Quantum Thermodynamics and Work Fluctuations with Applications to Magnetic Resonance Wellington L

Quantum thermodynamics and work fluctuations with applications to magnetic resonance Wellington L. Ribeiro Universidade Federal do ABC, 09210-580 Santo Andre, Brazil Gabriel T. Landia) Universidade Federal do ABC, 09210-580 Santo Andre, Brazil and Instituto de Fısica da Universidade de Sao~ Paulo, 05314-970 Sao~ Paulo, Brazil Fernando L. Semiao~ Universidade Federal do ABC, 09210-580 Santo Andre, Brazil (Received 25 December 2015; accepted 21 September 2016) In this paper, we give a pedagogical introduction to the ideas of quantum thermodynamics and work fluctuations, using only basic concepts from quantum and statistical mechanics. After reviewing the concept of work as usually taught in thermodynamics and statistical mechanics, we discuss the framework of non-equilibrium processes in quantum systems together with some modern developments, such as the Jarzynski equality and its connection to the second law of thermodynamics. We then apply these results to the problem of magnetic resonance, where all calculations can be done exactly. It is shown in detailhowtobuildthestatisticsofthework,both for a single particle and for a collection of non-interacting particles. We hope that this paper will serve as a tool to bring the new student up to date on the recent developments in non-equilibrium thermodynamics of quantum systems. VC 2016 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4964111]

I. INTRODUCTION The random fluctuations present in small systems also affect thermodynamic quantities such as work and heat. A Thermodynamics was initially developed to deal with beautiful example is the use of optical tweezers to fold and 1,2 macroscopic systems and is thus based on the idea that a unfold individual RNA molecules.6 The molecules are handful of macroscopic variables, such as volume, pressure, immersed in water and therefore subject to the incessant fluc- and temperature, suffice to completely characterize a system. tuations of Brownian motion. Thus, the amount of work However, after the advent of the atomic theory, it became required to fold a molecule will be different each time we clear that the variables of the underlying microscopic world repeat the experiment and should therefore be interpreted as a are constantly fluctuating due to the inherent chaos and ran- random variable. The same is true of the work that is extracted domness of the micro-world. Statistical mechanics was thus from the molecule when it is unfolded. In some realizations, it developed as a theory connecting these microscopic fluctua- is even possible to extract work without any changes in the tions with the emergent macroscopic variables. Because one thermodynamic state of the system—something that would usually deals with a large number of particles, the relative contradict the second law of thermodynamics. fluctuations become negligible, so that thermodynamic This example introduces the idea that fluctuations in small measurements usually coincide very well with expectation systems could lead to local violations of the second law. values of the microscopic fluctuating quantities (a conse- These violations were first observed in fluid simulations in the quence of the law of large numbers3). beginning of the 1990s by Evans, Cohen, Gallavotti, and col- Equilibrium statistical mechanics is now a well- laborators.7,8 Afterwards, in 1997 and 1998 came two impor- established and successful theory. Its main result is the tant (and intimately related) breakthroughs by Jarzynski9,10 Gibbs formula for the canonical ensemble,4,5 which provides and Crooks.11,12 They showed that the work performed in a a fundamental bridge between microscopic physics and ther- non-equilibrium (e.g., fast) process, when interpreted as a ran- modynamics for any equilibrium situation. Conversely, far dom variable, obeyed a set of exact relations that touched less is known about non-equilibrium processes. The reason is deeply on the nature of irreversibility and the second law. that in this case the handful of parameters used in thermody- Nowadays, these relations go by the generic name of fluctua- namics no longer suffices, forcing one to know the full tion theorems (not to be confused with the fluctuation- dynamics of the system; i.e., one must study Newton’s or dissipation theorem) and can be interpreted as generalizations Schr€odinger’s equation for all constituent particles, thus of the second law for fluctuating systems. These theoretical making the problem much more difficult. findings were later confirmed by experiments on diverse sys- These difficulties led researchers to look for non- tems such as trapped Brownian particles,13,14 mechanical equilibrium processes in the realm of small systems. On the oscillators,15 and biological systems.6,16 one hand, in these systems, the dynamics are somewhat eas- With time, researchers began to look for similar results in ier to describe because there are fewer particles. But on the quantum systems, both for unitary17,18 and for open19,20 other hand, fluctuations become important and must there- quantum dynamics. Here, in addition to the thermal fluctua- fore be included in the description. Substantial progress in tions, one also has intrinsically quantum fluctuations, leading experimental methods has made it possible for the first time to a much richer platform to work with. One may therefore to experiment with small systems and has therefore contrib- ask to what extent will these quantum fluctuations affect uted to this shift. thermodynamic quantities such as heat and work. This

948 Am. J. Phys. 84 (12), December 2016 http://aapt.org/ajp VC 2016 American Association of Physics Teachers 948 research was also motivated by the remarkable progress of Instead, thermodynamics usually focuses on quasi-static pro- the past decade in the experimental control of atoms and cesses, in which k changes very slowly in order to ensure that photons, particularly in areas such as magnetic resonance, throughout the process the system is always in thermal equilib- ultra-cold atoms, and quantum optics. In fact, the first experi- rium. Quasi-static processes have the advantage of being atem- mental verifications of the fluctuation theorems for quantum poral—we do not need to specify the function k t ,butmerely systems were done only recently, using nuclear magnetic res- its initial and final values. In this paper, our focusð Þ will be on onance21 and trapped ions.22 non-equilibrium processes. Butbeforewecangetthere,we It is a remarkable achievement of modern-day physics that must first have a solid understanding of quasi-static processes. we are now able to test the thermodynamic properties of sys- Perhaps the most important example of a quasi-static pro- tems containing only a handful of particles. But despite being cess is the isothermal process, in which the temperature of an active area of research, the basic concepts in this field can the system is kept constant throughout the protocol. Because be understood using only standard quantum and statistical work usually changes the temperature of an isolated system, mechanics. The purpose of this paper is to provide an introduc- to ensure a constant temperature the system must remain tion to some of these concepts in the realm of quantum sys- coupled to a heat reservoir kept at a constant temperature T. tems. Our main goal is to introduce the reader to the idea that It is also important to note that an isothermal process must work can be treated as a random variable. We then show how necessarily be quasi-static. For, if the process is not quasi- to construct all of its statistical properties such as the corre- static, the temperature will not remain constant or homoge- sponding probability distributionofworkorthecharacteristic neous. In fact, intensive quantities such as temperature and function. We discuss how these quantities can be used to derive pressure are defined only in thermal equilibrium, so any pro- Jarzynski’s equality,9,10 thence serving as a quantum mechani- cess where these quantities are kept fixed must be quasi- cal derivation of the second law. To illustrate these new ideas, static. Thus, henceforth whenever we speak of an isothermal we apply our results to the problem of magnetic resonance, an process, it will already be implied that it is quasi-static. elegant textbook example that can be worked out analytically. In classical mechanics, we learn that the work performed on a system can be stored as potential energy and that this II. WORK IN THERMODYNAMICS AND potential energy can be used to extract work from the sys- STATISTICAL MECHANICS tem. For a thermal system undergoing an isothermal process, this is no longer true since the system must always exchange A. Thermodynamic description some heat with the bath. The part of U available for perform- ing work is called the free energy, F, so that Consider any physical system described by a certain Hamiltonian H. When this system is placed in contact with a W DF DU Q; (2) heat reservoir, energy may flow between the system and the ¼ ¼ À bath. This change in energy is called heat. But the energy of where DF F T; kf F T; ki (cf. Ref. 1 or chapter 2 of a system may also change by means of an external agent, Ref. 23). The¼ energyð ÞÀ is “free”ð inÞ the sense that it is available which manually changes some parameter in the Hamiltonian to perform work. of the system. These types of changes are called work. Now suppose we try to repeat the same process, but we do Heat and work are not properties of the system. Rather, so too quickly, so that it cannot be considered quasi-static. they are the outcomes of processes that alter the state of the The initial state is still F T; k , but the final state will not be ð iÞ system. If during a certain interval of time an amount of heat F T; kf . Instead, the final state will be something compli- Q entered the system and work W was performed on the sys- catedð thatÞ depends on exactly how the process was per- tem, energy conservation implies that the total energy U of formed (see Fig. 1). Notwithstanding, because the system is the system must have changed by coupled to a bath, if we leave it alone after the protocol is over, it will eventually relax to the state F T; kf . Thus, over- DU Q W; (1) ð Þ ¼ þ all, a certain amount of work W was performed to take the system from F T; ki to F T; kf . But this work is not equal which is the first law of thermodynamics. When W > 0 we to DF, since Eq.ð (2) holdsÞ onlyð forÞ quasi-static processes. say the external agent performed work on the system, while Instead, according to the second law of thermodynamics, when W < 0 we say the system performed work on the exter- the work done in the non-equilibrium process must always nal agent (for instance, when a gas expands against a piston). be larger than DF, so that in general The process of performing work on a system can be described microscopically in a very general way through the change of some parameter k in the Hamiltonian. We shall call this the work parameter. Examples include the volume of a container, an electric or magnetic field, or the stiffness of a harmonic trap. In order to describe exactly how the work was performed we must also specify the protocol to be used. This means specifying precisely under what conditions are the changes being made and with what time dependence k t .We usually assume that the process lasts between a time t ð Þ0 and a time t s, during which k varies in some pre-defined¼ way Fig. 1. Diagram representing a non-equilibrium process. Through the proto- ¼ col k t , the system is taken from an initial state F T; ki to a final non- from an initial value ki k 0 to a final value kf k s . ð Þ ð Þ ¼ ð Þ ¼ ð Þ equilibrium state with parameter kf (solid line). After the process is done, Overall, describing an arbitrarily fast process can be a diffi- the system will eventually relax from the non-equilibrium state to the equi- cult task, because it requires detailed information about the librium state F T; kf (dashed line). Finally, the dotted line represents the dynamics of the system and how it is coupled to the bath. journey back toð the originalÞ state.

949 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 949 W DF; (3) and eigenvectors n . The probability of finding the system  in the state n is thereforej i with the equality holding only for the isothermal (i.e., quasi- j i bE static) process. This important result means that if we wish eÀ n P n q n : (5) to change the free energy of a system by DF, the minimum n ¼h j thj i¼ Z amount of work we need to perform is DF and will be accomplished in an isothermal (quasi-static) process. Any Moreover, the internal (average) energy of the system can be other protocol will require more work. The difference written as Wirr W DF 0, known as the irreversible work, there- tr (6) fore¼ representsÀ the extra work that had to be done due to the U H Hqth EnPn: ¼h i¼ ð Þ¼ n particular choice of protocol. X The inequality in Eq. (3) can be interpreted as a direct con- When we change k to k dk, both En and Pn will change; sequence of the Kelvin statement of the second law: “A trans- hence, U will change by þ formation whose only final result is to transform into work, heat extracted from a source which is at the same temperature dU d EnPn dEn Pn En dPn : (7) 1 throughout, is impossible.” The key part of this statement is ¼ n ð Þ¼ n ½ð Þ þ ð ފ the expression “only final result.” It means that it is impossi- X X This separation of in two terms allows for an interesting ble to extract work from a bath at a fixed temperature, without dU physical interpretation.11 changing anything else (such as the thermodynamic state of The change in k is infinitesimal and instantaneous, so the system). Extracting work while changing the thermody- immediately after the change, the system has not yet namic state of a system is not a problem. For instance, we can responded. This situation corresponds to the first term on the use a heat source to make a gas expand and thence extract right-hand side of Eq. (7): it is the average of the energy work from the expansion. But by the end of the process, we change dEn over the old (unperturbed) probabilities Pn. In the will have altered the state of the gas, so the extraction of work second term, the energies are fixed and the probabilities was not the only outcome. What the second law says is that it change. We interpret this as the second step, where the system is impossible to extract work from a single source at a fixed adjusts itself with the bath in order to return to equilibrium. temperature and keep the state of the system intact. Thus, each infinitesimal process can be separated into two To make the connection with Eq. (3), consider a process parts. The first part is the work performed, while the second is divided into three steps, represented by the three lines in the heat exchanged as the system relaxes to equilibrium. Fig. 1. In the first step, we perform a certain amount of work This decomposition motivates us to define W in a non-equilibrium process. In the second, we perform no work and allow the system to relax from the non- dW dEn Pn; (8) equilibrium state to F T; kf . Finally, in the third process ¼ n ð Þ (represented by a dottedð lineÞ in Fig. 1), we go back quasi- X statically from F T; k to F T; k . The amount of work dQ En dPn ; (9) f i ¼ ð Þ required for the returnð journeyÞ ð is Þ , because we n Wreturn DF X assume that this part is quasi-static. In the¼À end, we are back to the original state, having performed a total work so that Eq. (7) can be written as dU dQ dW (we use d instead of d simply to emphasize that¼ heat andþ work are not W Wreturn W DF. According to the second law, this 2 totalþ work cannot¼ À be negative, because that would mean we exact differentials ). In order to better understand the physi- cal meaning of these formulas, we will now explore them in would have extracted work from a reservoir at a fixed tem- more detail. perature, without any changes in the state of the system. We start with dW and show that it is related to the free Consequently, W DF 0, which is Eq. (3). À  energy of the system, defined as B. Isothermal processes in equilibrium statistical F T ln Z: (10) ¼À mechanics To understand this relationship, first note that since the E =T Let us now analyze the isothermal process quantitatively. temperature T is fixed, dF T dZ=Z. Since Z eÀ n , ¼À ¼ n Because it is a quasi-static process, we can decompose it into one can then readily show that dF n dEn Pn, which is a series of infinitesimal processes, where k is changed precisely Eq. (8); thus, ¼ ð Þ P slightly to k dk. The full process is then simply a succes- P sion of theseþ small steps. dW dF: (11) We assume that initially we had a system with ¼ Hamiltonian H k H in thermal equilibrium with a heat From this result, Eq. (2) is recovered by integrating over the bath at a temperatureð Þ¼ T. According to statistical mechanics, several infinitesimal steps. its state is then given by the Gibbs density operator Next we turn to dQ in Eq. (9). Instead of trying to manipu- late dPn, we can use the following trick: invert Eq. (5) to bH eÀ write En Tln ZPn . If we substitute this relation into Eq. qth ; (4) (9) we get¼À two terms,ð Þ one proportional to ln Z and the other ¼ Z ð Þ proportional to ln Pn . The term with ln Z will be bH ð Þ ð Þ where Z tr eÀ is the partition function and b 1=T in ¼ ð Þ ¼ units with Boltzmann’s constant equal to 1. We can also T ln Z d Pn T ln Z d Pn 0; (12) write this expression in terms of the energy eigenvalues E À n ð Þ ð Þ¼À ð Þ n ¼ n X  X 

950 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 950 since P 1 and d 1 0. Thus, we are left only with A. The Jarzynski equality n n ¼ ð Þ¼ dQP T dPn ln Pn: (13) Usually, our knowledge of non-equilibrium processes is ¼À n ð Þ restricted only to inequalities such as Eq. (3). The contribu- But now note that,X by the chain rule, tion of Jarzynski9,10 was to show that by interpreting W as a random variable, one can obtain an equality, even for a pro- Pn cess performed arbitrarily far from equilibrium. d PnlnPn dPn lnPn dPn ; (14) n ¼ n ðÞ þ n Pn ðÞ Consider several realizations of a non-equilibrium pro- X X X cess, such as that described by the solid line in Fig. 1. At and the last term is also zero for the same reason as above. each realization, we always prepare the system in the Hence, we conclude that same initial state. We then execute the protocol and measure the total work W performed. After repeating this process dQ Td P ln P : (15) many times, we can construct the probability distribution of ¼À n n n the work, P(W). From this probability, any average can be  X  We see that even though dQ is not a function of state, it is computed. For instance, the average work will be related to the variation of a quantity that is a function of state. We define the entropy as W P W dW; (18) h i¼ ð Þ ð S PnlnPn; (16) 2 ¼À n or we can study the average of other quantities such as W X and so on. h i so that we finally arrive at Jarzynski’s main result was to show that the statistical average of e bW should satisfy9,10 dQ T dS: (17) À ¼ bW bDF This relation, we emphasize, holds only for infinitesimal pro- eÀ eÀ ; (19) h i¼ cesses. For finite and irreversible processes, there may be additional contributions to the change in entropy. where DF F T; kf F T; ki , as described in Fig. 1. This ¼ ð ÞÀ ð Þ We therefore see that it is possible to give microscopic def- result is nothing short of remarkable. It holds for a process initions to thermodynamic quantities such as heat and work. performed arbitrarily far from equilibrium. And it is an equal- Moreover, it is possible to relate them to functions of state ity, which is a much stronger statement than the inequalities we are used to in thermodynamics. The appearance of that can be constructed from the initial density matrix qth. While these thermodynamic quantities can be defined inde- F T; kf in Eq. (19) is also surprising, since this is not the final ð Þ pendently of statistical mechanics, we believe that this micro- state of the process; it would only be the final state if the pro- scopic description helps to clarify their physical meanings. cess were quasi-static. Instead, as indicated in Fig. 1, the final state is a non-equilibrium state, which may differ substantially III. WORK AS A RANDOM VARIABLE from F T; kf . The appearance of F T; kf therefore reflects the stateð thatÞ the system wants to goð to,Þ but cannot do so In 1997, Jarzynski discovered that great insight into the because the process is not sufficiently slow. properties of non-equilibrium processes could be gained by It is possible to show that the inequality in Eq. (3), treating work as a random variable.9,10 For example, consider W DF, is contained within the Jarzynski equality. This can the process in which a movable piston is used to compress a be accomplished using Jensen’s inequality (see chapter 8 of bW b W gas contained in a cylinder. Since the molecules of the gas are Ref. 3), which states that eÀ eÀ h i. Combining this moving chaotically and hitting the walls of the piston in all with Eq. (19) then gives h i sorts of different ways, each time we press the piston the gas molecules will exert back on us a different force. This means W DF: (20) h i that the work needed to achieve a given compression will change each time we repeat the experiment. We therefore see that when we treat work as a random vari- Of course, one may object that in most systems these fluc- able the old results from thermodynamics are recovered for tuations are negligibly small. But that does not stop us from the average work. interpreting W in this way. And, as we will soon see, this In macroscopic systems, by the law of large numbers (see, treatment does lead to several advantages. On the other e.g., Ref. 3, chapter 8), individual measurements are usually hand, when dealing with microscopic systems this interpreta- very close to the average, so the distinction between the tion becomes essential because fluctuations become signifi- average work W and a single stochastic realization W is h i cant. A famous example is the work performed when folding immaterial. But for microscopic systems, this is usually not 6,16 true. In fact, although W DF, the individual realizations RNA molecules, already discussed in Sec. I. h i In addition to thermal fluctuations, some microscopic sys- W may very well be smaller than DF. These instances would tems also have a strong contribution from quantum fluctua- be local violations of the second law. For large systems, tions. These fluctuations are related to the fact that in order these local violations become extremely rare. But for small 6,16,21 to access the amount of work performed in a system, one systems, they can be measured experimentally. If we must measure its energy and therefore collapse the wave know the distribution P(W), then the probability of a local function. This measurement puts the system into different violation of the second law can easily be found as states with different probabilities [cf. Eq. (5)]. Thus, in quan- DF tum systems both thermal and quantum fluctuations must be Prob W < DF P W dW: (21) taken into consideration. ð Þ¼ ð Þ ðÀ1

951 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 951 f The demonstration of the Jarzynski equality in Eq. (19) random due to thermal fluctuations, and the second, Em, is requires detailed knowledge of the dynamics of the system, random due to quantum fluctuations. Consequently, W will i.e., if it is classical, quantum, unitary, stochastic, etc.24 also be a random variable, encompassing both thermal and Below we will give a demonstration for one particular case quantum fluctuations. of unitary quantum dynamics. C. Distribution of work and characteristic function B. Non-equilibrium unitary dynamics We will now obtain an expression for the probability dis- We will now consider in detail how to describe work in a tribution P(W) obtained by repeating the above protocol sev- non-equilibrium process. As mentioned above, this requires eral times. This can be accomplished by noting that we are detailed knowledge of the dynamics of the system and how it is dealing here with a two-step measurement process. From coupled to the heat bath. The last part is by far the most difficult probability theory, if A and B are two events, the total proba- because it requires concepts from open quantum systems such bility P(A, B) that both events occur can be written as as Lindblad dynamics or quantum Langevin equations.25 We P A; B P A B P B ; (26) will therefore make an important simplification and assume ð Þ¼ ð j Þ ð Þ that the coupling to the heat bath is so weak that during the pro- tocol no heat is exchanged. This situation is actually encoun- where P(B) is the probability that B occurs and P A B is the conditional probability that occurs given thatð j Þ has tered very often in experiments because many systems are only A B occurred. In our context, P A B is given in Eq. (24), weakly coupled to the bath. It also simplifies considerably the whereas P(B) is simply the initialð j probabilityÞ P . Hence, the description of the problem, because it allows us to use n probability that both events have occurred is Schr€odinger’s equation to describe the dynamics of the system. We shall consider the protocol described in Sec. II A. i f 2 Prob En protocol Em m U s n Pn: (27) Initially, the system had a Hamiltonian Hi H ki and was ð ! ! Þ¼jh j ð Þj ij ¼ ð Þ in thermal equilibrium with a bath at a temperature T. The Because we are interested in the work performed, we then initial state of the system is then given by the Gibbs thermal write density matrix in Eq. (4). As a first step, we measure the i 2 energy of the system. If we let En and n denote the eigen- f i j i i P W m U s n Pn d W Em En ; (28) values and eigenvectors of Hi, then the energy En will be ð Þ¼ n;m jh j ð Þj ij ½ Àð À ފ bEi obtained with probability P eÀ n =Z [cf. Eq. (5)]. X n ¼ Immediately after this measurement, we initiate the proto- where d x is the Dirac delta function. This formula is per- col, changing k from k 0 k to k s k according to ð Þ ð Þ¼ i ð Þ¼ f haps best explained in words: we sum over all allowed some pre-defined function k t . If we assume that during this events, weighted by their probabilities, and catalogue the process the contact with theð bathÞ is very weak, then the state f i terms according to the values of Em En. of the system will evolve according to Although it is exact, Eq. (28) isÀ not very convenient to work with. In most systems, there are a large number of w t U t n ; (22) allowed energy levels and therefore an even larger number j ð Þi¼ ð Þj i f i of allowed energy differences Em En. It is much more con- where U(t) is the unitary time-evolution operator, which sat- venient to work with the characteristicÀ function, defined as isfies Schr€odinger’s equation (with h 1) ¼ the Fourier transform of the original distribution @ ; 0 1: (23) i tU H t U U 1 ¼ ð Þ ð Þ¼ G r eirW P W eirW dW: (29) ð Þ¼h i¼ ð Þ (For a derivation of this equation, see Ref. 26, chapter 2.) ðÀ1 At the end of the process, we measure the energy of the From G(r) we can recover the original distribution from the system once again. The Hamiltonian is now Hf H kf and ¼ ð Þ f inverse Fourier transform therefore may have completely different energy levels Em and eigenvectors m . The probability that we now measure 1 1 irW an energy Ef is j i PW dr G r eÀ : (30) m ðÞ¼ 2p ðÞ ðÀ1 m w s 2 m U s n 2; (24) jh j ð Þij ¼jh j ð Þj ij Because P(W) and G(r) are Fourier transforms of each other, they contain the same information. which can be interpreted as the conditional probability that a With the help of Eq. (28), we can write system initially in n will be found in m after a time s. j i j i Because no heat is exchanged with the environment, any 2 ir Ef Ei G r m U n P e ð mÀ nÞ; change in the energy must necessarily be attributed to the ð Þ¼ jh j j ij n work performed by the external agent. The energy obtained n;m X f i i † irEm irEn in the first measurement was En, and the energy obtained in n U e m m UeÀ Pn n ; the second measurement was Ef . We then define the work ¼ n;m h j j ih j j i m X performed by the external agent as † irHf irHi n U e m m UeÀ qth n f i ¼ n;m h j j ih j j i W Em En: (25) X ¼ À † irHf irHi tr U s e U s eÀ qth ; (31) i f ¼ f ð Þ ð Þ g Both En and Em are fluctuating quantities that change during i each realization of the experiment. The first energy En is hence we conclude that

952 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 952 † irH irH G tr U s e f U s eÀ i q : (32) direction. The Hamiltonian of this interaction can be ¼ f ð Þ ð Þ thg 26 described using the Pauli matrix rz as The characteristic function does not have a particularly important physical meaning. But because it is written as the B0 H0 rz: (38) trace of a product of operators, it is usually much more con- ¼À 2 venient to work with than P(W). In many aspects, this func- For simplicity, we choose to measure the field in energy tion plays a role somewhat similar to the partition function Z units. Moreover, because we are using 1, the quantity in equilibrium statistical mechanics. One usually does not h B0 also represents the characteristic precession¼ frequency of the dwell on the physical meaning of Z, but rather uses it as a spin. The eigenvalues of H are B =2 and B =2, corre- convenient quantity from which observables such as the 0 0 0 sponding to spin up and spin down,À respectively. energy and entropy can be extracted. When the spin is coupled to a heat bath at temperature T, From G(r) we can also readily extract the statistical its state will be given by the thermal density matrix q in moments of . To see this, we expand Eq. (29) in a Taylor th W Eq. (4). The Hamiltonian H is already diagonal in the usual series in r to find 0 6 basis that diagonalizes rz. Whenever this is true, the cor- respondingj i matrix exponential e H0=T can be computed by r2 r3 À Gr eirW 1 ir W W2 i W3 : simply exponentiating the eigenvalues ðÞ¼h i¼ þ h iÀ 2 h iÀ 3! h iþÁÁÁ (33) B0=2T H0=T e 0 eÀ : (39) ¼ 0 e B0=2T Hence, Wn will be multiplied by the term of order rn in the À h i expansion. The partition function is the trace of this matrix, On the other hand, we can obtain quantum mechanical for- H0=T Z tr eÀ 2cosh B0=2T . The thermal density matrix mulas for the moments by doing a similar expansion in Eq. ¼ ð H0=T Þ¼ ð Þ q eÀ =Z can then be written in a convenient way as (32). The average work, for instance, is found to be th ¼ 1 W Hf Hi 0; (34) 1 f 0 h i¼h is Àh i 2 ðÞþ qth 0 1 1; (40) where, given any operator A, we define ¼ 0 1 f B 2 ðÞÀ C B C A tr U† t AU t q (35) @ A h it ¼ f ð Þ ð Þ thg where f is the equilibrium magnetization of the system as the expectation value of this operator at time t, a result B0 that follows directly from the fact that the state of the system f rz tanh ; (41) † ¼h ith ¼ T at time t is q t U t qthU t . We therefore see that W is  simply the differenceð Þ¼ ð Þ betweenð Þ the average energy ath timei s and the average energy at time 0, which is intuitive. One can corresponding to the paramagnetic response of a spin-1/2 continue with the expansion of Eq. (32) to obtain formulas particle. for higher-order moments. Unfortunately, they do not The work protocol is implemented by applying a very small acquire such a simple form. field of amplitude B1 rotating in the xy-plane with frequency x. The characteristic function can also be used to demonstrate That is, the work parameter k is described here by the field B1 B1 sin xt; B1 cos xt; 0 .Typically,B0 10 T and B1 the Jarzynski equality in Eq. (19) because, based on the defini- ¼ð Þ   bW 0:01 T, so we can always take as a good approximation that tion in Eq. (29), we should have G r ib eÀ . But ð ¼ Þ¼h i B1 B0.ThetotalHamiltonianofthesystemnowbecomes from Eqs. (4) and (32) we find that  B0 B1 1 † bH 1 bH Zf Ht rz rx sin xt ry cos xt : (42) Gib tr U eÀ f U tr eÀ f ; (36) ðÞ¼À 2 À 2 ðÞþ ðÞ¼ Zi ðÞ¼ Zi ðÞ¼ Zi

bF The oscillating field plays the role of a perturbation, which, and since Z eÀ , this yields Eq. (19) ¼ albeit extremely weak, may nonetheless promote transitions bW bDF between the up and down spin states. The transitions will be G ib eÀ eÀ : (37) ð Þ¼h i¼ most frequent at the resonance condition x B0, i.e., when the driving frequency x is the same as the natural¼ oscillation Notice that no assumptions have been made as to the speed frequency B0. of the process, so we conclude that the Jarzynski equality To make progress, we must now compute the time evolu- holds for a process arbitrarily far from equilibrium. tion operator U(t) defined in Eq. (23). Usually, accomplish- ing this for a time-dependent Hamiltonian is a very complicated task. Luckily, for the particular choice of H in IV. MAGNETIC RESONANCE Eq. (42) the task turns out to be quite simple. The first step is to define a new operator U~ t from the relation A. Statement of the problem ð Þ We will now apply the concepts of Sec. III to a magnetic U t eixtrz=2 U~ t : (43) resonance experiment. The typical setup consists of a sample ð Þ¼ ð Þ of non-interacting spin-1/2 particles (electrons, nucleons, Substituting this in Eq. (23), one finds that U~ must obey the etc.) placed under a strong static magnetic field B0 in the z modified Schr€odinger equation

953 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 953 i@tU~ H~U~; (44) But looking at Eq. (51) we see that U t v t . ¼ Therefore, v 2 represents the transition probabilityhÀj ð Þjþi per¼ unitð Þ where time for a jumpj j to occur. Moreover, the unitarity condition U†U 1 implies that v 2 u 2 1, so u 2 is the probabil- B0 x B1 ity that¼ no transition occurs.j j þj j ¼ j j H~ ðÞÀ r r : (45) ¼À 2 z À 2 y From Eq. (53), we also see that v sin h, which therefore attributes a physical meaning to the/ angle h [defined in Eq. We therefore see that U~ t evolves according to a time- (49)] as representing the transition probability. This proba- ð Þ independent Hamiltonian. Consequently, the solution of bility reaches a maximum precisely at resonance (x B0), iHt~ Eq. (44) is simply U~ t eÀ and the full time evolution as we intuitively expect. In fact, at resonance Eqs. (52)¼ and ð Þ¼ operator is (53) simplify to

~ ixtrz=2 iHt B t U t e eÀ : (46) ut eixt=2 cos 1 ; (54) ð Þ¼ ðÞ¼ 2  It is important to notice that because ry and rz do not com- i xr =2 H~ t ixt=2 B1t mute we cannot write U(t) as e ð z À Þ . v t e sin : (55) ðÞ¼ 2 It is also useful to have in hand an explicit formula for  iHt~ eÀ . This can be accomplished using the following trick. 2 Now that we have the initial density matrix [Eq. (40)] and Let be an arbitrary matrix such that I (where I is the time evolution operator [Eq. (51)], we can compute the the 2M 2 identity matrix). Then, if a is anM arbitrary¼ constant, evolution of any observable we want using Eq. (35). For a direct power series expansion of e ia yields A À M example, we could compute the evolution of the magnetiza-

ia tion components ri t. All calculations are reduced to the eÀ M I cos a i sin a: (47) h i ¼ À M multiplication of 2 2 matrices; for instance, the magnetiza- tion in the z-direction will be We can apply this idea to our problem by writing Eq. (45) as 2 2 2 rz f 1 2 v f cos h sin h cos Xt ; (56) X h it ¼ ð À j j Þ¼ ð þ Þ H~ rz cos h ry sin h ; (48) ¼À2 þ where f is given in Eq. (41) and we used the fact that ÀÁ u 2 v 2 1. where j j þj j ¼ B. Average work 2 2 B1 X B0 x B1; tan h : (49) ¼ ðÞÀ þ ¼ B0 x When computing the expectation values of quantities qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À related to the energy of the system, we can always use the Since r2 I it follows that r cos h r sin h 2 I. Thus, i ¼ ð z þ y Þ ¼ unperturbed Hamiltonian H0 in Eq. (38) instead of the full Eq. (47) applies and we get Hamiltonian H(t) in Eq. (42), which is justified since B1 B0. The energy of the system at any given time can  iHt~ Xt Xt therefore be found from Eq. (35) with A H0 eÀ I cos i rz cos h ry sin h sin : ¼ ¼ 2 þ þ 2   B0 B0f 2 ÀÁ(50) H0 r 1 v : (57) h it ¼À 2 h zit ¼À 2 Àj j ÀÁ Inserting this result into Eq. (46), we can finally write a The average work at time t is then simply the difference closed (exact) formula for the full time evolution operator between the energy at time t and the energy at time 0 U(t). After organizing the terms a bit, we get 2 2 B1 2 Xt u t v t W t fB0 v fB0 2 sin ; (58) U t ð Þ ð Þ ; (51) h i ¼ j j ¼ X 2 ð Þ¼ và t uà t  À ð Þ ð Þ 2 2 where we recall that X B0 x B . The average where ¼ ð À Þ þ 1 work therefore oscillates indefinitelyqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with frequency X=2, a Xt Xt consequence of the fact that the time evolution is unitary. ut eixt=2 cos i cos h sin ; (52) ðÞ¼ 2 þ 2 The amplitude multiplying the average work is propor-   2 tional to the initial magnetization f and to the ratio B1= 2 2 Xt B0 x B . This ratio is a Lorentzian function; it has a v t eixt=2 sin h sin : (53) ½ð À Þ þ 1Š ðÞ¼ 2 sharp peak at the resonance frequency x B0, which  ¼ becomes sharper with smaller values of B1. The maximum We see that, apart from a phase factor eixt=2, the final result possible work therefore occurs at resonance and has the depends only on X and h, which in turn depend on B0, B1, value fB0. and x [cf. Eq. (49)]. The equilibrium free energy, Eq. (10), is F T ln Z, ¼À To understand the physics behind u(t) and v(t), suppose where Z 2cosh B0=2T . Thus, the free energy of the initial the system starts in the pure state . The probability that state (at time¼ t ð0) andÞ the final state (at any arbitrary time after a time t it will be found in statejþi is U t 2. t) are the same,¼ giving DF 0. This is a consequence of the jÀi jhÀj ð Þjþij ¼

954 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 954 fact that B1 B0. According to Eq. (20), we should then violation of the second law. On the other hand, notice in Eq.  expect W 0, which is indeed observed in Eq. (58). (62) that P W 6B0 is proportional to 16f , where h i ð ¼ Þ f tanh bB0 . Thus, up-down flips are always more likely C. Characteristic function and distribution of work than¼ down-upð Þ flips, ensuring that W 0. That is to say, vio- lations to the second law are alwaysh i the exception, never the Next we turn to the characteristic function G(r) given in rule. Eq. (32), with both Hi and Hf replaced by H0 in Eq. (38). It is also possible to express Eq. (62) in terms only of the After carrying out the matrix multiplications, we get the fol- magnetization rz [Eq. (56)]. This is interesting because lowing very simple formula: h it rz t is a quantity that can be directly accessed experimen- htally.i Substituting v 2 f r =2f in Eq. (62) gives 1 1 j j ¼ð Àh zitÞ 2 2 f iB0r f iB0r Gr ut v t ðÞþ e ðÞÀ eÀ : ðÞ¼j ðÞj þj ðÞj 2 þ 2 f rz t 16f Prob W 6B0 Àh i : (63) (59) ðÞ¼¼ 2f 2  If we set r ib and recall the definition of f in Eq. (41), we This formula shows that by measuring the average magneti- find that the¼ term inside the square brackets becomes 1. zation of a system, which is a macroscopic observable, we bW 2 2 Thus, we are left with eÀ G ib u v 1, can extract the full distribution of work for a single spin-1/2 which is the Jarzynski equality,h Eq.i¼(37)ð , sinceÞ¼jDjFþj0.j ¼ particle. The Jarzynski equality for a quantum system was Expanding G(r) in a power series, as in Eq. (33)¼, we also first confirmed experimentally in Ref. 21 using magnetic res- obtain the statistical moments of the work. The first-order onance. However, in their experiment, it was necessary to term will give W exactly as in Eq. (58). Similarly, the sec- use two interacting spins, which turns out to be a conse- 2 ond moment canh bei found to be W2 B2 v . As a conse- quence of the fact that in their case Hi; Hf 0. In our case, 0 ½ Š 6¼ quence, the variance of the work ish i¼ j j since B1 B0, the initial and final Hamiltonians commute and thus we are able to relate the distribution of work to the var W W2 W 2 B2 v 2 1 f 2 v 2 : (60) properties of a single spin. ð Þ¼h iÀh i ¼ 0j j ð À j j Þ Finally, we can compute the full distribution of work D. Statistics of the work performed on a large number of P(W). The simplest way to do so is through the characteristic particles function. Recall from Eq. (30) that P(W) is the inverse So far we have studied the work performed by an external Fourier transform of G(r). To carry out the computation, we magnetic field on a single spin-1/2 particle. It is a remarkable must use the integral representation for the Dirac delta fact that with recent advances in experimental techniques, it function is now possible to experiment with just a single particle. Notwithstanding, in most situations, one is still usually faced 1 1 ia b r e ðÞÀ dr d a b : (61) with a system containing a large number of particles. The 2p ¼ ðÞÀ ðÀ1 next natural step is therefore to consider the work performed on spin-1/2 particles. For simplicity, we will assume that Using this in Eq. (59), we then find N the particles do not interact (otherwise the problem would be much more difficult). 2 2 1 f PW u d W v þ d W B0 The work corresponds to energy differences and for non- ðÞ ðÞ 2 ¼j j þj j ðÞÀ interacting systems, energy is an additive quantity. Hence, 2 1 f the total work performed during a certain process will be v À d W B0 : (62) þj j 2 ðÞþ the sum of the workW performed on each individual particle We therefore see that the work, interpreted as a random vari- W1 WN: (64) able, can take on three distinct values: W 0, B0,or B0. W¼ þÁÁÁþ ¼ þ À The physics behind this result is the following. Looking Because all spins are independent, it follows from this result back at the original Hamiltonian in Eq. (38), we see that B0 is that N W , where W is the average work in Eq. the energy spacing between the up and down states. The event (58);hWi this ¼ is ah manifestationi h ofi the fact that work, as with where W B0 corresponds to the situation where the spin energy, is an extensive quantity. ¼þ was originally up and then flipped down (“up-down flip”). The problem has thus been reduced to the sum of indepen- The change in energy in this case is B0=2 B0=2 B0. dent and identically distributed random variables, something À ðÀ Þ¼ Similarly, W B0 corresponds to a down-up flip. And, that is discussed extensively in introductory probability ¼À finally, W 0 corresponds to no flip at all. courses (e.g., Ref. 3, chapter 6). It is in problems such as this ¼ We can also find the distribution P(W) “by hand,” using that the characteristic function shows its true power. Because Eq. (28). For instance, the value W B0 corresponds to the variables are statistically independent, we have from Eq. ¼þ the up-down flip. The initial probability to have a particle up (64) that is 1 f =2 and the transition rate is U t 2 v 2. ð þ Þ 2 jhÀj ð Þjþij ¼j j ir ir W1 W irW1 irW Thus, P W B0 v 1 f =2, which agrees with Eq. e W e ð þÁÁÁþ N Þ e e N : (65) (62). Theð other¼ twoÞ¼ probabilitiesj j ð þ Þ can be computed in an iden- h i¼h i¼h iÁÁÁh i tical way. Moreover, since all spins are identical each term on the From the second law, we expect that W > 0. But our results right-hand side corresponds exactly to the characteristic show that in a down-up flip we should have W B0. function G(r) in Eq. (59). Thus, the characteristic function of ¼À Hence, P W B0 is the probability of observing a local the total work will be ð ¼À Þ W

955 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 955 ir N r e W G r : (66) Gð Þ¼h i¼ ð Þ Referring back to Eq. (59), let us introduce momentarily 2 2 the notation b0 u and b6 v 16f =2. Then Eq. (66) can be written as¼j j ¼j j ð Þ

iB0r iB0r N r b0 b e b eÀ : (67) Gð Þ¼ð þ þ þ À Þ The importance of this formula lies in its connection with the probability distribution P , established via the inverse Fourier transform in Eq. (30)ðW. IfÞ we expand the product in Eq. (67), we will get

N irB0k r Cke ; (68) Gð Þ¼k N X¼À where the factors Ck are complicated combinations of the b coefficients that result from expanding Eq. (67). When we take the inverse Fourier transform, this is mapped into

N Fig. 2. The distribution P computed from Eq. (69) for different system P C d B0k : (69) ðWÞ ðWÞ ¼ k ðWÀ Þ sizes N. The parameters used in this plot were B0 1; x 0:8; B1 0:1, k N f 0.5, and t = . The solid line in panel (d) is¼ a Gaussian¼ distribution¼ ¼À p X X with¼ the same¼ mean and variance. We therefore see that can take on values between NB0 W À and NB0. A work of NB0, for instance, corresponds to an This area of research is very active and lies at the boundary event where all spins have flipped from up to down. between many well established areas, such as non- Similarly, a work of N 1 B0 corresponds to N 1 spins ð À Þ À equilibrium statistical mechanics, quantum physics, and con- flipping. And there are N possibilities for which one of the densed matter; it also has deep connections with quantum spins did not flip. For other values the situation becomes information and quantum computing. Notwithstanding, even more complicated. unlike most frontier areas, the basic ideas can be understood The characteristic function in Eq. (67) can also be inter- using only concepts learned in standard quantum and statisti- preted as a random walk with N steps. In a single step, one cal mechanics courses, thus providing an opportunity to can think of b and b as the probabilities of taking a step to bring the student up to speed with the current research. the right or toþ the leftÀ (while b is the probability of not mov- 0 We have aimed to give an introduction that is as simple as ing). If we repeat this N times, we get a characteristic func- possible while still being useful to students. In this regard, tion of the form (67). Moreover, P plays the role of the distribution of discrete positions ofð theWÞ random walk. we would like to comment on the use of the characteristic Equation (69) is illustrated in Fig. 2 for an arbitrary choice function. This is a concept that is not necessary, per se, to of parameters, as explained in the figure caption. The impor- understand the main ideas in this paper. But it is such a use- tant point to be drawn from this analysis is that there is a cer- ful concept that we feel every student in this area should tain finite probability to observe a negative work . Since know how to work with it. Even though the use of the char- DF 0, these instances would then correspond toW local vio- acteristic function may have complicated the analysis a bit, lations¼ of the second law. However, notice also that as the we strongly believe that it was worth the effort. size N increases, the relative probability that < 0 dimin- In this paper, we have made the assumption that the ishes quickly. In fact, a more detailed analysisW shows that motion of the system is unitary. In other words, during the time evolution we have assumed that the system is not con- N Prob < DF eÀ : (70) nected to a heat reservoir. This is certainly true for many ðW Þ systems, including nuclear magnetic resonance experi- Thus, if the sample is macroscopic, it becomes extremely ments. However, in many other scenarios, it becomes unlikely to observe such a violation. This is why, in our important to consider open quantum systems, that is, sys- everyday experience, the second law is always satisfied. tems whose dynamics evolve coupled to a heat bath. The Lastly, we should mention that when N is large we can tools for working with open quantum systems are already approximate P(W) by a Gaussian distribution, as a conse- extensively used in many areas, but the subject is still in its quence of the central limit theorem.3 This distribution will infancy. We believe that in the future these systems will have mean N W and variance var Nvar W , play a particularly important role in the developments of which are quantitieshWi ¼ h wei already know fromðW Eqs.Þ ¼ (58)ðandÞ this area. (60). This distribution is plotted as a solid line in Fig. 2(d), to be compared with the exact solution. ACKNOWLEDGMENTS The authors would like to thank Professor Mario Jose de V. CONCLUSIONS   Oliveira, Professor Ivan Santos Oliveira, and Professor The goal of this paper is to introduce the student to the Andre Timpanaro for fruitful discussions. For their financial concepts of quantum thermodynamics and work fluctuations. support, the authors would like to acknowledge the Brazilian

956 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 956 funding agencies FAPESP (2014/01218-2) and CNPq 16D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. (307774/2014-7). Support from the INCT of Quantum Bustamante, “Verification of the Crooks fluctuation theorem and recovery Information is also acknowledged. of RNA folding free energies,” Nature 437, 231–234 (2005). 17S. Mukamel, “Quantum extension of the Jarzynski relation: analogy with

a) stochastic dephasing,” Phys. Rev. Lett. 90, 170604 (2003). Electronic mail: [email protected] 18P. Talkner, E. Lutz, and P. H€anggi, “Fluctuation theorems: Work is not an 1Enrico Fermi, Thermodynamics (Dover, Mineola, NY, 1956). 2 observable,” Phys. Rev. E 75, 050102 (2007). Herbert B. Callen, Thermodynamics and an Introduction to 19G. E. Crooks, “On the Jarzynski relation for dissipative quantum dynami- Thermostatistics, 2nd ed. (Wiley, New York, 1985). 3 cs,” J. Stat. Mech. 2008, P10023. Sheldon Ross, A First Course in Probability, 8th ed. (Pearson, Upper 20P. Talkner, M. Campisi, and P. H€anggi, “Fluctuation theorems in driven Saddle River, NJ, 2010). 4 open quantum systems,” J. Stat. Mech. 2009, P02025. Josiah W. Gibbs, Elementary Principles of Statistical Mechanics, reprinted 21T. B. Batalh~ao, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. ed. (Dover, Mineola, NY, 2014). 5 S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Richard P. Feyman, Statistical Mechanics: A Set of Lectures, 2nd ed. “Experimental reconstruction of work distribution and study of fluctuation (Westview Press, Boulder, CO, 1998). 6 relations in a closed quantum system,” Phys. Rev. Lett. 113, 140601 J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, (2014). “Equilibrium information from nonequilibrium measurements in an exper- 22S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H. T. imental test of Jarzynski’s equality,” Science 296, 1832–1835 (2002). Quan, and K. Kim, “Experimental test of the quantum Jarzynski equality 7D. J. Evans, E. G. D. Cohen, and G. P. Morriss, “Probability of second law with a trapped-ion system,” Nat. Phys. 11, 193–199 (2014). violations in shearing steady states,” Phys. Rev. Lett. 71, 2401–2404 (1993). 23 E. Schr€odinger, Statistical Thermodynamics (Dover, Mineola, NY, 8G. Gallavotti and E. G. D. Cohen, “Dynamical ensembles in nonequilib- 1989). rium statistical mechanics,” Phys. Rev. Lett. , 2694–2697 (1995). 74 24For completeness, we mention that it is very easy to demonstrate Eq. (19) 9C. Jarzynski, “Nonequilibrium equality for free energy differences,” Phys. in the case of infinitesimal processes as discussed in Sec. II B. But it is Rev. Lett. 78, 2690–2693 (1997). important to clarify that this case is physically of no interest at all: The 10C. Jarzynski, “Equilibrium free-energy differences from nonequilibrium importance of the Jarzynski equality is that it holds for non-equilibrium measurements: A master-equation approach,” Phys. Rev. E 56, 5018–5035 processes. For infinitesimal processes, we do not need such an equality. In (1997). any case, the demonstration is as follows. Based on Eq. (8), we can define 11G. E. Crooks, “Nonequilibrium measurements of free energy differences the random work performed in an infinitesimal process as for microscopically reversible Markovian systems,” J. Stat. Phys. 90, dW dE E k dk E k . Equation (8) can then be interpreted 1481–1487 (1998). ¼ n ¼ nð þ ÞÀ nð Þ 12G. E. Crooks, “Path-ensemble averages in systems driven far from equi- instead as the definition of dW . In other words, Pn is the probability dis- tribution of the random variableh i . We can then compute any type of librium,” Phys. Rev. E 61, 2361–2366 (2000). dW 13 bdW bdW bdW I. A. Martınez, E. Roldan, L. Dinis, D. Petrov, and R. A. Rica, “Adiabatic average we want. In particular, eÀ n eÀ Pn n eÀ 1 h i¼ ¼ processes realized with a trapped Brownian particle,” Phys. Rev. Lett. bEn k bEn k dk bdF eÀ ð Þ=Z k eÀ ð þ Þ Z kP dk = Z k P eÀ ; 114, 120601 (2015). ð ð ÞÞ ¼ Z k n ¼ ðð ð þ ÞÞ ð ð ÞÞÞ ¼ 14 ð Þ bF k dk J. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, “Heat fluctuations in a where we have used theX fact that Z k dk eÀ ð þ Þ. 25 ð þ Þ¼ nonequilibrium bath,” Phys. Rev. Lett. 106, 200602 (2011). H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems 15F. Douarche, S. Ciliberto, A. Petrosyan, and I. Rabbiosi, “An experimental (Oxford U.P., New York, 2007). test of the Jarzynski equality in a mechanical experiment,” Europhys. Lett. 26J. J. Sakurai and J. J. Napolitano, Modern Quantum Mechanics, 2nd ed. 70, 593–599 (2005). (Addison-Wesley, Boston, 2010).

957 Am. J. Phys., Vol. 84, No. 12, December 2016 Ribeiro, Landi, and Semi~ao 957