Quantum Otto cycle with inner friction: finite time and disorder effects
Abstract
The concept of inner friction, by which a quantum heat engine is unable to follow adiabatically its strokes and thus dissipates useful energy, is illustrated in an exact physical model where the working substance consists of an ensemble of misaligned spins interacting with a magnetic field and performing the Otto cycle. The effect of this static disorder under a finitetime cycle gives a new perspective of the concept of inner friction under realistic settings. We investigate the efficiency and power of this engine and relate its performance to the amount of friction from misalignment and to the temperature difference between heat baths. Finally we propose an alternative experimental implementation of the cycle where the spin is encoded in the degree of polarization of photons.

March 2015
1 Introduction
The recent boosting interest in the study of the quantum counterpart of classical wellknown heat engines such as Otto, Carnot, Stirling and Szilard ones [1, 2, 3, 4, 5, 6, 7, 8] has been motivated both by the need of a fundamental understanding of the limits imposed by quantum mechanics on the thermodynamic performances of small devices (in terms of both efficiency and power output), and by the growing experimental ability to control various types of quantum systems with a high degree of accuracy. There have been, indeed, many proposals aimed at implementing thermodynamic transformations and cycles with many different quantum working substances, ranging from trapped ions to magnetic materials [9], with the prospect of building quantum heat engines, exploring the abilities and limitations of quantum machines in converting heat into work, and, on more general ground, to build a self contained description of thermodynamics in the quantum regime. As a specific example, an interesting proposal in this respect have been made for implementing a nanoheat engine with a single trapped ion, performing a quantum Otto cycle [10]. Besides its specific applications, this is an important example as the quantum Otto cycle constitutes a useful test ground to study irreversibility in the quantum realm.
Indeed, the cycle consists of two isochoric thermalization branches (with a fixed system Hamiltonian) and two isentropic branches in which the system is detached from the thermal baths and its evolution is generated by a parametric timedependent Hamiltonian. Every practical realization of these latter transformations has to face the general problem of understanding and describing the (unwanted) irreversible entropy production which can occur in nonideal, finite time quantum parametric processes. This general problem has been variously analyzed through the use of fluctuation relations [11, 12], and has attracted a lot of attention in recent years [13, 14].
In this paper, we explicitly address the study of the Otto cycle by focusing on the finite time case and discussing the implications of finite time transformations as opposed to ideal infinitely lasting ones. In this respect, in a series of papers, Kosloff and Feldmann [15, 16, 17] introduced the concept of intrinsic/inner friction, whereby the engine is never able to accomplish a frictionless adiabatic transformation and thus looses power. This concept has been then extended and applied to various contexts [12, 18, 19].
Inner friction is a fully quantum phenomenon, whose consequences are similar to those of the mechanical friction occurring when displacing a piston in compressing/expanding a gas in a classical thermodynamic setting. Its origin, however,, is completely different: when the external control Hamiltonian does not commute with the internal one, the states of the working fluid cannot follow the instantaneous energy levels, leading to additional energy stored in the working medium. Inner friction is thus associated to diabatic transitions, i.e. changes of populations which occur during the time dependent adiabatic (here referring to closed system) strokes if they are performed at finite speed.
So far inner friction occurring in specific cycles and transformations has been analyzed by adopting phenomenological and physically motivated assumptions about the explicit time dependence of the population changes (e.g., in Ref. [15], a friction coefficient is introduced, giving rise to a constant dissipated power). Our treatment, instead, does not rely on any ad hoc assumption, but rather on the exact dynamics of the working substance. This is important because it has been shown, [12], that inner friction is not only an indicator of irreversibility of a quantum process, but also a quantitative measure of its amount. It is therefore crucial to identify and highlight its role in the efficiency reduction of finite time cycles by analyzing the full quantum dynamics that produces it.
In particular, we will explore the quantum friction arising from disorder within the sample playing the role of working substance. We will consider an ensemble of qubits in a setting in which their Hamiltonian parameters are not homogeneous, and connect the presence of these static errors to the appearance of friction and losses during the implementation of the Otto cycle. Explicitly, we provide a quantitative analysis of the amount of losses due to the inner friction as a function of the degree of disorder.
Indeed, the performance of the heat machine are negatively affected by inner friction and cycle’s outputs such as extracted work, power and efficiency are gradually suppressed as disorder and friction increase.
The remainder of the paper organized as follows. In Secs. 2.1 and 2.2 we introduce and review the concept of inner friction by focusing on the particular case of a spin system in presence of misalignments and disorder, which will then be of interest for the rest of the present paper. In Sec. 2.3 we introduce the quantum Otto cycle and its constituent transformations specifying the assumptions about the model we use to describe the working substance. In Sec. 3 we present and discuss our main results, while in Sec. 4 we propose a feasible experimental implementation of the quantum Otto cycle in order to test our findings. Finally, Sec. 5 is devoted to some concluding remarks and to a discussion of possible future developments.
2 Model and methodology
In this section we introduce the model, and give a possible explanation of the origin of inner friction. We then introduce the quantum Otto cycle (QOC) and the figures of merit through which the cycle will be characterized.
2.1 Misalignment and disordered samples
In order to understand what we mean by losses and friction in a closed quantum system, and in particular in the case of one qubit, let us focus on the dynamics generated by a Hamiltonian of the form:
(1) 
The analysis reported here applies to the general case of qubit dynamics (1) and in the following we will consider the case of a spin interacting with a magnetic field (Sections 2 and 3) as well as of a qubit encoded in photon polarization (optical implementation in Section 4).
An adiabatic transformation is obtained by the unitary time evolution generated by the Hamiltonian (1), with a linear driving of the external field at a fixed rate , which we allow to be misaligned by an angle with respect to the static field . The misalignment affects the energy spacing as well as the eigenstates and the populations.
We assume that at the qubit in a thermal state at inverse temperature . For a very slow driving, ideally taking an infinite time to complete the transformation in the quantum adiabatic regime, the qubit populations would remain unchanged while the the energy spacing increases/decreases and the system remains in a thermal state with a lower/higher temperature. The same occurs in absence of misalignment, in Eq. (1), as in this simple scenario the adiabatic transformation
(2) 
reduces to a compression/expansion of the energy spacing of the qubit, thus preserving the initial thermal populations even in presence of fast driving.
Interesting dynamical and thermodynamical implications arise, instead, when considering deviations from the limit of perfect alignment (). This is the case that we are going to explore in this work in order to characterize inner friction and its effects on the efficiency of quantum thermal machines, resulting from the simultaneous presence of the static field and of the misaligned time depending part . Our aim it to apply this analysis to an ensemble of (independent) spins, considering some degree of disorder and looking at average effects across the sample. In particular, this can correspond to different situations as represented in Figure 1. A condensed system on a lattice, with embedded magnetic dipoles having disordered orientations, can be modelled by randomly oriented spins with tilting angles () with respect to the direction of a uniform external field. We assume that the distribution of the spin orientations in the sample is given by a function . Alternatively, all sample dipoles could be perfectly aligned in an ordered configuration but the inner friction could be due to inhomogeneity of the external fields in space (Figure 1b). The field orientation across the sample would be given, in this case, again by the function .
2.2 Inner friction and irreversibility
In order to have a simple physical picture for the behavior of our quantum machine, let us first consider the simpler case of a driven quantum two level system undergoing the unitary dynamics generated by a parametric time dependent Hamiltonian . If the parameter changes slowly enough (in the sense of the quantum adiabatic theorem, [20, 21]) the system evolves without its energy population ever changing at all, even if the instantaneous energy eigenvalues and eigenstates do change in time. If the system has been prepared in equilibrium with a thermal bath, which is then removed, such an ideal adiabatic parameter change keeps the system in an equilibrium state at every stage. In particular, if the parameter gets back to its initial value after some time, the final result is that the system is brought back to its initial state. On the other hand, if the cycle is performed in finite time, the final state of the system will differ from the equilibrium state it started off because nonadiabatic transition have taken place [22]. The difference between the two states, if properly quantified, can be regarded as a measure of the deviation from an ideal adiabatic transformation. The quantum nonadiabaticity has the same effects as friction has in a classical context: an extra energy is needed to complete the process (indeed, the work done in the ideal adiabatic is always smaller than the one performed in finite time, see [21, 12]), which is then dissipated if the system equilibrates at the end of the process.
With this picture in mind, let us now address the dynamics generated by Hamiltonian (1) on an initial thermal state given by where is the inverse temperature in units of the Boltzmann constant. By changing very slowly from at up to at and then going back from to the system will be brought back into its initial state. To discuss what happens in the general case, namely when these changes are performed at finite rates, we consider the following protocols:
(3) 
The forward protocol, defined by the unitary operator ( being the time ordering operator), is generated by the Hamiltonian in Eq. (1) such that . It takes the initial density matrix to . The backward protocol is again generated by the Hamiltonian in Eq. (1) where now with the condition that . This consists just in ramping up and down the field with different rates and , respectively.
In order to characterize the above protocol, we first look at the time dependent polarization defined as where is the energy level spacing at time for both the forward and backward protocols. The result is shown in Fig. 2, where we notice that finitetime evolution introduces deviations with respect to the quantum adiabatic case as expected. Moreover, as it can be seen in Fig. 2, by applying the forward and backward protocols defined above, the system does not get back to its initial state, but reaches a different polarization (green line) at the end of the protocol.
This already gives a quantitative indication that finite time control leads to an irreversible behavior. Here, we use the word “irreversibility” in the thermodynamic sense: because of the occurrence of nonadiabatic transitions, the system is driven out of the manifold of equilibrium states and application of the same protocol in reverse does not bring it back to the initial state.
A more precise way of quantifying the irreversibility of such a transformation is through the distance of the final state from the initial one, expressed in terms of the relative entropy , where . As shown in [12], this quantity has a well defined thermodynamical interpretation as it precisely gives the nonadiabatic part of the work performed on the system by the driving agent, i.e. the inner friction.
Indeed, for an adiabatic transformation, the quantum relative entropy between the actual final state and the ideal thermal equilibrium one is proportional to the difference between the work done on the system during the parametric change and the same quantity taken in the infinitely slow limit. This is precisely the definition of the inner friction, hereafter called , [12]. Furthermore, the same quantity is linked to the generation of extra heat; that is to say, to the irreversible production of ‘waste energy’. This extra energy is exactly the energy that needs to be dissipated if, at the end of the protocol, we were to thermalize the system to the initial temperature. Specifically, the following relations hold:
(4) 
where is the heat the system takes to thermalize at the initial inverse temperature . This is what we shall refer to as inner friction in the following.
The inner friction for the time evolution described above is reported in Fig. 3, where we can clearly see that when both transformations are either very slow (quantum adiabatic case) or very fast (‘diabatic’ or sudden case), at the end of the protocol the system is found to be in (or very close to) its initial state. For finite time transformations, however, the system does not get back to its initial state. From a dynamical point of view, this is not surprising; however, if interpreted from a thermodynamical perspective, this fact suggests that transformations done in finite time are, in general, irreversible ones.
2.3 Model for the quantum Otto cycle (time scales assumptions)
The Otto cycle is the simplest cycle for our purposes as it allows for a clear separation between dissipative steps (thermalization processes, in contact with a thermal bath) from nondissipative ones (in which work is done or extracted), as opposed, for instance, to the Carnot cycle, which contains two isotherms in which one has to perform (extract) work while the system is attached to a thermal bath. This separation will be very useful in order to identify finite time effects on the single adiabatic transformations and thus on the total cycle.
The quantum version of the Otto cycle is the composition of two adiabatic transformations, in which the systems evolves unitarily, and two isochoric branches corresponding to thermalization in contact with a hot (and, respectively, a cold) heat bath at temperature ( ).
In the next subsections we better specify the assumptions we employ to describe the different branches of the quantum Otto cycle and the physical quantities we investigate to characterize it. The ideal Otto cycle is represented by the dashed (yellow) rectangle in Fig. 4. The blue line, instead, describes a finite time cycle, in which the end points of the adiabatic strokes are moved towards larger values of (which just means that there is more population than expected in the excited states) because of the presence of inner friction.
2.3.1 Adiabatic transformation 
As already mentioned above, the adiabatic transformations can be described by the unitary operator generated by the Hamiltonian in Eq. (1). For simplicity, in the following we will consider the case where the two adiabatic branches ( and ) last equally long, namely and have the same rate of change for the field .
2.3.2 Isochoric transformations 
For the isochoric transformations, we assume perfect thermalization at the given temperatures , (hot and cold, respectively). To study the relation between inner friction and the finite time of the adiabatic branches, we will assume that perfect thermalization is achieved very fast with respect to all other time scales, and also that the isochoric branch will be assigned with a fixed short time duration (to be eventually neglected) with respect to adiabatic ones but long with respect to thermalization time of the system:
(5) 
where , and are the typical time scales for the thermalization process, isochoric and adiabatic transformations, respectively.
2.4 Figures of merit
In order to characterize the quantum Otto cycle, we will look at the extractable work, , at the power , at the efficiency , and at their averages over disorder.
To properly define these quantities, let us start by defining the work done on an adiabatic branch as:
(6) 
where and are the Hamiltonian and the density matrix of the system at the beginning (end) of each transformation. In particular, both adiabatic transformations start with a Gibbslike state since we assume perfect thermalization to occur at the end of each isochores. In the adiabatic transformations, the work defined in Eq. (6) does coincide with the first moment of the work distribution for closed but nonautonomous systems [25]. On the other hand such a work distribution allows to define a fluctuation relation and thus its moments have a clear thermodynamical meaning.
In the isochoric branches, we have that the initial and final Hamiltonians are the same and the final state is thermal, and thus diagonal in energy eigenbasis. The amount of energy exchanged between the reservoir and the system in each isochoric transformation is given by:
(7) 
Thus, the energy absorbed from the bath equals the energy spacing , times the change in the population of the lowest energy state (we denoted the ground and excited state populations as and , respectively).
Since the change of the total internal energy along the cycle vanishes, the total work done on the system is given by where () is the amount of energy exchanged with the reservoir at inverse temperature (), given by Eq. (7) for the isochores (). The first quantity we will use to characterize the cycle is the extractable work given by the relation:
(8) 
where is the energy level spacing of the Hamiltonian at point in the  diagram of Fig. 4.
For the quantum Otto cycle, and by means of the definition of , we can then write the following condition
(9) 
ensuring that the work extracted is strictly positive and we are actually using the engine to perform work. This is in agreement with Carnot theorem, as shown in Ref. [15], and for ideal quantum adiabatic branches (i.e. infinitely slow transformations) it reduces to where are the energy spacings at the end of quantum adiabatic branches [23]. Notice that, in this quantum adiabatic limit, a breakdown of the positive work condition coincides with the saturation of Carnot inequality.
Let us now define the power and the efficiency of a cycle on a single qubit as:
(10)  
(11) 
where we have made explicit the dependence upon the angle in both and and we explicitly included in the above definitions, despite the assumption that it is small compared to the characteristic times scale of the cycle, just to avoid that is illdefined in the formal limit .
In addressing the disordered case we average all quantities over assuming it to have a Gaussian distribution with zero mean and variance . The averaged extractable work, power and efficiency are thus given by:
(12)  
(13)  
(14) 
3 Results and discussion
In this section we characterize the QOC by looking at the extractable work , its power and its efficiency , paying particular attention to the role of the inner friction in limiting the performances of such an heat engine. In the first subsection we will look at these figures of merit for different sets of parameters of our modelsystem. In the second one, we study the behavior of the efficiency at maximum power, .
3.1 Extractable work, power and efficiency
In what follows we will study the performance of the QOC in various cases of equal rates for both of the adiabatic branches. In Figure 5(a), 5(c) and 5(e) we plot , and as function of the total time of the cycle for different values of the misalignment angle at a fixed value of the ratio between the temperatures of the hot and cold reservoirs . It can be seen that the extractable work becomes negative if the exceeds a maximum time , which is a function of the misalignment . This means that if the cycle lasts too long we are actually doing work on the system. Moreover there exists a critical value of such that the extractable work is negative for any value of . Under these conditions, the cycle is not a heat engine but rather a refrigerator, which uses external work to cool the cold reservoir. An analogous behavior is shown by power and efficiency.
In Figure 5(b), 5(d) and 5(f), we show the dependence of , and on the total time of the cycle, for a fixed misalignment angle and for different values of the ratio . We can see that as the ratio increases the extractable work increases too. This is something which is expected, nevertheless we can clearly see that the finiteness in time of the cycle introduces again negative works for . This is again due to the generation of inner friction, which comes along with the finite time condition.
Inner friction is explicitly shown in Figure 6, where the sum of the friction produced in the two adiabatic strokes is shown as a function of the total cycle time for various misalignment angles . As discussed above, the case is very special as no friction is generated, whatever rate of variation is considered for the driving field . On the other hand, increases with increasing the tilting angle and decreases with decreasing the driving rate . The behavior of should be compared to that of shown in Figure 5: the more friction is present, the less work can be extracted from the engine.
So far we have considered QOCs with given values of the misalignment angle . We now consider the effect of disorder and assume that is a Gaussian random variable with mean value and variance . In Figure 7(a), 7(c) and 7(e) we show the behavior of extractable work, power and efficiency for different values of the variance and given temperature ratio . We can see that at a given total time , the best performance is always obtained with sharper distributions (smaller ). Thus, if the disorder of the system grows, the capability of the latter of providing work and of doing it in a more efficient way decreases. Again we mention the fact that there exist a maximum total time above which the QOC is not an heat engine anymore. We also notice that even a small disorder has quite dramatic effect in reducing the efficiency for long enough times (upper curve in Fig.7(e)).
In Figure 7(b), 7(d) and 7(f) we plotted the behavior of the same quantities for different values of the ratio at a given variance . Again, all of the quantities increase as the difference in temperatures increases.
We plotted all of the quantities as a function of the total cycle time because, from an operational point of view, this is the quantity one can control once the working substance is prepared and the stage is set for the thermal machine to operate. However, it has to be mentioned that some care should be payed when comparing the values of the efficiency at different operating times. Indeed, since varies with time, each different gives rise to a different value of the final frequency (called , above) at which the isochoric takes place, see Figure (4). The ideal cycle with infinitely slow adiabatic branches, corresponding for us also to the absence of misalignment () and shown for comparison in each plot in Figure (5) and (6), has efficiency . The dependence of on implies that the efficiency should be compared with a different at each different . To avoid any confusion in this respect, and to better display the role of finitetime induced friction in the machine performance, we show this comparison in Figure (8), where the ratio is displayed as a function of the operating time. Once the efficiency is renormalized in this way, its residual dependence on can be fully ascribed to the presence of inner friction.
3.2 Efficiency at maximum power
Let us now consider the relation between and (see Figure9) and then extract the value of the efficiency at maximum power . Two sets of these data are reported in Tab. 1 and 2. They refer to averaged power and efficiency considering in the first case various temperature ratios at a fixed width, and the other way round for the second case. The effects of disorder and temperature difference continue to stand: our analysis provides larger values of (averaged) power and (averaged) efficiency at maximum power for smaller and smaller and for larger and larger , with the following onesentence summary: We obtain considerably larger values of when the temperatures ratio is large and for very picked misalignment distributions, that is, when the inner friction is smaller.
4 Experimental implementation with an optical setup
In this Section we propose one possible experimental setup by means of which it is possible to realize the Otto cycle discussed so far and test our findings. The Otto cycle is made up of two different types of branches, namely adiabatic and isochoric transformation. Thus the proposed setup has to be able to implement both of them.
The physical system we have in mind is an optical one and in particular we propose to encode the qubit into the polarization degree of freedom of a single photon. In the following we address the implementation of the two types of branches separately, stressing the key points for both of them.
4.1 Implementation of the adiabatic transformation
The adiabatic branch in the Otto cycle is achieved by means of a time dependent effective magnetic field and its time evolution is given by the unitary operator:
(15) 
For a fixed the above operator can be written as a rotation in the Hilbert space of the qubit using the Euler decomposition as:
(16) 
This expression is helpful for our purposes because the single rotations appearing in it can be easily implemented in an optical setup as rotation of the polarization degrees of freedom of a single photon.
Therefore by encoding the qubit into the polarization degree of freedom of a photon and in particular by choosing the basis of horizontal and vertical polarization, we can perform the wanted rotations by means of properly chosen phase retarders.
4.2 Implementation of the isochoric transformation
The isochoric transformation requires more care. By definition it amounts to attaching the system to a thermal bath, which makes the system thermalize into a Gibbslike state. The latter is characterized by a density matrix with no coherences between different eigenstates of the Hamiltonian of the system whereas the diagonal ones are given by the Boltzmann factors , where or is the inverse temperature we want the state to thermalize at and are the eigenenergies of the final Hamiltonian of the adiabatic transformation preceding the isochoric one we are addressing. In the case of a qubit the thermalization process leads to a final state which is diagonal in a given basis (determined by the form of the bathspin coupling) and whose populations are related to the temperature of the thermal bath by the relation
(17) 
where, is the population of the lowest state of the qubit after thermalization has occurred.
In order to implement an isochoric transformation we propose to exploit the experimental setup used in Ref. [24]. The idea is to exploit the spatial degrees of freedom of the photon as an effective bath for its polarization. The coupling between the two is achieved by exploiting the birefringent property of a quartz plate. The effect of the latter on a photon passing through it is to phaseshift the horizontal and vertical component of the polarization by an amount proportional to the number of photons per mode. Once the spatial part of the photon is traced out, the dynamics of the polarization turns out to be driven by the following dynamical map between an initial state to a final state , which describes decoherence:
(18) 
where the parameter can be tuned from (identity map) to (complete decoherence, namely the final density matrix has vanishing offdiagonal terms). Because of our assumption of complete thermalization we will always assume .
We can thus exploit this mechanism in order to engineer thermalization in the following way. Let us assume that the inverse temperature of the bath we want to mimic is . Through Eq. (17) we can determine the population of the lowest energy level after the system completely thermalized. Let us write the initial state (which in turn correspond to the final state of the adiabatic transformation preceding the isochoric one) as:
(19) 
Since the decoherence mapping in Eq. (18) has the effect of making the off diagonal elements vanish we first need to perform a rotation on the initial state to turn it into a state of the form:
(20) 
where is calculated through relation in Eq. (17). The application Eq. (18) has now the effect of making , thus leaving us with the desired state:
(21) 
It is easy to see that in order to get from the state in Eq. (19) to the one in Eq. (20) we can apply a specific rotation, which has to be chosen by taking into account both states. For instance in the case and , that is, if we are ‘heating’ our system, the right rotation to perform is:
(22) 
with . For we have solutions for only if .
5 Conclusions
We have worked out an exact dynamical model with which we discussed the performance of a Quantum Otto cycle in presence of inner friction. With respect to previous approaches to the same problem, we obtained the growth of polarization in the adiabatic branches of the cycle without any adhoc assumption, but rather by following the dynamics generated by the system Hamiltonian. In this way we have been able to deal with the irreversibility of such transformations when they are performed in a finite time and so to better characterize the whole cycle. Finite time evolution leads to a decrease of values of thermodynamical figures of merit for the heat engine, and we have concentrated on the extractable work, the power and efficiency which are partially quenched if inner friction is present.
The friction is related to the noncommutativity of the system and control Hamiltonian, due to some misalignment between the internal and control magnetic field axes. After explicitly studying its effects for a fixed (and controlled) case, we turned to the more realistic case in which such a misalignment is an unwanted sideeffect of the lack of control in the system, ultimately due to the presence of disorder in the sample or inhomogeneity of the magnetic field. The limiting cases are the completely ordered and disordered samples; in the first one, we obtain an ideal quantum Otto engine with efficiency given by ; while for the completely disordered case we showed that no positive work can be extracted from the system, which cannot behave as an heat engine at all. In the more general case of a finitewidth Gaussian distribution of tilting angles , describing a given degree of disorder as quantified by its variance, we obtained a quantitative description of the efficiency reduction due to the disorderinduced inner friction.
Finally, we proposed an optical experimental implementation of such Otto cycle using the effective polarization qubit of a photon, whose thermalization can be obtained by the coupling with the spatial degree of freedom in a birefringent crystal.
6 Acknowledgements
We acknowledge funding from FIRB 2012 RBFR12NLNA, EU, MICINN, CSIC, FEDER funding under Grants FIS201123526 (TIQS), FIS201460343P (NOMAQ), postdoctoral JAE program (ESF), FETproactive project QuProCS and COST Action MP1209, and the UIB for Invited Professors program.
References
 [1] Quan HT, Yuxi Liu, Sun CP, Nori F 2007 Phys. Rev. E 76, 031105
 [2] Thomas G, and Johal R S 2011 Phys. Rev. E 83 031135.
 [3] Huang X L, Liu Y, Huang Z, Niu X Y 2014 Eur. Phys. J. Plus 129 4.
 [4] Esposito M, Kawai R, Lindenberg K, and Van den Broeck C 2010 Phys. Rev. E 81, 041106; Polettini M, Verley G, and Esposito M 2010 Phys. Rev. Lett. 114, 050601
 [5] Scully M 2010 Phys. Rev. Lett. 104, 207701.
 [6] Huang XL, Niu XY, Xiu XM, and Yi XX 2014 Eur. Phys. J. D 68 32.
 [7] Kim S W, Sagawa T, De Liberato S, and Ueda M 2011 Phys. Rev. Lett. 106 070401.
 [8] Campisi M 2014 J. Phys. A: Math. Theor. 47 245001; Campisi M, Pekola J, Fazio R 2015 New J. Phys. 17 035012.
 [9] Azimi M, Chotorlishvili L, Mishra S K, Vekua T, Hübner W, and Berakdar J 2014, New J. Phys. 16 063018.
 [10] Abah O, Roßnagel J, Jacob G, Deffner S, SchmidtKaler F, Singer K, Lutz E 2014, Phys. Rev. Lett. 109 203006; Roßnagel J, Abah O, SchmidtKaler F, Singer K, Lutz E 2014, Phys. Rev. Lett. 112 030602.
 [11] Jarzynski C 1997 Phys. Rev. Lett. 78 2690; Mukamel S 2003 Phys. Rev. Lett. 90 170604; Esposito M and Van den Broeck C 2010 Phys. Rev. Lett. 104 090601; Deffner S and Lutz E 2010 Phys. Rev. Lett. 105 170402.
 [12] Plastina F, Alecce A, Apollaro T J G, Falcone G, Francica G, Galve F, Lo Gullo N, and Zambrini R 2014 Phys. Rev. Lett. 113, 260601.
 [13] Esposito M, Harbola U, and Mukamel S 2009 Rev. Mod. Phys. 81 1665; Jarzynski C, Annu. Rev. Condens. Matter Phys. 2 329 Campisi M, Hänggi P, Talkner P 2011 Rev. Mod. Phys. 83, 771; Seifert U 2012 Rep. Prog. Phys. 75 126001.
 [14] Dorner R, Clark S R, Heaney L, Fazio R, Goold J, Vedral V 2013 Phys. Rev. Lett. 110 230601; Mazzola L, De Chiara G, Paternostro M 2013 Phys. Rev. Lett. 110 230602; Batalhão T B, Souza A M, Mazzola L, Auccaise R, Sarthour R S, Oliveira I S, Goold J, De Chiara G, Paternostro M, Serra R M 2014 Phys. Rev. Lett. 113 140601; Campisi M, Blattmann R, Kohler S, Zueco D, Hänggi P (2013) New J. Phys. 15 105028.
 [15] Kosloff R, and Feldmann T 2000 Phys. Rev. E 61 4774.
 [16] Kosloff R, and Feldmann T, 2010 Phys. Rev. E 82 011134.
 [17] Kosloff R, and Feldmann T 2002 Phys. Rev. E 65 055102; Kosloff R, and Feldmann T 2003 Phys. Rev. E 68 016101; Kosloff R, and Feldmann T 2004 Phys. Rev. E 70 046110; Kosloff R, and Feldmann T 2010 Phys. Rev. E 82 011134; Kosloff R, and Feldmann T 2012 Phys. Rev. E 85 051114.
 [18] Wang J, Wu Z, He J 2012 Phys. Rev. E 85 041148.
 [19] Wang R, Wang J, He J, and Ma Y 2013 Phys. Rev. E 87 042119.
 [20] Messiah A, Quantum Mechanics 1970 North Holland, Amsterdam.
 [21] Allahverdyan A E, and Nieuwenhuizen Th M 2005 Phys. Rev. E 71 046107.
 [22] see, e.g., Polkovnikov A 2008 Phys. Rev. Lett. 101 220402.
 [23] Kieu T D 2004 Phys. Rev. Lett. 93 140403.
 [24] Liu et al. 2011 Nat. Phys. 7 931.
 [25] Talkner P, Lutz E, and Hänggi P 2007 Phys. Rev. E 75 050102(R); Roncaglia A J, Cerisola F, and Paz J P 2014 Phys. Rev. Lett. 113 250601.