quantum heat engines and refrigerators: continuous...

PC65CH17-Kosloff ARI 10 March 2014 16:38

Quantum Heat Enginesand Refrigerators:Continuous DevicesRonnie Kosloff and Amikam LevyInstitute of Chemistry, The Hebrew University, Jerusalem 91904, Israel;email: [email protected]

Annu. Rev. Phys. Chem. 2014. 65:365–93

The Annual Review of Physical Chemistry is online atphyschem.annualreviews.org

This article’s doi:10.1146/annurev-physchem-040513-103724

Copyright c© 2014 by Annual Reviews.All rights reserved

Keywords

quantum thermodynamics, quantum tricycle, quantum amplifier, lasercooling, absolute zero temperature

Abstract

Quantum thermodynamics supplies a consistent description of quantum heatengines and refrigerators up to a single few-level system coupled to the en-vironment. Once the environment is split into three (a hot, cold, and workreservoir), a heat engine can operate. The device converts the positive gaininto power, with the gain obtained from population inversion between thecomponents of the device. Reversing the operation transforms the deviceinto a quantum refrigerator. The quantum tricycle, a device connected bythree external leads to three heat reservoirs, is used as a template for enginesand refrigerators. The equation of motion for the heat currents and powercan be derived from first principles. Only a global description of the cou-pling of the device to the reservoirs is consistent with the first and secondlaws of thermodynamics. Optimization of the devices leads to a balancedset of parameters in which the couplings to the three reservoirs are of thesame order and the external driving field is in resonance. When analyzingrefrigerators, one needs to devote special attention to a dynamical versionof the third law of thermodynamics. Bounds on the rate of cooling whenTc → 0 are obtained by optimizing the cooling current. All refrigerators asTc → 0 show universal behavior. The dynamical version of the third lawimposes restrictions on the scaling as Tc → 0 of the relaxation rate γc andheat capacity cV of the cold bath.

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1. INTRODUCTION

Our cars, refrigerators, air conditioners, lasers, and power plants are all examples of heat engines.The trend toward miniaturization has not skipped the realm of heat engines, leading to devices onthe nano- or even on the atomic scale. Typically, in the practical world, all such devices operatefar from the maximum efficiency conditions set by Carnot (1). Real heat engines are optimized forpowers sacrificing efficiency. This trade-off between efficiency and power is the focus of the fieldof finite-time thermodynamics. The field was initiated by the seminal paper of Curzon & Ahlborn(2). From everyday experience, the irreversible phenomena that limit the optimal performanceof engines can be identified as losses due to friction, heat leaks, and heat transport (3). Is therea unifying fundamental explanation for these losses? Is it possible to trace the origin of thesephenomena to quantum mechanics? The tradition of thermodynamics is followed by the studyof hypothetical quantum heat engines and refrigerators to address these issues. Once understood,these models serve as a template for real devices.

Gedanken heat engines are an integral part of thermodynamical theory. In 1824, Carnot (1)set the stage by analyzing an ideal engine. His analysis preceded the systematic formulation thatled to the first and second laws of thermodynamics (4). Amazingly, thermodynamics was able tokeep its independent status despite the development of parallel theories dealing with the samesubject matter. Quantum mechanics overlaps with thermodynamics in that it describes the stateof matter. However, quantum mechanics additionally includes a comprehensive description ofdynamics. This suggests that quantum mechanics can originate a concrete interpretation of theword dynamics in thermodynamics, leading to a fundamental basis for finite-time thermodynamics.

Quantum thermodynamics is the study of thermodynamical processes within the context ofquantum dynamics. Historically, consistency with thermodynamics led to Planck’s law, the basicsof quantum theory. Following the ideas of Planck on blackbody radiation, Einstein (5) quantizedthe electromagnetic field in 1905. Einstein’s paper is the birth of quantum mechanics togetherwith quantum thermodynamics.

Quantum thermodynamics is devoted to unraveling the intimate connection between the lawsof thermodynamics and their quantum origin (6–34). The following questions come to mind: Howdo the laws of thermodynamics emerge from quantum mechanics? What are the requirementsof a theory to describe quantum mechanics and thermodynamics on a common ground? Whatare the fundamental reasons for a trade-off between power and efficiency? Do quantum devicesoperating far from equilibrium follow thermodynamical rules? Can quantum phenomena affectthe performance of heat engines and refrigerators?

We address these issues below by analyzing quantum models of heat engines and refrigerators.Extreme care has been taken to choose a model that can be analyzed from first principles. Weconsider two types of models: continuous operating models resembling turbines and discretefour-stroke reciprocating engines. The present review focuses on continuous devices.

2. THE CONTINUOUS ENGINE

An engine is a device that converts one form of energy to another: heat to work. In this conversion,part of the heat from the hot bath is ejected to the cold bath, limiting the efficiency of powergeneration. This is the essence of the second law of thermodynamics.

A continuous engine operates in an autonomous fashion, attaining a steady-state mode ofoperation. The analysis therefore requires an evaluation of steady-state energy currents. Theoperating part of the device is connected simultaneously to the hot, cold, and power output leads.The primary macroscopic example is a steam or gas turbine. The primary quantum heat engine is

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ν

Th

κ↑ Є2

Є1

ω10

ω20

Є0

κ↓

κ↑

κ↓

Tc

P

Figure 1The quantum three-level amplifier as a Carnot heat engine. The system is coupled to a hot bath withtemperature Th and to a cold bath with temperature Tc. The output P is a radiation field with frequency ν.(In the figure, � = 1.) Power is generated, provided there is population inversion between level ε2 andε1 : p2 > p1. The hot bath equilibrates levels ε0 and ε2 via the rates k ↑ and k ↓ such that k ↑ /k ↓ =exp(�ω20/kB Th ). The efficiency η = ν/ω20 ≤ 1 − Tc /Th . Reversing the direction of operation using powerto drive the population from level ε1 to ε2 generates a heat pump; then p2 < p1.

the laser. These devices share a universal aspect exemplified by the equivalence of the three-levellaser with the Carnot engine.

Reversing the operation of a heat engine generates a heat pump or a refrigerator. We reviewthe basic principles of quantum continuous engines and then invert their operation and studyquantum refrigerators.

2.1. The Quantum Three-Level System as a Heat Engine

A contemporary example of a Carnot engine is the three-level amplifier. The principle of oper-ation is to convert population inversion into output power in the form of light. Heat gradientsare employed to achieve this goal. Figure 1 shows its schematic construction. A hot reservoircharacterized by temperature Th induces transitions between the ground state ε0 and the excitedstate ε2. When equilibrium is reached, then the population ratio between these levels becomes

p2

p0= e− �ωh

kB Th ,

where ωh ≡ ω20 = (ε2 − ε0)/� is the Bohr frequency, and kB is the Boltzmann constant. The coldreservoir at temperature Tc couples level ε0 and level ε1, meaning that

p1

p0= e− �ωc

kB Tc ,

where ωc ≡ ω10 = (ε1 − ε0)/�. The amplifier operates by coupling the energy levels ε3 and ε2 tothe radiation field, generating an output frequency with an on resonance of ν = (ε3 − ε2)/�. Thenecessary condition for amplification is positive gain or population inversion, defined by

G = p2 − p1 ≥ 0. (1)

From this condition, by dividing by p0, we obtain e− �ωhkB Th − e− �ωc

kB Tc ≥ 0, which leads to

ωc

ωh≡ ω10

ω20≥ Tc

Th. (2)

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The efficiency of the amplifier is defined by the ratio of the output energy �ν to the energyextracted from the hot reservoir, �ω20:

ηo = ν

ω20= 1 − ωc

ωh. (3)

Equation 3 is termed the quantum Otto efficiency (23). Inserting the positive gain condition givenin Equations 1 and 2, one finds that the efficiency is limited by Carnot efficiency (1):

ηo ≤ ηc ≡ 1 − Tc

Th. (4)

This result connecting the efficiency of a quantum amplifier to the Carnot efficiency was firstobtained by Scuville and colleagues (6, 7).

The above description of the three-level amplifier is based on a static quasi-equilibrium view-point. Real engines that produce power operate far from equilibrium conditions. Typically, theirperformance is restricted by friction, heat transport, and heat leaks. A dynamical viewpoint istherefore the next required step.

2.2. The Quantum Tricycle

A quantum description enables one to incorporate dynamics into thermodynamics. The tricyclemodel is the template for almost all continuous engines (Figure 2). This model is employed toincorporate the quantum dynamics of the devices. Surprisingly, very simple models exhibit thesame features of engines generating finite power. Their efficiency at operating conditions is lowerthan the Carnot efficiency. In addition, heat leaks restrict performance, meaning that reversibleoperation is unattainable.

The tricycle engine has the generic structure displayed in Figure 2. The basic model consistsof three thermal baths: a hot bath with temperature Th, a cold bath with temperature Tc, and a

Tw

Tw TcTc

Th

Hs

JJc + Jh + P = 0

Jc Jh Jw

JwP

Jh

Th

Jc

≥ 0–––

Figure 2The quantum tricycle, a quantum device coupled simultaneously to a hot, cold, and power reservoir.A reversal of the heat currents constructs a quantum refrigerator.

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work bath with temperature Tw. Each bath is connected to the engine via a frequency filter, whichwe model by three oscillators:

HF = �ωh a†a + �ωc b†b + �ωw c†c, (5)

where ωh , ωc , and ωw are the filter frequencies on resonance ωw = ωh −ωc . The device operates asan engine by removing an excitation from the hot bath and generating excitations on the cold andwork reservoirs. In second quantization formalism, the Hamiltonian describing such an interactionbecomes

HI = �ε(ab†c† + a†bc), (6)

where ε is the coupling strength.The device operates as a refrigerator by removing an excitation from the cold bath as well as

from the work bath and generating an excitation in the hot bath. The term a†bc in the Hamiltonianof Equation 6 describes this action (see Section 3).

One can employ different types of heat baths, including bosonic baths comprising phonons orphotons and fermonic baths comprising electrons. From the continuous spectrum of the bath, thefrequency filters select the working component to be employed in the tricycle. These frequencyfilters can also be constructed from two-level systems or can be formulated as qubits (35–37).

The interaction term is strictly nonlinear, incorporating three heat currents simultaneously.This crucial fact has important consequences. A linear device cannot operate as a heat engineor refrigerator (38). A linear device is constructed from a network of harmonic oscillators withlinear connections of the type �μi j (ai a

†j + a†i a j ) with some connections to harmonic heat baths. In

such a device, the hottest bath always cools down and the coldest bath always heats up. Thus, thisconstruction can transport heat but not generate power because power is equivalent to transportingheat to an infinitely hot reservoir. Another flaw in a linear model is that the different bath modesdo not equilibrate with each other. A generic bath should equilibrate any system Hamiltonianirrespective of its frequency.

Many nonlinear interaction Hamiltonians of the type HI = A ⊗ B ⊗ C can lead to a workingheat engine. One can reduce these Hamiltonians to the form of Equation 6, which captures theessence of such interactions.

The first law of thermodynamics represents the energy balance of heat currents originatingfrom the three baths and collimating on the system:

d Es

d t= Jh + Jc + Jw. (7)

At steady state, no heat is accumulated in the tricycle; thus (d Es )/(dt) = 0. In addition, in steadystate, the entropy is generated only in the baths, leading to the second law of thermodynamics:

ddt

�Su = −Jh

Th− Jc

Tc− Jw

Tw

≥ 0. (8)

This version of the second law is a generalization of Clausius’s statement; heat does not flowspontaneously from cold to hot bodies (39). When the temperature Tw → ∞, no entropy isgenerated in the power bath. An energy current with no accompanying entropy production isequivalent to generating pure power: P = Jw, where P is the output power.

The evaluation of the currentsJ j in the tricycle model requires dynamical equations of motion.The major assumption is that the total system is closed, and its dynamics is generated by the globalHamiltonians:

H = H0 + HSB. (9)


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This Hamiltonian H0 includes the bare system and the heat baths:

H0 = Hs + HH + HC + HW , (10)

where the system Hamiltonian Hs = HI + Hh + Hc + Hw consists of three energy-filteringcomponents and an interaction part with an external field. The reservoir Hamiltonians are thehot HH , cold HC , and work HW . The system-bath interaction Hamiltonian, HSB, is equal toHs H + Hs C + Hs W .

A thermodynamical idealization assumes that the tricycle system and the baths are uncorrelated,meaning that the total state of the combined system becomes a tensor product at all times (31):

ρ = ρ s ⊗ ρH ⊗ ρC ⊗ ρW . (11)

Under these conditions, the dynamical equations of motion for the tricycle become

ddt

ρ s = Lρ s , (12)

where L is the Liouville superoperator described in terms of the system Hilbert space, where thereservoirs are described implicitly. Within the formalism of a quantum open system,L can take theform of the Gorini-Kossakowski-Sudarshan-Lindblad (GKS-L) Markovian generator (40, 41).

Thermodynamics is notorious for employing a very small number of variables. In equilibriumconditions, the knowledge of the Hamiltonian is sufficient. When deviating from equilibrium,additional observables are added. This suggests presenting the dynamical generator in Heisenbergform for arbitrary system operators Os :

ddt

Os = L∗Os = i�

[Hs , Os ] +∑

k

VkOs V†k − 1

2{VkV†

k, Os }. (13)

The operators Vk are system operators still to be determined. The task of evaluating the modifiedsystem Hamiltonian Hs and the operators Vk is made extremely difficult owing to the nonlinearinteraction in Equation 6. Any progress from this point requires a specific description of the heatbaths and approximations to deal with the nonlinear terms.

2.3. The Quantum Amplifier

The quantum amplifier is the most elementary quantum continuous heat engine converting heatto work. Its purpose is to generate power from a temperature difference between the hot and coldreservoirs. The output power is described by a time-dependent external field. The general deviceHamiltonian is therefore time dependent:

Hs (t) = H0 + V(t). (14)

When the system is coupled to the hot and cold baths, the Markovian master equation in Heisen-berg form for the system operator Os is

ddt

Os = i�

[H(t), Os (t)] + L∗h(Os (t)) + L∗

c (Os (t)). (15)

One can obtain the change in energy of the device replacing Os with Hs :

d Es

d t= 〈L∗

h(Hs )〉 + 〈L∗c (Hs )〉 +

⟨∂Hs

∂t

⟩. (16)

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Equation 16 can be interpreted as the time derivative of the first law of thermodynamics (8, 31,42, 43) based on the Markovian GKS-L generator. Power is identified as

P =⟨∂ H s

∂t

⟩(17)

and the heat current as

Jh = 〈L∗h(Hs )〉, Jc = 〈L∗

c (Hs )〉. (18)

The partition between the Hamiltonian and the dissipative part in the GKS-L generator is notunique (31). A unique derivation of the master equation based on the weak coupling limit leads todynamics consistent with the first and second laws of thermodynamics (44).

The template for the tricycle model is employed to describe the dynamics of the amplifier.The interaction Hamiltonian is modified to become

HI (t) = �ε(a b†e+iνt + a†be−iνt), (19)

where ν ≡ ωw is the frequency of the time-dependent driving field, and ε is the coupling amplitude.The modification of Equation 6 eliminates the nonlinearity; it amounts to replacing the operatorc and c† by a c-number. The amplifier output power becomes

P = �εν(〈ab†〉e+iνt − 〈a†b〉e−iνt). (20)

After the nonlinearity has been eliminated, one can derive the quantum master equation for theamplifier from first principles based on the weak system-bath coupling expansion. This approxi-mation is a thermodynamic idealization equivalent to an isothermal partition between the systemand baths (31).

The interaction with the baths is given by

HSB = λa (a + a†) ⊗ Rh + λb (b + b†) ⊗ Rc , (21)

where R are reservoir operators and λ is the small system-bath coupling parameter. A crucialstep has to be performed before this procedure is applied. The system Hamiltonian has to berediagonalized with the interaction before the system is coupled to the baths. This diagonaliza-tion modifies the frequencies of the system, resulting in a splitting of the filter frequencies. Theprediagonalization is crucial for the master equations to be consistent with the second law ofthermodynamics (9, 37, 45).

The operators in the diagonal form d+ = (a + b)/(√

2), d− = (a − b)/(√

2), are determinedfrom the interaction representation: a(t) = Us(t, 0)†aUs(t, 0) and b(t) = Us (t, 0)†bUs (t, 0). Themain ingredients of the derivation of the master equation can be seen in the SupplementalAppendix (follow the Supplemental Material link from the Annual Reviews home page athttp://www.annualreviews.org).

2.3.1. Solving the equations of motion for the engine. In thermodynamic tradition, the engineis well described by a small set of observables. They in turn are represented by operators definingthe heat currents in the engine. To exploit this property, we transform the Hamiltonian to theinteraction frame:

HI (t) = U†s (t, 0)H(t)Us (t, 0) = �

ωh + ωc

2W + �

ωh − ωc

2X + �εZ, (22)

where the operators are closed to commutation relations, forming the SU(2) Lie algebra:

W = (d†+d+ + d†

−d−), X = (d†+d−e i2εt + d†

−d+e−i2εt),

Y = i (d†+d−e i2εt − d†

−d+e−i2εt), Z = (d†+d+ − d†

−d−).


Supplemental Material

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http://www.annualreviews.org/doi/suppl/10.1146/annurev-physchem-040513-103724


The dynamical description of the engine is completely determined by the expectation values ofthe operators constituting the SU(2) algebra and can be found in the Supplemental Appendix.One finds that the commutator [Y, HI ] �= 0; therefore, Y is related to the nondiagonal elementsof the Hamiltonian that define the coherence between energy levels.

The solution of the equation of motion leads to the thermodynamical observables at steady-state conditions; the power and heat flows become

P = �εν〈Y〉 = −2�(ωh − ωc )ε2G1

4ε2 + κhκc

(κhκc

κh + κc

),

Jh =L†h(HI ) =

(�εG2

2+ 2�ωhε

2G1

4ε2 + κhκc

) (κhκc

κh + κc

),

Jc =L†c (HI ) = −

(�εG2

2+ 2�ωc ε

2G1

4ε2 + κhκc

) (κhκc

κh + κc

),

(23)

where the generalized gain is

G1 = (N +h + N −

h ) − (N +c + N −

c ),

G2 = (N +h − N −

h ) − (N +c − N −

c ),(24)

and

N ±h(c ) =

[exp

(�ω±

h(c )

kB Th(c )

)− 1

]−1

, (25)

with ω±h(c ) = ωh(c ) ± ε. The effective heat conductivities κh/c are defined in the Supplemental

Appendix.The first law of thermodynamics is satisfied such thatJh +Jc +P = 0, as well as the second law:

−Jh/Th − Jc /Tc ≥ 0. The power P is proportional to the expectation value of the coherence〈Y〉. As a consequence, additional pure dephasing originating from external noise will degradethe power. Such noise generated by a Gaussian random process is described by the generatorLD = −γ 2[HI , [HI , •]] (46). In the regime where the external driving amplitude is larger thanthe heat conductivity (ε2 � κhκc ), Equation 23 converges to the results of Reference 35 when thetricycle operates as a heat engine.

Optimal power is obtained when the heat conductances from the hot and cold baths are bal-anced, � ≡ κc = κh ; then the power and the heat flows from the reservoirs become

P = �εν〈Y〉 = −�νε2�G1

4ε2 + �2,

Jh =L†h(HI ) = �ε�G2

4+ �ωhε

2�G1

4ε2 + �2,

Jc =L†c (HI ) = −�ε�G2

4− �ωc ε

2�G1

4ε2 + �2.

(26)

Figure 3 shows the power as a function of the heat conductivity coefficient � and the coupling tothe external field ε. A clear global maximum is obtained for � = 2ε.

The efficiency of the amplifier is defined as η = −P/Jh . For the present model, it becomes

η = ν

ωh + (�2+ε2)G24|ε|G1

. (27)

In the limit of ε → 0 and � → 0, the efficiency approaches the Otto value, ηo = 1 − ωc /ωh , andfor zero gain, ωc /ωh = Tc /Th , we obtain the Carnot limit, ηc = 1 − Tc /Th .

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1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

0 0.05

Є

Γ

0.1 0.15

Figure 3The normalized power P/Pmax as a function of the coupling amplitude to the external field ε and the heatconductivity �. A clear maximum is obtained for � = 2ε.

For finite fixed �, the efficiency in the limit ε → 0 becomes

η = ν

ωh + �2(

Th (1−cosh(ωhTh

))−Tc(

1−cosh(

ωcTc

)))4Th Tc

(sinh

( ωhTh

)−sinh

(ωcTc

)+sinh

(ωcTc

− ωhTh

)). (28)

By examining Equation 28, one can see that the Carnot limit is unattainable. When the Ottoefficiency approaches the Carnot limit ωc /ωh = Tc /Th , the amplifier’s efficiency becomes zero.This comparison between the two limits can be seen in Figure 4.

The efficiency of Equation 27 can be either smaller or greater than the Otto efficiency de-pending on the sign of G2. In the low-temperature limit, G2 is less than zero; thus ηo ≤ η ≤ ηc .Increasing ε will increase the efficiency up to a critical point εcrit, at which both G1 → 0 andG2 → 0 because N +

c → 0 and N +h → 0 and N −

h ∼ N −c . At this point, the engine operates at the

Carnot limit, and all currents vanish, such that the process becomes isoentropic (Figure 5). In thehigh-temperature limit for harmonic oscillators, while increasing ε, the gain G2 will change signand become positive; thus η ≤ ηo (Figure 6).

2.3.2. Optimizing the amplifier’s performance. Further optimization for power is obtainedwhen the pumping rate � is optimized for maximum power; then �max = 2ε, and the powerbecomes

P = −12

�ν|ε|G1. (29)

At the limit of high temperature and small ε, the gain becomes G1 ≈ (KB Th)/(�ωh)−(KB Tc )/(�ωc ).Then optimizing the power given in Equation 29 with respect to the output frequency ν leads to

ωc

ωh=

√Tc

Th,


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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

η/η c

P/Pmax

Figure 4The normalized efficiency η/ηc versus the normalized power P/Pmax for fixed ε, �, and ωc while varying ωh.The blue line is for finite �, and the dashed red line is optimized at each point � = 2ε. In this case, ε � 1.

0 1 2 3−5

0

5

10 × 10−7Currents

0 1 2 3

0.4

0.5

0.6

0.7Efficiency

0 1 2 30

1

2

3

4 × 10−7Entropy production

0 1 2 3−2

−1

0

1

2 × 10−5G1 and G2

Otto

G1

G2

Carnot

P

Jh

Jc

Figure 5(a) The heat Jc and Jh and power P currents as a function of ε. (b) Efficiency η as a function of ε. (c) Entropyproduction as a function of ε. (d ) The gain G1 and G2 as a function of ε. The harmonic tricycle engineoperates in the limit of low temperature. kB Tc = 0.1 and kB Th = 0.3, �ωc = 4 and �ωh = 6, and � = 0.05.

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Currents Efficiency

Entropy production G1 and G2

0 1 2 3−4

−2

0

2

4

0 1 2 3

0.2

0.4

0.6

0 1 2 30

0.005

0.01

0.015

0 1 2 3

0

20

40

60

G1

G2

P

Jh

JcOtto

Carnot

Figure 6(a) The heat Jc and Jh and power P currents as a function of ε. (b) Efficiency η as a function of ε. (c) Entropyproduction as a function of ε. (d ) The gain G1 and G2 as a function of ε. The harmonic tricycle engineoperates in the limit of high temperature. kB Tc = 100 and kB Th = 300, �ωc = 4 and �ωh = 6, and � = 0.05.

resulting in the efficiency at maximum power:

ηCA = 1 −√

Tc

Th, (30)

which is the well-established endoreversible result of Curzon & Ahlborn (2, 8). The optimal poweris obtained when all the characteristic parameters are balanced: ε ∼ � ∼ κc ∼ κh . Figure 7 showsa trajectory of efficiency with respect to power with changing field coupling strength ε for differentfrequency ratios ωc /ωh = (Tc /Th)α . The power vanishes with the coupling ε = 0, and then whenε = εcrit, the splitting in the dressed energy levels nulls the gain.

2.4. Dynamical Model of a Three-Level Engine

The dynamical description of the three-level engine is closely related to the tricycle model (9). Themodel is a template for the three-level laser shown in Figure 1 with the inclusion of a dynamicaldescription (47, 48).

The Hamiltonian of the device has the form

Hs = H0s + V(t) =

⎛⎜⎝ε0 0 0

0 ε1 εe iνt

0 εe−iνt ε2

⎞⎟⎠ , (31)

where H0s = ∑

εi Pi i , Pi j = |i〉〈 j |, and V(t) = ε(P12e iνt + P21e−iνt). The state of the three-level system is fully characterized by the expectation values of any eight independent operators,excluding the identity operator.


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0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

P/Pmax−CAη/η c

Figure 7The normalized efficiency η/ηc versus the normalized power P/Pmax for different optimal values of � = 2ε.Pmax is obtained for the Curzon-Ahlborn ratio ωc /ωh = √

Tc /Th shown in blue. The orange line is forωc /ωh = (Tc /Th )1/4, and the green line is for ωc /ωh = (Tc /Th )3/4. The red line is for the Carnot ratioωc /ωh ∼ Tc /Th , where the power is multiplied by 103.

The final equations of motion for these observables are given in the Supplemental Ap-pendix. The expressions for the power and efficiency become up to numerical factors identical tothe expressions obtained for the driven quantum tricycle. Figure 8 shows a schematic view of thesplitting of the energy levels of the three-level system owing to the driving field. As a result, theengine also splits into two parts, the upper and lower manifolds.

One can evaluate the steady-state power and heat flows by solving the generalized Lambequations. The results are represented by Equation 23 in which the definition of the populationis changed from that of a harmonic oscillator to a two-level system: N ±

h(c ) = 1/[eβh(c )ω±h(c ) + 1].

The numerator of the expressions for the power and heat flows consists of a sum of twocontributions: One, proportional to the gain N +

h −N +c , is associated with the population inversion

ν

Th

κ↑–

κ↑–

κ↑+

κ↑+

+Є

–Є

κ↓κ↓

Tc

P

ω10

ω20

ν

+Є

–Є

Figure 8The three-level amplifier. The interaction with the external field splits the two upper levels. As a result, theheat transport terms are modified. Two parallel engines emerge, which can operate in opposite directions,producing zero power for a certain choice of parameters.

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in the upper manifold, whereas the other, proportional to the gain N −h − N −

c , is associated withthe population inversion in the lower manifold.

At low temperatures, the three-level amplifier model is equivalent to that of the harmonictricycle. At the high-temperature limit, the gain G2 is positive for all driving conditions ε. Themaximum power point is the result of competition between the upper and lower manifolds. As ε

increases, N +h − N +

c increases, whereas N −h − N −

c decreases. Hence, the power production ofthe upper manifold increases (i.e., becomes more negative), whereas that of the lower manifolddecreases (i.e., becomes less negative).

From a thermodynamic point of view, each manifold is associated with a separate heat engine.As the coupling with the work reservoir (ε) increases, the engine associated with the upper manifoldoperates faster, while that associated with the lower one operates slower. In addition, the energyis leaking from the former to the latter, thereby diminishing the net power production.

Not only does the power decrease as a function of ε, it also changes sign at a cer-tain finite value of the field amplitude, denoted by εcrit. This results from the fact that atsome point, the lower manifold starts operating backward as a heat pump, rather than aheat engine. At ε = εcrit, the power consumption by the lower manifold is exactly bal-anced by the power production of the upper manifold such that the net power production iszero.

An examination of the steady-state heat fluxes reveals that they do not vanish at zero-poweroperating conditions. The entropy generated exclusively by the heat leak from the upper to lowermanifold is obtained by eliminating the power in Equation 26:

ddt

�S l = �ε�G2

4

(1

Tc− 1

Th

)≥ 0. (32)

Zero-power operating conditions correspond to the short-circuit limit. Heat is effectively trans-ferred 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 �ε/kB � Tc , Th . For example, thislimit is realistic in the optical domain where ν � Tc , Th . In such a case, N +

h − N +c is negligible

relative to N −h − N −

c , and one needs to account for only the lower manifold.The value of εcrit in this limit, denoted by εlT

crit, is that for which N −h − N −

c = 0. It is generallygiven by

εlTcrit = Thωc − Tc ωh

Th − Tc. (33)

The condition Thωc − Tc ωh > 0 is necessary and sufficient for lasing in the limit of weak drivingfields. Population inversion becomes increasingly more difficult in this manifold as the field in-tensifies. It is therefore no longer a sufficient condition for lasing in intense driving fields. Finally,zero-power operating conditions in the low-temperature limit, obtained at ε = εlT

crit, asymptoticallyimply zero heat flows and hence the reversible limit. Indeed, substituting εlT

crit from Equation 33for ε reduces to the Carnot efficiency, ηc = 1 − Tc /Th .

The difference between a three-level amplifier and the tricycle with harmonic oscillator filtersat the high-temperature limit is observed in Figure 9. The source of the difference is the saturationof finite levels. The efficiency at maximum power is lower than the Curzon-Ahlborn efficiency,approximated by

ηpmax ≈ ηc −√

kB Tc

�ωhη

12c . (34)


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P/Pmaxη/η c

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 HOTLS

Figure 9The normalized efficiency η/ηc versus the normalized power P/Pmax for ωc /ωh ∼ Tc /Th for varying� = 2ε. The high-temperature limit is shown where �ω � kb T . The red line is the three-level amplifier,and the blue line is the harmonic tricycle engine. Saturation limits the performance of the three-level engine.Abbreviations: HO, harmonic oscillator; TLS, two-level system.

2.5. The Four-Level Engine and Two-Level Engine

The four-level engine is a dynamical model of the four-level laser (49). In the four-level engine,the pumping excitation step is isolated from the coupling to the external field by employing anadditional cold reservoir. Figure 10 shows a schematic view of the engine’s structure. Analyzing

κ↑

κ↓

Th P

Tc

Tc

κ↑–

κ↑+

κ↓

κ↑–κ↑+

κ↓

+Є

–Єω10

ω32+Є

–Є

ω30

ν

ν

Figure 10The four-level engine. The interaction with the external field splits the two upper levels. As a result, the heattransport terms are modified. Two parallel engines emerge, which can operate in opposite directions,producing zero power for a certain choice of parameters. Coherence between levels 0 and 1 can enhance theperformance by synchronizing the engines. ωh = ω30, ωc 1 = ω32, and ωc 2 = ω10.

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a static viewpoint, one can see that positive gain is obtained when

G = p3 − p2 = p0

(e− �ωh

K B Th e�ωc 2kB Tc − e− �ωc 1

kB Tc

)≥ 0, (35)

where ωh = ω30, ωc 1 = ω32, and ωc 2 = ω10. The optimal output frequency at resonance isν = ωh − ωc 1 − ωc 2; therefore, the efficiency becomes ηo = ν/ωh . If the additional cold reservoiris assumed to have temperature Tc, then the Carnot restriction is obtained: ηo ≤ ηc .

A dynamical analysis reveals a splitting of the energy levels driven by the external field, leadingto a similar performance as the three-level amplifier. The advantage of the four-level engine isthat the hot bath thermal pumping is isolated from the coupling to the power extraction, thusreplacing the decoherence associated with the hot bath with a quieter operation associated withthe cold bath. In addition, one can optimize performance by balancing the rates between the upperand lower manifold by ωc 1 = ωc 2.

The four-level engine has been employed to study the influence of initial coherence on theengine’s performance (50–56). In this case, the power output is connected to the two upperlevels, and the coherence is generated by splitting the ground state. The basic idea is that theinitial coherence present between energy levels associated with the entangling operator Y can beexploited to generate additional power, even from a single bath. There is no violation of the secondlaw because the initial coherence reduces the initial entropy. Contact with a single bath will increasethis entropy. A unitary manipulation can then exploit this entropy difference to generate work.

An extreme exploitation of the dynamical splitting of the energy levels owing to the externaldriving field results in the two-level engine. Supplemental Figure 1 presents a schematic opera-tion of the 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, and a model ofsuch an engine has been worked out (57, 58). The maximum efficiency of such an engine becomesη ≤ 1/2 ≤ ηc as two-pump steps are required for one output step.

2.6. Power Storage

An integral part of an engine is a device that allows one to store the power and retrieve it ondemand, a flywheel. A model based on the tricycle of such a device just disconnects the power bathand stores the energy in the power oscillator, ωw c†c. Once there is positive gain generated by thehot and cold baths, energy will flow to this oscillator. A model based on the three-level amplifierused this approach (21, 34, 59). The drawback of these studies is that they use a local descriptionof the master equation. When the flywheel oscillator has a large amplitude, it will modify thesystem-bath coupling in analogy to the dressed state picture (9).

Other studies also considered coupling to a cavity mode (54, 56, 60). The issue to be addressed isthe amount of energy that can be stored and then extracted. Because the storage mode is entangledwith the rest of the engine, this issue is delicate (60). The amount of possible work to be extractedis defined by an entropy-preserving unitary transformation to a passive state (61).

3. CONTINUOUS REFRIGERATORS

In a nutshell, refrigerators are just inverted heat engines. They employ power to pump heat froma cold to a hot bath. As with heat engines, the first and second laws of thermodynamics imposethe same restrictions on the refrigerator’s performance. The distinction between the static anddynamical viewpoints also applies. The key elements in a refrigerator are entropy extraction anddisposal. This entropy disposal problem is enhanced at low temperatures. The unique feature ofrefrigerators is therefore the third law of thermodynamics.



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3.1. The Third Law of Thermodynamics

Two independent formulations of the third law of thermodynamics have been presented, bothoriginally stated by Nernst (62–64). The first is a purely static (equilibrium) one, also known as theNernst heat theorem, phrased as follows: The entropy of any pure substance in thermodynamicequilibrium approaches zero as the temperature approaches zero. The second formulation isknown as the unattainability principle: It is impossible by any procedure, no matter how idealized,to reduce any assembly to absolute zero temperature in a finite number of operations (64, 65).

There is an ongoing debate on the relations between the two formulations and their relationto the second law, regarding which, and if at all, one of these formulations implies the other (4,66–68). Quantum considerations can illuminate these issues. The tricycle model is the templateused to analyze refrigerator performance as Tc → 0.

We first analyze the implications of the Nernst heat theorem and its relations to the secondlaw of thermodynamics. At steady state, the second law implies that the total entropy productionis non-negative (see Equation 8):

ddt

�Su = −Jc

Tc− Jh

Th− Jw

Tw

≥ 0.

This behavior can be quantified by a scaling exponent; as Tc → 0, the cold current should scalewith temperature as

Jc ∝ T 1+αc ,

with an exponent α.When the cold bath approaches the absolute zero temperature, it is necessary to eliminate the

entropy production divergence at the cold side; when Tc → 0, the entropy production scales as

�Sc ∼ −T αc , α ≥ 0. (36)

For the case in which α = 0, the fulfillment of the second law depends on the entropy productionof the other baths, −Jh/Th − Jw/Tw > 0, which should compensate for the negative entropyproduction of the cold bath. The first formulation of the third law modifies this restriction. Insteadof α ≥ 0, the third law imposes α > 0, guaranteeing that at absolute zero the entropy productionat the cold bath is zero: �Sc = 0. The Nernst heat theorem then leads to the scaling condition ofthe heat current with temperature (11):

Jc ∼ T α+1c and α > 0. (37)

We now examine the unattainability principle. Quantum mechanics enables a dynamical in-terpretation of the third law, modifying the definition to the following: No refrigerator can cool asystem to absolute zero temperature at finite time. This form of the third law is more restrictive,imposing limitations on the system-bath interaction and the cold bath properties when Tc → 0(35). To quantify the unattainability principle, one finds that the rate of temperature decrease ofthe cooling process should vanish according to the characteristic exponent ζ :

d Tc (t)dt

= −c T ζc , Tc → 0, (38)

where c is a positive constant. Solving Equation 38, one obtains

Tc (t)1−ζ = Tc (0)1−ζ − c t, for ζ < 1,

Tc (t) = Tc (0)e−c t, for ζ = 1,1

Tc (t)ζ−1= 1

Tc (0)ζ−1+ c t, for ζ > 1.

(39)

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From Equation 39, it is apparent that the cold bath can be cooled to zero temperature at finitetime for ζ < 1. The third law therefore requires ζ ≥ 1. The two third-law scaling exponents canbe related by accounting for the heat capacity c V (Tc ) of the cold bath:

Jc (Tc (t)) = −c V (Tc (t))d Tc (t)

dt, (40)

where c V (Tc ) is determined by the behavior of the degrees of freedom of the cold bath at lowtemperatures at which c V ∼ T η

c when Tc → 0. Therefore, the scaling exponents are related,ζ = 1 + α − η (31).

The third-law scaling relations given in Equations 37 and 40 can be used as an independentcheck for quantum refrigerator models (69). Violation of these relations points to flaws in thequantum model of the device, typically in the derivation of the master equation.

3.2. The Quantum Power–Driven Refrigerator

Laser cooling is a crucial technology for realizing quantum devices. When the temperature isdecreased, degrees of freedom freeze out, and systems reveal their quantum character. Inspiredby the mechanism of solid state lasers and their analogy with Carnot engines (6), investigatorsrealized that inverting the operation of the laser at the proper conditions will lead to refrigeration(70–73). A few years later, a different approach for laser cooling was initiated based on the Dopplershift (74, 75). In this scheme, translational degrees of freedom of atoms or ions were cooled bylaser light detuned to the red of the atomic transition. Unfortunately, the link to thermodynamicswas forgotten. A flurry of activity then followed with the realization that the actual temperaturesachieved were below the Doppler limit, kB TDoppler = hγ /2, where γ is the natural line width(76–78).

Elaborate quantum theories were devised to unravel the discrepancy. The recoil limit is basedon the assumption that the photon momentum is in quasi-equilibrium with the momentum of theatoms, p = �k. As a result, the temperature is related to the average kinetic energy:

kB T recoil = �2k2

2M, (41)

where �k is the photon momentum and M the atomic mass of the particle being cooled. Forsodium, the Doppler limit is TDoppler = 235 μK and the recoil limit T recoil = 2.39 μK (79).

Thermodynamically, these limits do not bind; the only hard limit is absolute zero, Tc = 0.To approach this limit, one has to optimize the cooling power to match the cold bath temper-ature. Unoptimized refrigerators become restricted by a minimum temperature above absolutezero.

A reexamination of the elementary three-level model stresses this point. By examining theheat engine model of Figure 1, one finds that if the power direction is reversed, a refrigerator isgenerated, provided the gain is negative: G = p2− p1 < 0. Assuming quasi-equilibrium conditions,this leads to

ωc

ωh= ω10

ω20≤ Tc

Th, (42)

which translates to a minimum temperature of Tc (min) ≥ (ωc /ωh)Th . We can consider typicalvalues for laser cooling of sodium based on the D line of 589.592 nm, which translates to ωh =508.838×1013 Hz. If the translational cold bath is coupled to the hyperfine structure splitting, thenone finds that F2 → F1 of the ground state 3S1/2, meaning ωc = 1.7716 × 1010 Hz. The coolingratio becomes ωc /ωh = 3.48 × 10−6, and the minimum temperature, assuming room temperatureof the hot bath, is Tc (min) ∼ 1.510−3 K. With the use of a magnetic field, the hyperfine levels F1


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split in three, leading to further splittings in the megahertz limit, increasing the cooling ratio bythree orders of magnitude. When the cooling is in progress, one can reduce the hyperfine splittingby changing the magnetic field such that it follows the cold bath temperature. In principle, anytemperature above Tc = 0 is reachable.

3.3. Dynamical Refrigerator Models

A quantum dynamical framework of the refrigerator is based on the tricycle template. The workbath is replaced by a time-dependent driving term, and the cooling current can be calculatedanalytically. One can convert the dynamical equations of the heat engine (Equation 23) to thoseof a refrigerator by inverting the sign of all heat currents. Consequently, one finds that the gainG1 ≤ 0 (Equation 24). For example, when ε > κ , the cooling current becomes (35)

Jc = 12

(�ω−

c�−

c �−h

�−c + �−

h(N −

c − N −h ) + �ω+

c�+

c �+h

�+c + �+

h(N +

c − N +h )

). (43)

The phenomenon of the refrigerator splitting into two parts is also found here. In a similarfashion, the three-level amplifier can be reversed to become a refrigerator (9). The heat currentsare identical to Equation 23 with the two-level-system definition of N ±

c /h .In a refrigerator, the object of optimization is the cooling power Jc compared to the output

power P in an engine. The efficiency is defined by the coefficient of performance (COP), the ratiobetween Jc and the input power P :

COP = Jc

P ≤ COPo ≤ COPc , (44)

where COPo = ωc /ν and COPc = Tc /(Th − Tc ). All power-driven refrigerators are restricted bythe Otto COPo . In Section 3.5, we analyze the performance of power-driven refrigerators at lowtemperatures.

3.4. The Quantum Absorption Refrigerator

An absorption chiller is a refrigerator that employs a heat source to replace mechanical work fordriving a heat pump (80). The first device, developed in 1850 by the Carre brothers, becamethe first useful refrigerator. In 1926, Einstein & Szilard (81) invented an absorption refrigeratorwith no moving parts. This invention is an inspiration for miniaturizing the device to solve thegrowing problem of heat generated in integrated circuits. Even a more challenging proposalis to miniaturize to a few-level quantum system. Such a device could be incorporated into aquantum circuit. The first quantum version was based on the three-level refrigerator (12) drivenby thermal noise. A more recent model of an autonomous quantum absorption refrigerator withno external intervention based on three qubits (36) has renewed interest in such devices. A setup ofoptomechanical refrigerators powered by incoherent thermal light was introduced in Reference82. The study showed that cooling increases while the photon number increases up to the pointthat the fluctuation of the radiation pressure becomes dominant and heats the mechanical mode.Studies of cooling based on electron tunneling have also been considered. The working mediumcomprises quantum dots (83, 84) or a normal metal insulator superconductor junction (85). Heatis removed from the reservoir by electron transport induced by thermal or shot noise.

3.4.1. The three-level absorption refrigerator. In the three-level absorption refrigerator, thepower source has a finite temperature Tw. The coefficient of performance becomes COP = Jc /Jw.

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From the second and first laws of thermodynamics (Equations 7 and 8) using steady-state condi-tions, one obtains

COP ≤ (Tw − Th)Tc

(Th − Tc )Tw

, (45)

and Tw > Th > Tc (80). Under the assumption that the three baths are uncorrelated, the three-level state is diagonal in the energy representation and determined by the heat conductivities(12, 86). From these values, one can evaluate the heat currents. In this refrigerator, no internalcoherence is required for operation.

3.4.2. Tricycle absorption refrigerator. The template to understand the absorption refrigeratoris the tricycle model, a refrigerator connected to three reservoirs with the Hamiltonians given inEquations 5 and 6. A thermodynamical consistent description requires that one first diagonalizesthe Hamiltonian and then obtains the generalized master equations for each reservoir. The finalstep is to obtain the steady-state heat currents Jc , Jh , and Jw. The nonlinearity in Equation 6hampers this task.

One remedy to this issue is to replace the harmonic oscillators in the model by qubits (37).The other approach is to consider the high-temperature limit of the power reservoir or a noise-driven refrigerator (35, 87). This three-qubit refrigerator model was introduced as the ultimateminiaturization model (36, 88, 89). The free Hamiltonian of the device has the form

HF = �ωhσhz + �ωc σ

cz + �νσ

w

z , (46)

and the interaction Hamiltonian is

HI = �ε(σ

h+ ⊗ σ

c− ⊗ σ

w

− + σh− ⊗ σ

c+ ⊗ σ

w

+)

, (47)

where σ are two-level operators. The diagonal energy levels are described in the SupplementalAppendix.

Of major importance is the distinction between the complete nonlocal description of thetricycle and a local description. In the local case, the dissipative terms Lh/c are set to equilibrateonly the local qubit. In the nonlocal case, the local qubit is mixed with the other qubits owingto the nonlinear coupling. As a result, in the nonlocal approach it is impossible to reach theCarnot coefficient of performance, COPc , regardless of how small the internal coupling ε is. Thiscan be observed in Figure 11, which shows the coefficient of performance versus the coolingcurrent Jc for different coupling strengths, ε. Conversely, COPc is reachable in the local model.Interestingly, a universal limit for the coefficient of performance at the maximum cooling powerwas found applicable for all absorption refrigerator modes of COPc ≤ (d/d + 1)COPc , where d isthe dimensionality of the phonon cold bath (86).

3.4.3. Noise-driven refrigerator. An ideal power source generates zero entropy: �Sw =−Jw/Tw. A pure mechanical external field achieves this goal. Another possibility is a thermalsource; when Tw → ∞, it also generates zero entropy. An unusual power source is noise. Gaus-sian white noise also carries with it zero entropy production. Analysis demonstrates that theperformance of refrigerators driven by all these power sources is very similar when approachingthe absolute zero temperature (87).

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 interaction nonlinearterm in Equation 6 is replaced with

Hint = f (t)(a†b + ab†) = f (t)X, (48)



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1

0.8

0.6

0.4

0.2

00

Jc /J*c

Є = 0.01Є = 0.03

Є = 0.06

Є = 0.09

5.0 × 10–7 1.0 × 10–6 1.5 × 10–6 2.0 × 10–6

COP/CO

P c

Figure 11The coefficient of performance, COP/COPc, of the three-qubit refrigerator versus the normalized coldcurrent Jc /J ∗

c for different values of the internal coupling constant, ε. The coefficient of performance atmaximum power is found to be bound by 3/4COPc, independent of ε. Figure courtesy of L.A. Correa andJ.P. Palao.

where f (t) is the noise field; X = (a†b + ab†) is the generator of a swap operation between the

two oscillators and is part of a set of SU(2) operators, Y = i (a†b − ab†), Z = (a†a − b†b), and the

Casimir N = (a†a + b†b).A Gaussian source of white noise is characterized by zero mean 〈 f (t)〉 = 0 and delta time

correlation 〈 f (t) f (t′)〉 = 2ηδ(t − t′). The Heisenberg equation for a time-independent operatorO is reduced to

ddt

O = i [Hs , O] + L∗n(O) + L∗

h(O) + L∗c (O), (49)

where Hs = �ωh a†a + �ωc b†b. The noise dissipator for Gaussian noise is L∗n(O) = −η[X, [X, O]]

(46). The bath relaxation generators Lh/c are the standard local energy relaxation terms (see theSupplemental Appendix) (90).

The equations of motion including the dissipative part are closed to the SU(2) set of operators.To derive the cooling current Jc = 〈Lc (�ωc b†b)〉, we solve for stationary solutions of N and Z.The cooling current becomes

Jc = �ωc2η�

2η + �(N c − N h), (50)

where the effective heat conductance is � = (�c �h)/(�c + �h). Cooling occurs for N c > N h ⇒ωh/Th > ωc /Tc . The coefficient of performance for the absorption chiller is defined by the relationCOP = Jc /Jn; with the help of Equation 50, we obtain the Otto COPo (23) (see Equation 44).

We now show the equivalence of the noise-driven refrigerator with the high-temperature limitof the work bath TW . Based on the weak coupling limit, the dissipative generator of the power

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bath becomes

L∗w(O) = �w(N w + 1)

(a†bOb

†a − 1

2{a†abb

†, O}

)

+ �w N w

(ab

†Oa

†b − 1

2{aa†b†b, O}

),

(51)

where N w = (exp((�ωw)/(kTh)) − 1)−1. At finite temperature, Lw(O) does not lead to a close setof equations. But in the limit of Tw → ∞, it becomes equivalent to the Gaussian noise generator:L∗

w(O) = −η/2([X, [X, O]] + [Y, [Y, O]]), where η = �w N w. This noise generator leads to thesame current Jc and coefficient of performance as in Equations 44 and 50. We conclude thatGaussian noise represents the singular bath limit equivalent to Tw → ∞.

One can employ Poisson white noise as a power source. This noise is typically generated bya sequence of independent random pulses with exponential interarrival times (91, 92). Theseimpulses drive the coupling between the oscillators in contact with the hot and cold baths, leadingto

dOdt = (i/�)[H, O] − (i/�)λ〈ξ〉[X, O]

+ λ(∫ ∞

−∞ dξ P (ξ )e (i/�)ξXOe (−i/�)ξX − O)

,(52)

where H is the total Hamiltonian including the baths, λ is the rate of events, and ξ is the impulsestrength averaged over a distribution P (ξ ). Using the Hadamard lemma and the fact that theoperators form a closed SU(2) algebra, we can separate the noise contribution into its unitary anddissipation parts, leading to the master equation

dOdt

= (i/�)[H, O] + (i/�)[H′, O] + L∗n(O). (53)

The unitary part is generated with the addition of the Hamiltonian H′ = �ε X and with theinteraction

ε = −λ

2

∫dξ P (ξ )(2ξ/� − sin(2ξ/�)). (54)

This term can cause a direct heat leak from the hot to the cold bath. The noise generator Ln(ρ)can be reduced to the form L∗

n(O) = −η[X, [X, O]], with a modified noise parameter:

η = λ

4

(1 −

∫dξ P (ξ ) cos(2ξ/�)

). (55)

The Poisson noise generates an effective Hamiltonian comprising H and H′, modifying the energylevels of the working medium. One must incorporate this new Hamiltonian structure into thederivation of the master equation; otherwise, the second law will be violated. The first step isto rewrite the system Hamiltonian in its dressed form (93). A new set of bosonic operators isdefined

A1 = a cos(θ ) + b sin(θ ),A2 = b cos(θ ) − a sin(θ ).

(56)

The dressed Hamiltonian is given by

Hs = ��+A†1A1 + ��−A†

2A2, (57)

where �± = (ωh +ωc )/2±√

((ωh − ωc )/2)2 + ε2 and cos2(θ ) = (ωh −�−)/(�+ −�−). Equation 57imposes the restriction �± > 0, which can be translated to ωhωc > ε2. The master equation inthe Heisenberg representation becomes

dOdt

= (i/�)[Hs , O] + L∗h(O) + L∗

c (O) + L∗n(O), (58)


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where the details can be found in Reference 35. The noise generator becomes

L∗n(O) = −η[W, [W, O]], (59)

where W = sin(2θ )Z + cos(2θ )X. Again, the SU(2) algebra is employed to define the operatorsX = (A†

1A2 +A†2A1), Y = i (A†

1A2 −A†2A1), and Z = (A†

1A1 −A†2A2). The total number of excitations

is accounted for by the operator N = (A†1A1 + A†

2A2). Once the set of linear equations is solved,one can extract the exact expressions for the heat currents: Jh = 〈L∗

h(Hs )〉, Jc = 〈L∗c (Hs )〉, and

Jn = 〈L∗n(Hs )〉.

The distribution of impulse determines the performance of the refrigerator. For a normal

distribution of impulses in Equation 52, P (ξ ) = (1/√

2πσ 2)e− (ξ−ξ0)2

2σ 2 . The energy shift is controlledby

ε = −λ

2

(2ξ0/� − e− 2σ 2

�2 sin(2ξ0/�))

. (60)

The effective noise strength becomes

η = λ

4

(1 − e− 2σ 2

�2 cos(2ξ0/�))

. (61)

In the limit of σ → 0, one finds that P (ξ ) = δ(ξ − ξ0). Another possibility is an exponential

distribution: P (ξ ) = (1/ξ0)e− ξ

ξ0 . Then

ε = −λ(ξ0/�)3

4(1 + (ξ0/�)2)(62)

and

η = λ(ξ0/�)2

1 + 4(ξ0/�)2. (63)

The Poissonian noise plays two opposing roles. On the one hand, it increases the cooling currentJc , by increasing η. On the other hand, it decreases ε (becomes more negative) and by thatdecreases Jc . Both parameters η and ε depend linearly on λ, which can be interpreted as the rateof photon absorption in the system that enhances the cooling process. Supplemental Figure 2shows that Jc increases with λ until a point at which ε dominates and Jc decreases.

The coefficient of performance for the Poisson-driven refrigerator is restricted by the Ottoand Carnot coefficients of performance:

COP = �−�+ − �−

≤ ωc

ωh − ωc≤ Tc

Th − Tc. (64)

3.5. Refrigerators Operating as Tc → 0 and the Third Law of Thermodynamics

The performance of all types of refrigerators at low temperature has universal properties. Inpower-driven refrigerators, the lower split manifold is dominant, leading to the cold current

Jc ≈ �ω−c

2ε2�

4ε2 + �c �h· G, (65)

where the gain G = N −c − N −

h and � = (�c �h)/(�c + �h).In the three-level absorption refrigerator, the cold current becomes

Jc = �ωc�c �h�w

�· G, (66)

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where � is a combination of kinetic constants and G = e− �ωw

kB Tw e− �ωckB Tc − e− �ωh

kB Th (86).In the Gaussian noise-driven refrigerator, Equation 50 becomes

Jc = �ωc2η�

2η + �· G, (67)

where G = N c − N h .In the Poisson-driven refrigerator, we obtain

Jc ≈ ��−2η�

2η + �· G, (68)

with G = (N −c − N +

h ) and �− ≈ ωc − ε2/(ωh − ωc ). All expressions for the cooling current Jc area product of an energy quant �ωc , the effective heat conductance, and the gain G.

To approach the absolute zero temperature, we require a further optimization. Optimizing the

gain G is obtained when ωh → ∞, which leads to G ∼ e− �ωckB Tc . In addition, the driving amplitude

should be balanced with the heat conductivity. One can also express the cooling current in terms of

the relaxation rate to the bath via the relation γ (ω)e− �ω

kB T = �(ω)N (ω). The universal optimizedcooling current as Tc → 0 becomes

Jc = �ωc · γc · e− �ωckB Tc . (69)

This form is correct both in the weak coupling limit and in the low density limit. One can interpretthe heat currentJc as the quant of energy �ωc extracted from the cold bath at the rate γ c multipliedby a Boltzmann factor. Further optimization with respect to ωc is dominated by the exponentialBoltzmann factor (optimizing the function xa e−x/b leads to x∗ ∝ b). As a result, one finds thatω∗

c ∝ Tc , obtaining Jc ∝ ω∗c · γc (ω∗

c ). The linear relation between the optimal frequency ωc andTc allows one to translate the temperature scaling relations to the low-frequency scaling relationsof γc (ω) ∼ ωμ and c V (ω) ∼ ωη when ω → 0.

To fulfill the Nernst heat theorem (Equation 37), one restricts the scaling of the relaxation rateto γc (ω) ∼ ωα and α > 0. The fulfillment of the unattainability principle (Equation 40) dependson the ratio between the relaxation rate and the heat capacity γc /c V ∼ ωζ−1, where ζ > 1.

We now examine the low-frequency properties of the relaxation rate γ c and the heat ca-pacity cV . For three-dimensional ideal degenerate Bose gases, cV scales as T 3/2

c . For degenerateFermi gases, cV scales as Tc. In both cases, the fraction of the gas that can be cooled decreaseswith temperature. Based on a collision model, when cooling occurs owing to inelastic scatter-ing, we have found the scaling exponent ζ = 3/2 for both degenerate Bose and Fermi gases(35).

The most studied generic bath is the harmonic bath. It includes the electromagnetic field(a photon bath), a macroscopic piece of solid (a phonon bath), or Bogoliubov excitations in aBose-Einstein condensate. The standard form of the bath’s Hamiltonian is

Hint = (b + b†)

(∑k

(g(k)a(k) + g(k)a†(k))

), HB =

∑k

ω(k)a†(k)a(k), (70)

where a(k) and a†(k) are the annihilation and creation operators for a mode k. For this model, theweak coupling limit procedure leads to the GKS-L generator with the cold bath relaxation rategiven by (35)

γc ≡ γc (ωc ) = π∑

k

|g(k)|2δ(ω(k) − ωc )[

1 − e− �ω(k)kB Tc

]−1

. (71)


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For the bosonic field in d-dimensional space, with the linear low-frequency dispersion law(ω(k) ∼ |k|), one obtains the following scaling properties for the cooling rate at low frequen-cies: γc ∼ ωκ

c ωd−1c , where ωκ

c represents the scaling of the form factor |g(ω)|2, and ωd−1c is the

scaling of the density of modes. For ωc ∼ Tc , the final current scaling becomes J optc ∼ T d+κ

c orα = d + κ − 1.

At low temperatures, the heat capacity of the bosonic systems scales as follows: c V (Tc ) ∼ T dc ,

which produces the scaling ζ = κ . This means that κ ≥ 1 to fulfill the third law. To rationalizethe scaling, cV is a volume property and γc is a surface property, so ωc γc scales the same as cV . Thescaling of the form factor κ is related to the speed at which the excitation can be carried away fromthe surface.

The exponent κ = 1 when ω → 0 is typical in systems such as electromagnetic fields or acousticphonons that have a linear dispersion law, ω(k) = v|k|. In these cases, the form factor becomesg(k) ∼ |k|/√ω(k); therefore, |g(ω)|2 ∼ |k|. The condition κ ≥ 1 excludes exotic dispersion laws,ω(k) ∼ |k|δ with δ < 1, which produce infinite group velocity forbidden by relativity theory.Moreover, the popular choice of Ohmic coupling is excluded for systems in dimension d > 1 (31).We note that challenges to the third law have been published (94). Our viewpoint is that thesediscrepancies are the result of flaws in the reduction of the spin bath to a harmonic spectral densityformulation.

4. SUMMARY

Thermodynamics represents physical reality with an amazingly small number of variables. Inequilibrium, the energy operator H is sufficient to reconstruct the state of the system, from whichall other observables can be deduced. Dynamical systems out of equilibrium require more variables.Quantum thermodynamics advocates that only a few additional variables are required to describea quantum device: a heat engine or a refrigerator. A description based on the Heisenberg equationof motion lends itself to this viewpoint. A sufficient condition for the set of operators to be closedto the Hamiltonian is that they form a Lie algebra. In addition, the Hamiltonian needs to bea linear combination of operators from the same algebra (95). Finally, the operators have to beclosed to the dissipative part Lh/c . Most of the solvable models in this review are based on theSU(2) Lie algebra with four operators. We chose them to represent the energy H; the identityI; and two additional operators, X and Y. These operators can associate with coherence because[H, X] �= 0, [H, Y] �= 0. Generically, in these models, the power is proportional to the coherenceP ∝ 〈Y〉. We can speculate that the steady-state operation of a working engine or refrigerator isminimally characterized by the SU(2) algebra of three noncommuting operators. This is also thecase in reciprocating engines (96–98).

The Hamiltonian can be decomposed to noncommuting local operators that couple to thebaths. The nonlocal structure influences the derivation of the master equation. A local approachbased on equilibrating Hs leads to equations of motion that can violate the second law of thermody-namics (9, 31, 35, 69). A thermodynamically consistent derivation requires a prediagonalization ofthe system Hamiltonian. This Hamiltonian defines the system-bath weak coupling approximationleading to the GKS-L completely positive generator.

Quantum mechanics introduces dynamics into thermodynamics. As a result, the laws of ther-modynamics have to be reformulated, replacing heat with heat currents and work with power. Thefirst law at steady state requires that the sum of these currents is zero. For the devices considered,the entropy production is exclusively generated in the baths and the sum of all contributions is pos-itive. All quantum devices should be consistent with the first and second laws of thermodynamics.Apparent violations point to erroneous derivations of the dynamical equations of motion.

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The tricycle template is a universal description for continuous quantum heat engines andrefrigerators. This model incorporates basic thermodynamic ideas within a quantum formalism.The tricycle combines three energy currents from three sources by a nonlinear interaction.A nonlinear construction is the minimum requirement for all heat engines or refrigerators.This universality means that the same structure can describe a wide range of quantum devices.The performance characteristics are given by the choice of working medium and reservoirs.In the derivation, the reservoirs are characterized by their temperature and correlation functions.The working medium filters out the channels of heat transfer and power.

The trade-off between power and efficiency is a universal characteristic of the quantum devices.The output power or the cooling current exhibits a turnover with any control parameter (99). Forexample, when the coupling parameter to the external field is increased first, the power or coolingcurrent increases with it, but then beyond a critical parameter, because of the splitting of the energylevels, a decrease in performance appears. Another turnover phenomenon is observed when thecoupling parameter to the baths is increased. Initially increasing the coupling will allow more heatto flow to the device. Above a critical value, dissipation will degrade the coherence and with it theperformance. Optimal power and optimal cooling currents are obtained in a balanced operationalpoint, far from reversible conditions of maximum efficiency.

The quantum version of the third law originates from the existence of a ground state. Onecan think of the third law as a limit to the distillation of a pure ground state in a subsystemdisentangling it from the environment. We can generalize and apply the third law to any pure stateof the subsystem, as such a state is accessible from the ground state by a unitary transformation.Unitary transformations are isoentropic and therefore allowed by the Nernst heat theorem atTc = 0. The quantum dilemma of cooling to a pure state stems from the requirement of a system-bath interaction to induce a change of the entropy of the system. However, such an interactiongenerates system-bath entanglement so that the system cannot be in a pure state. Approaching theabsolute zero temperature is a delicate maneuver. The system-bath coupling has to be reducedwhile the cooling takes place, eventually vanishing when Tc → 0. As a result, the means to coolare exhausted before the target is reached (100, 101). The universal behavior of all quantumrefrigerators as Tc → 0 emphasizes this point.

How do quantum phenomena such as coherence and entanglement influence performance?This issue can be separated into global effects, which include the reservoirs, and internal quantumeffects within the device. The description of heat transport into the device by the GKS-L masterequation is the thermodynamic equivalent of an isothermal partition. In this description, the sys-tem and bath are not entangled (31). The issue of internal quantum effects is more subtle. Internalcoherence is necessary for operation in quantum-driven devices. For quantum autonomous ab-sorption refrigerators, it has been recently claimed that, although present, coherence is not a deter-minant in engine operation (86). Other papers claim the opposite (50, 84, 102). Entanglement hasbeen related to the extraction of additional work (60, 103, 104). These issues require further study.

For completeness, here we mention what was excluded from the review above: reciprocatingheat engines and refrigerators. Significant insight can be gained from the analysis of such devices.The template is the quantum Otto four-stroke cycle in which the unitary adiabatic branches areseparated from the heat transport (22–24, 33, 98, 105–110). The analysis of reciprocating engineis simplified owing to this separation. The disadvantage is that the timing of the branches isdetermined externally. A review of reciprocating engines and refrigerators will be treated in aseparate publication.

Finally, we can conclude from this review that, when analyzed according to the laws of quan-tum mechanics, quantum devices are consistent with a dynamical interpretation of the laws ofthermodynamics.


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DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings thatmight be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

This review was supported by the Israel Science Foundation. The authors thank Tova Feldmann,Eitan Geva, Jose P. Palao, Jeff Gordon, Lajos Diosi, Peter Salamon, Gershon Kuritzky, and RobertAlicki for sharing their wisdom. Part of this work was supported by the COST Action MP1209“Thermodynamics in the quantum regime.”

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