Performance of continuous quantum thermal devices indirectly connected to environments

0.62 MB·

by  Javier Onam González

Performance of continuous quantum thermal devices indirectly connected to
environments
J. Onam Gonz´alez,1,2 Daniel Alonso,1,2 and Jos´e P. Palao1,2
1IUdEA Instituto Universitario de Estudios Avanzados, Universidad de La Laguna, 38203 Spain
2Dpto. F´ısica Universidad de La Laguna, 38203 Spain
Ageneralquantumthermodynamicsnetworkiscomposedofthermaldevicesconnectedtotheen-
vironmentsthroughquantumwires. Thecouplingbetweenthedevicesandthewiresmayintroduce
additionaldecaychannelswhichmodifythesystemperformancewithrespecttothedirectly-coupled
device. Weanalyzethiseffectinaquantumthree-leveldeviceconnectedtoaheatbathortoawork
source through a two-level wire. The steady state heat currents are decomposed into the contri-
butions of the set of simple circuits in the graph representing the master equation. Each circuit is
associatedwithamechanisminthedeviceoperationandthesystemperformancecanbedescribed
7 by a small number of circuit representatives of those mechanisms. Although in the limit of weak
1 couplingbetweenthedeviceandthewirethenewirreversiblecontributionscanbecomesmall,they
0 prevent the system from reaching the Carnot efficiency.
2
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I. INTRODUCTION
Continuous quantum thermal devices are quantum systems connected to several baths at different temperatures
and to work sources [1]. Their operation is necessarily irreversible when the heat currents are non-negligible. One of
thepossibleirreversibleprocessesistheubiquitousfinite-rateheattransfereffectconsideredinendoreversiblemodels.
In these models the control parameters can be tunned to reach the reversible limit but at vanishing energy flows.
Examples are the three-level and the two-qubit absorption refrigerators [2–5]. In other models, as the power-driven
three-levelmaser[1,6]andthethree-qubitabsorptionrefrigerators[3,7],additionalirreversibleprocessesappearsuch
as heat leaks and internal dissipation [1, 8], which are detrimental to the device performance. Several experimental
realizationsofthesecontinuousquantumthermaldeviceshavebeenproposed,forexamplenano-mechanicaloscillators
oratomsinteractingwithopticalresonators[9,10],atomsinteractingwithnonequilibriumelectromagneticfields[11],
superconducting quantum interference devices [12], and quantum dots [13]. Besides, the coupling between artificial
atomsandharmonicoscillatorsisexperimentallyfeasiblenowadays[14],openingtheperspectiveofconnectingthermal
devices to environments through quantum systems.
Although the most general design of a quantum thermal network is composed of thermal devices and wires [15],
the device performance has been usually analyzed assuming a direct contact with the environments. The coupling
between the device and a quantum probe has been suggested to characterize the device irreversible processes [16]. In
this paper we adopt a different perspective and study the additional irreversible processes induced by the coupling
between the device and the wire. In particular we will analyze in detail the performance of a system composed of
a three-level device and a two-level wire connected to a work bath or source. When the system is weakly coupled
to the baths, its evolution is described by a quantum master equation in which the dynamics of the populations can
be decoupled from the coherences choosing an appropriate basis [17]. This property implies the positivity of the
entropy production along the system evolution [18, 19], and is broken when some uncontrolled approximations are
considered in the derivation of the quantum master equation [20]. The Pauli master equation for the populations
(in the following simply the master equation) is a particular example of the general master equations considered in
stochasticthermodynamicsforsystemsconnectedtomultiplereservoirs[21]. Forlongenoughtime,thesystemreaches
a non-equilibrium steady state where the heat currents Q˙ describe the energy transfer between the system and the
α
baths, and the power P characterizes the energy exchange with the work source [15]. Although the heat currents can
be obtained directly from the steady solution of the master equation, to identify the different irreversible processes
contributing to them is in general a very complicated task.
An alternative approach to analyze non-equilibrium processes is the decomposition of the steady state fluxes and
the entropy production in the contribution due to simple circuits [22–24], fundamental circuits [25–27] or cycles
[28] of the graph representation of the master equation. We will consider a decomposition in the full set of simple
circuits that combined with the all minors matrix-tree theorem [29] lead to very simple expressions for the steady
stateheatcurrents. Moreimportantly, eachcircuitcanbeinterpretedasathermodynamicallyconsistentunitandits
contribution to the different irreversible processes can be easily identified [30]. Although the number of circuits may
be very large, we will show that the system performance can be described by means of a reduced number of circuit
representatives [31].
The paper is organized as follows: Section II presents a brief review of the derivation of the quantum master
equation for a device coupled to a heat bath through a quantum wire. Next the graph representation of the master
equation and the decomposition in simple circuits is discussed, with special emphasis on the characterization of the
steady state heat currents. Some procedures to determine the set of simple circuits are described in Appendix A.
The absorption refrigerator composed of a three-level device connected to a work bath through a two-level wire is
studied in Section III and some circuit representatives are suggested to describe the system performance. The same
analysis is applied to a system driven by a periodic classical field in Section IV, which includes a brief discussion of
the derivation of the master equation for the time dependent Hamiltonian. In this case we study the performance
operating as a refrigerator or as an engine. Finally, we draw our conclusions in Section V.
II. CIRCUIT DECOMPOSITION OF THE STEADY STATE HEAT CURRENTS AND ENTROPY
PRODUCTION
A. The master equation
We consider a system composed of a quantum thermal device with Hamiltonian Hˆ directly coupled with a cold
D
bath and a hot bath, at temperatures T and T , and a quantum wire with Hamiltonian Hˆ which connects the
c h wire
device to an additional bath at temperature T (work bath). The situation in which the system is driven instead by
w
a work source will be discussed in section IV. The total Hamiltonian reads
3
Hˆ = Hˆ + Hˆ + Hˆ + Hˆ + Hˆ + (cid:88) (cid:16)Hˆ + Hˆ (cid:17) , (1)
D D,wire wire wire,w w D,α α
α=c,h
where Hˆ is the coupling between the device and the wire, Hˆ and Hˆ the coupling terms of the device
D,wire D,α wire,w
and the wire with the baths, and Hˆ the bath Hamiltonians. We assume that the coupling terms of the system
√ α
with the baths are γ (cid:126)Sˆα⊗Bˆα, where Sˆα is a device or wire Hermitian operator, Bˆα is a bath operator and γ
α α
characterizes the coupling strength.
If the system is weakly coupled with the baths and its relaxation time scale is slow compared with the correlation
times of the baths, the system evolution can be described by a Markovian quantum master equation for its reduced
density operator ρˆ. The procedure to obtain this quantum master equation is described for example in [17]. Here
we just comment on the final result. Let Uˆ (t)=exp(−iHˆ t/(cid:126)) denotes the evolution operator corresponding to the
S S
system Hamiltonian Hˆ = Hˆ + Hˆ + Hˆ . The essential elements in the quantum master equation can be
S D D,wire wire
identified from the following decomposition of the operators Sˆα in interaction picture
Uˆ†(t)SˆαUˆ (t) = (cid:88)Sˆαexp(−iωt) + Sˆα†exp(iωt), (2)
S S ω ω
ω>0
where(cid:80) denotesthesummationoverthepositivetransitionfrequenciesω =ω −ω betweeneigenstatesofHˆ .
ω>0 ij j i S
The difference between the spectrum of Hˆ and Hˆ makes the frequencies and terms in the previous decomposition
S D
different from the one corresponding to the device directly coupled with the baths, and leads to new decay channels.
This is the origin of the additional irreversible processes.
When system intrinsic dynamics is fast compared to the relaxation dynamics, the rotating wave approximation
applies and the Lindbland-Gorini-Kossakovsky-Sudarshan (LGKS) generators of the irreversible dynamics associated
with each bath can be written as
(cid:18) (cid:19) (cid:18) (cid:19)
L [ρˆ(t)] = (cid:88) Γα SˆαρˆSˆα†− 1{Sˆα†Sˆα,ρˆ} + Γα Sˆα†ρˆSˆα− 1{SˆαSˆα†,ρˆ} , (3)
α ω ω ω 2 ω ω −ω ω ω 2 ω ω
ω>0
We have introduced the anticommutators {SˆSˆ†,ρˆ}=SˆSˆ†ρˆ+ρˆSˆSˆ†. In the following we will consider bosonic baths
ofphysicaldimensionsd andcouplingoperatorsBˆα ∝(cid:80) √ω (ˆbα+ˆbα†). Thesummationisoverallthebathmodes
α µ µ µ µ
of frequencies ω and annhilation operatorsˆb . With this choice the rates Γα are [17]
µ µ ±ω
Γα =γ (ω/ω )dα[Nα(ω)+1],
ω α 0
Γα =Γαexp(−ω(cid:126)/k T ), (4)
−ω ω B α
with Nα(ω)=[exp(ω(cid:126)/k T )−1]−1, k the Boltzmann constant, and the frequency ω depends on the physical
B α B 0
realization of the coupling with the bath. Finally, assuming that the Lamb shift of the unperturbed energy levels is
small enough to be neglected, the quantum master equation in the Shr¨odinger picture is given by
dρˆ(t) = −i[Hˆ ,ρˆ(t)] + (cid:88) L [ρˆ(t)]. (5)
dt (cid:126) S α
α=c,w,h
ThisquantummasterequationisinthestandardLindbladformanddefinesageneratorofadynamicalsemigroup.
IfthespectrumofHˆ isnon-degenerated,equation(5)forthepopulationsoftheN eigenstates|i(cid:105)ofHˆ ,p =(cid:104)i|ρˆ|i(cid:105),
S S i
reduces to [17]
N N
d (cid:88) (cid:88) (cid:88)
p (t) = Wαp (t) = W p (t), (6)
dt i ij j ij j
j=1 α=c,w,h j=1
whereWα isthetransitionratefromthestatej tothestateiduetothecouplingwiththebathα. Inthefollowing
ij
W will denote the matrix with elements W =(cid:80) Wα. The diagonal elements satisfy
ij α=c,w,h ij
4
(cid:88)
Wα = − Wα, (7)
ii ji
j(cid:54)=i
implyingtheconservationofthenormalization. Besides,asaconsequenceoftheKubo-Martin-Schwingercondition
in (4), the forward and backward transition rates are related by
Wα (cid:18) ω (cid:126) (cid:19)
ji = exp − ij . (8)
Wα k T
ij B α
Equation (6) is the starting point of our analysis. When the system is driven by a work source we will arrive to an
equation with a similar structure, and the results described below will also apply to that case.
B. Circuit fluxes and affinities
Here we describe how to determine the heat currents Q˙ and the entropy production S˙ in the steady state. In the
α
following we assume that the currents are positive when the energy flows towards the system. The method is based
ontherepresentationofthemasterequation(6)byaconnectedgraphG(V,E),being|V|=N thenumberofvertices,
representingthesystemstates,and|E|thenumberofundirectededges,representingthetransitionsbetweendifferent
states. AsimplecircuitC ofG isaclosedpathwithnorepetitionofverticesoredges. Someprocedurestodetermine
ν
the set of circuits in a graph are discussed in appendix A. Each one of the two possible different orientations of a
simplecircuit,denotedbyC(cid:126) and−C(cid:126) ,isacycle. Acyclethenconsistsofasequenceofdirectededgeswithtransition
ν ν
rates Wα, and it has an associated algebraic value [25]
ij
A(C(cid:126) ) = (cid:89) Aα(C(cid:126) ), (9)
ν ν
α=c,w,h
with
Aα(C(cid:126) ) = (cid:89) Wα, (10)
ν ij
ij∈ν
where (cid:81) denotes the product of all the transition rates due to the bath α in the cycle C(cid:126) . If the cycle does not
ij∈ν ν
involve the bath α, Aα(C(cid:126) )=1 for consistency.
ν
The cycle affinity [25] is defined by
(cid:32) (cid:33)
X(C(cid:126) ) = (cid:88) Xα(C(cid:126) ) = k ln A(C(cid:126)ν) , (11)
ν ν B A(−C(cid:126) )
α=c,w,h ν
wheretheaffinityassociatedwitheachbathisXα(C(cid:126) )=k ln[Aα(C(cid:126) )/Aα(−C(cid:126) )]. Whenthesystemisonlycoupled
ν B ν ν
with thermal baths, the same amount of energy is taken and transferred to them in a complete cycle, implying
(cid:80) T Xα(C(cid:126) )=0. However,whenthesystemisinadditioncoupledtoaworksourcethesummationmaydifferfrom
α α ν
zero, indicating the net exchange of energy between the work source and the baths. The cycle flux is defined by [30]
I(C(cid:126) )=D−1det(−W|C )[A(C(cid:126) ) − A(−C(cid:126) )], (12)
ν ν ν ν
whereD = |det(W(cid:102))|. ThematrixW(cid:102) isobtainedfromtheratematrixWreplacingtheelementsofanarbitraryrow
byones,and(−W|C )denotesthematrixresultingfromremovingfrom−W alltherowsandcolumnscorresponding
ν
to the vertices of the circuit C . Considering the relation between the diagonal and non-diagonal elements of W, the
ν
determinant of (−W|C ) is always positive. The opposite cycle affinities and flux change according to X(α)(−C(cid:126) )=
ν ν
−X(α)(C(cid:126) ) and I(−C(cid:126) )=−I(C(cid:126) ).
ν ν ν
5
Using these definitions, the steady state heat current between the system and a bath associated with a simple
circuit is
Q˙ (C ) = −T I(C(cid:126) )Xα(C(cid:126) ), (13)
α ν α ν ν
and the steady state entropy production is given by S˙(C ) = I(C(cid:126) )X(C(cid:126) ). At this point it is important to notice
ν ν ν
thatalthoughtheaffinityandthefluxaredefinedforeachcycle,thesteadystateheatcurrentsandentropyproduction
are independent of the cycle orientation and can then be assigned to the circuit without any ambiguity. Besides each
circuit is consistent with the first and second law of thermodynamics as (cid:80) Q˙ (C ) = 0 and S˙(C ) ≥ 0 [22–
α=c,w,h α ν ν
24, 30]. Finally the heat currents and the entropy production in the steady state can be obtained by adding the
contribution of all the simple circuits of the graph, Q˙ =(cid:80) Q˙ (C ) and S˙ =(cid:80) S˙(C ).
α ν α ν ν ν
The relative importance of the contribution due to a simple circuit to the heat current (13) is determined by both
its affinity Xα(C(cid:126) ) and flux I(C(cid:126) ) (12). When the system is coupled with thermal baths, the circuits can be classified
ν ν
as trivial circuits (all the affinities Xα = 0), circuits associated with heat leaks (one of the affinities is zero) and
tricycles (the three affinities are different from zero) [30]. Trivial circuits do not contribute to the steady state heat
currentsorentropyproduction. Circuitsassociatedwithheatleaksonlyconnecttwobaths,andtheheatalwaysflows
from the higher to the lower temperature bath. Tricycles [15] are circuits connecting the three baths, independently
of the number of edges involved. When the system is coupled instead with a work source, the circuits associated with
heat leaks are identified from the condition T Xc+T Xh =0 (as there is not net energy exchange with the source),
c h
and the tricycles from T Xc+T Xh (cid:54)=0 [30].
c h
The analysis of the circuit flux is more complicated as the term A(C(cid:126) ) − A(−C(cid:126) ) strongly depends on the system
ν ν
parameters. Inanycase,thenumberoftermsinthedeterminantofthematrix(−W|C )decreaseswhenthenumberof
ν
edgesinthecircuitincreases. Thenthenon-trivialcircuitswithalowernumberofedgesarethedominantcontribution
to the heat currents in a large range of the parameters, and the system operation as a thermal machine is mainly
determined by tricycles with a low number of edges.
III. QUANTUM THREE-LEVEL DEVICE COUPLED THROUGH A TWO-LEVEL SYSTEM TO A
WORK BATH
FIG.1. (a)Schematicillustrationofathree-leveldevicecoupledtoaworkbathorsourcethroughatwo-levelwire. (b)Graph
representationofthemasterequationwhenthewireconnectsaworkbath. ThesixverticesrepresenttheeigenstatesofHˆ and
S
the eleven edges the transitions assisted by the cold (blue lines), work (green lines) and hot (dashed red lines) baths, labeled
by c, w and h respectively. (c) Graph representation of the master equation when the system is driven by a periodic classical
field. NowtheverticescorrespondtotheeigenstatesofHˆ , andthereareeightpairsofparalleledgesassociatedwiththecold
2
(solid lines) and hot (dashed lines) baths.
Anabsorptionrefrigeratorisathermaldeviceextractingheatfromacoldbathandrejectingittoahotbathatrates
Q˙ and Q˙ respectively. This process is assisted by the heat Q˙ extracted from a work bath at higher temperature.
c h w
Its coefficient of performance (COP) is given by ε=Q˙ /Q˙ . The simplest model of quantum absorption refrigerator
c w
is a three-level system directly coupled with the heat baths [2, 3]. When T < T < T the refrigerator operates in
c h w
the cooling window ω < ω = ω T (T −T )/[T (T −T )], with ω , ω and ω = ω −ω the frequencies of
c c,rev h c w h h w c c h w h c
the transitions coupled with the cold, hot and work baths respectively. In the limit of ω approaching from below
c
to ω the COP reaches the Carnot limit ε = T (T −T )/[T (T −T )], as the only source of irreversibility is
c,rev C c w h w h c
6
the finite heat transfer rate through the thermal contacts. To analyze the effect of the indirect coupling we consider
a system consisting of the three-level device now connected through a two-level wire to the work bath, schematically
shown in Figure 1(a). The device and wire Hamiltonians read
Hˆ =ω (cid:126)|2 (cid:105)(cid:104)2 |+ω (cid:126)|3 (cid:105)(cid:104)3 |, (14)
D c D D h D D
and
Hˆ =ω (cid:126)|2 (cid:105)(cid:104)2 |. (15)
wire w W W
The operators in the coupling terms with the baths are taken as Sˆα =(Sˆα +Sˆα), with Sˆα =Sˆα† and
− + + −
Sˆc =|1 (cid:105)(cid:104)2 |; Sˆh =|1 (cid:105)(cid:104)3 |; Sˆw =|1 (cid:105)(cid:104)2 |. (16)
− D D − D D − W W
The interaction between the device and the wire is described by
Hˆ =g(cid:126)(|3 1 (cid:105)(cid:104)2 2 |+|2 2 (cid:105)(cid:104)3 1 |), (17)
D,wire D W D W D W D W
where the parameter g is the coupling strength. The eigenfrequencies of Hˆ =Hˆ + Hˆ + Hˆ are ω =0,
S D D,wire wire 1
ω =ω , ω =ω , ω =[2ω −∆−(∆2+4g2)1/2]/2, ω =[2ω −∆+(∆2+4g2)1/2]/2 and ω =ω +ω . We have
2 w 3 c 4 h 5 h 6 w h
introduced the detuning ∆=ω −ω −ω . Using the procedure of section II to determine the master equation (6),
h c w
we obtain the following non-zero transition rates Wα with indexes j >i,
ij
Wc =Γc , Wc =|c |2Γc , Wc =|c |2Γc ,
13 ωc 24 − ω4−ωw 25 + ω5−ωw
Wh =|c(cid:48) |2Γh , Wh =|c(cid:48) |2Γh , Wh =Γh ,
W1w4 =Γ−w , ω4 W1w5 =|c+|2Γωw5 , W2w6 =|cωh|2Γw , (18)
W1w2 =|cω(cid:48)w|2Γw , W3w4 =|c−(cid:48) |2Γωw4−ωc , 35 + ω5−ωc
46 − ωw+ωh−ω4 56 + ωw+ωh−ω5
where the coefficients are given by
[−∆±(∆2+4g2)1/2]d
c = ± ,
± 4g(∆2+4g2)1/2
d
c(cid:48) = ± , (19)
± 2(∆2+4g2)1/2
with d2 = 4g2 +[∆±(∆2 +4g2)1/2]2. The remaining elements can be obtained using (7) and (8). The graph
±
representationofthemasterequationisshowninfigure1(b)whereweidentified38simplecircuitsusingthemethods
described in appendix A. As each pair of vertices is connected by only one edge, the sequence of E ≤ 6 vertices
{i ,i ,...,i ,i } will denote in the following both a circuit containing these vertices and the corresponding cycle
1 2 E 1
with orientation i → i ··· → i → i . As we are interested in the system operating as a refrigerator we will focus
1 2 E 1
our analysis on the steady state heat current with the cold bath. Now we will assume ∆=0, for which ω =ω −g,
4 h
ω =ω +g and |c |=|c(cid:48) |2 = 1. The non-resonant case will be discussed later.
5 h ± ± 2
The simplest circuits in the graph are the three-edge tricycles C = {1,3,4,1}, C = {1,3,5,1}, C = {1,2,5,1},
1 2 3
C = {1,2,4,1}, C = {2,5,6,2} and C = {2,4,6,2}, with affinities Xc(C(cid:126) ) = −ω (cid:126)/T , Xc(C(cid:126) ) = −(ω +g)(cid:126)/T
4 5 6 1,2 c c 3,5 c c
and Xc(C(cid:126) ) = −(ω −g)(cid:126)/T . In all cases the leading term of the affinity is proportional to the frequency of the
4,6 c c
transition coupled with the cold bath when g (cid:28) ω . From (13) we find that the upper limit of the cooling window
c
(Q˙ >0) for the circuits C and C is given by
c 1 2
T (T −T )
ω (C ) = ω + (−1)νg c w h . (20)
c,rev ν c,rev T (T −T )
h w c
A similar analysis gives ω (C ) = ω + (−1)νgT (T −T )/T (T −T ) for ν = 3,4 and ω (C ) =
c,rev ν c,rev w h c h w c c,rev ν
ω +(−1)νg forν =5,6. ThetricyclesreachtheCarnotCOPwhenapproachingtoω (C ),butwithvanishing
c,rev c,rev ν
circuit heat currents.
7
0.06 0.3
c
w (a) 0.2 (b)
c 0.04 c w
T T
B B 0.1
k 0.02 k
0.0
γ c γ
/ /
) 0.00 ) −0.1
7 8
C C
(α−0.02 (α−0.2 h
˙Q h 3 2 ˙Q−0.3 3 2
4 −0.04 4 ω
0 4 1 0−0.4 4 1 c,rev
1 1
−0.06 5 6 −0.5 5 6
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
ω /ω ω /ω
c h c h
6 1.0
(c) (d)
5
0.8
c
T 4
B
k 3 (3) c0.6
γ ε
/ ω /
c 2 c,rev ε
˙Q 0.4
4 1
0 (5)
1
0.2
0 (6)
−1 0.0
0.0 0.2 0.4ω /ω 0.6 0.8 1.0 0.0 0.2 0˙.4/ ˙ m0ax.6 0.8 1.0
c h Qc Qc
FIG.2. Steadystateheatcurrentsasafunctionofthefrequencyω for(a)acircuitassociatedwithaheatleakand(b)afive-
c
edgetricycle. Thecircuitsareshownintheinsets. (c)HeatcurrentwiththecoldbathQ˙ forthethree-leveldeviceconnected
c
through the two-level wire (solid line), the three-level directly coupled with the baths (dashed-dotted line). The labeled thin
dashed lines show the contribution of tricycles with 3, 5 and 6 edges. The vertical line indicates the frequency ω . (d)
c,rev
Performance characteristics obtained from (c) in the cooling range. The dashed line depicts the performance characteristic of
the circuit representatives C and C . Each curve has been normalized with respect to its maximum cooling rate. Parameters
1 2
were chosen as T =9, T =10, T =20, ω =1, g=0.05, d =3 and γ =γ =10−6 in units for which (cid:126)=k =ω =1.
c h w h α α B 0
Weidentify10four-edgecircuitsassociatedwithheatleaks,4involvingthecoldandworkbaths,4theworkandhot
bathsand2thecoldandhotbaths. Inallcasesthetwonon-zeroaffinitiesXα areproportionaltothecouplingstrengh
g. An example is the circuit C ≡ {1,4,6,5,1} shown in figure 2(a). In this case the affinities are Xh(C(cid:126) ) = 2g(cid:126)/T
7 7 h
and Xw(C(cid:126) ) = −2g(cid:126)/T . From the circuit heat currents (13), the condition T < T and the relation (8), one can
7 w h w
easilydeterminethatQ˙ (C )<0andQ˙ (C )>0,resultinginadirectenergytransferfromtheworkbathtothehot
h 7 w 7
bath. There is also a trivial four-edge circuit involving only the edges associated with the work bath. We found 14
five-edge tricycles, as for example C ={1,4,6,2,5,1} shown in figure 2(b). In this case Xc(C(cid:126) )=−(ω +g)(cid:126)/T and
8 8 c c
thecircuitcoolingwindowisgivenbytheconditionω <ω (C )=ω +g(2T T −T T −T T )/(T T −T T ).
c c,rev 8 c,rev c w c h w h w h h c
Finally there are 7 six-edge circuits, 2 tricycles and 5 associated to heat leaks. All the tricycles have affinities Xc
with a leading term proportional to ω and cooling windows in the interval ω −g ≤ω (C )≤ω +g, whereas the
c c c,rev λ c
non-zero affinities Xα of circuits associated with heat leaks are proportional to the coupling constant g.
For a given choice of the system parameters the cooling power is determined by the positive contribution of
the tricycles operating in their cooling window and the negative contribution of the other tricycles and the circuits
associatedwithheatleaks. Theoptimalcouplingconstantg satisfiesγ (cid:28)g (cid:28)ω ,aregimewherethecontributionof
α c
theheatleaksisverysmall,thetricyclescoolingwindowsapproximatelycoincideandtherotatingwaveapproximation
is still valid. An example is shown in 2(c). As expected, the larger contributions correspond to the tricycles with a
lower number of edges. The maximum cooling rate is slightly greater and displaced to higher frequencies when the
device is connected through the two-level wire. This effect is the result of the evaluation of the rate functions (4) at
8
the displaced frequencies ω ±g, and increases with the bath physical dimension d . However, it cannot be further
α α
exploited as a larger g would increase the heat leaks, and in any case the COP would not improve.
If we examine the system performance characteristic, see figure 2(d), a closed curve is found for the indirectly-
coupledthree-leveldeviceindicatingtheexistenceofadditionalirreversibleprocesses: heatleaks,withsmallinfluence
for small g, and the internal dissipation appearing when approaching the upper limit of the cooling window. The
internaldissipationresultsfromthecompetitionofpositiveandnegativeheatcurrentsassociatedwithtricycleshaving
slightly different values of ω (C ) [8] and only works for very small ω (where the finite heat transfer rate effects
c,rev ν c
dominate) and in the interval ω −g <ω <ω −g. These irreversible contributions can be reduced decreasing
c,rev c c,rev
the coupling strength g, but cannot be avoided, making the reversible limit unattainable for the device connected
through a quantum wire.
Foroptimalcouplingconstantg,themainfeaturesofthesystemperformanceresultsfromthetricyclecontributions
and can be described with a small number of circuit representatives. We choose as circuit representatives C and C ,
1 2
with ω (C ) below and above ω respectively, see (20). Their fluxes are given by
c,rev 1,2 c,rev
I(C(cid:126) )=D−1det(−W|C )(cid:0)Wc WwWh −Wc WwWh(cid:1) ,
1 1 31 43 14 13 34 41
I(C(cid:126) )=D−1det(−W|C )(cid:0)Wc WwWh −Wc WwWh(cid:1) , (21)
2 2 31 53 15 13 35 51
from which the steady state heat currents of the circuit representatives Q˙R = Q˙ (C ) + Q˙ (C ) can be easily
α α 1 α 2
obtained. The currents Q˙R incorporate the main features of the system performance such as the frequency at which
α
the maximum cooling rate is reached and the essential irreversible processes, and might be renormalized to account
for the total heat currents [31]. Figure 2(d) compares the performance characteristic of the system and its circuit
representatives. For a more accurate description, or for larger values of g, a larger number of circuit representatives,
including for example heat leaks, might be needed.
Theanalysisofthenon-resonantcaseleadstothesamequalitativeresults,asthestructureofthegraphrepresenta-
tion of the master equation is not modified and the same simple circuits and irreversible mechanisms are found. The
maindifferenceisthedependenceofthetransitionsratesWα onthedetuning∆. Whenγ (cid:28)g (cid:28)∆,thecoefficients
ij α
c and c(cid:48) vanish and three circuits associated with heat leaks dominate the heat currents: {1,2,4,3,1} (from the
+ −
work bath to the cold one) {1,5,6,2,1} (from the work to the hot) and {1,3,4,2,6,5,1} (from the hot to the cold).
IV. QUANTUM THREE-LEVEL DEVICE COUPLED THROUGH A TWO-LEVEL SYSTEM TO A
WORK SOURCE
The three-level device coupled with a work source modeled by a periodic classical field is the simplest model of
driven quantum thermal machines [1]. The engine efficiency is given by η = −P/Q˙ and the refrigerator COP by
h
ε=Q˙ /P. The operating mode of the device is determined by the frequency of the transition coupled with the cold
c
bath. Whenω <ω −λη ,thedeviceworksasarefrigeratorwhereasforω >ω +λη itworksasanengine.
c c,max C c c,max C
Here we have introduced the coupling strength with the field λ and the engine Carnot efficiency η = 1−T /T .
C c h
At the limit frequency ω = ω T /T , an idealized device (λ = 0) would reach the engine Carnot efficiency or
c,max h c h
the refrigerator Carnot COP, ε = T /(T −T ). However, when ω −λη < ω < ω +λη the operating
C c h c c,max C c c,max C
modeofthethree-leveldeviceisgivenbythecompetitionbetweentheheatcurrentsassociatedwiththetwomanifold
resulting from the splitting of the system energy levels due to the field interaction. The competition of those heat
currents is the origin of the internal dissipation preventing the system to reach the Carnot performance in any of the
two working modes [1]. In this section we will analyze the three-level device connected to a classical driving field
through the two-level wire. The system Hamiltonian is
Hˆ = Hˆ + Hˆ + Hˆ + Hˆ (t) + (cid:88) (cid:16)Hˆ + Hˆ (cid:17) . (22)
D D,wire wire wire,w D,α α
α=c,h
All these terms were already introduced in the previous section except for Hˆ (t), which describes the coupling
wire,w
of the two-level system with the classical field,
Hˆ (t) = λ(cid:126)[|2 (cid:105)(cid:104)1 |exp(−iω t) + |1 (cid:105)(cid:104)2 |exp(iω t)]. (23)
wire,w W W w W W w
We will assume the resonant case in which the field frequency is equal to ω =ω −ω . As the Hamiltonian (22)
w h c
depends on time, the derivation of the quantum master equation described in section II requires some modifications
[4, 32, 33]. Let us define the operators Hˆ =Hˆ +Hˆ and
1 D wire
9
Hˆ = Hˆ + λ(cid:126)(|1 2 (cid:105)(cid:104)1 1 |+|2 2 (cid:105)(cid:104)2 1 |+|3 2 (cid:105)(cid:104)3 1 | + h.c.), (24)
2 D,wire D W D W D W D W D W D W
where h.c. stands for the Hermitian conjugate of the preceding terms. The eigenfrequencies of Hˆ , Hˆ |i(cid:105)=ω (cid:126)|i(cid:105),
2 2 i
are ω = −λ, ω = λ, ω = −[g +(4λ2 +g2)1/2]/2, ω = [g −(4λ2 +g2)1/2]/2, ω = [−g +(4λ2 +g2)1/2]/2 and
1 2 3 4 5
ω =[g+(4λ2+g2)1/2]/2. One can easily probe [33] that the propagator associated with Hˆ (t)=Hˆ + Hˆ +
6 S D D,wire
Hˆ + Hˆ (t) is given by Uˆ (t) = Uˆ (t)Uˆ (t) . In the following we assume that the Lamb shifts of the energy
wire wire,w S 1 2
levels of H can be neglected. The coupling operators with the cold and hot baths (16) are then decomposed into
S
5
Uˆ†(t)SˆαUˆ (t) = (cid:88)(cid:88) Sˆαexp[−i(ω +ω )t] + Sˆα†exp[i(ω +ω )t], (25)
S S ij α ij ij α ij
i=1 j>i
where Sˆα = cα|i(cid:105)(cid:104)j| and cα = (cid:104)i|Sˆα|j(cid:105). With these ingredients the LKGS generators for each bath (3) can be
ij ij ij −
obtained as the summation of the terms corresponding to the frequencies ω +ω , leading to the following quantum
α ij
master equation in the interaction picture under the unitary transformation associated with Uˆ†,
S
d (cid:88)
ρˆ (t) = L [ρˆ (t)]. (26)
dt I α I
α=c,h
The steady state properties can then be derived from the diagonal part of the equation (26) in the eigenbasis
of Hˆ , which resembles (6). Now the steady state populations p and energy currents must be interpreted as the
2 i
corresponding time-averaged quantities over a period τ = 2π/ω of the driving. The transition rates with indexes
w
j >i are given by
Wα =|c |2Γα , (27)
ij ij ωα+ωij
where the non-zero coefficients are
(1−u )2 (1+u )2
|c |2 =|c |2 = − ; |c |2 = |c |2 = − ;
13 26 4(1+u2) 23 16 4(1+u2)
− −
(1+u )2 (1−u )2
|c |2 =|c |2 = + ; |c |2 = |c |2 = + , (28)
14 25 4(1+u2) 24 15 4(1+u2)
+ +
with u =2λ/[g±(4λ2+g2)1/2]. The remaining transition rates can be obtained using (7) and (8).
±
The graph representation of the master equation is shown in figure 1(c). Each pair of vertices is simultaneously
connectedbytwoedges,oneassociatedwiththecoldbathandtheotherwiththehotbath. Thetopologicalstructure
of the graph is very simple as the states 1 and 2 are only coupled with 3, 4, 5 and 6. With this structure only simple
circuits with two or four edges can be found. We have identified 104 circuits, 12 of them being trivial circuits and
92 contributing to the steady state heat currents. The energy exchange between the system and the work source is
described by the power P =(cid:80) P(C ), being P(C )=−Q˙ (C )−Q˙ (C ) the contribution of each circuit.
ν ν ν c ν h ν
The simplest circuits are 8 two-edge tricycles C = {i,j,i}, where i = 1,2 and j = 3,4,5,6. In the following we
i,j
will assume that the two-edge circuits are oriented choosing i → j as the edge corresponding to the cold bath. The
affinities can be easily calculated to yield Xc(C(cid:126) )=−(cid:126)(ω +ω −ω )/T and Xh(C(cid:126) )=(cid:126)(ω +ω −ω )/T . When
i,j c j i c i,j h j i h
g,λ(cid:28)ω , the leading term in the affinities is proportional to the transition frequencies. The circuit fluxes (12) are
c,h
given by
I(C(cid:126) ) = D−1det(−W|C )(cid:0)WcWh −WcWh(cid:1) . (29)
i,j ij ji ij ij ji
With this expression the limit frequency of each circuit can be obtained imposing I(C(cid:126) )=0 to yield
i,j
ω (C ) = ω −(ω −ω )η . (30)
c,max i,j c,max j i C
10
4 30
(a) (b)
c 3
T h 20
kB 2 Tc (2)
B 10
γ 1 k
/ P (4)
) γ
9 0 / 0
C c
˙(αQ−1 3 2 c ˙4Q−10
810 −−32 4 1 10−20 ωc,max
−4 5 6
−30
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
ω /ω ω /ω
c h c h
1.0 1.0
(c) (d)
0.8 0.8
0.6 0.6
c c
ε η
/ /
ε η
0.4 0.4
0.2 0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
˙ / ˙ max / min
Qc Qc P P
FIG. 3. (a) Steady state heat currents as a function of the frequency ω for the circuit shown in the inset. (b) Heat current
c
with the cold bath Q˙ for the three-level device connected through the two-level wire (solid line) and directly coupled to the
c
classical field (dashed-dotted line). The labeled thin dashed lines show the contribution due to tricycles with two and four
edges. The vertical line indicates the frequency ω . (c) Performance characteristics for the system when working as a
c,max
refrigerator (ω < ω ) and (d) as an engine (ω > ω ) for the case in (b). The dashed lines depict the performance
c c,max c c,max
characteristicsofthecircuitrepresentativesC andC . Wehavefixedg=0.25,λ=0.05andtheotherparametersarethose
1,6 2,3
in figure 2.
When ω (C ) is approached from below the circuit reaches the refrigerator Carnot COP ε , and from above
c,max i,j C
the engine Carnot efficiency η .
C
We have also identified 96 four-edge circuits {i,j,i(cid:48),j(cid:48),i}: 12 trivial circuits, 60 tricycles (T Xc+T Xh (cid:54)= 0) and
c h
24 four-edge circuits associated with heat leaks (T Xc+T Xh =0). The tricycles can be classified into two different
c h
groups, one involving circuits with two edges associated with each bath, for which the affinities are proportional to
2ω +(ω +ω ), and the other with three edges associated to one of the baths, for which ω +ω −ω . The limit
α j j(cid:48) α j i
frequencies for the first group are ω (C) = ω −(ω +ω )η /2 and for the second are given by (30). The
c,max c,max j j(cid:48) C
circuits associated with heat leaks have affinities proportional to ω −ω or ω −ω . An example is the circuit
i i(cid:48) j j(cid:48)
C ={1,3,2,5,1} shown in figure 3(a) for which the affinities are Xc(C(cid:126) )=−2λ(cid:126)/T and Xh(C(cid:126) )=2λ(cid:126)/T .
9 3 c 3 h
The optimal coupling constants now satisfy γ (cid:28)g,λ(cid:28)ω , as the heat leaks are minimized and the energy flows
α c
aremainlydeterminedbythecontributionduetothetwo-edgetricycles. Figure3(b)showstheheatcurrentwiththe
coldbathforasignificantvalueofg, wherethecontributionofthefour-edgetricyclesbecomesrelevant. Asexplained
before for the absorption refrigerator, the device coupled through the wire reaches a larger maximum cooling rate.
Althougheachtricyclecanreachthereversiblelimit,theircombinationleadagaintointernaldissipation,nowworking
in the interval ω −f(λ,g)η <ω <ω +f(λ,g)η , with f(λ,g)=[λ+g+(4λ2+g2)1/2]/2, that depends
c,max C c c,max C
also on g.
We have found that the best choice of circuit representatives between the two-edge tricycles is determined by the
ratio g/λ: C , C when g/λ<1 and C , C when g/λ>1. For g (cid:28)λ the system performance is well described
1,4 2,5 1,6 2,3