Multi-stage quantum absorption heat pumps

1.3 MB·

by  Luis A. Correa

Multi-stagequantumabsorptionheatpumps
Luis A. Correa1,2,3,∗
1SchoolofMathematicalSciences,TheUniversityofNottingham,UniversityPark,NottinghamNG72RD,UK
2IUdEAInstitutoUniversitariodeEstudiosAvanzados, UniversidaddeLaLaguna, 38203Spain
3Dpto. F´ısicaFundamental,Experimental,Electro´nicaySistemas,UniversidaddeLaLaguna,38203Spain
(Dated:January17,2014)
Itiswellknownthatheatpumps,whilebeingalllimitedbythesamebasicthermodynamiclaws,mayfind
realizationonsystemsas‘small’and‘quantum’asathree-levelmaser.Inordertoquantitativelyassesshowthe
performanceofthesedevicesscaleswiththeirsize,wedesigngeneralizedN-dimensionalidealheatpumpsby
mergingN−2elementarythree-levelstages. Wesetthemtooperateintheabsorptionchillermodebetween
givenhotandcoldbaths,andstudytheirmaximumachievablecoolingpowerandthecorrespondingefficiencyas
afunctionofN. Whiletheefficiencyatmaximumpowerisroughlysize-independent,thepoweritselfslightly
increases with the dimension, quickly saturating to a constant. Thus, interestingly, scaling up autonomous
4 quantumheatpumpsdoesnotrenderasignificantenhancementbeyondtheoptimaldouble-stageconfiguration.
1
0 PACSnumbers:03.65.-w,03.65.Yz,05.70.Ln,03.67.-a
2

n
a I. INTRODUCTION becauseoftheirinherentsimplicity,thatallowstogaininsight
J intotheemergenceofthelawsofthermodynamicsfromindi-

6 Anabsorptionheatpumpisamulti-purposethermaldevice vidualopenquantumsystems[33].
1 inwhichthedrivingforceismostlyheat,ratherthanmechan- Inmanypracticalsituations,absorptionchillersmightalso
icalwork[1]. ThefirstdesignthereofwaspatentedbyFerdi- be a preferable alternative to power-driven quantum devices.
]
h nand Carre´ in 1860 and played a prominent role in the early Nonetheless, as in the classical case, these are usually much
p development of refrigerators. Anecdotally, in 1927, Albert lessefficientandpowerfulthanthebestreversedheatengines
- Einstein and Leo´ Szila´rd presented an absorption fridge run- [34], which turns the optimization of their performance into
t
n ningsolelyonheat,thatoperatedatnearlyconstantpressure, amatterofparamountimportance. Somestepshavebeenal-
a requiringnomovingparts[2]. Inspiteoftheobviousadvan- ready undertaken in this direction, by identifying the upper
u
tages of absorption technologies in terms of safety, reduced bounds on a suitable figure of merit for the cooling perfor-
q
environmentalimpactorbetterexploitationofthefreelyavail- mance, and the design prescriptions for their saturation [35],
[
able thermal resources, mechanical compression cycles are andbyproposingreservoirengineeringtechniques[36,37]in

1 preferredformostindustrialandcommercialapplications,as ordertopushthoseboundsfurther[34].
v theseareusuallymuchmoreefficient. Consequently,consid- Notwithstanding the important differences between clas-
1
erable effort has been devoted over the last decades to im- sical and quantum absorption heat pumps, it is most natu-
1
0 provetheperformanceofabsorptionheatpumps[3]inorder ral to ask whether significant enhancement can be expected
4 tomakethemcompetewiththeirpower-drivenanalogues.For fromscalingupquantumthermodynamiccyclesinto“larger”
. instance,onecouldthinkofscalingupabsorptionsystemsinto multi-stage networks. In order to quantitatively assess the
1
largermulti-stageconfigurationscapableofreusingtheresid- sizescalingontheperformanceofheatpumps,weconstruc-
0
4 ualoutputheatforfurthercooling,thusboostingtheiroverall tively build generalized N-level ideal absorption devices by
1 performance[4–7]. couplingtogetherN−2elementarythree-levelcycles,some-
: Itwasalreadyacknowledgedinthelate1950sthatathree what in the spirit of parallel multi-stage classical absorption
v
i levelmaserselectivelycoupledtotwoheatbathsandacoher- devices [1]. We then compute the efficiency at maximum
X ent driving field, is a valid embodiment for a quantum heat cooling power and the maximum power itself as a function
r engine [8–10], which running in reverse ultimately amounts of N for 3 ≤ N ≤ 10. While the efficiency proves to be
a
to a power-driven fridge. Ever since, quantum heat engines roughly size-independent, the corresponding power only in-
andrefrigeratorshavebeenobjectofextensivestudy[11–20], creases mildly with the number of stages, quickly saturating
including detailed experimental proposals [21, 22], and the to a constant asymptotic value. Therefore, and again in to-
developmentofstrategiestoovercometheirultimatethermo- tal correspondence with the classical scenario, double-stage
dynamicslimitations[19,20,23,24]. absorption heat pumps appear as the most reasonable com-
On the other hand, a three-level maser operating between promise between simplicity and performance. Interestingly,
heat baths only (i.e. without driving) realizes a quantum ab- thiscaseof N = 4happenstocorrespondwiththetwo-qubit
sorption heat pump [25], that can function either as a fridge modelrecentlyintroducedin[26],thusmarkingitasatarget
or as a ‘heat transformer’. In particular, quantum absorption forexperimentalimplementations.
fridges have attracted a lot of attention [20, 26–32], mostly We also carry out an extensive numerical investigation
thatsupportsthegeneralvalidityandtightnessofthemodel-
independent performance bound established in [34], for all
ideal multi-stage absorption refrigerators. Finally, we com-
∗Electronicaddress:[email protected] parethenon-idealeight-levelabsorptionfridgeof[27]withall
2
its ideal counterparts, when operating under the same condi- Notethatthefrequencyoftheworktransition|2(cid:105)↔|3(cid:105)isjust
tions. Wefindthatidealfridgeslargelyoutperformnon-ideal ω ≡ ω − ω . We will take Eq. (1) as the defining prop-
w h c
ones, not only when it comes to their maximum achievable erty of an ideal heat pump. More generally, the asymptotic
efficiencybut,especially,intermsoftheirmaximumcooling stationarityofenergy,translatesinto
power.
This paper is structured as follows: We start by providing Q˙w+Q˙h+Q˙c =0, (2)
a brief review of quantum absorption heat pumps in Sec. II.
whichholdsforbothidealandnon-idealdevices. Sincethere
In Sec. III, we introduce our constructive scheme to build
is no work involved in the operation of the pump, Eq. (2) is
generalized multi-stage parallel cycles, discuss their micro-
just a statement of the first law of thermodynamics. On an-
scopicmodelandcorrespondingmasterequation, andobtain
other note, and always provided that the dissipation is suffi-
the steady-state heat currents flowing across the system. We
cientlyweak, thepositivityoftherateofirreversibleentropy
shall be then, in Sec. IV, in the position to quantitatively
production [38, 39] makes it possible to write down the sec-
comparethemaximumcoolingpowerandthecorresponding
efficiency of devices with increasing N and fixed input ondlawofthermodynamicsintheform[33]
thermal resources. Finally, in Sec. V, we summarize and
Q˙ Q˙ Q˙
drawourconclusions. w + h + c ≤0. (3)
T T T
w h c
Eqs.(2)and(3)encodemostoftherelevantinformationabout
absorptionheatpumps. Letustakeforinstance,anidealde-
II. IDEALQUANTUMABSORPTIONHEATPUMPS vice,assumingthatitworksinthechiller/heatermode,sothat
Q˙ ,Q˙ >0andQ˙ <0. Then,combiningEqs.(1)and(3)one
c w h
Letusconsiderathree-levelsystemwithHamiltonianHˆ = arrivesto
3
(cid:126)ωc|2(cid:105) + (cid:126)ωh|3(cid:105) in its energy eigenbasis {|1(cid:105),|2(cid:105),|3(cid:105)}. We (T −T )T
shall allow for a very weak dissipative interaction between 0<ωc <ωc,max ≡ (Tw−Th)Tcωh, (4)
the transition |1(cid:105) ↔ |2(cid:105) and a ‘cold’ heat bath at tempera- w c h
ture Tc. Likewise, we couple the transitions |2(cid:105) ↔ |3(cid:105) and whichdefinesthecoolingwindow,thatis,theregioninparam-
|3(cid:105) ↔ |1(cid:105)veryweaklytoahightemperature‘work’reservoir eterspaceinwhichcooling(andheating)ispermittedbythe
anda‘hot’bathrespectively, suchthatTw > Th > Tc. Once secondlaw. Forωc,max <ωc <ωh,itistheheat-transforming
itsasymptoticstatebuildsup,thissystemoperatesasthemost cycle the one that dominates, thus rendering a quantum ab-
fundamental quantum absorption heat pump [25], relying on sorptionheattransformer.
the imbalance between the steady-state rates associated with From the cooling point of view, the ratio of the incom-
thecoolingcycle|1(cid:105)→−c |2(cid:105) −→w |3(cid:105)→−h |1(cid:105),anditscomplement, ing cold heat Q˙c and the energetic cost Q˙w of its process-
h w c ing,definestheefficiencyorcoefficientofperformance(COP)
theheat-transformingcycle|1(cid:105)→− |3(cid:105)−→|2(cid:105)→− |1(cid:105). ε≡Q˙ /Q˙ ,whichbenchmarkstheusefuleffectofanabsorp-
c w
Whentheasymptoticrateofthecoolingcycleexceedsthat
tionchiller. CombiningEqs.(2)and(3),onegenerallyfinds
oftheheat-transformingcycle, net‘heat’perunittimeisex-
tracted from the cold and work reservoirs and dumped into (T −T )T
ε≤ w h c ≡ε , (5)
the hot bath. That is, net energy is moved between the hot (T −T )T C
h c w
andcoldbathsagainst thetemperaturegradient, withtheas-
sistance of the input heat coming from the work bath. De- that is, that the maximum efficiency allowed is the cor-
pendingonthewhethertheusefuleffectissoughtinthecold responding Carnot COP ε [1]. It is easy to see that the
C
orthehotbath,wespeakofaquantumabsorptionchillerora Carnot efficiency is indeed saturated by an ideal fridge at
quantum absorption heater. If on the contrary, the stationary ω → ω , where each transition equilibrates with its
c c,max
rate of the heat-transforming cycle exceeds that of the cool- localbathandconsequently, allheatcurrentsvanish[10]. In
ing cycle, low-quality heat coming from the hot bath would general, for a non-ideal absorption chiller [i.e. a system not
be upgraded to high-temperature heat leaking into the work obeying Eq. (1)], Eq. (5) would be a strict inequality [35],
bath, only by means of the coupling to the cold bath. These whileEq.(4)wouldceasetohold.
arethetwomodesofoperationofbothclassicalandquantum
absorptionheatpumps[1].
At the steady state and given that each reservoir interacts
locally with its corresponding transition [35], it is intuitively III. GENERALIZEDN-LEVELIDEALHEATPUMPS
clearthatallthreeheatcurrents(work,hotandcold)mustflow
at the same rate, in order to keep the average energy asymp- An autonomous thermal device based on two non-
totically stationary [10, 28], i.e. one must have: Q˙ = ω q, interacting qubits was put forward in Ref. [26], which ulti-
α α
whereQ˙ standsforthesteady-stateheatcurrentflowingfrom matelyamountstotwoelementarythree-levelcyclessharing
α
bathα∈{w,h,c}intothesystem. Therefore the work transition. It can be seen that it realizes a parallel
double-stageidealheatpumpandthatthepartialheatcurrents
(cid:12) (cid:12)
(cid:12)(cid:12)Q˙α/Q˙α(cid:48)(cid:12)(cid:12)=ωα/ωα(cid:48). (1) from each stage just add up to the total [34]. The overlap
3

|5 …
2𝜔 |5
ℎ
|4
|4 |3 𝜔ℎ + 𝜔𝑐 |4
|3 𝜔 |3
ℎ
|3
|2 𝜔 |2
𝑐
|2
0 |1
|1
FIG.1:Ontherighthandside,weshowthedetailofthefirstfivelevelsofageneralizedN-dimensionalidealquantumabsorptionheatpump.
Thecoloredarrowsrepresentdissipativeinteractionbetweenthecorrespondingtransitionandoneofthethreeheatreservoirs: Bluestands
for the cold bath, orange, for the hot bath, and red, for the high temperature work reservoir. Note that the energy difference between two
consecutivelevelskeepsalternatingbetweenωc andωw = ωh−ωc. Ontheleft ha n d𝑻side,wes howthequtritbreakupofa5-levelfridge: It
consistsofthreeelementarythree-levelcyclesoperatinginparallel. Thefirstandthesecondsharetheworktransition,whilethesecondand
𝒄 𝝎
thethird,shareacoldtransition. Thenextelementaryblocktobeaddedfora6-levelheatpump,wouldsharetheworktransition|4(cid:105) ↔ |5(cid:105)
𝒄 𝑻
withtheprecedingblock,while|5(cid:105)↔|6(cid:105)and|4(cid:105)↔|6(cid:105)wouldbeconnectedwiththecoldandhotbathsrespectively.
𝒉
𝑻 𝝎
𝒉
betweenthem,however,makestheirindividualcontrib𝝎utions 𝝎HˆS-𝒘B = (cid:80)αΣˆ(αN)⊗𝒘Bˆα,w𝝎here𝒘
𝑻
smallerthanwhatwouldbeexpectablefromtwoindepende𝒉nt
three-level heat pumps. Once more, the corres𝒉pondence be- Σˆ(N) =(cid:88)(cid:100)N/2(cid:101)−1|2n(cid:105)(cid:104)2n+1|+|2n+1(cid:105)(cid:104)2n| (7a)
tweentheclassicalandthequantumcaseisapparent[1,3,4], w𝑻 n= 1
andthequestionnaturallyarisesaboutthesizescalingofthe 𝝎 Σˆ(N) =(cid:88)𝒄N−2|n(cid:105)(cid:104)n+2|+|n+2(cid:105)(cid:104)n| (7b)
coolingpowerQ˙ andtheCOPεofmulti-stagequantumab- 𝒄 h n=1
sorptionrefrigeractors. Doesthesizereallymatterinquantum Σˆ(N) =(cid:88)(cid:98)N/2(cid:99)|2n−1(cid:105)(cid:104)2n|+|2n(cid:105)(cid:104)2n−1|. (7c)
c n=1
absorptioncooling?
The reservoirs may consist, as usual, of an infinite collec-
𝝎 = 𝝎 − 𝝎
tion of uncoupled harmonic modes in three dimensions. We
𝒘 𝒉 𝒄
choosethebathcouplingoperatorsBˆ tobe
α
(cid:88) (cid:16) (cid:17)
A. Microscopicmodelandmasterequation Bˆ = g bˆ +bˆ† . (8)
α αµ αµ αµ
µ
A generalized N-level ideal heat pump may be built by
mergingN−2three-levelsystems,alternativelysharingawork Withbˆαµandbˆ†αµ,wedenotetheannihilationandcreationop-
oracoldtransition. InFig.1,thecaseN = 5(triple-stage)is eratorsformodeωµ√inthereservoirα,anddefinethecoupling
schematically illustrated (see figure caption for details). The constantsasgαµ ≡ γαωµ,whichyieldsflatspectraldensities
(cid:80)
totalHamiltonianofsuchN-levelsystemreads (Jα(ω)∝ µg2αµ/ωµ).Thedissipationstrengthsγαarechosen
soastodefinethelargestdynamicaltime-scaleintheproblem
(cid:98)(cid:88)N/2(cid:99) (cid:100)N(cid:88)/2(cid:101)−1 (cid:40) (cid:126) (cid:41)
HˆN = [(n−1)ωh+ωc]|2n(cid:105)+ nωh|2n+1(cid:105), (6) γα−1 ≫ |ωα−ωα(cid:48)|−1,k T . (9)
n=1 n=1 B α
This allows us to perform the Born-Markov and secular ap-
where(cid:98)·(cid:99)and(cid:100)·(cid:101)standforthefloorandceilingfunctions. Let proximations in deriving a master equation for the system
usmodelthesystem-bathsinteractionswithatermoftheform [40]. Under the further assumption of a preparation with
4
totally uncorrelated system and equilibrium reservoirs, one Ingeneral,forarbitraryN,onewouldbeleftwith
readilyobtains
Q˙(N) =ω (cid:88)(cid:100)N/2(cid:101)−1(cid:104)2n+1|L(N)(cid:37)ˆ(∞)|2n+1(cid:105) (15a)
d (cid:16) (cid:17) w w n=1 w
dt(cid:37)ˆN(t)= L(wN)+L(hN)+L(cN) (cid:37)ˆN(t), (10) Q˙(N) =ω (cid:88)N ((cid:100)n/2(cid:101)−1)(cid:104)n|L(N)(cid:37)ˆ(∞)|n(cid:105) (15b)
h h n=3 h
where(cid:37)ˆN(t)isthestateofthesystemintheinteractionpicture Q˙(N) =ω (cid:88)(cid:98)N/2(cid:99)(cid:104)2n|L(N)(cid:37)ˆ(∞)|2n(cid:105), (15c)
andthedissipatorsL(N)aregivenby c c n=1 w
α
which provides, together with Eqs. (7) and (11), closed for-
(cid:32) (cid:33)
L(N)(cid:37)ˆ =Γ σˆ− (cid:37)ˆσˆ+ − 1σˆ+ σˆ− (cid:37)ˆ− 1(cid:37)ˆσˆ+ σˆ− + mulasfortheevaluationofthestationaryheatcurrents. From
α α,ωα N,α N,α 2 N,α N,α 2 N,α N,α nowon,weshallfocusonthechilleroperationmode. Atfirst
(cid:32) (cid:33) glance,onecouldintuitivelyexpectadifferentscalingofe.g.
1 1
Γα,−ωα σˆ+N,α(cid:37)ˆσˆ−N,α− 2σˆ−N,ασˆ+N,α(cid:37)ˆ− 2(cid:37)ˆσˆ−N,ασˆ+N,α . (11) |Q˙(cN)| for even and odd N: The addition of an even level to
the heat pump provides it with an extra cold transition, so
Here, the positive decay rates Γ are the diagonal ele- that a new term contributes to the cold current in Eq. (15c)
α,ωα while, whentheaddedlevelisodd, thenewcontributionap-
ments of the spectral correlation tensor [40], which for a
pears instead in the work current of Eq. (15a). It must be
three-dimensional free bosonic field in thermal equilibrium
are given by Γ ∝ ω3[1 + n (ω)] > 0, with n (ω) = noted, however, that each of the already contributing terms
(exp(cid:126)ω/kBTα−α1,ω)α−1. The mutuaαlly adjoint jump opαerators will always decrease as N grows, e.g. |(cid:104)2|L(c3)(cid:37)ˆ3(∞)|2(cid:105)| <
σˆ+ =(σˆ− )†arejust |(cid:104)2|L(4)(cid:37)ˆ (∞)|2(cid:105)|whengoingfromEq.(13c)toEq.(14c).
N,α N,α c 4
ItcanbeseenbynumericalinspectionofEqs.(15)thatin-
σˆ−N,w =(cid:88)n(cid:100)N=/12(cid:101)−1|2n(cid:105)(cid:104)2n+1| (12a) cmreeanstainlgfothreanllucmubrreernotsf,sit.aeg.e|sQ˙f(rNo+m1)|ev≤en|Q˙t(oNo)|d,dwihseinndNeeidsdeveterni-.
σˆ− =(cid:88)N−2|n(cid:105)(cid:104)n+2| (12b) Still for arbitrary N, one alwaαys finds |Qα˙(αN+2)|−|Q˙(αN)| ≥ 0,
N,h n=1 thoughtheenhancementdecreasesasthesystemscalesup.
σˆ− =(cid:88)(cid:98)N/2(cid:99)|2n−1(cid:105)(cid:104)2n| (12c) For practical purposes, one would wish to optimize the
N,c n=1 incoming cold heat current to cool as quickly as possible.
Given a set of heat baths T and dissipation rates γ , one
ThedissipatorsinEq.(11)areinthestandardLindbladform α α
has to find the cold frequency 0 ≤ ω ≤ ω that renders
[39],whichguaranteesthatthereduceddynamicsofthesys- c c,max
the maximum cooling power Q˙ for every fixed ω . The
temiscompletelypositiveandtrace-preserving.Eachofthem c,max h
correspondingCOPε isasensiblefigureofmeritforcooling
individually,generatesafullycontractivedynamicsofthesys- ∗
tem towards its ‘local’ equilibrium states ∝ exp{−Hˆ /k T } and, asrigorouslyprovenin[34], itistightlyupperbounded
N B α by d ε both in the single and double-stage cases, where
[38],whichinturn,impliesthesecondlawofthermodynam- d+1 C
d stands for the spatial dimensionality of the cold bath, i.e.
icsasexpressedbyEq.(3)[30,33].
in our case d = 3. Once parameter optimization has taken
placeatthesingle-stagelevel,weshallbeconcernedwiththe
potentialenhancementinbothε andQ˙ asthenumberof
B. Stationaryheatcurrents ∗ c,max
stages,andthusthesystem’scomplexity,grows.
In the light of Eqs. (10) and (11), our definition of
steady-state heat current from Sec. II translates into Q˙(N) ≡
α
tr{HˆNL(αN)(cid:37)ˆN(∞)}[25], wheretheasymptoticstate(cid:37)ˆN(∞)re- IV. RESULTSANDDISCUSSION
sultsfromequatingtozerotheright-handsideofEq.(10).
InthesimplestcaseofN =3,oneeasilyfinds
A. MaximumpowerandCOPatmaximumpower
Q˙(3) =ω (cid:104)3|L(3)(cid:37)ˆ (∞)|3(cid:105) (13a)
w w w 3 The dependence of the maximum cooling power on the
Q˙(h3) =ωh(cid:104)3|L(h3)(cid:37)ˆ3(∞)|3(cid:105) (13b) numberofstagesisillustratedinFig.2(a). Thetemperatures
Q˙(3) =ω (cid:104)2|L(3)(cid:37)ˆ (∞)|2(cid:105). (13c) Tα,dissipationratesγαandthehotfrequencyωhwerechosen
c c c 3
atrandomsoastofixthetimescalesforthermalfluctuations,
Theadditionofasecondstagetotheheatpumpintroducesthe dissipation and system’s free evolution. Then, the optimiza-
further cold and hot transitions |3(cid:105) ↔c |4(cid:105) and |2(cid:105) ↔h |4(cid:105) that tioAnsincoωucldwabseceaxrprieecdteoduftrwomithtihnetheveecno/oodlidngoswciinlldaotiwon.softhe
contributetothetotalsteady-stateheatcurrentstoyield stationaryheatcurrents, Q˙ (N)scalesdifferentlyforeven
c,max
Q˙(4) =ω (cid:104)3|L(4)(cid:37)ˆ (∞)|3(cid:105) (14a) andodd N. Whileintheevencase,thecoolingpowerbarely
w w w 4 increases, odd heat pumps significantly benefit from scaling
(cid:16) (cid:17)
Q˙(4) =ω (cid:104)3|L(4)(cid:37)ˆ (∞)|3(cid:105)+(cid:104)4|L(4)(cid:37)ˆ (∞)|4(cid:105) (14b) up. Onealsoseesthatanyevenconfigurationoutperformsall
h h h 4 h 4
(cid:16) (cid:17) theoddones.Incontrast,thecorrespondingCOPatmaximum
Q˙(4) =ω (cid:104)2|L(4)(cid:37)ˆ (∞)|2(cid:105)+(cid:104)4|L(4)(cid:37)ˆ (∞)|4(cid:105) . (14c)
c c c 4 c 4 power ε proves to be roughly size-independent, as shown
∗
5
0.10 a b
0.75
0.08
(cid:72) (cid:76) (cid:72) (cid:76)
(cid:182)
x (cid:42)
ma 0.06 (cid:182) (cid:144) (cid:144)
c, C (cid:144) (cid:144)
Q
(cid:160)
0.744795
0.04
0.744792
0.02
2 4 6 8 10 2 4 6 8 10
N N
FIG.2: (a)(Squares)Maximumcoolingpowerasafunctionof N forgeneralizedmulti-stageheatpumpsinthechillerconfiguration,with
parameters: T = 7.1×103,T = 1.57×103,T = 54.25,γ = 3.5×10−3,γ = 5.1×10−3,γ = 8.8×10−3 andω = 102.6((cid:126) = k = 1).
w h c w h c h B
(Rhombs)MaximumcoolingpowerforincreasingNwhenthemodesoftheworkreservoirareallsqueezedby7dB,whicheffectivelyraises
theworktemperaturetoT (cid:39)1.8×104.(Triangles)MaximumcoolingpowerversusNforreversedmulti-stageCarnotengines,i.e.heatpumps
w
withsaturatedworktransitions. Thesolidorange,dottedredanddashedbluecurvesaremerelyaguidetotheeye,highlightingthedifferent
sizescalingofdeviceswithevenandoddnumberofparallelcycles. Thedomainreachablebyabsorptionchillersisdepictedinshadedgray.
(b)EfficiencyatmaximumpowerasafunctionofN,forthesameparametersasin(a). Theshadedgrayregioncorrespondthethedomainof
ε allowedbyabosoniccoldbathinthreedimensionswithflatspectraldensity[34].
∗
in Fig. 2(b). Consequently, the double-stage absorption heat sible, to benchmark the operation of realistic devices. The
pump [26] tuned to operate at maximum power, appears as paradigmaticexampleistheCurzon-Ahlbornbound[41]: Al-
thebestpossiblechoiceintermsofsystemoptimization. Our thoughobtainedforaspecificphenomenologicalmodelingof
extensivenumericsshowthatthesepropertiesarecompletely the sources of irreversibility, i.e. the endoreversible approx-
generalintherangeofapplicationofEqs.(10)and(11). imation, it emerges naturally as a limiting case in different
Itisalwayspossibletoenhanceboththecoolingpowerand instances of heat engine [42, 45, 46] and describes well the
thecorrespondingCOP,bysuitablyengineeringthespectrum performanceofactualpowerplants.
of the environments [22–24]. Perhaps the simplest thing to The bound ε < 3ε , derived from first principles for the
∗ 4 C
try is squeezing the modes of the work bath, which effec- three-level maser and the four-level double stage chiller in
tively raises its temperature T and boosts the output power [34], has been also seen to hold [35] in the non-ideal eight-
w
of the heat pump [34]. The Carnot bound ε is also effec- levelrefrigeratormodelof[27],andevensucceedsinlimiting
C
tivelyincreased,whichallowsforlargerefficienciesatmaxi- from above the performance of classical endoreversible ab-
mumpower. Theeffectofreservoirsqueezinginthecooling sorptionfridges[47]. Inordertoestablishwhetheritcontin-
power is also illustrated in Fig. 2(a) for a squeezing of 7 dB ues to hold for multi-stage ideal absorption refrigerators be-
or r (cid:39) 0.8 (see figure caption for details). As T increases, yondthecasesN =3andN =4,weperformedglobalnumer-
w
the corresponding contact transitions tend to saturate. In the ical optimization of ε over ω , T , γ and N ∈ [3,10]. An
∗ h α α
limitofT →∞,onerealizesgeneralizedmulti-stageCarnot histogram on the resulting COP at maximum power for 105
w
enginesrunninginreverse[10]andthus,abandonstherealm randomlydrawnrefrigeratorsispresentedinFig.3(a),clearly
of absorption chillers. The limiting cooling powers are also showingthattheboundholdstightlyregardlessofthesizeof
depictedinFig.2(a). thesystem.
B. BoundontheCOPatmaximumpower C. Idealvs.non-idealabsorptionfridges
ThequestfortightboundstotheCOPatmaximumpower Alotofinteresthasbeengeneratedbythenon-idealeight-
for heat engines and refrigerators has attracted a lot of inter- level (three-qubit) absorption fridge, first introduced in [27],
est over the last decades, both in the classical [41–43] and including experimental proposals for its implementation on
quantum [16, 35, 44, 45] scenarios. By making minimal as- super-conductingqubits[48]andquantumdots[49]. Itisle-
sumptionsaboutthedominatingsourcesofirreversibility,the gitimatetoaskhowdoesitcompare,intermsofabsolutecool-
aimistoderivepracticalbounds,aswidelyapplicableaspos- ingpower,withitsidealeight-levelcounterpart. Interestingly,
6
aa bb
0.001
33 44
(cid:72)(cid:72) (cid:76)(cid:76)
(cid:72)(cid:72) (cid:76)(cid:76)
(cid:144)(cid:144) 10(cid:45)5
(cid:160)
Q
c
10(cid:45)7
10(cid:45)9
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(cid:182) (cid:182) (cid:182) (cid:182)
(cid:42) C C
FIG.3: (a)HistogramoftheCOPatmaximumpowerfor105 multi-stageabsorptionfridgeswithω ,T ,γ andN chosenatrandom. The
h α α
blueverticallinemarksthe 43εCultimate(cid:144)boundonε∗[34].(b)Coolingpowerversusnormalizedefficien(cid:144)cyforthethree-qubitnon-idealfridge
of[27](solidorange),andforaneight-levelidealchiller(dottedblue)forω =60,T =130,T =60,T =5andγ =10−3.Thethree-body
w w h c α
interactionstrengthbetweenthequbitsinthenon-idealmodelwassettog=0.1.Again,naturalunitsareassumed.
allidealmulti-stagefridges(eventhethree-levelmaser), can thesizeofthesystem.
beseentolargelyoutperformtheeightlevelnon-idealchiller, Even if the heat currents were formally given in Eqs. (11)
typicallybyseveralordersofmagnitude. and (15), the explicit formulas for any N are quite involved
In Fig. 3(b) this is illustrated by plotting together the per- and nothing but insightful. The whole analysis is therefore
formancecharacteristicsofboththenon-idealthree-qubitand based on extensive numerics. It must be acknowledged as
theideal‘sextuple-stage’(N =8)chiller,forthesamechoice well, that the working substance of an ideal multi-stage ab-
of parameters. A closed curve in the Q˙ – ε plane is indica- sorption refrigerator is generally a rather artificial quantum
c
tive of irreversibility and prevents the refrigerator from ever system. Interestinglyenough,thecaseofN = 4,thatappears
realizingtheCarnotefficiency. TheCOPatmaximumpower as the optimal compromise between performance and com-
is roughly the same in both cases (close to its ultimate 3ε plexity,alsoturnsouttocorrespondtothetwo-qubitmodelof
4 C
bound), but the power Q˙ of the ideal fridge exceeds that [26]. This suggests that it should be considered for practical
c,max
ofthenon-idealonebyoverthreeordersofmagnitude. applicationsofabsorptioncoolingtoquantumtechnologies.
This suggests that any realistic application of absorption The double-stage fridge (more generally, any ideal heat
technologies to quantum cooling, should rely upon physical pump) was seen to outperform by several orders of magni-
implementations of ideal (multi-stage) heat pumps, out of tudethemaximumcoolingpowerdeliveredbytheonlyavail-
which the double-stage design [26] stands out with its opti- able (non-ideal) alternative model: The three-qubit refriger-
malbalancebetweenperformanceandsimplicity. ator [27]. Arrangements of superconducting qubits [48] or
coupled quantum dots [49], similar to those proposed to re-
alizethethree-qubitdevice,couldalsobegoodcandidatesto
supportthemorepromisingdouble-stageabsorptionchillers.
V. CONCLUSIONS
Further enhancement of both the power and the efficiency
can always be achieved by applying quantum reservoir
Wehavestudiedthesizescalingofthepowerandtheeffi-
engineering techniques [36, 37] meant to render exploitable
ciencyofparallelmulti-stagequantumabsorptionheatpumps.
nonequilibrium environmental fluctuations, thus raising
Whenworkinginthechillerconfiguration, heatpumpscom-
absorptiontechnologiestothelevelofthebestcompression-
prisedofanoddandevennumberstages,scaledifferentlyin
basedquantumthermodynamiccycles.
terms of their maximum achievable cooling power: Devices
with even N always provide a larger power, which is almost
size-independent,whileinheatpumpswithoddN,themaxi-
mumpowerincreaseswiththenumberofstages,quicklysat-
urating to a constant asymptotic value. Contrarily, the effi- Acknowledgements
ciency at maximum power does not significantly scale with
N and, upon global optimization over all free parameters, it TheauthorisgratefultoG.Adesso,J.P.Palao,D.Alonso,
saturatestothesamemodel-independentbound,regardlessof K. Hovhannisyan, P. Skrzypczyk, N. Brunner and P. Ramos-
7
Garc´ıa for fruitful discussions and useful comments. This Canary Islands Government through the ACIISI fellowships
workwassupportedbytheCOSTActionMP1006andbythe (85%cofundedbyEuropeanSocialFund).
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