Quantum heat engines and refrigerators: Continuous devices
aa r X i v : . [ qu a n t - ph ] O c t Quantum heat engines and refrigerators: Continuous devices
Ronnie Kosloff, Amikam Levy
Institute of Chemistry, the Hebrew University, Jerusalem 91904, Israel
Quantum thermodynamics supplies a consistent description of quantum heat en-gines and refrigerators up to the level of a single few level system coupled to theenvironment. Once the environment is split into three; a hot, cold and work reser-voirs a heat engine can operate. The device converts the positive gain into power;where the gain is obtained from population inversion between the components ofthe device. Reversing the operation transforms the device into a quantum refrigera-tor. The quantum tricycle, a device connected by three external leads to three heatreservoirs is used as a template for engines and refrigerators. The equation of motionfor the heat currents and power can be derived from first principle. Only a globaldescription of the coupling of the device to the reservoirs is consistent with the firstand second laws of thermodynamics. Optimisation of the devices leads to a balancedset of parameters where the couplings to the three reservoirs are of the same orderand the external driving field is in resonance. When analysing refrigerators specialattention is devoted to a dynamical version of the third law of thermodynamics.Bounds on the rate of cooling when T c → T c → T c → γ c and heat capacity c V of the cold bath. uantum heat engines and refrigerators Contents
I. Introduction II. The continuous engine P Representation 21E. The four-level engine and two-level engine 23F. Power storage 25
III. Continuous refrigerators T c → IV. Summary References uantum heat engines and refrigerators I. INTRODUCTION
Our cars, refrigerators, air-conditioners, lasers and power plants are all examples of heatengines. The trend toward miniaturisation has not skipped the realm of heat engines leadingto devices on the nano or even on the atomic scale. Typically, in the practical world all suchdevices operate far from the maximum efficiency conditions set by Carnot [1]. Real heatengines are optimised for powers scarifying efficiency. This trade-off between efficiency andpower is the focus of ”finite time thermodynamics”. The field was initiated by the seminalpaper of Curzon and Ahlboron [2]. From everyday experience the irreversible phenomenathat limits the optimal performance of engines can be identified as losses due to friction,heat leaks, and heat transport [3]. Is there a unifying fundamental explanation for theselosses? Is it possible to trace the origin of these phenomena to quantum mechanics? Toaddress these issues the tradition of thermodynamics is followed by the study of hypotheticalquantum heat engines and refrigerators. Once understood, these models serve as a templatefor real devices.
Gedanken heat engines are an integral part of thermodynamical theory. Carnot in 1824set the stage by analysing an ideal engine [1]. Carnot’s analysis preceded the systematicformulation that led to the first and second laws of thermodynamics [4]. Amazingly, ther-modynamics was able to keep its independent status despite the development of paralleltheories dealing with the same subject matter. Quantum mechanics overlaps thermody-namics in that it describes the state of matter. However, in addition, quantum mechanicsinclude a comprehensive description of dynamics. This suggests that quantum mechanicscan originate a concrete interpretation of the word dynamics in thermodynamics leading toa fundamental basis for finite time thermodynamics.
Quantum thermodynamics is the study of thermodynamical processes within the contextof quantum dynamics. Historically, consistency with thermodynamics led to Planck’s law,the basics of quantum theory. Following the ideas of Planck on black body radiation, Einsteinin (1905) quantized the electromagnetic field [5]. This paper of Einstein is the birth ofquantum mechanics together with quantum thermodynamics.
Quantum thermodynamics is devoted to unraveling the intimate connection between thelaws of thermodynamics and their quantum origin [6–35]. The following questions come tomind: uantum heat engines and refrigerators • How do the laws of thermodynamics emerge from quantum mechanics? • What are the requirements of a theory to describe quantum mechanics and thermo-dynamics on a common ground? • What are the fundamental reasons for tradeoff between power and efficiency? • Do quantum devices operating far from equilibrium follow thermodynamical rules? • Can quantum phenomena affect the performance of heat engines and refrigerators?These issues are addressed by analyzing quantum models of heat engines and refrigerators.Extreme care has been taken to choose a model which can be analyzed from first principles.Two types of models are considered, continuos operating models resembling turbines anddiscrete four stroke reciprocating engines. The present review will focus on continuousdevices.
II. THE CONTINUOUS ENGINE
An engine is a device that converts one form of energy to another: heat to work. In thisconversion, part of heat from the hot bath is ejected to the cold bath limiting the efficiencyof power generation. This is the essence of the second-law of thermodynamics.A heat engine employs the natural current from a hot to a cold bath to generate power.The Carnot engine is a model of such a device. Carnot was able to incorporate the practicalknowledge on steam engines of his era into a universal scientific statement on maximumefficiency [1]. Out of this insight the laws of thermodynamics were later formulated. Thistheme of learning from an example is typical in thermodynamics and will be employed toobtain insight from analysis of quantum devices.A continuous engine operates in an autonomous fashion attaining steady state mode ofoperation. The analysis therefore requires an evaluation of steady sate energy currents.The operating part of the device is connected simultaneously to the hot, cold and poweroutput leads. The primary macroscopic example is a steam or gas turbine. The primaryquantum heat engine is the laser. These devices share a universal aspect exemplified by theequivalence of the 3-level laser with the Carnot engine. uantum heat engines and refrigerators
A. The quantum 3 level system as a heat engine
A contemporary example of a Carnot engine is the 3-level amplifier. The principle ofoperation is to convert population inversion into output power in the form of light. Heatgradients are employed to achieve this goal. Fig 1 shows its schematic construction. A hotreservoir characterised by temperature T h induces transitions between the ground state ǫ and the excited state ǫ . When equilibrium is reached, the population ratio between theselevels becomes p p = e − ¯ hωhkBTh where ω h ≡ ω = ( ǫ − ǫ ) / ¯ h is the Bohr frequency and k B is the Boltzmann constant. Thecold reservoir at temperature T c couples level ǫ and level ǫ meaning that: p p = e − ¯ hωckBTc , where ω c ≡ ω = ( ǫ − ǫ ) / ¯ h . The amplifier operates by coupling the energy levels ǫ and ǫ to the radiation field generating an output frequency which on resonance is ν = ( ǫ − ǫ ) / ¯ h .The necessary condition for amplification is positive gain or population inversion defined by: G = p − p ≥ . (1)From this condition, by dividing by p we obtain e − ¯ hωhkBTh − e − ¯ hωckBTc ≥
0, which leads to: ω c ω h ≡ ω ω ≥ T c T h , (2)The efficiency of the amplifier is defined by the ratio of the output energy ¯ hν to the energyextracted from the hot reservoir ¯ hω : η o = νω = 1 − ω c ω h . (3)Eq. (3) is termed the quantum Otto efficiency [23]. Inserting the positive gain conditionEq. (1) and Eq. (2) the efficiency is limited by Carnot [1]: η o ≤ η c ≡ − T c T h . (4) uantum heat engines and refrigerators ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( T h (((((((((((((((((((((((((((((((((((( T c e e e P nw w k › k fl k fl k › FIG. 1: The quantum 3-level amplifier as a Carnot heat engine. The system is coupled to a hotbath with temperature T h and to a cold bath with temperature T c . The output P is a radiationfield with frequency ν . (In the figure ¯ h = 1). Power is generated provided there is populationinversion between level ǫ and ǫ : p > p . The hot bath equilibrates levels ǫ and ǫ via therates k ↑ and k ↓ such that k ↑ /k ↓ = exp(¯ hω /k B T h ). The efficiency η = ν/ω ≤ − T c /T h .Reversing the direction of operation using power to drive population from level ǫ to ǫ generatesa heat pump then p < p . This result connecting the efficiency of a quantum amplifier to the Carnot efficiency wasfirst obtained by Scuville et al. [6, 7].The above description of the 3-level amplifier is based on a static quasi-equilibrium view-point. Real engines which produce power operate far from equilibrium conditions. Typically,their performance is restricted by friction, heat transport and heat leaks. A dynamical view-point is therefore the next required step.
B. The quantum tricycle
A quantum description enables to incorporate dynamics into thermodynamics. The tri-cycle model is the template for almost all continuous engines Cf. Fig. 2. This model willbe employed to incorporate the quantum dynamics of the devices. Surprisingly very sim-ple models exhibit the same features of engines generating finite power. Their efficiency atoperating conditions is lower than the Carnot efficiency. In addition, heat leaks restrict thepreformance meaning that reversible operation is unattainable. uantum heat engines and refrigerators ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( T c T h T w H s P J c J h J c +J h +P =0 J c J h J w T c T h T w − − − ‡ J w FIG. 2: The quantum tricycle: A quantum device coupled simultaneously to a hot, cold and powerreservoir. Reversing the heat currents constructs a quantum refrigerator.
The tricycle engine has a generic structure displayed in Fig. 2. • The basic model consists of three thermal baths: a hot bath with temperature T h , acold bath with temperature T c and a work bath with temperature T w . • Each bath is connected to the engine via a frequency filter which we will model bythree oscillators: ˆH F = ¯ hω h ˆa † ˆa + ¯ hω c ˆb † ˆb + ¯ hω w ˆc † ˆc , (5)where ω h , ω c and ω w are the filter frequencies on resonance ω w = ω h − ω c . • The device operates as an engine by removing an excitation from the hot bath and gen-erating excitations on the cold and work reservoirs. In second quantization formalismthe Hamiltonian describing such an interaction becomes: ˆH I = ¯ hǫ (cid:18) ˆaˆb † ˆc † + ˆa † ˆbˆc (cid:19) , (6)where ǫ is the coupling strength. • The device operates as a refrigerator by removing an excitation from the cold bath aswell as from the work bath and generating an excitation in the hot bath. The term ˆa † ˆbˆc in the Hamiltonian of Eq. (6) describes this action. (Cf. Section III).Some comments are appropriate: uantum heat engines and refrigerators
81. Different types of heat baths can be employed which can include bosonic baths com-posed of phonons or photons, or fermonic baths composed of electrons. The frequencyfilters select from the continuous spectrum of the bath the working component tobe employed in the tricycle. These frequency filters can be constructed also fromtwo-level-systems (TLS) or formulated as qubits [36–38].2. The interaction term Eq. (6), is strictly non-linear, incorporating three heat currentssimultaneously. This crucial fact has important consequences. A linear device cannotoperate as a heat engine or refrigerator [39]. A linear device is constructed from anetwork of harmonic oscillators with linear connections of the type ¯ hµ ij (cid:16) ˆa i ˆa † j + ˆa † i ˆa j (cid:17) with some connections to harmonic heat baths. In such a device the hottest bathalways cools down and the coldest bath always heats up. Thus, this construction cantransport heat but not generate power since power is equivalent to transporting heatto an infinitely hot reservoir. Another flaw in a linear model is that the differentbath modes do not equilibrate with each other. A generic bath should equilibrate anysystem Hamiltonian irrespective of its frequency.3. Many nonlinear interaction Hamiltonians of the type ˆH I = ˆA ⊗ ˆB ⊗ ˆC can lead to aworking heat engine. These Hamiltonians can be reduced to the form of Eq. (6) whichcaptures the essence of such interactions.The first-law of thermodynamics represents the energy balance of heat currents originat-ing from the three baths and collimating on the system: dE s dt = J h + J c + J w . (7)At steady state no heat is accumulated in the tricycle, thus dE s dt = 0. In addition, in steadystate the entropy is only generated in the baths, leading to the second-law of thermodynam-ics: ddt ∆ S u = − J h T h − J c T c − J w T w ≥ . (8)This version of the second-law is a generalisation of the statement of Clausius; heat doesnot flow spontaneously from cold to hot bodies [40]. When the temperature T w → ∞ , noentropy is generated in the power bath. An energy current with no accompanying entropyproduction is equivalent to generating pure power: P = J w , where P is the output power. uantum heat engines and refrigerators J j in the tricycle model requires dynamical equationsof motion. The major assumption is that the total systems is closed and its dynamics isgenerated by the global Hamiltonians. ˆH = ˆH + ˆH SB . (9)This Hamiltonian ˆH includes the bare system and the heat baths: ˆH = ˆH s + ˆH H + ˆH C + ˆH W , (10)where the system Hamiltonian ˆH s = ˆH I + ˆH h + ˆH c + ˆH w consists of three energy filteringcomponents and an interaction part with an external field. The reservoir Hamiltonians arethe hot ˆH H , cold ˆH C and work ˆH W . The system-bath interaction Hamiltonian: ˆH SB = ˆH sH + ˆH sC + ˆH sW .A thermodynamical idealisation assumes that the tricycle system and the baths are un-correlated, meaning that the total state of the combined system becomes a tensor productat all times [32]: ˆ ρ = ˆ ρ s ⊗ ˆ ρ H ⊗ ˆ ρ C ⊗ ˆ ρ W . (11)Under these conditions the dynamical equations of motion for the tricycle become: ddt ˆ ρ s = L ˆ ρ s , (12)where L is the Liouville superoperaor described in terms of the system Hilbert space, wherethe reservoirs are described implicitly. Within the formalism of quantum open system, L cantake the form of the Gorini-Kossakowski-Sudarshan-Lindblad (GKS-L) Markovian generator[41, 42].Thermodynamics is notorious in employing a very small number of variables. In equi-librium conditions the knowledge of the Hamiltonian is sufficient. When deviating fromequilibrium additional observables are added. This suggest presenting the dynamical gener-ator in Heisenberg form for arbitrary system operators ˆO s : ddt ˆO s = L ∗ ˆO s = i ¯ h [ ˆH s , ˆO s ] + X k ˆV k ˆO s ˆV † k − { ˆV k ˆV † k , ˆO s } . (13)The operators ˆV k are system operators still to be determined. The task of evaluating themodified system Hamiltonian ˆH s and the operators ˆV k is made extremely difficult due to thenonlinear interaction in Eq. (6). Any progress from this point requires a specific descriptionof the heat baths and approximations to deal with the nonlinear terms. uantum heat engines and refrigerators C. The quantum amplifier
The quantum amplifier is the most elementary quantum continuous heat engine convert-ing heat to work. The purpose is to generate power from a temperature difference betweenthe hot and cold reservoirs. The output power is described by a time dependent externalfield.The general device Hamiltonian is therefore time dependent: ˆH s ( t ) = ˆH + ˆV ( t ) . (14)The Markovian master equation in Heisenberg form for the system operator ˆO s when thesystem coupled to the hot and cold bath: ddt ˆO s = i ¯ h [ ˆH ( t ) , ˆO s ( t )] + L ∗ h ( ˆO s ( t )) + L ∗ c ( ˆO s ( t )) . (15)The change in energy of the device is obtained by replacing ˆO s by ˆH s : dE s dt = hL ∗ h ( ˆH s ) i + hL ∗ c ( ˆH s ) i + h ∂ ˆH s ∂t i . (16)Equation (16) can be interpreted as the time derivative of the first law of thermodynamics[8, 32, 43, 44] based on the Markovian GKS-L generator. Power is identified as: P = h ∂H s ∂t i , (17)and the heat current as: J h = hL ∗ h ( ˆH s ) i , J c = hL ∗ c ( ˆH s ) i . (18)The partition between the Hamiltonian and the dissipative part in the GKS-L generator isnot unique [32]. A unique derivation of the Master equation based on the weak couplinglimit leads to dynamics which is consistent with the first and second laws of thermodynamics[45].The template of the tricycle model is employed to describe the dynamics of the amplifier.The interaction Hamiltonian is modified to become: ˆH I ( t ) = ¯ hǫ (cid:18) ˆa ˆb † e + iνt + ˆa † ˆb e − iνt (cid:19) , (19) uantum heat engines and refrigerators ν ≡ ω w is the frequency of the time dependent driving field and ǫ is the couplingamplitude. The modification of Eq. (6) eliminates the nonlinearity, it amounts to replacingthe operator ˆc and ˆc † by a c-number. The amplifier output power becomes: P = ¯ hǫν (cid:18) h ˆaˆb † i e + iνt − h ˆa † ˆb i e − iνt (cid:19) . (20)After the nonlinearity has been eliminated the quantum Master equation for the amplifiercan be derived from first principles based on the weak system bath coupling expansion. Thisapproximation is a thermodynamic idealisation equivalent to an isothermal partition betweenthe system and baths [32].The interaction with the baths is given by ˆH sb = λ a ( ˆa + ˆa † ) ⊗ ˆR h + λ b ( ˆb + ˆb † ) ⊗ ˆR c . (21)where ˆR are reservoir operators and λ is the small system-bath coupling parameter. Acrucial step has to be performed before this procedure is applied. The system Hamiltonianhas to be rediagonalized with the interaction before the system is coupled to the baths. Thisdiagonalization modifies the frequencies of the system resulting in a splitting of the filterfrequencies. The prediagonalization is crucial for the master equations to be consistent withthe second-law of thermodynamics [9, 38, 46].The main ingredients of the derivation:I) Transformation to interaction picture. The reservoir coupling operators ˆR transformaccording to the free baths Hamiltonian, and the system operators are subject to the unitarypropagator which under resonance conditions becomes: ˆU s ( t,
0) = T exp n − i ¯ h Z t ˆH ( s ) ds o = e − i ¯ h ˆH t e − i ¯ h ˆV t , (22)where ˆH = ¯ hω h ˆa † ˆa + ¯ hω c ˆb † ˆb , ˆV = ¯ hǫ ( ˆa † ˆb + ˆaˆb † ) . (23)II) Fourier decomposition of the interaction part. The operators in the interaction picturetake the form ˜a ( t ) = ˆU s ( t , ) † ˆa ˆU s ( t , ) = e i ¯ h ˆV t h e i ¯ h ˆH t ˆa e − i ¯ h ˆH t i e − i ¯ h ˆV t = cos( ǫt ) e − iω h t ˆa − i sin( ǫt ) e − iω h t ˆb , (24) uantum heat engines and refrigerators ˜b ( t ). The Fourier decomposition becomes: ˜a ( t ) = √ ( e − i ( ω + h ) t ˜d + + e − i ( ω − h ) t ˜d − ) (25)and ˜b ( t ) = √ ( e − i ( ω + c ) t ˜d + − e − i ( ω − c ) t ˜d − ) (26)where ˜d + = ˜a + ˜b √ , ˜d − = ˜a − ˜b √ and ω ± h ( c ) = ( ω h ( c ) ± ǫ ). Similarly ˜a † ( t ) , ˜b † ( t ) are evaluated.III) Performing the system-bath weak coupling approximation. This approximation in-volves averaging over fast oscillating terms with typical frequencies ∼ ω c , ω h , ǫ . Thisapproximation is restricted to conditions that the relaxation time of the open system ismuch longer than the intrinsic time scale ω − c , ω − h and ǫ − . Thus, terms oscillating rapidlyover the relaxation time average out. Such equations were derived in Ref.[36]. If the cou-pling to the external field is weak, such that ω h , ω c ≫ ǫ and the relaxation time of the opensystem is comparable with ǫ − , the derivation is modified accordingly. In this case there isno justification to neglect terms oscillating with frequency ∼ ǫ . Keeping such terms, thetotal time-dependent (interaction picture) generator has the form: L ( t ) = L (+) h + L ( − ) h + L (+) c + L ( − ) c , (27)where L (+) h ˆ ρ = γ (+) h (cid:16) [ ˜d + , ˆ ρ ˜d † + ] + [ ˜d − , ˆ ρ ˜d † + ] e i ǫt + e − β h ω + h (cid:16) [ ˜d † + , ˆ ρ ˜d + ] + [ ˜d †− , ˆ ρ ˜d + ] e − i ǫt (cid:17) + h.c (cid:17) L (+) c ˆ ρ = γ (+) c (cid:16) [ ˜d + , ˆ ρ ˜d † + ] − [ ˜d − , ˆ ρ ˜d † + ] e i ǫt + e − β c ω + c (cid:16) [ ˜d † + , ˆ ρ ˜d + ] − [ ˜d †− , ˆ ρ ˜d + ] e − i ǫt (cid:17) + h.c (cid:17) L ( − ) h ˆ ρ = γ ( − ) h (cid:16) [ ˜d − , ˆ ρ ˜d †− ] + [ ˜d + , ˆ ρ ˜d †− ] e − i ǫt + e − β h ω − h (cid:16) [ ˜d †− , ˆ ρ ˜d − ] + [ ˜d † + , ˆ ρ ˜d − ] e i ǫt (cid:17) + h.c (cid:17) L ( − ) c ˆ ρ = γ ( − ) c (cid:16) [ ˜d − , ˆ ρ ˜d †− ] − [ ˜d + , ˆ ρ ˜d †− ] e − i ǫt + e − β c ω − c (cid:16) [ ˜d †− , ˆ ρ ˜d − ] − [ ˜d † + , ˆ ρ ˜d − ] e i ǫt (cid:17) + h.c (cid:17) (28)and the inverse temperature is β = ¯ h/k B T . The relaxation rates γ ( ± ) h ( c ) = γ h ( c ) ( ω h ( c ) ± ǫ ), havethe structure: γ ( ω ) = λ Z ∞−∞ e iωt T r ( ˆ ρ R e i ¯ h ˆH R t ˆR e − i ¯ h ˆH R t ˆR ) dt (29)and can be calculated explicitly for different types of heat baths.Note that if we neglect the time dependent terms in Eq. (28) the master equation derivedin Ref. [36] is recovered. The generators in Eq.(28) are not in GKS-L completely positive uantum heat engines and refrigerators L D ˆ ρ = P i,j C i,j (cid:18) ˆF i ˆ ρ ˆF † j − { ˆF † j ˆF i , ˆ ρ } (cid:19) . Next, weinsure that these terms will not contribute twice by rescaling the kinetic coefficients. Inorder to verify that the map is completely positive, it is sufficient that the matrix ( C i,j ) ispositive definite. The modified GKS-L master equations now become: L ( ± ) h ˆ ρ = γ ( ± ) h (cid:16) ˜d ± ˆ ρ ˜d †± − { ˜d †± ˜d ± , ˆ ρ } + e − β h ω ± h (cid:16) ˜d †± ˆ ρ ˜d ± − { ˜d ± ˜d †± , ˆ ρ } (cid:17)(cid:17) + γ ( ± ) h (cid:16) ˜d − ˆ ρ ˜d † + − { ˜d † + ˜d − , ˆ ρ } + e − β h ω ± h (cid:16) ˜d † + ˆ ρ ˜d − − { ˜d − ˜d † + , ˆ ρ } (cid:17)(cid:17) e i ǫt + γ ( ± ) h (cid:16) ˜d + ˆ ρ ˜d †− − { ˜d †− ˜d + , ˆ ρ } + e − β h ω ± h (cid:16) ˜d †− ˆ ρ ˜d + − { ˜d + ˜d †− , ˆ ρ } (cid:17)(cid:17) e − i ǫt L ( ± ) c ˆ ρ = γ ( ± ) c (cid:16) ˜d ± ˆ ρ ˜d †± − { ˜d †± ˜d ± , ˆ ρ } + e − β c ω ± c (cid:16) ˜d †± ˆ ρ ˜d ± − { ˜d ± ˜d †± , ˆ ρ } (cid:17)(cid:17) − γ ( ± ) c (cid:16) ˜d − ˆ ρ ˜d † + − { ˜d † + ˜d − , ˆ ρ } + e − β c ω ± c (cid:16) ˜d † + ˆ ρ ˜d − − { ˜d − ˜d † + , ˆ ρ } (cid:17)(cid:17) e i ǫt − γ ( ± ) c (cid:16) ˜d + ˆ ρ ˜d †− − { ˜d †− ˜d + , ˆ ρ } + e − β c ω ± c (cid:16) ˜d †− ˆ ρ ˜d + − { ˜d + ˜d †− , ˆ ρ } (cid:17)(cid:17) e − i ǫt (30)Note that the rotating term e − i ǫt can be absorbed in ˜d by a second rotation of the frame.The derivation of the Master equation, Eq. (30), for a driven system is a delicate task. Thepre-diagonalization step ensures consistency with the second-law [9, 46, 47].
1. Solving the equations of motion for the engine
In thermodynamic tradition, the engine is well described by a small set of observables.They in turn are represented by operators defining the heat currents in the engine. Toexploit this property the Hamiltonian is transformed to the interaction frame: ˆH I ( t ) = ˆU † s ( t, ˆH ( t ) ˆU s ( t,
0) = ¯ h ω h + ω c ˆW + ¯ h ω h − ω c ˆX + ¯ hǫ ˆZ (31)where the operators are closed to commutation relations, forming the SU(2) Lie algebra: ˆW = ( ˜d † + ˜d + + ˜d †− ˜d − ) , ˆX = ( ˜d † + ˜d − e i ǫt + ˜d †− ˜d + e − i ǫt ), ˆY = i ( ˜d † + ˜d − e i ǫt − ˜d †− ˜d + e − i ǫt ) , ˆZ = ( ˜d † + ˜d + − ˜d †− ˜d − ).The dynamical description of the engine is completely determined by the expectation values uantum heat engines and refrigerators d ˆW dt = − Γ T ˆW − (Γ + − Γ − ) ˆZ − (Γ h − Γ c ) ˆX + (Γ + h N + h + Γ − h N − h + Γ + c N + c + Γ − c N − c ) d ˆX dt = − Γ T ˆX − (Γ h − Γ c ) ˆW + (Γ + h N + h + Γ − h N − h − Γ + c N + c − Γ − c N − c ) + 2 ǫ ˆY d ˆY dt = − Γ T ˆY − ǫ ˆX d ˆZ dt = − Γ T ˆZ − (Γ + − Γ − ) ˆW + (Γ + h N + h − Γ − h N − h + Γ + c N + c − Γ − c N − c ) , (32)where the equilibrium populations of the dressed filter operators and heat transport coeffi-cients become: N ± h ( c ) = (cid:18) ¯ hω ± h ( c ) kBTh ( c ) (cid:19) − , Γ ± h ( c ) = γ ± h ( c ) (1 − exp (cid:18) − ¯ hω ± h ( c ) k B T h ( c ) (cid:19) ) (33)with ω ± h ( c ) = ω h ( c ) ± ǫ andΓ T = Γ + h + Γ − h + Γ + c + Γ − c , Γ ± = Γ ± h + Γ ± c , Γ h ( c ) = Γ + h ( c ) + Γ − h ( c ) . (34)A simplifying limit is obtained when the relaxation rate of the upper and lower manifold areapproximately equal: Γ + h = Γ − h ≡ κ h and Γ + c = Γ − c ≡ κ c . For a typical harmonic bath whereΓ( ω ) ∼ ω d , and d stands for the bath dimension, this is a good approximation ( ω h , ω c ≫ ǫ ).The equations of motion for the amplifier relax to a steady state operational mode. Theexpectation values for the operators in steady state d ˆX dt = d ˆY dt = d ˆZ dt = d ˆW dt = 0 become: D ˆX E = κ h κ c ( N + h + N − h − N + c − N − c )2(4 ǫ + κ h κ c ) D ˆY E = − ǫ ( N + h + N − h − N + c − N − c )4 ǫ + κ h κ c (cid:16) κ h κ c κ h + κ c (cid:17)D ˆZ E = ( N + h − N − h ) κ h +( N + c − N − c ) κ c κ h + κ c D ˆW E = ( N + h + N − h + N + c + N − c ) κ h κ c ǫ + κ h κ c ) + λ (( N + h + N − h ) κ h +( N + c + N − c ) κ c )(4 ǫ + κ h κ c )( κ h + κ c ) . (35)The commutator [ ˆY , ˆH I ] = 0 therefore ˆY is related to the non diagonal elements of theHamiltonian which define the coherence between energy levels. uantum heat engines and refrigerators P = ¯ hǫν D ˆY E = − h ( ω h − ω c ) ǫ G ǫ + κ h κ c (cid:16) κ h κ c κ h + κ c (cid:17) J h = L † h ( ˆH I ) = (cid:16) ¯ hǫG + hω h ǫ G ǫ + κ h κ c (cid:17) (cid:16) κ h κ c κ h + κ c (cid:17) J c = L † c ( ˆH I ) = − (cid:16) ¯ hǫG + hω c ǫ G ǫ + κ h κ c (cid:17) (cid:16) κ h κ c κ h + κ c (cid:17) (36)where the generalised gain becomes: G = ( N + h + N − h ) − ( N + c + N − c ) G = ( N + h − N − h ) − ( N + c − N − c ) (37)The first law of thermodynamics is satisfied such that J h + J c + P = 0 as well as thesecond law: − J h T h − J c T c ≥
0. The power P is proportional to the expectation value of thecoherence D ˆY E . As a consequence additional pure dephasing originating from external noisewill degrade the power. Such noise generated by a Gaussian random process is described bythe generator L D = − γ [ ˆH I , [ ˆH I , • ]] [48].In the regime where the external driving amplitude is larger than the heat conductivity ǫ ≫ κ h κ c , Eq.(36) converges to the results of Ref. [36]. P = ¯ hǫν D ˆY E = − ¯ hνG (cid:16) κ h κ c κ h + κ c (cid:17) J h = L † h ( ˆH I ) = ¯ h ( ǫG + ω h G ) (cid:16) κ h κ c κ h + κ c (cid:17) J c = L † c ( ˆH I ) = − ¯ h ( ǫG + ω c G ) (cid:16) κ h κ c κ h + κ c (cid:17) (38)Returning to Eq. (36), optimal power is obtained when the heat conductances from thehot and cold bath are balanced: Γ ≡ κ c = κ h , then the power and the heat flows from thereservoirs become: P = ¯ hǫν D ˆY E = − ¯ hνǫ Γ G ǫ +Γ J h = L † h ( ˆH I ) = ¯ hǫ Γ G + ¯ hω h ǫ Γ G ǫ +Γ J c = L † c ( ˆH I ) = − ¯ hǫ Γ G − ¯ hω c ǫ Γ G ǫ +Γ (39) uantum heat engines and refrigerators FIG. 3: The normalized power P / P max as a function of the coupling amplitude to the externalfield ǫ and the heat conductivity Γ. A clear maximum is obtained for Γ = 2 ǫ . Figure 3 shows the power as a function of the heat conductivity coefficient Γ and the couplingto the external field ǫ . A clear global maximum is obtained for Γ = 2 ǫ .The efficiency of the amplifier, is defined as η = − PJ h . For the present model it becomes: η = νω h + (Γ + ǫ ) G | ǫ | G . (40)In the limit of ǫ → → η o = 1 − ω c ω h andfor zero gain, ω c ω h = T c T h , we obtain the Carnot limit η c = 1 − T c T h .For finite fixed Γ the efficiency in the limit ǫ → η = νω h + Γ (cid:16) T h (1 − cosh( ωhTh )) − T c (1 − cosh( ωcTc )) (cid:17) T h T c (cid:16) sinh( ωhTh ) − sinh( ωcTc )+sinh( ωcTc − ωhTh ) (cid:17) . (41)Examining Eq. (41) shows that the Carnot limit is unattainable. When the Otto efficiencyapproached the Carnot limit ω c ω h = T c T h the amplifiers efficiency becomes zero. This comparisonbetween the two limits can be seen in Fig. 4.The efficiency of Eq. (40) can be either smaller or greater than the Otto efficiencydepending on the sign of G . In the low temperature limit G <
0, thus η o ≤ η ≤ η c .Increasing ǫ will increase the efficiency up to a critical point ǫ crit which both both G → uantum heat engines and refrigerators P/P max h / h C FIG. 4: The normalised efficiency η/η c vs. the normalised power P / P max for fixed ǫ and Γ and ω c while varying ω h . The blue line is for finite Γ while the dashed red line is optimised at each pointΓ = 2 ǫ . In this case ǫ ≪ and G →
0, since N + h ( c ) → N − h ∼ N − c . At this point the engine operates at theCarnot limit and all currents vanish, such that the process becomes isoentropic, Cf. Fig. 5.In the high temperature limit for harmonic oscillators, while increasing ǫ , the gain G willchange sign and become positive, thus η ≤ η o , Cf. Fig. 6.
2. Optimizing the amplifier’s performance
Further optimisation for power is obtained when the pumping rate Γ is optimised formaximum power then Γ max = 2 ǫ and the power becomes: P = −
12 ¯ hν | ǫ | G . (42)At the limit of high temperature and small ǫ the gain becomes G ≈ K B T h ¯ hω h − K B T c ¯ hω c , thenoptimising the power Eq. (42) with respect to the output frequency ν leads to: ω c ω h = s T c T h , resulting in the efficiency at maximum power: η CA = 1 − s T c T h , (43)which is the well established endoreversible result of Curzon and Ahlborn [2, 8]. The optimalpower is obtained when all the characteristic parameters are balanced: ǫ ∼ Γ ∼ κ c ∼ κ h . uantum heat engines and refrigerators −7 Currents −7 Entropy Production 0 1 2 3−2−1012 x 10 −5 G1 and G2
OttoG1G2CarnotJ c PJ h FIG. 5: Top left: The heat J c , J h and power P currents as a function of ǫ . Top Right: Efficiency η as a function of ǫ . Bottom left: Entropy production as a function of ǫ . Bottom right: The gain G and G as a function of ǫ . The harmonic tricycle engine operates in the limit of low temperature. k B T c = 0 . k B T h = 0 .
3, ¯ hω c = 4 . and ¯ hω h = 6 and Γ = 0 . Figure 7 shows a trajectory of efficiency with respect to power with changing field couplingstrength ǫ for different frequency ratios ω c ω h = ( T c T h ) α . The power vanishes with the coupling ǫ = 0 and then when ǫ = ǫ crit , at this point the splitting in the dressed energy levels nullsthe gain. D. Dynamical model of a 3-level engine
The dynamical description of the 3-level engine is closely related to the tricycle model [9].The model is a template for the 3-level laser shown in Fig. 1 with inclusion of a dynamicaldescription.The Hamiltonian of the device has the form: ˆH s = ˆH s + ˆV ( t ) = ǫ ǫ ǫe iνt ǫe − iνt ǫ (44)where ˆH s = P ǫ i ˆP ii , and ˆP ij = | i ih j | . ˆV ( t ) = ǫ (cid:16) ˆP e iνt + ˆP e − iνt (cid:17) . The state of the uantum heat engines and refrigerators CarnotOttoG1G2PJ h J c FIG. 6: Top left: The heat J c , J h and power P currents as a function of ǫ . Top Right: Efficiency η as a function of ǫ . Bottom left: Entropy production as a function of ǫ . Bottom right: The gain G and G as a function of ǫ . The harmonic tricycle engine operates in the limit of high temperature. k B T c = 100 and k B T h = 300, ¯ hω c = 4 . and ¯ hω h = 6 and Γ = 0 . three-level system is fully characterised by the expectation values of any eight independentoperators, excluding the identity operator. Different choices of the eight independent oper-ators corresponds to different viewpoints. • The P representation , is based on the eigen-representation of the free Hamiltonian, ˆH s , in the rotating frame. ˜P i,j = e − iν ˆP z t ˆP i,j e iν ˆP z t . (45)The following notations are also used [9]: ˆP i = ˆP ii , ˆP + = ˆP , ˆP − = ˆP , ˆP x = ( ˆP + + ˆP − ), ˆP y = i ( ˆP + − ˆP − ), ˆP z = ( ˆP u − ˆP l ). • The Π representation , which is the eigen-representation of the full Hamiltonian in therotating frame, ˜H s = ˆH + ˜V : ˜Π i , j = e − iθ ˜P y ˜P i , j e iθ ˜P y , (46)where tan( θ ) = 2 ǫ/ ∆ ω . Note that ˆH + ˜V = ∆ ω ˆP z + 2 ǫ ˜P x = ν ˜Π z where ∆ ω = ω h − ω c − ν . Further notations are introduced by ˜Π i , ˜Π ± , ˜Π x , ˜Π y , ˜Π z , defined in a uantum heat engines and refrigerators P/P maxC−A h / h C FIG. 7: The normalised efficiency η/η c vs the normalised power P / P max for different optimalvalues of Γ = 2 ǫ . P max is obtained for the Curzon-Ahlborn ratio ω c ω h = q T c T h shown in blue. Theorange line is for ω c ω h = (cid:16) T c T h (cid:17) / , the green line is for ω c ω h = (cid:16) T c T h (cid:17) / . The red line is for the Carnotratio ω c ω h ∼ T c T h where the power is multiplied by 10 . similar manner to the analogous operators in the P representation, provided that each ˆP is replaced by a ˆΠ . The Π representation coincides with the atom-field dressed staterepresentation [49]. This representation is employed to obtain the master equation forthe engine [9].The power and heat fluxes only depend on a reduced subset of SU(2) operators, whichare decoupled from the rest of the operators. As a result, the equations of motion of the3-level amplifier can be represented by the set: ˜P + , ˜P − , ˜P u , ˜P l . The final equations ofmotion for these observables and the expressions for the power and efficiency become upto numerical factors identical to the expressions obtained for the driven quantum tricycle.Figure 8 shows a schematic view of the splitting of the energy levels of the 3-level systemdue to the driving field. As a result, the engine also splits into two parts, the upper andlower manifolds. uantum heat engines and refrigerators ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( T h (((((((((((((((((((((((((((((((((((( T c +e P nw w k › + k fl k fl k › - k › + k › - n -e+e-e FIG. 8: The three-level-amplifier: The interaction with the external field splits the two upperlevels. As a result the heat transport terms are modified. Two parallel engines emerge which canoperated in opposite direction producing zero power for a certain choice of pararmeters.
1. The Generalized Lamb Equations in the P Representation
The basic equations describing the operation of a 3-level laser have been formulated byLamb [50, 51]. These equations are modified to include the dynamical effects induced bythe driving field. The equations of motion for the closed set of operators characterising theheat currents become: ddt ˆP + ˆP − ˆP u ˆP l = i ∆ ω + Γ + − iǫ + Γ + u iǫ + Γ + l − i ∆ ω + Γ − iǫ + Γ − u − iǫ + Γ − l − iǫ + Γ u + iǫ + Γ u − Γ u Γ ul iǫ + Γ l + − iǫ + Γ l − Γ lu Γ l ˆP + ˆP − ˆP u ˆP l + γ + γ − γ u γ l (47) The relaxation coefficients { Γ i,j , Γ i , γ i } consist of linear combinations of the bath parametersand can be found explicitly in Ref. [9].The steady-state power and heat flows can be evaluated by solving the generalised Lambequations. The results are represented by Eq. (36) where the definition of population ischanged from that of a harmonic oscillator to a two-level-system: N ± h ( c ) = e βh ( c ) ω ± h ( c ) +1 .The numerator of the expressions for the power and heat flows consist of a sum of twocontributions: one, which is proportional to the gain N + h − N + c , is associated with thepopulation inversion in the upper manifold, while the other, which is proportional to thegain N − h − N − c , is associated with the population inversion in the lower manifold. uantum heat engines and refrigerators G is positive for all driving conditions ǫ . The maximum power point is the result of competition between the upper and lowermanifolds. As ǫ increases , N + h − N + c increases , whereas N − h − N − c decreases . Hence, thepower production of the upper manifold increases , (i.e. becomes more negative) whereasthat of the lower manifold decreases (i.e. becomes less negative).From a thermodynamic point of view, each manifold is associated with a separate heatengine. As the coupling with the work reservoir ( ǫ ) increases, the engine associated with theupper manifold operates faster while that associated with the lower one operates slower . Inaddition, the energy is leaking from the former to the latter, thereby diminishing the netpower production.Not only does the power decrease as a function of ǫ , it also changes sign at a certain finite value of the field amplitude, denoted by ǫ crit . This results from the fact that at some pointthe lower manifold starts operating backwards as a heat pump, rather than a heat engine.At ǫ = ǫ crit , the power consumption by the lower manifold is exactly balanced by the power production of the upper manifold, such that the net power production is zero.An examination of the steady-state heat fluxes, reveals that they do not vanish at zero-power operating conditions. Although both manifolds operate at the same rate and inopposite directions, the upper manifold absorbs more heat from the hot bath than thatrejected by the lower one. Similarly, the upper manifold rejects more heat into the coldbath than that absorbed by the lower one. The net heat absorbed from the hot bath andrejected into the cold bath in zero-power operating conditions is therefore proportional tothe difference in the energy gaps associated with these transitions. The entropy generatedexclusively by the heat leak from the upper to lower manifold is obtained by eliminating thepower in Eq. (39): ddt ∆ S l = ¯ hǫ Γ G k B (cid:18) T c − T h (cid:19) ≥ . (48)Zero-power operating conditions correspond to the short circuit limit. Heat is effectivelytransferred from the hot bath into the cold bath, such that no net work is involved.An interesting limit is that of low temperatures, such that ¯ hǫ/k B ≫ T c , T h . This limitis realistic in the optical domain where ν ≫ T c , T h . In such a case N + h − N + c is negligiblerelative to N − h − N − c , and only the lower manifold needs be accounted for.The value of ǫ crit in this limit, denoted by ǫ lTcrit , is that for which N − h − N − c = 0. It is uantum heat engines and refrigerators ǫ lTcrit = T h ω c − T c ω h T h − T c . (49)The condition T h ω c − T c ω h > ǫ = ǫ lTcrit , asymptotically implies zero heat flows and hence the reversible limit. Indeed,substituting ǫ lTcrit from Eq. (49) for ǫ reduces to the Carnot efficiency, η c = 1 − T c /T h .The difference between a 3-level-amplifier and the tricycle with harmonic oscillator filtersat the high temperature limit is observed in Fig. 9. The source of the difference is thesaturation of finite levels. The efficiency at maximum power is lower than the Curzon-Ahlborn efficiency approximated by: η pmax ≈ η c − s k B T c ¯ hω h η c . (50) E. The four-level engine and two-level engine
The four level-engine is a dynamical model of the 4-level laser [52]. In the four-levelengine the pumping excitation step is isolated from the coupling to the external field byemploying an additional cold reservoir. Figure 10 shows a schematic view of the structureof the engine. Analysing a static viewpoint, positive gain is obtained when G = p − p = p (cid:18) e − ¯ hωhKBTh e ¯ hωc kBTc − e − ¯ hωc kBTc (cid:19) ≥ , (51)where: ω h = ω , ω c = ω , ω c = ω . The optimal output frequency at resonance is ν = ω h − ω c − ω c , therefore the efficiency becomes: η o = νω h . If the additional cold reservoiris assumed to have temperature T c , then the Carnot restriction is obtained: η o ≤ η c .A dynamical analysis reveals a splitting of the energy levels that are driven by the externalfield, leading to a similar performance as the three-level amplifier. The advantage of thefour-level engine is that the hot bath thermal pumping is isolated from the coupling to thepower extraction, thus replacing the decoherence associated with the hot bath with a quieteroperation associated with the cold bath. In addition, optimising performance by balancingthe rates between the upper and lower manifold can be achieved by ω c = ω c . uantum heat engines and refrigerators P/P max h / h C HOTLS
FIG. 9: The normalised efficiency η/η c vs the normalised power P / P max for ω c ω h ∼ T c T h for varyingΓ = 2 ǫ . The high temperature limit is shown where ¯ hω ≪ k b T . The red line is the 3-level amplifierwhile the blue is the harmonic tricycle engine. Saturation limits the performance of the 3-levelengine. The four level engine has been employed to study the influence of initial coherence on theperformance of the engine [53–59]. In this case the power output is connected to the twoupper levels and the coherence is generated by splitting the ground state. The basic idea isthat initial coherence present between energy levels associated with the entangling operator ˆY , Eq. (35) can be exploited to generate additional power even from a single bath. Thereis no violation of the second-law since initial coherence reduces the initial entropy. Contactwith a single bath will increase this entropy. A unitary manipulation can then exploit thisentropy difference to generate work.An extreme exploitation of the dynamical splitting of the energy levels due to the externaldriving field results in the two-level engine. Figure 11 sketches the schematic operation ofthe engine. Two excitations from the hot bath are required to generate a gain in the uppermanifold. A clever choice of coupling to the bath is required to generate this gain. A modelof such an engine has been worked out [60]. The maximum efficiency of such an enginebecomes η ≤ ≤ η c since two-pump steps are required for one output step. uantum heat engines and refrigerators ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( T h (((((((((((((((((((((((((((((((((((( T c +e P nw w k › + k fl k fl k › - k › n -e+e-e (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( T c k › + k fl k › - w FIG. 10: The four-level-engine: The interaction with the external field splits the two upper levels.As a result the heat transport terms are modified. Two parallel engines emerge which can operatedin opposite direction producing zero power for a certain choice of pararmeters. Coherence betweenlevels 0 and 1 can enhance the performance by synchronising the engines. ω h = ω , ω c = ω and ω c = ω . ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( T h (((((((((((((((((((((((((((((((((((( T c P nw=n k fl - k › +e-e +e -e FIG. 11: The two-level-engine: The interaction with the external field splits the lower and upperlevels. If the coupling to the hot and cold bath are engineered properly output power can beproduced due to the positive gain between the upper split levels. Only the transitions leading tothis mechanism are indicated.
F. Power storage
An integral part of an engine is a device which allows to store the power and retrieve iton demand, a flywheel. A model based on the tricycle of such a device just disconnects the uantum heat engines and refrigerators ω w ˆc † ˆc . Once there is positive gaingenerated by the hot and cold bath energy will flow to this oscillator. A model based on the3-level amplifier suggested this approach [21, 35, 61]. The drawback of these studies is thatthey use a local description of the master equation. When the flywheel oscillator has largeamplitude it will modify the system bath coupling in analogy to the dressed state picture[9].Other studies also considered coupling to a cavity mode [57, 59, 62]. The issue to beaddressed is the amount of energy that can be stored and then extracted. Since the storagemode is entangled to the rest of the engine this issue is delicate [62]. The amount of possiblework to be extracted is defined by an entropy preserving unitary transformation to a passivestate [63]. III. CONTINUOUS REFRIGERATORS
In a nutshell, refrigerators are just inverted heat engines. They employ power to pumpheat from a cold to a hot bath. As in heat engines, the first and second law of thermo-dynamics imposes the same restrictions on the refrigerator’s performance. The destinctionbetween the static and dynamical viewpoints also applies. The key element in a refrigeratoris entropy extraction and disposal. This entropy disposal problem is enhanced at low tem-perature where the entropy production in the cold bath can diverge. The unique feature ofrefrigerators is therefore the third-law of thermodynamics.
A. The third law of thermodynamics
Two independent formulations of the third-law of thermodynamics have been presented,both originally stated by Nernst [64–66]. The first is a purely static (equilibrium) one, alsoknown as the ”Nernst heat theorem”, phrased: • The entropy of any pure substance in thermodynamic equilibrium approaches zero asthe temperature approaches zero.The second formulation is known as the unattainability principle: • It is impossible by any procedure, no matter how idealised, to reduce any assembly toabsolute zero temperature in a finite number of operations [66, 67]. uantum heat engines and refrigerators T c → ddt ∆ S u = − J c T c − J h T h − J w T w ≥ . This behaviour can be quantified by a scaling exponent, as T c → J c ∝ T αc , with an exponent α .When the cold bath approaches the absolute zero temperature, it is necessary to eliminatethe entropy production divergence at the cold side, when T c → S c ∼ − T αc , α ≥ . (52)For the case when α = 0 the fulfilment of the second law depends on the entropy produc-tion of the other baths − J h T h − J w T w >
0, which should compensate for the negative entropyproduction of the cold bath. The first formulation of the third-law modifies this restriction.Instead of α ≥ α > S c = 0. Nernst’s heat theorem then leads to thescaling condition of the heat current with temperature [11] : J c ∼ T α +1 c and α > . (53)We now examine the unattainability principle. Quantum mechanics enables a dynamicalinterpretation of the third-law modifying the definition to: No refrigerator can cool a system to absolute zero temperature at finite time .This form of the third-law is more restrictive, imposing limitations on the system bathinteraction and the cold bath properties when T c → uantum heat engines and refrigerators ζ : dT c ( t ) dt = − c T ζc , T c → . (54)where c is a positive constant. Solving Eq. (54), leads to: T c ( t ) − ζ = T c (0) − ζ − ct , f or ζ < ,T c ( t ) = T c (0) e − ct , f or ζ = 1 , T c ( t ) ζ − = T c (0) ζ − + ct , f or ζ > , (55)From Eq. (55) it is apparent that the cold bath can be cooled to zero temperature at finitetime for ζ <
1. The third-law requires therefore ζ ≥
1. The two third-law scaling relationscan be related by accounting for the heat capacity c V ( T c ) of the cold bath: J c ( T c ( t )) = − c V ( T c ( t )) dT c ( t ) dt . (56) c V ( T c ) is determined by the behaviour of the degrees of freedom of the cold bath at lowtemperature where c V ∼ T ηc when T c →
0. Therefore the scaling exponents are related ζ = 1 + α − η [32].The third-law scaling relations Eq. (56) and Eq. (53) can be used as an independentcheck for quantum refrigerator models [71]. Violation of these relations points to flaws inthe quantum model of the device, typically in the derivation of the master equation. B. The quantum power driven refrigerator
Laser cooling is a crucial technology for realising quantum devices. When the temperatureis decreased, degrees of freedom freeze out and systems reveal their quantum character.Inspired by the mechanism of solid state lasers and their analogy with Carnot engines [6] itwas realised that inverting the operation of the laser at the proper conditions will lead torefrigeration [72–75]. A few years later a different approach for laser cooling was initiatedbased on the doppler shift [76, 77]. In this scheme, translational degrees of freedom of atomsor ions were cooled by laser light detuned to the red of the atomic transition. Unfortunatelythe link to thermodynamics was forgotten. A flurry of activity then followed with therealisation that the actual temperature achieved were below the Doppler limit k B T doppler = hγ/ γ is the natural line width [78–80]. uantum heat engines and refrigerators p = ¯ hk . As a result, the temperature is related to the averagekinetic energy: k B T recoil = ¯ h k M , (57)where ¯ hk is the photon momentum and M the atomic mass of the particle being cooled.For sodium the doppler limit is T doppler = 235 µK and the recoil limit T recoil = 2 . µK [81].Thermodynamically, these limits do not bind, the only hard limit is the absolute zero T c = 0. To approach this limit the cooling power has to be optimised to match the coldbath temperature. Unoptimised refrigerators become restricted by a minimum temperatureabove the absolute zero.A reexamination of the elementary 3-level model stresses this point. Examining the heatengine model of Fig 1 shows that if the power direction is reversed a refrigerator is generatedprovided the gain is negative: G = p − p <
0. Assuming quasi equilibrium conditions.This leads to: ω c ω h = ω ω ≤ T c T h , (58)which translated to a minimum temperature of T c ( min ) ≥ ω c ω h T h . We can consider typicalvalues for laser cooling of Na based on the D line of 589 . nm which translates to ω h =508 . · Hz . If the translational cold bath is coupled to the hyperfine structure splitting F → F of the ground state 3 S / , meaning ω c = 1 . · Hz . The cooling ratio becomes: ω c ω h = 3 . · − and the minimum temperature assuming room temperature of the hot bathis T c ( min ) ∼ . − K . By employing a magnetic field the hyperfine levels F split in three,leading to further splittings in the MHz limit increasing the cooling ratio by three ordersof magnitude. When the cooling is in progress, the hyperfine splitting can be reduced bychanging the magnetic field such that it follows the cold bath temperature. In principle anytemperature above T c = 0 is reachable. C. Dynamical refrigerator models
A quantum dynamical framework of the refrigerator is based on the tricycle template.The work bath is replaced by a time dependent driving term. The cooling current can becalculated analytically. The derivation of the equation of motion are identical to the heat uantum heat engines and refrigerators G ≤
0, Eq. (37). For example, when ǫ > κ the cooling current becomes [36]: J c = 12 ¯ hω − c Γ − c Γ − h Γ − c + Γ − h ( N − c − N − h ) + ¯ hω + c Γ + c Γ + h Γ + c + Γ + h ( N + c − N + h ) ! . (59)The phenomena that the refrigerator splits into two parts is also found here. In a similarfashion the 3-level amplifier can be reversed to become a refrigerator [9]. The heat currentsare identical to Eq. (36) with the TLS definition of N ± c/h .In a refrigerator the object of optimisation is the cooling power J c compared to the outputpower P in an engine. The efficiency is defined by the coefficient of performance COP theratio between J c and the input power P . COP = J c P ≤
COP o ≤ COP c (60)where COP o = ω c ν and COP c = T c T h − T c . All power driven refrigerators are restricted by theOtto COP o . In section III E the performance of power driven refrigerators at low tempera-ture will be analysed. D. The quantum absorption refrigerator
The absorption chiller is a refrigerator which employs a heat source to replace mechani-cal work for driving a heat pump [82]. The first device was developed in 1850 by the Carr´ebrothers which became the first useful refrigerator. In 1926 Einstein and Szil´ard invented anabsorption refrigerator with no moving parts [83]. This fact is an inspiration for miniatur-izing the device to solve the growing problem of heat generated in integrated circuits. Evena more challenging proposal is to miniaturize to the level of a few-level quantum system.Such a device could be incorporated into a quantum circuit. The first quantum version wasbased on the 3-level refrigerator [12] driven by thermal noise. A more recent model of anautonomous quantum absorption refrigerator with no external intervention based on three-qubits [37] has renewed interest in such devices. A setup of opto-mechanical refrigeratorspowered by incoherent thermal light was introduced by [84]. The study showed that coolingincreases while increasing the photon number up to the point that fluctuation of the radia-tion pressure becomes dominant and heats the mechanical mode. Studies of cooling based uantum heat engines and refrigerators
1. The 3-level absorption refrigerator
In the 3-level absorption refrigerator the power source has a finite temperature T w . Thecoefficient of performance becomes COP = J c J w . From the second and first law Eqs. (7), (8)using steady state conditions one obtains: COP ≤ ( T w − T h ) T c ( T h − T c ) T w (61)and T w > T h > T c [82]. Under the assumption that the three baths are uncorrelated, the 3-level state is diagonal in the energy representation and determined by the heat conductivities[12, 88]. From these values the heat currents can be evaluated. This refrigerator emphasisesthe fact that no internal coherence is required for operation.
2. Tricycle absorption refrigerator
The template to understand the absorption refrigerator is the tricycle model, a refrigeratorconnected to three reservoirs with the Hamiltonian Eq. (5) and Eq. (6). A thermodynamicalconsistent description requires to first diagonalize the Hamiltonian and then to obtain thegeneralized master equations for each reservoir. The final step is to obtain the steady stateheat currents J c , J h and J w . The nonlinearity in Eq. (6) hampers this task.One remedy to this issue is to replace the harmonic oscillators in the model by qubits[38]. The other approach is to consider the high temperature limit of the power reservoir ora noise driven refrigerator [36, 89]. This three qubit refrigerator model was introduced asthe ultimate miniaturization model [37, 90, 91]. The free Hamiltonian of the device has theform: ˆH F = ¯ hω h ˆ σ hz + ¯ hω c ˆ σ cz + ¯ hν ˆ σ wz (62)and the interaction Hamiltonian: ˆH I = ¯ hǫ (cid:16) ˆ σ h + ⊗ ˆ σ c − ⊗ ˆ σ w − + ˆ σ h − ⊗ ˆ σ c + ⊗ ˆ σ w + (cid:17) , (63) uantum heat engines and refrigerators ˆ σ are two-level operators. Assuming ω h = ω c + ν the eigenstates of the Hamiltonian ˆH = ˆH F + ˆH I become: N umber Energy state | , , i hω c | , , i hν | , , i h ( ν + ǫ ) √ ( | , , i + | , , i )5 ¯ h ( ν − ǫ ) √ ( | , , i − | , , i )6 ¯ h ( ω c + ω h ) | , , i h ( ω h + ν ) | , , i hω h | , , i (64)The coupling to the bath has the form: ˆH SR = X j = c,h λ j ( ˆ σ j + ⊗ ˆB j − + ˆ σ j − ⊗ ˆB j + ) , (65)where j is the bath index. In this model, the coupling to the cold bath connects levels1 → → , , →
8. The master equation for the refrigerator can be derivedusing the weak system bath coupling limit [38]: L = i ¯ h [ ˆH , • ] + L h + L c + L w . (66)The cooling and power currents can be solved for this model. Of major importance is thedistinction between the complete non-local description of the tricycle and a local description.In the local case the dissipative terms L h/c are set to equilibrate only the local qubit. In thenonlocal case the local qubit is mixed with the other qubits due to the nonlinear coupling.As a result in the nonlocal approach it is impossible to reach the Carnot COP c no matterhow small is the internal coupling ǫ . This can be observed in Fig. 12, showing the COP vsthe cooling current J c for different coupling strength ǫ . On the other hand the Carnot limit COP c is reachable in the local model. An interesting observation is that a universal limitfor the COP at the maximum cooling power was found [88] applicable for all absorptionrefrigerator modes of
COP ∗ ≤ dd +1 COP c where d is the diemsionality of the phonon coldbath. uantum heat engines and refrigerators e e e e J c / J c C O P / C O P c * FIG. 12: The
COP/COP c of the 3-qubit refrigerator vs the normalised cold current J c / J ∗ c fordifferent values of the internal coupling constant ǫ . The COP at maximum power is found to bebound by 3 / COP c independent of ǫ . Figure in courtesy of L.A. Correa and J.P. Palao.
3. Noise driven refrigerator
An ideal power source generates zero entropy: ∆ S w = − J w T w . A pure mechanical externalfield achieves this goal. Another possibility is a thermal source, when T w → ∞ it alsogenerates zero entropy. An unusual power source is noise. Gaussian white noise also carrieswith it zero entropy production. Analysis will show that the performance of refrigeratorsdriven by all these power sources are very similar even when approaching the absolute zerotemperature [89].Employing the tricycle as a template we examine the option of employing noise as a powersource. The simplest option is a Gaussian white noise source. As a result the interactionnonlinear term Eq. (6) is replaced with: ˆH int = f ( t ) (cid:18) ˆa † ˆb + ˆaˆb † (cid:19) = f ( t ) ˆX , (67)where f ( t ) is the noise field. ˆX = ( ˆa † ˆb + ˆaˆb † ) is the generator of a swap operation between thetwo oscillators and is part of a set of SU (2) operators , ˆY = i ( ˆa † ˆb − ˆaˆb † ), ˆZ = (cid:18) ˆa † ˆa − ˆb † ˆb (cid:19) and the Casimir ˆN = (cid:18) ˆa † ˆa + ˆb † ˆb (cid:19) . uantum heat engines and refrigerators
34A Gaussian source of white noise is characterized by zero mean h f ( t ) i = 0 and deltatime correlation h f ( t ) f ( t ′ ) i = 2 ηδ ( t − t ′ ). The Heisenberg equation for a time independentoperator ˆO reduced to: ddt ˆO = i [ ˆH s , ˆO ] + L ∗ n ( ˆO ) + L ∗ h ( ˆO ) + L ∗ c ( ˆO ) , (68)where ˆH s = ¯ hω h ˆa † ˆa + ¯ hω c ˆb † ˆb . The noise dissipator for Gaussian noise is L ∗ n ( ˆO ) = − η [ ˆX , [ ˆX , ˆO ]] [48].The next step is to derive the quantum Master equation of each reservoir. It is assumedthat the reservoirs are uncorrelated and also uncorrelated with the driving noise. Theseconditions simplify the derivation of L h which become the standard energy relaxation terms,driving oscillator ¯ hω h ˆa † ˆa to thermal equilibrium with temperature T h and L c drives oscillator¯ hω c ˆb † ˆb to equilibrium T c [92]. L ∗ h ( ˆO ) = Γ h ( N h + 1) (cid:16) ˆa † ˆOˆa − n ˆa † ˆa , ˆO o(cid:17) + Γ h N h (cid:16) ˆa ˆOˆa † − n ˆaˆa † , ˆO o(cid:17) L ∗ c ( ˆO ) = Γ c ( N c + 1) (cid:18) ˆb † ˆOˆb − (cid:26) ˆb † ˆb , ˆO (cid:27)(cid:19) + Γ c N c (cid:18) ˆb ˆOˆb † − (cid:26) ˆbˆb † , ˆO (cid:27)(cid:19) . (69)The kinetic coefficients Γ h/c are determined from the system bath coupling and the spectralfunction [9, 36].The equations of motion including the dissipative part are closed to the SU (2) set ofoperators. To derive the cooling current J c = hL c (¯ hω c ˆb † ˆb ) i , we solve for stationary solutionsof ˆN and ˆZ . The cooling current becomes: J c = ¯ hω c η ¯Γ2 η +¯Γ ( N c − N h ) . (70)where the effective heat conductance is ¯Γ = Γ c Γ h Γ c +Γ h . Cooling occurs for N c > N h ⇒ ω h T h > ω c T c .The coefficient of performance ( COP ) for the absorption chiller is defined by the relation
COP = J c J n , with the help of Eq. (70) we obtain the Otto COP o [23], Cf. Eq. (60).We now show the equivalence of the noise driven refrigerator with the high temperaturelimit of the work bath T w . Based on the weak coupling limit the dissipative generator of thepower bath becomes: L ∗ w ( ˆO ) = Γ w ( N w + 1) (cid:18) ˆa † ˆb ˆOˆb † ˆa − (cid:26) ˆa † ˆaˆbˆb † , ˆO (cid:27)(cid:19) + Γ w N w (cid:18) ˆaˆb † ˆOˆa † ˆb − (cid:26) ˆaˆa † ˆb † ˆb , ˆO (cid:27)(cid:19) . (71) uantum heat engines and refrigerators N w = (exp( ¯ hω w kT h ) − − . At finite temperature L w ( ˆO ) does not lead to a close set ofequations. But in the limit of T w → ∞ it becomes equivalent to the Gaussian noise generator: L ∗ w ( ˆO ) = − η/ (cid:16) [ ˆX , [ ˆX , ˆO ]] + [ ˆY , [ ˆY , ˆO ]] (cid:17) , where η = Γ w N w . This noise generator leads tothe same current J c and COP as Eq. (70) and (60). We conclude that Gaussian noiserepresents the singular bath limit equivalent to T w → ∞ .Poisson white noise can be employed as a power source. This noise is typically generatedby a sequence of independent random pulses with exponential inter-arrival times [93, 94].These impulses drive the coupling between the oscillators in contact with the hot and coldbath leading to: d ˆO dt = ( i/ ¯ h )[ ˜H , ˆO ] − ( i/ ¯ h ) λ h ξ i [ ˆX , ˆO ]+ λ (cid:16)R ∞−∞ dξP ( ξ ) e ( i/ ¯ h ) ξ ˆX ˆO e ( − i/ ¯ h ) ξ ˆX − ˆO (cid:17) , (72)where ˜H is the total Hamiltonian including the baths. λ is the rate of events and ξ is theimpulse strength averaged over a distribution P ( ξ ). Using the Hadamard lemma and thefact that the operators form a closed SU (2) algebra, we can separate the noise contributionto its unitary and dissipation parts, leading to the master equation, d ˆO dt = ( i/ ¯ h )[ ˜H , ˆO ] + ( i/ ¯ h )[ ˆH ′ , ˆO ] + L ∗ n ( ˆO ) . (73)The unitary part is generated with the addition of the Hamiltonian ˆH ′ = ¯ hǫ ˆX and with theinteraction ǫ = − λ Z dξP ( ξ )(2 ξ/ ¯ h − sin (2 ξ/ ¯ h )) . (74)This term can cause a direct heat leak from the hot to the cold bath. The noise generator L n ( ˆ ρ ), can be reduced to the form L ∗ n ( ˆO ) = − η [ ˆX , [ ˆX , ˆO ]] , with a modified noise parameter: η = λ (cid:18) − Z dξP ( ξ ) cos (2 ξ/ ¯ h ) (cid:19) . (75)The Poisson noise generates an effective Hamiltonian which is composed of ˜H and ˆH ′ ,modifying the energy levels of the working medium. This new Hamiltonian structure has tobe incorporated in the derivation of the master equation otherwise the second law will beviolated. The first step is to rewrite the system Hamiltonian in its dressed form. A new setof bosonic operators is defined ˆA = ˆa cos( θ ) + ˆb sin( θ ) ˆA = ˆb cos( θ ) − ˆa sin( θ ) , (76) uantum heat engines and refrigerators ˆH s = ¯ h Ω + ˆA † ˆA + ¯ h Ω − ˆA † ˆA , (77)where Ω ± = ω h + ω c ± q ( ω h − ω c ) + ǫ and cos ( θ ) = ω h − Ω − Ω + − Ω − Eq.(77) impose the restriction,Ω ± > ω h ω c > ǫ . The master equation in the Heisenbergrepresentation becomes: d ˆO dt = ( i/ ¯ h )[ ˆH s , ˆO ] + L ∗ h ( ˆO ) + L ∗ c ( ˆO ) + L ∗ n ( ˆO ) , (78)where the details can be found in ref [36]. The noise generator becomes: L ∗ n ( ˆO ) = − η [ ˆW , [ ˆW , ˆO ]] , (79)where ˆW = sin(2 θ ) ˆZ + cos(2 θ ) ˆX . Again, the SU (2) algebra is employed to define theoperators: ˆX = ( ˆA † ˆA + ˆA † ˆA ) , ˆY = i ( ˆA † ˆA − ˆA † ˆA ) and ˆZ = ( ˆA † ˆA − ˆA † ˆA ). Thetotal number of excitations is accounted for by the operator ˆN = ( ˆA † ˆA + ˆA † ˆA ).Once the set of linear equations is solved the exact expression for the heat currents isextracted, J h = D L ∗ h ( ˆH s ) E , J c = D L ∗ c ( ˆH s ) E and J n = D L ∗ n ( ˆH s ) E .The distribution of impulse determines the performance of the refrigerator. For a normaldistribution of impulses in Eq. (72), P ( ξ ) = √ πσ e − ( ξ − ξ σ . The energy shift is controlledby: ǫ = − λ ξ / ¯ h − e − σ h sin (2 ξ / ¯ h )) . (80)The effective noise strength becomes: η = λ − e − σ h cos (2 ξ / ¯ h )) . (81)In the limit of σ → P ( ξ ) = δ ( ξ − ξ ). Another possibility is an exponential distribution: P ( ξ ) = ξ e − ξξ then: ǫ = − λ ( ξ / ¯ h ) ξ / ¯ h ) ) (82)and η = λ ( ξ / ¯ h ) ξ / ¯ h ) . (83)The Poissonian noise plays two opposing roles. On the one hand it increases the coolingcurrent J c , by increasing η . On the other hand it decreases ǫ (becomes more negative) and by uantum heat engines and refrigerators −15 −10 −5 −16 l J c FIG. 13: The noise driven refregerator. The cooling current J c as a function of λ . The refrigeratoris powered by a Poissonian noise source. The plot is for the limit of σ → that decreases J c . Both parameters η and ǫ depend linearly on λ , which can be interpretedas the rate of photon absorption in the system which enhances the cooling process. Fig. 13shows that J c increases with λ until a point where ǫ dominates and J c decreases.The COP for the Poisson driven refrigerator is restricted by the Otto and Carnot
COP : COP = Ω − Ω + − Ω − ≤ ω c ω h − ω c ≤ T c T h − T c . (84) E. Refrigerators operating close to the limit T c → and the third-law ofthermodynamics The performance of all types of refrigerators at low temperature have universal properties.In the power driven refrigerators the lower split manifold is dominant, leading to the coldcurrent: J c ≈ ¯ hω − c ǫ ¯Γ4 ǫ + Γ c Γ h · G , (85)where the gain G = N − c − N − h and ¯Γ = Γ c Γ h Γ c +Γ h .In the 3-level absorption refrigerator the cold current becomes [88]: J c = ¯ hω c Γ c Γ h Γ w ∆ · G (86)where ∆ is a combination of kinetic constants and G = e − ¯ hωwkBTw e − ¯ hωckBTc − e − ¯ hωhkBTh .In the Guassian noise driven refrigerator Eq. (70) becomes: J c = ¯ hω c η ¯Γ2 η +¯Γ · G . (87) uantum heat engines and refrigerators G = N c − N h .In the Poisson driven refrigerator we obtain: J c ≈ ¯ h Ω − η ¯Γ2 η + ¯Γ · G , (88)with G = ( N − c − N + h ) and Ω − ≈ ω c − ǫ ω h − ω c . All expressions for the cooling current J c area product of an energy quant ¯ hω c , the effective heat conductance and the gain G .To approach the absolute zero temperature a further optimisation is required. Optimisingthe gain G is obtained when ω h → ∞ , which leads to G ∼ e − ¯ hωckBTc . In addition the drivingamplitude should be balanced with the heat conductivity. The cooling current can also beexpressed in terms of the relaxation rate to the bath via the relation γ ( ω ) e − ¯ hωkBT = Γ( ω ) N ( ω ).The universal optimised cooling current as T c → J c = ¯ hω c · γ c · e − ¯ hωckBTc . (89)This form is correct both in the weak coupling limit and the low density limit. The heatcurrent J c can be interpreted as the quant of energy ¯ hω c extracted from the cold bath atthe rate γ c multiplied by a Boltzmann factor. Further optimisation with respect to ω c isdominated by the exponential Boltzmann factor (Optimising the function x a e − x/b leads to x ∗ ∝ b ). As a result ω ∗ c ∝ T c , obtaining: J c ∝ ω ∗ c · γ c ( ω ∗ c ). The linear relation between theoptimal frequency ω c and T c allows to translate the temperature scaling relations to the lowfrequency scaling relations of γ c ( ω ) ∼ ω µ and c V ( ω ) ∼ ω η when ω → γ c ( ω ) ∼ ω α and α >
0. The fulfilment of the unattainability principle Eq. (56) dependson the ratio between the relaxation rate and the heat capacity γ c /c V ∼ ω ζ − where ζ > γ c and the heat capacity c V are exam-ined. For three-dimentional ideal degenerate Bose gases c V scales as T / c . For degenerateFermi gas c V scales as T c . In both cases the fraction of the gas that can be cooled decreaseswith temperature. Based on a collision model when cooling occurs due to inelastic scatteringwe have found the scaling exponent ζ = 3 / ˆH int = ( ˆb + ˆb † ) X k ( g ( k ) ˆa ( k ) + ¯ g ( k ) ˆa † ( k )) ! , ˆH B = X k ω ( k ) ˆa † ( k ) ˆa ( k ) (90) uantum heat engines and refrigerators ˆa ( k ) and ˆa † ( k ) are the annihilation and creation operators for a mode k . For thismodel, the weak coupling limit procedure leads to the GKS-L generator with the cold bathrelaxation rate given by [36]: γ c ≡ γ c ( ω c ) = π X k | g ( k ) | δ ( ω ( k ) − ω c ) (cid:20) − e − ¯ hω ( k ) kBTc (cid:21) − . (91)For the bosonic field in d -dimensional space, with the linear low-frequency dispersion law( ω ( k ) ∼ | k | ), the following scaling properties for the cooling rate at low frequencies are ob-tained: γ c ∼ ω κc ω d − c where ω κc represents the scaling of the form-factor | g ( ω ) | , and ω d − c is the scaling of the density of modes. For ω c ∼ T c , the final current scaling becomes: J optc ∼ T d + κc or α = d + κ − c V ( T c ) ∼ T dc ,which produces the scaling ζ = κ . This means that to fulfil the third law κ ≥
1. Torationalise the scaling, c V is a volume property and γ c is a surface property, so ω c γ c scalesthe same as c V . The scaling of the form factor κ is related to the speed the excitation canbe carried away from the surface.The exponent κ = 1 when ω →
0, is typical in systems such as electromagnetic fieldsor acoustic phonons which have linear dispersion law, ω ( k ) = v | k | . In these cases theform-factor becomes, g ( k ) ∼ | k | / q ω ( k ), therefore | g ( ω ) | ∼ | k | . The condition κ ≥ ω ( k ) ∼ | k | δ with δ <
1, which produce infinite groupvelocity forbidden by relativity theory. Moreover, the popular choice of Ohmic coupling isexcluded for systems in dimension d >
IV. SUMMARY
Thermodynamics represents physical reality with an amazingly small number of variables.In equilibrium, the energy operator ˆH is sufficient to reconstruct the state of the system, fromwhich all other observables can be deduced. Dynamical systems out of equilibrium requiremore variables. Quantum thermodynamics advocates that only a few additional variablesare required to describe a quantum device; a heat engine or a refrigerator. A descriptionbased on Heisenberg equation of motion lends itself to this viewpoint. A sufficient condition uantum heat engines and refrigerators L h/c . Most of thesolvable models in this review were based on the SU(2) Lie algebra with four operators. Wecan chose them to represent the energy ˆH , the identity ˆI , and two additional operators ˆX and ˆY . These operators can associate with coherence since [ ˆH , ˆX ] = 0 , [ ˆH , ˆY ] = 0. Genericallyin these models the power is proportional to the coherence P ∝ h ˆY i . We can speculatethat steady state operation of a working engine or refrigerator is minimally characterised bythe SU(2) algebra of three non commuting operators. This is also the case in reciprocatingengines [97–99].The Hamiltonian can be decomposed to non-commuting local operators which couple tothe baths. The non-local structure influences the derivation of the master equation. A localapproach based on equilibrating ˆH s leads to equations of motion that can violate the second-law of thermodynamics [9, 32, 36, 71]. A thermodynamically consistent derivations requiresa pre-diagonalization of the system Hamiltonian. This Hamiltonian defines the system-bathweak coupling approximation leading to the GKS-L completely positive generator.Quantum mechanics introduces dynamics into thermodynamics. As a result the lawsof thermodynamics have to be reformulated, replacing heat to heat currents and work topower. The first law at steady state requires that the sum of these currents is zero. Forthe devices considered, the entropy production is exclusively generated in the baths and thesum of all contributions is positive. All quantum devices should be consistent with the firstand second laws of thermodynamics. Apparent violations point to erroneous derivations ofthe dynamical equations of motion.The tricycle template is a universal description for continuous quantum heat enginesand refrigerators. This model incorporates basic thermodynamic ideas within a quantumformalism. The tricycle combines three energy currents from three sources by a nonlinearinteraction. A nonlinear construction is the minimum requirement for all heat engines orrefrigerators. This universality means that the same structure can describe a wide rangeof quantum devices. The performance characteristics are given by the choice of workingmedium and reservoirs. In the derivation, the reservoirs are characterised by their tempera-ture and correlation functions. The working medium filters out the channels of heat transferand power. uantum heat engines and refrigerators T c = 0. The quantum dilemma of cooling to a pure statestems from the fact that a system-bath interaction is required to induce a change of entropyof the system. However, such an interaction generates system-bath entanglement so that thesystem cannot be in a pure state. Approaching the absolute zero temperature is a delicatemanoeuvre. The system-bath coupling has to be reduced while the cooling takes place,eventually vanishing when T c →
0. As a result the means to cool are exhausted before thetarget is reached [101, 102]. The universal behaviour of all quantum refrigerators as T c → uantum heat engines and refrigerators a. Acknowledgments: This review was supported by the Israel Science Foundation.We want to thank Tova Feldmann, Eitan Geva, Jose P. Palao, Jeff Gordon, Lajos Diosi,Peter Salamon, Gershon Kuritzky and Robert Alicki for sharing their wisdom. uantum heat engines and refrigerators [1] Carnot S. 1824. R´eflections sur la Puissance Motrice du Feu et sur les Machines propres `aD´evelopper cette Puissance . Paris: Bachelier[2] Curzon F, Ahlborn B. 1975. Efficiency of a carnot engine at maximum power output.
Am.J. Phys.
Energy
Rev. Mod. Phys.
Annalen der Physik
Phys. Rev.Lett.
Phys. Rev.
J.Chem. Phys.
J. Chem. Phys.
Physics Letters A
J. Appl. Phys.
Phys. Rev.E
Phys. Rev. A systems as workingsubstance. Phys. Rev. E
Phys. Rev. uantum heat engines and refrigerators engine. Proceedings of the Royal Society of London. Series A: Mathematical, Physical andEngineering Sciences
Phys. Rev. Lett.
Phys. Rev. E
Phys. Rev. Lett.
Phys. Rev. E
Phys.Rev. A
Eur. Phys. J. B
Ann.Phys.
Phys. Rev. E
J.Chem. Phys.
Phys. Rev. E
Rev. Mod. Phys.
Rev. Mod. Phys. to appear.[29] Gemmer J, Michel M, Mahler G. 2009.
Quantum Thermodynamics . Springer[30] Olshanii M. 2012. Geometry of quantum observables and thermodynamics of small systems.
ArXiv e-prints:1208.0582 [31] Anders J, Giovannetti V. 2013. Thermodynamics of discrete quantum processes.
New Jour.of Phys.
Entropy uantum heat engines and refrigerators [33] Mandal D, Quan HT, Jarzynski C. 2013. Maxwells Refrigerator: An Exactly Solvable Model. Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. A
Phys. Rev. E
Phys. Rev. Lett.
Phys. Rev. E
Phys. Rev. Lett.
Annalen der Physik
Comm. Math. Phys.
J. Math. Phys.
Adv. Chem. Phys.
J. Chem. Phys.
Comm. Math. Phys.
J. Chem. Phys.
Phys. Rev. E
J. Math.Phys. uantum heat engines and refrigerators [49] C. Cohen-Tannoudji, J. A. Dupont-Roc, G. Grynberg. 1987. ”Atom-Photon Interaction” .Wiely[50] Lamb W. 1964. Theory of an Optical Maser. Phys. Rev.
Quantum Statistical Properties of Radiation . Wiley[52] Digonnet MJF. 1990. Closed-Form Expressions for the Gain in Three- and Four-Level LaserFibers .
IEEE J. Quantum Electrinics
Science
Phys. Rev. Lett.
Phys. Rev. A
Proc. Natl. Acad. Sci. U S A
Eur. Phys. Lett.
Phys. Rev. A
Phys. Rev. A
Phys. Rev. E
Phys. Rev. Lett.
Phys. Rev. E
ArXiv e-prints: 1302.3468 [64] Nernst W. 1906. Ueber die Berechnung chemischer Gleichgewichte aus thermischen Messun-gen.
Nachr. Kgl. Ges. Wiss. G¨ott. uantum heat engines and refrigerators kondensierten Systemen. er. Kgl. Pr. Akad. Wiss. The theoretical and experimental bases of the New Heat Theorem Ger., Dietheoretischen und experimentellen Grundlagen des neuen Wa¨rmesatzes . Halle: W. Knapp[67] Fowler RH, Guggenheim EA. 1939.
Statistical Thermodynamics . Cambridge university press[68] Landsberg PT. 1989. A comment on nernst’s theorem.
J. Phys A: Math.Gen.
J. Phys A: Math.Gen.
J. Phys A: Math.Gen.
Phys. Rev. Lett.
J. App. Phys.
J. Phys. Soc. Jpn.
Phys. Rev.
Phys. Rev. Lett.
Bull. Am. Phys. Soc.
Opt. Commun.
Phys. Rev. Lett.
Phys. Rev. Lett.
J. Opt. Soc. Am. B steck.us/alkalidata/sodiumnumbers.1.6.pdf [82] Gordon JM, Ng KC. 2000.
Cool Thermodynamics . Cambridge International Science Pub- uantum heat engines and refrigerators lishing[83] Einstein A, Szil´ard L. 1930. ”refrigeration”. ”US patent No 1,781,541” [84] Mari A, Eisert J. 2012. Cooling by heating: Very hot thermal light can significantly coolquantum systems. Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. Lett.
ArXiv e-prints:1308.4174 [89] Levy A, Kosloff R. 2012. Quantum absorption refrigerator.
Phys. Rev. Lett.
J. Phys A: Math.Gen.
Phys. Rev. E
Open quantum systems . Oxford university press[93] ´Luczka J, Niemeic M. 1991. ”a master equation for quantum systems driven by poisson whitenoise”.
J. Phys A: Math.Gen.
Phys. Rev. A
Phys. Rev. Lett.
Phys. Rev. A
Phys. Rev. E
Phys. Rev. E uantum heat engines and refrigerators [99] Rezek Y, Kosloff R. 2006. Irreversible performance of a quantum harmonic heat engine. NewJ. Phys.
J. Am. Chem. Soc.
J. Chem. Phys.
Sci Rep.
ArXive-prints: 1302.2811 [104] Hovhannisyan KV, Perarnau-Llobet M, Huber M, Ac´ın A. 2013. The role of entanglementin work extraction.
ArXiv e-prints:1303.4686 [105] Brunner N, Huber M, Linden N, Popescu S, Silva R, Skrzypczyk P. 2013. Entanglementenhances performance in microscopic quantum fridges.
ArXiv e-prints: 1305.6009 [106] Geva E, Kosloff R. 1992. A Quantum Mechanical Heat Engine Operating in Finite Time. AModel Consisting of Spin half
Systems as The Working Fluid.
J. Chem. Phys.
J. Chem. Phys.
Am. J. Phys.
Science in China Series G-Phys. Mech. & Ast.
Physica Scripta