctua- form of the response function . Note that the tions). Johnson (1928) and Nyquist (1928)1 discovered the fluctuation-dissipation theorem is valid also for many-particle corresponding relation between the resistance of a circuit and systems independent of the respective particle statistics. the spontaneous current fluctuations occurring in the absence Besides offering a unified and rigorous picture of the of an applied electric potential. fluctuation-dissipation theorem, the theory of Kubo also in- The next step was taken by Callen and Welton (1951) who cluded other important advancements in the field of nonequi- derived the previous results within a general quantum me- librium thermodynamics. Specifically, we note the celebrated chanical setting. The starting point of their analysis was a Onsager-Casimir reciprocity relations (Onsager, 1931a, quantum mechanical system described by a Hamiltonian 1931b; Casimir, 1945). These relations state that, as a con- sequence of microreversibility, the matrix of transport coef- H 0. Initially this system stays in a thermal equilibrium state 1 ficients that connects applied forces to so-called fluxes in a at the inverse temperature ðkBTÞ , wherein kB is the Boltzmann constant. This state is described by a density system close to equilibrium consists of a symmetric and an matrix of canonical form; i.e., it is given by a Gibbs state antisymmetric block. The symmetric block couples forces and fluxes that have same parity under time reversal and the H 0 %0 ¼ e =Z0; (2) antisymmetric block couples forces and fluxes that have different parity. Z Tr H 0 where 0 ¼ e denotes the partition function of the Most importantly, the analysis of Kubo (1957) opened the unperturbed system and Tr denotes trace over its Hilbert possibility for a systematic advancement of response theory, 0 space. At later times t> , the system is perturbed by the allowing, in particular, one to investigate the existence of higher order fluctuation-dissipation relations beyond linear 1Nyquist already discussed in his Eq. (8) a precursor of the regime. This task was soon undertaken by Bernard and quantum fluctuation-dissipation theorem as developed later by Callen (1959), who pointed out a hierarchy of irreversible Callen and Welton (1951). He only missed the correct form by thermodynamic relationships. These higher order fluctuation- omitting the zero-point energy contribution in his result; see also dissipation relations were investigated by Stratonovich for Ha¨nggi and Ingold (2005). the Markovian system, and later by Efremov (1969) for
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... 773 non-Markovian systems; see Stratonovich (1992), Chap. I, II. NONLINEAR RESPONSE THEORY AND CLASSICAL and references therein. FLUCTUATION RELATIONS Even for arbitrary systems far from equilibrium the linear response to an applied force can likewise be related to A. Microreversibility of nonautonomous classical systems tailored two-point correlation functions of corresponding sta- Two ingredients are at the heart of fluctuation relations. tionary nonequilibrium fluctuations of the underlying unper- The first one concerns the initial condition of the system turbed, stationary nonequilibrium system (Ha¨nggi, 1978; under study. This is supposed to be in thermal equilibrium Ha¨nggi and Thomas, 1982). They coined the expression described by a canonical distribution of the form of Eq. (2). ‘‘fluctuation theorems’’ for these relations. As in the near It hence is of statistical nature. Its use and properties are thermal equilibrium case, in this case higher order nonlinear discussed in many textbooks on statistical mechanics. The response can also be linked to corresponding higher order other ingredient, concerning the dynamics of the system, is correlation functions of those nonequilibrium fluctuations the principle of microreversibility. This point needs some (Ha¨nggi, 1978; Prost et al., 2009). clarification since microreversibility is customarily under- At the same time, in the late 1970s Bochkov and Kuzovlev stood as a property of autonomous (i.e., nondriven) systems (1977) provided a single compact classical expression that described by a time-independent Hamiltonian (Messiah, contains fluctuation relations of all orders for systems that are 1962, Vol. 2, Ch. XV). On the contrary, here we are concerned at thermal equilibrium when unperturbed. This expression, with nonautonomous systems, governed by explicitly time- Eq. (14), can be seen as a fully nonlinear, exact, and universal dependent Hamiltonians. In the following we analyze this fluctuation relation. The Bochkov and Kuzovlev formula, principle for classical systems in a way that at first glance Eq. (14), soon turned out useful in addressing the problem may appear rather formal but will prove indispensable later of connecting the deterministic and the stochastic descrip- on. The analogous discussion of the quantum case is given tions of nonlinear dissipative systems (Bochkov and next in Sec. IV.A. Kuzovlev, 1978; Ha¨nggi, 1982). We deal here with a classical system characterized by a As often happens in physics, the most elegant, compact, Hamiltonian that consists of an unperturbed part H0ðzÞ and a and universal relations are consequences of general physical perturbation tQðzÞ due to an external force t that couples symmetries. In the case of Bochkov and Kuzovlev (1977) the to the conjugate coordinate QðzÞ. Then the total system fluctuation relation follows from the time-reversal invariance Hamiltonian becomes2 of the equations of microscopic motion, combined with the assumption that the system initially resides in thermal equi- Hðz; tÞ¼H0ðzÞ tQðzÞ; (7) librium described by the classical analog of the Gibbs state, z ¼ðq pÞ Eq. (2). Bochkov and Kuzovlev (1977, 1979, 1981a, 1981b) where ; denotes a point in the phase space of the proved Eq. (14) for classical systems. Their derivation will be considered system. In the following we assume that the force acts within a temporal interval set by a starting time 0 and a reviewed in the next section. The quantum version, Eq. (55), final time . The instantaneous force values are specified was not reported until recently (Andrieux and Gaspard, t by a function , which we refer to as the force protocol. In the 2008). In Sec. III.C we discuss the fundamental obstacles sequel, it turned out necessary to clearly distinguish between that prevented Bochkov and Kuzovlev (1977, 1979, 1981a, the function and the value that it takes at a particular 1981b) and Stratonovich (1994), who also studied this very t instant of time t. quantum problem, from obtaining Eq. (55). For these systems the principle of microreversibility holds A new wave of activity in fluctuation relations was ini- in the following sense. The solution of Hamilton’s equations tiated by Evans et al. (1993) and Gallavotti and Cohen (1995) of motion assigns to each initial point in phase space z0 ¼ on the statistics of the entropy produced in nonequilibrium ðq0; p0Þ a point z at the later time t 2½0; , which is steady states, and by Jarzynski (1997) on the statistics of t specified by the values of the force in the order of their work performed by a transient, time-dependent perturbation. appearance within the considered time span. Hence, the Since then, the field has generated much interest and flour- position ished. The existing reviews on this topic mostly cover clas- sical fluctuation relations (Jarzynski, 2008, 2011; Marconi zt ¼ ’t;0½z0; (8) et al., 2008; Rondoni and Mejı´a-Monasterio, 2007; Seifert, z ; 2008), while the comprehensive review by Esposito et al. at time t is determined by the flow ’t;0½ 0 which is a (2009) provided a solid, though in parts technical account of function of the initial point z0 and a functional of the force 3 the state of the art of quantum fluctuation theorems. With this protocol . In a computer simulation one can invert the work we want to present a widely accessible introduction to direction of time and let the trajectory run backward without quantum fluctuation relations, covering as well the most a problem. Although, as experience evidences, it is impos- recent decisive advancements. Particularly, our emphasis sible to actively revert the direction of time in any experi- will be on (i) their connection to the linear and nonlinear ment, there is yet a way to run a time-reversed trajectory in response theories (Sec. II), (ii) the clarification of fundamen- real time. For simplicity we assume that the Hamiltonian H0 tal issues that relate to the notion of ‘‘work’’ (Sec. III), (iii) the derivation of quantum fluctuation relations for both 2The generalization to the case of several forces coupling via closed and open quantum systems (Secs. IV and V), and also different conjugate coordinates is straightforward. 3 (iv) their impact for experimental applications and validation Because of causality, ’t;0½z0; may, of course, depend only on (Sec. VI). the part of the protocol including times from t ¼ 0 up to time t.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 774 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... R H0ðz0Þ with Z0 ¼ dz0e . Consequently the trajectory Bt becomes a random quantity. Next we introduce the quantity Z _ W0½z0; ¼ dt tQt; (12) 0 _ where Qt is the time derivative of Qt ¼ Qð’t;0½z0; Þ. From Hamilton’s equations it follows that (Jarzynski, 2007):
W0½z0; ¼H0ð’ ;0½z0; Þ H0ðz0Þ: (13)
Therefore, we interpret W0 as the work injected in the system 4 described by H0 during the action of the force protocol. The central finding of Bochkov and Kuzovlev (1977) is a formal relation between the generating functional for multitime correlation functions of the phase-space functions Bt and Qt and the generating functional for the time-reversed multi- time autocorrelation functions of Bt, reading R R dtu B dtu~ " B he 0 t t e W0 i ¼he 0 t B t i ~; (14) "Q
where u is an arbitrary test function, u~t ¼ u t is its tem- poral reverse, and the average denoted by h i is taken with FIG. 2. Microreversibility for nonautonomous classical respect to the Gibbs distribution 0 of Eq. (11). On the left- (Hamiltonian) systems. The initial condition z0 evolves, under the hand side, the time evolutions of B and Q are governed by protocol , from time t ¼ 0 until time t to z ¼ ’ 0½z0; and until t t t t; the full Hamiltonian (7) in the presence of the forward time t ¼ to z . The time-reversed final condition "z evolves, ~ protocol as indicated by the subscript , while on the right- under the protocol "Q from time t ¼ 0 until t to z ; ~ z ; hand side the dynamics is determined by the time-reversed ’ t;0½" "Q ¼"’t;0½ 0 , Eq. (10), and until time t ¼ to ~ the time-reversed initial condition "z0. protocol indicated by the subscript "Q . The derivation of Eq. (14), which is based on the microreversibility principle, Eq. (10), is given in Appendix A. The importance of Eq. (14) is time reversal invariant, i.e., that it remains unchanged if the lies in the fact that it contains the Onsager reciprocity rela- signs of momenta are reverted. Moreover, we restrict our- tions and fluctuation relations of all orders within a single selves to conjugate coordinates QðzÞ that transform under compact formula (Bochkov and Kuzovlev, 1977). These re- time reversal with a definite parity "Q ¼ 1. Stratonovich lations may be obtained by means of functional derivatives (1994, Sec. 1.2.3) showed that the flow under the backward of both sides of Eq. (14), of various orders, with respect protocol ~, with to the force field and the test field u at vanishing fields 0 ~ ¼ u ¼ . The classical limit of the Callen and Welton t ¼ t; (9) (1951) fluctuation-dissipation theorem, Eq. (6), for instance, is related to the flow under the forward protocol via is obtained by differentiation with respect to u, followed by a differentiation with respect to (Bochkov and Kuzovlev, ~ ’t;0½z0; ¼"’ t;0½"z ; "Q ; (10) 1977), both at vanishing fields u and . Another remarkable identity is achieved from Eq. (14)by " z where maps any phase-space point on its time-reversed setting u ¼ 0, but leaving the force finite. This yields the "z ¼ "ðq; pÞ¼ðq; pÞ image . Equation (10) expresses the Bochkov-Kuzovlev equality, reading principle of microreversibility in driven systems. Its meaning W0 is shown in Fig. 2. Particularly, it states that in order to trace he i ¼ 1: (15) back a trajectory, one has to reverse the sign of the velocity, as well as the temporal succession of the force values , and the In other words, for any system that initially stays in thermal ~ 1 sign of force ,if"Q ¼ 1. equilibrium at a temperature T ¼ =kB , the work, Eq. (12), done on the system by an external force is a random quantity W0 with an exponential expectation value he i that is inde- B. Bochkov-Kuzovlev approach pendent of any detail of the system and the force acting on it. n We consider a phase-space function BðzÞ which has a This, of course, does not hold for the individual th moments definite parity under time reversal " ¼ 1, i.e., Bð"zÞ¼ of work. Since the exponential function is concave, a direct B consequence of Eq. (15)is "BBðzÞ. Bt ¼ Bð’t;0½z0; Þ denotes its temporal evolution. Depending on the initial condition z0 different trajectories Bt 4 are realized. Under the above stated assumption that at time Following Jarzynski (2007) we refer to W0 as the exclusive work, t ¼ 0 the system is prepared in a Gibbs equilibrium, the to distinguish from the inclusive work W ¼ Hðz ; Þ Hðz0; 0Þ, initial conditions are randomly sampled from the distribution Eq. (19), which accounts also for the coupling between the external source and the system. We will come back later to these two H0ðz0Þ 0ðz0Þ¼e =Z0; (11) definitions of work in Sec. III.A.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... 775
0 hW0i : (16) In terms of the force t and the conjugate coordinate Qt, the inclusive work is expressed as5: That is, on average, a driven Hamiltonian system may only Z z Z absorb energy if it is perturbed out of thermal equilibrium. _ @Hð t; tÞ _ W½z0; ¼ dt t ¼ dt tQt This does not exclude the existence of energy releasing events 0 @ t 0 which, in fact, must happen with certainty in order that ¼ W0½z0; Q j : (20) Eq. (15) holds if the average work is larger than zero. t t 0 Equation (16) may be regarded as a microscopic manifesta- For simplicity we confined ourselves to the case of an even tion of the second law of thermodynamics. For this reason conjugate coordinate Q. In a corresponding way, as described Stratonovich (1994, Sec. 1.2.4) referred to it as the in Appendix A, we obtained the following relation between H theorem. We recapitulate that only two ingredients, initial generating functionals of forward and backward processes in Gibbsian equilibrium and microreversibility of the dynamics, analogy to Eq. (14): have led to Eq. (14). In conclusion, this relation not only R R Zð Þ t ~ contains linear and nonlinear response theories, but also the 0 dtutBt W 0 dtut"BBt he e i ¼ he i ~: (21) second law of thermodynamics. Zð 0Þ The complete information about the statistics is contained While on the left-hand side the time evolution is controlled ; in the work PDF p0½W0 . The only random element enter- by the forward protocol and the average is performed with ing the work, Eq. (13), is the initial phase point z0 which is respect to the initial thermal distribution ðz; 0Þ, on the ; distributed according to Eq. (11). Therefore, p0½W0 may right-hand side the time evolution is governed by the reversed formally be expressed as protocol ~ and the average is taken over the reference equi- Z ~ librium state ðz; 0Þ¼ ðz; Þ. Here p0½W0; ¼ dz0 0ðz0Þ ðW0 H0ðz ÞþH0ðz0ÞÞ; Hðz; tÞ ðz; tÞ¼e =Zð tÞ (22) (17) formally describes the thermal equilibrium of a system with where denotes Dirac’s delta function. The functional de- the Hamiltonian Hðz; tÞ at the inverse temperature . The pendence of p0½W0; on the force protocol is contained in partitionR function Zð tÞ is defined accordingly as Zð tÞ¼ z z ; z the term ¼ ’ ;0½ 0 . Using the microreversibility prin- dze Hð ; tÞ. Note that in general the reference state ciple, Eq. (10), one obtains the following fluctuation relation: ðz; tÞ is different from the actual phase-space distribution p0½W0; reached under the action of the protocol at time t, i.e., ¼ W0 z 1 z; 1 z; ~ e ; (18) ð ;tÞ¼ ð’t;0 ½ ; 0Þ, where ’t;0 ½ denotes the p0½ W0; " Q point in phase space that evolves to z in the time 0 to t under in a way analogous to the derivation of Eq. (14). We refer to the action of . this relation as the Bochkov-Kuzovlev work fluctuation rela- Setting u 0 we obtain tion, although it was not explicitly given by Bochkov and he W i ¼ e F; (23) Kuzovlev, but was only recently obtained by Horowitz and Jarzynski (2007). This equation has a profound physical where meaning. Consider a positive work W0 > 0. Then Eq. (18) Zð Þ says that the probability that this work is injected into the F ¼ Fð Þ Fð 0Þ¼ ln (24) system is larger by the factor e W0 than the probability that Zð 0Þ the same work is released under the reversed forcing: Energy is the free energy difference between the reference state consuming processes are exponentially more probable than ðz; tÞ and the initial equilibrium state ðz 0Þ.Asa energy releasing processes. Thus, Eq. (18) expresses the consequence of Eq. (23) we have second law of thermodynamics at a detailed level which quantifies the relative frequency of energy releasing pro- hWi F; (25) cesses. By multiplying both sides of Eq. (18)by which is yet another expression of the second law of thermo- ; ~ W0 p0½ W0 "Q e and integrating over W0, one recovers dynamics. Equation (23) was first put forward by Jarzynski the Bochkov-Kuzovlev identity, Eq. (15). (1997) and is commonly referred to in the literature as the ‘‘Jarzynski equality.’’ C. Jarzynski approach In close analogy to the Bochkov-Kuzovlev approach the PDF of the inclusive work can be formally expressed as An alternative definition of work is based on the compari- Z ; z z ð z z Þ son of the total Hamiltonians at the end and the beginning of a p½W ¼ d 0 ð 0; 0Þ W Hð ; ÞþHð 0; 0Þ : force protocol, leading to the notion of ‘‘inclusive’’ work in contrast to the ‘‘exclusive’’ work defined in Eq. (13). The (26) latter equals the energy difference referring to the unper- Its Fourier transform defines the characteristic function of turbed Hamiltonian H0. Accordingly, the inclusive work is work: the difference of the total Hamiltonians at the final time t ¼ and the initial time t ¼ 0: 5For a further discussion of inclusive and exclusive work we refer z ; z z W½ 0 ¼Hð ; Þ Hð 0; 0Þ: (19) to Sec. III.A.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 776 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... Z literature refer to such general functional identities as G½u; ¼ dWeiuW p½W; Eq. (21). We introduced them here to emphasize the connec- Z tion between the recent results, Eqs. (23) and (28), with the iu½Hðz ; Þ Hðz0; 0Þ Hðz0; 0Þ ¼ dz0e e =Zð 0Þ older results of Bochkov and Kuzovlev (1977), Eqs. (15) and (18). The latter ones were practically ignored, or sometimes Z Z z Hðz0; 0Þ _ @Hð t; tÞ e misinterpreted as special instances of the former ones for the ¼ dz0 exp iu dt t : 0 @ t Zð 0Þ case of cyclic protocols ( F ¼ 0), by those working in the (27) field of nonequilibrium work fluctuations. Only recently Jarzynski (2007) pointed out the differences and analogies Using the microreversibility principle, Eq. (10), we obtain between the inclusive and exclusive approaches. in a way similar to Eq. (18) the (inclusive) work fluctuation relation: III. FUNDAMENTAL ISSUES ; p½W ðW FÞ ¼ e ; (28) A. Inclusive, exclusive, and dissipated work p½ W; ~ where the probability p½ W; ~ refers to the backward pro- As evidenced in the previous section, the studies of cess which for the inclusive work has to be determined with Bochkov and Kuzovlev (1977) and Jarzynski (1997) are based on different definitions of work, Eqs. (13) and (19) reference to the initial thermal state ðz; Þ. First put forward by Crooks (1999), Eq. (28) is commonly referred reflecting two different viewpoints (Jarzynski, 2007). From to in the literature as the ‘‘Crooks fluctuation theorem.’’ The the exclusive viewpoint of Bochkov and Kuzovlev (1977) the Jarzynski equality, Eq. (23), is obtained by multiplying both change in the energy H0 of the unforced system is considered, sides of Eq. (28)byp½ W; ~ e W and integrating over W. thus the forcing term ( tQ) of the total Hamiltonian is not Equations (21), (23), and (28) continue to hold also when Q is included in the computation of work. From the inclusive point odd under time reversal, with the provision that ~ is replaced of view the definition of work, Eq. (19), is based on the by ~. change of the total energy H including the forcing term We here point out the salient fact that, within the inclusive ( tQ). In experiments and practical applications of fluc- approach, a connection is established between the nonequi- tuation relations, special care must be paid in properly iden- librium work W and the difference of free energies F, of the tifying the measured work with either the inclusive (W)or exclusive (W0) work, bearing in mind that represents the corresponding equilibrium states ðz; Þ and ðz; 0Þ. Most remarkably, Eq. (25) says that the average (inclusive) prescribed parameter progression and Q is the measured work is always larger than or equal to the free energy conjugate coordinate. difference, no matter the form of the protocol ; even more The experiment of Douarche et al. (2005) is well suited to surprising is the content of Eq. (23) saying that the equilib- illustrate this point. In that experiment a prescribed torque Mt rium free energy difference may be inferred by measurements was applied to a torsion pendulum whose angular displace- of nonequilibrium work in many realizations of the forcing ment t was continuously monitored. The Hamiltonian of the experiment (Jarzynski, 1997). This is similar in spirit to the system is fluctuation-dissipation theorem, also connecting an equilib- p2 Hðy;p ; ;MÞ¼H ðyÞþH ðy;p ; Þþ rium property (the fluctuations) to a nonequilibrium one (the t B SB 2I linear response), with the major difference that Eq. (23)isan I!2 2 exact result, whereas the fluctuation-dissipation theorem þ 2 Mt ; (29) holds only to first order in the perturbation. Note that as a consequence of Eq. (28) the forward and backward PDFs of where p is the canonical momentum conjugate to , HBðyÞ is exclusive work take on the same value at W ¼ F.This the Hamiltonian of the thermal bath to which the pendulum is property has been used in experiments (Liphardt et al., coupled via the Hamiltonian HSB, and y is a point in the bath 2002; Collin et al., 2005; Douarche et al., 2005) in order phase space. Using the definitions of inclusive and exclusive to determine free energy differences from nonequilibrium work, Eqs. (12) and (20), and noticing that M playsR the role of _ measurements of work. Equations (23) and (28) have further and R that of Q, we find in this case W ¼ Mdt and _ been employed to develop efficient numerical methods for the W0 ¼ M dt. R estimation of free energies (Jarzynski, 2002; Minh and Adib, Note that the work W ¼ Mdt_ , obtained by monitor- 2008; Vaikuntanathan and Jarzynski, 2008; Hahn and Then, ing the pendulum degree of freedom only, amounts to the 2009, 2010). energy change of the total pendulum þ bath system. This is Both the Crooks fluctuation theorem, Eq. (28), and the true in general (Jarzynski, 2004). Writing the total Jarzynski equality, Eq. (23), continue to hold for any time- Hamiltonian as dependent Hamiltonian Hðz; Þ without restriction to t x y x x y y Hamiltonians of the form in Eq. (7). Indeed no restriction Hð ; ; tÞ¼HSð ; tÞþHBSð ; ÞþHBð Þ; (30) of the form in Eq. (7) was imposed in the seminal paper by with HSðx; tÞ being the Hamiltonian of the system of inter- Jarzynski (1997). In the original works of Jarzynski (1997) est, one obtains and Crooks (1999), Eqs. (23) and (28) were obtained directly, Z Z Z without passing through the more general formula in Eq. (21). @HS _ @H dH dt t ¼ dt ¼ dt ¼ W; (31) Notably, neither these seminal papers nor the subsequent 0 @ t 0 @t 0 dt
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... 777 because HBS and HB do not depend on time, and as to the laboratory frame. The dynamics of the pendulum are a consequence of Hamilton’s equations of motion now described by the Hamiltonian dH=dt ¼ @H=@t. 2 2 2 p I! 2 Introducing the notation Wdiss ¼ W F, for the dissi- H ¼ H þH þ þ I! þgð Þ: (35) B SB 2I 2 t t pated work, one deduces that the Jarzynski equality can be R reexpressed in a way that looks exactly like the Bochkov- If the work W ¼ N dt_ done by the elastic torque N ¼ Kuzovlev identity, namely, I!2ð Þ on the support is recorded then the requirement @H=@ ¼ N singles out the gauge gð Þ¼I!2 2=2 þ he Wdiss i ¼ 1: (32) t t const, leaving only the freedom to chose the unimportant 2 This might lead one to believe that the dissipated work constant. Note that when Mt ¼ I! t, the pendulum obeys coincides with W0. This, however, would be incorrect. As exactly the same equations of motion in the two examples discussed by Jarzynski (2007) and explicitly demonstrated above, Eqs. (29) and (35). The gauge is irrelevant for the law by Campisi et al. (2011a), W0 and Wdiss constitute distinct of motion but is essential for the energy-Hamiltonian 6 stochastic quantities with different PDFs. The inclusive, ex- connection. clusive, and dissipated work coincides only in the case of The issue of gauge freedom was first pointed out by Vilar and Rubi (2008c), who questioned whether a connection cyclic forcing ¼ 0 (Campisi et al., 2011a). between work and Hamiltonian may actually exist. Since then this topic has been highly debated,7 but neither the gauge B. The problem of gauge freedom invariance of fluctuation relations nor the fact that different experimental setups imply different gauges was clearly rec- We pointed out that the inclusive work W, and free energy ognized before. difference F, as defined in Eqs. (19) and (24) are, to use the expression coined by Cohen-Tannoudji et al. (1977),not ‘‘true physical quantities.’’ That is to say they are not invari- C. Work is not a quantum observable ant under gauge transformations that lead to a time-dependent shift of the energy reference point. To elucidate this, consider Thus far we have reviewed the general approach to work a mechanical system whose dynamics are governed by the fluctuation relations for classical systems. The question then naturally arises of how to treat the quantum case. Obviously, Hamiltonian Hðz; tÞ. The new Hamiltonian the Hamilton function Hðz; tÞ is to be replaced by the 0 z z H ð ; tÞ¼Hð ; tÞþgð tÞ; (33) Hamilton operator H ð tÞ, Eq. (3). The probability density ðz; tÞ is then replaced by the density matrix %ð tÞ, reading where gð tÞ is an arbitrary function of the time-dependent H ð tÞ force, generates the same equations of motion as H. However, %ð tÞ¼e =Zð tÞ; (36) 0 0 z 0 z the work W ¼ H ð ; Þ H ð 0; 0Þ that one obtains from H Z Tr ð tÞ this Hamiltonian differs from the one that follows from H, where ð tÞ¼ e is the partition function and Tr 0 denotes the trace over the system Hilbert space. The free Eq. (19): W ¼ W þ gð Þ gð 0Þ. Likewise we have for 0 the free energy difference F ¼ F þ gð Þ gð 0Þ. energy is obtained from the partition function in the same way 1 lnZ Evidently the Jarzynski equality, Eq. (23), is invariant under as for classical systems, i.e., Fð tÞ¼ ð tÞ. Less obvious is the definition of work in quantum mechanics. such gauge transformations, because the term gð Þ gð 0Þ appearing on both sides of the identity in the primed gauge, Originally, Bochkov and Kuzovlev (1977) defined the exclu- would cancel; explicitly this reads sive quantum work, in analogy with the classicalR expression, _ H Eqs. (12) and (13), as the operator W 0 ¼ 0 dt tQt ¼ W0 F0 W F H H H he i ¼ e ,he i ¼ e : (34) 0 0, where the superscript H denotes the Heisenberg picture: Thus, there is no fundamental problem associated with the BH y B gauge freedom. t ¼ Ut;0½ Ut;0½ : (37) However, one must be aware that, in each particular ex- B periment, the way by which the work is measured implies a Here is an operator in the Schro¨dinger picture and Ut;0½ specific gauge. Consider, for example, the torsion pendulum is the unitary time evolution operator governed by the Schro¨dinger equation experiment of Douarche etR al. (2005). The inclusive work W ¼ Mdt_ was computed as . The condition that this @Ut;0½ iℏ ¼ H ð ÞU 0½ ;U0 0½ ¼1; (38) measured work is related to the Hamiltonian of Eq. (29) @t t t; ; z z viaRW ¼ Hð R; Þ Hð 0; 0Þ, Eq. (19), is equivalent to _ _ with 1 denoting the identity operator. We use the notation 0 Mdt ¼ 0ð@H=@MÞMdt; see Eq. (31). If this is re- quired for all then the stricter condition @H=@M ¼ is Ut;0½ to emphasize that, similar to the classical evolution z; implied, restricting the remaining gauge freedom to the ’t;0½ of Eq. (8), the quantum evolution operator is a choice of a constant function g. This residual freedom, how- ever, is not important as it affects neither work nor free 6See also Kobe (1981), in the context of nonrelativistic energy. We now consider a different experimental setup electrodynamics. where the support to which the pendulum is attached is 7See Horowitz and Jarzynski, 2008, Peliti, 2008a, 2008b, Vilar rotated in a prescribed way according to a protocol t, and Rubi, 2008a, 2008b, 2008c, Adib, 2009, Chen, 2008a, 2008b, specifying the angular position of the support with respect Chen, 2009, Crooks, 2009, and Zimanyi and Silbey, 2009.
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8 _ H functional of the protocol . The time derivative Qt next embark on the study of work fluctuation relations in is determined by the Heisenberg equation. In the case quantum systems. As in the classical case, in the quantum _ H of a time-independent operator Q it becomes Qt ¼ case one also needs to be careful in properly identifying the H H i½H t ð tÞ; Qt =ℏ. exclusive and inclusive work, and must be aware of the gauge Bochkov and Kuzovlev (1977) were not able to provide freedom issue. In the following we adopt, except when any quantum analog of their fluctuation relations, otherwise explicitly stated, the inclusive viewpoint. The two Eqs. (14) and (15), with the classical work W0 replaced by fundamental ingredients for the development of the theory the operator W 0. are, in the quantum case as in the classical case, the canonical Yukawa (2000) and Allahverdyan and Nieuwenhuizen form of equilibrium and microreversibility. (2005) arrived at a similar conclusion when attempting to W H H define an inclusive work operator by ¼ ð Þ A. Microreversibility of nonautonomous quantum systems H ð 0Þ. According to this definition the exponentiated H W W ½H ð Þ H ð 0Þ work he i¼Tr%0e ¼he i agrees The principle of microreversibility is introduced and dis- F with e only if H ð tÞ commutes at different times cussed in quantum mechanics textbooks for autonomous ½H ð tÞ; H ð Þ ¼ 0 for any t, . This could lead to the (i.e., nondriven) quantum systems (Messiah, 1962). As in premature conclusion that there exists no direct quantum the classical case, however, this principle continues to hold analog of the Bochkov-Kuzovlev and the Jarzynski identities, in a more general sense also for nonautonomous quantum Eqs. (15) and (23)(Allahverdyan and Nieuwenhuizen, 2005). systems. In this case it can be expressed as Based on the works by Kurchan (2000) and Tasaki (2000), y ~ U ½ ¼ U 0½ ; (40) Talkner et al. (2007) demonstrated that this conclusion is t; t; based on an erroneous assumption. They pointed out that where is the quantum mechanical time-reversal operator work characterizes a process, rather than a state of the system; (Messiah, 1962).10 Note that the presence of the protocol ~ this is also an obvious observation from thermodynamics: and its time-reversed image distinguishes this generalized unlike internal energy, work is not a state function (its version from the standard form of microreversibility for differential is not exact). Consequently, work cannot be autonomous systems. The principle of microreversibility, represented by a Hermitian operator whose eigenvalues can Eq. (40), holds under the assumption that at any time t the be determined in a single projective measurement. In con- Hamiltonian is invariant under time reversal,11 that is, trast, the energy H ð Þ (or H 0, when the exclusive view- t H ð tÞ ¼ H ð tÞ: (41) point is adopted) must be measured twice, first at the initial time t ¼ 0 and again at the final time t ¼ . A derivation of Eq. (40) is presented in Appendix B. See also The difference of the outcomes of these two measurements Andrieux and Gaspard (2008) for an alternative derivation. then yields the work performed on the system in a particular In order to better understand the physics behind Eq. (40) ½ ¼ y ½ ~ ½ realization (Talkner et al., 2007). That is, if at time t ¼ 0 the we rewrite it as Ut;0 U t;0 U ;0 , where we 0 H used the concatenation rule Ut; ½ ¼Ut;0½ U0; ½ , and eigenvalue En of ð 0Þ and later, at t ¼ , the eigenvalue 1 the inverse U0 ½ ¼U ½ of the propagator U ½ . E of H ð Þ were obtained,9 the measured (inclusive) work ; ;0 t;s m jii becomes Applying it to a pure state , and multiplying by from the left, we obtain 0 w ¼ Em En : (39) ~ jc ti¼U t;0½ jfi; (42) Equation (39) represents the quantum version of the clas- where we introduced the notations jc i¼U 0½ jii and sical inclusive work, Eq. (19). In contrast to the classical case, t t; jfi¼U 0½ jii. Equation (42) says that, under the evolution this energy difference, which yields the work performed in a ; generated by the reversed protocol ~ the time-reversed final single realization of the protocol, cannot be expressed in the state jfi evolves between time 0 and t,to jc i. This form of an integrated power, as in Eq. (20). t is shown in Fig. 3. As for the classical case, in order to trace a The quantum version of the exclusive work, Eq. (13), is nonautonomous system back to its initial state, one needs not w0 ¼ e e (Campisi et al., 2011a), where now e are the m n l only to invert the momenta (applying ), but also to invert the eigenvalues of H 0. As demonstrated in the next section, with temporal sequence of Hamiltonian values. these definitions of work straightforward quantum analogs of the Bochkov-Kuzovlev results, Eqs. (14), (15), and (18) and of their inclusive viewpoint counterparts, Eqs. (21), (23), and B. The work probability density function (28) can be derived. We consider a system described by the Hamiltonian H ð tÞ initially prepared in the canonical state IV. QUANTUM WORK FLUCTUATION RELATIONS 10 Armed with all the proper mathematical definitions of Under the action of coordinates transform evenly, whereas nonequilibrium quantum mechanical work, Eq. (39), we linear and angular momenta, as well as spins change sign. In the coordinate representation, in absence of spin degrees of freedom, the operator is the complex conjugation c ¼ c . 8 11 Because of causality Ut;0½ may, of course, depend only on the In the presence of external magnetic fields the direction of these part of the protocol including times from 0 up to t. fields has also to be inverted in the same way as in the autonomous 9For a formal definition of these eigenvalues see Eq. (43). case.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... 779
At time a second measurement of H ð Þ yielding the eigenvalue Em with probability
pmjn½ ¼Tr m %nð Þ (47) is performed. The PDF to observe the work w is thus given by X 0 0 p½w; ¼ ðw ½Em En Þpmjn½ pn: (48) m;n The work PDF has been calculated explicitly for a forced harmonic oscillator (Talkner, Burada, and Ha¨nggi, 2008, 2009) and for a parametric oscillator with varying frequency (Deffner and Lutz, 2008; Deffner et al., 2010).
C. The characteristic function of work
The characteristic function of work G½u; is defined as in FIG. 3 (color online). Microreversibility for nonautonomous the classical case as the Fourier transform of the work quantum systems. The normalized initial condition jii evolves, probability density function H under the unitary time evolution generated by ð tÞ, from time Z 0 c t ¼ until t to j ti¼Ut;0½ jii and until time t ¼ to jfi. G½u; ¼ dweiuwp½w; : (49) The time-reversed final condition jfi evolves, under the unitary evolution generated by ~ from time t ¼ 0 until t to ~ Similar to p½w; , G½u; contains full information regard- U t;0½ jfi¼ jc ti, Eq. (42), and until time t ¼ to the time-reversed initial condition jii. The motion occurs on the ing the statistics of the random variable w. Talkner et al. hypersphere of unitary radius in the Hilbert space. (2007) showed that the work PDF has the form of a two-time nonstationary quantum correlation function, i.e.,
H H H ½ ; ¼h iu ð Þ iu ð 0Þi H ð 0Þ G u e e %ð 0Þ¼e =Zð 0Þ. The instantaneous eigenvalues of H H H t Tr iu ð Þ ðiuþ Þ ð 0Þ Z H ð tÞ are denoted by En , and the corresponding instanta- ¼ e e = ð 0Þ; (50) t neous eigenstates by jc n; i: where the average symbol stands for quantum expectation t t t H ð tÞjc n; i¼En jc n; i: (43) over the initial state density matrix %ð 0Þ,Eq.(36), i.e., hBi¼Tr%ð 0ÞB, and the superscript H denotes the The symbol n labels the quantum number specifying the H y Heisenberg picture, i.e., H ð Þ¼U 0½ H ð ÞU 0½ . energy eigenvalues and denotes all further quantum num- ; ; Equation (50) was derived first by Talkner et al. (2007) in bers, necessary to specify an energy eigenstate in case of the case of nondegenerate H ð Þ and later generalized by g -fold degeneracy. We emphasize that the instantaneous t n Talkner, Ha¨nggi, and Morillo (2008) to the case of possibly eigenvalue Eq. (43) must not be confused with the degenerate H ð Þ.12 Schro¨dinger equation iℏ@ jc ðtÞi ¼ H ð Þjc ðtÞi. The instan- t t t The product of the two exponential operators taneous eigenfunctions resulting from Eq. (43), in particular, H H H eiu ð Þe iu ð 0Þ can be combined into a single exponent are not solutions of the Schro¨dinger equation. T 0 H under the protection of the time ordering operator to yield At t ¼ the first measurement of ð 0Þ is performed, H H iuH ð Þ iuH ð 0Þ iu½H ð Þ H ð 0Þ 0 e e ¼ T e . In this way one with outcome E . This occurs with probability n can convert the characteristic function of work to a form 0 E 0 that is analogous to the corresponding classical expression, p ¼ g e n =Zð 0Þ: (44) n n Eq. (27), According to the postulates of quantum mechanics, immedi- H H H H 0 ; TrT iu ð Þ ð 0Þ ð 0Þ Z ately after the measurement of the energy En the system is G½u ¼ e e = 0 Z found in the state @H Hð Þ _ t t H ð 0Þ ¼ TrT exp iu dt t e =Z0: 0 0 0 0 @ t %n ¼ n %ð 0Þ n =pn; (45) P (51) 0 c 0 c 0 where n ¼ j n; ih n; j is the projector onto the eigen- space spanned by the eigenvectors belonging to the eigen- 12The derivation in Talkner, Ha¨nggi, and Morillo (2008) is more value E 0 . The system is assumed to be thermally isolated n general in that it does not assume any special form of the initial t 0 at any time , so that its evolution is determined by state, thus allowing the study, e.g., of microcanonical fluctuation the unitary operator Ut;0½ , Eq. (38); hence it evolves ac- relations. The formal expression, Eq. (50), remains valid for any cording to initial state %,P with the provision that the average is taken with 0 0 y respect to % ¼ n n % n representing the diagonal part of in % ðtÞ¼U 0½ % U ½ : (46) n t; n t;0 the eigenbasis of H ð 0Þ.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 780 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ...
The second equality follows from the fact that the total time Heisenberg representation, Eq. (37), ut is a real function, derivative of the Hamiltonian in the Heisenberg picture and u~t ¼ u t. This can be proved by using the quantum coincides with its partial derivative. microreversibility principle, Eq. (40), in a similar way as in As a consequence of quantum microreversibility, Eq. (40), the classical derivation, Eq. (A1). the characteristic function of work obeys the following im- The derivation of Eq. (55) was provided by Andrieux and portant symmetry relation (see Appendix C): Gaspard (2008), who also recovered the formula of Kubo, Eq. (4), and the Onsager-Casimir reciprocity relations.13 Z ; Z ; ~ ð 0ÞG½u ¼ ð ÞG½ u þ i : (52) These relations are obtained by means of functional de- rivatives of Eq. (55) with respect to the force fields and test By applying the inverse Fourier transform and using Zð tÞ¼ t H ð Þ ð Þ fields u at ¼ u ¼ 0 (Andrieux and Gaspard, 2008). Tre t ¼ e F t one ends up with the quantum version t t t of the Crooks fluctuation theorem in Eq. (28): Relations and symmetries for higher order response functions follow in an analogous way as in the classical case, Eq. (14), ; p½w ðw FÞ by means of higher order functional derivatives with respect ~ ¼ e : (53) p½ w; to the force field t. Such relations were investigated experi- mentally by Nakamura et al. (2010, 2011); see Sec. V, This result was first accomplished by Kurchan (2000) and Eq. (89). Tasaki (2000). Later Talkner and Ha¨nggi (2007) gave a Within the exclusive viewpoint approach, the counterparts systematic derivation based on the characteristic function of of Eqs. (50) and (52)–(55) are obtained by replacing H H work. The quantum Jarzynski equality, H H 0 with H 0 , H ð 0Þ with H 0, Zð tÞ with Z0 ¼ Tre , and w F he i ¼ e ; (54) setting accordingly F to 0 (Campisi et al., 2011a). follows by multiplying both sides by p½ w; e w and integrating over w. Given the fact that the characteristic E. Microreversibility, conditional probabilities, and entropy function is determined by a two-time quantum correlation H function rather than by a single time expectation value is For a Hamiltonian ð tÞ with nondegenerate instanta- another clear indication that work is not an observable but neous spectrum for all times t and instantaneous eigenvectors c t instead characterizes a process. j n i, the conditional probability pmjn½ , Eq. (47), is given c c 0 2 As discussed by Campisi et al. (2010a) the Tasaki-Crooks by the simple expression pmjn½ ¼jh m jU ;0½ j n ij .As relation, Eq. (53), and the quantum version of the Jarzynski a consequence of the assumed invariance of the Hamiltonian, t equality, Eq. (54), continue to hold even if further projective Eq. (41), the eigenstates jc n i are invariant under the action t measurements of any observable A are performed within of the time-reversal operator up to a phase jc n i¼ t i’n t the protocol duration (0, ). These measurements, however, e jc n i. Then, expressing microreversibility as U ;0½ ¼ y ~ do alter the work PDF (Campisi et al., 2011b). U0; ½ , see Eq. (40), one obtains the following sym- metry relation for the conditional probabilities: D. Quantum generating functional ~ pmjn½ ¼pnjm½ : (56) The Jarzynski equality can also immediately be obtained from the characteristic function by setting u ¼ i , in Eq. (50) Note the exchanged position of m and n in the two sides of (Talkner et al., 2007). In order to obtain this result it is this equation. From Eq. (56) the Tasaki-Crooks fluctuation important that the Hamiltonian operators at initial and final theorem is readily obtained for a canonical initial state, using times enter into the characteristic function, Eq. (50), as argu- Eq. (48). ments of two factorizing exponential functions, i.e., in the Using instead an initial microcanonical state at energy E, H H H 14 form e ð Þe ð 0Þ. In general, this, of course, is differ- described by the density matrix H H H ent from a single exponential e ½ ð Þ ð 0Þ . In the defi- 0ðEÞ¼ ðE H ð 0ÞÞ=!ðE; 0Þ; (57) nitions of generating functionals, Bochkov and Kuzovlev (1977, 1981a) and Stratonovich (1994) employed yet differ- where !ðE; tÞ¼Tr ðE H ð tÞÞ, we obtain (Talkner, ent ordering prescriptions which do not lead to the Jarzynski Ha¨nggi, and Morillo, 2008) equality. In order to maintain the structure of the classical generating functional, Eq. (21), for quantum systems the p½E; W; ¼ e½SðEþW; Þ SðE; 0Þ =kB ; (58) classical exponentiated work e W also has to be replaced p½E þ W; W; ~ by the product of exponentials as it appears in the character- istic function of work, Eq. (50). This then leads to a desired generating functional relation 13Andrieux and Gaspard (2008) also allowed for a possible Z dependence of the Hamiltonian on a magnetic field H ¼ H H H ð tÞ H ð 0Þ H ð ; BÞ. Then it is meant that the dynamical evolution of B on exp dsutBt e e t 0 H ~ B the right-hand side is governed by the Hamiltonian ð t; Þ, i.e., Z besides inverting the protocol, the magnetic field needs to be exp ~ BH F ¼ dtut"B t e ; (55) inverted as well. 0 ~ 14Strictly speaking, in order to obtain well-defined expressions the where B is an observable with definite parity "B (i.e., function has to be understood as a sharply peaked function with y H B ¼ "BB), Bt denotes the observable B in the infinite support.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... 781 where SðE; tÞ¼kB ln!ðE; tÞ denotes Boltzmann’s thermo- a superbath at inverse temperature ; see Fig. 4. It is then dynamic equilibrium entropy. The corresponding classical assumed either that the contact to the superbath is removed derivation was provided by Cleuren et al. (2006). A classical for t 0 or that the superbath is so weakly coupled to the microcanonical version of the Jarzynski equality was put compound system that it bears no influence on its dynamics forward by Adib (2005) for non-Hamiltonian isoenergetic over the time span 0 to . dynamics. It was recently generalized to energy controlled Because the system and the environmental Hamiltonians systems by Katsuda and Ohzeki (2011). commute with each other, their energies can be simulta- neously measured. We denote the eigenvalues of H Sð tÞ as E t , and those of H as EB. In analogy with the isolated case F. Weak-coupling case i B we assume that at time t ¼ 0 a joint measurement of H Sð 0Þ H 0 B In Secs. IV.B–IV.D we studied a quantum mechanical and B is performed, with outcomes En and E . A second H H system at canonical equilibrium at time t ¼ 0. During the joint measurement of Sð Þ and B at t ¼ yields the B subsequent action of the protocol it is assumed to be com- outcomes Em and E . pletely isolated from its surrounding apart from the influence In analogy to the energy change of an isolated system, of the external work source and hence to undergo a unitary the differences of the eigenvalues of system and bath time evolution. The quality of this approximation depends on Hamiltonians yield the energy changes of system and bath, the relative strength of the interaction between the system and E and EB, respectively, in a single realization of the its environment, compared to typical energies of the isolated protocol, i.e., system as well as on the duration of the protocol. In general, 0 though, a treatment that takes into account possible environ- E ¼ Em En ; (61) mental interactions is necessary. As will be shown, the inter- B B B action with a thermal bath does not lead to a modification of E ¼ E E : (62) the Jarzynski equality, Eq. (54), nor of the quantum work fluctuation relation, Eq. (53), in both the cases of weak and In the weak-coupling limit, the change of the energy strong coupling (Campisi et al., 2009a; Talkner, Campisi, content of the total system is given by the sum of the energy and Ha¨nggi, 2009), a main finding which holds true as well changes of the system and bath energies apart from a negli- for classical systems (Jarzynski, 2004). In this section we gibly small contribution due to the interaction Hamiltonian H address the weak-coupling case, while the more intricate case SB. The work w performed on the system coincides with of strong coupling is discussed in the next section. the change of the total energy because the force is assumed to We consider a driven system described by the time- act only directly on the system. For the same reason, the dependent Hamiltonian H Sð tÞ, in contact with a thermal change of the bath energy is due solely to an energy exchange bath with time-independent Hamiltonian H ; see Fig. 4. The with the system and hence can be interpreted as negative heat B 15 Hamiltonian of the compound system is EB ¼ Q. Accordingly we have E ¼ w þ Q: (63) H ð tÞ¼H Sð tÞþH B þ H SB; (59) Following the analogy with the isolated case, we consid- where the energy contribution stemming from H SB is as- sumed to be much smaller than the energies of the system and ered the joint probability distribution function p½ E; Q; that the system energy changes by E and the heat Q is bath resulting from H Sð tÞ and H B. The parameter that is manipulated according to a protocol solely enters in the exchanged, under the protocol : system Hamiltonian H ð Þ. X S t p½ E; Q; ¼ ð E E þ E 0 Þ The compound system is assumed to be initially (t ¼ 0)in m n m;n; ; the canonical state B B 0 ðQ þ E E Þpm jn ½ pn ; (64) H ð 0Þ %ð 0Þ¼e =Yð 0Þ; (60) where pm jn ½ is the conditional probability to obtain the H ð Þ where Yð Þ¼Tre t is the corresponding partition B 0 B t outcome Em , E at , provided that the outcome En , E was function. This initial state may be provided by contact with 0 obtained at time t ¼ 0, whereas pn is the probability to find 0 B the outcome En , E in the first measurement. The condi- tional probability pm jn ½ can be expressed in terms of the projectors on the common eigenstates of H Sð tÞ, H B, and the unitary evolution generated by the total Hamiltonian H ð tÞ (Talkner, Campisi, and Ha¨nggi, 2009).
15 By use of the probabilityR distribution in Eq. (64), the averaged FIG. 4 (color online). Driven open system. A driven system quantity h Ei ¼ dð EÞdQp½ E; Q; E ¼ Tr% H Sð Þ [represented by its Hamiltonian H Sð tÞ] is coupled to a bath Tr%ð 0ÞH Sð 0Þ cannot, in general, be interpreted as a change in (represented by H B) via the interaction Hamiltonian H SB. The thermodynamic internal energy. The reason is that the final state, % , compound system is in vanishingly weak contact with a superbath reached at the end of the protocol is typically not a state of that provides the initial canonical state at inverse temperature , thermodynamic equilibrium; hence its thermodynamic internal en- Eq. (60). ergy is not defined.
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By taking the Fourier transform of p½ E; Q; with re- reduced system dynamics S.16 With the above derivation spect to both E and Q, one obtains the characteristic we followed Talkner, Campisi, and Ha¨nggi (2009) in which function of system energy change and heat exchange, reading one does not rely on a Markovian quantum evolution and Z consequently the results hold true as well for a general G½u; v; ¼ dð EÞdQeiðu EþvQÞp½ E; Q; ; (65) non-Markovian reduced quantum dynamics of the system S. which can be further simplified and cast, as in the isolated G. Strong-coupling case case, in the form of a two-time quantum correlation function (Talkner et al., 2009): In the case of strong coupling, the system-bath interaction energy is non-negligible, and therefore it is no longer possible H H i½uH ð Þ vH i½uH ð 0Þ vH G½u;v; ¼he S B e S B i; (66) to identify the heat as the energy change of the bath. How to define heat in a strongly coupled driven system and whether it where the average is over the state %ð 0Þ that is the diagonal is possible to define it at all currently present open problems. part of %ð 0Þ,Eq.(60), with respect to fH Sð 0Þ; H Bg. This, however, does not hinder the possibility to prove that Notably, in the limit of weak coupling this state %ð 0Þ the work fluctuation relation, Eq. (72), remains valid also in approximately factorizes into the product of the equilibrium the case of strong coupling. For this purpose it suffices to states of system and bath with the deviations being of second properly identify the work w done on and the free energy FS order in the system-bath interaction (Talkner, Campisi, and of an open system, without entering the issue of what heat Ha¨nggi, 2009). means in a strong-coupling situation. As for the classical Using Eq. (66) in combination with microreversibility, case (see Sec. III.A), the system Hamiltonian H ð Þ is the Eq. (40), leads, in analogy with Eq. (52), to S t only time-dependent part of the total Hamiltonian H ð tÞ. ~ Therefore, the work done on the open quantum system co- ZSð 0ÞG½u; v; ¼ZSð ÞG½ u þ i ; v i ; ; incides with the work done on the total system, as in the (67) weak-coupling case treated in Sec. IV.F. Consequently, the where work done on an open quantum system in a single realization is H Sð tÞ ZSð tÞ¼TrSe ; (68) 0 w ¼ Em En ; (73) with TrS denoting the trace over the system Hilbert space. Upon applying an inverse Fourier transform of Eq. (67) one E t arrives at the following relation: where l are the eigenvalues of the total Hamiltonian H ð tÞ. ; p½ E; Q Regarding the proper identification of the free energy of an ¼ e ð E Q FSÞ; (69) p½ E; Q; ~ open quantum system, the situation is more involved because 0 H the bare partition function Z ð Þ¼Tr e Sð tÞ cannot where S t S take into account the full effect of the environment in any 1 FS ¼ ln½ZSð Þ=ZSð 0Þ (70) case other than the limiting situation of weak coupling. For strong coupling the equilibrium statistical mechanical de- denotes the system free energy difference. Equation (69) scription has to be based on a partition function of the open generalizes the Tasaki-Crooks fluctuation theorem, Eq. (53), quantum system that is given as the ratio of the partition to the case where the system can exchange heat with a functions of the total system and the isolated environment,17 thermal bath. i.e., Performing the change of the variable E ! w, Eq. (63), in Eq. (69) leads to the following fluctuation relation for the ZSð tÞ¼Yð tÞ=ZB; (74) joint probability density function of work and heat: H B H ð tÞ where ZB ¼ TrBe and Yð tÞ¼Tre with TrB, p½w; Q; ¼ e ðw FSÞ: (71) and Tr denoting the traces over the bath Hilbert space and the p½ w; Q; ~ total Hilbert space, respectively. It must be stressed that, in general, the partition function ZSð tÞ of an open quantum Notably, the right-hand side does not depend on the heat Q system differs from its partition function in absence of a bath: but depends on the work w only. This fact implies that the marginal probability density of work p½w; ¼ H R Z ð Þ Tr e Sð tÞ: (75) dQp½w; Q; obeys the Tasaki-Crooks relation: S t S
p½w; 16 ¼ e ðw FSÞ: (72) See Mukamel (2003), de Roeck and Maes (2004), Esposito and p½ w; ~ Mukamel (2006), and Crooks (2008). 17See Feynman and Vernon (1963), Caldeira and Leggett w F Subsequently the Jarzynski equality he i¼e S is also (1983), Grabert et al. (1984), Ford et al. (1985), Grabert et al. satisfied. Thus, the fluctuation relation, Eq. (53), and the (1988), Dittrich et al. (1998), Ingold (2002), Nieuwenhuizen and Jarzynski equality, Eq. (54), keep holding, unaltered, also in Allahverdyan (2002), Ha¨nggi and Ingold (2005, 2006), Ho¨rhammer the case of weak coupling. This result was originally found and Bu¨ttner (2008), Campisi et al. (2009a, 2009b), and Campisi, upon assuming a Markovian quantum dynamics for the Zueco, and Talkner (2010).
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The equality is restored, though, in the limit of a weak coupling. The free energy of an open quantum system follows ac- cording to the standard rule of equilibrium statistical mechan- ics as 1 F ð Þ¼Fð Þ F ¼ lnZ ð Þ: (76) S t t B S t
In this way the influences of the bath on the thermodynamic properties of the system are properly taken into account. Besides, Eq. (76) complies with all the grand laws of ther- modynamics (Campisi et al., 2009a). For a total system initially prepared in the Gibbs state, FIG. 5 (color online). Exchange fluctuation relation setup. Several Eq. (60), the Tasaki-Crooks fluctuation theorem, Eq. (53), reservoirs (large semicircles) interact via the coupling V t (sym- applies yielding bolized by the small circle), which is switched on at time t ¼ 0 and switched off at t ¼ . During the on period (0, ) the reservoirs p½w; Yð Þ ¼ e w: (77) exchange energy and matter with each other. The resulting net ; ~ p½ w Yð 0Þ energy change of the ith reservoir is Ei and its particle content changes by Ni. Initially the reservoirs prescribed temperatures Z 1 Since B does not depend on time, the salient relation Ti ¼ðkB iÞ , and chemical potentials i of the particle species that are exchanged, respectively. Yð Þ=Yð 0Þ¼ZSð Þ=ZSð 0Þ (78) Xs holds, leading to H ðV tÞ¼ H i þ V t; (80) i¼1 p½w; ZSð Þ w ðw FSÞ H V ~ ¼ e ¼ e ; (79) where i is the Hamiltonian of the ith system, and t p½ w; Z ð 0Þ S describes the interaction between the subsystems, which sets in at time t ¼ 0 and ends at time t ¼ . Consequently, where F ¼ F ð Þ F ð 0Þ is the proper free energy S S S V 0 0 V V 0 difference of an open quantum system. Because w coincides t ¼ for t=2ð ; Þ, and, in particular, 0 ¼ ¼ . with the work performed on the open system, both the Tasaki- As before, it is important to distinguish between the values V V Crooks relation, Eq. (72), and the Jarzynski equality, Eq. (54), t at a specific time and the whole protocol . are recovered also in the case of strong coupling (Campisi Initially, the subsystems are supposed to be isolated from each other and to stay in a factorized grand-canonical state et al., 2009a). Y Y i½H i iN i %0 ¼ %i ¼ e = i; (81) i i V. QUANTUM EXCHANGE FLUCTUATION RELATIONS iðH i iN iÞ with i, i, and i ¼ Trie the chemical poten- The transport of energy and matter between two reservoirs tial, inverse temperature, and grand potential, respectively, of N Tr that stay at different temperatures and chemical potentials subsystem i. Here i and i denote the particle number represents an important experimental setup (see also operator and the trace of the ith subsystem, respectively. Sec. VI), as well as a central problem of nonequilibrium We also assume that in the absence of interaction the thermodynamics (de Groot and Mazur, 1984). Here the particle numbers in each subsystem are conserved, i.e., H N 0 two-measurement scheme described above in conjunction ½ i; i ¼ . Since operators acting on Hilbert spaces of H N 0 with the principle of microreversibility leads to fluctuation different subsystems commute, we find ½ i; j ¼ , relations similar to the Tasaki-Crooks relation, Eq. (53), for ½N i; N j ¼0, and ½H i; H j ¼0 for any i, j. the probabilities of energy and matter exchanges. The result- Accordingly, one may measure all the H i’s and all the ing fluctuation relations have been referred to as ‘‘exchange N i’s simultaneously. Adopting the two-measurement fluctuation theorems’’ (Jarzynski and Wo´jcik, 2004), to dis- scheme discussed above in the context of the work fluctuation tinguish them from the ‘‘work fluctuation theorems.’’ relation, we make a first measurement of all the H i’s and all The first quantum exchange fluctuation theorem was put the N i’s at t ¼ 0. Accordingly, the wave function collapses forward by Jarzynski and Wo´jcik (2004). It applies to two onto a common eigenstate jc ni of all these observables with i i systems initially at different temperatures that are allowed to eigenvalues En and Nn. Subsequently, this wave function interact over the lapse of time (0, ), via a possibly time- evolves according to the evolution Ut;0½V generated by dependent interaction. This situation was later generalized by the total Hamiltonian, until time when a second measure- Saito and Utsumi (2008) and Andrieux et al. (2009) , to allow ment of all H i’s and N i’s is performed leading to a wave for the exchange of energy and particles between several function collapse onto an eigenstate jc mi, with eigenvalues i i interacting systems initially at different temperatures and Em and Nm. As in the case studied in Sec. IV.F, the joint chemical potentials; see Fig. 5. probability density of energy and particle exchanges The total Hamiltonian H ðV tÞ consisting of s subsystems p½ E; N; V completely describes the effect of the is interaction protocol V :
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 784 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... X Y E N; V i i E N; V Y p½ ; ¼ ð Ei Em þ EnÞ p½ ; c ! ið Ei i NiÞ m;n i ~ e : (86) p½ E; N; V i i i 0 ð Ni Nm þ NnÞpmjn½V pn; For those large times t c a nonequilibrium steady state (82) sets in under the condition that the reservoirs are chosen macroscopic. For this reason Eq. (86) is referred to as a where pmjn½V is the transition probability from state jc ni to 0 i i steady state fluctuation relation. This is in contrast to the c i½En iNn j mi and pn ¼ ie = i is the initial distribution other fluctuation relations discussed above, which instead are E N of energies and particles. Here the symbols and are valid for any observation time and are accordingly referred shorthand notations for the individual energy and particle to as transient fluctuation relations. Saito and Dhar (2007) ... number changes of all subsystems E1; E2; ; Es and provided an explicit demonstration of Eq. (86) for the quan- ... N1; N2; ; Ns, respectively. tum heat transfer across a harmonic chain connecting two Assuming that the total Hamiltonian commutes with the thermal reservoirs at different temperatures. Ren et al. (2010) time-reversal operator at any instant of time and using the reported on the breakdown of Eq. (86) induced by a non- time-reversal property of the transition probabilities, Eq. (56), vanishing Berry-phase heat pumping. The latter occurs when one obtains the temperatures of the two baths are adiabatically modulated E N; V Y in time. p½ ; ½ E N ~ ¼ e i i i i : (83) p½ E; N; V i VI. EXPERIMENTS This equation was derived by Andrieux et al. (2009) and expresses the exchange fluctuation relation for the case of A. Work fluctuation relations transport of energy and matter. In the case of a single isolated system (s ¼ 1), it reduces to Regarding the experimental validation of the work fluctua- the Tasaki-Crooks work fluctuation theorem, Eq. (53), upon tion relation, a fundamental difference exists between the classical and the quantum regime. In classical experiments rewriting E1 ¼ w and assuming that the total number of x particles is conserved also when the interaction is switched work is accessible by measuring the trajectory t of the possibly open system and integrating the instantaneous power on, i.e., ½H ðV tÞ; N ¼0, to obtain N ¼ 0. The free R energy difference does not appear in Eq. (83) because we according to W ¼ dt@HS=@t, Eq. (31). In clear contrast, in have assumed the protocol V to be cyclic. quantum mechanics the work is obtained as the difference of In the case of two weakly interacting systems (s ¼ 2, V two measurements of the energy, and an ‘‘integrated power’’ small) that do not exchange particles, it reduces to the expression does not exist for the work; see Sec. III.C. fluctuation theorem of Jarzynski and Wo´jcik (2004) for heat Closely following the prescriptions of the theory one exchange: should perform the following steps in order to experimentally verify the work fluctuation relation, Eq. (53): (i) Prepare a p½Q; V quantum system in the canonical state, Eq. (2), at time t ¼ 0. ð 1 2ÞQ ~ ¼ e ; (84) (ii) Measure the energy at t ¼ 0. (iii) Drive the system by p½ Q; V means of some forcing protocol t for times t between 0 and , and make sure that during this time the system is well where Q ¼ E1 ¼ E2, with the second equality follow- insulated from its environment. (iv) Measure the energy again ing from the assumed weak interaction. at and record the work w, according to Eq. (39). (v) Repeat In case of two weakly interacting systems (s ¼ 2, V this procedure many times and construct the histogram of the small) that do exchange particles and are initially at the statistics of work as an estimate of the work PDF p½w; .In same temperature, yielding Q ’ 0, the fluctuation relation order to determine the backward probability the same type of takes on the form experiment has to be repeated with the inverted protocol, p½q; V starting from an equilibrium state at inverse temperature ð 1 2Þq ~ ¼ e ; (85) and at those parameter values that are reached at the end of p½ q; V the forward protocol. where q N1 ¼ N2. 1. Proposal for an experiment employing trapped cold ions One basic assumption leading to the exchange fluctuation relation, Eq. (83), is that the initial state is a factorized state, Huber et al. (2008) suggested an experiment that exactly in which the various subsystems are uncorrelated from each follows the procedure described above. They proposed to other. In most experimental situations, however, unavoidable implement a quantum harmonic oscillator by optically trap- interactions between the systems would lead to some ping an ion in the quadratic potential generated by a laser correlations and a consequent deviation from the assumed trap, using the setup developed by Schulz et al. (2008).In factorized state, Eq. (81). The resulting deviation from the principle, the setup of Schulz et al. (2008) allows, on the one exchange fluctuation relation, Eq. (83), is expected to vanish hand, to drive the system by changing in time the stiffness of for observation times larger than some characteristic time the trap, and, on the other hand, to probe whether the ion is in scale c, determined by the specific physical properties of the a certain Fock state jni, i.e., in an energy eigenstate of the experimental setup (Andrieux et al., 2009; Esposito et al., harmonic oscillator. The measurement apparatus may be 2009; Campisi et al., 2010a): understood as a single Fock state ‘‘filter,’’ whose outcome
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... 785 is ‘‘yes’’ or ‘‘no,’’ depending on whether the ion is or is not in and readout as before, the matrix pmjn½ can be determined the probed state. Thus the experimentalist probes each pos- experimentally. Accordingly one can test the validity of the ~ sible outcome (n, m), where (n, m) denotes the Fock states at symmetry relation pmjn½ ¼pnjm½ ,Eq.(56), which in turn time t ¼ 0 and t ¼ , respectively. Then the relative fre- implies the work fluctuation relation; see Sec. IV.E. At vari- quency of the outcome (n, m) occurrence is recorded by ance with the proposal of Huber et al. (2008), in this case the repeating the driving protocol many times always preparing initial state would not be randomly sampled from a canonical the system in the same canonical initial state. In this way the state, but would be rather created deterministically by the 0 joint probabilities pmjn½ pn are measured. experimenter. The work histogram is then constructed as an estimate of The theoretical values of transition probabilities for this the work PDF, Eq. (48), thus providing experimental access to case corresponding to a displacement of the oscillator were the fluctuation relation, Eq. (53). Likewise the relative fre- first reported by Husimi (1953); see also Campisi (2008). quency of the outcomes having n as the initial state gives the Talkner, Burada, and Ha¨nggi (2008) provided an analytical 0 experimental initial population pn. Thus, with this experi- expression for the characteristic function of work and inves- ment one can actually check the symmetry relation of the tigated in detail the work probability distribution function and ~ conditional probabilities pmjn½ ¼pnjm½ , Eq. (56), and its dependence on the initial state, such as, for example, compare their experimental values with the known theoretical canonical, microcanonical, and coherent states. values (Husimi, 1953; Deffner and Lutz, 2008; Talkner, So far we addressed possible experimental tests of the Burada, and Ha¨nggi, 2008, 2009). Tasaki-Crooks work fluctuation theorem, Eq. (53), for iso- Another suitable quantum system to test quantum fluctua- lated systems. The case of open systems, interacting with a tion relations are quantum versions of nanomechanical oscil- thermal bath, poses extra difficulties related to the fact that in lator setups that with present day nanotechnology are at the order to measure the work in this case one should make two verge of entering the quantum regime.18 In these systems measurements of the energy of the total macroscopic system, work protocols can be imposed by optomechanical means. made up of the system of interest and its environment. This presents an extra obstacle that at the moment seems difficult 2. Proposal for an experiment employing circuit quantum to surmount except for a situation at (i) weak coupling and 0 electrodynamics (ii) Q , then yielding, together with Eq. (63) w E. Currently, the experiment proposed by Huber et al. (2008) has not yet been carried out. An analogous experiment could, B. Exchange fluctuation relations in principle, be performed in a circuit quantum electrody- Similar to the quantum work fluctuation relations, the namics (QED) setup as the one described by Hofheinz et al. quantum exchange fluctuation relations are understood in (2008, 2009). The setup consists of a Cooper pair box qubit terms of two-point quantum measurements. In an experimen- (a two state quantum system) that can be coupled to and tal test, the net amount of energy and/or number of particles decoupled from a superconducting 1D transmission line, [depending on which of the three exchange fluctuation rela- where the latter mimics a quantum harmonic oscillator. tions, Eqs. (83)–(85), is studied] has to be measured in each With this architecture it is possible to implement various subsystem twice, at the beginning and at the end of the functions with a very high degree of accuracy. Among them protocol. However, typically these are macroscopic reser- the following tasks are of special interest in the present voirs, whose energy and particle number measurement are context: (i) creation of pure Fock states jni, i.e., the energy 19 practically impossible. Thus, seemingly, the verification of eigenstates of the quantum harmonic oscillator in the reso- the exchange fluctuation relations would be even more prob- p nator; (ii) measurement of photon statistics m, i.e., measure- lematic than the validation of the quantum work fluctuation j i ments of the population of each quantum state m of the relations. Indeed, while experimental tests of the work fluc- oscillator; and (iii) driving the resonator by means of an tuation relations have not yet been reported, experiments external field. concerning quantum exchange fluctuation relations have al- Hofheinz et al. (2008) reported, for example, on the ready been performed. In the following we discuss two of j0i creation of the ground Fock state , followed by a driving them, one by Utsumi et al. (2010) and the other by Nakamura protocol (a properly engineered microwave pulse applied to et al. (2010). In doing so we demonstrated how the obstacle the resonator) that ‘‘displaces’’ the oscillator and creates a of energy or particle content measurement of macroscopic coherent state j i, whose photon statistics pmj0½ was mea- reservoirs was circumvented. sured with good accuracy up to nmax 10. In more recent experiments (Hofheinz et al., 2009) the accuracy was im- 1. An electron counting statistics experiment proved and nmax was raised to 15. The quantity pmj0½ is actually the conditional probability to find the state jmi at Utsumi et al. (2010) recently performed an experimental time t ¼ , given that the system was in the state j0i at time verification of the particle exchange fluctuation relation, t ¼ 0. Thus, by preparing the oscillator in the Fock state jni Eq. (85), using bidirectional electron counting statistics instead of the ground state j0i, and repeating the same driving (Fujisawa et al., 2006). The experimental setup consists of
18Note the exciting recent advancements obtained with the works 19See Andrieux et al. (2009), Esposito et al. (2009), and Campisi LaHaye et al. (2004), Kippenberg and Vahala (2008), Anetsberger et al. (2010a), and also Jarzynski (2000) regarding the classical et al. (2009), and O’Connell et al. (2010). case.
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The question, however, remains of how to connect this experiment in which the flux of electrons through an interface is monitored and the theory, leading to Eq. (85), which instead prescribes only two measurements of total particle numbers in the reservoirs. The answer was given by Campisi et al. (2010a), who showed that the exchange fluctuation relation, Eq. (83), remains valid, if in addition to the two FIG. 6 (color online). Scheme of a bidirectional counting statistics measurements of total energy and particle numbers occurring experiment. Two leads (large semicircles) with different electronic at 0 and , the evolution of a quantum system is interrupted by chemical potentials ( 1 2) and same temperature ( 1 ¼ 2) means of projective quantum measurements of any observ- are connected through a double quantum dot (small circles), whose able A that commutes with the quantum time-reversal op- quantum state is continuously monitored. The state (1,0), i.e., one erator . In other words, while the forward and backward electron in the left dot, no electrons in the right dot, is depicted. The probabilities are affected by the occurrence of intermediate transition from this state to the state (0,1) signals the exchange of measurement processes, their ratio remains unaltered. one electron from subsystem 1 to subsystem 2. H 1 2 and N 1 2 ; ; In the experiment by Utsumi et al. (2010) one does not denote the Hamiltonian and electron number operators of the need to measure the initial and final content of particles in the subsystems, respectively. reservoirs because the number of exchanged particles is inferred from the sequence of intermediate measurements outcomes fðl; rÞg . Thus, thanks to the fact that quantum two electron reservoirs (leads) at the same temperature. k measurements do not alter the fluctuation relation, one may The two leads are connected via a double quantum dot; see overcome the problem of measuring the energy and number Fig. 6. of particles of the macroscopic reservoirs, by monitoring When an electric potential difference V ¼ 1 2 is instead the flux through a microscopic junction. applied to the leads, a net flow of electrons starts transporting charges from one lead to the other, via lead-dot and dot-dot quantum tunnelings. The measurement apparatus consists of 2. Nonlinear response relations in a quantum coherent a secondary circuit in which a current flows due to an applied conductor voltage. Thanks to a properly engineered coupling between As discussed in the Introduction, the original motivation the secondary circuit and the double quantum dot, the current for the study of fluctuation relations was to overcome the in the circuit depends on the quantum state of the double dot. limitations of linear response theory and to obtain relations The latter has four relevant states, which we denote as j00i, connecting higher order response functions to fluctuation j01i, j10i, and j11i corresponding, respectively, to no elec- properties of the unperturbed system. As an indirect and trons in the left dot and no electrons in the right dot, no partial confirmation of the fluctuation relations higher electrons in the left dot and one electron in the right dot, etc. order static fluctuation-response relations can be tested Each of these states leads to a different value of the current in experimentally. the secondary circuit. In the experiment an electric potential Such a validation was recently accomplished in coherent difference is applied to the two leads for a time . During this quantum transport experiments by Nakamura et al. (2010, time the state of the double quantum dot is monitored by 2011), where the average current I and the zero-frequency registering the current in the secondary circuit. This current current noise power S generated in an Aharonov-Bohm ring was found to switch between the four values corresponding to were investigated as a function of an applied dc voltage V the four quantum states mentioned above. The outcome of the and magnetic field B. In the nonlinear response regime, the experiment is a sequence Ik of current values, with Ik taking current and noise power may be expressed as power series of only four possible values. In other words, the outcome of the the applied voltage: experiment consists of a sequence fðl; rÞg (with l, r ¼ 0,1)of k B B L R G2ð Þ 2 G3ð Þ 3 joint eigenvalues of two commuting observables , spec- IðV;BÞ¼G1ðBÞV þ V þ V þ ; (87) ifying the occupation of the left (l) and right (r) dots by single 2 3! electrons at the time of the kth measurement. The presence of an exchange of entries within one time step of the form S2ðBÞ 2 SðV;BÞ¼S0ðBÞþS1ðBÞV þ V þ ; (88) ð1; 0Þn, ð0; 1Þnþ1 signals the transfer of one electron from 2 left to right, and vice versa ð0; 1Þ , ð1; 0Þ 1 the transfer n nþ where the coefficients depend on the applied magnetic field from right to left. Thus, given a sequence fðl; rÞg , the total k B. The steady state fluctuation theorem, Eq. (86), then pre- number q½fðl; rÞg of electrons transferred from left to right is k dicts the following fluctuation relations (Saito and Utsumi, obtained by subtracting the total number of right-to-left trans- 2008): fers from the total number of left-to-right transfers. It was S S A A found that, for observation times larger than a characteristic S0 ¼ 4kBTG1;S1 ¼ 2kBTG2;S1 ¼ 6kBTG2 ; ½ ¼ ½ Vq time c, the fluctuation relation p q p q e , Eq. (85), (89) was satisfied with the actual temperature of the leads replaced S B B A B B by an effective temperature; see Fig. 1. The renormalization where Si ¼ Sið ÞþSið Þ, Si ¼ Sið Þ Sið Þ, and S A of temperature was explained as an effect due to an exchange analogous definitions for Gi and Gi . The first equation in of electrons occurring between the dots and the secondary (89) is the Johnson-Nyquist relation (Johnson, 1928; Nyquist, circuit (Utsumi et al., 2010). 1928). In the experiment by Nakamura et al. (2010) good
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... 787 quantitative agreement with the first and third expressions in Less understood are exchange fluctuation relations with Eq. (89) was established, whereas, for the time being, only their important applications to counting statistics (Esposito qualitative agreement was found with the second relation. et al., 2009). The theory there so far is restricted to situations The higher order static fluctuation-dissipation relations where the initial state factorizes into grand-canonical states of (89) were obtained from a steady state fluctuation theorem reservoirs at different temperatures or chemical potentials. for particle exchange under the simplifying assumption that The interaction between these reservoirs is turned on and it is no heat exchange occurs (Nakamura et al., 2010). Then the assumed that it will lead to a steady state within the duration probability of transferring q particles is related to the proba- of the protocol. Experimentally, it is in general difficult to bility of the reverse transfer by pðqÞ¼pð qÞeAq, where exactly follow this prescription and therefore a comparison of A ¼ V ¼ ð 1 2Þ is the so-called affinity. If both sides theory and experiment is only meaningful for the steady state. are multiplied by q and integrated over q a comparison of Alternative derivations of exchange relations for more real- equal powers of applied voltage V yields Eq. (89)(Nakamura istic, nonfactorizing initial states would certainly be of inter- et al., 2011). An alternative approach, that also allows one to est. In this context, the issue of deriving quantum fluctuation include the effect of heat conduction, is offered by the relations for open systems that initially are in nonequilibrium fluctuation theorems for currents in open quantum systems. steady quantum transport states constitutes an interesting This objective has been put forward by Saito and Utsumi challenge. Likewise, from the theoretical point of view little (2008) and also by Andrieux et al. (2009), based on a is known thus far about quantum effects for transport in generating function approach in the spirit of Eq. (55). presence of time-dependent reservoirs, for example, using a varying temperature and/or chemical potentials (Ren et al., 2010). VII. OUTLOOK The experimental applications and validation schemes In closing this Colloquium we stress that the known fluc- involving nonlinear quantum fluctuation relations still tuation relations are based on two facts: (a) microreversibility are in a state of infancy, as detailed in Sec. VI, so that there for nonautonomous Hamiltonian systems Eq. (40), and (b) the is plenty of room for advancements. The major obstacle special nature of the initial equilibrium states which is ex- for the experimental verification of the work fluctuation pressible in either microcanonical, canonical, or grand- relation is posed by the necessity of performing quantum canonical form, or products thereof. The final state reached projective measurements of energy. Besides the proposal of at the end of a protocol though is in no way restricted. It Huber et al. (2008) employing trapped ions, we suggested evolves from the initial state according to the governing here the scheme of a possible experiment employing dynamical laws under a prescribed protocol. In general, this circuit-QED architectures. In regard to exchange fluctuation final state may markedly differ from any kind of equilibrium relations instead, the main problem is related to the difficulty state. of measuring microscopic changes of macroscopic For quantum mechanical systems it also is of utmost quantities pertaining to heat and matter reservoirs. importance to correctly identify the work performed on a Continuous measurements of fluxes seemingly provide a system as the difference between the energy of the system at practical and efficient loophole for this dilemma (Campisi the end and the beginning of the protocol. In case of open et al., 2010a). systems the difference of the energies of the total system at The idea that useful work may be obtained by using the end and beginning of the protocol coincides with the information (Maruyama et al., 2009) has established a work done on the open system as long as the forces exclu- connection between the topical fields of quantum information sively act on this open system. With the free energy of an theory (Vedral, 2002) and quantum fluctuation relations. open system determined as the difference of free energies of Piechocinska (2000) and Kawai et al. (2007) used fluctuation the total system and that of the isolated environment the relations and information theoretic measures to derive quantum and classical Jarzynski equality and the Tasaki- Landauer’s principle. A generalization of the Jarzynski equal- Crooks theorem continue to hold true even for systems ity to the case of feedback controlled systems was provided in strongly interacting with their environment. Deviations the classical case by Sagawa and Ueda (2010), and in the from the fluctuation relations, however, must be expected if quantum case by Morikuni and Tasaki (2010). Recently protocol forces not only act on the system alone but as well Deffner et al. (2010) gave bounds on the entropy production directly on the environmental degrees of freedom, for ex- in terms of quantum information concepts. In a similar spirit, ample, if a time-dependent system-bath interaction protocol Hide and Vedral (2010) presented a method by relating is applied. relative quantum entropy to the quantum Jarzynski fluctua- The most general and compact formulation of quantum tion identity in order to quantify multipartite entanglement work fluctuation relations also containing the Onsager- within different thermal quantum states. A practical applica- Casimir reciprocity relations and nonlinear response to all tion of the Jarzynski equality in quantum computation was orders is the Andrieux-Gaspard relation, Eq. (55), which shown by Ohzeki (2010). represents the proper quantum version of the classical In conclusion, we are confident in our belief that this topic Bochkov-Kuzovlev formula (Bochkov et al., 1977), of quantum fluctuation relations will exhibit an ever growing Eq. (14). These relations provide a complete theoretical activity within nanosciences and further may invigorate read- understanding of those nonequilibrium situations that emerge ers to pursue their own research and experiments as this from arbitrary time-dependent perturbations of equilibrium theme certainly offers many more surprises and unforeseen initial states. applications.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 788 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ...
ACKNOWLEDGMENTS APPENDIX A: DERIVATION OF THE BOCHKOV-KUZVLEV RELATION We thank Y. Utsumi, D. S. Golubev, M. Marthaler, K. Saito, T. Fujisawa, and G. Scho¨n for providing the data for We report the steps leading to Eq. (14): Fig. 1. This work was supported by the cluster of excellence Nanosystems Initiative Munich (NIM) and the Volkswagen Foundation (Project No. I/83902).
Z Z z Z e ½H0ð 0ÞþW0 W0 exp dsusBs e ¼ dz0 exp dsusBð’s;0½z0; Þ 0 Zðt0Þ 0 Z Z ~ ¼ dz 0ðz Þ exp dsusBð"’ s;0½"z ; "Q Þ 0 Z Z 0 0 0 ~ ¼ dz 0ðz Þ exp dru r"BBð’r;0½z ; "Q Þ ; (A1) 0 where the first equality provides an explicit expression for for any complex number u. Using this equation and the fact the left-hand side of Eq. (14). In going from the second that " is real valued, " ¼ ", we obtain Eq. (40): to the third line we employed the expression of work in y ~ ði=ℏÞH ð N"Þ" ði=ℏÞH ð N"þ"Þ" U t;0½ ¼ lim e e Eq. (13), the microreversibility principle (10) and made the N!1 z change of variable 0 ! z . The Jacobian of this trans- ℏ H ði= Þ ð Þ" formation is unity, because the time evolution in classical e ℏ H ℏ H mechanics is a canonical transformation. A further change ¼ lim ½e ði= Þ ð Þ" e ði= Þ ð N"þ"Þ" of variables z ! z0 ¼ "z , whose Jacobian is unity as N!1 ℏ H well, and the change s ! r ¼ s, yields the expression e ði= Þ ð N"Þ" y in the last line that coincides with the right-hand side of y ¼ U ;t½ ¼Ut; ½ : (B5) Eq. (14). In the last line we used the property 0ðzÞ¼ H0 0ð"zÞ, inherited by 0 ¼ e =Z0 from the assumed time-reversal invariance of the Hamiltonian HðzÞ¼ Hð"zÞ. APPENDIX C: TASAKI-CROOKS RELATION FOR THE CHARACTERISTIC FUNCTION
APPENDIX B: QUANTUM MICROREVERSIBILITY From Eq. (50) we have
In order to prove the quantum principle of microreversi- y iuH ð Þ iuH ð 0Þ Zð 0ÞG½u; ¼TrU ½ e U 0½ e bility, we first discretize time and express the time evolution ;0 ; ~ H ð 0Þ operator Ut;0½ as a time ordered product (Schleich, 2001): e : (C1) ℏ H ~ ℏ H ~ ~ lim ði= Þ ð N"Þ" ði= Þ ð "Þ" U t;0½ ¼ e e t ¼ 0 N!1 For , the microreversibility principle, Eq. (40), becomes y y ~ ~ U0; ½ ¼U 0½ ¼ U ;0½ . Therefore, e ði=ℏÞH ð 0Þ"; (B1) ; H y Z ; Tr y ~ iu ð Þ y ~ where " ¼ t=N denotes the time step. Using Eq. (9), we ð 0ÞG½u ¼ U ;0½ e U ;0½ obtain e iuH ð 0Þe H ð 0Þ y ; (C2)
~ ði=ℏÞH ð N"Þ" ði=ℏÞH ð "Þ" U t;0½ ¼ lim e e y N!1 where we inserted ¼ 1 under the trace. Using Eq. (B4), ℏ H we obtain e ði= Þ ð Þ": (B2) H y H Z ; Tr y ~ iu ð Þ ~ iu ð 0Þ Thus, ð 0ÞG½u ¼ U ;0½ e U ;0½ e
ℏ H H ð 0Þ y ~ y ði= Þ ð N"Þ" y e : (C3) U t;0½ ¼ lim e N!1 ℏ H e ði= Þ ð N"þ"Þ" y The antilinearity of implies, for any trace class operator A Tr yA ¼ TrAy ℏ H , . Using this we can write e ði= Þ ð Þ" ; (B3) H ð 0Þ iuH ð 0Þ ~ iuH ð Þ Zð 0ÞG½u; ¼Tre e U 0½ e where we inserted y ¼ 1, N 1 times. Assuming that ; H y ~ ð tÞ commutes at all times with the time-reversal operator U ;0½ : (C4) , Eq. (41), we find Using the cyclic property of the trace one then obtains the y ði=ℏÞH ð Þu ði=ℏÞH ð Þu e t ¼ e t ; (B4) important result
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y ~ ið uþi ÞH ð 0Þ Campisi, M., P. Talkner, and P. Ha¨nggi, 2011a, ‘‘Quantum Bochkov- Zð 0ÞG½u; ¼TrU 0½ e ; Kuzovlev Work Fluctuation Theorems,’’ Phil. Trans. R. Soc. A ~ ið uþi ÞH ð 0Þ H ð 0Þ U ;0½ e e 369, 291. Campisi, M., P. Talkner, and P. Ha¨nggi, 2011b, ‘‘Influence of ¼ Zð ÞG½ u þ i ; ~ : (C5) measurements on the statistics of work performed on a quantum system,’’ Phys. Rev. E 83, 041114. Campisi, M., D. Zueco, and P. Talkner, 2010, ‘‘Thermodynamic anomalies in open quantum systems: Strong coupling effects in the isotropic XY model,’’ Chem. Phys. 375, 187. REFERENCES Casimir, H. B. G., 1945, ‘‘On Onsager’s Principle of Microscopic Reversibility,’’ Rev. Mod. Phys. 17, 343. Adib, A. B., 2005, ‘‘Entropy and density of states from isoenergetic Chen, L. Y., 2008a, ‘‘Nonequilibrium fluctuation-dissipation theo- nonequilibrium processes,’’ Phys. Rev. E 71, 056128. rem of Brownian dynamics,’’ J. Chem. Phys. 129, 144113. Adib, A. B., 2009, ‘‘Comment on ’On the Crooks fluctuation Chen, L. Y., 2008b, ‘‘On the Crooks fluctuation theorem and the theorem and the Jarzynski equality,’’’ [J. Chem. Phys. 129, Jarzynski equality,’’ J. Chem. Phys. 129, 091101. 091101 (2008)], J. Chem. Phys. 130, 247101. Chen, L. Y., 2009, Response to ‘‘Comment on ‘On the Crooks Allahverdyan, A. E., and T. M. Nieuwenhuizen, 2005, ‘‘Fluctuations fluctuation theorem and the Jaraynski equality’ and of work from quantum subensembles: The case against quantum ’Nonequilibrium fluctuation dissipation theorem of Brownian work-fluctuation theorems,’’ Phys. Rev. E 71, 066102. dynamics’,’’ [J. Chem. Phys. 130, 107101 (2009)] J. Chem. Andrieux, D., and P. Gaspard, 2008, ‘‘Quantum Work Relations and Phys. 130, 107102. Response Theory,’’ Phys. Rev. Lett. 100, 230404. Cleuren, B, C. Van den Broeck, and R. Kawai, 2006, ‘‘Fluctuation Andrieux, D., P. Gaspard, T. Monnai, and S. Tasaki, 2009, ‘‘The and Dissipation of Work in a Joule Experiment,’’ Phys. Rev. Lett. fluctuation theorem for currents in open quantum systems,’’ New 96, 050601. J. Phys. 11, 043014. Cohen-Tannoudji, C., B. Diu, and F. Laloe¨, 1977, Quantum Anetsberger, G., O. Arcizet, Q. P. Unterreithmeier, R. Rivie`re, A. Mechanics (Wiley, New York), Vol. 1, p. 318. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, 2009, Collin, D., F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. ‘‘Near-field cavity optomechanics with nanomechanical oscilla- Bustamante, 2005, ‘‘Verification of the Crooks fluctuation theo- tors,’’ Nature Phys. 5, 909. rem and recovery of RNA folding free energies,’’ Nature (London) Bernard, W., and H. B. Callen, 1959, ‘‘Irreversible Thermodynamics 437, 231. of Nonlinear Processes and Noise in Driven Systems,’’ Rev. Mod. Crooks, G. E., 1999, ‘‘Entropy production fluctuation theorem and Phys. 31, 1017. the nonequilibrium work relation for free energy differences,’’ Bochkov, G. N., and Y.E. Kuzovlev, 1977, ‘‘General theory of Phys. Rev. E 60, 2721. thermal fluctuations in nonlinear systems,’’ Zh. Eksp. Teor. Fiz. Crooks, G. E., 2008, ‘‘On the Jarzynski relation for dissipative 72, 238 [Sov. Phys. JETP 45, 125 (1977)]. quantum dynamics,’’ J. Stat. Mech. P10023. Bochkov, G. N., and Y.E. Kuzovlev, 1978, ‘‘Nonlinear stochastic Crooks, G. E., 2009, Comment regarding ‘‘On the Crooks fluctua- models of oscillator systems,’’ Radiophys. Quantum Electron. 21, tion theorem and the Jarzynski equality’’ [J. Chem. Phys. 129, 1019. 091101 (2008)] and ‘‘Nonequilibrium fluctuation-dissipation Bochkov, G. N., and Y.E. Kuzovlev, 1979, ‘‘Fluctuation-dissipation theorem of Brownian dynamics’’ [J. Chem. Phys. 129, 144113 relations for nonequilibrium processes in open systems,’’ Zh. (2008)]; J. Chem. Phys. 130, 107101. Eksp. Teor. Fiz. 76, 1071 [Sov. Phys. JETP 49, 543 (1979)]. Deffner, S., O. Abah, and E. Lutz, 2010, ‘‘Quantum work statistics Bochkov, G. N., and Y.E. Kuzovlev, 1981a, ‘‘Nonlinear fluctuation- of linear and nonlinear parametric oscillators,’’ Chem. Phys. 375, dissipation relations and stochastic models in nonequilibrium 200. thermodynamics I. Generalized fluctuation-dissipation theorem,’’ Deffner, S., and E. Lutz, 2008, ‘‘Nonequilibrium work distribution Physica (Amsterdam) 106A, 443. of a quantum harmonic oscillator,’’ Phys. Rev. E 77, 021128. Bochkov, G. N., and Y.E. Kuzovlev, 1981b, ‘‘Nonlinear fluctuation- Deffner, S., and E. Lutz, 2010, ‘‘Generalized Clausius Inequality for dissipation relations and stochastic models in nonequilibrium Nonequilibrium Quantum Processes,’’ Phys. Rev. Lett. 105, thermodynamics II. Kinetic potential and variational principles 170402. for nonlinear irreversible processes,’’ Physica (Amsterdam) 106A, de Groot, S. R., and P. Mazur, 1984, Non-equilibrium 480. Thermodynamics (Dover, New York). Caldeira, A. O., and A. J. Leggett, 1983, ‘‘Quantum tunnelling in a de Roeck, W., and C. Maes, 2004, ‘‘Quantum version of free- dissipative system,’’ Ann. Phys. (N.Y.) 149, 374. energy–irreversible-work relations,’’ Phys. Rev. E 69, 026115. Callen, H. B., and T. A. Welton, 1951, ‘‘Irreversibility and Dittrich, T., P. Ha¨nggi, G.-L. Ingold, B. Kramer, G. Scho¨n, and W. Generalized Noise,’’ Phys. Rev. 83,34. Zwerger, 1998, Quantum Transport and Dissipation (Wiley-VCH, Campisi, M., 2008, ‘‘Increase of Boltzmann entropy in a quantum Weinheim). forced harmonic oscillator,’’ Phys. Rev. E 78, 051123. Douarche, F., S. Ciliberto, A. Petrosyan, and I. Rabbiosi, 2005, ‘‘An Campisi, M., P. Talkner, and P. Ha¨nggi, 2009a, ‘‘Fluctuation experimental test of the Jarzynski equality in a mechanical Theorem for Arbitrary Open Quantum Systems,’’ Phys. Rev. experiment,’’ Europhys. Lett. 70, 593. Lett. 102, 210401. Efremov, G. F., 1969, ‘‘A fluctuation dissipation theorem for non- Campisi, M., P. Talkner, and P. Ha¨nggi, 2009b, ‘‘Thermodynamics linear media,’’ Zh. Eksp. Teor. Fiz. 55, 2322 [Sov. Phys. JETP 28, and fluctuation theorems for a strongly coupled open quantum 1232 (1969)]. system: an exactly solvable case,’’ J. Phys. A 42, 392002. Einstein, A., 1905, ‘‘U¨ ber die von der molekularkinetischen Theorie Campisi, M., P. Talkner, and P. Ha¨nggi, 2010a, ‘‘Fluctuation der Wa¨rme geforderte Bewegung von in ruhenden Flu¨ssigkeiten theorems for continuously monitored quantum fluxes,’’ Phys. suspendierten Teilchen,’’ Ann. Phys. (Leipzig) 322, 549, trans- Rev. Lett. 105, 140601. lated into English in Einstein, 1926.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 790 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ...
Einstein, A., 1906a, ‘‘Eine neue Bestimmung der Ho¨rhammer, C., and H. Bu¨ttner, 2008, ‘‘Information and Entropy in Moleku¨ldimensionen,’’ Ann. Phys. (Leipzig) 324, 289, translated Quantum Brownian Motion Thermodynamic Entropy versus von into English in Einstein, 1926. Neumann Entropy,’’ J. Stat. Phys. 133, 1161. Einstein, A., 1906b, ‘‘Zur Theorie der Brownschen Bewegung,’’ Horowitz, J., and C. Jarzynski, 2008, Comment on ‘‘Failure of the Ann. Phys. (Leipzig) 324, 371, translated into English in Einstein, work-Hamiltonian connection for free-energy calculations,’’ Phys. 1926. Rev. Lett. 101, 098901. Einstein, A., 1926, Investigations on the Theory of Brownian Horowitz, J., and C. Jarzynski, 2007, ‘‘Comparison of work fluc- Movement (Methuen, London), reprinted by Dover, New York, tuation relations,’’ J. Stat. Mech. P11002. 1956. Huber, G., F. Schmidt-Kaler, S. Deffner, and E. Lutz, 2008, Esposito, M., U. Harbola, and S. Mukamel, 2009, ‘‘Nonequilibrium ‘‘Employing trapped cold ions to verify the quantum Jarzynski fluctuations, fluctuation theorems, and counting statistics in quan- equality,’’ Phys. Rev. Lett. 101, 070403. tum systems,’’ Rev. Mod. Phys. 81, 1665. Husimi, K., 1953, ‘‘Miscellanea in Elementary Quantum Esposito, M., and S. Mukamel, 2006, ‘‘Fluctuation theorems for Mechanics, I,’’ Prog. Theor. Phys. 9, 238. quantum master equations,’’ Phys. Rev. E 73, 046129. Ingold, G.-L., 2002 Lect. Notes Phys. 611,1. Evans, D. J., E. G. D. Cohen, and G. P. Morriss, 1993, ‘‘Probability Jarzynski, C., 1997, ‘‘Nonequilibrium equality for free energy of second law violations in shearing steady states,’’ Phys. Rev. differences,’’ Phys. Rev. Lett. 78, 2690. Lett. 71, 2401. Jarzynski, C., 2000, ‘‘Hamiltonian derivation of a detailed fluctua- Feynman, R. P., and J. F. L. Vernon, 1963, ‘‘The theory of a general tion theorem,’’ J. Stat. Phys. 98,77. quantum system interacting with a linear dissipative system,’’ Jarzynski, C., 2002, ‘‘Targeted free energy perturbation,’’ Phys. Rev. Ann. Phys. (N.Y.) 24, 118. E 65, 046122. Ford, G. W., J. T. Lewis, and R. F. O’Connell, 1985, ‘‘Quantum Jarzynski, C., 2007, ‘‘Comparison of far-from-equilibrium work oscillator in a blackbody radiation field,’’ Phys. Rev. Lett. 55, relations,’’ C.R. Physique 8, 495. 2273. Jarzynski, C., 2008, ‘‘Nonequilibrium work relations: foundations Fujisawa, T., T. Hayashi, R. Tomita, and Y. Hirayama, 2006, and applications,’’ Eur. Phys. J. B 64, 331. ‘‘Bidirectional counting of single electrons,’’ Science 312, 1634. Jarzynski, C., 2011, ‘‘Equalities and inequalities: Irreversibility and Gallavotti, G., and E. G. D. Cohen, 1995, ‘‘Dynamical Ensembles in the second law of thermodynamics at the nanoscale,’’ Annu. Rev. Nonequilibrium Statistical Mechanics,’’ Phys. Rev. Lett. 74, 2694. Condens. Matter Phys. 2, 329. Grabert, H., P. Schramm, and G.-L. Ingold, 1988, ‘‘Quantum Jarzynski, C., 2004, ‘‘Nonequilibrium work theorem for a system Brownian motion: The functional integral approach,’’ Phys. strongly coupled to a thermal environment,’’ J. Stat. Mech. Rep. 168, 115. P09005. Grabert, H., U. Weiss, and P. Talkner, 1984, ‘‘Quantum theory of the Jarzynski, C., and D. K. Wo´jcik, 2004, ‘‘Classical and quantum damped harmonic oscillator,’’ Z. Phys. B 55,87. fluctuation theorems for heat exchange,’’ Phys. Rev. Lett. 92, Green, M. S., 1952, ‘‘Markoff Random Processes and the Statistical 230602. Mechanics of Time-Dependent Phenomena,’’ J. Chem. Phys. 20, Johnson, J. B., 1928, ‘‘Thermal agitation of electricity in conduc- 1281. tors,’’ Phys. Rev. 32,97. Green, M. S., 1954, ‘‘Markov Random Processes and the Statistical Katsuda, H., and M. Ohzeki, 2011, ‘‘Jarzynski Equality Mechanics of Time-Dependent Phenomena. II. Irreversible for an Energy-Controlled System,’’ J. Phys. Soc. Jpn. 80, Processes in Fluids,’’ J. Chem. Phys. 22, 398. 045003. Hahn, A. M., and H. Then, 2009, ‘‘Using bijective maps to improve Kawai, R., J. M. R. Parrondo, and C. V. den Broeck, 2007, free-energy estimates,’’ Phys. Rev. E 79, 011113. ‘‘Dissipation: The Phase-Space Perspective,’’ Phys. Rev. Lett. Hahn, A. M., and H. Then, 2010, ‘‘Measuring the convergence of 98, 080602. Monte Carlo free-energy calculations,’’ Phys. Rev. E 81, 041117. Khinchin, A., 1949, Mathematical foundations of statistical me- Ha¨nggi, P., 1978, ‘‘Stochastic Processes II: Response theory and chanics (Dover, New York). fluctuation theorems,’’ Helv. Phys. Acta 51, 202. Kippenberg, T. J., and K. J. Vahala, 2008, ‘‘Cavity optomechanics: Ha¨nggi, P., 1982, ‘‘Nonlinear fluctuations: The problem of deter- Back-action at the mesoscale,’’ Science 321, 1172. ministic limit and reconstruction of stochastic dynamics,’’ Phys. Kobe, D. H., 1981, ‘‘Gauge-invariant classical Hamiltonian formu- Rev. A 25, 1130. lation of the electrodynamics of nonrelativistic particles,’’ Am. J. Ha¨nggi, P., and G.-L. Ingold, 2006, ‘‘Quantum Brownian motion Phys. 49, 581. and the third law of thermodynamics,’’ Acta Phys. Pol. B 37, 1537 Kubo, R., 1957, ‘‘Statistical-mechanical theory of irreversible pro- [http://www.actaphys.uj.edu.pl/vol37/pdf/v37p1537.pdf]. cesses. I,’’ J. Phys. Soc. Jpn. 12, 570. Ha¨nggi, P., and G. Ingold, 2005, ‘‘Fundamental aspects of quantum Kurchan, J., 2000, ‘‘A quantum fluctuation theorem,’’ arXiv:cond- Brownian motion,’’ Chaos 15, 026105. mat/0007360. Ha¨nggi, P., and H. Thomas, 1982, ‘‘Stochastic processes: Time LaHaye, M. D., O. Buu, B. Camarota, and K. C. Schwab, 2004, evolution, symmetries and linear response,’’ Phys. Rep. 88, 207. ‘‘Approaching the Quantum Limit of a Nanomechanical Hide, J., and V. Vedral, 2010, ‘‘Detecting entanglement with Resonator,’’ Science 304,74. Jarzynski’s equality,’’ Phys. Rev. A 81, 062303. Liphardt, J., S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, Hofheinz, M., et al., 2009, ‘‘Synthesizing arbitrary quantum states 2002, ‘‘Equilibrium information from nonequilibrium measure- in a superconducting resonator,’’ Nature (London) 459, 546. ments in an experimental test of Jarzynski’s equality,’’ Science Hofheinz, M., E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, 296, 1832. M. Neeley, A. D. O’Connell, H. Wang, J. M. Martinis, and A. N. Marconi, U. M. B., A. Puglisi, L. Rondoni, and A. Vulpiani, 2008, Cleland, 2008, ‘‘Generation of Fock states in a superconducting ‘‘Fluctuation-dissipation: Response theory in statistical physics,’’ quantum circuit,’’ Nature (London) 454, 310. Phys. Rep. 461, 111.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 Michele Campisi, Peter Ha¨nggi, and Peter Talkner: Colloquium: Quantum fluctuation relations: ... 791
Maruyama, K., F. Nori, and V. Vedral, 2009, ‘‘Colloquium: The Schleich, W. P., 2001, Quantum Optics in Phase Space (Wiley- physics of Maxwell’s demon and information,’’ Rev. Mod. Phys. VCH, Berlin). 81,1. Schulz, S. A., U. Poschinger, F. Ziesel, and F. Schmidt-Kaler, 2008, Messiah, A., 1962, Quantum Mechanics (North-Holland, ‘‘Sideband cooling and coherent dynamics in a microchip multi- Amsterdam). segmented ion trap,’’ New J. Phys. 10, 045007. Minh, D. D. L., and A. B. Adib, 2008, ‘‘Optimized free energies Seifert, U., 2008, ‘‘Stochastic thermodynamics: Principles and from bidirectional single-molecule force spectroscopy,’’ Phys. perspectives,’’ Eur. Phys. J. B 64, 423. Rev. Lett. 100, 180602. Stratonovich, R. L., 1992, Nonlinear Nonequilibrium Morikuni, Y., and H. Tasaki, 2011, ‘‘Quantum Jarzynski-Sagawa- Thermodynamics I: linear and Nonlinear Fluctuation-dissipation Ueda relations,’’ J. Stat. Phys. 143,1. Theorems, Springer Series in Synergetics Vol. 57 (Springer- Mukamel, S., 2003, ‘‘Quantum extension of the Jarzynski Verlag, Berlin). relation: Analogy with stochastic dephasing,’’ Phys. Rev. Lett. Stratonovich, R. L., 1994, Nonlinear Nonequilibrium 90, 170604. Thermodynamics II: Advanced Theory, Springer Series in Nakamura, S., et al., 2010, ‘‘Nonequilibrium fluctuation relations in Synergetics Vol. 59 (Springer-Verlag, Berlin). a quantum coherent conductor,’’ Phys. Rev. Lett. 104, 080602. Sutherland, W., 1902, ‘‘Ionization, ionic velocities, and atomic Nakamura, S., et al., 2011, ‘‘Fluctuation Theorem and sizes,’’ Philos. Mag. 3, 161. Microreversibility in a Quantum Coherent Conductor,’’ Phys. Sutherland, W., 1905, ‘‘Dynamical theory of diffusion for non- Rev. B 83, 155431. electrolytes and the molecular mass of albumin,’’ Philos. Mag. Nieuwenhuizen, T. M., and A. E. Allahverdyan, 2002, ‘‘Statistical 9, 781. thermodynamics of quantum Brownian motion: Construction of Talkner, P., P.S. Burada, and P. Ha¨nggi, 2008, ‘‘Statistics of work perpetuum mobile of the second kind,’’ Phys. Rev. E 66, 036102. performed on a forced quantum oscillator,’’ Phys. Rev. E 78, Nyquist, H., 1928, ‘‘Thermal agitation of electric charge in con- 011115. ductors,’’ Phys. Rev. 32, 110. Talkner, P., P.S. Burada, and P. Ha¨nggi, 2009, ‘‘Erratum: Statistics O’Connell, A. D., et al., 2010, ‘‘Quantum ground state and single- of work performed on a forced quantum oscillator’’ [Phys. Rev. E phonon control of a mechanical resonator,’’ Nature (London) 464, 78, 011115 (2008)], Phys. Rev. E 79, 039902(E). 697. Talkner, P., M. Campisi, and P. Ha¨nggi, 2009, ‘‘Fluctuation theo- Ohzeki, M., 2010, ‘‘Quantum Annealing with the Jarzynski rems in driven open quantum systems,’’ J. Stat. Mech. P02025. Equality,’’ Phys. Rev. Lett. 105, 050401. Talkner, P., and P. Ha¨nggi, 2007, ‘‘The Tasaki-Crooks quantum Onsager, L., 1931a, ‘‘Reciprocal relations in irreversible processes. fluctuation theorem,’’ J. Phys. A 40, F569. I.,’’ Phys. Rev. 37, 405. Talkner, P., P. Ha¨nggi, and M. Morillo, 2008, ‘‘Microcanonical Onsager, L., 1931b, ‘‘Reciprocal relations in irreversible processes. quantum fluctuation theorems,’’ Phys. Rev. E 77, 051131. II.,’’ Phys. Rev. 38, 2265. Talkner, P., E. Lutz, and P. Ha¨nggi, 2007, ‘‘Fluctuation theorems: Peliti, L., 2008a, Comment on ‘‘Failure of the Work-Hamiltonian Work is not an observable,’’ Phys. Rev. E 75, 050102. Connection for Free-Energy Calculations,’’ Phys. Rev. Lett. 101, Tasaki, H., 2000, ‘‘Jarzynski relations for quantum systems and 098903. some applications,’’ arXiv:cond-mat/0009244. Peliti, L., 2008b, ‘‘On the work-Hamiltonian connection in manipu- Utsumi, Y., D. S. Golubev, M. Marthaler, K. Saito, T. Fujisawa, and lated systems,’’ J. Stat. Mech. P05002. G. Scho¨n, 2010, ‘‘Bidirectional single-electron counting and the Piechocinska, B., 2000, ‘‘Information erasure,’’ Phys. Rev. A 61, fluctuation theorem,’’ Phys. Rev. B 81, 125331. 062314. Vaikuntanathan, S., and C. Jarzynski, 2008, ‘‘Escorted free energy Prost, J., J.-F. Joanny, and J. M. R. Parrondo, 2009, ‘‘Generalized simulations: Improving convergence by reducing dissipation,’’ Fluctuation-Dissipation Theorem for Steady-State Systems,’’ Phys. Rev. Lett. 100, 190601. Phys. Rev. Lett. 103, 090601. Vedral, V., 2002, ‘‘The role of relative entropy in quantum infor- Ren, J., P. Ha¨nggi, and B. Li, 2010, ‘‘Berry-Phase-Induced Heat mation theory,’’ Rev. Mod. Phys. 74, 197. Pumping and Its Impact on the Fluctuation Theorem,’’ Phys. Rev. Vilar, J. M. G., and J. M. Rubi, 2008a, Comment on ‘‘Failure of Lett. 104, 170601. the work-Hamiltonian connection for free-energy calculations– Rondoni, L., and C. Mejı´a-Monasterio, 2007, ‘‘Fluctuations in Reply,’’ Phys. Rev. Lett. 101, 098902. nonequilibrium statistical mechanics: models, mathematical the- Vilar, J. M. G., and J. M. Rubi, 2008b, Comment on ‘‘Failure of ory, physical mechanisms,’’ Nonlinearity 20,R1. the work-Hamiltonian connection for free-energy calculations’’– Sagawa, T., and M. Ueda, 2010, ‘‘Generalized Jarzynski Equality Reply,’’ Phys. Rev. Lett. 101, 098904. under Nonequilibrium Feedback Control,’’ Phys. Rev. Lett. 104, Vilar, J. M. G., and J. M. Rubi, 2008c, ‘‘Failure of the work- 090602. Hamiltonian connection for free-energy calculations,’’ Phys. Saito, K., and A. Dhar, 2007, ‘‘Fluctuation Theorem in Quantum Rev. Lett. 100, 020601. Heat Conduction,’’ Phys. Rev. Lett. 99, 180601. Yukawa, S., 2000, ‘‘A quantum analogue of the Jarzynski equality,’’ Saito, K., and Y. Utsumi, 2008, ‘‘Symmetry in full counting J. Phys. Soc. Jpn. 69, 2367. statistics, fluctuation theorem, and relations among nonlinear Zimanyi, E. N., and R. J. Silbey, 2009, ‘‘The work-Hamiltonian transport coefficients in the presence of a magnetic field,’’ Phys. connection and the usefulness of the Jarzynski equality for free Rev. B 78, 115429. energy calculations,’’ J. Chem. Phys. 130, 171102.
Rev. Mod. Phys., Vol. 83, No. 3, July–September 2011 REVIEWS OF MODERN PHYSICS, VOLUME 83, OCTOBER–DECEMBER 2011 Erratum: Colloquium: Quantum fluctuation relations: Foundations and applications [Rev. Mod. Phys. 83, 771 (2011)]
Michele Campisi, Peter Ha¨nggi, and Peter Talkner
(published 19 December 2011) DOI: 10.1103/RevModPhys.83.1653 PACS numbers: 05.30. d, 05.40. a, 05.60.Gg, 05.70.Ln, 99.10.Cd
The first line of Eq. (51) contains some typos: it correctly reads
H iu½H ð Þ H ð 0Þ H ð 0Þ G½u; ¼TrT e e =Zð 0Þ: (51) This compares with its classical analog, i.e., the second line of Eq. (27). Quite surprisingly, notwithstanding the identity
Z H H H _ @ t ð tÞ H ð Þ H ð 0Þ¼ dt t ; (1) 0 @ t one finds that generally Z H H @H ð Þ iu½H ð Þ H ð 0Þ _ t t T e Þ T exp iu dt t : (2) 0 @ t R H _ H As a consequence, it is not allowed to replace H ð Þ H ð 0Þ, with 0 dt t@H t ð tÞ=@ t in Eq. (51). Thus, there is no quantum analog of the classical expression in the third line of Eq. (27). This is yet another indication that ‘‘work is not an observable’’ (Talkner, Lutz, and Ha¨nggi, 2007)). This observation also corrects the second line of Eq. (4) of the original reference (Talkner, Lutz, and Ha¨nggi, 2007). The correct expression is obtained from the general formula Z d T exp½Að Þ Að0Þ ¼ T exp dt eAðtÞ e AðtÞ ; (3) 0 dt H where AðtÞ is any time dependent operator [in our case AðtÞ¼iuH t ð tÞ]. Equation (3) can be proved by demonstrating that the operator expressions on either side of Eq. (3) obey the same differential equationR with the identity operator as the initial AðtÞ 1 sAðtÞ _ ð1 sÞAðtÞ condition. This can be accomplished by using the operator identity de =dt ¼ 0 dse AðtÞe . There are also a few minor misprints: (i) The symbol ds in the integral appearing in the first line of Eq. (55) should read dt. (ii) The correct year of the reference (Morikuni and Tasaki, 2010) is 2011 (not 2010). The authors are grateful to Professor Yu. E. Kuzovlev for providing them with this insight, and for pointing out the error in the second line of Eq. (51).
REFERENCES
Talkner P., E. Lutz, and P. Ha¨nggi, 2007, Phys. Rev. E 75, 050102(R).
0034-6861= 2011=83(4)=1653(1) 1653 Ó 2011 American Physical Society