Quantum thermodynamic Carnot and Otto-like cycles for a two-level system

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1 OFFPRINT Quantum thermodynamic Carnot and Otto-like cycles for a two-level system Gian Paolo Beretta EPL, 99 () 5 Please visit the new website

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5 July EPL, 99()5 doi:.9/95-575/99/5 Quantum thermodynamic Carnot and Otto-like cycles for a two-level system GianPaoloBeretta (a) Università di Brescia- via Branze 8, 5 Brescia, Italy, EU receivedjanuary;acceptedinfinalform6june published online 8 July PACS 5.7.-a Thermodynamics PACS.65.-w Quantum mechanics Abstract Within the recent revival of interest in quantum heat engines between two thermal reservoirs whereby the working substance is a two-level system, it has been suggested that the celebratedcarnotheat-to-workconversionefficiency (T low / )cannotbereached.contrary tothissuggestion,weshowthatreachingthecarnotboundnotonlyisnotimpossibleanddoes not require an infinite number of heat baths and infinitesimal processes, but it is also within reach of the current experimental techniques. It is sufficient to cycle smoothly(slowly) over at least three (ingeneralfour)valuesofthetunableenergylevelgapδofthesystem,byvaryingδnotonlyalong the isoentropics, but also along the isotherms. This is possible by means of the recently suggested maser-laser tandem technique. We base our proof on the general thermodynamic equilibrium properties of a two-level system together with a careful distinction between the Gibbs relation de=tds+(e/δ)dδandtheenergybalanceequationde=δq δw.wederivebounds to the net-work to high-temperature-heat ratio(energy efficiency) for a Carnot cycle and for the inscribed Otto-like cycle. By representing these cycles on useful thermodynamic diagrams, we infer and confirm important aspects of the second law of thermodynamics. Copyright c EPLA, Introduction. Recent studies [ 7] of Maxwell demons, quantum heat engines (often called Carnot engines even if the cycle is not a Carnot cycle), and quantum heat pump, refrigeration and cryogenic cycles operatingbetweentwoheatsourcesat andt low find maximal efficiencies lower than the celebrated Carnot network to high-temperature-heat ratio, (T low / ). In particular, ref. [], in studying a specific two-isoenergy-gap/two-isoentropic-processes Otto-type cycle for a spin-/ system, seems to hint that the quantum nature of the working substance implies a fundamental bound to the thermodynamic efficiency of heat-to-work conversion, lower than the celebrated Carnot bound. Pioneering studies [8] of quantum equivalents of the Carnot cycle for multilevel atomic and spin systems appeared soon after the association of negative temperatures with inverted population equilibrium states of pairs of energy levels[9,] and the experimental proof of the maser principle[]. Intunewiththeseearlystudies,hereweshow[]that acarnotcycleforatwo-levelsystemispossible,atleastin (a) beretta@ing.unibs.it principle, but requires cycling over a range of values of the energy level gap Δ. A critical and characteristic feature ofthiscycleisthatalongtheisothermsthevalueofδ must vary continuously and hence the two-level system must experience simultaneously a work and a heat interaction. Usually, the different typical time scales underlying mechanical and thermal interactions imply fundamental technological difficulties that are among the main reasons why the Carnot cycle has hardly ever been engineered with normal substances. In the framework of quantum thermodynamics the understanding and modeling of mechanical and thermal interactions is a current research topic, having to do with entanglement, decoherence[], relaxation[], adiabatic(unitary) accessibility[5], but a recentsuggestionbyscully[]indicatesthattheuseofa maser-laser tandem may provide an effective experimental means to implement the simultaneous heat and work interactionbysmooth(slow )continuouschangeofthe Theenergylevelgap mustbetunedslowlysoastomaintain the system along a sequence of thermodynamic-equilibrium states, thereby avoiding irreversible relaxation effects that would obtain if fast tuning of drives the system off equilibrium. Such kind of inefficiencies are discussed in refs.[5,6]. 5-p

6 Gian Paolo Beretta magnetic field necessary to realize the isotherms of our Carnot cycle: the maser serves as the incoherent(heat) energy and entropy exchange mechanism, the laser as the coherent(work) energy exchange. Consider a two-level system with a one-parameter Hamiltonian H(Δ) such that the energy levels are ε = Δ/andε =Δ/,forexampleaspin-/system, inamagneticfieldofintensityb,withδ=µ B B and µ B =e /m e =9.7 J/TBohr smagneton. Forourpurposeshereitsufficestoconsiderthecanonical Gibbs states(the stable equilibrium states of quantum thermodynamics), i.e., the two-parameter family of densityoperatorsρ(t,δ)witheigenvaluespand p, meanvalueoftheenergye,andentropysgivenbythe relations ρ(t,δ)= exp[ H(Δ)/k BT] Trexp[ H(Δ)/k B T], () p= +exp(δ/k B T) = +E Δ, () E=Trρ(T,Δ)H(Δ)= Δ [ tanh Δ/ ], () k B T S= k B Trρlnρ= k B [plnp+( p)ln( p)] [( ) ( ) ( ) ( )] = k B +E ln Δ +E + Δ E ln Δ E. () Δ We note that the thermodynamic-equilibrium fundamentalrelation S=S(E,Δ)forthissimplestsystemtakes theexplicitforms=s(e/δ)givenbythelastofeqs.(). As is well known, all equilibrium properties can be derived from the fundamental relation. We also note the following dependences on the two parameters(temperature T and energylevelgapδ):eq.()impliesthattheratioe/δ dependsonlyontheratioδ/t,therefore,eqs.()and() implythatalsopandsarefunctionsoftheratioδ/t only. It is clear from () and () that for an isoentropic process, S=const E Δ =const Δ T =const E T =const (5) and, hence, also the Massieu characteristic function M = S (E/T) is constant. More generally, by differentiating the second of eqs. () and using eqs. (), () and the identity ln[(+tanhx)/( tanhx)]=x with x= Δ/k B T,wefind ds= k B ln p p dp= k Bln +E/Δ E/Δ de Δ = Δ T de Δ = T de E dδ, (6) TΔ therefore, the following Gibbs relation holds for all processesinwhichtheinitialandfinalstatesofthetwolevel system are neighboring thermodynamic-equilibrium states: de=tds+(e/δ)dδ. (7) 5-p Next we write the energy balance equation assuming that the system experiences both net heat and work interactions [6] with other systems in its environment (typically a heat bath or thermal reservoir at some temperaturet Q,andaworksinkorsource,respectively), de=δq δw, (8) where we adopt the standard notation by which a left (right) arrow on symbolδq (δw) means heat (work) received by (extracted from) the system, when δq (δw )ispositive(negative). Comparing the right-hand sides of eqs. (7) and (8), thefollowingidentificationofaddenda,δq =TdSand δw =( E/Δ)dΔistempting,butnotvalidingeneral unless we make further important assumptions. To prove and clarify this last assertion, we consider two counterexamples, in both of which the system changes between neighboring thermodynamic-equilibrium states so that botheqs.(7)and(8)hold. As a first counterexample, consider a system which experiences a work interaction with no heat interaction (δq =).TheenergychangedEisprovidedbythework interaction only, while the entropy change ds,required to maintain the system at thermodynamic equilibrium, is generated within the system by irreversible relaxation and decoherence. In this case, the entropy balance equation is ds=δs gen,whereδs gen denotestheentropygeneratedby irreversibility. The work is δw = E Δ dδ TδS gen [ EΔ dδift> ], (9) and,ofcourse,theprocessispossibleonlyifds. As a second counterexample, consider a system which experiences no(net) work interaction and a heat interactionwithasourceattemperaturet Q sothattheentropy exchangedwiththeheatsourceisδs =δq /T Q.Inthis case,theenergychangedeisprovidedbytheheatinteraction only, while the entropy change ds required to maintain the system at thermodynamic equilibrium is partly provided by the heat source and partly generated within thesystembyirreversibility(ds=δq /T Q +δs gen ).The heat is δq =TdS+ E dδ [ TdSifdΔ ], () Δ andtheprocessispossible,fort>,onlyif ) E ( Δ dδ TTQ de. () The two examples show clearly that the correct association between the work and heat exchanged, and the energy and entropy changes, cannot be made by just comparing eqs.(7) and(8) without considering also the entropy balance equation ds= δq T Q +δs gen, withδs gen, ()

7 Quantum thermodynamic Carnot and Otto-like cycles for a two-level system written here assuming the system experiences heat interaction(s) with a single heat source, bath or thermal reservoirattemperaturet Q.Thisbalanceequationspecifies unambiguously what part of the entropy change is due to exchange via heat interaction(s) and what part is generated spontaneously within the system due to its internal dynamics(relaxation, decoherence). Eliminating de and dsfromeqs.(7),(8)and(),wefind δw = E ( Δ dδ+ T )δq TδS gen. () T Q ThisreducestoδW =( E/Δ)dΔifandonlyif δs gen = δq T δq T Q, () i.e., only when entropy generation is due exclusively to the heat interaction across the finite temperature difference between system and heat source, and not to other irreversible spontaneous processes induced in the system by other interactions that tend to pull the system offthermodynamicequilibrium(seefootnote ). Equation()andtheconditionδS gen implythat thesystemcanreceiveheatonlyif /T Q /T,which forpositivetemperaturesimpliest Q T.Asevidenced by Ramsey[], /T measures the thermodynamic equilibrium escaping tendency of energy by heat interaction, and is a better indicator of hotness than the temperaturet because it validly extends to negative-temperature states.figureshowsgraphsofenergye,entropys,and Massieufunctionplottedasfunctionsof /k B T andδ, aswellasgraphsofe,sandδvs.s. Each graph in fig. shows a Carnot cycle, i.e.,a sequence of an isothermal process - at a high temperaturet high,anisoentropic-,anotherisothermalprocess atatemperaturet low <,andanotherisoentropic-. BecauseS=S(Δ/T)(SdecreasingwithΔ/TforT>), itisclearthattheisoentropicchangesbetweent low and and the isothermal changes between S =S and S =S are possible only by changing the energy level gapδ.therefore,tohaves >S,weneedΔ / = Δ /T low <Δ /T low =Δ /,i.e., Δ high Δ low [ Tlow ] = Δ Δ > Δ Δ = T low = Δ Δ > Δ low Δ high, (5) where,notingthatδ <Δ <Δ andδ <Δ <Δ,we setδ low =Δ andδ high =Δ.Relations(5)imply(see also figure legend) general bounds on the net-work to hightemperature-heat ratio(carnot coefficient), Δ high Δ low [ Tlow ] < W net, Q = T low < Δ low Δ high. (6) Notice that Δ Δ depending on whether Δ low /Δ high (T low / ). Indeed, we may choose arbitrarily, T low <,Δ high =Δ,and Δ low =Δ <Δ high T low /. Then, we must set Δ =Δ low /T low and Δ =Δ high T low /.To obtainacarnotcyclebetweenonlythreevaluesofδ,we maysetδ low =Δ high (T low / ) sothatδ =Δ. Ofcourse,ifthecycleisreversed,weobtain,insteadof a heat-engine effect, a refrigeration or heat-pump effect. Eachgraphinfig.showsalsoanOtto-typecycle[,6] boundbythesame andt low,i.e.,asequenceofan iso-energy-gapprocess - atδ high,anisoentropic -, anotheriso-energy-gapprocess - atδ low,andanother isoentropic -. Here, the fact that S=S(Δ/T)isa decreasingfunctionofδ/tfort>,impliesthattohave S >S,weneedΔ high /=Δ low /T <Δ high /T = Δ low /T low,i.e., T low [ ] Δ low Δ = T high T > T = Δ low Δ = T low > T low. high T (7) Relations (7) imply (see also figure legend) general bounds on the net-work to high-temperature-heat ratio, T low [ Δ low Δ high ] < W net, Q = Δ low Δ < T low. high (8) Notice that T T depending on whether (Δ low / Δ high ) T low /.Thus,toobtainaspecialOtto-like cycle with T =T and efficiency (T low / ) /, we may set (Δ low /Δ high ) =T low / so that Δ =Δ. Notice also that in terms of the isoenergy-level gaps of the Carnot cycle in which the Otto cycle is inscribed, the case T >T obtains for (T low / ) 5/ <Δ low /Δ high <(T low / ) /.Inthis range, the Otto cycle cannot be run in reverse(refrigerationorheat-pump)modebetweentwoheatbaths,forin suchmodethehotbathtemperaturemustbeatmostt andthecoldbathatleastt. Because the iso-energy-gap processes(iso-magnetic field for spin-/ system) which characterize the Otto-type cycle are not isotherms, if they are obtained [] by contactswithheatbathsat andt low,respectively, they involve entropy generation due to irreversibility resulting from the heat exchange (see eq. ()) across alargetemperaturedifference(decreasingast andt T low ).Theseandotherrealisticirreversibilities are modeled in ref.[6] with a Kossakowski-Lindblad-type linear dissipative term in the quantum dynamical law, as a means to describe relaxation to equilibrium and decoherence, required for example to decouple the system from the heat source, i.e., to model dynamically the heat interactions. The only way to avoid these inefficiencies is the impractical sequence of infinitesimal contacts with an infinite collection of hot heat baths covering the temperaturerangebetweent and andaninfinite collection of cold heat baths covering the temperature rangebetweent andt low. 5-p

8 Gian Paolo Beretta.5 x.5 =.968 =.8 =.97 =.67.5 x.5.5 x.5 =.968 =.855 =.8 =.97.5 x.5 E [J] S [J/K] M = S E/T [J/K] x x 5 /k x B T [J ] E [J] T [K] Q.5 W W.5 [J] x W W Q S [J/K] x T = K T = 6 K T = K T = 5 K E [J] S [J/K] M = S E/T [J/K] x x 5 /k x B T [J ] E [J] T [K] [J] x T = K T = 6 K T = K T = 5 K S [J/K] x (a) <, T <T (b) <, T <T (c) >, T >T (d) >, T >T Fig. :(Color online) Graphs of thermodynamic equilibrium properties of a spin-/ system. Panels(a) and(c): graphs of energy E, entropy S, Massieu function M=S (E/T) vs. /k BT at four values of the energy level gap Δ. Panels (b) and(d):graphsofenergye andtemperaturet vs.entropys atfourvaluesofδ,andoftheenergylevelgapδvs.s at four values of temperature. The four values of Δ in (a) and (b) correspond to magnetic-field intensities of 6, 5, 5, and 8T, respectively; in (b) and (d) of 6,, 5, and 8T (these extremely large values are chosen for ease of visualization at ordinary temperatures and intermediate entropies). On each graph, the --- paths represents a Carnotcycle:isotherm-at =6K,isoentropic-,isotherm-atT low =K,isoentropic-.Forthesecycles, Q = (S S ),W =E E Q,W =E E,Q =T low (S S ),W =E E Q,W =E E and, therefore,w net,=w +W W W =( T low )(S S )=Q [ (T low / )].The paths( = and =in(c)and(d))representinsteadanotto-typecycleofthekindconsideredinrefs.[,6]:iso-energy-gapprocess - witht =6K,isoentropic -,iso-energy-gap - witht T low =K,isoentropic -.FortheseOtto-typecycles, Δ =Δ =Δ high=max(δ,δ ),Δ =Δ =Δ low=min(δ,δ ),Q =E E,W =E E =E [(Δ low/δ high) ],W =E E =E [(Δ low/δ high) ]and,therefore,w net, =W W =Q [ (Δ low/δ high)]. In this paper, instead, by showing the feasibility of acarnotcycleforatwo-levelsystem,withnoneedof sequences of infinitesimal heat exchanges with an infinite numberofheatbaths,weshowthatthequantumnature of the working substance does not impose any fundamental bound, other than the celebrated Carnot bound, to the thermodynamic efficiency of heat-to-work conversion when two different temperature thermal reservoirs are available. The possibility of engineering simultaneously heat and work interactions as needed for the isotherms of the Carnot cycle seems within the reach of the current experiments, e.g., via a maser-laser tandem technique[]. The Carnot cycle efficiency is higher, as it should, than that of the inscribed Otto-like cycle at the center of recent studies[ 6]. Only twenty years ago quantum thermodynamics and pioneering proposals to incorporate the second law of thermodynamics into the quantum level of description were considered adventurous schemes[7,8]. Discussions in quantum terms of old thermodynamic problems 5-p

9 Quantum thermodynamic Carnot and Otto-like cycles for a two-level system such as that of unitary accessibility [5] or of defining entropy for non-equilibrium states, were perceived as almost irrelevant speculations. Today s experimental techniques bring thermodynamics questions back to the forefront of quantum theory. Remarkably, the rigorous application of energy and entropy balances, provides ideas and guidance, and the second law remains a perpetual source of inspiration towards the discovery of new physics. Work done as part of the UniBS-MIT-MechE faculty exchange program under grant 8-9 by the CARIPLO Foundation, Italy. REFERENCES [] ScullyM.O.,Phys.Rev.Lett.,88()56. [] LloydS.,Phys.Rev.A,56(997)7;LloydS.and ZurekW.H.,J.Stat.Phys.,6(99)89. [] HeJ.,ChenJ.andHuaB.,Phys.Rev.E,65() 65; Bender C. M., Brody D. C. and Meister B.K.,J.Phys.A,()7. [] KieuT.D.,Phys.Rev.Lett.,9(). [5] Geva E.andKosloff R.,J.Chem.Phys.,96(99) 5; 97 (99) 98; (996) 768; Kosloff R., Geva E. and Gordon J., J. Appl. Phys., 87 () 89. [6] FeldmannT.andKosloffR.,Phys.Rev.E,7() 6 and references therein. [7] AbeS.andOkuyamaS.,Phys.Rev.E,8(); Abe S.,Phys. Rev. E,8() 7;Abe S.and OkuyamaS.,Phys.Rev.E,85(). [8] Scovil H. andschulz-dubois E. O., Phys. Rev. Lett., (959)6;GeusicJ.E.,Schulz-DuboisE.O.and Scovil H., Phys. Rev., 56(967) and references therein. [9] PurcellE.M.andPoundR.V.,Phys.Rev.,8(95) 79. [] RamseyN.F.,Phys.Rev.,(956). [] GordonJ.P.,ZeigerH.J.andTownesC.H.,Phys. Rev.,95(95)8. [] Beretta G. P., e-print arxiv:quant-ph/76(7). [] ZurekW.H.,Rev.Mod.Phys.,75()75. [] Gheorghiu-Svirschevski S., Phys. Rev. A, 6() 5 and references therein. [5] Allahverdyan A. E., Balian R. andnieuwenhuizen T. M., Europhys. Lett., 67 () 565; but the issue of unitary accessibility vs. what engineers call adiabatic availability is already addressed and solved in Hatsopoulos G. N. andgyftopoulos E. P., Found. Phys.,6(976)7. [6] For a precise definition of the termodynamic concept of a heat interaction see Gyftopoulos E. P. and Beretta G. P., Thermodynamics. Foundations and Applications (Dover, Mineola) 5, pp (first edition (Macmillan, New York) 99). [7] Maddox J., Nature, 6 (985) ; Beretta G. P., PhDThesis(98),e-printarXiv:quant-ph/596and references therein. [8] Beretta G. P., Phys. Rev. E, 7 (6) 6 and references therein. See also Beretta G. P., Rep. Math. Phys.6(9)9. 5-p5