Light-matter quantum Otto engine in finite time
G. Alvarado Barrios, F. Albarrán-Arriagada, F. J. Peña, E. Solano, J. C. Retamal
LLight-matter quantum Otto engine in finite time
G. Alvarado Barrios, ∗ F. Albarr´an-Arriagada, F. J. Pe˜na, E. Solano,
1, 3, 4, 5, † and J. C. Retamal
6, 7, ‡ International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist)and Physics Department, Shanghai University, 200444 Shanghai, China Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110V, Valpara´ıso 2340000, Chile Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain IQM, Nymphenburgerstr. 86, 80636 Munich, Germany Departamento de F´ısica, Universidad de Santiago de Chile (USACH), Avenida Ecuador 3493, 9170124, Santiago, Chile Center for the Development of Nanoscience and Nanotechnology 9170124, Estaci´on Central, Santiago, Chile (Dated: February 23, 2021)We study a quantum Otto engine at finite time, where the working substance is composed of a two-levelsystem interacting with a harmonic oscillator, described by the quantum Rabi model. We obtain the limitcycle and calculate the total work extracted, efficiency, and power of the engine by numerically solving themaster equation describing the open system dynamics. We relate the total work extracted and the efficiencyat maximum power with the quantum correlations embedded in the working substance, which we considerthrough entanglement of formation and quantum discord. Interestingly, we find that the engine can overcomethe Curzon-Ahlborn efficiency when the working substance is in the ultrastrong coupling regime. This high-efficiency regime roughly coincides with the cases where the entanglement in the working substance experiencesthe greatest reduction in the hot isochoric stage. Our results highlight the efficiency performance of correlatedworking substances for quantum heat engines.
I. INTRODUCTION
A quantum heat engine (QHE) is a quantum device thatgenerates power from the heat flow between a hot and a coldreservoir interacting with a quantum system. Therefore, theworking mechanism of the engine is described by the lawsof quantum mechanics. Quantum reciprocating heat enginesoperate under the quantum versions of well known thermody-namical cycles, such as Carnot or Otto cycles [1–7]. The per-formance of these devices depend strongly on the choice of theworking substance and the thermodynamic cycle which gov-ern the dynamics. This has led to several cases being consid-ered, for instance, noninteracting spin 1/2 particles, harmonicoscillators, atoms in harmonic traps, among others [1–12].For a heat engine to achieve maximum performance, i.e. maximum efficiency and energy extracted, it must employinfinitely long cycles since it demands thermalization of theworking substance. As a natural consequence, the energy ex-tracted per unit time, i.e. the power, becomes vanishinglysmall. From a more realistic point of view, we would like toconsider the operation of quantum heat engines for a finite cy-cle time, and find the optimal conditions where the energy ex-tracted per unit time is maximum. For operation in finite time,four-stroke engines show a transient behavior from initializa-tion to a periodic steady state, characterized by a sequence ofstates, referred to as the limit cycle [1, 2, 13], which is usu-ally achieved after a few cycles. The optimal performance ofa cycle in finite time is associated with the maximum powerit can yield, which is obtained for an optimal cycle duration.The efficiency of the cycle in the optimal state is of particular ∗ G. Alvarado Barrios [email protected] † E. Solano [email protected] ‡ J. C. Retamal [email protected] relevance, it is simply termed efficiency at maximum power(EMP) and, classically, it is limited by the Curzon-Ahlbornefficiency [14, 15]. This limit is also fulfilled for some quan-tum working substances [16–18], but it is interesting to con-sider under what conditions a QHE can exceed this classicalbound [19]. The finite-time operation of the quantum Otto cy-cle has been studied for interacting and noninteracting work-ing substances, such as, two-level systems [20, 21], harmonicoscillators [16, 22], coupled harmonic oscillators [18] amongothers.The possibility of extracting work from quantum correla-tions [23, 24] has highlighted the thermodynamic value ofcorrelations. This has motivated the study of the role of quan-tum correlations in the performance of QHEs whose workingsubstance is composed by coupled quantum systems. For anappropriately strong coupling it is possible to have non-zeroquantum correlations even when the global system is in ther-mal equilibrium with a reservoir at a given temperature. Thisproblem has been studied in the context of QHEs in which theheat strokes act for a sufficiently long time so as to allow fora full thermalization [25–31]. From these works, it is clearthat the effect of quantum correlations on the performance ofquantum heat machines are model dependent. On the otherhand, the role played by quantum correlations in finite timeoperation of QHEs is not yet completely understood.In this work, we consider a quantum Otto engine in finitetime with a working substance composed by a two-level sys-tem (TLS) interacting with a harmonic oscillator, described bythe quantum Rabi model (QRM), a paradigmatic light-matterinteraction. Such working substance can be realized in su-perconducting circuits [32–37] where the interaction can beengineered to have access to different coupling regimes. Weobtain the limit cycle and calculate the total work extracted,efficiency, and power of the engine by numerically solving themaster equation describing the open system dynamics. After- a r X i v : . [ qu a n t - ph ] F e b wards, we consider the relation between the light-matter quan-tum correlations and the total work extracted, efficiency andpower, by considering entanglement of formation and quan-tum discord. We consider the reduction of quantum correla-tions during the isochoric stages and find that the engine canovercome the Curzon-Ahlborn efficiency when the workingsubstance is in the ultrastrong coupling regime. In addition,we find that quantum discord, rather than entanglement actsas an indicator of the total work extracted by the engine. Thismay suggest that for this model, quantum correlations otherthan entanglement affect the performance of the engine. II. THE MODEL
We consider a QHE where a TLS is coupled to a singlequantized mode, described by the QRM dipolar coupling H = (cid:126) ω q σ z + (cid:126) ω r a † a + (cid:126) gσ x ( a + a † ) . (1)Here, the operators σ α , α = { x, z } , are the Pauli matricescorresponding to the TLS, and a ( a † ) is the annihilation (cre-ation) bosonic operator for the harmonic mode. The parame-ters, ω r , ω q and g , describe the resonator frequency, qubit fre-quency, and qubit-resonator coupling strength, respectively.Throughout this work we will consider the resonance condi-tion ω r = ω q = ω .The dynamics of the QRM can be separated into three dif-ferent regimes which are governed by the ratio g/ω [38, 39].When g is much larger than any decoherence or dephasingrate in the system, and g/ω (cid:46) − we can define the strongcoupling (SC) regime. In this regime, the rotating wave ap-proximation (RWA) can be performed, obtaining the Jaynes-Cummings model [40], in this model the number of excita-tions is preserved, thus, the dynamics involve only states withthe same number of excitations as in the initial state. For . < g/ω ≤ is considered the ultrastrong coupling regime(USC) [32–34]. In the USC regime, the RWA cannot be per-formed, then, the dynamics of the QRM no longer preserve thenumber of excitations. Finally, when the coupling strength islarger than the frequency of the system, i.e. g (cid:29) ω we ob-tain the deep-strong coupling regime (DSC) [35, 41]. In thisregime again the TLS degrees of freedom decouple from theharmonic oscillator degrees of freedom [42].One characteristic of the spectrum of the QRM is that theenergy levels start to degenerate when the coupling parameteris increased from the SC regime to the DSC regime. Also,in the USC the eigenstates of the model correspond to highlycorrelated states of the TLS and resonator mode, ideal to studythe role of quantum correlation of the working substance inthe performance of a QHE. III. QUANTUM OTTO CYCLE
In what follows we study a QHE operating under a quantumOtto cycle in finite time. This cycle is composed of two adia-batic processes and two isochoric processes. In the latter, the T c T h FIG. 1. Diagram of the considered quantum Otto cycle. working substance interacts with a cold(hot) reservoir at tem-perature T c ( T h ). We will consider that the coupling strength g is kept constant throughout the cycle.For the adiabatic processes, we consider a change in the res-onator frequency between the values { ω h , ω c } with ω h > ω c .The timescale, τ ad of this process must be sufficiently largeas to satisfy the adiabatic theorem [43]. This, in general, issatisfied if the duration is much larger than the dynamicaltimescale, that is, τ ad (cid:29) (cid:126) /ε [43], where ε is a character-istic transition energy of the system, in this case, it can beconsidered to be of the order of GHz. This choice is justifiedas long as the smallest relevant transition frequency remainsin the range of GHz.For the isochoric processes, we consider that the systemundergoes a Markovian evolution, described by [44] ˙ ρ ( t ) = − i [ H, ρ ( t )] + (cid:88) m L m ( ρ ( t )) . (2)Where L m ( ρ ( t )) = (cid:88) j,k>j Γ jkm ¯ n (∆ kj , T ) D ( | k (cid:105)(cid:104) j | ) ρ ( t )+ (cid:88) j,k>j Γ jkm (¯ n (∆ kj , T ) + 1) D ( | j (cid:105)(cid:104) k | ) ρ ( t ) . (3)Where {| j (cid:105)} j =0 , , .. are the eigenvectors of the Hamil-tonian H of Eq.(1), obeying H | j (cid:105) = (cid:15) j | j (cid:105) , and D ( O ) ρ = 1 / OρO † − O † Oρ − ρO † O ) . The decay rates Γ jkm are taken as [45] Γ jkm = κ m ∆ kj ω | C mkj | , (4)where m = 1 refers to the qubit m = 2 to the res-onator, ∆ kj = (cid:15) k − (cid:15) j , ¯ n (∆ kj , T ) = 1 / (cid:0) exp( (cid:126) ∆ kj k B T ) − (cid:1) and C mkj = − i (cid:104) k | C m | j (cid:105) , with C m =1 = σ x and C m =2 = a + a † .Here we have only considered energy relaxation in the masterequation.For both the qubit and resonator dissipation channels, weconsider the decay rates as κ m ∼ − ω , { m = 1 , } [45].Notice that the relaxation timescale of the system is of theorder of MHz. On the other hand, the dynamical timescale ofthe system, (cid:126) /E , is in the order of nanoseconds. Therefore,the total duration of the cycle is dominated by the relaxationtimescale.The thermodynamic cycle that we consider is shown inFig.1 and proceeds as follows:Stage 1: Quantum isochoric process (hot bath stage)[6].The system, with frequency ω = ω h and Hamiltonian H h ,prepared in some initial state ρ i , is brought into contact witha hot thermal reservoir at temperature T h for a time duration τ . At the end of this process the state becomes ρ ( τ ) . Dur-ing this process only the populations change while the energylevel structure remains invariant.Stage 2: Quantum adiabatic (expansion) process [5, 6]. Thesystem is isolated from the hot reservoir, and the resonatorfrequency is changed from ω h to ω c , with ω h > ω c , to sat-isfy the quantum adiabatic theorem [43, 46], the duration ofthis process, τ , is chosen as τ (cid:29) (cid:126) / ( (cid:15) h − (cid:15) h ) , where (cid:15) h ( c ) i is the i-th eigenenergy of Hamiltonian H h ( c ) . During this pro-cess only the energy level structure changes. We denote theHamiltonian at the end of this process by H c , and the state ofthe system is ρ ( τ ) .Stage 3: Quantum isochoric process (cold bath stage). Theworking substance, with ω = ω c and Hamiltonian H c , isbrought into contact with a cold thermal reservoir at tempera-ture T c for a time duration τ . At the end of this process, thestate of the system is ρ ( τ ) . Since the resonator frequency haschanged to ω c due to the adiabatic process, in this stage theratio g/ω c for a given value of g is different than in stage 1 asa consequence of the adiabatic process.Stage 4: Quantum adiabatic (compression) process. Thesystem is isolated from the cold reservoir, and the resonatorfrequency is changed from ω c to ω h . During this process thepopulations remain constant while the energy level structurereturns to its configuration in Stage 1. The duration of thisprocess, τ , is chosen as τ (cid:29) (cid:126) / ( (cid:15) c − (cid:15) c ) . At the end of thisprocess the state of the system is given by ρ ( τ ) .The total cycle time, τ , is the sum of the time duration ofeach stage τ = (cid:80) i τ i . We will consider equal time durationfor the isochoric stages, τ = τ , then τ = 2 τ + τ + τ .Since we are considering finite duration for the thermody-namic cycle, the initial state of the system has an effect onthe performance of the QHE. We will consider that the systemstarts in a thermal state at temperature T c , and furthermore,we study the QHE under several consecutive iterations of thecycle. Since the isochoric stages involve a Markovian evolu-tion, we can expect that after several iterations, there shouldbe a limit cycle which is independent of the initial state of thesystem.To describe a thermodynamical cycle in finite time requiresthe formulation of thermodynamic laws in a dynamical con-text [47, 48]. Let us consider a dynamical system with a dis-crete time-dependent Hamiltonian H ( t ) . The average energyof the system, U ( t ) , for a state ρ ( t ) is given by U ( t ) = (cid:104) H ( t ) (cid:105) = T r { ρ ( t ) H ( t ) } . (5) The rate of change of the average energy of the system isgiven by dUdt = dWdt + dQdt . (6)This expression is a dynamical form of the first law of ther-modynamics. The quantities Q and W are associated to heatand work, respectively, and can be written as Q ( t ) = (cid:90) t dt (cid:48) T r { H ( t (cid:48) ) ˙ ρ ( t (cid:48) ) } , (7)and W ( t ) = (cid:90) t dt (cid:48) T r { ˙ H ( t (cid:48) ) ρ ( t (cid:48) ) } . (8)Furthermore, we can write the power, P , as P ( t ) = dWdt = T r { ˙ H ( t ) ρ ( t ) } . (9)Notice that for an isochoric process we have H ( t ) = H ,which means W ( t ) = 0 . On the other hand, in an adiabaticprocess the system undergoes a unitary evolution with a time-dependent Hamiltonian, which naturally means Q ( t ) = 0 .The quantum Otto cycle consists of four stages, in whichthe energy change of the system is accounted purely by workor purely by heat. Since the cycle is completed by returning tothe initial configuration of the system, we can write the totalwork extracted as W ( t ) = Q ( t )+ Q ( t ) , where the subindexindicates the stage of the cycle as explained above.The efficiency of the engine is defined as the ratio of thetotal work extracted and the heat enters the system η ( t ) = W ( t ) Q ( t ) . (10)The Carnot efficiency η C = 1 − T c T h represents a uni-versal upper-bound for all heat engines. However, the effi-ciency bound for a working cycle at maximum power is usu-ally of greater practical importance than the Carnot efficiency.The efficiency at maximum power obeys the Curzon-Ahlborn(CA) bound [14, 15, 19], given by η CA = 1 − (cid:114) T c T h . (11) IV. LIMIT CYCLE
Let us first consider the behavior of the engine over sev-eral iterations of the cycle. We will denote by ρ ( N ) ( τ ) thestate of the system at the end of the hot isochoric stage for FIG. 2. (Fidelity between ρ ( N ) ( τ ) and ρ ( N − ( τ ) in terms of τ for different iterations, where N = 2 (blue), N = 3 (yellow), N = 4 (green), N = 5 (red) and N = 6 (black). We have chosen g/ω c = 0 . , T c = 20 mK and T h = 8 T c . the Nth iteration of the cycle. We consider that each itera-tion of the cycle is identical. In Fig. 2 we plot the fidelity, F h = F ( ρ ( N − ( τ ) , ρ ( N ) ( τ )) , of state ρ ( τ ) between con-secutive iterations. As can be seen, when τ → the stateof the system experiences almost no change during the cycle,and therefore we have F h → . For < τ (cid:28) , the evolutionthrough one full cycle changes the state of the system leadingto a decrease in the fidelity between consecutive cycles. Aswe consider increasing values of τ , the interaction with thehot reservoir drives ρ ( τ ) closer to the stationary thermal state,and the fidelity increases with τ . For times comparable withthe relaxation time, F h → which indicates that the engineoperates in the limit cycle from the first iteration. As expected,the minimum cycle time required to achieve maximal fidelitydiminishes as we increase the number of iterations.We want to characterize the performance of the engine re-gardless of the state in which it is initialized, to this end, weconsider the thermodynamic figures of merit in the limit cycle.Now, let us consider, for the limit cycle, the dependenceof the total work extracted on the coupling strength g/ω c , fordifferent cycle times. This is shown in Fig. 3, where the cy-cle times differ mainly in the time assigned to the isochoricstages, τ and τ ( τ = τ ). As can be seen in the figure, thetotal work extracted increases with the cycle time until the sys-tem closely approaches thermalization with the heat reservoirsin the isochoric stages. This is because when the durationof the isochoric stages are large compared to the relaxationtimes, the thermodynamic quantities reach their correspond-ing values for a stationary Otto cycle [31].In Fig. 4, we show the total work extracted, efficiency andpower as a function of the total cycle time τ for different val-ues of g/ω c . Since the adiabatic processes have finite dura-tion to satisfy the adiabatic approximation, the total cycle timecannot be less than τ = τ + τ . Where τ and τ depend on g/ω c . This can be seen in Fig. 4, where we see that for eachquantity the smallest values of τ increase with g/ω c . We seethat when the adiabatic times do not change significantly with g/ω c and for very small cycle times, the coupling strength has FIG. 3. Total Work extracted as a function of the couplingstrength g/ω c . The dots denote the total work extracted for finitecycle times for τ = 0 . (blue), τ = 0 . (purple), τ = 0 . (yel-low) and τ = 1 . (red) in units of π/κ h . The black solid line isthe total work extracted for a stationary cycle (thermalizing workingsubstance). We have considered T c = 20 mK and T h = 8 T c . little effect on the total work extracted. On the other hand,we can see that for large cycle times the total work extractedapproaches its stationary value, matching the amount in Fig. 3for corresponding values of g/ω c . In Fig. 4(b) we see thatthe efficiency is greater for increasing values of the couplingstrength. In Fig. 4(c) we can see that there is an optimal cycletime that maximizes the energy extracted per unit time. Thisoptimal performance depends on the cycle duration and thestrength of the coupling parameter. Thus, the more relevantquantities that characterize the quantum Otto cycle in finitetime are the power and the EMP. In the following sections wewill analyze the relationship between these quantities and thequantum correlations between the TLS and the resonator, em-bedded in the working substance. V. QUANTUM CORRELATIONS FOR FINITE-TIMEOPERATION
An interesting issue is to consider the effect of quantum cor-relations between the components of the working substancein the performance of the QHE. This effect has been stud-ied previously in this context for an Otto cycle in which thesystem thermalizes during the isochoric stages (infinite timeQHE) [30, 31]. These studies have shown that the changein quantum correlations during the hot isochoric stage are in-dicative of the behavior of the total work extracted. However,the role that these quantum correlations may play in the per-formance of finite-time Otto cycles has not been addressed.We will study the quantum correlations between the two-level system and the resonator. We will quantify two kinds ofquantum correlations, namely, the entanglement of formation(EoF) [49] and quantum discord (QD) [50].For a general mixed state ρ AB of a bipartite system A and B , we can write a specific pure state decomposition as ρ AB = (cid:80) i p i | Φ i (cid:105)(cid:104) Φ i | . We can define the entanglement of this pure (a)(b)(c) FIG. 4. (a) Total work extracted, (b) efficiency and (c) power as afunction of the total cycle time τ for g/ω c = 0 . (blue), g/ω c = 0 . (yellow), g/ω c = 0 . (green) and g/ω c = 0 . (red). state decomposition as E p i , Φ i = (cid:88) i p i E ( | Φ i (cid:105) ) , (12)where E ( | Φ (cid:105) ) = S ( ρ A ( B ) ) is the entanglement of the purestate | Φ i (cid:105) , S ( A ) = − tr ( A ln A ) is the von Neumann entropy,and ρ A ( B ) is the reduced density matrix of the subsystem A ( B ) . Now, the EoF of the density matrix ρ is given by aminimization over all the possible pure state decompositions p i , Φ i of Eq. (12), that is E ( ρ ) = min p i , | Φ i (cid:105) (cid:88) i p i E ( | Φ i (cid:105) ) , (13) (a)(b) FIG. 5. Difference of entanglement δE = E ( ρ ( τ )) − E ( ρ ( τ )) as afunction of the interaction time with the hot thermal reservoir, τ , forthe limit cycle, for g/ω c = 0 . (blue), g/ω c = 0 . (yellow), g/ω c =0 . (green) and g/ω c = 0 . (red). We have chosen T h = 160 mK. it should be noted that EoF does not capture all forms ofquantum correlations. On the other hand, quantum discordcan quantify all the quantum correlations (other than entan-glement).The quantum discord in a bipartite AB system can be cal-culated as follows [50]: D A = S ( ρ A ) − S ( ρ AB ) + min { Π Aj } S ( ρ B |{ Π Aj } ) , (14)where ρ B |{ Π Aj } is the state of the bipartite system after aprojective measurement Π Aj has been performed on subsys-tem A . In our case, subsystem A will denote the two-levelsystem and subsystem B the resonator mode. Due to the sizeof the Hilbert space of the working substance we only con-sider projective measurements on the two-level system.To calculate either of the two measures it is necessary tosolve a minimization problem, for which we employ the sim-ulated annealing technique used in ref. [51].Now, we analyze the relation between quantum correla-tions and the thermodynamic figures of merit of the cycle.We define the following quantities; we denote the differenceof quantum correlations between the initial and final statesof the hot isochoric stage as measured by entanglement by δE = E ( ρ ( τ )) − E ( ρ ( τ )) , and as measured by quantum dis-cord by δD = D ( ρ ( τ )) − D ( ρ ( τ )) . In Fig. 5(a) and Fig. 6(a) (a)(b) FIG. 6. Difference of quantum discord δD = D ( ρ ( τ )) − D ( ρ ( τ )) as a function of the interaction time with the hot thermal reservoir, τ , for the limit cycle, for g/ω c = 0 . (blue), g/ω c = 0 . (yel-low), g/ω c = 0 . (green) and g/ω c = 0 . (red). We have chosen T h = 160 mK. we show δE and δD as a function of τ , respectively, for dif-ferent values of g/ω c . As can be seen by comparing Fig. 4(a)with Fig. 6(a) and 5(a), the total work extracted follows thetime dependence of the difference of quantum correlations ofthe isochoric stages. Notice that while both correlation mea-sures capture the optimal behavior of the engine in terms ofthe coupling strength g/ω , it is quantum discord which bet-ter captures the behavior of the total work extracted, as canbe seen for the cases of g/ω = 0 . and g/ω = 0 . . Thisindicates that for this model, quantum correlations other thanentanglement have an effect on the performance of the en-gine. This result can be regarded as generalizing the resultsobtained for cycles in which the system thermalizes duringthe heat strokes [29–31].In Fig. 5(b) and Fig. 6(b) we plot δE/τ and δD/τ , respec-tively, which are the difference of quantum correlations in theisochoric stages over the total cycle time. As can be seen bycomparing Figures 4(b) with 5(b) and 6(b), the quantities δE/τ and δD/τ are indicative of the behavior of the power P , where again, quantum discord acts as a better indicator.This indicates that the performance of the engine is affectedby quantum correlations other than entanglement. FIG. 7. (a) Efficiency at maximum power η mp (solid line) and effi-ciency of Curzon-Ahlborn (dashed line), (b) difference of entangle-ment δE mp and (c) difference of quantum discord δD mp as a functionof the coupling strength g/ω c , at cycle time τ opt . We have considered T h = 160 mK. VI. EFFICIENCY AT MAXIMUM POWER
Finally, we are interested in analyzing the EMP, η mp , andits relation with the change in quantum correlations in thehot isochoric stage. For each value of the coupling pa-rameter g/ω c we search for the time duration of the cycle, τ opt = 2 τ opt + τ opt + τ opt , at which the power is maximum,and calculate the efficiency, η mp , for that cycle duration. InFig. 7(a) we show the EMP (solid line) in terms of the cou-pling strength g/ω c , together with the value of the CA ef-ficiency, η CA (dashed line). Remarkably, we notice that forcoupling strength values . (cid:46) g/ω c (cid:46) . , belonging toroughly the USC regime and near DSC regime, the EMP sur-passes the standard bound of CA efficiency, which highlightsthe capabilities of highly correlated working substances.We wish to study if there is a relation between the EMPand quantum correlations, to do so we define the follow-ing quantities; we denote the difference of quantum correla-tions between the initial and final states of the hot isochoricstage by δ Λ mp = Λ( ρ ( τ opt )) − Λ( ρ ( τ opt )) , where Λ( ρ ) = { D ( ρ ) , E ( ρ ) } . The subindex ‘mp’ indicates that the quan-tum correlations are calculated at the times τ opt i for which thepower is maximum. In Fig. 7(b) we show the difference ofquantum discord δD mp and Fig. 7(c) the difference of entan-glement δE mp as a function of the coupling strength g/ω c .We have plotted these quantities for the range g/ω c ∈ [0 , since it is where the total work extracted is positive and hencethe efficiency is well defined. As can be seen by comparingFig. 7(a), Fig. 7(b) and Fig. 7(c), the difference of entangle-ment acts as a better indicator of the behavior of the EMP,rather than the difference of quantum discord.The quantity δE mp (and similarly for δD mp ) compares theamount of entanglement in the states before and after inter-acting with the reservoir in the hot isochoric stage. A biggervalue of δE mp means the interaction with the reservoir de-stroyed a greater amount of quantum correlations in the work-ing substance. Since the parameter range of high efficiencyroughly coincides with the range where δE mp is greatest, wemay interpret that the environment must spend energy to re-duce the quantum correlations in the working substance whichcan later be harvested by the engine and improve its perfor-mance.This suggests that while quantum discord may act as anindicator of total work extracted and power, entanglement actsas an indicator of the efficiency of the engine in the optimumregime (maximum energy extracted per unit of time). VII. CONCLUSIONS
We have studied the finite-time operation of a quantum Ottocycle embedding a working substance composed of a two-level system interacting with a resonator, described by thequantum Rabi model. We obtained the total work extracted,efficiency and power for the limit cycle. We focused our studyon the relation between quantum correlations and the totalwork extracted, as well as the efficiency at maximum power,by considering entanglement of formation and quantum dis-cord. We found that when the hot isochoric stage reducesthe quantum correlations in the working substance, it can leadto enhanced positive work extraction. Furthermore, quantumdiscord, rather than entanglement, acts as a better indicator ofthe behavior of the total work extracted. On the other hand,we studied the behavior of the engine for a cycle durationwhich yields the maximum power, and its relation with thequantum correlations in the working subtance. Interestingly,when the working substance is roughly in the range of ultra-strong coupling, it overcomes the Curzon-Ahlborn efficiency.Consequently, it roughly coincides with the cases where theentanglement in the working substance experiences the great-est reduction in the hot isochoric stage. Our results suggestthat quantum correlations, i.e. , entanglement and quantumdiscord, can act as indicators of the performance of a QHEworking at finite time, and possibly both should be consideredwhen searching for the optimal configuration. Our study high-light the capabilities of highly correlated quantum systems forenhancing the performance of QHEs for finite time.
VIII. ACKNOWLEDGMENTS
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