Quantum thermodynamics and work fluctuations with applications to magnetic resonance
Abstract
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 nonequilibrium 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 may be done exactly. It is shown in detail how to build the statistics of the work, both for a single particle and for a collection of noninteracting particles. We hope that this paper may serve as a tool to bring the new student up to date on the recent developments in nonequilibrium thermodynamics of quantum systems.
I Introduction
Thermodynamics was initially developed to deal with macroscopic systemsFermi1956 (); Callen1985 () and is thus based on the idea that a handful of macroscopic variables, such as volume, pressure and temperature, suffice to completely characterize a system. However, after the advent of the atomic theory, it became clear that the variables of the underlying microscopic world are constantly fluctuating due to the inherent chaos and randomness of the microworld. Statistical mechanics was thus developed as a theory connecting these microscopic fluctuations with the emergent macroscopic variables. Since one usually deals with a large number of particles, the relative fluctuations become negligible, so that thermodynamic measurements usually coincide very well with expectation values of the microscopic fluctuating quantities (which is a consequence of the law of large numbersRoss2010 ()).
Equilibrium statistical mechanics is now a very well established and successful theory. Its main result is the Gibbs formula for the canonical ensembleGibbs (); Feyman1998 () which provides a fundamental bridge between microscopic physics and thermodynamics for any equilibrium situation. Conversely, far less is known about nonequilibrium processes. The reason is that in this case the handful of parameters used in thermodynamics no longer suffice, forcing one to know the full dynamics of the system; i.e., one must study Newton’s or Schrödinger’s equation for all constituent particles, thus making the problem much more difficult.
These difficulties led researchers to look for nonequilibrium processes in the realm of small systems. On the one hand, in these systems the dynamics are somewhat easier to describe since there are fewer particles. But on the other hand, fluctuations become important and must therefore be included in the description. Substantial progress in experimental methods also made possible for the first time to experiment with small systems, and therefore also contributed to this shift.
The random fluctuations present in small systems also affect thermodynamic quantities such as work and heat. A beautiful example is the use of optical tweezers to fold and unfold individual RNA molecules.Liphardt2002 () The molecules are immersed in water and therefore subject to the incessant fluctuations of brownian motion. Thus, the amount of work required to fold a molecule will be different each time we repeat the experiment and should therefore be interpreted as a random variable. The same is true of the work that is extracted from the molecule when it is unfolded. In some realizations, it is even possible to extract work without any changes in the thermodynamic state of the system, something which would contradict the second law of thermodynamics.
This introduces the idea that fluctuations in small systems could lead to local violations of the second law. These violations were first observed in fluid simulations in the beginning of the 1990s by Evans, Cohen, Gallavotti and collaborators. Evans1993 (); Gallavotti1995b () Afterwards, in 1997 and 1998 came two very important (and intimately related) breakthroughs by JarzynskiJarzynski1997 (); Jarzynski1997a () and Crooks.Crooks1998 (); Crooks2000 () They showed that the work performed in a nonequilibrium (e.g. fast) process, when interpreted as a random variable, obeyed a set of exact relations that touched deeply into the nature of irreversibility and the second law. Nowadays these relations go by the generic name of fluctuation theorems (not to be confused with the fluctuationdissipation theorem) and can be interpreted as generalizations of the second law for fluctuating systems. These theoretical findings were later confirmed by experiments in diverse areas, such as trapped brownian particles,Martinez2015 (); GomezSolano2011 () mechanical oscillatorsDouarche2005 () and biological systems. Collin2005 (); Liphardt2002 ()
With time, researchers began to look for similar results in quantum systems, both for unitaryMukamel2003 (); Talkner2007 () as well as for openCrooks2008 (); Talkner2009 () quantum dynamics. Here, in addition to the thermal fluctuations, one also has intrinsically quantum fluctuations, leading to a much richer platform to work with. One may therefore ask to what extent will these quantum fluctuations affect thermodynamic quantities such as heat and work. This research was also motivated by the remarkable progress of the past decade in the experimental control of atoms and photons, particularly in areas such as magnetic resonance, ultracold atoms and quantum optics. In fact, the first experimental verifications of the fluctuation theorems for quantum systems were done only recently, using nuclear magnetic resonanceBatalhao2014 () and trapped ions. An2014 ()
It is a remarkable achievement of modern day physics that we are now able to test the thermodynamic properties of systems containing only a handful of particles. But despite being an active area of research, the basic concepts in this field can be understood using only undergraduate quantum and statistical mechanics. The purpose of this paper is to provide an introduction to some of these concepts in the realm of quantum systems. Our main goal is to introduce the reader to the idea that work may be treated as a random variable. We then show how to construct all of its statistical properties, such as the corresponding probability distribution of work or the characteristic function. We discuss how these quantities may be used to derive Jarzynski’s equality, Jarzynski1997 (); Jarzynski1997a () thence serving as a quantum mechanical derivation of the second law. To illustrate these new ideas, we apply our results to the problem of magnetic resonance, an elegant textbook example that can be worked out analytically.
Ii Work in thermodynamics and statistical mechanics
ii.1 Thermodynamic description
Consider any physical system described by a certain Hamiltonian . When this system is placed in contact with a heat reservoir, energy may flow between the system and the bath. This change in energy is called heat. But the energy of a system may also change by means of an external agent, which manually changes some parameter in the Hamiltonian of the system. These types of changes are called work.
Heat and work are not properties of the system. Rather, they are the outcomes of processes which alter the state of the system. If during a certain interval of time an amount of heat entered the system and a work was performed on the system, energy conservation implies that the total energy of the system must have changed by
(1) 
which is the first law of thermodynamics. When we say the external agent performed work on the system. Conversely, when we say the system performed work on the external agent (for instance, when a gas expands a piston).
The process of performing work on a system may be described microscopically in a very general way through the change of some parameter 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 exactly under what conditions are the changes being made and with what timedependence is changing. We usually assume that the process lasts between a time and a time , during which varies in some predefined way from an initial value to a final value .
Overall, describing an arbitrarily fast process can be a very difficult task since it requires detailed information about the dynamics of the system and how it is coupled to the bath. Instead, thermodynamics usually focuses on quasistatic processes, in which changes very slowly in order to ensure that throughout the process the system is always in thermal equilibrium. Quasistatic processes have the advantage of being atemporal: we do not need to specify the function , but merely its initial and final value. In this paper our focus will be on nonequilibrium processes. But before we can get there, we must first have a solid understanding of quasistatic processes.
Perhaps the most important example of a quasistatic process is the isothermal process, in which the temperature of the system is kept constant throughout the protocol. Since work usually changes the temperature of an isolated system, to ensure a constant temperature the system must remain coupled to a heat reservoir kept at a constant temperature . It is also important to note that an isothermal process must necessarily be quasistatic. For, if the process is not quasistatic, the temperature will not remain constant nor homogeneous. In fact, intensive quantities such as temperature and pressure are only defined in thermal equilibrium, so any process where these quantities are kept fixed must be quasistatic. Thus, henceforth whenever we speak of an isothermal process, it will already be implied that it is quasistatic.
According to Eq. (1), in an isothermal process not all work will be converted into useful energy for the system, since part is consumed as heat. The remainder is called the free energy:
(2) 
where (cf. chapter 2 of Ref. Schrodinger, or Ref. Fermi1956, ). The energy is “free” because it is the part of that may be used to perform work.
Now suppose we try to repeat the process above, but we do so too quickly, so that it cannot be considered quasistatic. The initial state is still , but the final state will not be . Instead, the final state will be something complicated which depends exactly on how the process was performed (see Fig. 1). Notwithstanding, since the system is coupled to a bath, if we leave it alone after the protocol is over, it will eventually relax to the state . Thus, overall, a certain amount of work was performed to take the system from to . But this work is not equal to , since Eq. (2) holds only for quasistatic processes.
Instead, according to the second law of thermodynamics, the work done in the nonequilibrium process must always be larger or equal to :
(3) 
with the equality holding only for the isothermal (i.e. quasistatic) process. This is a very important result. It means that if we wish to change the free energy of a system by , the minimum amount of work we need to perform is and will be accomplished in an isothermal (quasistatic) process. Any other protocol will require more work. The difference , known as the irreversible work, therefore represents the extra work that had to be done due to the particular choice of protocol.
The inequality (3) can be interpreted as a direct consequence of the Kelvin statement of the second law: “A transformation whose only final result is to transform into work, heat extracted from a source which is at the same temperature throughout, is impossible”.Fermi1956 () The key part of this statement is the expression “only final result”. It means that it is impossible to extract work from a bath at a fixed temperature, without changing anything else (such as e.g. the thermodynamic state of the system). Extracting work while changing the thermodynamic state of a system is not a problem. For instance, we can use a heat source to make a gas expand and thence extract work from the expansion. But by the end of the process we will have altered the state of the gas, so the extraction of work was not the only outcome. What the second law says is that it is impossible to extract work from a single source at a fixed temperature and keep the state of the system intact.
To make the connection with Eq. (3), consider a process divided in 3 steps, represented by the three lines in Fig. 1. In the first step we perform a certain amount of work in a nonequilibrium process. In the second, we perform no work and allow the system to relax from the nonequilibrium state to . Finally, in the third process (represented by a dotted line in Fig. 1), we go back quasistatically from to . The amount of work required for the return journey is because we assume that this part was quasistatic. In the end, we are back to the original state, having performed a total work . According to the second law, this total work cannot be negative, because that would mean we would have extracted work from a reservoir at a fixed temperature, without any changes in the state of the system. Consequently, , which is Eq. (3).
ii.2 Isothermal processes in equilibrium statistical mechanics
Let us now analyze the isothermal process quantitatively. Since it is a quasistatic process, we may always decompose it into a series of infinitesimal processes, where is changed slightly to . The full process is then simply a succession of these small steps.
We assume that initially we had a system with Hamiltonian in thermal equilibrium with a heat bath at a temperature . According to statistical mechanics, its state is then given by the Gibbs density operator
(4) 
where is the partition function and (we choose Boltzmann’s constant equal to unity). We may also write this expression in terms of the energy eigenvalues and eigenvectors . The probability of finding the system in the state is therefore,
(5) 
Moreover, the internal (average) energy of the system may be written as
(6) 
When we change to , both and will change. Hence, will change by:
(7) 
This separation of in two terms allows for an interesting physical interpretation.Crooks1998 ()
The change in is infinitesimal and instantaneous. So immediately after this change, the system has not yet responded to it. This corresponds to the first term: it is the average of the energy change over the old (unperturbed) probabilities . In the second term, the energy is fixed and the probabilities change. We interpret this as the second step, where the system adjusts itself with the bath in order to return to equilibrium. Thus, each infinitesimal process may be separated in two parts. The first part is the work performed and the second is the heat exchanged as the system relaxes to equilibrium.
This motivates us to define:
so that Eq. (7) may be written as (we use instead of simply to emphasize that heat and work are not exact differentials.Callen1985 ()) In order to better understand the physical meaning of these formulas, we will now explore them in more detail.
We start with and show that it is related to the free energy of the system, defined as
(9) 
To see this, first note that since the temperature is fixed, . Since , one may then readily show that , which is precisely Eq. (II.2). Thus
(10) 
From this result, Eq. (2) is recovered by integrating over the several infinitesimal steps.
Next we turn to in Eq. (II.2). Instead of trying to manipulate , we can use the following neat trick: invert Eq. (5) to write . If we substitute this in Eq. (II.2) we get two terms, one proportional to and the other to . The term with will be
(11) 
since and . Thus, we are left only with
(12) 
But now note that, by the chain rule,
(13) 
and the last term is also zero for the same reason as above. Hence, we conclude that
(14) 
We see that even though is not a function of state, it is related to the variation of a quantity which is a function of state. We define the entropy as
(15) 
so that we finally arrive at
(16) 
This relation, we emphasize, holds only for infinitesimal processes. For finite and irreversible processes, there may be additional contributions to the change in entropy.
We therefore see that it is possible to give microscopic definitions to thermodynamic quantities such as heat and work. Moreover, it is possible to relate them to functions of state which can be constructed from the initial density matrix . While these thermodynamic quantities may be defined independently of statistical mechanics, we believe that this microscopic description helps to clarify their physical meanings.
Iii Work as a random variable
In 1997 Jarzynksi discovered that great insight into the properties of nonequilibrium processes could be gained by treating work as a random variable.Jarzynski1997 (); Jarzynski1997a () The idea is as follows. Consider the process where a movable piston is used to compress a gas contained in a cylinder. Since the molecules of the gas are moving chaotically and hitting the walls of the piston in all sorts of different ways, each time we press the piston, the gas molecules will exert back on us a different force. This means that the work needed to achieve a given compression will change each time we repeat the experiment. Hence, work may be treated as a random variable.
Of course, one may object that in most systems these fluctuations are negligibly small. But that does not stop us from interpreting in this way. And, as we will soon see, this does bring several advantages. On the other hand, when dealing with microscopic systems, this interpretation becomes essential since fluctuations become significant. A famous example is the work performed when folding RNA molecules,Liphardt2002 (); Collin2005 () already discussed in Sec. I.
In addition to thermal fluctuations, some microscopic systems also have a strong contribution from quantum fluctuations. These fluctuations are related to the fact that in order to access the amount of work performed in a system, one must measure its energy and therefore collapse the wavefunction. As is known from quantum mechanics, when this is done the system may tend to different states with different probabilities [cf. Eq. (5)]. Thus, in quantum systems both thermal and quantum fluctuations must be taken into consideration.
iii.1 The Jarzynski equality
Usually, our knowledge of nonequilibrium processes is restricted only to inequalities such as Eq. (3). The contribution of JarzynskiJarzynski1997 (); Jarzynski1997a () was to show that by interpreting as a random variable, one may obtain an equality, even for a process performed arbitrarily far from equilibrium.
Consider several realizations of a nonequilibrium process, such as that described by the solid line in Fig. 1. At each realization we always prepare the system in the same initial state. We then execute the protocol and measure the total work performed. After repeating this process many times, we may construct the probability distribution of the work, . From this probability any average may be computed. For instance the average work will be
(17) 
Or we may study the average of other quantities such as and so on.
Jarzynski’s main result was to show that the statistical average of should satisfy:Jarzynski1997 (); Jarzynski1997a ()
(18) 
where , as described in Fig. 1. This result is nothing short of remarkable. It holds for a process performed arbitrarily far from equilibrium. And it is an equality, which is a much stronger statement than the inequalities we are used to in thermodynamics. The appearance of in Eq. (18) is also quite surprising, since this is not the final state of the process. It would only be the final state if the process were quasistatic. Instead, as indicated in Fig. 1, the final state is a nonequilibrium state which may differ substantially from . The appearance of therefore reflects the state that the system wants to go to, but cannot do so because the process is not sufficiently slow.
It is possible to show that the inequality (3) [ie, ] is contained within the Jarzynski equality. This can be accomplished using Jensen’s inequality (see chapter 8 of Ref. Ross2010, ), which states that Combining this with Eq. (18) then gives
(19) 
We therefore see that, when we treat work as a random variable, the old results from thermodynamics are recovered for the average work.
In macroscopic systems, by the law of large numbers (Ref. Ross2010, , chapter 8), individual measurements are usually very close to the average, so the distinction between the average work and a single stochastic realization is immaterial. But for microscopic systems this is usually not true. In fact, although , the individual realizations may very well be smaller than . This therefore corresponds to local violations of the second law. For large systems, these local violations become extremely rare. But for small systems, they may be measured experimentally.Liphardt2002 (); Collin2005 (); Batalhao2014 () If we know the distribution , then the probability of a local violation of the second law can easily be found as
(20) 
iii.2 Nonequilibrium unitary dynamics
We will now consider in detail how to describe work in a nonequilibrium process. As mentioned above, this requires detailed knowledge of the dynamics of the system and how it is coupled to the heat bath. The last part is by far the most difficult since it requires concepts from open quantum systems, such as Lindblad dynamics or quantum Langevin equations.Breuer2007 () We will therefore make an important simplification and assume that the coupling to the heat bath is so weak that, during the protocol, no heat is exchanged. This situation is actually encountered very often in experiments since many systems are only weakly coupled to the bath. It also simplifies considerably the description of the problem since it allows us to use Schrödinger’s equation to describe the dynamics of the system.
We shall consider the protocol described in the previous section. Initially the system had a Hamiltonian and was in thermal equilibrium with a bath at a temperature . The initial state of the system is then given by the Gibbs thermal density matrix in Eq. (4). As a first step, we measure the energy of the system. If we let and denote the eigenvalues and eigenvectors of , then the energy will be obtained with probability [cf. Eq. (5)].
Immediately after this measurement we initiate the protocol, changing from to according to some predefined function . If we assume that during this process the contact with the bath was very weak, we may assume that the state of the system will evolve according to:
(21) 
where is the unitary timeevolution operator, which satisfies Schrödinger’s equation (with ):
(22) 
For a derivation of this equation, see Ref. Sakurai2010, , chapter 2.
At the end of the process we measure the energy of the system once again. The Hamiltonian is now and therefore may have completely different energy levels and eigenvectors . The probability that we now measure an energy is
(23) 
which may be interpreted as the conditional probability that a system initially in will be found in after a time .
Since no heat may be exchanged with the environment, any change in the energy must necessarily be attributed to the work performed by the external agent. The energy obtained in the first measurement was and the energy obtained in the second measurement was . We then define the work performed by the external agent as
(24) 
Both and are fluctuating quantities which change during each realization of the experiment. The first energy is random due to thermal fluctuations and the second, , is random due to quantum fluctuations. Consequently, will also be a random variable, encompassing both thermal and quantum fluctuations.
iii.3 Distribution of work and characteristic function
We will now obtain an expression for the probability distribution obtained by repeating the above protocol several times. This can be accomplished by noting that we are dealing here with a 2step measurement processes. From probability theory, if and are two events, the total probability that both events occur may be written as
(25) 
where is the probability that occurs and is the conditional probability that occurs given that has occurred. In our context, is given by Eq. (23) whereas is simply the initial probabilities . Hence, the probability that both events have occurred is
(26) 
Since we are interested in the work performed, we may then write
(27) 
where is Dirac’s delta function. This formula is perhaps best explained in words: we sum over all allowed events, weighted by their probabilities, and catalogue the terms according to the values of .
Albeit exact, Eq. (27) is not very convenient to work with. In most systems there are a large number of allowed energy levels and therefore an even larger number of allowed energy differences . It is much more convenient to work with the characteristic function, defined as the Fourier transform of the original distribution
(28) 
From we may recover the original distribution from the inverse Fourier transform:
(29) 
Since and are Fourier transforms of each other, they both contain the same amount of information.
With the help of Eq. (27) we may write
Hence, we conclude that
(30) 
The characteristic function does not have a particularly important physical meaning. But, since it is written as the trace of a product of operators, it is usually much more convenient to work with than . In many aspects, it plays a role somewhat similar to the partition function in equilibrium statistical mechanics. One usually does not question the physical meaning of , but rather uses it as a convenient quantity from which observables such as the energy and entropy may be extracted.
From we may also readily extract the statistical moments of . To see this, we expand Eq. (28) in a Taylor series in to find
(31) 
Hence will be proportional to the term of order in the expansion.
On the other hand, we may obtain quantum mechanical formulas for the moments by doing a similar expansion in Eq. (30). The average work, for instance, is found to be
(32) 
where, given any operator , we define
(33) 
as the expectation value of this operator at time , a result which follows directly from the fact that the state of the system at time is . We therefore see that is simply the difference between the average energy at time and the average energy at time , which is intuitive. One may continue with the expansion of eq. (30) to obtain formulas for higher order moments. Unfortunately, they do not acquire such a simple form.
The characteristic function may also be sued to demonstrate the Jarzynski equality (18) since, based on the definition (28), we should have . But from Eqs. (4) and (30) we find that
(34) 
Since , this yields Eq. (18):
(35) 
Notice that no assumptions have been made as to the speed of the process, so we conclude that the Jarzynski equality holds for a process arbitrarily far from equilibrium.
Iv Magnetic resonance
iv.1 Statement of the problem
We will now apply the concepts of the previous section to a magnetic resonance experiment. The typical setup consists of a sample of noninteracting spin 1/2 particles (electrons, nucleons, etc.) placed under a strong static magnetic field in the direction. The Hamiltonian of this interaction can be described using the Pauli matrix asSakurai2010 ()
(36) 
For simplicity, we choose to measure the field in energy units. Moreover, since , the quantity also represents the characteristic precession frequency of the spin. The eigenvalues of are and , corresponding to spin up and spin down respectively.
When the spin is coupled to a heat bath at a temperature , its state will be given by the thermal density matrix in Eq. (4). The Hamiltonian is already diagonal in the usual basis which diagonalizes . Whenever this is true, the corresponding matrix exponential may be easily computed by exponentiating the eigenvalues
The partition function is the trace of this matrix, , and reads . The thermal density matrix may then be written in a convenient way as
(37) 
where is the equilibrium magnetization of the system
(38) 
corresponding to the paramagnetic response of a spin 1/2 particle.
The work protocol is implemented by applying a very small field of amplitude rotating in the plane with frequency . That is, the work parameter is described here by the field . Typically, T and T so we may always take as a good approximation that . The total Hamiltonian of the system now becomes
(39) 
The oscillating field plays the role of a perturbation which, albeit extremely weak, may nonetheless promote transitions between the up and down spin states. The transitions will be the most frequent at the resonance condition, which corresponds to ; ie, when the driving frequency is the same as the natural oscillation frequency .
To make progress, we must now compute the time evolution operator defined in Eq. (22). Usually, accomplishing this for a timedependent Hamiltonian is a very complicated task. Luckily, for the particular choice of in Eq. (39), this turns out to be quite simple. The first step is to define a new operator from the relation
(40) 
Substituting this in Eq. (22) one finds that must obey a modified Schrödinger equation
(41) 
where
(42) 
We therefore see that evolves according to a timeindependent Hamiltonian. Consequently, the solution of Eq. (41) will be simply and the full time evolution operator will be given by
(43) 
It is important to notice that, since and do not commute, we cannot write as .
It is also useful to have in hand an explicit formula for . This can be accomplished using the following trick. Let be an arbitrary matrix, which is such that (where is the identity matrix). Then, if is an arbitrary constant, a direct power series expansion of yields
(44) 
We can apply this idea to our problem by writing Eq. (42) as
(45) 
where
(46) 
Since , it follows that . Hence, Eq. (44) applies and we get
(47) 
Inserting this result in Eq. (43) we can finally write a closed (exact) formula for the full time evolution operator . After organizing the terms a bit, we get
(48) 
where
(49)  
(50) 
We see that, apart from a phase factor , the final result depends only on and , which in turn depend on , and [cf. Eq. (46)].
To understand the physics behind and , suppose the system starts in the pure state . The probability that after a time it will be found in state is . But looking at Eq. (48) we see that . Therefore, represents the transition probability per unit time for a jump to occur. Moreover, the unitarity condition implies that , so is the probability that no transition occurs.
From Eq. (50) we also see that . This therefore attributes a physical meaning to the angle [defined in Eq. (46)] as representing the transition probability. It reaches a maximum precisely at resonance (), as we intuitively expect. In fact, at resonance Eqs. (49) and (50) simplify to
(51)  
(52) 
Now that we have the initial density matrix [Eq. (37)] and the time evolution operator [Eq. (48)], we may compute the evolution of any observable we wish using Eq. (33). For instance, we could compute the evolution of the magnetization components . All calculations are reduced to the multiplication of matrices. For instance, the magnetization in the direction will be
(53) 
where is given by Eq. (38) and we used the fact that .
iv.2 Average work
When computing the expectation values of quantities related to the energy of the system, we may always use the unperturbed Hamiltonian in Eq. (36), instead of the full Hamiltonian in Eq. (39), which is justified since . The energy of the system at any given time may therefore be found from Eq. (33) with . It reads
The average work at time is then simple the difference between the energy at time and the energy at time :
(54) 
where we recall that . The average work therefore oscillates indefinitely with frequency . This is a consequence of the fact that the time evolution is unitary.
The amplitude multiplying the average work is proportional to the initial magnetization and to the ratio . This is known as a Lorentzian function. It presents a sharp peak at the resonance frequency , which becomes sharper the smaller is the value of . The maximum possible work therefore occurs at resonance and has the value .
iv.3 Characteristic function and distribution of work
Next we turn to the characteristic function given in Eq. (30), with both and replaced by in Eq. (36). After carrying out the matrix multiplications, we get the following very simple formula:
(55) 
If we set and recall the definition of in Eq. (38), we find that the term inside brackets becomes 1. Thus, we are left with . This is the Jarzynski equality, Eq. (35), since .
Expanding in a power series, as in Eq. (31), we also obtain the statistical moments of the work. The first order term will give exactly as in Eq. (54). Similarly, the second moment can easily be found to be . As a consequence, the variance of the work is
(56) 
Finally, we may compute the full distribution of work . The simplest way to do so is through the characteristic function. Recall from Eq. (29) that is the inverse Fourier transform of . To carry out the computation, we must use the following representation for the Dirac delta function:
(57) 
Using this in Eq. (55) we then find
We therefore see that the work, interpreted as a random variable, may take on three distinct values: or .
The physics behind this result is the following. Looking back at the original Hamiltonian in Eq. (36), we see that is the energy spacing between the up and down states. The event where corresponds to the situation where the spin was originally up and then flipped down (“updown flip”). The change in energy in this case is . Similarly, corresponds to a downup flip. And, finally, corresponds to no flip at all.
We may also find the distribution “by hand”, using Eq. (27). For instance, the value corresponds to the updown flip. The initial probability to have a particle up is and the transition rate is . Thus, , which agrees with Eq. (IV.3). The other two probabilities may be computed in an identical way.
From the second law we expect that . But our results show that in a downup flip we should have . Hence, is the probability of observing a local violation of the second law. On the other, hand, notice in Eq. (IV.3) that is proportional to where . Thus, updown flips are always more likely than downup flips, ensuring that . That is to say, violations to the second law are always the exception, never the rule.
It is also possible to express Eq. (IV.3) in terms only of the magnetization , Eq. (53). This is interesting because is a quantity that can be directly accessed experimentally. Substituting in Eq. (IV.3) gives
(59) 
This formula shows that by measuring the average magnetization of a system, which is a macroscopic observable, we may extract the full distribution of work for a single spin 1/2 particle. The Jarzynski equality for a quantum system was first confirmed experimentally in Ref. Batalhao2014, also using magnetic resonance. However, in their experiment it was necessary to use two interacting spins, which turns out to be a consequence of the fact that in their case . In our case, since , the initial and final Hamiltonians commute, thus we are able to relate the distribution of work to the properties of a single spin.
iv.4 Statistics of the work performed on a large number of particles
So far we have studied the work performed by an external magnetic field on a single spin 1/2 particle. It is a remarkable fact that with recent advances in experimental techniques it is now possible to experiment with just a single particle. Notwithstanding, in most situations one is still usually faced with a system containing a large number of particles. The next natural step is therefore to consider the work performed on spin 1/2 particles. For simplicity, we will assume that the particles do not interact (otherwise the problem would be much more difficult to deal with).
The work corresponds to energy differences and for noninteracting systems, energy is an additive quantity. Hence, the total work performed during a certain process will be the sum of the work performed on each individual particle:
(60) 
Since all spins are independent, it follows from this result that , where is the average work in Eq. (54). This is a manifestation of the fact that work, as with energy, is an extensive quantity.
The problem has thus been reduced to the sum of independent and identically distributed random variables, something which is discussed extensively in introductory probability courses (Ref. Ross2010, , chapter 6). It is in problems such as this that the characteristic function shows its true power. Since the variables are statistically independent, we have from Eq. (60) that
(61) 
Moreover, since all spins are identical, each term on the righthand side corresponds exactly to the characteristic function in Eq. (55). Thus, the characteristic function of the total work will be
(62) 
Referring back to Eq. (55), let us introduce momentarily the notation and . Then Eq. (62) may be written as
(63) 
The importance of this formula lies in its connection with the probability distribution , established via the inverse Fourier transform [Eq. (29)]. If we expand the product in Eq. (63) we will get
(64) 
where are complicated combinations of the coefficients, which result from expanding Eq. (63). When we take the inverse Fourier transform this is mapped into
(65) 
We therefore see that may take on values between and . A work of for instance, corresponds to an event where all spins have flipped from up to down. Similarly, a work of corresponds to spins flipping. And there are possibilities for which of the spins did not flip. For other values the situation becomes even more complicated.
The characteristic function (63) may also be interpreted as a random walk with steps. In a single step, one may think of and as the probability of taking a step to the right or to the left (while is the probability of not moving). If we repeat this times, we get a characteristic function of the form (63). Moreover, plays the role of the distribution of discrete positions of the random walk.
Eq. (65) is illustrated in Fig. 2 for an arbitrary choice of parameters, as explained in the figure caption. The important point to be drawn from this analysis is that there is a certain finite probability to observe a negative work . Since , these would then correspond to local violations of the second law. However, notice also that as the size increases, the relative probability that diminishes quickly. In fact, a more detailed analysis shows that
(66) 
Thus, if the sample is macroscopic, it becomes extremely unlikely to observe such a violation. This is why, in our everyday experience, the second law is always satisfied.
Lastly, we should mention that when is large we may approximate by a Gaussian distribution, as a consequence of the central limit theorem.Ross2010 () This distribution will have mean and variance , which are quantities we already know [Eqs. (54) and (56)]. This distribution is plotted as a solid line in Fig. 2(d), to be compared with the exact solution.
V Conclusions
The goal of this paper was to introduce the student to the concepts of quantum thermodynamics and work fluctuations. This area of research is very active and lies at the boundary between many well established areas, such as nonequilibrium statistical mechanics, quantum physics and condensed matter. It also has deep connections with quantum information and quantum computing. Notwithstanding, unlike most frontier areas, its basic ideas can be understood using only concepts learned in undergraduate quantum and statistical mechanics courses. This makes for a unique opportunity to put the student up to speed with the current research, which was the goal of this paper.
In this paper we have aimed to give an introduction which was as simple as possible but, at the same time, useful for students entering this area and for a general audience. We would like to take as an example the characteristic function. This is a concept which is not necessary, per se, to understand the ideas involved in this area. But it is such a useful concept that, we feel, every student in this area should know how to work with it. Hence, even though the use of the characteristic function may have complicated the analysis a bit, we strongly believe that it was worth it.
In this paper we have made the assumption that the motion of the system is unitary. That is, that during the time evolution we may assume that the system is not connected to a heat reservoir. This is certainly true for many systems, including nuclear magnetic resonance experiments. However, in many other scenarios it becomes important to consider also open quantum systems. That is, systems whose dynamics evolve coupled to a heat bath. The tools of open quantum systems are already extensively used in many areas, but its role in quantum thermodynamics is still in its infancy. We believe that in the future it will play a particularly important role in the developments of this area.
Acknowledgements.
The authors would like to thank Prof. Mário José de Oliveira, Prof. Ivan Santos Oliveira and Prof. André Timpanaro for fruitful discussions. For their financial support, the authors would like to acknowledge the Brazilian funding agencies FAPESP (2014/012182) and CNPq (307774/20147). Support from the INCT of Quantum Information is also acknowledged.References
 (1) Enrico Fermi, Thermodynamics. (Dover Publications Inc., 1956).
 (2) Herbert B. Callen, Thermodynamics and an introduction to Thermostatistics. (Wiley, 2nd ed., 1985).
 (3) Sheldon Ross, A first course in probability. (Upper Saddle River: Pearson, 8th ed., 2010).
 (4) Josiah W. Gibbs, Elementary principles of statistical mechanics. (Dover, reprint ed., 2014).
 (5) Richard P. Feyman, Statistical Mechanics: a set of lectures. (Westview press, 2nd ed., 1998).
 (6) J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, “Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality.,” Science, 296, 1832–1835 (2002).
 (7) D. J. Evans, E. G. D. Cohen, and G. P. Morriss, “Probability of second law violations in shearing steady states,” Physical Review Letters, 71, 2401–2404 (1993).
 (8) G. Gallavotti and E. G. D. Cohen, “Dynamical ensembles in nonequilibrium statistical mechanics,” Physical Review Letters, 74, 2694–2697 (1995).
 (9) C. Jarzynski, “Nonequilibrium Equality for Free Energy Differences,” Physical Review Letters, 78, 2690–2693 (1997).
 (10) C. Jarzynski, “Equilibrium freeenergy differences from nonequilibrium measurements: A masterequation approach,” Physical Review E, 56, 5018–5035 (1997).
 (11) G. E. Crooks, “Nonequilibrium Measurements of Free Energy Differences for Microscopically Reversible Markovian Systems,” Journal of Statistical Physics, 90, 1481–1487 (1998).
 (12) G. E. Crooks, “Pathensemble averages in systems driven far from equilibrium,” Physical Review E, 61, 2361–2366 (2000).
 (13) I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, and R. A. Rica, “Adiabatic Processes Realized with a Trapped Brownian Particle,” Physical Review Letters, 114, 120601 (2015).
 (14) J. R. GomezSolano, A. Petrosyan, and S. Ciliberto, “Heat Fluctuations in a Nonequilibrium Bath,” Physical Review Letters, 106, 200602 (2011).
 (15) F. Douarche, S. Ciliberto, a. Petrosyan, and I. Rabbiosi, “An experimental test of the Jarzynski equality in a mechanical experiment,” Europhysics Letters, 70, 593–599 (2005).
 (16) D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature, 437, 231–234 (2005).
 (17) S. Mukamel, “Quantum extension of the Jarzynski relation: analogy with stochastic dephasing,” Physical Review Letters, 90, 170604 (2003).
 (18) P. Talkner, E. Lutz, and P. Hänggi, “Fluctuation theorems: Work is not an observable,” Physical Review E, 75, 050102 (2007).
 (19) G. E. Crooks, “On the Jarzynski relation for dissipative quantum dynamics,” Journal of Statistical Mechanics: Theory and Experiment, 2008, P10023 (2008).
 (20) P. Talkner, M. Campisi, and P. Hänggi, “Fluctuation theorems in driven open quantum systems,” Journal of Statistical Mechanics: Theory and Experiment, 2009, P02025 (2009).
 (21) T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, “Experimental Reconstruction of Work Distribution and Study of Fluctuation Relations in a Closed Quantum System,” Physical Review Letters, 113, 140601 (2014).
 (22) S. An, J.N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.Q. Yin, H. T. Quan, and K. Kim, “Experimental test of the quantum Jarzynski equality with a trappedion system,” Nature Physics, 11, 193–199 (2014).
 (23) E. Schrödinger, Statistical thermodynamics, (Dover Publications Inc., 1989).

(24)
For completeness, we mention that it is very easy to demonstrate Eq. (18) in the case of infinitesimal processes discussed in Sec. II.2. But it is important to clarify that this case is physically of no interest at all: the importance of the Jarzynski equality is that it holds for nonequilibrium processes. For infinitesimal processes we do not need such an equality.
In any case, the demonstration is as follows.
Based on Eq. (II.2) we may define the random work performed in an infinitesimal process as .
Eq. (II.2) may then be interpreted instead as the definition of .
In other words, is the probability distribution of the random variable .
We may then compute any type of average we want. In particular,
 (25) H.P. Breuer and F. Petruccione, The Theory of Open Quantum Systems. (Oxford University Press, USA, 2007).
 (26) J. J. Sakurai and J. J. Napolitano, Modern Quantum Mechanics. (AddisonWesley, 2nd ed., 2010).