PHYSICAL REVIEW A 88, 013842 (2013)

Thermodynamics of quantum heat engines

Himangshu Prabal Goswami and Upendra HarbolaInorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India

(Received 4 May 2013; published 26 July 2013)

We consider a recently proposed four-level quantum heat engine (QHE) model to analyze the role of quantumcoherences in determining the thermodynamic properties of the engine, such as flux, output power, and efficiency.A quantitative analysis of the relative effects of the coherences induced by the two thermal baths is brought out.By taking account of the dissipation in the cavity mode, we define useful work obtained from the QHE andpresent some analytical results for the optimal values of relative coherences that maximizes flux (hence outputpower) through the engine. We also analyze the role of quantum effects in inducing population inversion (lasing)between the states coupled to the cavity mode. The universal behavior of the efficiency at maximum power (EMP)is examined. In accordance with earlier theoretical predictions, to leading order, we find that EMP ∼ ηc/2, whereηc is Carnot efficiency. However, the next higher order coefficient is system dependent and hence nonuniversal.

DOI: 10.1103/PhysRevA.88.013842 PACS number(s): 42.50.Ct, 88.40.hj, 05.70.Ln, 07.20.Pe

I. INTRODUCTION

Heat engines have been a subject of intense research sincethe days of the industrial revolution due to their great practicalapplications. In recent times, various efforts have been made tounderstand the working mechanism of heat engines, where theworking engine is a quantum system. These can be technicallytermed as quantum heat engines (QHEs). Unlike a classicalheat engine, in QHE, the energy exchange between the systemand thermal reservoirs occurs in quantized fashion. QHEs aretherefore modeled as sets having discrete energy levels. Thisrealization was first studied by Scovil and Dubois [1] usinga three-level model where they showed that the system couldact as a heat engine when coupled to two thermal reservoirswith Carnot bound on the efficiency. The thermodynamicsof quantum heat engines has been well studied in the lastdecade [2–7].

One of the most peculiar properties of a quantum systemis to exhibit quantum coherences: existence of the system insuperposition states, resulting in mutual correlation betweenthe states. This aspect of quantum systems has been exploredonly lately in the context of QHEs. The quantum effects in suchsystems may lead to surprising thermodynamic behaviors. Forexample, it has been shown that the quantum effects can beexploited to extract work from a single heat bath [8,9], toreduce recombination in photocells [10], and produce lasingwithout population inversion [11].

Recently, it was shown [12–14] that the output power andthe efficiency of a four-level QHE model can be increasedbeyond their classical values (without quantum effects) by asuitable manipulation of quantum coherences. The four-levelquantum system is coupled to two thermal baths with atemperature gradient and an additional cavity mode. Thecreation of coherent photons in the cavity mode is interpretedas work. Here we consider the same model and extend theanalysis by computing the useful work that can be extracted ineach cycle. We define a useful work obtained from the engineby taking proper account of the dissipation in the cavity modeand compute the engine efficiency and the output power. Whenthe engine is working infinitesimally close to the equilibrium,that is, the net flux is close to zero, efficiency reduces to theCarnot result. We show how relative coherences induced dueto coupling to thermal baths can influence the current and the

power of the QHE. In addition to the earlier results, we obtaina condition on the relative values of induced coherences forwhich the quantum effects do not contribute to thermodynamicproperties. The optimal value of quantum coherences thatmaximizes the output power is derived. The optimal coherenceinduced by the hot (cold) bath can be manipulated by changingthe coherence induced by the cold (hot) bath. We investigatethis interdependence of coherences and show that, for mostcases, only the linear dependence is important.

It is well known that the efficiency of a heat engine, definedas the ratio of the work extracted from the engine to theinput energy, is maximum when the work is done reversibly.For an engine working between two thermal reservoirsat temperatures Th and Tc(< Th), the maximum (Carnot)efficiency (ηc = 1 − Tc/Th) is universal in that it does notdepend on the properties of the working substance. From theperspective of fundamental understanding, the Carnot enginehas always been the mainstay in the development of heatengines. However, from a practical point of view, the Carnotengine is not very useful since the work is done infinitesimallyslowly and therefore the output power is zero. In real engines,one likes to increase the output power but it is achieved onlyat the cost of the machine efficiency. The interdependence ofpower and efficiency is therefore a crucial aspect of any realengine. Following this, Curzon and Ahlborn [15] studied theefficiency of a heat engine in the endoreversible regime whenthe output power is maximum and expressed the efficiencyat maximum power as a function of the Carnot efficiency:EMP = 1 − √

1 − ηc. Close to equilibrium (Th ≈ TC), theuniversality of the expansion EMP ≈ ηc/2 + η2

c/8 has beenstudied in recent years [2,3,16–19]. The linear coefficient1/2 is shown [2] to be universal while its been argued thatfor systems with left-right symmetry, the next higher ordercoefficient 1/8 is also universal [3]. We discuss this universalbehavior in context to the QHE and show that, although thelinear coefficient is 1/2, the quadratic coefficient turns outto be system dependent and vanishes logarithmically as theoccupation in the cavity mode is decreased.

In the next section, we briefly illustrate the QHE model andthen present the model equations. In Sec. III we point out thethermodynamic properties of the QHE and show how quantumeffects can regulate these properties. In Sec. IV, we illustrate

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HIMANGSHU PRABAL GOSWAMI AND UPENDRA HARBOLA PHYSICAL REVIEW A 88, 013842 (2013)

the population inversion in the cavity mode and the leveragefrom the quantum effects. In Sec. V we define useful workextracted from the QHE, scrutinize output power, and studythe universality in the EMP. We then conclude the work inSec. VI.

II. QHE MODEL

We consider a QHE model that has been studied inRefs. [12,13,20]. This model represents a prototype for manynatural and man-made devices. It has recently been used (withsmall modifications) to model photosynthetic reaction centersas QHEs [21]. The model consists of two thermal baths attemperatures Th and Tc(< Th). The four levels in the QHE arecoupled to these baths and also to a single cavity mode (Fig. 1).The total Hamiltonian is HT = H0 + V + V ′,

H0 =∑

ν=1,2,a,b

EνBνν +∑

k

εka†kak + εla

†l al . (1)

Here, ν = 1,2,a,b is the Hamiltonian for the four-level system,k represents the heat reservoirs, and l represents the unimodalcavity. The coupling between the (working) system andthermal baths is

V =∑

X=a,b

∑k,i=1,2

gikakB†iX + H.c. (2)

The coupling between the cavity mode and the system is givenby

V ′ = g(a†l Bba + B

†baal), (3)

where Bνν ′ = |ν〉〈ν ′| denotes the excitation operator betweenstates |ν〉 and |ν ′〉, a† (a) stands for the creation (annihilation)operator for the harmonic modes in thermal baths and in theelectromagnetic field. g (real number) is the coupling strengthbetween the system and the radiation field.

A. Equations of motion

The time evolution of the full system is given by Liouville–von Neumann equation

∂

∂t|ρT (t)〉〉 = −iL|ρT (t)〉〉, (4)

FIG. 1. (Color online) Level scheme of the model quantum heatengine. A pair of degenerate levels |1〉,|2〉 is resonantly coupled to twoexcited levels |a〉 and |b〉 by two thermally populated field modes withhot (Th) and cold (Tc) temperatures. Levels |a〉 and |b〉 are coupledthrough a nonthermal (cavity) mode of frequency νl . Emission ofphotons into this mode is the work done by the QHE.

where vector |ρT 〉〉 corresponds to the Hilbert space densitymatrix of the full system and L = HT L − HT R with HT L(R) isthe left (right) superoperator [22,23] corresponding to the totalHamiltonian, HT . We expand the density matrix perturbativelyto second order in coupling (lowest contributing order) usingthe Liouville space formalism. Standard approximations oflarge reservoirs whose equilibrium state is not perturbed by thesystem-reservoir coupling and Weiskopf-Wigner approxima-tion are invoked to generate the time evolution of the reduceddensity matrix vector in the Liouville space. Details are givenin Refs. [12,20].

We consider the system to be symmetrically coupled toreservoirs, i.e., all couplings between the QHE and thermalbaths are taken to be equal and denoted by r . FollowingRef. [13], we also add a dephasing rate denoted by adimensionless parameter τ . This parameter τ is includedphenomenologically so as to take care of the dephasing arisingfrom the environmental effects, such as electron-phononinteractions, fluctuations, etc. The coherence ρ12 betweenstates |1〉 and |2〉 arise due to interactions with the hot andthe cold baths. This coherence couples to populations due totransition involving the states |1〉 and |2〉. However, unlikecoupling between the populations of two states, which isproportional to the square of the modulus of the dipole fortransition between the states; the coupling of coherence (ρ12)to population of a state depends on the relative orientation ofthe transition dipoles between that state and states |1〉 and |2〉since it involves an intermediate state |a〉 or |b〉 [see Feynmandiagrams in Fig. 2]. The coupling between coherence andpopulations can therefore be manipulated by changing therelative orientation of the two transition dipole vectors. Whenthe dipole vectors are perpendicular, the coupling vanishes andit is maximum when dipoles are parallel. To account for theserelative angles, we introduce two dimensionless parameters,ph and pc with 0 � ph,pc � 1, where subscripts c and h areused to keep track of contributions coming from couplingsto the hot and the cold baths, respectively. The net couplingbetween the populations and coherence is then parametrizedusing two parameters, γ12c = rpc and γ12h = rph [12,13].

Time evolution of the density matrix of the QHEis described by Linblad equation ρ = Lρ, where ρ ={ρaa,ρ22,ρaa,ρbb,ρaa,ρ12} is a vector containing the steady-state populations ρii = 〈〈i,i|ρ(t)〉〉, where i = 1,2,a,b, and

FIG. 2. Two-sided Feynman diagrams corresponding to the pro-cesses that contribute to the evolution of state |1〉 according to (6).Rate processes such as shown in (i) involve evolution of both theket and the bra of the density matrix. Contribution from the mixedstate ρ12 involves processes that contain evolution of either the ket orthe bra of the density matrix through an intermediate state as shownin (ii).

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THERMODYNAMICS OF QUANTUM HEAT ENGINES PHYSICAL REVIEW A 88, 013842 (2013)

the (real part) coherences, ρ12. Under symmetric coupling,Lindblad operator L is

L = r

⎛⎜⎜⎜⎜⎝

−n 0 nh nc −y

0 −n nh nc −y

nh nh−g2nl−2rnh

r

g2nl

r2phnh

nc ncg2nl

r

−g2nl−2rnc

r2pcnc

y

2y

2 phnh pcnc −n − τ

⎞⎟⎟⎟⎟⎠ . (5)

Here, n = nc + nh, y = ncpc + nhph, and ni = ni + 1 are theoccupation factors for the hot, cold, and unimodal cavities writ-ten as, nc = 1/(e(Eb−E1)/kBTc − 1) and nh = 1/(e(Ea−E1)/kBTh −1). The occupation of the cavity mode, nl , is assumed to beheld constant by means of external manipulations [12,13,20]).Thus any coherent photons created (consumed) in (from)the cavity is removed (added) by an external source is notincluded explicitly in the present calculations. Thus nl is agiven parameter.

The populations of the states can be obtained by solvingthe following equations:

ρ11 = −rnρ11 + rnhρaa + ncρbb − ryρ12, (6)

ρaa = rnhρ11 + rnhρ22 − (g2nl + 2rnh)ρaa + g2nlρbb

+ 2rphnhρ12, (7)

ρbb = rncρ11 + rncρ22 + g2nlρaa − (g2nl + 2rnc)ρbb

+ 2rpcncρ12, (8)

ρ12 = −ry

2ρ11 − ry

2ρ22 + rphnhρaa + rpcncρbb

− r(n + τ )ρ12, (9)

with

ρ11 + ρ22 + ρaa + ρbb = 1. (10)

Setting ρ = 0, at the steady state, we can solve for the steady-state values of ρaa , ρbb, ρ11, ρ22, and ρ12. From now on wewill consider only the steady-state properties.

Each term on the right-hand side of Eqs. (6)–(9) can beinterpreted easily using double-sided Feynman diagrams. Forexample, in Eq. (6), the first three terms on the right-handside show how populations of various states contribute to theevolution of the population of state |1〉. This has a simpleinterpretation of rate equation. The physical processes that areinvolved in these can be represented by Feynman diagramssimilar to those shown in Fig. 2. The system starts from apure state and jumps into (out of) state |1〉, which involves twointeractions with thermal baths. This involves evolution of boththe ket and the bra of the reduced density matrix. However, inthe last term in Eq. (6), the system starts from a mixed state andpasses through an intermediate state in order to populate state|1〉. These processes involve evolution of only either ket or braof the density matrix. It is also obvious that the intermediatestate involves a coherent evolution between states |1〉 and |a〉or |b〉, depending on which thermal bath is interacting withthe QHE. Such processes thus involve transitions between twodifferent sets of states. It is the interference between thesetwo transitions that is captured by the parameters ph and pc

introduced earlier.

III. THERMODYNAMIC QUANTITIES

We first consider the net flux j through the QHE. This canbe defined either with respect to the heat exchanged betweenthe QHE and baths or the photon exchanged between the QHEand the cavity mode. We define the flux to be the rate ofchange of photons within the cavity, that is, j = d

dt〈a†

l al〉; tosecond order in the coupling we get j = g2(nlρaa − nlρbb).Substituting steady-state values for ρaa and ρaa , we obtain

j = 2g2rncnhnl(ζ − 1)(n − ncp

2c − nhp

2h + τ

)Aj − Bj

, (11)

where

Aj = g2y[2ncnlpc + y(nl + nl) + 2phnhnl]

+ 2r[ncnh + nc(2 + 3nh)](ncp2c + nhp

2h − n)

Bj = (n + τ )[g2(nl(2 + 3nh) + nlnc(2nc + nh)(1 + 2nl)]

− 2r[ncnh + nc(2 + 3nh)], (12)

and ζ = nc nlnh

ncnhnl. The function F = ln(ζ ) represents the net

thermodynamic force (affinity) that drives the QHE out ofequilibrium [20]. Obviously, j = 0 for ζ = 1 (F = 0) bring-ing the QHE to equilibrium. Note that this condition is verydifferent from the equilibrium condition (Tc = Th) for classicalCarnot-like heat engines. This difference is due to the fact that,unlike Carnot-like engines, the energy exchange between theQHE and its surroundings is quantized. For ζ > 1 (ζ < 1),the system works as a heat engine (refrigerator). The inducedcoherence is obtained as,

ρ12 = g2ncnhnl(ζ − 1)(pc − ph)

B12 − A12, (13)

where

A12 = g2y[(2 + 3nc + 2nl + 4ncnl)pc

+ph(nh + 2nl + 4nhnl)] + 2p2c

{n2

c(3 + 4nh)

+ nh(2 + 3nh)p2h + nc

[(2 + 3nh)p2

c (14)

+ nhp2h(3 + 4nh)

]}r,

B12 = (n + τ ){g2[nhnl + 2nhnl + 2ncnl + nhnl

+ nc(1 + 2nl)] + 2[ncnh + nc(nh + 2nh)]r}.Thus at equilibrium (ζ = 1), coherence vanishes.

When pc = ph = 0, coherences do not couple to popula-tions and decay exponentially to zero with the rate n + τ , thesteady-state flux is then j = jo,

jo = 2rg2ncnhnl(ζ − 1)

g2[n + 2nl + 4nl(1 + nl)] + 2r(2 + 3n + 4ncnh).

(15)

Next, we define a coherent function (CF), j ′(pc,ph) = j/jo,which contains all the quantum effects through pc,ph.

As was observed in an earlier study [20], for symmetriccoupling and (ph = pc), the CF reduces to unity, that is,the quantum coherences do not affect the thermodynamics.We observe that, in addition to the above condition, CF alsoreduces to unity when the ratio of the induced coherences

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HIMANGSHU PRABAL GOSWAMI AND UPENDRA HARBOLA PHYSICAL REVIEW A 88, 013842 (2013)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.980

0.985

0.990

0.995

1.000

ph

CF

0.8

0.5

Tc 0.4

FIG. 3. (Color online) The variation of the CF with ph showingthe maxima at ph,max for different Tc values. The other parametersused are Ea = 1.5, Eb = 0.4, E1 = 0.1, nl = 0.5, g = 1, r = 0.7,and τ = 0.5. The horizontal line at unity is the result for CF whencoherences are zero. All energy values are in the unit of kBTh. Thefirst three dots (from the left) on the line correspond to the case whenEq. (16) is satisfied, while pc = ph at the last dot.

satisfies the following relation:

ph

pc

= nc(nhnl + 2nhnl + nhnl)

nh(ncnl + ncnl + 2ncnl). (16)

Thus there are two values of the ratio ph/pc = 1 and (16) forwhich coherence does not affect populations. The ratio in (16)approaches unity as ζ → 1, i.e, the two points merge togetheras the equilibrium is reached.

We next focus again on the flux j . This is an importantquantity since it defines the rate with which the output poweris generated or the heat energy is taken in by the QHE. Welook for the optimization of j using quantum effects (ph,pc).Since CF contains all the coherent effects, in order to optimizethe flux, it is enough to optimize the CF with respect to ph orpc. The variation on CF with respect to ph is shown in Fig. 3.It shows a maximum at ph = ph,max. The optimal value of ph

is obtained as

ph,max = 1

dppcnh

[Bpnh + App2

c −√

B2pn2

h + Ah

], (17)

where

Bp = [2nl nc + nc(1 + 2nl)](n + τ ),

Ap = ncnh[2nhnl − 2ncnl + (nh − nc)(1 + 2nl)],

Ah = Xhp2c + Yp4

c ,

Xh = −2nh

{2n2

cn2hn

2l + n3

cnh(1 + 2nl)2

+ 2ncncnlnh[2ncnl + nh(1 + 2nl)]

+ n2c

[2n2

hnl + n2h(1 + 2nl) (18)

+ 2nh(1 + 2nl)(nhnl + 2ncnl)]}

(n + τ ),

Y = nncnh

{4n2

c n2l nh + n2

cnh(1 + 2nl)

+ nc

[4n2

hn2l + n2

h(1 + 2nl)2

+ 4nh(1 + 2nl)(nhnl + ncnl)]}

,

dp = 2{ncnhnl + nc[nhnl + nh(2nl + 1)]}.The dependence of the optimal value ph,max on pc allows usto coherently control (by maneuvering pc) the optimal valueof ph. Since pc � 1, we can expand ph,max in powers of pc.Keeping the first two leading terms, we obtain a simplified

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

pc

p h,max

FIG. 4. (Color online) Variation in ph,max with pc. Parameters arethe same as in Fig. 3. Temperature increases from bottom to top (0.03,0.1, 0.2, 0.4, 0.5, and 1). Dashed lines represent the approximation,Eq. (19). As we can see, the bottommost line represents the variation atTc = 0.03 (in units of Th). This line actually represents the minimumvalue of ph,max that is possible under any parameter regime. And thisvalue is equal to 1

2 + 18[1+(τ/nh)] ∀nh,nl,nc.

expression,

ph,max ≈ Uhpc + Vhp3c , Uh = Ap

dpnh

− Xh

2Bpdpn2h

,

Vh = X 2h

8B3pdpn3

h

− Y2Bpdpn2

h

. (19)

The variation in ph,max with pc is shown in Fig. 4. Theapproximate result, Eq. (19), compares extremely well withthe exact result in (17). The simplicity in this equation allowsus to gain more quantitative insight into coherently controllingthe QHE system. We have used the expression in Eq. (19)to study the temperature dependence of the optimal valueof ph. A plot of Uh and Vh with Tc is shown in Fig. 5. Thecoefficient Vh is much smaller than Uh for all values of Tc anddecays with increasing Tc, while Uh increases with increasingTc values. The cubic dependence in Eq. (19) can be ignored athigh temperatures (Tc → Th). The minimum value of ph,max

is always zero, which occurs at pc = 0. Note that both Uh andVh converge to a constant as Tc → 0, implying that curves inFig. 4 converge onto a single curve as Tc → 0. This restrictsthe maximum value of ph,max, X, to a certain range overwhich it can be varied coherently using pc, as is evident fromFig. 4. Using (19), we find that

1

2+ 1

8(1 + τ

nh

) � X � Min{1,m}, (20)

0.0 0.2 0.4 0.6 0.8 1.00.4

0.5

0.6

0.7

0.8

0.9

1.0

Tc

Uh

0.0 0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

Tc

V h

FIG. 5. (Color online) The values of Uh and Vh as calculatedanalytically. Tc is in units of Th.

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THERMODYNAMICS OF QUANTUM HEAT ENGINES PHYSICAL REVIEW A 88, 013842 (2013)

where

m = limTc→Th

Uh = 1 − nc(nh − nl)

2nhnl(1 + 2nc). (21)

If nl < nh, m < 1 and vice versa. Thus increasing ordecreasing the cavity mode we can actually change the rangeover which ph,max is varied. But this trend is not indefinite. Thevalue of m is bounded with respect to the cavity occupation.

1 − nc

2(1 + 2nc)� m � 1 + nc

2nh(1 + 2nc). (22)

The flux can also be maximized with respect to pc.Performing the same analysis as done above with respect toph, we obtain an optimal value for pc.

p(2)c,max = Ucpc + Vcp

3c , Uc = −Ap

dpnc

+ Xc

2Bpdpn2c

,

(23)

Vc = X 2c

8B3pdpn3

c

− Y2Bpdpn2

c

.

However, this expression is valid if B2pn2

c � Xcp2h + Yp4

h,where

Xc = −2nc

{2n2

cn2hn

2l + n2

h(1 + 2nl)2

+ n2c

[2n2

hn2l + 2nhnhnl(1 + 2nl)

]+ ncnh

[4n2

hn2l + n2

h(1 + 2nl)2

+ 2nh(1 + 2nl)(nhnl + 2ncnl)]}

(n + τ ). (24)

IV. COHERENT CONTROL OF LASING

Population inversion is the most significant phenomenonin the production of lasers. Lasing without inverting thepopulation has been demonstrated in QHEs [24], which is aquantum effect. Here we are interested in population inversionbetween states |a〉 and |b〉, and controlling it coherently. Fromthe solution of Eqs. (6)–(10), the ratio ρaa/ρbb is obtained as

ρaa

ρbb

= (g2nln + 2ncnhr)(n + τ ) − Aaa

(g2nln + 2ncnhr)(n + τ ) − Abb

, (25)

where

Aaa = g2nly2 + 2rncnh

(ncp

2c + nhp

2h

),

(26)Abb = g2nly

2 + 2rnhnc

(ncp

2c + nhp

2h

).

In Fig. 6 we plot the ratio of populations in states |a〉 and|b〉 as a function of occupation of the cold bath. Populationinversion is seen at smaller values of nc. We ignore theenvironmental dephasing (τ = 0) to simplify the analysis. Allour previous conclusions are valid even without the dephasing.Note that in previous work [20], it was observed that in order tomaximize the output power, τ must be nonzero. However thatis valid only if the induced coherence due to the cold bath ismaximum, pc = 1. For a more general situation (pc < 1), as isthe case here, the output power can be maximized coherentlyeven if τ = 0. This is obvious from Eq. (19).

In order to create population inversion, we obtain that theratio between the occupations in the two thermal baths must

0.0 0.1 0.2 0.3 0.4 0.50.70.80.91.01.11.21.3

nc

ρ aaρ bb

FIG. 6. (Color online) A plot of the ratio of the populations ofstates |a〉 and |b〉 with nc, showing how the population inversion istaking place as the occupation of the cold bath nc is decreased. Thedotted curves show results without coherences. The three curves arefor nh = 1,1.5, and 2 (bottom to top). Required ratio nc/nh for lasingdecreases in the presence of coherences.

satisfy the following condition:

x− � nc

nh

� x+, (27)

where

x± = −B ±√B2 − (g2 − 2r)

(1 + p2

h

)(g2 + 2r)

(1 + p2

c

) ,

(28)

B = r(p2

h − p2c

) − g2(1 − pcph)

(g2 + 2r)(1 − p2

c

) .

From Eq. (28), we first make some observations.Case I. Since nc/nh must be real and positive, we have

the following conclusions. For g2 <r(p2

h−p2c )

1−pcph, B > 0 and

g2 < 2r , which implies that x− < 0 and x+ > 0. In this case,condition (27) for the lasing becomes

nc

nh

� x+, for g2 <r(p2

c − p2h

)1 + pcph

. (29)

Case II. When B < 0, i.e., g2 >r(p2

h−p2c )

1−pcph, there are two

conditions under which inversion can happen. (i) If g2 < 2r

then the condition (27) reduces to nc

nh� x+. (ii) If g2 > 2r , we

must have x− � nc

nh� x+.

Combining the two cases, we conclude that if g2 < 2r , thelasing is possible only if nc

nh� x+, otherwise the necessary

condition is x− � nc

nh� x+.

For the classical situation (no coherence effects), x− < 0and the lasing condition (27) becomes nc

nh� 2r−g2

2r+g2 , and there

is no lasing if g2 > 2r . It can be shown that the quantumeffects always tend to reduce the required value of the rationc/nh, that is, x+ <

2r−g2

2r+g2 . Thus for fixed Eb − E1 and nh,the above result can also be interpreted that for lasing, thetemperature of the cold bath is effectively reduced due to thepresence of coherences.

V. USEFUL WORK, OUTPUT POWER, AND EFFICIENCY

Coherent photons in the cavity mode are generated eachtime the system relaxes between states |a〉 and |b〉. This

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HIMANGSHU PRABAL GOSWAMI AND UPENDRA HARBOLA PHYSICAL REVIEW A 88, 013842 (2013)

relaxation is stimulated by the cavity photons which arekept at a constant intensity corresponding to the cavity modeoccupation nl . Thus each stimulated emission provides acoherent photon with energy Ea − Eb. However, some amountof energy, Wdiss, is dissipated (due to spontaneous processes)in the cavity which is proportional to ln(nl/nl) [20]. Thus,at steady state, in each forward cycle of the heat engine,the total dissipation is due to the entropy change in the twobaths plus the energy dissipation in the cavity mode, ln ζ =Sh + Sc + Wdiss [20], where Sh(Sc) is the change inthe entropy of the hot (cold) bath. Using the first and secondlaws of thermodynamics, the maximum useful work that canbe obtained per cycle is

W = Ea − Eb − kBTc lnnl

nl

. (30)

A similar expression for the useful work was used previouslyin Refs. [13,14]; the two expressions are the same in the limitζ → 1. Using the definitions of nh and nc, Eq. (30) can berecast into a more useful form:

W = Qh

(1 − Tc

Th

)− kBTc ln ζ, (31)

where Qh = Ea − E1 is the amount of energy taken from thehot bath. It is clear that the work obtained is maximum whenthe engine works infinitesimally close to equilibrium (ζ →1). The efficiency, η = W/Qh, increases and approachesthe Carnot efficiency as ζ → 1 (however, the power thenapproaches zero). When the cavity occupation is very large,nl 1, the second term on the right-hand side in (30) isnegligible and the maximum useful work can be approximatedby Ea − Eb. For Th = Tc and ζ > 1 (for the heat engine),the useful work is always negative, indicating that no usefulwork can be obtained from the QHE. In order to have W > 0,we fix nl = 0.5 > (e(Ea−Eb)/(kBTc) − 1)−1 for the parametersconsidered here. The output power is defined as P = jW .Since W is independent of coherences, the coherent control ofthe output power is done only by changing the flux, which wehave discussed earlier.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

ηC

EMP 0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.000.050.100.150.200.250.30

nl

b

FIG. 7. (Color online) Variation in EMP with ηC . The variouscurves represent EMPs simulated at different nl values. The nl valuesincrease from bottom to top (0.05, 0.1, 0.5, and 10). The dashedline represents the EMP derived by Curzon and Ahlborn [15]. Theinset shows how the quadratic term b varies logarithmically withchange in nl . Points represent data values of b mentioned in theTable I. The curve in the inset represents the logarithmic fit, f (nl) =p ln[1 + qnl].

TABLE I. Linear and quadratic coefficients of EMP.

Cavity occupation (nl) a,O(ηC) b,O(η2C)

0.01 0.49 0.020.05 0.49 0.090.1 0.49 0.140.2 0.49 0.180.4 0.49 0.220.5 0.49 0.230.6 0.49 0.251 0.49 0.2710 0.48 0.28100 0.48 0.28

We next analyze interdependency of the output power andthe efficiency. In 1975, Curzon and Ahlborn [15] studiedthe efficiency of a heat engine at maximum power (EMP).Quantitatively, this EMP is ηCA = 1 − √

Tc/Th which, nearequilibrium, can be expanded in Carnot efficiency, ηCA =ηC/2 + η2

C/8 + On(ηC). It is well accepted that, in the linearregime, the 1/2 dependence is universal [2] and can beregarded as an upper bound on the efficiency of a real engine.The quadratic term 1/8 has been obtained in many heatengines, for example, in a nanomaser [3] under the condition ofleft-right symmetry, on a quantum dot engine [16] under weakdissipation, in a Brownian particle Carnot cycle, and weaklyinteracting gas, etc. [17,18]. However, in a Brownian heatengine [19], for systems far from equilibrium, the quadraticcoefficient is shown to be less than 1/8.

We analyze the efficiency at maximum power and itsbehavior as function of Carnot’s efficiency for our QHE.We numerically compute the maximum in output power withrespect to Ea and, using this optimal value of Ea in (31), weevaluate the EMP. The results are shown in Fig. 7 for variousvalues of the cavity occupation. For each nl we look for theoptimal value of Ea only in the range over which W > 0.The linear and quadratic coefficients are obtained by fittingthe numerical results over small values of ηc � 0.3. We findthat the universality of the linear term is well preserved but thecoefficient of the quadratic term (b) is not 1/8 but increases asthe cavity occupation number is increased.

The values of the expansion coefficients are reported inthe Table I. We find that the coefficient b approaches zerologarithmically as the cavity occupation is decreased. This isshown in the inset of Fig. 7. Note that the left-right symmetry[3] in our model is broken due to the presence of cavity mode.Thus, in view of the analysis in Ref. [3], we should not expectto find b = 1/8. This symmetry is, however, recovered as theoccupation in the cavity mode becomes large (nl 1) and wefind that the coefficient saturates to b ≈ 0.28. Our numericalresults show that the coefficients a and b are robust againstthe quantum effects, that is, they do not show any significantchange (less than 0.1%) with ph and pc.

VI. CONCLUSIONS

We have performed a thorough investigation of coherenteffects in a quantum heat engine which has been previ-ously studied in Refs. [12,13,20]. We extended the previous

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THERMODYNAMICS OF QUANTUM HEAT ENGINES PHYSICAL REVIEW A 88, 013842 (2013)

density-matrix-based analysis to explore the effects of relativecoherences induced by the two thermal baths. The flux throughthe QHE can be maximized beyond classical (zero coherence)values with respect to the individual coherences inducedby baths. These optimal values of coherences can be tunedcoherently over a certain range which is computed analytically.A condition in terms of the ratio of occupations in the hotand cold baths is derived for the lasing in the cavity mode.The useful work obtained from the QHE depends on theoccupation in the cavity mode. This is shown to be due tothe dissipation in the cavity which depends logarithmicallyon the cavity mode occupation. The efficiency of the QHEdefined in terms of the useful work approaches the Carnotefficiency as the equilibrium is approached (output powergoes to zero). Near the equilibrium, the efficiency when thepower is maximum is expressed perturbatively in terms of the

Carnot efficiency. The efficiency at maximum power revealsthe universal behavior of the linear coefficient as predicted bythe theory [2]. The next-order (quadratic) coefficient, however,is nonuniversal and decays logarithmically to zero as the cavitymode occupation is decreased. Our analysis shows that theseexpansion coefficients are almost unaffected by the quantumeffects.

ACKNOWLEDGMENTS

We thank Professor Shaul Mukamel, Professor Saar Rahav,and Dr. Konstantin Dorfman for helpful discussions. H.P.G.acknowledges financial support from the University GrantsCommission, India. U.H. is supported by the start-up grant(Grant No. 11-0201-0591-01-412/415/433) from the IndianInstitute of Science, Bangalore, India.

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