TThe Quantum Harmonic Otto Cycle
Ronnie Kosloff , ∗ , † and Yair Rezek , † Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel Computer Science Department, Technion,Haifa 32000, Israel; [email protected] Tel Hai College, Kiryat Shmona, 1220800 Israel a r X i v : . [ qu a n t - ph ] M a r bstract The quantum Otto cycle serves as a bridge between the macroscopic world of heat enginesand the quantum regime of thermal devices composed from a single element. We compile recentstudies of the quantum Otto cycle with a harmonic oscillator as a working medium. This modelhas the advantage that it is analytically trackable. In addition, an experimental realization hasbeen achieved, employing a single ion in a harmonic trap. The review is embedded in the fieldof quantum thermodynamics and quantum open systems. The basic principles of the theory areexplained by a specific example illuminating the basic definitions of work and heat. The rela-tion between quantum observables and the state of the system is emphasized. The dynamicaldescription of the cycle is based on a completely positive map formulated as a propagator for eachstroke of the engine. Explicit solutions for these propagators are described on a vector space ofquantum thermodynamical observables. These solutions which employ different assumptions andtechniques are compared. The tradeoff between power and efficiency is the focal point of finite-time-thermodynamics. The dynamical model enables the study of finite time cycles limiting timeon the adiabatic and the thermalization times. Explicit finite time solutions are found which arefrictionless (meaning that no coherence is generated), and are also known as shortcuts to adi-abaticity.The transition from frictionless to sudden adiabats is characterized by a non-hermitiandegeneracy in the propagator. In addition, the influence of noise on the control is illustrated. Theseresults are used to close the cycles either as engines or as refrigerators. The properties of the limitcycle are described. Methods to optimize the power by controlling the thermalization time are alsointroduced. At high temperatures, the Novikov–Curzon–Ahlborn efficiency at maximum power isobtained. The sudden limit of the engine which allows finite power at zero cycle time is shown.The refrigerator cycle is described within the frictionless limit, with emphasis on the cooling ratewhen the cold bath temperature approaches zero. ontents I. Introduction II. The Quantum Otto Cycle
III. Quantum Thermodynamics
IV. The Dynamics of the Quantum Otto Cycle
V. Closing the Cycle
VI. Overview Acknowledgments References . INTRODUCTION
Quantum thermodynamics is devoted to the link between thermodynamical processes andtheir quantum origin. Typically, thermodynamics is applied to large macroscopic entities.Therefore, to what extent is it possible to miniaturize. Can thermodynamics be applicableto the level of a single quantum device? We will address this issue in the tradition ofthermodynamics, by learning from an example: Analysis of the performance of a heat engine[1]. To this end, we review recent progress in the study of the quantum harmonic oscillatoras a working medium of a thermal device. The engine composed of a single harmonicoscillator connected to a hot and cold bath is an ideal analytically solvable model for aquantum thermal device. It has therefore been studied extensively and inspired experimentalrealisation. Recently, a single ion heat engine with an effective harmonic trap frequency hasbeen experimentally realised [2]. This device could roughly be classified as a reciprocatingOtto engine operating by periodically modulating the trap frequency.Real heat engines operate far from reversible conditions. Their performance resides be-tween the point of maximum efficiency and maximum power. This has been the subject offinite time thermodynamics [3, 4]. The topic has been devoted to the irreversible cost ofoperating at finite power. Quantum engines add a twist to the subject, as they naturallyincorporate dynamics into thermodynamics [5, 6].Quantum heat engines can be classified either as continuous or reciprocating. The proto-type of a continuous engine is the three-level amplifier pioneered by Scovil and Schulz-DuBois[7]. This device is simultaneously coupled to three input currents. It is therefore termed atricycle, and can operate either as an engine or as a refrigerator. A review of continuousquantum heat engines has been published recently [8] and therefore is beyond the scope ofthis review.Reciprocating engines are classified according to their sequence of strokes. The moststudied cycles are Carnot [9–13] and Otto cycles [14–19]. The quantum Otto cycle is eas-ier to analyze, and therefore it became the primary example of a reciprocating quantumheat engine. The pioneering studies of quantum reciprocating engines employed a two-level system—a qubit—as a working medium [9, 10, 14, 20]. The performance analysis ofthe quantum versions of reciprocating engines exhibited an amazing resemblance to macro-scopic counterparts. For example, the efficiency at maximum power of the quantum version4f the endoreversiable engine converges at high temperature to the Novikov–Curzon–Ahlbornmacroscopic Newtonian model predictions [21, 22]. The deviations were even small at lowtemperature, despite the fact that the heat transport law was different [9]. The only quan-tum feature that could be identified was related to the discrete structure of the energylevels.Heat engines with quantum features require a more complex working medium than asingle qubit weakly coupled to a heat bath. This complexity is required to obtain quantumanalogues of friction and heat leaks. A prerequisite for such phenomena is that the externalcontrol part of the Hamiltonian does not commute with the internal part. This generatesquantum non-adiabatic phenomena which lead to friction [23]. A working medium composedof a quantum harmonic oscillator has sufficient complexity to represent generic phenomena,but can still be amenable to analytic analysis [24].The quantum Otto cycle is a primary example of the emerging field of quantum thermody-namics. The quest is to establish the similarities and differences in applying thermodynamicreasoning up to the level of a single quantum entity. The present analysis is based on thetheory of quantum open systems [25, 26]. A dynamical description based on the weak systembath coupling has been able to establish consistency between quantum mechanics and thelaws of thermodynamics [25]. These links allow work and heat to be defined in the quantumregime [27]. This framework is sufficient for the present analysis.In the strong coupling regime where the partition between system and bath is not clear,the connection to thermodynamics is not yet established—although different approacheshave been suggested [28, 29]. A different approach to quantum thermodynamics termedquantum thermodynamics resource theory follows ideas from quantum information resourcetheory, establishing a set of rules [30, 31]. We will try to show how this approach can belinked to the Otto cycle under analysis.
II. THE QUANTUM OTTO CYCLE
Nicolaus August Otto invented a reciprocating four stroke engine in 1861, and won a goldmedal in the 1867 Paris world fair [32]. The basic components of the engine are hot andcold reservoirs, a working medium, and a mechanical output device. The cycle of the engineis defined by four segments: 5. The hot isochore : heat is transferred from the hot bath to the working medium withoutvolume change.2. The power adiabat : the working medium expands, producing work, while isolated fromthe hot and cold reservoirs.3. The cold isochore : heat is transferred from the working medium to the cold bathwithout volume change.4. The compression adiabat : the working medium is compressed, consuming power whileisolated from the hot and cold reservoirs, closing the cycle.Otto determined that the efficiency η of the cycle is limited to η o ≤ − ( V h V c ) CpCv − , where V c/h and T c/h are the volume and temperature of the working medium at the end of the hotand cold isochores , respectively. C p and C v are the heat capacities under constant pressureand constant volume [33]. As expected, Otto efficiency is always smaller than the efficiencyof the Carnot cycle η o ≤ η c = 1 − T c T h .The first step in learning from an example is to establish a quantum version of the Ottocycle. This is carried out by seeking analogues for each segment of the cycle. What makesthe approach unique is that it is applicable to a small quantum system such as a single atomin a harmonic trap. The description is embedded in the theory of open quantum systems.Each of these segments is defined by a completely positive (CP) propagator [34] describingthe change of state in the working medium: ˆ ρ f = U i → f ˆ ρ i , where the density operator ˆ ρ describes the state of the working medium.The quantum engine Otto cycle is therefore described as:1. The hot isochore : heat is transferred from the hot bath to the working medium withoutchange in the external parameter ω h . The stroke is described by the propagator U h .2. The expansion adiabat : the working medium reduces its energy scale. The harmonicfrequency changes from ω h to ω c , with ω h > ω c , producing work while isolated fromthe hot and cold reservoirs. The stroke is described by the propagator U hc .3. The cold isochore : heat is transferred from the working medium to the cold bathwithout change in the external parameter ω c . The stroke is described by the propagator U c . 6. The compression adiabat : the working medium increases its energy scale. The har-monic frequencies increase from ω c to ω h , consuming power while isolated from thehot and cold reservoirs. The stroke is described by the propagator U ch .The cycle propagator becomes the product of the segment propagators: U cyc = U ch U c U hc U h . (1)The cycle propagator is a completely positive (CP) map of the state of the workingmedium [34]. The order of propagators is essential, since the segment propagators do notcommute; for example, [ U h , U hc ] (cid:54) = 0. The non-commuting property of the segment propaga-tors is not an exclusive quantum property. It is also present in stochastic descriptions of theengine where the propagators operate on a vector of populations of the energy eigenvalues.Nevertheless, it can have a quantum origin for engines with propagators with small action[35, 36]. The same operators but with different parameters (such as different frequencies) canbe used to describe an Otto refrigeration cycle. Figure 1 shows a schematic representationof the Otto cycle in phase space.In the adiabatic limit when the population N stays constant in the expansion and com-pression segments, the work per cycle becomes: W cyc = (cid:126) ∆ ω ∆ N , (2)where ∆ ω = ω h − ω c and ∆ N is the population difference. ∆ N = ∆ N c = ∆ N h , since thecycle is periodic. Under these conditions, the efficiency becomes: η o = 1 − ω c ω h ≤ η c , (3)where η c is the Carnot efficiency η c = 1 − T c T h . At this stage, it is also useful to define thecompression ratio C = ω h ω c . A. Quantum Dynamics of the Working Medium
The quantum analogue of the Otto cycle requires a dynamical description of the workingmedium, the power output, and the heat transport mechanism.A particle in a harmonic potential will constitute our working medium. This choice isamenable to analytic solutions and has sufficient complexity to serve as a generic example.Even a single specimen is sufficient to realize the operation of an engine.7 C DB U ch U h U c U hc FIG. 1: Otto cycle in phase space. The blue and red bowls represent the energy value in positionand momentum. The compression ratio is C = ω h ω c = 2. Expansion adiabat A → B . Cold isochore B → C . Compression adiabat C → D . Hot isochore D → A . The Wigner distribution inphase space is shown in green. The state in A is a thermal equilibrium state with the hot bathtemperature. The state in B is squeezed with respect to the cold bath frequency ω c . The state in C is an equilibrium state with the cold bath temperature. The state in D shows position momentumcorrelation (cid:104) ˆC (cid:105) (cid:54) = 0. We can imagine a single particle in a harmonic trap V ( Q ) = k Q . Expansion and com-pression of the working medium is carried out by externally controlling the trap parameter k ( t ). The energy of the particle is represented by the Hamiltonian operator: ˆH = 12 m ˆP + k ( t )2 ˆQ , (4)where m is the mass of the system and ˆP and ˆQ are the momentum and position operators.All thermodynamical quantities will be intensive; i.e., normalized to the number of particles.In the macroscopic Otto engine, the internal energy of the working medium during theadiabatic expansion is inversely proportional to the volume. In the harmonic oscillator, theenergy is linear in the frequency ω ( t ) = (cid:112) k ( t ) /m [37]. This therefore plays the role ofinverse volume V . 8he Hamiltonian (4) is the generator of the evolution on the adiabatic segments. Thefrequency ω changes from ω h to ω c in a time period τ hc in the power adiabat ( ω h > ω c ) andfrom ω c to ω h in a period τ ch in the compression adiabat . The dynamics of the state ˆ ρ duringthe adiabatic segments is unitary and is the solution of the Liouville von Neumann equation[38]: ddt ˆ ρ ( t ) = − i (cid:126) [ ˆH ( t ) , ˆ ρ ( t )] , (5)where ˆH is time dependent during the evolution. Notice that [ ˆH ( t ) , ˆH ( t (cid:48) )] (cid:54) = 0, since thekinetic energy does not commute with the varying potential energy. This is the origin ofquantum friction [23, 39]. The formal solution to Equation (5) defines the propagator:ˆ ρ ( t ) = U ( t ) ˆ ρ (0) = ˆU ˆ ρ (0) ˆU † , (6)where ˆU satisfies the equation: i (cid:126) ddt ˆU = ˆH ( t ) ˆU (7)with the initial condition ˆU (0) = ˆI .The dynamics on the hot and cold isochores is a thermalization process of the workingmedium with a bath at temperature T h or T c . The dynamics is of an open quantum system,where the working medium is described explicitly and the bath implicitly [40–42]: ddt ˆ ρ ( t ) = − i (cid:126) [ ˆH , ˆ ρ ] + L D ( ˆ ρ ) , (8)where L D is the dissipative term responsible for driving the working medium to thermalequilibrium, while the Hamiltonian ˆH = ˆH ( ω h/c ) is static. The equilibration is not completein typical operating conditions, since only a finite time τ h or τ c is allocated to the hot orcold isochores . The dissipative “superoperator” L D must conform to Lindblad’s form for aMarkovian evolution [40, 41], and for the harmonic oscillator can be expressed as [43–45]: L D ( ˆ ρ ) = k ↑ ( ˆa † ˆ ρ ˆa − { ˆaˆa † , ˆ ρ } ) + k ↓ ( ˆa ˆ ρ ˆa † − { ˆa † ˆa , ˆ ρ } ) , (9)where anticommutator { ˆA , ˆB } ≡ ˆA ˆB + ˆB ˆA . k ↑ and k ↓ are heat conductance rates obeyingdetailed balance k ↑ k ↓ = e − (cid:126) ωkbT , and T is either T h or T c . The operators ˆa † and ˆa are theraising and lowering operators, respectively. Notice that they are different in the hot andcold isochores , since ˆa = √ ( (cid:112) mω (cid:126) ˆQ + i (cid:113) (cid:126) mω ˆP ) depends on ω . Formally for the isochore U h/c = exp( L t ) where L = − i/ (cid:126) [ ˆH , · ] + L D . 9quation (9) is known as a quantum Master equation [42] or L-GKS [40, 41]. It is anexample of a reduced description where the dynamics of the working medium is soughtexplicitly while the baths are described implicitly by two parameters: the heat conductivityΓ = k ↓ − k ↑ and the bath temperature T . The Lindblad form of Equation (9) guaranteesthat the density operator of the extended system (system + bath) remains positive (i.e.,physical) [40]. Specifically, for the harmonic oscillator, Equation (9) has been derived fromfirst principles by many authors [44, 46–49].To summarize, the quantum model of the Otto cycle is composed of a working fluidof harmonic oscillators (4). The power stroke is modeled by the Liouville von Neumannequation (5), while the heat transport via a Master equation (8) and (9). III. QUANTUM THERMODYNAMICS
Thermodynamics is notorious for its ability to describe a process employing an extremelysmall number of variables. In scenarios where systems are far from thermal equilibrium,further variables have to be added. The analogue description in quantum thermodynamicsis based on a minimal set of quantum expectations (cid:104) ˆX n (cid:105) , where (cid:104) ˆX n (cid:105) = T r { ˆX n ˆ ρ } . Thedynamics of this set is generated by the Heisenberg equations of motion ddt ˆX = ∂ ˆX ∂t + i (cid:126) [ ˆH , ˆX ] + L ∗ D ( ˆX ) , (10)where the first term addresses an explicitly time-dependent set of operators, ˆX ( t ).The dynamical approach to quantum thermodynamics [25] seeks the relation betweenthermodynamical laws and their quantum origin. The first law of thermodynamics is equivalent to the energy balance relation. The energyexpectation E is obtained when ˆX = ˆH ; i.e., E = (cid:104) ˆH (cid:105) . The quantum energy partitiondefining the first law of thermodynamics, dE = d W + d Q , is obtained by inserting ˆH into(10) [5, 6, 25, 50]: ddt E = ˙ W + ˙ Q = (cid:104) ∂ ˆH ∂t (cid:105) + (cid:104) L ∗ D ( ˆH ) (cid:105) . (11)The power is identified as P = ˙ W = (cid:104) ∂ ˆH ∂t (cid:105) . ddt Q = (cid:104) L ∗ D ( ˆH ) (cid:105) . The analysis of the Otto cycle benefits from the simplification that power is produced orconsumed only on the adiabats and heat transfer takes place only on the isochores .The thermodynamic state of a system is fully determined by the thermodynamical vari-ables. Statistical thermodynamics adds the prescription that the state is determined bythe maximum entropy condition subject to the constraints set by the thermodynamicalobservables [51–53]. Maximizing the von Neumann entropy [38] S V N = − k B T r { ˆ ρ ln( ˆ ρ ) } (12)subject to the energy constraint leads to thermal equilibrium [53]ˆ ρ eq = 1 Z e − ˆH kBT , (13)where k B is the Boltzmann constant and Z = T r { e − ˆH kBT } is the partition function.In general, the state of the working medium is not in thermal equilibrium. In order togeneralize the canonical form (13), additional observables are required to define the state ofthe system. The maximum entropy state subject to this set of observables [54 ? ] (cid:104) ˆX j (cid:105) = tr { ˆX j ˆ ρ } becomes ˆ ρ = 1 Z exp (cid:32)(cid:88) j β j ˆX j (cid:33) , (14)where β j are Lagrange multipliers. The generalized canonical form of (14) is meaningful onlyif the state can be cast in the canonical form during the entire cycle of the engine, leadingto β j = β j ( t ). This requirement is called canonical invariance [55]. It implies that if aninitial state belongs to the canonical class of states, it will remain in this class throughoutthe cycle.A necessary condition for canonical invariance is that the set of operators ˆX in (14)is closed under the dynamics generated by the equation of motion. If this condition issatisfied, then the state of the system can be reconstructed from a small number of quantumobservables (cid:104) ˆX j (cid:105) ( t ). These become the thermodynamical observables, since they define thestate under the maximum entropy principle.The condition for canonical invariance on the unitary part of the evolution taking placeon the adiabats is as follows: if the Hamiltonian is a linear combination of the operators in11he set ˆH ( t ) = (cid:80) m h m ˆX m ( h m ( t ) are expansion coefficients), and the set forms a closed Liealgebra [ ˆX j , ˆX k ] = (cid:80) l C jkl ˆX l (where C jkl is the structure factor of the Lie algebra), then theset ˆX is closed under the evolution [56].For a closed Lie algebra, the generalized Gibbs state Equation (14) can always be writtenin a product form: ˆ ρ = (cid:89) k e λ k ˆX k , (15)where there is a one-to-one relation between λ and β , depending on the order of the productform. Multiplying the equation of motion by ˆ ρ − leads to ddt ˆ ρ ˆ ρ − = L ( ˆ ρ ) ˆ ρ − . Using theproduct form and the Backer–Housdorff relation, the l.h.s. ddt ˆ ρ ˆ ρ − decomposes to a linearcombination of the operator algebra. This is also true for the r.h.s [ ˆH , ˆ ρ ] ˆ ρ − , which alsobecomes a linear combination of the operator algebra. Comparing both sides of the equationof motion, one obtains a set of coupled differential equations for the coefficients λ k . Theirsolution guarantees that canonical invariance prevails [54].For the harmonic Otto cycle, the set of the operators ˆP , ˆQ , and ˆD = ( ˆQ ˆP + ˆP ˆQ ) forma closed Lie algebra. Since the Hamiltonian is a linear combination of the first two operatorsof the set ( ˆP and ˆQ ), canonical invariance will prevail on the adiabatic segments.On the isochores , the set of operators also has to be closed to the operation of L D . Theset ˆP , ˆQ , and ˆD is closed to L D , defined by (9). For canonical invariance of ˆ ρ , L D ˆ ρ ˆ ρ − should also be a linear combination of operators in the algebra. For the harmonic workingmedium and L D defined in (9), this condition is fulfilled. As a result, canonical invariancewith the set of operators ˆP , ˆQ , and ˆD = ( ˆQ ˆP + ˆP ˆQ ) prevails for the whole cycle [24].The significance of canonical invariance is that a solution of the operator dynamics allowsthe reconstruction of the state of the working medium during the whole cycle. As a result,all dynamical quantities become functions of a very limited set of thermodynamic quantumobservables (cid:104) ˆX j (cid:105) . The choice of a set of operators { ˆX j } should reflect the most essentialthermodynamical variables. The operator algebra forms a vector space with the scalarproduct (cid:16) ˆX j · ˆX k (cid:17) = T r { ˆX † j ˆX k } . This vector space will be used to describe the state ˆ ρ anddefine the cycle propagators U l . This description is a significant reduction in the dimensionof the propagator U from N , where N is the size of Hilbert space to M the size of theoperator algebra.Explicitly, variables with thermodynamical significance are chosen for the harmonic os-12illator. These variables are time-dependent and describe the current state of the workingmedium: • The Hamiltonian ˆH ( t ) = m ˆP + mω ( t ) ˆQ . • The Lagrangian ˆL ( t ) = m ˆP − mω ( t ) ˆQ . • The position momentum correlation ˆC ( t ) = ω ( t )( ˆQ ˆP + ˆP ˆQ ) = ω ( t ) ˆD .These operators are linear combinations of the same Lie algebra as ˆQ , ˆP , and ˆD . Atypical cycle in terms of these variables is shown in Figure 2.In the algebra of operators, a special place can be attributed to the Casimir operator ˆG .This Casimir commutes with all the operators in the algebra [57, 58]. Explicitly, it becomes: ˆG = ˆH − ˆL − ˆC (cid:126) ω = − (cid:16) ˆP ˆQ + ˆQ ˆP (cid:17) + 2 ˆP ˆP ˆQ ˆQ + 2 ˆQ ˆQ ˆP ˆP (cid:126) . (16)Since [ ˆH , ˆG ] = 0, ˆG is constant under the evolution of the unitary segments generatedby ˆH . The Casimir for the harmonic oscillator is a positive operator with a minimum valuedetermined by the uncertainty relation: (cid:104) ˆG (cid:105) ≥ [59].A related invariant to the dynamics is the Casimir companion [59], which for the harmonicoscillator is defined as: X = (cid:104) ˆH (cid:105) − (cid:104) ˆL (cid:105) − (cid:104) ˆC (cid:105) (cid:126) ω . (17)Combining Equations (16) and (17), an additional invariant to the dynamics can be defined:1 (cid:126) ω (cid:16) V ar ( ˆH ) − V ar ( ˆL ) − V ar ( ˆC ) (cid:17) = const , (18)where V ar ( ˆA ) = (cid:104) ˆA (cid:105) − (cid:104) ˆA (cid:105) .Coherence is an important quantum feature. The coherence is characterised by the devi-ation of the state of the system from being diagonal in energy [60, 61], and it can be definedas: C o = 1 (cid:126) ω (cid:113) (cid:104) ˆL (cid:105) + (cid:104) ˆC (cid:105) . (19)From Equation (16), we can deduce that increasing coherence has a cost in energy ∆ E = (cid:126) ω C o .For the closed algebra of operators, the canonical state of the system ˆ ρ can be cast intothe product form [24, 62]. This state ˆ ρ is defined by the parameters β , γ , and γ ∗ :ˆ ρ = 1 Z e γ ˆa e − β ˆH e γ ∗ ˆa † , (20)13 - - -
20 0 20 40C
D CBA
FIG. 2: Otto refrigeration cycle displayed in the thermodynamical variables ˆH , ˆL , ˆC . When theworking medium is in contact with a hot bath, the system exhausts heat and equilibrates, spirallingdownward from a high (cid:104) ˆH (cid:105) (energy) value and towards zero correlation (cid:104) ˆC (cid:105) and Lagrangian (cid:104) ˆL (cid:105) (i.e., towards thermal equilibrium). The hot ishochore is marked by the red dotted line A → D .On the expansion adiabat , the system spirals downwards, losing energy as it cools down—markedby the green line D → C . It then spirals upwards (blue line), gaining energy from the cold bath C → B . In addition , it spirals towards zero (cid:104) ˆC (cid:105) and (cid:104) ˆL (cid:105) . Then, the compression adiabat (blackline) takes it back to the top of the hot (red) spiral B → A . where ˆH = (cid:126) ω ( ˆaˆa † + ˆa † ˆa ), ˆC = − i (cid:126) ω ( ˆa − ˆa † ) , ˆL = − (cid:126) ω ( ˆa + ˆa † ), and Z = e β (cid:126) ω ( e β (cid:126) ω − (cid:113) − γγ ∗ ( e β (cid:126) ω − . (21)From (20), the expectations of ˆH and ˆa are extracted, leading to (cid:68) ˆH (cid:69) = (cid:126) ω ( e β (cid:126) ω − γγ ∗ − e β (cid:126) ω − − γγ ∗ ) and (cid:68) ˆa (cid:69) = 2 γ ∗ ( e β (cid:126) ω − − γγ ∗ . (22)Equation (22) can be inverted, leading to γ = (cid:126) ω ( (cid:68) ˆL (cid:69) + i (cid:68) ˆC (cid:69) ) (cid:68) ˆL (cid:69) + (cid:68) ˆC (cid:69) − ( (cid:126) ω − (cid:68) ˆH (cid:69) ) (23)14nd the inverse temperature β : e β (cid:126) ω = (cid:68) ˆL (cid:69) + (cid:68) ˆC (cid:69) − (cid:68) ˆH (cid:69) + (cid:126) ω (cid:68) ˆL (cid:69) + (cid:68) ˆC (cid:69) − (cid:16) (cid:126) ω − (cid:68) ˆH (cid:69)(cid:17) . (24)Equations (23) and (24) relate the state of the system ˆ ρ (by Equation 20) to the thermo-dynamical observables (cid:104) ˆH (cid:105) , (cid:104) ˆL (cid:105) , and (cid:104) ˆC (cid:105) .The generalized canonical state of the system Equation (20) is equivalent to a squeezedthermal state [63]: ˆ ρ = ˆS ( γ ) 1 Z e − β ˆH ˆS † ( γ ) , (25)with the squeezing operator ˆS ( γ ) = exp( ( γ ∗ ˆ a − γ ˆ a † )). This state is an example of ageneralized Gibbs state subject to non-commuting constraints [64, 65]. Figure 1 showsexamples of such states, which all have a Gaussian shape in phase space. Entropy Balance
In thermodynamics, the entropy S is a state variable. Shannon introduced entropy asa measure of missing information required to define a probability distribution p [66]. Theinformation entropy can be applied to a complete quantum measurement of an observablerepresented by the operator ˆO with possible outcomes p j : S ˆO = − k B (cid:88) j p j ln p j , (26)where p j = T r { ˆP j ˆ ρ } . The projections ˆP j are defined using the spectral decompositiontheorem ˆO = (cid:80) j λ j ˆP j , where λ j are the eigenvalues of the operator ˆO . S ˆO is then themeasure of information gain obtained by the measurement.The von Neumann entropy [38] is equivalent to the minimum entropy S ˆY n associated witha complete measurement of the state ˆ ρ by the observable ˆY n , where the set of operators ˆY n includes all possible non-degenerate operators in Hilbert space. The operator that minimizesthe entropy commutes with the state [ ˆ ρ, ˆY min ] = 0. This leads to a common set of projectorsof ˆY min and ˆ ρ ; therefore, S V N = − tr { ˆ ρ ln ˆ ρ } , which is a function of the state only. Obviously, S V N ≤ S ˆO . This provides the interpretation that S V N is the minimum information requiredto completely specify the state ˆ ρ . 15he primary thermodynamic variable for the heat engine is energy. The entropy asso-ciated with the measurement of energy S E = S ˆH in general differs from the von Neumannentropy S E ≥ S V N . Only when ˆ ρ is diagonal in the energy representation—such as inthermal equilibrium (13)— S E = S V N .The relative entropy between the state and its diagonal representation in the energyeigenfucntions is an alternative measure of coherence [67]: D ( ˆ ρ || ˆ ρ ed ) = T r { ˆ ρ (ln ˆ ρ − ln ˆ ρ ed ) } , (27)where ˆ ρ ed is the state composed of the energy projections which has the same populations ofthe energy levels as state ˆ ρ . The conditional distance D ( ˆ ρ || ˆ ρ ed ) is equivalent to the differencebetween the energy entropy S E and the von Neumann entropy S V N : D ( ˆ ρ || ˆ ρ ed ) = S E −S V N ≥
0. The von Neumann entropy is invariant under unitary evolution [54]. This is the result ofthe property of unitary transformations, where the set of eigenvalues of ˆ ρ (cid:48) = ˆU ˆ ρ ˆU † is equalto the set of eigenvalues of ˆ ρ . Since the von Neumann entropy S V N is a functional of theeigenvalues of ˆ ρ , it becomes invariant to any unitary transformation.When the unitary transformation is generated by members of the Lie algebra, the Casimiris invariant. The von Neumann entropy of the generalized Gibbs state (20) is a function ofthe Casimir (cid:104) ˆG (cid:105) [68] so that in this case it also becomes constant: S V N = ln (cid:32)(cid:114) (cid:104) ˆG (cid:105) − (cid:33) + (cid:113) (cid:104) ˆG (cid:105) asinh (cid:113) (cid:104) ˆG (cid:105)(cid:104) ˆG (cid:105) − . (28)An alternative expression for the S V N entropy is calculated from the covariance matrixof Gaussian canonical states [69–71]: S V N = ν + 12 ln( ν + 12 ) − ν −
12 ln( ν −
12 ) , (29)where ν = (cid:126) √ σ , σ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ pp σ pq σ qp σ qq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and σ ij is the covariance. 16he energy entropy S E of the oscillator (not in equilibrium) is found to be equivalent tothe entropy of an oscillator in thermal equilibrium with the same energy expectation value: S E = 1 (cid:126) ω (cid:18)(cid:68) ˆH (cid:69) + (cid:126) ω (cid:19) ln (cid:68) ˆH (cid:69) + (cid:126) ω (cid:68) ˆH (cid:69) − (cid:126) ω − ln (cid:126) ω (cid:68) ˆH (cid:69) − (cid:126) ω . (30) S E in (30) is completely determined by the energy expectation E = (cid:68) ˆH (cid:69) . As an extremeexample, for a squeezed pure state, S V N = 0 and S E ≥ /T int = (cid:0) ∂ S ∂E (cid:1) V at constant volume. For the quantum Ottocycle, S E is used to define the inverse internal temperature 1 /T int = (cid:0) ∂ S E ∂E (cid:1) ω . T int is ageneralized temperature appropriate for non equilibrium density operators ˆ ρ . Using thisdefinition, the internal temperature T int of the oscillator working medium can be calculatedimplicitly from the energy expectation: E = 12 (cid:126) ω coth (cid:18) (cid:126) ω k B T int (cid:19) , (31)which is identical to the equilibrium relation between temperature and energy in the har-monic oscillator. This temperature defines the work required to generate the coherence: W c = k B T int ( S E − S V N ) [39].
IV. THE DYNAMICS OF THE QUANTUM OTTO CYCLE
A quantum heat engine is a dynamical system subject to the tradeoff between efficiencyand power. The dynamics of the reciprocating Otto cycle can be partitioned to the fourstrokes and later combined to generate the full cycle. Each of the segments influences thefinal performance: power extraction or refrigeration. The performance of the cycle can beoptimized with respect to efficiency and power. Each segment can be optimized separately,and finally a global optimization is performed. The first step is to describe the dynamics ofeach segment in detail.
A. Heisenberg Dynamics of Thermalisation on the Isochores
The task of the isochores is to extract and reject heat from thermal reservoirs. Thedynamics of the working medium is dominated by an approach to thermal equilibrium. In17he Otto cycle, the Hamiltonian ˆH is constant ( ω = ω h/c is constant). The Heisenbergequations of motion generating the dynamics for an operator ˆX become: ddt ˆX = i (cid:126) [ ˆH , ˆX ] + k ↓ ( ˆa † ˆXˆa − { ˆa † ˆa , ˆX } ) + k ↑ ( ˆa ˆXˆa † − { ˆaˆa † , ˆX } ) . (32)Equation (32) is the analogue of (8) and (9) in the Schr¨odinger frame.For the dynamical set of observables, the equations of motion become: ddt ˆHˆLˆCˆI ( t ) = − Γ 0 0 Γ (cid:104) ˆH (cid:105) eq − Γ − ω
00 2 ω − Γ 00 0 0 0 ˆHˆLˆCˆI ( t ) , (33)where Γ = k ↓ − k ↑ is the heat conductance and k ↑ /k ↓ = e − (cid:126) ω/k B T obeys detailed balancewhere ω = ω h/c and T = T h/c are defined for the hot or cold bath, respectively. From (11),the heat current can be identified as:˙ Q = − Γ( (cid:104) ˆH (cid:105) − (cid:104) ˆH (cid:105) eq ) = Γ (cid:126) ω (cid:18) coth( (cid:126) ω k B T B ) − coth( (cid:126) ω k B T int ) (cid:19) , (34)where T B is the bath temperature. In the high temperature limit, the heat transport lawbecomes Newtonian [22]: ˙ Q = Γ( k B T B − K B T int ).The solution of isochore dynamics (33) generates the propagator defined on the vectorspace of the observables ˆH , ˆL , ˆC , ˆI : The propagator on the isochore has the form [68, 72]: U h/c = R H eq (1 − R )0 Rc − Rs Rs Rc
00 0 0 1 , (35)where R = e − Γ t . c = cos(2 ωt ), s = sin(2 ωt ), and H eq = (cid:126) ωe (cid:126) ωkT − . It is important to note thatthe propagator on the isochores does not generate coherence from energy ˆH . The coherenceEquation (19) is a function of the expectations of ˆL , ˆC , which are not coupled to ˆH . B. The Dynamics on the Adiabats and Quantum Friction
The dynamics on the adiabats is generated by a time-dependent Hamiltonian. The taskis to change the energy scale of the working medium from one bath to the other. The18scillator frequency changes from ω h to ω c on the power expansion segment and from ω c to ω h on the compression segment. The Hamiltonian—which is explicitly time-dependent—does not commute with itself at different times [ ˆH ( t ) , ˆH ( t (cid:48) )] (cid:54) = 0. As a result, coherence isgenerated with an extra cost in energy.The Heisenberg equations of motion (10) for the dynamical set of operators are expressedas [68, 72]: ddt ˆHˆLˆCˆI ( t ) = ω ( t ) µ − µ − µ µ − µ
00 0 0 0 ˆHˆLˆCˆI ( t ) , (36)where µ = ˙ ωω is a dimensionless adiabatic parameter. In general, all operators in (36)are dynamically coupled. This coupling is characterized by the non-adiabatic parameter µ .When µ →
0, the energy decouples from the coherence and the cycle can be characterizedby p n —the probability of occupation of energy level (cid:15) n .Power is obtained from the first-law (11) as: P = µω (cid:16) (cid:104) ˆH (cid:105) − (cid:104) ˆL (cid:105) (cid:17) . (37)Power on the adiabats (37) can be decomposed to the “useful” external power P ex = µω (cid:104) ˆH (cid:105) and to the power invested to counter friction P f = − µω (cid:104) ˆL (cid:105) if (cid:104) ˆL (cid:105) >
0. Underadiabatic conditions µ → (cid:104) ˆL (cid:105) = 0, since no coherence is generated; therefore, P f = 0.Generating coherence consumes power when the initial state is diagonal in energy [ ˆ ρ, ˆH ] =0 [73, 74].Insight on the adiabatic dynamics can be obtained from the closed-form solution of thedynamics when the non-adiabatic parameter µ = ˙ ωω is constant. This leads to the explicittime dependence of the control frequency ω : ω ( t ) = ω (0)1 − µω (0) t . Under these conditions,the matrix in Equation (36) becomes stationary. This allows a closed-form solution to beobtained by diagonalizing the matrix. Under these conditions, the adiabatic propagator U a has the form: U a = ω ( t ) ω (0) 1Ω − µ c − µ Ω s − µ ( c −
1) 0 − µ Ω s Ω c − s µ ( c −
1) 2Ω s c − µ
00 0 0 1 , (38)19here Ω = (cid:112) − µ and c = cos(Ω θ ( t )), s = sin(Ω θ ( t )), and θ ( t ) = − µ log( ω ( t ) ω (0) ) [68]. For | µ | <
2, the solutions are oscillatory. For | µ | >
2, the sin and cos functions become sinh andcosh. More on the transition point from damped to over-damped dynamics is presented inSection IV E.The difference between the expansion adiabats U hc and compression adiabats U hc is inthe sign of µ and the ratio ω ( t ) /ω (0). The propagator U a can be viewed as a product ofa changing energy scale by the factor ω ( t ) ω (0) and a propagation in a moving frame generatedby a constant matrix. The fraction of additional work on the adiabats with respect to the adiabatic solution is causing extra energy invested in the woking medium, which is definedas: δ f = ω i ω f ( U a (1 , − ω f ω i ) . (39)For the case of µ constant: δ f = 2 µ sin( θ Ω2 ) − µ (40)and δ f ≥ ψ ( x, t ) = exp (cid:18) i (cid:126) ( a ( t ) x + b ( t ) x + c ( t )) (cid:19) . (41) a ( t ) can be mapped to a time-dependent classical harmonic oscillator: a ( t ) = M ˙ X/X ,where: m d dt X + ω ( t ) X = 0 . (42)The local adiabatic parameter is defined as: Q ∗ ( t ) = 12 ω i ω f (cid:16) ω i ( ω f X ( t ) + ˙ X ( t ) ) + ( ω f Y ( t ) + ˙ Y ( t ) ) (cid:17) , (43)where X ( t ) and Y ( t ) are the solution of Equation (42) with the boundary conditions X (0) =0, ˙ X (0) = 1 and Y (0) = 1, ˙ Y (0) = 0 for a constant frequency Q ∗ = 1. In general, theexpectation value of the energy at the end of the adiabats becomes: (cid:104) ˆH (cid:105) f = ω f ω i Q ∗ (cid:104) ˆH (cid:105) i , (44)where i/f correspond to the beginning and end of the stroke. In general, Q ∗ ( t ) can beobtained directly from the solution of Equation (42). Q ∗ ( t ) is related to δ f by: Q ∗ ( t ) = 1+ δ f .20or the case of µ , constant Q ∗ ( t ) can be obtained from Equation (40). In addition, Q ∗ ( t )can be obtained by the solution of the Ermakov equation (Equation (49)) [76].The general dynamics described in Equation (38) mixes energy and coherence. As canbe inferred from the Casimir Equation (16), generating coherence costs energy. This extracost gets dissipated on the isochores , and is termed quantum friction [39, 77]. The energycost scales as µ ; therefore, slow operation (i.e., | µ | (cid:28)
1) will eliminate this cost. Thedrawback is large cycle times and low power. Further analysis of Equation (38) shows asurprising result. Coherence can be generated and consumed, resulting in periodic solutionsin which the propagator becomes diagonal. As a result, mixing between energy and coherenceis eliminated. These solutions appear when cos(Ω θ ( τ a )) = 1, where τ a is the expansionor compression stroke time allocation. These periodic solutions can be characterised bya quantization relation [68]: µ ∗ = − C ) (cid:112) π l + log( C ) , (45)where C = ω c ω h is the engine’s compression ratio, and l the quantization number l = 1 , , , ... ,accompanied by the time allocation τ ∗ hc : τ ∗ hc = 1 − C µ ∗ ω c . (46)A frictionless solution with the shortest time is obtained for l = 1, and it scales as τ hc ∝ /ω c .This observation raises the question: are there additional frictionless solutions in finitetime? What is the shortest time that can achieve this goal?The general solution of the dynamics depends on an explicit dependence of ω ( t ) ontime. ω ( t ) can be used as a control function to optimise the performance, obtaining astate ˆ ρ diagonal in energy at the interface with the isochores . Such a solution will generatea frictionless performance. Operating at effective adiabatic conditions has been termed shortcut to adiabaticity [78–82].The search for frictionless solutions has led to two main directions. The first is based ona time-dependent invariant operator ˆI ( t ) [82]: ddt ˆI ( t ) = ∂∂t I ( t ) + i (cid:126) [ ˆH ( t ) , ˆI ( t )] = 0 . (47)For the harmonic oscillator, the invariant is [78]: ˆI ( t ) = 12 (cid:18) b ˆQ mω + 1 m ˆ π (cid:19) , (48)21here ˆ π = b ˆP − m ˙ b ˆQ . The invariant must satisfy [ ˆH , ˆI ] = 0 for the initial τ i and final τ f times, then at these times the eigenstates of the invariant at the initial and final time of the adiabat are identical to those of the Hamiltonian [83, 84]. This is obtained if b (0) = 1 and˙ b (0) = 0, ¨ b (0) = 0, as well as b ( τ f ) = (cid:112) ω /ω f and ˙ b ( τ f ) = 0, ¨ b ( τ f ) = 0. In addition, thefunction b ( t ) satisfies the Ermakov equation:¨ b + ω ( t ) b = ω /b . (49)The instantaneous frequency becomes ω ( t ) = ω /b for ¨ b (0) = 0. There are many so-lutions to the Ermakov equation, and additional constraints must be added—for example,that the frequency is at all times real and positive. These equations can be used to search forfast frictionless solutions. To obtain a minimal time τ a , some constrains have to be imposed.For example, limiting the average energy stored in the oscillator. In this case, τ hc scales as τ ∗ hc ∝ / √ ω c [80]. Other constraints on ω ( t ) have been explored. For example, the use ofimaginary frequency corresponding to an inverted harmonic potential. These schemes allowfaster times on the adiabat [80, 85]. If the peak energy is constrained, τ hc scales logarith-mically with 1 /ω c ; however, if the average energy is constrained, then the scaling becomes τ ∗ hc ∝ / √ ω c .The second approach to obtain frictionless solutions is based on optimal control theory:finding the fastest frictionless solution where the control function is ω ( t ) [68, 85–90]. Optimalcontrol theory reveals that the problem of minimizing time is linear in the control, which isproportional to ω ( t ) [68]. As a result, the optimal control solution depends on the constraints ω max and ω min . If these are set as ω max = ω h and ω min = ω c , then the optimal time scalesas τ ∗ a ∝ √ ω c √ ω c ) . Other constraints will lead to faster times, but their energetic cost willdiverge. This scaling is consistent considering the cost of the counter-adiabatic terms infrictionless solutions leading to the same scaling [91].The optimal solution can be understood using a geometrical description [87]. The deriva-tive of the change of (cid:104) ˆQ (cid:105) with respect to the change in (cid:104) ˆP (cid:105) becomes: d (cid:104) ˆQ (cid:105) d (cid:104) ˆP (cid:105) = − ω ( t ) ≡ v . (50)The time allocated to the change τ becomes: τ = (cid:90) (cid:104) ˆP (cid:105) f (cid:104) ˆP (cid:105) i d (cid:104) ˆP (cid:105) (cid:113) (cid:104) ˆP (cid:105)(cid:104) ˆQ (cid:105) − (cid:104) ˆG (cid:105) , (51)22here ˆG is the Casimir defined in Equation (16). In addition, the control v is constrainedby ω c ≤ v ≤ ω h . The initial and final (cid:104) ˆP (cid:105) i/f = mE i/f , since the initial and final (cid:104) ˆL (cid:105) and (cid:104) ˆC (cid:105) are zero. The minimum time is obtained by maximizing the product (cid:104) ˆP (cid:105)(cid:104) ˆQ (cid:105) alongthe trajectory. The minimum time optimization leads to a bang-bang solution where thefrequency is switched instantly from ω h to ω c , as in the sudden limit Equation (58) is followedby a waiting period then switched back to ω h until the target is reached and switched finallyto ω c . The relation between the geometric optimization and the Ermakov equation of theshortcuts to adiabaticity has been obtained based on the geometrical optimization [92, 93].To summarize, frictionless solutions can be obtained in finite time. As a result, theengine can be completely described by the population of the energy eigenvalues or for theharmonic working medium by the expectation value of number operator ˆN . Employingreasonable constraints on the control function ω ( t ) results in the minimum time τ ∗ a scallingas O ( √ ω c √ ω h ). C. The Influence of Noise on the Adiabats
The frictionless adiabat requires a very accurate protocol of ω ( t ) as a function of time. Forany realistic devices, such a protocol will be subject to fluctuations in the external control.The controllers are subject to noise, which will induce friction-like behaviour. Can thisadditional friction be minimized? Insight on the effects of noise on the performance of theOtto cycle can be obtained by analysing a simple model based on the frictionless protocolwith constant µ [94]. The obvious source of external noise is induced by fluctuations in thecontrol frequency ω ( t ). This noise is equivalent to Markovian random fluctuations in thefrequency of the harmonic oscillator. These errors are modelled by a Gaussian white noise.The dissipative Lindbland term generating such noise has the form [42, 95]: L N a ( ˆA ) = − γ a ω [ ˆB , [ ˆB , ˆA ]] , (52)where ˆB = mω ˆQ / (2 (cid:126) ).The influence of the amplitude noise generated by L N a ( ˆA ) = − γ a ω [ ˆB , [ ˆB , ˆA ]] is obtainedby approximating the propagator by the product form U hc = U a U an . The equations of motion23or the amplitude noise U an are obtained from the interaction picture in Liouville space: dωdt U an ( t ) = U a ( − t ) N a ( t ) U a ( t ) U an ( t )= W an ( t ) U an ( t ) , (53)where W an is the interaction propagator in Liouville space [94] and U a is the adiabatic propagator, Equation (38). A closed-form solution is obtained in the frictionless limit µ → W a is expanded up to zero order in µ : W a ( t ) ≈ γ a ω − c s c − c cs − s cs − s
00 0 0 0 , (54)where s = sin(ΩΘ) c = cos(ΩΘ). The Magnus expansion [96] is employed to obtain the l period propagator U a ( X = 2 lπ ), where the periods are of the adiabatic propagator U a ofEquation (38): U a ( X = 2 lπ ) ≈ e B + B + ... , (55)where B = (cid:82) nπ dX W a ( X ), B = (cid:82) nπ (cid:82) X dXdX (cid:48) [ W a ( X ) , W a ( X (cid:48) )], and so on. Thefirst-order Magnus term leads to the propagator U an ( X = 2 lπ ) B = e γ a F /µ e − γ a F / (2 µ ) e − γ a F / (2 µ )
00 0 0 1 , where F = (cid:16) ω ( − µ ) (cid:17) (cid:0) e πlµ Ω − (cid:1) . For large l , in Equation (45) the limit from hot to coldsimplifies to: F = ( ω h − ω c ). The solution of Equation (53) shows that the fraction of workagainst friction δ f will diverge when l → ∞ or µ →
0, nulling the adiabatic solution foreven a very small γ a . The best way to eliminate amplitude noise is to choose the shortestfrictionless protocol. Nevertheless, some friction-like behaviors will occur.Next, phase noise is considered. It occurs due to errors in the piecewise process used forcontrolling the scheduling of ω in time. For such a procedure, random errors are expectedin the duration of the time intervals. These errors are modeled by a Gaussian white noise.24athematically, the process is equivalent to a dephasing process on the adiabats [97]. Thedissipative operator L N has the form given by [42, 95]: L N p ( ˆA ) = − γ p (cid:126) [ ˆH , [ ˆH , ˆA ]] . (56)In this case, the interaction picture for the phase noise U p becomes dωdt U pn ( t ) = U a ( − t ) N p ( t ) U a ( t ) U pn ( t ) = W p ( t ) U pn ( t ) , which at first order in µ can be approximated as W p ( t ) ≈ γ p ω × µs µ (1 − c ) 0 µs − (2 + µX ) 0 0 µ ( c −
1) 0 − (2 + µX ) 00 0 0 0 . Again, using the Magnus expansion for one period of X leads to U p ( X = 2 π ) B = − e πγ p ω ) µ/ e πγ p ω (1 − π µγ p ω ) 0 0( − e πγ p ω ) µ/ e πγ p ω (1 − π µγ p ω ) 00 0 0 1 . At first order in µ , this evolution operator maintains δ f (1) = 0, so the frictionless caseholds. The second-order Magnus term leads to the noise correction U p ( X = 2 π ) B = cosh β − sinh β − sinh β cosh β , (57)where β = ω γ p µ (cid:0) e πlµ Ω − (cid:1) . In the limit of l → ∞ , β = 4 γ ( ω h − ω c ). The propagator U p ( X = 2 π ) B mixes energy and coherence, even at the limit µ → τ a → ∞ , whereone would expect frictionless solutions.We can characterize the fraction of additional energy generated by a parameter δ . Asymp-totically for amplitude noise: δ a = e γ a F /µ >
0, and for phase noise δ f (1) = e γ p F µ − ≈ δ f (2) = cosh( β ) − >
0. Imperfect control on the adiabats will always lead to δ f > . The Sudden Limit The limit of vanishing time on the adiabats τ a (cid:28) /ω c leads to the sudden propagator;therefore, µ → ±∞ . Such dynamics is termed sudden quench . The propagator U a has anexplicit expression: U a = (1 + α ) (1 − α ) 0 0 (1 − α ) (1 + α ) 0 00 0 1 00 0 0 1 , (58)where α = ( ω f ω i ) is related to the compression ratio α ch = C and α hc = C − . The propagatormixes ˆH and ˆL when the compression ratio deviates from 1. As a result coherence isgenerated. The sudden propagator is an integral part of the frictionless bang-bang solutions[68, 87]. Equation (58) can be employed as part of a bang-bang adiabat or as part of acomplete sudden cycle. E. Effects of an Exceptional Point on the Dynamics on the Adiabat
Exceptional points (EPs) are degeneracies of non-Hermitian dynamics [98, 99] associatedwith the coalescence of two or more eigenstates. The studies of EPs have substantially growndue to the observation of (space-time reflection symmetry) PT symmetric Hamiltonians[100]. These Hamiltonians have a real spectrum, which becomes complex at the EP. Themain effect of EPs (of any order) on the dynamics of PT-symmetric systems is the suddentransition from a real spectrum to a complex energy spectrum [101, 102].The adiabatic strokes are generated by a time-dependent Hamiltonian Equation (4). Wetherefore expect the propagator U a to be unitary, resulting in eigenvalues with the property | u j | = 1. These properties are only true for a compact Hilbert space. We find surprisingexceptions for the non-compact harmonic oscillator with an infinite number of energy levels.We can remove the trivial scaling ω ( t ) ω (0) in Equation (36) which originates from the diagonalpart. The propagator can be written as U a = U U , where U = ω ( t ) ω (0) I is a rescaling of the26nergy unit. The equation of motion for U becomes: ddθ U ( θ ) = − µ − µ −
20 2 0 U ( θ ) , (59)where the trivial propagation of the identity is emitted and the time is rescaled θ ( t ) = (cid:82) t ω ( t (cid:48) ) dt (cid:48) . Diagonalising Equation (59) for constant µ , we can identify three eigenvalues: λ = 0 and λ = ± i (cid:112) − µ . For µ ≤
2, as expected, Equation (59) generates a unitarypropagator. The three eigenvalues become degenerate when µ = 2, and become real for µ ≥ λ [103]. This is possible because the generator Equation (59) is non-Hermitian. Atthe exceptional point, the matrix in Equation (59) has a single eigenvector correspondingto λ = λ , which is self-orthogonal. To show this property, it is necessary to multiply theright and left eigenvectors of the non-symmetric matrix at the EP. Their product is equalto zero, showing that the eigenvector is self-orthogonal [104]. The propagator Equation (38)changes character at the EP; Ω = (cid:112) − µ changes from a real to an imaginary number.As a result, the dynamics at the EP changes from oscillatory to exponential [103].This effect can also be observed in the classical parametric oscillator Equation (42). Bychanging the time variable ddt = ω ( t ) ddτ , and for constant µ , the equation of motion becomes (cid:18) d dτ + µ ddτ + 1 (cid:19) X ( τ ) = 0 , (60)which is the well-known equation of motion of a damped harmonic oscillator. Note thatthe original model (given by Equation (42)) does not involve dissipation, and a priori onewould not expect the appearance of an EP. The rescaling of the time coordinate allows usto identify an EP at | µ | = 2, corresponding to the transition between an underdamped andan over-damped oscillator [105].Exceptional points are also expected in the eigenvalues and eigenvectors of the totalpropagator U cyc , which posses complex eigenvalues. Such points will indicate a drasticchange in the cycle performance. V. CLOSING THE CYCLE
Periodically combining the four propagators leads to the cycle propagator. Dependingon the choice of parameters, we get either an engine cycle where heat flow is converted to27ower: U ecyc = U ch U c U hc U h where ω c ω h > T c T h , (61)or a refrigerator cycle where power drives a heat current from the cold to the hot bath: U rcyc = U ch U c U hc U h where ω c ω h < T c T h . (62)In both cases, ω h > ω c . Frictionless cycles are either refrigerators or engines. Frictionadds another possibility. When the internal friction dominates both, the engine cycle andthe refrigeration cycle will operate in a dissipative mode, where power is dissipated to boththe hot and cold baths. For an engine, this dissipative mode will occur when the internaltemperature of the oscillator Equation (31) after the expansion adiabat (cf. Figure 3 pointD) will exceed T h and in a refrigerator cycle when the internal temperature exceeds T c (cf.Figure 4 point C). A. Limit Cycle
When a cycle is initiated, after a short transient time it settles to a steady-state operationmode. This periodic state is termed the limit cycle [14, 106]. An engine cycle converges to alimit cycle when the internal variables of the working medium reach a periodic steady state.As a result, no energy or entropy is accumulated in the working medium. Figure 2 is anexample of a periodic limit cycle. Subsequently, a balance is obtained between the externaldriving and dissipation. When the cycle time is reduced, friction causes additional heat to beaccumulated in the working medium. The cycle adjusts by increasing the temperature gapbetween the working medium and the baths, leading to increased dissipation. Overdrivingleads to a situation where heat is dissipated to both the hot and cold bath and power isonly consumed. When this mechanism is not sufficient to stabilise the cycle, one can expecta breakdown of the concept of a limit cycle, resulting in catastrophic consequences [72].The properties of a completely positive (CP) map can be used to prove the existence of alimit cycle. Lindblad [107] has proven that the conditional entropy decreases when applyinga trace-preserving completely positive map Λ to both the state ˆ ρ and the reference stateˆ ρ ref : D (Λ ˆ ρ || Λ ˆ ρ ref ) ≤ D ( ˆ ρ || ˆ ρ ref ) , D ( ˆ ρ || ˆ ρ (cid:48) ) = T r ( ˆ ρ (log ˆ ρ − log ˆ ρ (cid:48) )) is the conditional entropy distance. A CP map reducesthe distinguishability between two states. This can be employed to prove the monotonicapproach to steady-state, provided that the reference state ˆ ρ ref is the only invariant of theCP map Λ (i.e., Λ ˆ ρ ref = ˆ ρ ref ) [108–110]. This reasoning can prove the monotonic approachto the limit cycle. The mapping imposed by the cycle of operation of a heat engine is aproduct of the individual evolution steps along the segments composing the cycle propagator.Each one of these evolution steps is a completely positive map, so the total evolution U cyc Equation (1) that represents one cycle of operation is also a CP map. If then a state ˆ ρ lc isfound that is a single invariant of U cyc (i.e., U cyc ˆ ρ lc = ˆ ρ lc ), then any initial state ˆ ρ init willmonotonically approach the limit cycle.The largest eigenvalue of U cyc with a value of is associated with the invariant limit cyclestate U cycr ˆ ρ lc = 1 ˆ ρ lc , the fixed point of U cyc . The other eigenvalues determine the rate ofapproach to the limit cycle.In all cases studied of a reciprocating quantum heat engine, a single non-degenerateeigenvalue of was the only case found. The theorems on trace preserving completelypositive maps are all based on C ∗ algebra, which means that the dynamical algebra of thesystem is compact. Can the results be generalized to discrete non-compact cases such as theharmonic oscillator? In his study of the Brownian harmonic oscillator, Lindblad conjectured:“ in the present case of a harmonic oscillator, the condition that L is bounded cannot hold.We will assume this form for the generator with ˆH and L unbounded as the simplest way toconstruct an appropriate model ” [45]. The master equation in Lindblad’s form Equation (9)is well established. Nevertheless, the non-compact character of the resulting map has notbeen challenged.A nice demonstration is the study of Insinga et al. [72], which shows conditions where alimit cycle is not obtained. This study contains an extensive investigation of the limit cyclesas a function of the parameters of the system [72]. B. Engine Operation and Performance
The engine’s cycle can operate in different modes, which are: adiabatic, frictionless,friction-dominated, and the sudden cycle. In addition, one has to differentiate between twolimits: high temperature k B T (cid:29) (cid:126) ω , where the unit of energy is k B T , to low temperature29 ω (cid:29) k B T , where the unit of energy is (cid:126) ω .In the adiabatic and frictionless cycles [111], the performance can be completely deter-mined by the value of energy at the switching point between strokes.
1. Optimizing the Work per Cycle
The adiabatic limit with infinite time allocations on all segments maximises the work.No coherence is generated, and therefore the cycle can be described by the change in energy.On the expansion adiabat E B = ω c ω h E A , and on the compression adiabat E D = ω h ω c E C . As aresult, when the cycle is closed, the heat transferred to the hot bath Q h = E A − E D and tothe cold bath Q c = E B − E C are related: Q c Q h = ω c ω h .The efficiency for an engine becomes the Otto efficiency: η = WQ h = 1 − ω c ω h ≤ − T c T h . (63)Choosing the compression ratio C = ω h ω c = T h T c maximises the work and leads to Carnotefficiency η o = η c . Since for this limit the cycle time τ cyc is infinite, the power P = W /τ cyc of this cycle is obviously zero.
2. Optimizing the Performance of the Engine for Frictionless Conditions
Frictionless solutions allow finite time cycles with the same efficiency η o = 1 − ω c ω h asthe adiabatic case. A different viewpoint is to account as wasted work the average energyinvested in achieving the frictionless solution, termed superadiabatic drive [112]: η = WQ h + (cid:104) H ch (cid:105) + (cid:104) H hc (cid:105) , (64)where (cid:104) H ch (cid:105) is the average additional energy during the adiabatc stroke. Using for exampleEquation (38), the average additional energy becomes (cid:104) H ch (cid:105) = ω h ω c E C µ − µ , which vanishes asthe non-adiabatic parameter µ →
0. This additional energy (cid:104) H ch (cid:105) in the engine is the pricefor generating coherence. Coherence is exploited to cancel friction. This extra energy is notdissipated, and can therefore be viewed as a catalyst. For this reason, we do not accept theviewpoint of [112]. 30n our opinion, what should be added to the accounting is the additional energy generatedby noise on the controls: η = WQ h + E C δ ch + E A δ hc ≤ η o , (65)where δ hc = δ a + δ f for the power adiabat and δ a and δ f are generated by amplitude and phasenoise on the controller (cf. Section IV C). A similar relation is found for the compression adiabat .Optimizing power requires a finite cycle time τ cyc . Optimisation is carried out withrespect to the time allocations on each of the engine’s segments: τ h , τ hc , τ c , and τ ch . Thissets the total cycle time τ cyc = τ h + τ hc + τ c + τ ch . The time allocated to the adiabats isconstrained by the frictionless solutions τ ∗ hc and τ ∗ ch . The resulting optimization is very closeto the unconstrained optimum [72], especially in the interesting limit of low temperatures.The frictionless conditions are obtained either from Equation (45) or from other shortcutsto adiabaticity methods Equation (47). In the frictionless regime, the number operator isfixed at both ends of the adiabat . The main task is therefore to optimize the time allocatedto thermalisation on the isochores . This heat transport is the source of entropy production.The time allocations on the isochores determine the change in the number operator N = (cid:104) ˆN (cid:105) = (cid:126) ω (cid:104) ˆH (cid:105) (cf. Equation (33)): N B = e − Γ h τ h (cid:0) N A − N heq (cid:1) + N heq on the hot isochore ,where N B is the number expectation value at the end of the hot isochore , N A at thebeginning, and N heq is the equilibrium value point E . A similar expression exists for the cold isochore .Work in the limit cycle becomes W q = E C − E B + E A − E D = (cid:126) ( ω c − ω h )( N B − N D ) , (66)where the convention of the sign of the work for a working engine is negative, in correspon-dance with Callen [33], and we use the convention of Figure 3 to mark the population andenergy at the corners of the cycle.The heat transport from the hot bath becomes Q h = E B − E D = (cid:126) ω h ( N B − N D ) . (67)In the limit cycle for frictionless conditions, N B = N A , which leads to the relation N B = ( e Γ c τ c − − e Γ c τ c +Γ h τ h ( N heq − N ceq ) + N heq . (68)31 h ω c ABC D T h T c ω S E FIG. 3: Typical engine cycle S E vs. ω . Expansion adiabat A → B . Cold isochore B → C .Compression adiabat C → D . Hot isochore D → A . The hot and cold isotherms are indicated.The cycle parameters are ω c = 0 . , T c = 5 , ω h = 2 , T h = 200 , τ c = τ h = 2 . , | µ | = 0 . In the periodic limit cycle, the number operator change N B − N D is equal on the hotand cold isochores , leading to the work per cycle: W q = (cid:126) ( ω h − ω c )( N heq − N ceq ) ( e x c − e x h − − e x c + x h (69) ≡ − G W ( T c , ω c , T h , ω h ) F ( x c , x h ) , where the scaled time allocations are defined x c ≡ Γ c τ c and x h ≡ Γ h τ h . The work W q Equation (69) becomes a product of two functions: G W , which is a function of the staticconstraints of the engine, and F , which describes the heat transport on the isochores . Ex-plicitly, the function G W is G W ( T c , ω c , T h , ω h ) = (cid:126) ω h − ω c ) (cid:18) coth (cid:18) (cid:126) ω h k B T h (cid:19) − coth (cid:18) (cid:126) ω c k B T c (cid:19)(cid:19) . (70)The function F in Equation (69) is bounded 0 ≤ F ≤
1; therefore, for the engine toproduce work, G W ≥
0. The first term in (70) is positive. Therefore, G W ≥ ω c ω h ≥ T c T h , or in terms of the compression ratio, 1 ≤ C ≤ T h T c . This is equivalent tothe statement that the maximum efficiency of the Otto cycle is smaller than the Carnotefficiency η o ≤ η c .In the high temperature limit when (cid:126) ωk B T (cid:28) G W simplifies to G W = k B T c (1 − C ) + k B T h (1 − C − ) . (71)In this case, the work W q = − G W F can be optimized with respect to the compressionratio C = ω h ω c for fixed bath temperatures. The optimum is found at C = (cid:113) T h T c . As a result,32he efficiency at maximum power for high temperatures becomes η q = 1 − (cid:114) T c T h , (72)which is the well-known efficiency at maximum power of an endo-reversible engine [6, 9,13, 21, 22, 113]. Note that these results indicate greater validity to the Novikov–Curzon–Ahlbourn result from what their original derivation [22] indicates.The function F defined in (69) characterizes the heat transport to the working medium.As expected, F maximizes when infinite time is allocated to the isochores . The optimalpartitioning of the time allocation between the hot and cold isochores is obtained when:Γ h (cosh(Γ c τ c ) −
1) = Γ c (cosh(Γ h τ h ) − . (73)If (and only if) Γ h = Γ c , the optimal time allocations on the isochores becomes τ h = τ c .Optimising the total cycle power output P is equivalent to optimizing F/τ cyc , since G W is determined by the engine’s external constraints. The total time allocation τ cyc = τ iso + τ adi is partitioned to the time on the adiabats τ adi , which is limited by the adiabatic frictionlesscondition, and the time τ iso allocated to the isochores .Optimising the time allocation on the isochores subject to (73) leads to the optimalcondition Γ c τ cyc (cosh(Γ h τ h ) −
1) = sinh(Γ h τ h + Γ c τ c ) − sinh(Γ c τ c ) − sinh(Γ h τ h ) . (74)When Γ h = Γ c ≡ Γ, this expression simplifies to:2 x + Γ τ adi = 2 sinh( x ) , (75)where x = Γ c τ c = Γ h τ h . For small x , Equation (75) can be solved, leading to the optimaltime allocation on the isochores : τ c = τ h ≈ (Γ τ adi / / Γ. Considering the restriction due tofrictionless condition [86], this time can be estimated to be: τ c = τ h ≈ (cid:16) Γ √ ω c ω h (cid:17) . Whenthe heat transport rate Γ is sufficiently large, the optimal power conditions lead to the bang-bang solution where vanishingly small time is allocated to all segments of the engine [14]and τ cyc ≈ τ adi .The entropy production ∆ S U reflects the irreversible character of the engine. In friction-less conditions, the irreversibility is completely associated with the heat transport. ∆ S U canalso be factorized to a product of two functions:∆ S u = G S ( T c , ω c , T h , ω h ) F ( x c , x h ) , (76)33here F is identical to the F function defined in (69). The function G S becomes: G S ( T c , ω c , T h , ω h ) = 12 (cid:18) (cid:126) ω h k B T h − (cid:126) ω c k B T c (cid:19) (cid:18) coth (cid:18) (cid:126) ω c k B T c (cid:19) − coth (cid:18) (cid:126) ω h k B T h (cid:19)(cid:19) . (77)Due to the common F ( x c , x h ) function, the entropy production has the same dependenceon the time allocations τ h and τ c as the work W [114]. As a consequence, maximizingthe power will also maximize the entropy production rate ∆ S u /τ cyc . Note that entropyproduction is always positive, even for cycles that produce no work, as their compressionratio C is too large, which is a statement of the second law of thermodynamics.The dependence of the G s function on the compression ratio can be simplified in the hightemperature limit, leading to: G S = C T c T h + C − T h T c − , (78)which is a monotonic decreasing function in the range 1 ≤ C ≤ T h T c that reaches a minimumat the Carnot boundary when C = T h T c . When power is generated, the entropy productionrate in the frictionless engine is linearly proportional to the power: S u = (cid:18) (cid:126) ω h k B T h − (cid:126) ω c k B T c (cid:19) (cid:18) ω h − ω c (cid:19) P . (79)Frictionless harmonic cycles have been studied under the name of superadiabatic driving [111]. The frictionless adiabats are obtained using the methods of shortcut to adiabaticity [82]and the invariant Equation (47). An important extension applies shortcuts to adiabaticityto working mediums composed of interacting particles in a harmonic trap [76, 115–117].A variant of the Otto engine is an addition of projective energy measurements beforeand after each adiabat . This construction is added to measure the work output [118]. As aresult, the working medium is always diagonal in the energy basis. In the frictionless case,the cycle is not altered by this projective measurement of energy.
3. The Engine in the Sudden Limit
The extreme case of the performance of an engine with zero time allocation on the adiabats is dominated by the frictional terms. These terms arise from the inability of the workingmedium to adiabatically follow the external change in potential. A closed-form expressionfor the sudden limit can be derived based on the adiabatic branch propagator U hc and U ch in Equation (58). 34o understand the role of friction, we demand that the heat conductance terms Γ arevery large, thus eliminating the thermalisation time. In this limiting case, the work per cyclebecomes: W s = ( ω c − ω h )( ω c + ω h )4 ω c ω h (cid:18) (cid:126) ω c coth( (cid:126) ω h k B T h ) − (cid:126) ω h coth( (cid:126) ω c k B T c ) (cid:19) . (80)The maximum produced work −W s can be optimised with respect to the compressionratio C . At the high temperature limit: W s = 12 k B T h ( C − T c T h − C ) . (81)For the frictionless optimal compression ratio C = (cid:113) T h T c , W s is zero. The optimal com-pression ratio for the sudden limit becomes: C = (cid:16) T h T c (cid:17) / , leading to the maximal work inthe high temperature limit W s = − k B T c (cid:32) − (cid:114) T h T c (cid:33) . (82)The efficiency at the maximal work point becomes: η s = 1 − (cid:113) T hT c (cid:113)
T hT c . (83)Equation (83) leads to the following hierarchy of the engine’s maximum work efficiencies: η s ≤ η q ≤ η c . (84)Equation (84) leads to the interpretation that when the engine is constrained by frictionits efficiency is smaller than the endo-reversible efficiency, where the engine is constrainedby heat transport that is smaller than the ideal Carnot efficiency. At the limit of T c → η s = and η q = η c = 1 [119].An upper limit to the work invested in friction W f is obtained by subtracting the max-imum work in the frictionless limit Equation (69) from the maximum work in the suddenlimit Equation (80). In both these cases, infinite heat conductance is assumed, leading to N B = N heq and N D = N ceq . Then, the upper limit of work invested to counter frictionbecomes: W f = (cid:126) ω h ( C − (1 + C + 2 C N ceq + 2 N heq )4 C . (85)35t high temperature, Equation (85) changes to: W f = 12 k b T h ( C − ( C − + T c T h ) . (86)The maximum produced work at the high temperature limit of the frictionless and suddenlimits differ by the optimal compression ratio. For the frictionless case, C ∗ = (cid:113) T h T c , and forthe sudden case, C ∗ = ( T h T c ) / .The work against friction W f (Equation (86)) is an increasing function of the temperatureratio. For the compression ratio that optimises the frictionless limit, the sudden work iszero. At this compression ratio, all the useful work is balanced by the work against friction W f = W q . Beyond this limit, the engine transforms to a dissipator, generating entropyat both the hot and cold baths. This is in contrast to the frictionless limit, where thecompression ratio C = T h T c leads to zero power.The complete sudden limit assumes short time dynamics on all segments including the isochores . These cycles with vanishing cycle times approach the limit of a continuous engine.The short time on the isochores means that coherence can survive. Friction can be partiallyavoided by exploiting this coherence, which—unlike the frictionless engine—is present in thefour corners of the cycle. The condition for such cycles is that the time allocated is muchsmaller than the natural period set by the frequency τ c , τ h (cid:28) π/ω and by heat transfer τ c , τ h (cid:28) / Γ. The heat transport from the hot and cold baths in each stroke becomes verysmall. For simplicity, Γ h τ h = Γ c τ c is chosen to be balanced. Under these conditions, thecycle propagator becomes: U cyc = (1 − g ) g (1 − g ) (1 + C ) (cid:126) ω c N eqc + g (cid:126) ω h N eqh ]0 (1 − g ) − g ) g (1 − C ) (cid:126) ω c N eqc − g )
00 0 0 1 , (87)where the degree of thermalisation is g = 1 − R ≈ Γ h τ h . Observing Equation (87), it is clearthat the limit cycle vector contains both ˆH and ˆL .The work output per cycle becomes: W S = − (cid:126) ω h g − g C − C ( N eqh − C N eqc ) . (88)Extractable work is obtained in the compression range of 1 < C < N eqh N eqc . The maximumwork is obtained when C ∗ = (cid:16) N eqh N eqc (cid:17) / . At high temperature, the work per cycle simplifies36o: W S ≈ − k B T c Γ τ C − C (cid:18) T h T c − C (cid:19) . (89)The work vanishes for the frictionless compression ratio C = (cid:113) T h T c . The optimal compres-sion ratio at the high temperature limit becomes: C ∗ = (cid:16) T h T c (cid:17) / .The entropy production becomes:∆ S u = (cid:126) ω h g − g C − C (cid:18) N eqh ( 1 + C C T c − C T h ) + N eqc ( 1 + C T h − T c ) (cid:19) . (90)Even for zero power (e.g., C = 1), the entropy production is positive, reflecting a heatleak from the hot to cold bath.The power of the engine for zero cycle time τ h → τ c → P S = − (cid:126) ω h Γ2 C − C ( N eqh − C N eqc ) . (91)This means that we have reached the limit of a continuously operating engine. Thisobservation is in accordance with the universal limit of small action on each segment [35, 36].When additional dephasing is added to Equation (87), no useful power is produced and thecycle operates in a dissipator mode.The efficiency of the complete sudden engine becomes: η S = C − C − C N eqc N eqh − (1 + C ) C N eqc N eqh . (92)The extreme sudden cycle is a prototype of a quantum phenomenonan engine that requiresglobal coherence to operate. At any point in the cycle, the working medium state is non-diagonal in the energy representation.
4. Work Fluctuation in the Engine Cycle
Fluctuations are extremely important for a single realisation of a quantum harmonicengine. The work fluctuation can be calculated from the fluctuation of the energy at thefour corners of the cycle [120, 121]. The energy fluctuations for a generalised Gibbs state(Equation (20)) is related to the internal temperature (Equation (31))
V ar ( E ) = ( k B T int ) .For frictionless cycles, the variance of the work becomes: V ar ( W ) = ( k B T hint ) (1 + 1 C ) + ( k B T cint ) (1 + C ) , (93)37here T hint and T cint are the internal temperatures at the end of the hot and cold thermali-sation. For the case of complete thermalisation when the oscillator reaches the temperatureof the bath, the work variance is smallest for the Carnot compression ratio C = T h /T c .Generating coherence will increase the energy variance (cf. Equation (18)), and with it thework variance [120].
5. Quantum Fuels: Squeezed Thermal Bath
Quantum fuels represent a resource reservoir that is not in thermal equilibrium due toquantum coherence or quantum correlations. The issue is how to exploit the additionalout-of-equilibrium properties of the bath. The basic idea of quantum fuels comes from theunderstanding that coherence can reduce the von Neumann entropy of the fuel. In principle,this entropy can be exploited to increase the efficiency of the engine without violating thesecond-law [122]. An example of such a fuel is supplied by a squeezed thermal bath [123–130]. Such a bath delivers a combination of heat and coherence. As a result, work canbe extracted from a single heat bath without violating the laws of thermodynamics. Anadditional suggestion for a quantum fuel is a non-Markovian hot bath [131].The model of this engine starts from a squeezed boson hot bath where ˆH B = (cid:80) k (cid:126) Ω k ˆb † k ˆb k .This bath is coupled to the working medium by the interaction ˆH SB = (cid:80) k ig k ( ˆaˆb † k − ˆa † ˆb k ).As a result, the master equation describing thermalisation (Equation (9)) is modified to[126, 132]: L D ( ˆ ρ ) = k ↑ ( ˆs † ˆ ρ ˆs − { ˆsˆs † , ˆ ρ } ) + k ↓ ( ˆs ˆ ρ ˆs † − { ˆs † ˆs , ˆ ρ } ) , (94)where ˆs = ˆa cosh( γ ) + ˆa † sinh( γ ) = ˆSˆaˆS † . ˆS is the squeezing operator (Equation (25)) and γ the squeezing parameter.Under squeezing, the equation of motion of the hot isochore thermalisation (Equation(33)) is modified to: ddt ˆHˆLˆCˆI ( t ) = − Γ 0 0 Γ (cid:104) ˆH (cid:105) sq − Γ − ω
00 2 ω − Γ Γ (cid:104) ˆC (cid:105) sq ˆHˆLˆCˆI ( t ) , (95)where Γ = k ↓ − k ↑ is the heat conductance and k ↑ /k ↓ = e − (cid:126) ω h /k B T h obeys detailed balance.The difference from the normal thermalisation dynamics Equation (33) is in the equilibrium38alues: (cid:104) ˆH (cid:105) sq = cosh ( γ ) (cid:104) ˆH (cid:105) eq + sinh ( γ ) (cid:126) ω h k ↓ Γ , where (cid:104) ˆH (cid:105) eq is the equlibrium value of theoscillator at temperature T h . In addition, the invariant state of Equation (95) containscoherence: (cid:104) ˆC (cid:105) sq = − sinh(2 γ ) k ↑ + k ↓ Γ . This coherence is accompanied by additional energythat is transferred to the system. The squeezed bath delivers extra energy to the workingfluid as if the hot bath has a higher temperature, since (cid:104) ˆH (cid:105) sq ≥ (cid:104) ˆH (cid:105) eq . This temperaturecan be calculated from Equation (31). The thermalization to the squeezed bath generatesmutual correlation between the system and bath [132].The coherence transferred to the system (cid:104) ˆC (cid:105) ≤ (cid:104) ˆC (cid:105) sq can be cashed upon to increase thework of the cycle. This requires an adiabatic protocol which is similar to the frictionless case.In the frictionless case, the protocol of ω ( t ) was chosen to cancel the coherence generatedduring the stroke and to reach a state diagonal in energy. This protocol can be modified toexploit the initial coherence and to reach a state diagonal in energy but with lower energy,thus producing more work. The coherence thus serves as a source of quantum availability,allowing more work to be extracted from the system [130, 133, 134]. For example, using thepropagator on the adiabat Equation (38) based on µ = constant , the stroke period τ hc canbe increased from the frictionless value to add a rotation cos(ΩΘ( t )) = µ , which will nullthe coherence and reduce the final energy. Other frictionless solutions could be modified toreach the same effect. C. Closing the Cycle: The Performance of the Refrigerator
A refrigerator or heat pump employs the working medium to shuttle heat from the coldto hot reservoir. A prerequisite for cooling is that the expansion adiabat should cause thetemperature of the working medium to be lower than the cold bath. In addition, at the endof the compression adiabat , the temperature should be hotter than the hot bath (cf. Figure4). To generate a refrigerator, we use the order of stroke propagators in Equation (62). Theheat extracted from the cold bath becomes: Q c = E B − E C = (cid:126) ω c ( N B − N C ) . (96)The interplay between efficiency and cooling power is the main theme in the performanceanalysis. The efficiency of a refrigerator is defined by the coefficient of performance (COP): COP = Q c W = ω c ω h − ω c ≤ T c T h − T c . (97)39 h ω c DCB A T h T c ω S E FIG. 4: Typical frictionless refrigerator cycle S E vs. ω . Expansion adiabat D → C . Cold isochore C → B . Compression adiabat B → A . Hot isochore A → D . The hot and cold isotherms areindicated. The cycle parameters are ω c = 0 . , T c = 1 . , ω h = 3 , T h = 3 , τ c = τ h = 2 . , | µ | =0 . , Γ = 1.
The cooling power R c is defined as: R c = Q c τ cyc . (98)Optimising the performance of the refrigerator can be carried out by a similar analysis tothe one employed for the heat engine. Insight into the ideal performance can be gained byexamining the expansion adiabat . The initial excitation should be minimized, requiring thehot bath to cool the working medium to its ground state. This is possible if (cid:126) ω h (cid:29) k B T h .Next, the expansion should be as adiabatic as possible so that at the end the workingmedium is still as close as possible to its ground state E C ≈ (cid:126) ω c . The frictionless solutionsfound in Section IV B can be employed to achieve this task in minimum time.
1. Frictionless Refrigerator
The adiabatic refrigerator is obtained in the limit of infinite time µ →
0, leading toconstant population N and S E . Then, E C = ω c ω h E D . At this limit, since τ → ∞ , the coolingrate vanishes R c = 0. The Carnot efficiency can be obtained when C = T h T c .Frictionless solutions require that the state ˆ ρ is diagonal in energy in the beginning andat the end of the adiabat . The analytic propagator on the expansion adiabat (Equation4038)) describes the expansion adiabat: D → C : E C = 1 C (cid:0) − µ c (cid:1) · E D , (99)where c = cos(Ω θ c ) and θ c = − µ log ( C ).Frictionless points are obtained whenever N C = N D . The condition is c = 1 in Equation(99). Then, µ <
2, leading to the critical frictional points (Equation (45)). These solutionshave optimal efficiency Equation (97) with finite power. The optimal time allocated to the adiabat becomes (cf. Equation (46)) τ ∗ hc = (1 − C ) / ( µ ∗ ω h ).This frictionless solution with a minimum time allocation τ ∗ hc scales as the inverse fre-quency ω − c , which outperforms the linear ramp solution ω ( t ) = ω i + gt .Other faster frictionless solutions can be obtained using the protocols of Section IV B,such as the superadiabatic protocol or by applying optimal control theory [86]. Both caseslead to the scaling of the adiabatic expansion time as τ hc ∝ √ ω c ω h .Once the time allocation on the adiabats is set, the time allocation on the isochores isoptimised for the thermalisation using the method of [24], and the optimal cooling powerbecomes: R ∗ c = e z (1 + e z ) Γ (cid:126) ω c ( N eqc − N eqh ) , (100)where z = Γ h τ h = Γ c τ c . The optimal z is determined by the solution of the equation2 z + Γ( τ hc + τ ch ) = 2 sinh( z ).
2. The Sudden Refrigerator
Short adiabats generally lead to the excitation of the oscillator and result in friction (cf.Section IV B). Nevertheless, a refrigerator can still operate at the limit of vanishing cycletime. In a similar fashion to the sudden engine, coherence can be exploited and leads to afinite cooling power when τ cyc →
0. The cooling power for the sudden limit becomes: R c = − (cid:126) ω c Γ (cid:18) N eqc − C (1 + C ) N eqh (cid:19) . (101)Note that the cooling rate in this sudden-limit becomes zero at a sufficiently low coldbath temperature so that N eqc = (1 + C ) N eqh . This formula loses its meaning and shouldnot be used below this temperature. 41 . The Quest to Reach Absolute Zero The quantum harmonic refrigerator can serve as a primary model to explore coolingat very low temperatures. A necessary condition is that the internal temperature of theoscillator (Equation (31)) should be lower than the cold bath. T int < T c when T c →
0. Thiscondition imposes very high compression ratios
C (cid:29) T h T c so that at point D (cf. Figure 4),at the end of the hot thermalization, the oscillator is very close to the ground state.An important feature of the model is that the cooling power vanishes as T c approacheszero. Qualitatively, R c → D → C for highcompression ratios requires a significant amount of time. Another issue is the rate of coldthermalisation C → B and its scaling with T c when the oscillator extracts heat from thecold bath. These issues can be made quantitative by exploring the scaling exponent α ofthe optimal cooling power with the cold bath temperature T c : R c = Q c τ cyc ∝ T αc . (102)The vanishing of the cooling power R c as T c → • The entropy of any pure substance in thermodynamic equilibrium approaches zero asthe temperature approaches zero.The second formulation is dynamical, known as the unattainability principle [135, 139–142]: • It is impossible by any procedure—no matter how idealised—to reduce any assemblyto absolute zero temperature in a finite number of operations [138].The second law of thermodynamics already imposes a restriction on α [25, 135, 143]. Insteady-state, the entropy production rate is positive. Since the process is cyclic, it takesplace only in the baths: σ = ˙ S c + ˙ S h ≥
0. When the cold bath approaches the absolute zerotemperature, it is necessary to eliminate the entropy production divergence at the cold sidebecause ˙ S c = R c T c . Therefore, the entropy production at the cold bath when T c → S c ∼ − T αc , α ≥ . (103)For the case when α = 0, the fulfillment of the second-law depends on the entropyproduction of the other baths, which should compensate for the negative entropy productionof the cold bath. The first formulation of the third-law slightly modifies this restriction.Instead of α ≥
0, the third-law imposes α >
0, guaranteeing that the entropy productionat the cold bath is zero at absolute zero: ˙ S c = 0. This requirement leads to the scalingcondition of the heat current R c ∼ T α +1 c , α > no refrigerator can cool a system to absolute zero temperature at finite time . Thisformulation is more restrictive, imposing limitations on the system bath interaction and thecold bath properties when T c → ζ : dT c ( t ) dt ∼ − T ζc , T c → . (104)In order to evaluate Equation (104), the heat current can be related to the temperaturechange: J c ( T c ( t )) = − c V ( T c ( t )) dT c ( t ) dt . (105)This formulation takes into account the heat capacity c V ( T c ) of the cold bath. c V ( T c ) isdetermined by the behaviour of the degrees of freedom of the cold bath at low temperature.Therefore, the scaling exponents can be related ζ = 1 + α − η , where c V ∼ T ηc when T c → adibatic expansion will lead to insight on the coolingrate R c and the exponent α .The frictionless solutions lead to an upper bound on the optimal cooling rate (Equation(98)). For the limit T c →
0, Γ τ hc is large; therefore, z is large, leading to: R ∗ c ≈ Γ( τ hc + τ ch )(1 + Γ τ hc ) Γ (cid:126) ω c ( N eqc − N eqh ) . (106)At high compression ratio, N eqh →
0, and in addition ω c (cid:28) Γ one obtains: R ∗ c ≈ τ hc Γ (cid:126) ω c N eqc . (107)43ptimizing R c with respect to ω c leads to a linear relation between ω c and T c , (cid:126) ω c = k B T c ;therefore: R c ≤ Aω ν N eqc , (108)where A is a constant and the exponent ν is either ν = 2 for the µ = const solution or ν = for the optimal control solution. Therefore: R ∗ c ≈ (cid:126) ω c N eqc (109)for the µ = const frictionless solution, and R ∗ c ≈ (cid:126) ω c √ ω h N eqc (110)for the optimal control frictionless solution. Due to the linear relation between ω c and T c , Equations (109) and (110) determine the exponent α , where α = 1 for the frictionlessscheduling with constant µ , and α = for the optimal control frictionless scheduling. In allcases, the dynamical version of Nernst’s heat law is observed based only on the adiabaticexpansion.If one is forced to spend less time on the adiabat than the minimal time required fora shortcut solution, the oscillator cannot reach arbitrarily low energies or temperatures atthe end of the expansion [92, 93]. At the limit, one approaches the sudden adiabat limit.In this case, the refrigerator cannot cool below a minimal ( T ∗ c >
0) temperature, and therefrigerator thus satisfies the unattainability principle trivially.The unattainability principle is related to the scaling of the heat transport Γ c with T c .This issue has been explored in [25, 135], and is related to the scaling of the heat conduc-tivity with temperature. The arguments of [135] are applicable to the quantum harmonicrefrigerator. VI. OVERVIEW
Learning from example has been one of the major sources of insight in the study ofthermodynamics. A good example can bridge the gap between concrete and abstract theory.The harmonic oscillator quantum Otto cycle serves as a primary example of a quantumthermal device inspiring experimental realisation [2, 144]. On the one hand, the model is44ery close to actual physical realisations in many scenarios [19]. On the other hand, manyfeatures of the model can be obtained as closed-form analytic solutions.Many of the features obtained for the quantum harmonic Otto engine have been ob-served in stochastic thermodynamics [121, 144–147]. The analytic properties of the har-monic oscillator—in particular, the Gaussian form of the state—have motivated studies ofclassical stochastic models of harmonic heat engines [144, 148, 149]. When comparing thetwo theories, the results seem identical in many cases. Observing the Heisenberg equa-tions of motion for the thermodynamical variables, (cid:126) does not appear. Planck’s constantin the commutators is cancelled by the inverse Planck constant in the equation of motion.This raises the issue of what is quantum in the quantum harmonic oscillator, or a relatedissue— what is quantum in quantum thermodynamics [31]?In this review, we emphasized the power of the generalized Gibbs state in allowing aconcise description of an out-of-equilibrium situation of non-commuting operators. Usingproperties of Lie algebra of operators, we could obtain a dynamical description of the statebased on only three variables: ˆH , ˆL , and ˆC . In the spirit of open quantum systems, wecould describe the cycle propagator as a catenation of stroke propagators. All these propa-gators were cast in the framework of the operator algebra, showing the power of Heisenbergrepresentation. The quantum variables were chosen to have direct thermodynamical rele-vance as energy and coherence. In this review, we emphasized the connections between thealgebraic approach and other popular methods that have been employed to obtain insighton the harmonic engine.This formalism allows the cycles to be classified according to the role of coherence. Ifthe coherence vanishes at the points where the strokes meet, frictionless cycles are obtained.Such cycles require special scheduling of ω ( t ) so that the coherence generated at the begin-ning of the stroke can be cashed upon at the end. We reviewed the different approachesto obtain such scheduling and the minimum time that such moves can be generated. Thisperiod is related to quantum speed limits [150–154], which are in turn related to the energyresources available to the system. We chose the geometric mean τ a ∝ / √ ω h ω c to representthe minimum time allocation. Faster scheduling requires unreasonable constraints on thestored energy in the oscillator during the stroke. We also assume that this extra energyrequired to achieve the fast control is not dissipated and can be accounted for as a catalyst.For these frictionless solutions on the adiabats , the optimal time allocation for thermalisa-45ion is finite, leading to incomplete thermalisation. This allows the minimum cycle time forfrictionless cycles to be estimated. Avoiding friction completely is an ideal that practicallycannot be obtained. Using a simple noise model, we show that some friction will always bepresent.The model demonstrates the fundamental tradeoff between efficiency and power. Thefrictionless solutions are a demonstration that quantum coherence which is related to frictioncan be cashed upon, using interference to cancel this friction. As a result, the maximumefficiency of the engine can be obtained in finite cycle time. Nevertheless, the Otto efficiencyis smaller than the reversible Carnot efficiency η o = 1 − ω c ω h ≤ η c = 1 − T c T h , and operating atthe Carnot efficiency will lead to zero power. The entropy production can be associated withthe heat transport, and for this case the entropy production is linearly related to the power.Maximum power also implies maximum entropy production. This finding is consistent withthe study of Shiraishi et al. [155]. Any finite power cycle requires out-of-equilibrium setupsthat lead to dissipation. In the sudden limit, there is no reversible choice. Even at zeropower the entropy production is positive. This could be the cost of maintaining coherence.Beyond a minimum time allocation on the adiabats τ a , friction cannot be avoided. Thetransition point is the exceptional point of the non-hermitian degeneracy on the adiabaticpropagator [103]. These short time cycles are in the realm of the sudden cycles. Thesudden cycles are an example of an engine or refrigerator with no classical analogue. Powerproduction requires coherence. A sudden model without coherence operates as a dissipatorgenerating entropy on both the hot and cold baths. The sudden cycle is composed of non-commuting propagators with small action. Such cycles are universal and have a commoncontinuous limit [35, 36]. In the continuous limit, friction and heat leaks cannot be avoided[39, 156].An obvious direction to look for quantum effects is to go to low temperatures wherethe unit of energy changes from k B T to (cid:126) ω . The adiabatic expansion is the bottleneck forcooling to extremely low temperatures. The zero point energy plays an important role. Wecan approach the ground state on the hot side by increasing the frequency, leading to theminimal initial energy E A = (cid:126) ω h . This is a large amount of energy compared to the coldside, which has to be eliminated adiabatically or by using frictionless protocols. Any smallerror in these protocols will null the cooling.The Otto quantum refrigerator is a good example for gaining insight into the lim-46ts of cooling when operating at extremely low temperatures. Such refrigerators arean integral part of any quantum technology. In the reciprocating Otto cycle, thecooling power is restricted either by the adiabatic expansion or by vanishing of theheat transport when T c → adiabatic expansion time is an intrin-sic property of the working medium. For optimal frictionless solutions, it scales as τ hc = O ( T c ), which gives a maximum rate of entropy production σ = O ( T c ), thus van-ishing when T c →
0. This is a demonstration of a dynamical version of the Nernst heat law[25, 68, 135].The quantum harmonic Otto cycle has been a template for many models of quantum heatdevices due to its analytic properties—for example, Otto cycles with interacting particles[115, 116] or operating with many modes [157]. The protocols developed for the harmoniccase are generalised to eliminate friction in many-body dynamics.The quantum harmonic Otto cycle has been a source of inspiration for theory and ex-periment. The model incorporates generic features of irreversible operation which includesfriction and heat transport. The system can bridge the conceptual gap between a singlemicroscopic device to a macroscopic heat engine.
Acknowledgments
We thank Amikam Levy, Tova Feldmann, Raam Uzdin and Erik Torrontegui for sharingtheir ideas and insights. We also thank Peter Salamon, Gonzales Muga, Robert Alicki forfruitful discussions. We acknowledge funding by the Israel Science Foundation, and COSTAction MP1209
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