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A single Dirac-fermion Quantum Heat Engine Enrique Munõz1 and Francisco J. Penã2 1Facultad de F´ısica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile. 2Instituto de F´ısica, Pontificia Universidad Católica de Valpara´ıso, Av. Brasil 2950, Valpara´ıso, Chile. (Dated: July 27, 2012) We studied the efficiency of two different schemes for a quantum heat engine, by considering a single Dirac fermion trapped in a one-dimensional potential well as the ”working substance”. The 2 first scheme is a cycle, composed of two adiabatic and two iso-energetic reversible trajectories in 1 configuration space. The trajectories are driven by a quasi-static deformation of the potential well 0 dueto an external applied force. The second scheme is a variant of theformer, where iso-energetic 2 trajectories arereplaced byisothermal ones,alongwhichthesystem isin contactwith macroscopic thermostats. ThissecondschemeconstitutesaquantumanalogueoftheclassicalCarnotcycle. Our l u expressions, as obtained from the Dirac single-particle spectrum, converge in the non-relativistic J limit to some of theexisting results in theliterature for the Schrödingerspectrum. 6 2 PACSnumbers: 05.30.Ch,05.70.-a ] h I. INTRODUCTION thepotentialwell. Thenon-relativisticlimitcorresponds c to the regime where λ/L 1, while evidence of the un- e ≪ m A classical heat engine consists of a cyclic sequence of derlying relativistic nature of the spectrum manifests in terms of finite corrections in powers of λ/L. Another reversible transformations over a ”working substance”, - t typically a macroscopicmass of fluid enclosedin a cylin- limitoftheoreticalinterestisthe”ultra-relativistic”case a of massless Dirac fermions, where λ/L . An impor- t der with a mobile piston at one end [1, 2]. The two →∞ s tant realizationof this former case in solid state systems most famous examples are the Otto and Carnot cycles. . at In particular, the classical Carnot cycle comprises four is providedby conductionelectrons in the vicinity of the so called Dirac point in graphene [11–14]. m stages, two isothermal and two adiabatic (iso-entropic) ones. Ideal quasi-static and reversible conditions are - d achievedbyassumingthatanexternalforce,whichdiffers II. A DIRAC FERMION TRAPPED IN A n only infinitesimally from the force exerted by the inter- o ONE-DIMENSIONAL INFINITE POTENTIAL nalpressureofthe fluid, is appliedto the pistonin order c WELL to let itmoveextremely slowly[1, 2]. Onthe otherhand, [ theisothermaltrajectoriesareperformedbybringingthe 1 fluid contained by the cylinder into thermal equilibrium Wedefineaone-dimensionalpotentialwellbythefunc- v tion with external reservoirs at temperatures T < T , re- 9 C H spectively. V(x)= lim V [Θ(x L)+Θ( x)], (1) 4 0 1 A quantum analogue of a heat engine involves a V0→∞ − − 6 sequence of transformations (trajectories) in Hilbert’s with Θ(x) the Heaviside step function. The eigenvalue . space, where the ”working substance” is of quantum problemforaDiracfermiontrappedinthispotentialwell 7 mechanical nature[3–10]. One of the simplest concep- is stated as follows 0 2 tual realizations of this idea is a system composed by Hˆψˆ=Eψˆ, (2) 1 a single-particle trapped in a one-dimensional potential : well [3–5, 8]. The different trajectories are driven by a where the Dirac Hamiltonian in 1+1 dimensions is [15] v quasi-staticdeformationof the potentialwell, due to the Xi application of an external force. Two different schemes Hˆ =−i~σˆ1∂x+mc2σˆ3+V(x)1ˆ. (3) r of this process have been discussed in the literature, in Here, σˆ and σˆ are Pauli matrices, while ψˆ(x) = a 1 3 the context of a non-relativistic particle whose energy (φ(x),χ(x)) is a two-component spinor. eigenstates are determined by the Schrödinger spectrum The solution, considering the boundary condition at [3, 4, 9]. In this paper, we shallrevisit these approaches, the infinite barrier interfaces φ(0) = φ(L) = 0, is given and we will study the performance of the correspond- by ing heat engine for a single Dirac fermion. Since Dirac’s sin(nπx/L) ereqsuualttisonobdtaeisncerdibefosrtthheisspcaecsterashoofulrdelraetdivuicsetictoptahreticcloers-, ψˆn(x)=A i~c(nπ/L)cos(nπx/L) , (4) (cid:18)− En+mc2 (cid:19) responding ones from Schrödinger’sequation in the non- with associated energy eigenvalues organized in pairs relativistic limit. As we shall discuss below, the transi- with opposite signs [15], tion between the relativistic and non-relativistic regimes is determined by the ratio λ/L, with λ = 2π~/mc the ED(L)= mc2 1+(nλ/2L)2 1 . (5) Compton wavelength of the particle, and L the width of n ± − (cid:18)q (cid:19) 2 FIG.2: (Coloronline)PictorialdescriptionoftheEnergyver- sus Entropy(von Neumann) for the E-Carnot cycle. FIG.1: (Coloronline)PictorialdescriptionoftheForceversus sizeofthepotentialwell,fortheE-CarnotCycledescribedin thetext. particle system, where each copy may be in any of the possible different energy eigenstates. We therefore say Here, we have substracted the rest energy, and λ = that the single-particle system is in a statistically mixed 2π~/(mc)istheComptonwavelength. Thepositive(neg- quantumstate[16]. Thecorrespondingdensitymatrixop- ative) sign corresponds to the particle (anti-particle) so- erator is ρˆ= p (L)ψ (L) ψ (L), with ψ (L) an n n | n ih n | | n i lution,respectively[15]. Inwhatfollows,weshallassume eigenstate of the single-particleHamiltonian Eq.(3), cor- P thatasingleparticleistrappedbythepotentialwell,and respondingtothespinorsdefinedbyEq.(4). Thisdensity hence we shall keep the positive eigenvalue. matrix operator is stationary, since in the absence of an The spectrum predicted by Eq.(5) can be compared external perturbation[16] i~∂ ρˆ= [Hˆ,ρˆ] = 0. Here, the t with the corresponding Schrödinger problem, whose coefficient 0 p (L) 1 represents the probability for n ≤ ≤ eigenvalues are given by the system, within the statistical ensemble, to be in the particular state ψ (L) . Therefore, the p (L) satisfy n2π2~2 | n i { n } ES(L)= . (6) the normalization condition n 2mL2 Here, a single eigenvalue for the energy is obtained and, Trρˆ= pn(L)=1. (9) moreover,it scales as n2, in contrastwith the Dirac par- n X ticle case where a richer scaling with n is observed. In In the context of Quantum Statistical Mechanics, en- particular, in the regime λ/L 1, we have: ≪ tropy is defined according to von Neumann [16, 17] as ED(L) mc2 (nλ/2L)2 =ES(L), (7) S =−kBTrρˆlnρˆ. Sinceintheenergyeigenbasistheden- n → 2 n sity matrix operator is diagonal, the entropy reduces to the explicit expression which corresponds to the non-relativistic limit. Beyond this regime, relativistic corrections depending on the fi- S(L)= k p (L)ln(p (L)). (10) nite ratio λ/L are observed. Another interesting limit − B n n n of Eq.(5) corresponds to a massless Dirac particle with X λ , where the spectrum reduces to the expression In our notation, we emphasize explicitly the dependence →∞ nπ~c of the energy eigenstates ψn(L) , as well as the prob- EnD(L) m=0 = L . (8) abilitycoefficients{pn(L)}{,|onthei}widthofthepotential well L. The ensemble-average energy of the quantum This situation may be (cid:12)of interest in graphene systems, (cid:12) single-particle system is where conduction electrons in the vicinity of the so called Dirac point can be described as effective mass- E Hˆ =Tr(ρˆHˆ)= p (L)E (L). (11) n n less chiral particles, satisfying Dirac’s equation in two ≡h i n dimensions[11–14]. X For the statistical ensemble just defined, we conceive two different schemes for a quantum analogue of a ther- III. A SINGLE-PARTICLE QUANTUM HEAT modynamic heat engine. The first one, that we shall ENGINE refer to as the E-Carnot cycle, consists on four stages of reversible trajectories: two iso-entropic and two iso- Asthe”workingsubstance”foraquantumheatengine, energetic ones. During the iso-energetic trajectories, letusconsiderastatisticalensembleofcopiesofasingle- the ensemble-average energy Eq.(11) is conserved, while 3 during the iso-entropic ones, the von Neumann entropy 0.8 α = 2.0 defined by Eq.(10) remains constant. We distinguish α = 1.8 this first scheme from the quantum Carnot cycle to α = 1.6 0.6 be discussed next, where the iso-energetic trajectories in Hilbert’s space are replaced by isothermal processes. α = 1.4 During these stages, the system is brought into thermal D 0.4 equilibrium with macroscopic reservoirsat temperatures η T T , respectively. α = 1.2 C H ≤ 0.2 IV. THE E-CARNOT CYCLE 0 ThesystemtrajectoriesinHilbert’sspaceareassumed 0 1 2 3 4 5 L /λ tobedrivenbyreversiblequasi-staticprocesses,inwhich A thewallsofthepotentialwellaredeformed”veryslowly” FIG.3: (Coloronline)EfficiencyoftheE-Carnotcycle,calcu- by an applied external force, such that the distance L is lated after Eq.(28),as afunction of theexpansion parameter modified accordingly. Along these trajectories, the total α=LC/LB. change in the ensemble average energy of the system is given by energy is decreasing during expansion, as in a classical dE = p (L)dE (L)+ dp (L)E (L) n n n n ideal gas. n n X X Letus nowconsideraniso-energeticprocess,thatis, a = (δE) +(δE) (12) pn(L) =cnt. En(L) =cnt. trajectoryinHilbertspacedefinedbytheequationdE = { } { } 0. Thesolutiontothisequation,forL [L ,L ],isgiven The first term in Eq.(12) represents the total energy ∈ 1 2 by the path changeduetoaniso-entropicprocess,whereasthesecond term represents a trajectory where the energy spectrum remains rigid. ∞ ∞ p (L)E (L)= p (L )E (L ), (15) n n n 1 n 1 Let us consider first the iso-entropic process, where n=1 n=1 p (L) =cnt. We remark that this represents a strong X X n { } sufficient condition for the entropy to remain constant along with the normalization condition Eq.(9). Clearly, along the trajectory, but is not a necessary one[18]. Un- by definition an iso-energetic process satisfies derquasi-staticconditions,theexternalforcedrivingthe change in the width of the potential well is equal to dE =δW +δQ =0, (16) 1 2 1 2 the internal ”pressure” of the one-dimensional system, → → Fsys=te−m(∂agEa/in∂sLt)tSh.eTehxeterernfoarlef,otrhcee,wwohrkenptehrfeowrmidetdhboyf tthhee with δW1→2 ≡ (δE){pn(L)}=cnt. and δQ1→2 ≡ (δE) . In the former equation, the integral potential well expands from L = L1 to L = L2, is given along{Ethn(eLt)}r=ajcenct.tory L L gives 1 2 by → ∆E =W +Q =0. (17) L2 ∂E 1→2 1→2 W = dL 1 2 → ZL1 (cid:18)∂L(cid:19){pn(L)}={pn(L1)}=cnt. TwohrekfipresrtfortmeremdbWy1t→he2sycostrermespaognadinssttothetheextmerencahlafonriccael, ∞ = pn(L1)[En(L2) En(L1)]. (13) whichdrivesthechangeinthewidthofthepotentialwell − nX=1 acotrcreosnpsotanndtsetnoetrhgye.amThoeunstecoofnednetregrmy eQxc1h→a2ng=ed−bWy1t→h2e For the case of a Dirac fermion, the work performed system with the environment, in order to rearrange its under iso-entropic conditions is given after Eq.(13) and internal level occupation. This equation is in analogy Eq.(5) by withthefirstlawofthermodynamicsformacroscopicsys- tems, when considering a reversible process over a clas- W = mc2 ∞ p (L ) 1+(nλ/2L )2 sical ideal gas which is being compressed/expanded at 1 2 n 1 2 → n=1 (cid:18)q constant internal energy conditions. The first term has X a precise correspondence with the mechanical work for 1+(nλ/2L1)2 . (14) expansion/compression, whereas the second is in corre- − q (cid:19) spondence with the heat exchanged by the gas with the Notice that our sign convention is such that, for an ex- environmentinordertosatisfytotalenergyconservation. pansionprocessL >L ,theworkperformedbythesys- Accordingtothepreviousanalysis,theheatexchanged 2 1 temisnegative[2],indicatingthattheensemble-averaged by the system with the environment along the iso- 4 energetic process is given by The heat released to the environment along this first stage of the cycle is calculated after Eq.(20), ∞ L2 dpn(L) Q1→2 =nX=1ZL1 En(L) dL dL. (18) −QA→B = E1D(LA)ln(cid:20)EE2D2D(2(LLAA))−−EE11DD((2LLAA))(cid:21) Evidently,Eq.(15)combinedwiththenormalizationcon- 1 mc2+ED(L ) + mc2ln 2 A . (23) dition Eq.(9) are not enough to uniquely define the co- 4 mc2+ED(2L ) efficients p (L) along the iso-energetic trajectory. An (cid:20) (cid:26) 1 A (cid:27)(cid:21) n exceptionisthecasewhentheenergyscaleofallthepro- The next process is an iso-entropic expansion, charac- cesses involved is such that only transitions between the terized by the condition p2(LB)=p2(LC)=1. We shall ground state (n = 1) and the first excited state (n = 2) define the expansion parameter α LC/LB > 1. The ≡ are possible. In this effective two-level spectrum, com- work performed during this stage, with LB = 2LA (as bining Eq.(15) with the normalization condition Eq.(9), discussed before), is calculated from Eq.(14), the trajectory for the iso-energetic process is described by the following relation λ 2 λ 2 W = mc2 1+ 1+ . B C E2(L1) E2(L) E1(L1) E2(L1) → s (cid:18)2LA(cid:19) −s (cid:18)2αLA(cid:19)  p (L) = − + − p (L ), 1 E1(L) E2(L) E1(L) E2(L) 1 1  (24) − − (19) The cycle continues with a maximal compression pro- withp2(L)=1 p1(L)afterthe normalizationcondition cess from LC = 2αLA to LD = αLA under iso-energetic Eq.(9). Thehea−texchangedbythesystemwiththeenvi- conditions. The energy conservation condition is in this ronmentduringaniso-enegetictrajectoryconnectingthe case similar to Eq.(22), with p2(LC) = p1(LD) = 1. initial and final states L L , for the case of a Dirac Theheatexchangedbythesystemwiththeenvironment 1 2 particle, is given by the ex→pression along this process, applying Eq.(20), is given by the expression Q = ED(L )+ ED(L ) ED(L ) −×p11(→L21)]ln(cid:2) 2EE1DD((1LL2))−(cid:0) EE12DD((LL12))−+22(m1c2(cid:1))ln LL2 −QC→D = E2D(2αLA)ln(cid:20)EE2D2D(2(ααLLAA))−−EE11DD((α2αLLAA))(cid:21) m(cid:20)c21+E1D−(L )2mc12+(cid:21)ED(L ) (cid:18) 1(cid:19) +mc2ln 1 mc2+E2D(αLA) (25) +(mc2)ln mc2+E2D(L2)mc2+E1D(L2) (20) (cid:20)4(cid:26)mc2+E1D(2αLA)(cid:27)(cid:21) (cid:20) 2 1 1 1 (cid:21) whereED(L)wasdefined inEq.(5). The lastpathalong n where ED(L) was defined in Eq.(5). the cycle is an adiabatic process, which returns the sys- n Fortheeffectivetwo-levelsystempreviouslydescribed, temtoitsinitialgroundstatewithp (L )=p (L )=1. 1 D 1 A we shall conceive a cycle, as depicted in Figure 1, which The work performed during this final stage, as obtained starts in the ground state with p1(LA) = 1. Then, by applying Eq.(14), is given by the system experiences an iso-energetic expansion from LA LB > LA. Then, it experiences an iso-entropic λ 2 λ 2 ecxompa→pnrseisosniofnroLm LBL→<LCL>, aLnBd,fitnhaenllyaint gisooe-senbearcgkettioc WD→A = mc2s1+(cid:18)2αLA(cid:19) −s1+(cid:18)2LA(cid:19) . C D C → the initialgroundstatethroughaniso-entropiccompres-  (26) sion L L . D A We sh→all assume that the final state after the iso- Itisinterestingtocheckthattheworkalongthe twoiso- penaenrsgioenti,ctphraotceisss, LthAe→sysLtBemcoernredsspocnomdsptleotemlyaxliomcaallizeexd- eTnhterroepfiocrter,atjhecetoerffiiecsiecnacnyceolsf,tthhaetciyscWleBi→sCd+efiWneDd→bAy=th0e. in the excited state n = 2. In this condition, Eq.(19) ratio reduces to Q ηD =1 C→D. (27) − Q p1(LB)=0, p2(LB)=1. (21) A→B When substituting the corresponding expressions from The condition of total energy conservation between the Eq.(23) and Eq.(25) into Eq.(27), we obtain the explicit initialandfinalstatesconnectedthroughaniso-energetic analytical expression process, for maximal expansion, leads to the equation p1(LA)E1(LA)=p2(LB)E2(LB), (22) ηD = 1 ln(cid:20)4111++θθ((α2LαAL/A2))nθθ((ααLLAA/)2−)−θ(θ2(ααLLAA))oθ(αLA)(cid:21). where p1(LA) = p2(LB) = 1 for maximal expansion. − ln 11+θ(LA/2) θ(LA/2)−θ(LA) θ(LA) Therefore Eq.(22), given the Dirac spectrum Eq.(5), im- (cid:20)4 1+θ(2LA) n θ(LA)−θ(2LA) o (cid:21) plies that L /L =2. (28) B A 5 Neumannentropyofthe systemachievesamaximumfor the Boltzmann distribution [16, 17] pn(L,βH)=[Z(L,βH)]−1e−βHEnD(L), (31) with β =(k T) 1, andthe normalizationfactor is given B − by the partition function Z(L,β)= ∞ e−βEnD(L) 2Leβmc2K1(βmc2). (32) ∼ λ n=0 X Here, the second expression, as shown in Appendix, is the continuum approximation to the discrete sum, valid in the physically relevant regime λ L. Here, K (x) is 1 ≪ a Bessel function of the second kind. FIG.4: (Coloronline)PictorialdescriptionoftheForceversus From a similar analysis as in the previous section, we width of the potential well, for the Quantum Carnot Cycle described in the text. conclude that the heat exchanged by the system to the thermal reservoir is given by Here, we have defined the function θ(L) = Q = LB ∞ E (L)dpn(L,βH)dL A B n 1+(λ/2L)2, with λ = 2π~/(mc) the Compton → ZLA n=0 dL X wavelength. It is important to remark that in the p ∂ln Z(LB,βH) nreodnu-rceelsattoivitshteicklnimowitnλS/cLho¨→din0,gethrelimexiptressionin Eq.(27) = − (cid:16)Z∂(βLA,βH)(cid:17) +βH−1ln ZZ((LLB,,ββH)) H (cid:18) A H (cid:19) L lim ηD =1 1/α2. (29) = mc2ln B . (33) λ/L→0 − (cid:18)LA(cid:19) In the second line, we have done integration by parts, In the ”ultra-relativistic” case of a massless Dirac and we made direct use of the definition Eq.(30) of the fermion,λ/L ,theefficiencyconvergestothelength →∞ partition function. The final result follows from substi- independent limit tuting the explicit expression for the partition function lim ηD =1 1/α. (30) Eq.(32). λ/L − Similarly, during the third stage of the cycle, the sys- →∞ tem is again brought into contact with a thermal reser- Thetrendoftheefficiencyisshowninbetweenbothlim- voir, but at a lower temperature T < T . Therefore, C H its in Figure 3. It is worth to remark that, since the the probability distribution of states in the ensemble is expansion parameter α = LC/LB > 1, the efficiency in pn(L,βC), as defined in Eq.(31), but with TC instead of the strict non-relativistic (Schrödinger) regime λ/L 0 T . The heat released to the reservoir during this stage → H Eq.(29)isthehighestpossibleone. Thisisalsoclearfrom is given by the expression the asymptotics of the curves displayed in Fig. 3, where L the ”ultra-relativistic” limit corresponding to massless Q =mc2ln D . (34) C D fermions (λ/L ) indeed represents the less efficient → (cid:18)LC(cid:19) → ∞ regime for a fixed expansion parameter α. The second and fourth stages of the cycle consti- tute iso-entropic trajectories. In order to analyze these stages, we shall derive the ”equation of state” for the V. THE QUANTUM CARNOT CYCLE statistical ensemble of single-particle systems. When substituting the Boltzmann distribution p (β,L) = n In this section, we shall discuss the quantum version [Z(β,L)]−1exp(−βEnD(L)) into the expression for the of the Carnot cycle, as applied to the statistical ensem- von Neumann entropy Eq.(10), we obtain the relation ble of Dirac single-fermion systems under consideration. S =T 1E+k lnZ(β,L). (35) The thermodynamic cycle which defines the correspond- − B ingheatengineis composedoffourstagesortrajectories Here, E = Hˆ is the ensemble-average energy, as de- in Hilbert’s space: Two isothermal and two iso-entropic fined by Eq.(h11i). The equation of state is obtained from processes. Eq.(35) as In the first stage, the system is brought into contact with a thermal reservoirat temperature T . By keeping ∂E ∂ H F = =k T lnZ(β,L) B isothermal conditions, the width of the potential well is − ∂L ∂L (cid:18) (cid:19)S ewxipthantdheedrefsreormvoiLrAis→assuLmBe.d aSlionnceg tthhiesrmpraolceesqsu,itlihberivuomn = kBT. (36) L 6 In the last line, we have used the explicit analytical expression Eq.(32) for the partition function to calculate thederivative. TheequationofstateEq.(36)reflectsthat the ensemble of systems behaves as a one-dimensional ideal gas. This is not surprising, since the ensemble- average energy is given by ∂ E = Hˆ = lnZ(β,L) h i −∂β d = mc2 1 lnK (z) , (37) 1 − − dz (cid:18) (cid:19) FIG.5: (Coloronline)PictorialdescriptionoftheForceversus where we have substituted explicitly the expression for width of thepotential well for theQuantumCarnot Cycle. the partition function, and in the final step we defined z =βmc2. Here,K (x)isaBesselfunctionofthesecond 1 kind. Eq.(37)showsthatthe ensemble averageenergyof Dirac particle with z 0, the iso-entropic trajectory is the system is a function of the temperature solely, from given by the equation→LT−1 =cnt. whichthe idealgasequationof state followsasa natural We are now in conditions to discuss the second and consequence. We can thus define the ”specific heat” at fourth stages of the Carnot cycle. The second stage of constant length, which after Eq.(37) is given by theprocesscorrespondstoaniso-entropictrajectory,pa- rameterized in differential and integral form by Eqs.(40) ∂E dE d2 and (41), respectively. The work performed by the sys- C = = =k z2 lnK (z), (38) L ∂T dT B dz2 1 tem during this process is given by (cid:18) (cid:19)L where z = βmc2. It is interesting to remark that, W = LCFdL= LCk TdL B C B based on the asymptotic behaviour of the Bessel func- → −ZLB −ZLB L tions Kn(z) ∼ π/2z−1/2exp(−z)+ O(z−1), the spe- = mc2 zC d2 lnK (z)dz (42) cificheatdefinedinEq.(38)presentstheasymptoticlimit − dz2 1 C k /2 whepn k T mc2. This is the well known ZzH L B B → ≪ result for a classic non-relativistic ideal gas in one di- Here, in the second line we have used the differential mension. This feature and the general temperature de- equation defining the iso-entropic trajectory, Eq.(40). pendence of the ensemble specific heat is displayed in Evaluating the integral in Eq.(42), we explicitly obtain Fig. 6. The change in the ensemble averaged energy of Sthinecseytshteeme,nsfoermableg-eanveerraalgpereonceersgs,yisisdaEfu=ncTtidoSn−ofFtedmL-. WB→C =kB(TC −TH)−mc2(cid:20)KK10((zzCC)) − KK10((zzHH))(cid:21)(43) peratureonly,thedifferentialequationforaniso-entropic trajectory (dS =0) is[19] Thefourthandfinalstageofthecyclealsocorresponds toaniso-entropictrajectoryL L ,andtheworkper- dL D → A dE =C dT = FdL= k T . (39) formed by the system against the external applied force L B − − L is obtained similarly as in Eq.(42), Separating variables, after some algebra we obtain LA W = FdL (44) z d2 lnK (z)= dL. (40) D→A −ZLD dz2 1 L K (z ) K (z ) = k (T T ) mc2 0 H 0 C . B H C − − K (z ) − K (z ) Integrating Eq.(40) between initial conditions (z0,L0) (cid:20) 1 H 1 C (cid:21) and final conditions (z,L), we have Clearly, after Eqs.(43) and (45), we have W + B C W =0,andhencethecontributionofthework→along L = K1(z0)ezKK01((zz))−z0KK10((zz00)). (41) theD→isoA-entropic trajectories vanishes. L K (z) 0 1 From the equation for the iso-entropic trajectory Eq.(41), we conclude that the length ratios are deter- Here, we made use of the Bessel function identity K′(z) = K (z) z 1K (z), with z = βmc2. It is inter- mined by the temperatures of the thermal reservoirs, 1 0 − − 1 gesivtienngtthoecahseycmkptthoatticinbethhaevnioounr-roeflatthiveisBtiecssliemliftunzc≫tion1s, LC = LD = K1(zH)ezCKK10((zzCC))−zHKK10((zzHH)). (45) L L K (z ) K (z) π/2z 1/2exp( z)+O(z 1), Eq.(41) reduces B A 1 C n − − ∼ − to LT 1/2 =cnt. for the iso-entropic trajectory. On the From Eq.(45), we also obtain L /L =L /L . Sub- − p A B D C other hand, in the ”ultra-relativistic”limit of a massless stituting this relation in the expression for the efficiency 7 tively. Therefore, the statistical ensemble under consideration is described by the density matrix ρˆ= e βHˆ/Z, − with Z[β,L] = Tre βHˆ the partition function. We − showedthatthestatisticalpropertiesoftheensembleare suchthatanequationofstatecanbedefined,aswellasa specific heat, in analogy with a classical ideal gas in one dimension. Weobtainedtheequationfortheiso-entropic trajectory,whichinthenon-relativisticlimitk T mc2 B ≪ reduces to the classical result LT 1/2 = cnt. On the − otherhand,wealsoshowedthatinthe”ultra-relativistic” limitofamasslessDiracfermion,asforinstanceconduc- tion electrons in graphene, the iso-entropic trajectory is defined by the equation LT 1 = cnt. We also showed − that the efficiency for the Quantum Carnot cycle satis- fiesthesamerelationthattheclassicaloneintermsofthe FIG.6: (Coloronline)Specificheatofthestatisticalensemble temperaturesofthe thermostats,thatis η =1 T /T . C H of single Dirac-fermion systems, after Eq.(38), as a function − of temperature. Acknowledgements of the cycle, from Eq.(33) and Eq.(34) we have The authors wish to thank Dr. P. Vargas for inter- Q T ln(L /L ) esting discussions. E.M. acknowledges financial support ηC = 1 C→D =1 C D C from Fondecyt Grant 11100064. F.J.P. acknowledges fi- − Q − T ln(L /L ) A B H B A → nancial support from a Conicyt fellowship. T C = 1 . (46) − T H Appendix Therefore, we have recovered the expression for the efficiency identical to the classical Carnot cycle. We here discuss the continuum approximation to the discrete partition function Eq.(30). VI. CONCLUSIONS Z(L,β)= ∞ e−βmc2(cid:18)q1+(n2Lλ)2−1(cid:19) By considering as a ”working substance” the statisti- calensembleforaDiracsingle-fermionsystemtrappedin nX=0 a one-dimensional potential well, we have analyzed two Here, let us define the discrete variable x = nλ/(2L). n different schemes for a quantum heat engine. The first, The spacing between two consecutive values is given by that we have referred to as the E-Carnot cycle, consists ∆x=x x =λ/(2L), and hence the expression for n+1 n oftwoiso-entropicandtwoiso-energetictrajectories. We − the partition function can be written as obtained an explicit expression for the efficiency of this cycle and showed that our analytical result, in the non- relativistic limit λ/L 0, reduces to the corresponding Z(L,β)= 2L ∞ ∆xe−βmc2(cid:16)√1+x2n−1(cid:17) → λ one for a Schrödinger particle, as reported in the litera- n=0 X ture [3]. Our results also indicate that the efficiency for Forphysicallyrelevantsizesofthepotentialwell,weshall thisE-Carnotcycleishigherinthenon-relativisticregion haveλ/L 1,whichallowustotakethecontinuumlimit ofparameters,that is for L λ, whencomparingatthe ≪ ≫ inthesenseofaRiemannsumforthepreviousequation, same compressibility ratios α = L /L > 1. An excep- C B tion is the case of massless Dirac fermions, with λ= , where it is not possible to achieve non-relativistic con∞di- Z(L,β) 2L ∞dxe−βmc2(√1+x2−1) → λ tions. This is of potential practical interest for graphene Z0 systems,whereconductionelectronsareindeeddescribed = 2L ∞dte βmc2(cosh(t) 1)cosh(t) − − as massless chiral Dirac fermions[11–14]. λ Z0 As a second candidate for a quantum heat engine, we 2L discussed a version of the Carnot cycle, composed by = eβmc2K1 βmc2 . λ two iso-thermal and two iso-entropic trajectories. In or- (cid:0) (cid:1) der to achieve iso-thermal conditions, we consider that Here, in the second line we have made the substitution the single-fermion system is in thermal equilibrium with x=sinh(t), andK (x) is a Besselfunctionofthe second 1 macroscopicreservoirsattemperaturesT <T ,respec- kind. C H 8 [1] E. Fermi, Thermodynamics (Dover,1936). [13] E. Munõz, J. Lu, and B. I. Yakobson, Nano Lett. 10, [2] H. B. Callen, Thermodynamics and an Introduction to 1652 (2010). Thermostatistics (John Wiley and Sons, 1985), 2nd ed. [14] E.Munõz,J.Phys.: Condens.Matter24,195302(2012). [3] C.M. Bender,D.C. Brody,andB. K.Meister, Proc. R. [15] J.D.Bjorken andS.D.Drell,Relativistic Quantum Me- Soc. Lond. A 458, 1519 (2002). chanics (Mc Graw-Hill, 1964). [4] C. M. Bender, D. C. Brody, and B. K. Meister, [16] J.vonNeumann,MathematicalFoundationsofQuantum arXiv:quant-ph/0007002v1 (2000). Mechanics (Princeton University Press, 1955). [5] J. Wang and J. He, J. Appl. Phys.111, 043505 (2012). [17] R. C. 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