Quantum statistics and the performance of engine cycles
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Quantum statistics and the performance of engine cycles
Yuanjian Zheng and Dario Poletti
1, 2 Singapore University of Technology and Design, 8 Somapah Road, 487372 Singapore MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore
We study the role of quantum statistics in the performance of Otto cycles. First, we showanalytically that the work distributions for bosonic and fermionic working fluids are identical forcycles driven by harmonic trapping potentials. Subsequently, in the case of non-harmonic potentials,we find that the interplay between different energy level spacings and particle statistics stronglyaffects the performances of the engine cycle. To demonstrate this, we examine three trappingpotentials which induce different (single particle) energy level spacings: monotonically decreasingwith the level number, monotonically increasing, and the case in which the level spacing does notvary monotonically.
I. INTRODUCTION
A major technological challenge today is the design ofdevices at the nanoscale that can efficiently and reliablyconvert energy between different forms. At such smallscales, the thermal and quantum fluctuations of thermo-dynamic quantities become especially significant, and thestochastic nature plays a dominant role in determiningthe performance of quantum devices [1–5].More recently, major theoretical developments havealso been made specifically towards the understandingand implementation of quantum thermodynamics. Theseinclude for instance, advances in fluctuations theorems[1, 2, 6], and on the use of external control protocols toproduce adiabatic dynamics in finite time [7–14].At the same time, properties of non-equilibrium workdistributions have also been investigated experimentallyin classical systems [15, 16], culminating with the realiza-tion of a colloidal micro-scale stochastic heat engine[17],and quantum systems [18]. Due to the high degree oftunability and control, ultra-cold ion setups have alsopositioned themselves as promising candidates for exper-imental realizations of quantum heat engine cycles [19].In addition, a proposal for the realization of an Otto cyclewithin a solid state set-up was recently studied [20].More recently, it has also been shown numerically, thatthe exchange symmetry of particles do play a significantrole in the work distribution of quantum many-body sys-tems [21]. As such, here we address the pertinent issue ofmany-body statistics in quantum heat engines. In par-ticular, we elucidate the role of exchange statistics in theperformance of quantum cycles by providing an intuitiveunderstanding of the interesting connection between ex-change statistics and the geometry of the trapping po-tential.We consider a quantum equivalent of the Otto-cycle[22] in which the working fluid is a quantum gas of N indistinguishable and non-interacting particles. The col-lective system undergoes a cyclic sequence of adiabaticand iso-parametric processes, where the heat received byexternal reservoirs can be in part, converted into usefulwork, while the rest is transferred to a cold reservoir. Im-portant characteristics and figures of merit of the engine x ε j ε j ω a h E i ( c ) ( d )( a ) [B][C][D][A] ( b ) j = 0 j = 1 j = 2 j = 3 j = 0 j = 1 j = 2 j = 3 j = 1 j = 0 j = 2 j = 3 FIG. 1. (Color online) (a-c) Different potential wells withcorresponding single particle energy spectrum ε j . The darkblue (light red) squares represent the occupation of energylevels at 0 temperature for the case of 4 bosonic (fermionic)atoms. (a) triangular potential ( a = 1), (b) infinite squarewell potential ( a = ∞ ) and (c) double well potential. Thehorizontal thin grey lines indicate the position of the j -thsingle particle energy level. (d) Schematics of a quantumOtto cycle in an average energy h E i vs trapping parameter ω a . cycle’s performances are the probability distributions ofboth its net work output and its efficiency. In this workwe study how the statistics, either bosonic or fermionic,of the gas’ particles affects the engine’s performance. Weshow that this is strongly dependent on the single particleenergy level structure of the system and indicate strate-gies to improve the performance of the engine cycle by acareful choice of both the trapping potential and the typeof gas. It is only when considered in conjunction, thattrap geometry [23] and particle exchange statistics [21]can be utilized to optimize the performance of quantumheat engines.We consider non-interacting particles in different trap-ping potentials: with the energy level spacings eitherconstant (harmonic potential), monotonically decreasing(triangular potential, see Fig.1(a)), increasing (squarepotential, see Fig.1(b)) or change non-monotonically(double well, see Fig.1(c)). To gain some preliminaryintuitive insights to the problem, in Fig.1 we show how4 atoms are distributed in the respective potentials attemperature T = 0 depending on the given symmetry.We note that the energy difference between the groundstate and the first excited state can be very different de-pending on whether the gas is bosonic or fermionic innature and on whether the gap between energy levels isincreasing (e.g. square well) or decreasing (e.g. triangu-lar potential). In Fig.1(a) we observe that for bosons(blue squares) the gap is larger than for that of thefermions (red circles); Whereas in the case of a poten-tial in which the energy level distance is monotonicallyincreasing (square well in Fig.1(b), the gap will be largerfor the fermions and smaller for that of the bosons. Fora potential in which the distances between energy lev-els do not feature a monotonous behavior (double wellin Fig.1(c)), the excitation gap not only depends on theparticle statistics, but also on the number of atoms.This paper is organized as follows: In section II, weprovide general relations and results for quantum Ottocycles with scaling potentials; In section III, we showour main results and lastly, in section IV we draw ourconclusions. II. THE OTTO CYCLE IN SCALINGPOTENTIALS
In classical thermodynamics, the Otto cycle is com-posed of two processes in which only heat is exchangedwith the environment, and two processes in which onlywork is exchanged (and no heat is transferred). The for-mer is usually referred to as an isochoric process for aclassical ideal gas. Analogously, a quantum Otto cycleis composed of two quantum adiabatic processes, repre-sented in Fig.1(d) by the steps [ A ] → [ B ] and [ C ] → [ D ],as well as two iso-parametric processes, represented inFig.1(d) by the steps [ B ] → [ C ] and [ D ] → [ A ], in whichthe Hamiltonian of the system is kept constant and theenvironment is (weakly) coupled to the system [24]. Sim-ilar to their classical counterparts, no heat is exchangedwith the working fluid in quantum adiabatic processes,while only heat is exchanged in an iso-parametric pro-cess. T [ A ] corresponds to the lowest temperature in thecycle while T [ C ] is the largest. As we are interested inthe governing principles of quantum thermodynamic cy-cles involving non-interacting many-body quantum gases,we will only study quasi-static cycles, leaving the cycleswith finite power for a future study.To gain deeper analytical insights, we also focus onHamiltonians with scaling properties [13, 22, 23, 25], ormore generally on driving protocols in which the energy level structure after a unitary process is simply multi-plied by a scale-factor. The harmonic oscillator is theprototypical example of such scaling Hamiltonians, as theenergy levels are simply proportional to the trapping fre-quency.In a more general sense, for every potential of the form V ( x i ) = f ( x i /λ ) /λ (where λ is a control parameter withunits of length and x i is the position coordinate of the i -th atom), the Hamiltonian H has a scaling energy spec-trum. In fact H = X i (cid:20) − ~ m ∂ ∂x i + 1 λ f (cid:16) x i λ (cid:17)(cid:21) = 1 λ X i (cid:20) − ~ m ∂ ∂X i + f ( X i ) (cid:21) (1)where X i = x i /λ is a dimensionless variable [13]. Thisapplies to all power-law potentials of the form V ( x i ) = m ( ω a | x i | ) a since H = X i − ~ m ∂ ∂x i + m ω a | x i | ) a = X i ζ a ( ω a ) (cid:18) − ∂ ∂X i + 12 | X i | a (cid:19) (2)where ζ a ( ω a ) = h ( ~ ω a ) a /m a − i a +2 is the energy scale[23, 25]. It is thus now clear that that the energy levelsare simply scaled by the prefactor ζ a ( ω a ). For instance,the ratio of the energy of the j -th level for Hamiltonianparameter ω ′ a (i.e. ε j ( ω ′ a )) divided by that for parameter ω a (i.e. ε j ( ω a )) does not depend on j and it is given by ε j ( ω ′ a ) ε j ( ω a ) = ζ a ( ω ′ a ) ζ a ( ω a ) = ν (3)where ν is a parameter that only depends on ω a and ω ′ a .In the following we will use the lighter notation ζ a ( ω ′ a ) = ζ ′ a and ζ a ( ω a ) = ζ a .In addition, a more general polynomial like the double-well potential may also be made to possess a scaling prop-erty in its energy levels, albeit in the presence of certainrestrictions. For example, the double well potential V ( x i ) = m (cid:0) − ω x i + ω x i (cid:1) (4)has a scaling Hamiltonian H = ζ (cid:18) − ∂ ∂X i − X i + ~ m ω ω X i (cid:19) (5)so long as the anharmonic parameter γ ≡ ~ ω /mω re-mains fixed throughout the Hamiltonian evolution. A. Mean Work h W p i Let us consider a process from Hamiltonian parameter ω a to ω ′ a . The respective single particle j -th energy eigen-values will be ε j and ε ′ j . Let n j represent the number ofparticles in the j -th single energy eigenstate.A single N − body state is thus given by the orderedset of occupation numbers { n j } . A process is performedquasi-statically by changing the trap parameter adiabat-ically, and the corresponding N − body microstate at theend of the compression is then given by { m k } . For a sin-gle realization of the protocol p the work is given by atwo-times energy measurement [1, 26] W p = E ′{ n j } − E { m k } (6)The energy E { m k } (resp. E ′{ n j } ) is given by the sumover the energy ε l (resp. ε ′ l ) of the l -th single particleeigenstate of the Hamiltonian (e.g. E { m k } = P k ε k m k ).Hence, the ensemble average of work done for the processbetween two trap parameters is given by: h W p i = X { n j } X { m k } ( E ′{ m k } − E { n j } ) δ { m k } , { n j } P { n j } = X { n j } (cid:16) E ′{ n j } − E { n j } (cid:17) P { n j } (7)where the Kronecker delta δ { m k } , { n j } is due to the adi-abaticity of the process. The average h . . . i is computedover the intial probability distribution P { n j } which, atthis stage, can be general. Because of the scaling prop-erty of the Hamiltonians we consider, the ratio of ini-tial to final energy of each single particle eigenstate( ε ′ j /ε j ) = ν (see Eq.(3) and [13, 23]). This results inthe work of an adiabatic process to be h W p i = ( ν − h E i (8)where h E i = X { n j } X j ε j n j P { n j } (9)is the energy of the system at the beginning of the pro-cess. B. Work Standard Deviation σ W p The standard deviation of the work probability distri-bution function of a process p , σ W p = q h W p i − h W p i ,is also important to characterize a process as it quanti-fies the consistency and reliability of the output. For anadiabatic unitary process in a scaling potential we have h W p i = X { n j } X { m k } ( E ′{ m k } − E { n j } ) δ { m k }{ n j } P { n j } = ( ν − X { n j } X j ε j n j P { n j } = ( ν − h E i (10) It follows that the scaled standard deviation of work,defined as the standard deviation of the work distributiondivided by its mean work is given by σ W p h W p i ≡ s h W p ih W p i − s h E ih E i − ν , i.e. independent of whether (i)the process is a compression or an expansion, and (ii) theextent of the process (provided it is quantum adiabatic). C. Net work output of a cycle: average andvariance
The ensemble moments of the cyclic work distributionis derived from the work distribution of the constitutivesingle adibatic processes. The cycle work distribution iscomposed of two statistically independent work distri-butions on the compression and expansion arms of thecycle because of the nature of the ideal quantum Ottocycle. The combined work distribution is thus given bythe product distribution of two statistically independentprocesses P ( W ) = δ [ W − ( W c + W e )] P ( W c ) P ( W e ) (12)where W c ( W e ) is the work in the compression (expan-sion) process. For the cycle depicted in Fig.1(d) betweenparameters ω a and ω ′ a the net work output is given by h W i α = ( ν − h E [ A ] i α + (cid:18) ν − (cid:19) h E [ C ] i α (13)where here again, ν = ζ ′ a /ζ a and h E [ V ] i α is the averageenergy at the vertex [ V ] of the cycle and h . . . i α meansaverage over the thermal canonical state for fermions( α = F ) or bosons ( α = B ).Consequentially, the mean and variance of the cycleare also just the linear sum of the mean and variance ofthe individual processes. The scaled work fluctuationsare thus given by σ W α h W i α = p (∆ W c α ) + (∆ W e α ) h W c i α + h W e i α (14)In addition, we also note that if the adiabatic processes ofan Otto cycle are peformed within scaling potentials thenthe efficiency is a non-stochastic function of ν [23] inde-pendent of the quantum statistics of the working fluid.At each realization of the cycle η = − WQ [ B ] → [ C ] = (1 − /ν ) E [ C ] , { m k } + (1 − ν ) E [ A ] , { n j } E [ C ] , { m k } − νE [ A ] , { n j } = 1 − ν (15)where indeed ν > Q [ B ] → [ C ] is the heat transferredinto the system between states [ B ] and [ C ]. In Eq.(15) E [ V ] , { m k } represents the energy of the eigenstate { m k } at vertex [ V ] of the cycle. III. CYCLES PERFORMANCE FOR BOSONICAND FERMIONIC GASES
The average work output and its standard deviationcan thus be computed from Eq.(8) and (10). The av-erage energy and variance of a canonical thermal stateis computed using h E i α = − ∂ (ln Z αN ) /∂β and σ E α = ∂ (ln Z αN ) /∂ β where β = 1 /k B T is the inverse tem-perature and k B is the Boltzmann constant. To obtainaccurate results for low number of particles and low tem-peratures, we use the canonical parition function Z αN for N (fermionic or bosonic) atoms [27]. It can be computedusing the following recursion relation for non-degeneratespectra Z αN ( β ) = 1 N N X n =1 ( − γ Z α ( nβ ) Z αN − n ( β ) (16)where γ = 2 n for bosons ( α = B ) [28, 29] and γ = n + 1 for fermions ( α = F ) [30] and where Z α ( β ) = P j e − βǫ j is the single particle partition function (whichis obviously independent of the value of α ). Note alsothat Z α = 1. Eq.(16) (derived in detail in the AppendixA), will be used to compute Z αN ( β ) numerically. For theharmonic case, exact analytical results will be describedin the following section (III-A). A. Harmonic potential
For a simple harmonic oscillator of frequency ω , it ispossible to compute the partition function with the useof the q − shifted factorials ( z ; q ) n ≡ Q Nn =1 (1 − zq n − ).The partition function for N bosons is given by Z BN = e − Nβ ~ ω / ( q ; q ) N (17)and for N fermions [30] by Z FN = e − N β ~ ω / ( q ; q ) N (18)where q = e β ~ ω (see Appendix B for a detailed deriva-tion). This results in average energies, h E i α , given by h E i B = ~ ω N N X n =1 ~ ω ne β ~ ω n − h E i F = h E i B + ~ ω N ( N − . (20) T / Θ ∞ T / Θ ∞ S α /k B S α /k B ( d )( c ) ( b )( a ) [A][B] [C][D] FIG. 2. (Color online) Temperature T vs entropy S α diagramfor an Otto cycle in an infinite square well potential ( a = ∞ ) with fermion (red-dashed line) or bosons (blue-continuousline) as working substances. The trapping parameter variesbetween ζ ∞ and ζ ′∞ = 2 ζ ∞ while the minimum temperature T [ A ] = 2 Θ ∞ . The particle number N and the maximumtemperature T [ C ] takes the following values: (a) N = 2 and T [ C ] = 5 Θ ∞ , (b) N = 2 and T [ C ] = 100 Θ ∞ , (c) N = 5 and T [ C ] = 5 Θ ∞ , (d) N = 5 and T [ C ] = 100 Θ ∞ . T / Θ S α /k B T / Θ S α /k B [A][B] [C][D] FIG. 3. (Color online) Temperature T vs entropy S α dia-gram for an Otto cycle in a triangular potential ( a = 1) withfermions (red-dashed line) or bosons (blue-continuous line) asworking substances. The trapping parameter varies between ζ and ζ ′ = 2 ζ while the minimum temperature T [ A ] = Θ / N and the maximum temperature T [ C ] take the following values: (a) N = 2 and T [ C ] = Θ , (b) N = 2and T [ C ] = 10 Θ , (c) N = 5 and T [ C ] = Θ , (d) N = 5 and T [ C ] = 10 Θ . All higher moments of the energy are identical for bosonsand fermions. In particular, the standard deviation forbosons σ E B is equal to that of fermions σ E F σ E B = σ E F = N X n =1 ( ~ ω n ) e β ~ ω n ( e β ~ ω n − (21)The constant mean energy difference between fermionsand bosons ensures that the average work transfer is al-ways greater for that of fermions, h W p i F > h W p i B , whilethe rescaled standard deviation is instead, always largerfor bosons σ WpF h W p i F < σ WpB h W p i B [31]. This of course assumesthat the comparisons are between given initial canonicalstates of the same temperatures and identical quantumadiabatic protocols in a harmonic potential.For the work output of the full cycle, with Hamiltonianparameter that changes between ω and ω ′ , using Eq.(13)one can also easily show that h W i F = h W i B + (cid:20) ( ν − ω + (cid:18) ν − (cid:19) ω ′ (cid:21) ∆= h W i B + (cid:20)(cid:18) ω ′ ω − (cid:19) ω + (cid:18) ω ω ′ − (cid:19) ω ′ (cid:21) ∆= h W i B (22)where ∆ = ~ N ( N − B. Triangular and square well potentials
The previous results for the harmonic oscillator are adirect consequence of the constant energy level spacings.For non-harmonic potentials, where the level spacing isnot constant, the results will be very different. For in-stance, from Fig.1(a) it is clear that, at zero temperature,a fermionic system requires less energy to be excited com-pared to a bosonic one (because the gap between consec-utive energy levels monotonically decreases, hence theenergy gap from the last filled state to the first unfilledone is larger for bosons than for fermions). The converseis true for a square well potential in which the energylevel spacing monotonically increases, see Fig.1(b). Aswe will show later, this has strong consequences on theperformance of an engine cycle.To describe this in a clear and pictorial manner, seeFigs.2 and 3, we use a representation of a cycle typical ofclassical thermodynamics, the temperature vs entropy T - S diagram (where T is divided by a scale of temperaturegiven by Θ a = ζ a /k B while in the cycle the Hamiltonianparameter varies between ω a and ω ′ a > ω a ). To computethe entropy S α we use the thermodynamic relation F α = h E i α − T S α , where the free energy F α = − k B T ln( Z αN ).In this diagram, the area within the cycle represent boththe net heat and net work exchanges. Θ a /T [ C ] h W i F h W i B h W i F h W i B h W i F h W i B (b)(a)(c) FIG. 4. (Color online) Fermionic to bosonic ratio of the meanwork output of an Otto cycle vs. high temperature T [ C ] fordifferent particle numbers in the following potentials: (a) Tri-angular potential ( a = 1), (b) Infinite square well potential( a = ∞ ) and (c) Double well potential ( a = 2 , γ = 0 . ζ ′ a = 2 ζ a . In (a) T [ A ] = Θ /
3, while in (b) and(c) T [ A ] = 2Θ a . Different numbers of atoms are representedby the different (colored) lines: N = 2 for the dashed greenline, N = 3 for the dot-dashed red line, N = 4 for the dot-ted black line and N = 5 for the continuous blue line and N = 10 for the dotted brown line in (c). The insets depictthe potentials used. Using Eq.(16) we are able to compute Z αN for differentparticle numbers and cycle parameters.We consider only cases in which the coldest temper-ature T [ A ] is cold enough to see the combined effect ofboth energy level spacings and particle statistics [32]. Wethen consider different cases in which the value of thehighest temperature T [ C ] , varies from (relatively) cold tohot, and also with working fluids that consist of differentnumber of particles. For low T [ C ] we expect large differ-ences in the distribution for bosonic or fermionic gases.In Fig.2 we depict the case of an infinite square well,while in Fig.3 we study the case of a triangular poten-tial. In both figures the fermionic cycle is representedwith dashed red lines and the bosonic cycle by continu-ous blue lines. Whether a bosonic or a fermionic workingfluid produces more average work output depends on thedifference between the adiabatic work in the cold (com-pression from [ A ] → [ B ]) and in the hot arm (expansionfrom [ C ] → [ D ]).As clearly shown in Fig.2(c), when the number of par-ticles are large and both temperatures T [ A ] and T [ C ] arecold enough, there can be remarkable differences between Θ a /T [ C ] σ F σ B σ F σ B σ F σ B (a)(b)(c) FIG. 5. (Color online) Fermionic to bosonic ratio of the stan-dard deviation in the work distribution of an Otto cycle vs.high temperatures T [ C ] for different particle numbers. Valuesfor the cold temperature T [ A ] , particle number N and Hamil-tonian parameter values ω a are as given in Fig.4. The insetsdepict the potentials used. the work output of a fermionic or a bosonic system. Thearea enclosed in the fermionic cycle is much smaller. Infact, for fermions the energy level separation is too largefor entropy to change significantly. A similar scenario,although in the opposite direction, emerges for a trian-gular potential as depicted in Fig.3(c). In this case theenergy level spacing is larger for the bosons and the en-tropy will be lower (and less heat is absorbed). For smallnumber of atoms and large enough highest temperatures,the ratio of the net work output h W i F / h W i B tends to 1,see Figs.2(b) and 3(b).The case of two atoms and cold temperatures are de-picted in Figs.2-3(a). In this scenario the number ofatoms is too small to result in a marked difference of thecycles due to the quantum statistics. Cycles for largernumber of atoms and hotter temperatures are shown inFigs.2-3(d). It should be noted that for the triangularpotential, Fig.3(d), almost no difference can be notedbetween bosons and fermions in the high temperatureregime. However, in the cold temperature portion of thecycle, important and large differences in the amount ofentropy exchanged can be noticed. In particular, the en-tropy is lower for the species experiencing the largest gap(bosons in this case).To gain deeper understanding we revert to Figs.4-5(a-b). The triangular and the square well potential havequalitatively opposite behaviors. When the highest tem- perature T [ C ] is small, the total entropy and the entropychange for the bosons in the triangular potential (re-spectively for the fermions in the square well potential)are much smaller than that of the fermions (respectivelybosons), see Figs.2(c) and 3(c). It is easier to under-stand the qualitative behavior for large T [ C ] by consider-ing the work exchanges. In this case the ratio of the workexchanges h W [ C ] → [ D ] i F / h W [ C ] → [ D ] i B = h E [ C ] i F / h E [ C ] i B and thus it will eventually converge towards 1 as the tem-perature increases. However the difference in the workprocesses h W [ C ] → [ D ] i F − h W [ C ] → [ D ] i B behaves differentlydepending on the type of potential. It increases indefi-nitely for the square well, while it goes to 0 for the trian-gular potential. Hence, for the square well potential theratio h W i F / h W i B will tend to 1 from above, while forthe triangular potential it will tend to 1 from below [33].The same analysis also explains the qualitatively similarbehavior of the variance of the work probability distribu-tion, Fig.5. For this quantity, however, the crossings ofthe line σ F /σ B = 1 occur at a different values of T [ C ] . C. Double well potentials
The double well (Fig.1(c)), is a typical example of asystem in which the energy level spacing does not varymonotonically. Hence, we will use this to study the in-terplay of geometry and particle statistics in the perfor-mance of the engine cycle. As shown earlier in sectionII, the double well potential can have a scaling propertyso long as γ ≡ ~ ω /mω is a constant. For ease of ana-lytical and numerical analysis, we will focus on this case.As shown in Fig.4(c) the fermionic to bosonic ratio ofthe work output h W F i / h W B i has a pronounced oscilla-tory behavior compared to potentials with monotonicallychanging energy level structure as the (inverse) temper-ature of the hot reservoir β h .The same behavior is expected and observed also forthe ratio of the standard deviation of the work donefor a fermionic over a bosonic working fluid σ F /σ B , seeFig.5(c). IV. CONCLUSIONS
In this work we have studied how quantum statisticsof particles in a working fluid can affect the performanceof an engine cycle. We have shown that the interplayof statistics with energy level structure of a system hasremarkable and non-trivial effects on the engine cycle’soutput. We have analyzed systems with monotonical-lly increasing (square well potential) and decreasing (tri-angular potential) energy level structure, which possessqualitatively opposite behaviors, both in the low andhigh temperature regimes. For potentials with a non-monotonic energy level spacing (for example, a doublewell potential), the engine cycle’s peformance is charac-terized by the presence of oscillations in the fermionic tobosonic ratio of a given figure of merit, and the proper-ties of these oscillations are closely associated to the un-derlying energy level structure of the Hamiltonian, theparticle number and the temperature regimes. As a finalremark, systems driven in finite time, and many-bodyinteractions will be the subject of future studies.We aknowledge insightful discussions with C. Kollath at an early stage of the work. We are also grateful toU. Bissbort, J. Gong, C. Guo, P. H¨anggi, R. Tan, andG. Xiao for fruitful conversations. We are supportedby SUTD Start-up grant EPD2012-045 and by theSUTD-MIT International Design Centre (IDC). [1] M. Campisi, P. H¨anggi, P. Talkner, Rev. Mod. Phys. ,771 (2011).[2] U. Seifert, Rep. Prog. Phys. , 126001 (2012).[3] R. Kosloff, Entropy , 2100 (2013).[4] G. Benenti, G. Casati, T. Prosen, K. Saito,arXiv:1311.4430 (2013).[5] D. Gelbwaser-Klimovsky, W. Niedenzu, G. Kurizki,arXiv:1503.01195 (2015).[6] P. H¨anggi, P. Talkner, Nature Physics , 108 (2015).[7] M. Demirplak, S.A. Rice, J. Chem. Phys. B , 6838(2005).[8] M. V. Berry, J. Phys. A: Math. Gen. , 062122 (2013).[10] M. Palmero, E. Torrontegui, D. Gury-Odelin, J. G.Muga, Phys. Rev. A , 117(2013).[12] S. Campbell, G. De Chiara, M. Paternostro, G.M. Palma,R. Fazio, arXiv:1410.1555 (2014).[13] S. Deffner, C. Jarzynski, A. del Campo, Phys. Rev. X ,021013 (2014).[14] A. del Campo, J. Goold, M. Paternostro, Sci. Rep. , 6208(2014).[15] J. R. Gomez-Solano, C. July., J. Mehl, C. Bechinger,http://arxiv.org/abs/1501.02568 (2015).[16] V. Blickle, T. Speck, L. Helden, U. Seifert, C. Bechinger,Phys. Rev. Lett. 96, 070603 (2006).[17] V. Blicke C. Bechinger, Nature Physics , 143 (2012).[18] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang,Z.-Q. Yin, H.T. Quan, K. Kim, Nature Physics , 193(2015).[19] O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, E. Lutz, Phys. Rev. Lett., , 203006(2012).[20] M. Campisi, J. Pekola, R. Fazio, New J. Phys. , 035012(2015).[21] Z. Gong, S. Deffner, H. T. Quan, Phys. Rev. E , 062121(2014).[22] H. T. Quan, Y.-X. Liu, C. P. Sun, F. Nori, Phys. Rev. E , 031105 (2007).[23] Y. Zheng, D. Poletti, Phys. Rev. E , 012145 (2014).[24] A strong coupling significantly complicates the distinc-tion between heat and work transferred to the system.[25] C. Jarzynski, Phys. Rev. A , 040101 (2013).[26] P. Talkner, E. Lutz, P. H¨anggi, Phys. Rev. E. , 1801 (1997).[29] C. Weiss, M. Wilkens, Opt. Express , 272 (1997).[30] V. Kashcheyevs arXiv:1110.6264 (2011).[31] Note that for the Harmonic oscillator h W i F and h W i B approach the same value in the high temperature limit.[32] If T [ A ] was too high, the difference between the fermionicand bosonic statistics will be much reduced.[33] In this case it plays particular importance the fact thatat cold temperatures the work done in the cold arm is,in modulus, larger for the fermions than for the bosons.Hence, while the difference in work becomes negligible inone process, in the other process there is more work ex-change for the fermions. The combination of these two ef-fects results in a net work which is lower for the fermions. Appendix A: Derivation of recursive relation for thecanonical partition function
Here, we derive Eq.(16) following the argument as pre-sented in [30]. To do so we start from the grand canonicalpartition function Ξ α , where α = B or F , is the subscriptthat denotes bosonic and fermionic statistics respectively.The grand partition function can be written asΞ α = 1 + ∞ X N =1 Z αN z N (A1)where Z αN is the canonical partition function for N atoms.In equation (A1) z = e βµ is commonly known as thefugacity, where µ is the chemical potential associated tothe ensemble.It is also possible to write the logarithm of Ξ α asln (Ξ α ) = ∞ X N =1 κ α,N N ! z N (A2)where κ α,N in general, assumes different functional formsdepending on the given type of particle statistics con-sidered. By exponentiating (A2) and using the Fa`a diBruno’s formula, which states that the power series ex-pansion of the exponential of a polynomial P N a N x N /N !is exp X N a N x N /N ! ! = ∞ X N =0 B ( a , ...a N ) x N /N ! (A3)we can deriveΞ α = e ( P ∞ n =1 κα,NN ! z N ) = ∞ X N =0 B N ( κ α, , ..., κ α, N ) z N N !(A4)where B N is a Bell polynomial. Next, by equating corre-sponding terms of the polynomial expansion in (A4) and(A1), we get Z αN = B N ( κ α, , ...κ α, N ) N ! . (A5)Note that up to this point in the derivation, we havenot made any distinction between bosonic or fermionicstatistics. The grand partition function for the fermionscan be written as a formal power seriesln(Ξ F ) = X k ln (cid:0) ze − βε k (cid:1) = X k X N ( − N +1 ( N − e − Nβε k z N N != X N ( − N +1 ( N − X k e − Nβε k ! z N N != X N ( − N +1 ( N − Z ( N β ) z N N ! (A6)where we have used the expansion of ln(1 + x ) and where Z ( nβ ) is the single particle canonical partition functionwith inverse temperature nβ (of course Z = Z F = Z B ).Finally, using equations (A5) and (A6) together with theidentity for Bell polynomials: B n ( κ , ...κ n ) = κ n + n − X m =1 (cid:18) n − m − (cid:19) κ m B ( κ , ...κ n − m )(A7)we derive Eq.(16) for fermions. Similarly for bosonsln(Ξ B ) = − X k ln (cid:0) − ze − βε k (cid:1) = − X k X n ( − n +1 ( n − e − nβε k z n n != X n ( n − Z ( nβ ) z n n ! (A8)and again using Eq.(A5), (A7) and (A8) yields Eq.(16)for bosons. Appendix B: Partition function for the simpleharmonic oscillator