Reversing the Landauer's erasure: single-electron Maxwell's demon operating at the limit of thermodynamic efficiency

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by  Dmitri V. Averin

physicastatussolidi
Reversing the Landauer’s
erasure: single-electron Maxwell’s
demon operating at the limit of
thermodynamic efficiency
DmitriV.Averin*,1 andJukkaP.Pekola2
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1 DepartmentofPhysicsandAstronomy,StonyBrookUniversity,SUNY,StonyBrook,NY11794-3800
1
2 LowTemperatureLaboratory,DepartmentofAppliedPhysics,AaltoUniversitySchoolofScience,P.O.Box13500,00076Aalto,
0
Finland
2

n ReceivedXXXX,revisedXXXX,acceptedXXXX
a
PublishedonlineXXXX
J

9
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Keywords: Maxwell’sdemon,single-chargetunneling,nanostructurethermodynamics,fluctuation-dissipationrelations

] ∗Correspondingauthor:[email protected]
l
l
a
h
-
s
e According to Landauer’s principle, erasure of informa- Inthecaseoftheeffectivelytwo-levelsystem,whichhas
m tion is the only part of a computation process that un- been realized experimentally based on tunneling of in-
avoidablyinvolvesenergydissipation.Ifdonereversibly, dividual electron in a single-electron box [1], we also
.
t such an erasure generates the minimal heat of k T ln2 studiedandminimizedcorrectionstotheidealreversible
a B
m pererasedbitofinformation.Thegoalofthisworkisto operation of the demon. These corrections include, in
discuss the actual reversal of the optimal erasure which particular, the non-adiabatic terms which are described
-
d can serve as the basis for the Maxwell’s demon operat- by a version of the classical fluctuation-dissipation the-
n ing with ultimate thermodynamic efficiency as dictated orem. Theoverall reversibility ofthe Maxwell’s demon
o by the second law of thermodynamics. The demon ex- requires,besidethereversibilityoftheintrinsicworking
c tracts k T ln2 of heat from an equilibrium reservoir at bodydynamics,thereversibilityofthemeasurementand
[ B
temperatureT peronebitofinformationobtainedabout feedback processes. The single-electron demon can, in

1 the measured system used by the demon. We have ana- principle,bemadefullyreversiblebydevelopingather-
v lyzedthisMaxwell’sdemoninthesituationwhenituses modynamically reversible single-electron charge detec-
6 a general quantum system with a discrete spectrum of tor for measurements of the individual charge states of
6
energylevelsasitsworkingbody. thesingle-electronbox.
2
5
0
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.
1
0
7 1 Introduction Thermodynamics of nanostructures, Thefoundationforthephysicsofinformationwaspro-
1 in which quantum mechanics and statistical fluctuations vided by Rolf Landauer in the context of thermodynam-
: playanimportantrole,hasrecentlyattractedconsiderable ics of computation. He demonstrated [12] that erasure of
v
attention,see,e.g.,thereviews[2,3,4,5,6]andreferences informationistheonlypartofacomputationprocessthat
i
X therein. One of the most interesting issues in this field is unavoidablyinvolvesenergydissipation.Erasureofonebit
r realization of Maxwell’s demon [7,1,8,9], and more gen- ofinformationconsistsinbringingatwo-statesystemini-
a erally, understanding the role of information in thermo- tially in the most uncertain configuration (both states oc-
dynamicprocesses[10].Developmentofthermodynamics cupiedwithprobability1/2)tothepre-determinedconfig-
of information is also motivated by practical attempt to uration,whenthesystemisinoneofitsstateswithprob-
demonstrate thermodynamically reversible computation – ability 1. This process leads to generation of heat in the
see,e.g.,[11]. reservoir at temperature T, with the minimum of gener-
atedheat,k T ln2,achievediftheerasureisperformedin
B
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2 DmitriAverinetal.:ReversibleMaxwell’sdemon
theoptimal,i.e.reversible,way,aswasrecentlyconfirmed randomnessandthermalizationwhichrequirefurtherstud-
experimentally[13,14,15]. ies.
In this work, we illustrate the connection between
It has been understood for quite some time (see, e.g.,
the entropy and information by considering the reversible
[16])thattheprincipleofinformationerasure,andnotthe
Maxwell’s demon based on a general multi-level quan-
energydissipationinthemeasurementprocessasbelieved
tum system. The results here generalize to the multi-level
previously [17], provides the resolution of the Maxwell’s
situation the discussion in Refs. [1,19] of the Maxwell’s
demon paradox, reconciling operation of this device with
demon operation and thermodynamics properties of the
thesecondlawofthermodynamics.Whentheinformation
two-state state systems. The description of the Maxwell’s
collected by the demon in the process of its operation is
demon is also extended by developing the minimization
erasedinordertoreturnthesystemtotheinitialstate,this
scheme of the heat dissipated in its operation, which can
informationerasureprocessdissipatesbackintothereser-
beessentialforreachingtheultimatethermodynamiclimit
voir in the form of heat the same amount of free energy
oftheMaxwell’sdemonefficiency.
asextractedbythedemon.Thepurposeofthisworkisto
make this relation between the reversible Landauer’s in-
2 Thermodynamicsofthemasterequation
formationerasureandMaxwell’sdemonevenmoreclose,
2.1 First and second laws For completeness, we
by discussing the actual reversal of the optimal erasure
startourquantitativediscussionwithatheoreticaldescrip-
process,which,ifcomplementedwithameasurementand
tion of thermodynamic properties of the systems, evolu-
feedback, produces the Maxwells demon operating with
tion of which is governed by a rate equation – see, e.g.,
thehighestpossiblethermodynamicefficiency.Thedemon
[23,1]. Single-charge tunneling in tunnel junction struc-
extractsk T ln2ofheatfromanequilibriumreservoirat
B
tures [18], e.g., in a single-Cooper-pair [24] or single-
temperature T at the cost of creating one bit of informa-
electron box [25], provides one of the better-known and
tionaboutthestateofthesystemwhichservesasitswork-
precisely-controlled experimental examples of this case.
ingbody.OperationofsuchareversibleMaxwellsdemon
The system we consider is assumed to have a discrete set
wasdemonstratedrecently[1]usinganindividualelectron
of energy states |n(cid:105) with energies E controlled by some
chargeonthesingle-electronboxasthetwo-stateworking n
time-dependentparametersmakingtheenergiesE (t)also
substance.Thisdemonstrationgivesanexampleofagen- n
time-dependent,asillustratedinFig.(1).Ifthesystemisin
eraluseofthesingle-chargestructures[18]whichprovide
thestaten,anexternalsourceinducingasmallvariationof
a convenient setting for studying various aspects of non-
theparameterq,slowonthetimescalesetbytheenergies
equilibriumnanoscalethermodynamics[19,20,21].
E themselvessoasnottoinduceanytransitionsoutofthe
n
The main general view on the physics of information staten,doesworkdW =dE onthesystem.Theaverage
n
that emerged from the original theoretical studies of re- work done is then (cid:104)dW(cid:105) = (cid:80) p dE , where p is the
n n n n
versible computation and developed recently in details in probabilityforthesystemtobeinthestaten,andtheno-
the context of the nanoscale thermodynamics, is that the tation(cid:104)...(cid:105)=(cid:80) p ...willbeusedforallquantities.Asis
n n
informationcanbeviewedappropriatelyastheentropyof typical for experiments with many nanoscale systems, an
acomputingdevice[10].Althoughacomputingdeviceis equilibriumreservoirattemperatureT isassumedtointer-
quitedifferentfromagenericstatisticalsysteminthatthe actweaklywiththesystem,inducingstochastictransitions
main degrees of freedom in it are well-controlled and not amongtheenergystates|n(cid:105).Theevolutionoftheprobabil-
fluctuating in time, intrinsic randomness of bit strings in itiesp isgovernedthenbytheusualrateequation:
n
the computation process makes it necessary to view them
ascarryingentropyinthesamewayasthefluctuatingde- (cid:88)
p˙ = (Γ p −Γ p ), (1)
n nm m mn n
grees of freedom of a statistical system. This view of in-
m
formationasentropyunitesinasimplewayunderstanding
ofmanydifferentphenomenaininformation-assistedther- wheretherateΓ describesthetransitionfromstate|n(cid:105)
mn
modynamics,andhasmanyimplicationsforthephysicsof tostate|m(cid:105)–seeFig.(1).Intheabsenceofdegeneraciesin
computation.Oneexampleisthebasicnotionthatthelog- theenergyspectrumofthesystem,dynamicsofageneral
ically irreversible computation that does not conserve in- quantumsystemcanbedescribedwiththerateequationfor
formation,discardingapartofitduringthecomputational sufficientlyweakcouplingtothereservoirandsufficiently
steps,cannotbeachievedwithoutthedissipationofatleast slowevolutionofthesystem’senergies.Whenadegener-
k T ln2ofenergyperdiscardedbit,despiteclaimstothe acyispresent,validityoftherateequationrequiresinaddi-
B
contrary in the literature [22]. By the second law of ther- tionthattheoperatorcouplingthesystemtothereservoiris
modynamics,iftheinformationisentropy,reductionofin- diagonalinthebasisofdegenerateenergyeigenstates,con-
formationinthecomputingdeviceshouldbeinevitablyac- ditionthatissatisfiedintheexamplesofthesingle-electron
companiedbytheentropyincreaseinthesurroundingen- systems considered in this work. If some of these condi-
vironment,theprocessthatcannotbeachievedwithouten- tions is not satisfied, off-diagonal elements of the density
ergydissipation.Understandingofinformationasentropy matrix of the system starts playing role in its dynamics –
also poses new interesting questions about the nature of see,e.g.,[23].Forthereservoirinequilibrium,thetransi-
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TheentropySofthetotalstructure,includingthesystemof
interestandthereservoir,consistsofthetwoparts,Boltz-
mann’sentropyS oftheprobabilitydistribution{p }of
sys n
thesystemoverthestates|n(cid:105):
(cid:88)
S =−k p lnp , (5)
sys B n n
n
andtheentropyS oftheenvironment.WhileS isnot
env env
knownwithoutspecifyingexplicitlythemodeloftheenvi-
ronment, condition that the environment is in equilibrium
at temperature T provides sufficient information about it.
Under these conditions, the change of entropy S due
env
toeachtransition|n(cid:105) → |m(cid:105)canberelateddirectlytothe
Figure1Schematicsofageneralsystemwithdiscreteen-
heattransferredintotheenvironmentasaresultofthistran-
ergyspectrumwithenergiesE (t),E (t),...adiabatically
0 1 sition.Theaverageoverallpossibletransitionsthengives:
changing in time. Occupation probabilities of the energy
states of the system evolve according to the master equa- (cid:104)dQ(cid:105) 1 (cid:88)
dS = = (E −E )Γ p dt. (6)
tion(1)duetotransitionswithratesΓmninducedbyather- env T T n m mn n
malreservoirinequilibriumattemperatureT. n,m
CombiningthesetwopartsofthetotalentropyS,onecan
determine its dynamics. The time evolution of the system
tion rates Γ satisfy the detailed balance condition linking
entropyS followsdirectlyfromthemasterequation(1)
theratesofthedirectandreversetransitions: sys (cid:80)
andthenormalizationcondition p =1:
n n
Γmn/Γnm =e(En−Em)/kBT . (2) 1 ∂Ssys =−(cid:88)p˙ (1+lnp )
Each transition from state |n(cid:105) to state |m(cid:105) releases kB ∂t n n
n
the energy En −Em from the system into the reservoir. (cid:88)
= (Γ p −Γ p )lnp (7)
Since this process is not associated with any variation of mn n nm m n
themacroscopicparametersofthesystem,itrepresentsthe n,m
heat flow. This means that quantitatively, the contribution 1(cid:88)
= (Γ p −Γ p )(lnp −lnp ).
ofthetransitionsofthistypetotheheatdQflowingfrom 2 mn n nm m n m
n,m
thesystemintothereservoirduringasmalltimeintervaldt
canbewrittenasdQ = (En−Em)Γmndt,whereΓmndt Evolution equation for the environment entropy Senv
is the probability that the transition |n(cid:105) → |m(cid:105) happens which describes the evolution of Senv resulting from the
duringthesmallintervaldt.Takingintoaccountthatfora system dynamics, follows directly from Eq. (6), and can
given state n transition can take the system into any state betransformedasfollows:
m,wecanwritethetotalheatflowingfromthesysteminto ∂Senv = 1 (cid:88)(E −E )Γ p
thereservoir,whenthesystemisinthestaten,as ∂t T n m mn n
n,m
(cid:88)
dQ= (E −E )Γ dt. (3) 1 (cid:88)
n m mn = (E −E )(Γ p −Γ p ). (8)
m 2T n m mn n nm m
n,m
Theserelationsforheatandwork,togetherwiththemaster
AddingEqs.(7)and(8),wegettheequationthatgov-
equation(1),givethefirstlawofthermodynamicssatisfied
ernsthetimeevolutionofthetotalentropyS:
by the evolution of the system. Indeed, taking the incre-
(cid:80)
mentoftheinternalenergyU = nEnpn andusingthe ∂S = 1(cid:88)[k ln pn +En−Em](Γ p −Γ p ).
master equation to transform the increment of probability ∂t 2 B p T mn n nm m
m
p ,wehave n,m
n
(9)
(cid:88)
dU = (dEnpn+Endpn) Thisequationcanbesimplifiedfurtherusingthedetailed-
n balance condition (2) in the form (En − Em)/kBT =
=(cid:104)dW(cid:105)+(cid:88)E (Γ p −Γ p )dt (4) ln(Γmn/Γnm) to transform the last term in the brackets.
n nm m mn n
Inthisway,Eq.(9)givestherateofthetotalentropypro-
n,m
ductioninthecourseofthetimeevolutiondescribedbythe
(cid:88)
=(cid:104)dW(cid:105)+ (Em−En)Γmnpndt=(cid:104)dW(cid:105)−(cid:104)dQ(cid:105). masterequation(1):
n,m
Toformulatethesecondlawofthermodynamicsforthe ∂S = kB (cid:88)ln Γmnpn(Γ p −Γ p ). (10)
system considered here, one needs to define its entropy. ∂t 2 Γnmpm mn n nm m
n,m
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4 DmitriAverinetal.:ReversibleMaxwell’sdemon
This equation shows explicitly that ∂S/∂t ≥ 0, and pro- totheequationsfortheexpansionterms:p˙(k) = γp(k+1).
videstheexpressionofthesecondlawofthermodynamics Formally,theseequationscanbesolveddirectlytoproduce
fortheevolutiongovernedbythemasterequation. theseriesoftherecursionrelations:
2.2 Adiabatic evolution and dissipated heat and p(k+1) =γ−1p˙(k). (14)
work Adiabatic evolution, characterized by the vanish-
ingly small rate η of variation of the energies of the sys- However, since the matrix γ of the transition rates has a
tem, η ∼ E˙ /E → 0, conserves the total entropy S, non-vanishing kernel, spanned by p(0), its determinant is
n n zero,andthecalculationoftheinversematrixγ−1requires
and is of the main interest for all reversible processes, in
the introduction of a ”generalized” inverse (also some-
particular for the subsequent discussion of the reversible
times called ”pseudoinverse”) determined by some addi-
Maxwell’s demon. In this regime, the rate of the varia-
tional conditions. For the solution of the master equation,
tion of probabilities p is also small, p˙ ∝ η → 0 and
n n
the appropriate generalized inverse is the ”group” inverse
they in the main approximation are given by the station-
(see,e.g.,[26])definedbythefollowingrelations:
arysolutionofthemasterequation(1).Inthetypicalcase
when the energy spectrum {En} of the system includes γγ−1γ =γ, γ−1γγ−1 =γ−1, γγ−1 =γ−1γ. (15)
the state with the lowest energy, such a stationary state
corresponds to equilibrium, i.e., all the probability fluxes Combiningthesecondandthethirdoftheserelationsone
in the master equation vanish: Γ p = p Γ . For an sees,inparticular,thatthegroupinversehasthefollowing
mn n m nm
equilibrium reservoir characterized by Eq. (2), this rela- importantproperty:
tion implies that the probabilities p indeed maintain the
equilibrium form, pn/pm = e[Enn(t)−Em(t)]/T through- γ−1p(0) =0. (16)
out the evolution. Equation (10) shows then that the to- Expression for the average of the element of heat dQ
tal entropy production vanishes in this case, ∂S/∂t = 0, (3),asitappears,e.g.,inEq.(4),canbewritteninthema-
i.e., ∂Ssys/∂t = −∂Senv/∂t, implying that the changes trixformusingthevector-columnE = {En}oftheener-
∆Ssys of the entropy of the system are directly compen- giesEn:(cid:104)dQ(cid:105) = −E†γpdt = −E†p˙dt.Theaverageheat
sated by the average heat flow (cid:104)Q(cid:105)(rev) to (or from) the transferredintothereservoirtakesthenthefollowingform
reservoir:
(cid:90) (cid:90)
(cid:104)Q(cid:105)(rev) =−T∆Ssys. (11) (cid:104)Q(cid:105)= (cid:104)dQ(cid:105)=− E†p˙dt. (17)
Since the total entropy is conserved, this transfer of heat
into the reservoir represents reversible process which can We assume that the adiabatic evolution of the energies
beinverted,returningthesystemtotheinitialstateandre- En(t)startsandendsattimestiandtf,withenergiesstay-
movingthetransferredheatfromthereservoir. ingconstantoutsideofthistimeinterval.Thetimedepen-
While the relation (11) holds in the limit of vanishing denceofenergiesisalsoassumedtobesufficientlysmooth,
rateofvariationsinthesystem,η → 0,finiteη makesthe sothatE˙ (t)=0atbothendsoftheevolution,t=t ,t .
n i f
processirreversible,andgeneratesadditional“dissipated” Substituting the expansion (12) into Eq. (17) one can see
heat in the reservoir. To find such heat in a general situ- thatthefirst,equilibrium,termp(0)oftheexpansionrepro-
ation, it is convenient to write the master equation (1) in duces Eq. (11) for the reversible heat transfer. To do this,
thematrixform,p˙ = γp,wherepisthevectorcolumnof weintegrateEq.(17)bypartsandexpresstheequilibrium
the probabilities p , and γ is the matrix of the transition probabilitiesthroughthefreeenergyF = −T lnZ ofthe
n
rateswiththematrixelementsgivenbythetworelations: system with the help of the relation p(0) = ∂F/∂E to
(cid:80) n n
γnm = Γnm,forn (cid:54)= m,andγnn = − m(cid:54)=nΓmn.The obtain
vectorpcanbeexpandedtheneffectivelyinthepowersof
thevariationrateη: (cid:104)Q(cid:105)(rev) =−(cid:90) tf E†p˙(0)dt=(cid:90) tf E˙†p(0)dt
∞ ti ti
(cid:88)
p= p(k), (12) −E†p(0)(cid:12)(cid:12)tf =(cid:90) tf (cid:88) ∂F dE −(cid:88)E p(0)(cid:12)(cid:12)tf
k=0 (cid:12)ti ti n ∂En n n n n (cid:12)ti
where p(k) ∝ ηk, and p(0) is the vector of the in-
=∆F −∆U =−T∆S .
stantaneous equilibrium: p(n0) = e−En(t)/kBT/Z, Z = sys
(cid:80) e−En(t)/kBT. Substituting this expansion into the Thesecondtermintheexpansion(12)givesthemain
n contributiontothedissipatedheat(cid:104)Q(cid:105)(dis)intheadiabatic
equation p˙ = γp, using the fact that the instantaneous
evolution:
equilibriumsatisfiesthestationarymasterequation:
(cid:90) tf (cid:90) tf
(cid:104)Q(cid:105)(dis) =− E†p˙(1)dt= dtE˙†p(1)
γp(0) =0, (13)
ti ti
and that the time derivative adds one factor of the rate η, (cid:90) tf
= dtE˙†γ−1p˙(0), (18)
weseethattheexpansion(12)reducesthemasterequation
ti
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where we have used the fact that p(1) = γ−1p˙(0) = 0 at Thehigher-ordertermsintheexpansion(12)givespro-
t = t ,t ,becausep˙(0) vanishestogetherwiththederiva- gressively smaller corrections to the heat Q which vanish
i f
tivesE˙ ofthesystemenergiesattheendsoftheevolution. morerapidlywiththerateηofvariationoftheenergiesof
n
Calculatingthederivatives thesystem.Thismeansthatintheadiabaticlimit,onecan
takeintoaccountonlythefirsttwotermsintheexpansion
E˙ Z˙ Eq. (12), limiting the expression for the heat exchanged
p˙(0) =− n p(0)− p(0),
n k T n Z n betweenthesystemandthereservoirtothetwocontribu-
B
tions:
andmakinguseoftheproperty(16)ofthegroupinverse, (cid:104)Q(cid:105)=(cid:104)Q(cid:105)(rev)+(cid:104)Q(cid:105)(dis) (22)
we transform the expression for the heat dissipated irre-
which are given by Eqs. (11) and (19), respectively, and
versiblyintheadiabaticevolutionintothefinalform:
correspondtothereversibleandirreversibleheatgenerated
(cid:104)Q(cid:105)(dis) =− 1 (cid:90) tf dt(cid:88)E˙ [γ−1] E˙ p(0). iEnqt.h(e22re)siesrtvhoaitr.thOendeismsiopraeteudsehfeualtrceominacrkidtehsawtfiothllothwef“rdoims-
kBT ti n,m m m,n n n sipatedwork”(cid:104)W(cid:105)(dis),whichcanbedefinednaturallyas
(19) theworkdoneonthesysteminexcessofthechangeofthe
Equation (19) can be used for a system with arbitrary systemfreeenergy.Indeed,combiningEqs.(22)and(11)
spectrum E (t), as long as one can calculate the inverse withthefirstlaw(4)weseethat
n
γ−1 of the transition rate matrix, as defined by Eq. (15). (cid:104)Q(cid:105)(dis) =(cid:104)Q(cid:105)−(cid:104)Q(cid:105)(rev) =(cid:104)W(cid:105)
As an example of the application of Eq. (19), we calcu-
latethedissipatedheatforanadiabaticevolutionofatwo- −∆U +T∆Ssys =(cid:104)W(cid:105)−∆F =(cid:104)W(cid:105)(dis), (23)
statesystemwithstateenergiesE0(t)andE1(t).Thecal- asstatedabove.
culation is simplified by the general property of Eq. (19)
2.3 Classical fluctuation-dissipation theorem
whichfollowsfromEq.(16):thedissipatedheatdepends,
Another feature of the heat dissipated in the adiabatic
as should be, only on the energy differences in the en-
evolution that is important for operation of the reversible
ergy spectrum E (t), i.e., it does not change if all ener-
n Maxwell’s demon is the direct relation between the av-
gies are shifted by some time-dependent offset δ(t). For
erage dissipated heat given in general by Eq. (19), or by
atwo-statesystem,thismeansthatthedissipatedheatde-
Eq.(21)inthecaseofatwo-statesystem,andthemagni-
pends only on the time dependence of the energy differ-
tudeofthefluctuationsoftheheatexchangedbetweenthe
ence(cid:15)(t)=E (t)−E (t)andnotontheevolutionofthe
1 0 reservoir and the system during the evolution. This rela-
individualenergiesE separately.Thismeansthat,effec-
0,1 tion, derived below, shows that in the adiabatic limit, the
tively,wecantaketheenergiesofthetwostatestobe0and
dissipatedheatvanishesnotonlyonaverage,butforevery
(cid:15)(t),andtheEq.(19)isreducedasfollows:
instanceoftheevolutionimplyingthatinprinciple,itcan
bemadenegligibleineverycycleoftheMaxwell’sdemon
(cid:104)Q(cid:105)(dis) =− 1 (cid:90) tf dt(cid:15)˙(t)2[γ−1] p(0). operation. Qualitatively, the relation between the average
kBT ti 11 1 dissipatedheatanditsfluctuationscanbeseenasaversion
oftheclassicallimitofthefluctuation-dissipationtheorem
Foratwo-statesystem,transitionmatrixtakesthefol- connecting the linear heat conductance G and the zero-
th
lowingexplicitform: frequency spectral density S (0) of the fluctuating heat
J
fluxinthisconductance(see,e.g.,[27]):S (0)=2T2G .
(cid:32) (cid:33) J th
γ = −Γ10 Γ01 , Inthecaseofadiabaticevolution,thisrelationisreplaced
Γ −Γ bytherelationbetweenthedissipatedheat(cid:104)Q(cid:105)(dis),which
10 01
in this context is an analog of the thermal conductance,
andthegroupinversedefinedbytheconditions(15)canbe andthenoiseofthisheat,which(cid:104)Q2(cid:105)replacesthespectral
calculateddirectly: density SJ(0). This analogy stems from the fact that the
dissipated heat in the adiabatic evolution is produced as
γ−1 =γ/Γ2, Γ ≡Γ +Γ . (20) a response to a small deviation from the equilibrium be-
Σ Σ 10 01
tweenthesystemandthereservoir,createdbyafiniterate
Thisresultmeansthat[γ−1]11 = −Γ01/ΓΣ2,andcombin- ofchangeofenergiesofthesysteminthesamewayasthe
ing it with the detailed balance condition (2): Γ10/Γ01 = averageheatflowthroughthethermalconductanceGth is
e−(cid:15)/kBT andequilibriumprobabilityp(0) = 1/(e(cid:15)/kBT + producedbyasmalltemperaturebiasacrossit.
1
1) we obtain the heat dissipated in the adiabatic evolu- To derive the relation between the average dissipated
tion of the two-state system with energy difference (cid:15)(t) heat (19) and its noise quantitatively, we need to calcu-
betweenthestates: latetheheatnoise(cid:104)Q2(cid:105)intheadiabaticevolutiongoverned
bythemasterequation(1).Takingtheelementaryheat(3)
1 (cid:90) tf (cid:15)˙2 generatedwhenthesystemisinthestaten,andsumming
(cid:104)Q(cid:105)(dis) = dt . (21)
4kBT ti ΓΣcosh2((cid:15)/2kBT) it over all states with their probabilities, we get the total
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6 DmitriAverinetal.:ReversibleMaxwell’sdemon
elementaryheatexchangedbetweenthesystemandreser- assumptionofsmoothevolutionofthesystemenergies.In
voir addition,thecorrelationsamongdifferenttunnelingevents
(cid:88)
dQ= (E −E )Γ pˆ dt. (24) described by the master equation decay on the time scale
n m mn n
n,m Γ−1 set by the typical tunneling rate Γ, which is much
shorter than the evolution time interval t − t . Because
In this expression, pˆ denotes an arbitrary distribution of f i
n of this, the fluctuations of the initial and final energies of
occupation probabilities which satisfies the rate equation
thesystemandthefluctuationsoftheworkdoneduringthe
(1),butdoesnotnecessarilycoincidewiththeactualprob-
adiabaticevolutionareuncorrelated:
ability distribution in the evolution of the system because
oitfi,ese.pgˆ.,.dBifefceareunsteionfittihalisc,odnQdithioerneiimspnoostetdheonactthueaplraovbearbagile- (cid:104)Q˜2(cid:105)=(cid:104)E˜2(cid:105)+(cid:104)E˜2(cid:105)+(cid:104)W˜2(cid:105), W =(cid:90) tf (cid:88)E˙ pˆ dt.
n i f n n
heat as, e.g., in Eq. (4). Equation (24) can be simplified ti n
furtherasfollows: (27)
(cid:88) ThefluctuationsoftheworkW doneintheadiabaticevo-
dQ= (En−Em)(Γmnpˆn−Γnmpˆm)dt/2 lution can be found according to the standard prescrip-
n,m tion for the correlation functions in the master-equation-
=(cid:88)E (Γ pˆ −Γ pˆ )dt=−(cid:88)E pˆ˙ dt, (25) governedevolution:
n mn n nm m n n
whne,rme in the last step we used Eq. (1),nsince the time de- (cid:104)W˜2(cid:105)=(cid:90) tf dtdt(cid:48)(cid:88)E˙m(t(cid:48))pm,n(t(cid:48),t)E˙n(t)pn(t).
pendenceoftheotherwisearbitraryprobabilitiespˆnshould ti n,m
stillbegovernedbythisequation.UsingEq.(25)todefine (28)
anexpressionfortheheatQexchangedwiththereservoir Herepn(t)istheactualprobabilitiesintheevolutionpro-
duringthefullevolutionweget: cess, and pm,n(t(cid:48),t) is the distribution of probabilities
evolvingattimet(cid:48) outoftheinitialconfigurationatt(cid:48) = t
Q=(cid:90) tf dQ=−(cid:88)(cid:90) tf E pˆ˙ dt in which the system occupies with certainty the state n:
n n
p (t,t)=δ .SinceEq.(28)alreadycontainstherate
ti n ti m,n m,n
of change of energies squared, the probabilities in it can
=(cid:88)(cid:104)−E pˆ (cid:12)(cid:12)tf +(cid:90) tf E˙ pˆ dt(cid:105). (26) be calculated in the quasistatic approximation, in which
n n(cid:12) n n
n ti ti pn(t) = p(n0)(t), and the master equation that determines
theprobabilitiesp (t(cid:48),t):∂p/∂t(cid:48) = γ(t(cid:48))p,canbesim-
Equation (26) represents the heat generated in the m,n
plified by essentially neglecting the time dependence of
reservoir as the difference between the work W done on
the transition rates, i.e. reducing it to the following form:
the system and the change of the system energy in the
∂p/∂t(cid:48) =γ(t)p.Formalsolutionofthisequationcanthen
evolutionprocess,Q=W −(E −E ).Sincethesystem
f i
bewrittendownimmediately:
randomlyoccupiesdifferentenergylevelsintheevolution,
the work W is a fluctuating quantity similarly to the sys-
tem energies Ef,i. They all contribute to the fluctuations pm,n(t(cid:48),t)=(cid:2)eγ(t)|t(cid:48)−t|(cid:3)m,n. (29)
ofthegeneratedheatdefinedasusualas
The fact that this expression is the same for t(cid:48) < t as for
(cid:104)Q˜2(cid:105)=(cid:104)Q2(cid:105)−(cid:104)Q(cid:105)2. t(cid:48) > t is dictated by the fact that in the quasistationary
approximation, the correlation functions should be sym-
Similarly to the average dissipated heat (19), we find metric with respect to the interchange of the two times,
thesefluctuationsinthefirstorderinthesmalltypicalrate t ↔ t(cid:48).CombiningEq.(29)withEq.(28),andtakinginto
E˙ of the variation of the system energies. Note that al- accountthattheothertermsintheintegralsevolveonthe
n
thoughEq.(19)containsthetermsproportionaltoE˙2 un- timescalemuchlongerthanthatsetbythetunnelingrates,
n
dertheintegralovertime,areasonableadiabaticevolution we can take the integral over t(cid:48) in Eq. (28) by effectively
changesthesystemenergiesconsiderably,i.e.(cid:82)tf dtE˙ ∼ integratingonlytheprobabilities(29):
ti n
E , so effectively, the dissipated heat (19) is of the first
ornder in the evolution rate E˙n. In this approximation, the (cid:104)W˜2(cid:105)=−2(cid:90) tf dt(cid:88)E˙nE˙mp(n0)[γ−1]m,n. (30)
fluctuations of the initial and final energies of the sys- ti n,m
tem are given by their standard equilibrium expressions,
(cid:104)E˜2 (cid:105)=C T2,whereC aretheheatcapacitiesofthe Comparing Eq. (30) to Eqs. (19) and (23) of the pre-
i,f i,f i,f
systembeforeandaftertheadiabaticevolution,asfollows viousSubsection,weseethatthefluctuationsofthework
from the following considerations. First, corrections p(1) done on the system are directly related to the dissipated
totheequilibriumprobabilitiesp(0),whichcouldproduce partofthework:
anon-equilibriumcontributionstofluctuationsofE van-
i,f
ish at t = t ,t according to Eq. (14) under the adopted (cid:104)W˜2(cid:105)=2k T(cid:104)W(cid:105)(dis). (31)
i f B
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Theenergy-fluctuationtermscanberemovedfromEq.(27) E is well separated from the excited states, so that there
0
for the heat noise by averaging over the initial and final is a range of temperatures T in which the system occu-
statesoftheadiabaticevolution.Alternatively,insomesit- piesthegroundstatewithnearcertainty,sothattheentropy
uations,e.g.theonerelevantforthemostbasicformofthe (5)ofthesystemeffectivelyvanishes.Theadiabaticevolu-
reversible Maxwell’s demon discussed below, the energy tionshouldstartthenwiththesystemintheuniqueground
fluctuationsbecomenegligibleautomatically.Inthatcase, state and should bring it to another equilibrium configu-
fluctuationsoftheinitialenergyE vanishbecausetheevo- rationwiththeleveloccupationprobabilitiesp ,inwhich
i n
lutionofthesystemstartsfromthedefinite(ground)state theentropy(5)isincreasedtoanonvanishingvalueS .
sys
ofthesystemseparatedbylargeenergygap∆E (cid:29) k T AccordingtoEq.(11),inthecourseofthisevolution,the
B
from the excited states, which suppresses all transitions heat
(cid:88)
that could change the system energy. The fluctuations of (cid:104)Q(cid:105)=−k T p lnp (33)
B n n
thefinalenergyE vanishbecausetheevolutionendswith
f n
the equally occupied two degenerate states of the system.
isextractedonaveragefromthereservoir.Asfollowsfrom
Although the transitions between these two states are not
the discussion in the previous Section, in the adiabatic
suppressed,theydonotcauseanyexchangeofenergywith
limit,thefluctuationsoftheextractedheatinthiscaseare
thereservoir,sincetheenergiesofthetwostatesofthesys-
determinedbythefluctuationsofthesystemenergyatthe
temarethesame.Allotherstatesareagainseparatedfrom
final point of the evolution. If there are no energy fluctu-
thetwodegeneratestatesbylargeenergygapsuppressing
ations at the final point, the fluctuations of Q vanish and
the transitions to these states. For vanishing fluctuations
extracted heat Q is given by Eq. (33) for each individual
of the energies E , Eq. (31) can be extended directly
i,f
cycle, Q = (cid:104)Q(cid:105). The adiabatic evolution considered here
to the relation for the noise of the heat transferred to the
is reversible and generates entropy S in the system’s
reservoir: sys
(cid:104)Q˜2(cid:105)=2k T(cid:104)Q(cid:105)(dis), (32) states. If the system represents a computing device, this
B entropy corresponds to the information stored in this de-
which can be viewed as an analog of the classical vice.Thereversaloftheadiabaticevolutionjustdescribed
fluctuation-dissipation theorem for the thermal conduc- would remove this information from the device by bring-
tance in the situation of adiabatic evolution. The main ing it to the definite prescribed state, and therefore coin-
consequenceofthisrelationforthesubsequentdiscussion cides with the Landauer information erasure discussed in
ofthereversibleMaxwell’sdemonisthefactthatthestan- theIntroduction.Thus,thefirststepoftheoperationofour
darddeviationoftheheatnoisevanishestogetherwiththe Maxwell’sdemonisthereversaloftheinformationerasure
average dissipated heat, and as a result, in the adiabatic protocol.
limit,thedissipatedheatvanishesnotonlyonaverage,but The second step of the MD operation is the measure-
foreachrealizationoftheadiabaticevolution. mentoftheactualstateofthesystemattheendoftheevo-
lution.Sincewearelimitingthediscussionheretoclassi-
3 Reversible Maxwell’s demon The discussion of calsituations,theonlytwoimportantfeaturesofthemea-
adiabatic evolution in the previous Section illustrates ex- surement is whether it is precise, and whether it is done
plicitly the standard understanding that thermodynami- reversibly,withouttheavoidableenergydissipation.Ifthe
cally optimum processes require sufficiently slow, adia- measurementisidealinbothrespects,thenthestateofthe
batictransformations,whichminimizedissipatedheatand systemattheendoftheevolutionisdeterminedprecisely,
work and make the whole process effectively reversible. andtheonlythermodynamicpricepaidforthisistheneed
Here we consider “Maxwell’s demon” (for reviews, see to physically record this information I about the state of
[28]),adevicewhichextractsenergyfromthermalfluctu- thesystem.Foranidealmeasurement,thestatisticsofthe
ations in an equilibrium statistical system at temperature measurementoutcomesexactlyrepeatsthestatisticsofthe
T andtransformsthisthermalenergyintofreeenergy.All occupationoftheenergylevelsofthesystem,andischar-
this is achieved by means of a process the central part of acterized by the probabilities p . This means that in the
n
which is a measurement done on a fluctuating statistical limit of large number of MD operation cycles, each mea-
systeminthermalequilibrium,andapplicationofthefeed- surementgeneratesonaveragetheamountofinformation
back control pulses which depend on the outcome of this
(cid:88)
measurement.Inprinciple,Maxwell’sdemon(MD)canbe I =− p lnp (34)
n n
realizedusinganarbitraryquantumsystemwiththeenergy
n
spectrum E weakly interacting with a thermal reservoir,
n
as considered in the previous Section. The simplest cycle which coincides with the system entropy before the mea-
thatrealizesthedemonoperationconsistsofthreesteps. surement.Thisfactillustratesthegeneralunderstandingof
Thefirststepisanadiabaticevolutionofthesystemex- information as the entropy of the computing system that
tracting heat from a thermal reservoir in equilibrium. For wasdiscussedintheIntroduction.
simplicity, we assume that the system has a configuration ThelaststepoftheMDcycleistheapplicationofthe
of the energy levels E such that the ground state energy sequenceofpulsestothesystemthatdriveitquicklyfrom
n
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8 DmitriAverinetal.:ReversibleMaxwell’sdemon
theactualobservedstateattheendoftheadiabaticevolu-
tion into the configuration when the occupied state is the
unique ground state of the system, similar to the starting
point of the evolution. The pulses should be applied in a
time much shorter than the typical transition rates. Since
theenergyspectrumofthesystemiscontrolledbytheex-
ternalpulses,thegroundstatecaningeneralbeadifferent
stateineachcycle(dependingontheoutcomeofthemea-
surement) but can be made to have the same energy each
time. Such a rapid return to the ground states completes Figure 2 Typical qualitative time dependence of the en-
the MD cycle. It does not generate any heat in the reser- ergy difference between the two states of an effectively
voir,sincenotransitionshavetimetohappenduringit.The two-statesystem,e.g.single-electronbox,thatcorresponds
netresultofonecycleofMDoperationisthentheextrac- to several cycles of the operation of reversible Maxwell’s
tionoftheheat(33)fromthereservoir,which,bythefirst demon.Twopartsofeachcyclethatcanbeexplicitlyseen
law, was transformed into the free energy of the external in the diagram are: (1) adiabatic ramp of the energy dif-
source manipulating the energy states of the system. This ference starting from initial value that is much larger in
transformation was achieved at a cost of creation of the magnitudethanthethermalenergykBT andendingatthe
information (34) about the actual state of the fluctuating degeneracy point where the energy difference vanishes;
systemattheendoftheadiabaticevolution.Theseexpres- and (2) abrupt return from the degeneracy point to the
sionsshowsthattheoperationofsuchaoptimalreversible ground state. At the beginning of the ramp, the system is
demondoesnotcontradictthesecondlaw.Reversibleera- with certainty in the ground state, while at the degener-
sureoftheinformation(34)dissipatesbackintoheatpre- acy point both states are occupied with equal probability
cisely the energy k TI extracted by the demon from the 1/2.Whethertheappliedfeedbackmakesthestaten = 0
B
thermalreservoir. or n = 1 the ground state in a given cycle depends on
theactualstatethemeasurementfindsthesysteminatthe
Thedemonoperationjustdescribedreliesontheexis- degeneracy point in this cycle. The heat kBT ln2 is ex-
tenceoftheaccuratemeasurementsofthestatesofthesys- tractedfromthereservoirduringeachadiabaticrampand
tem that are distinguished by their energies E (t). While converted into the free energy transferred to the external
n
theoreticallytheenergyisanabsolutelylegitimateobserv- sourcewhichmanipulatestheenergiesofthesystem.
able,experimentally,thenon-invasivemeasurementsofen-
ergypresentsconsiderableproblems.Inthisrespect,prac-
ticalimplementationoftheMDcycleisonlypossiblewhen In the two-state regime, only the two charge states,
theenergystatesnofthesystemcanalsobedistinguished n=0andn=1,havethenon-vanishingoccupationprob-
bysomeotherphysicalquantitythatisdirectlymeasurable. abilities. These states differ by one extra electron charge
This is the case for the single-electron circuits, where the residing either on the left or on the right electrode of the
dominant energy of the system is the electrostatic energy, SEB. In this respect, the SEB Maxwell’s demon is simi-
andthedifferentenergystatesaredistinguishedalsobythe lar to the Szilard engine [35], in which the states of the
excess number of individual electron charges on the elec- Maxwell’sdemonworkingsubstancearedistinguishedby
trodes of the system. The charge states can be measured twopossiblepositionsofoneparticle.Thiscorrespondence
directly with any of the several existing single-charge de- betweentheenergyandchargestatesalsomakesitpossi-
tectors,DC[30,31,32]orRF[33]single-electrontransis- bletomeasuredirectlytheactualstateofthesystemwitha
tors,orquantum-pointcontacts[34].SuchaMaxwell’sde- chargedetector.TheenergiesE0andE1ofthetwocharge
monoperatingwithindividualelectronshasbeenrealized statescanbecontrolledbytheappliedtime-dependentgate
[1] using the most basic single-electron system, single- voltage(formoredetaileddescriptionofthesystem–see,
electron box (SEB) [24,25]: two electrodes connected by e.g.,[29].AspartlydiscussedinthepreviousSection,the
a small tunnel junction that allows electrons to tunnel be- thermodynamicpropertiesofatwo-statesystem,including
tween them. An advantage of this system is that at low now the characteristics of the MD cycle, depend only on
temperatures,kBT (cid:28) EC,whereEC isthecharacteristic theenergydifference(cid:15)=E1(t)−E0(t)andnotonthein-
electrostaticenergyassociatedwithoneelectroncharging dividualenergiesE0,1 separately.Thegate-voltage-driven
the system, its dynamics can be naturally reduced to that typical qualitative time dependence of the energy differ-
of a two-state system, simplifying the measurement and encethatcorrespondstoseveralcyclesofMDoperationas
controlpartoftheMDcycle.Moreover,theenergiesE describedaboveisshowninFig.2.
0,1
ofthetworemainingstatesofthetwo-statesystemcanbe Each cycle starts with an adiabatic ramp that slowly
gate-voltagedrivenback-and-forthbetweentheregimesof drives the system from the state with |(cid:15)| (cid:29) k T, when
B
large |E −E | (cid:29) k T, and small, |E −E | (cid:28) k T, the ground state is occupied with near absolute certainty,
0 1 B 0 1 B
energy separation, as needed for the MD operation - see tothedegeneracypoint(cid:15) = 0,wherethetwostatesofthe
Fig.2andthediscussionbelow. system are equally occupied. This means that the entropy
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pssheaderwillbeprovidedbythepublisher 9
ofthesystemisincreasedby∆S =k ln2andtheheat Integratingthisequationoverthetimeintervalofadiabatic
sys B
Q = k T ln2,asinEq.(11),isextractedfromthereser- evolution, from t = 0, when the energy difference ac-
B i
voir during each ramp. Note that under the conditions of quiressomelargevalue(cid:15)(0) = (cid:15) (cid:29) T,tot = τ,when
i f
thiscycle,therearenoequilibriumfluctuationsoftheheat (cid:15)(τ)=0,oneobtainstheconditionthatgivestheconstant
attheendpointoftheramp,andtherefore,noheatfluctu- heatfluxJ duringtheevolution:
ationsatallintheadiabaticlimit,i.e.,theextractedheatis
thesameineachcycle.ThecycleofMDoperationiscom- (cid:112) (cid:90) (cid:15)i d(cid:15)
4k TJτ = , (37)
pletedbythemeasurementoftheactualstateofthesystem B (cid:112)Γ ((cid:15))cosh((cid:15)/2k T)
0 Σ B
at degeneracy point, and application of the corresponding
feedbackpulseofgatevoltagewhichswiftlyturnsthemea- andtherefore,determinesthetotaldissipativeheat(cid:104)Q(cid:105)(dis)
sured state into the ground state of the system. Since the (21)intheadiabaticramp:
pulse is fast on the time scale set by the transition rates,
noheatisexchangedwiththereservoirinthisreturntothe (cid:104)Q(cid:105)(dis) =Jτ. (38)
groundstate,sothenetresultofthecycleistheextraction
of the heat during the adiabatic ramp, which is converted To obtain the explicit solution of Eq. (37), one needs
into the free energy transferred to the source of the con- to specify the energy dependence of the total tunneling
trol pulses. For instance, in the case of SEB, this means rate Γ ((cid:15)) in the two-state system. An important exam-
Σ
thatthebatteryproducingthegatevoltage,ideally,should ple is provided by the hybrid SEB based on the normal
bechargedbytheMDoperation.Thermodynamiccostof metal/insulator/superconductor (NIS) tunnel junction as
this transformation of heat into free energy is creation of usedintheexperiment([1]).Single-electrontunnelingrate
onebitofinformationabouttheactualstateofthesystem intheNISjunctionis(see,e.g.,[29])
at degeneracy; for the SEB, position of the extra electron
ontheleftorrightelectrodeofthebox.Ifthisinformation Γ ((cid:15))=2Γ cosh2{(cid:15)/2k T}, (39)
Σ m B
is erased in an optimal, reversible, fashion, precisely the
sameamountoffreeenergyasextractedbytheMaxwell’s whereΓmisthetunnelingrateatdegeneracy,when(cid:15)=0,
demonisconvertedbackintoheat,ensuringthatthesecond
1
lawofthermodynamicsissatisfied. Γ = (2π∆k T)1/2e−∆/kBT .
m R e2 B
3.1 Minimization of the dissipated heat Ideal op- T
erationoftheMaxwell’sdemondiscussedsofarisaffected
Here R is the normal-state resistance of the NIS tunnel
T
ingeneralbyseveralpossibleimperfection.Themostfun-
junctionand∆isthesuperconductingenergygapinthesu-
damentallimitationisimposedbythefiniterateoftheadi-
perconductingelectrodeofthejunction.Withtherate(39),
abaticrampwhich,inadditiontothereversibleextraction
integralinEq.(37)canbecalculatedexplicitlytofindthe
of the heat (11) from the reservoir, dissipates irreversibly minimum Q of the dissipated heat (cid:104)Q(cid:105)(dis) (21) for the
m
theheat(19).InthecaseoftheMDcyclebasedonatwo-
optimumadiabaticrampsatisfyingthecondition(36):
statesystem,e.g.theSEBdescribedabove,theirreversibly
dissipatedheatisgivenbyEq.(21)andcanbeminimized k T (cid:15)
Q =Jτ = B tanh2 i . (40)
by appropriate choice of the profile of the energy differ- m 2τΓ 2k T
m B
ence ramp (cid:15)(t). To do this, one needs to minimize the in-
tegral in Eq. (21) with respect to the function (cid:15)(t). Since Thisequationshowsthat,qualitatively,thedissipatedheat
theintegralinEq.(21)doesnotcontainanyderivativesof isindeedminimizedforslowadiabaticramps,whentheto-
(cid:15)(t)ofhigherthanthefirstorder,thisminimizationprob- talramptimeτ islarge,τ (cid:29)Γ−1,andthatinthislimit,the
m
lem is equivalent to the one encountered, e.g., in the La- dissipatedheatcaninprinciplebemadenegligibleincom-
grangian formulation of classical mechanics, and leads to parison with the reversible heat k T ln2 extracted from
B
theLagrangeequationforthefunction(cid:15)(t)minimizingthe thereservoirintheidealMDcycle.
dissipatedheat: It is instructive to compare Eq. (40) to the result that
would be obtained without the minimization of the dissi-
d ∂J ∂J (cid:15)˙2 pated heat, e.g., for the linear time dependence of the en-
= , J ≡ .
dt ∂(cid:15)˙ ∂(cid:15) 4k TΓ ((cid:15))cosh2((cid:15)/2k T) ergydifference(cid:15)(t)intheadiabaticevolutionshownqual-
B Σ B
itatively in Fig. 2. Evaluating the integral in Eq. (21) for
(35)
(cid:15)˙ = const = (cid:15) /τ and the total tunneling rate (39), one
Calculatingthederivatives,onecanseethattheLagrange i
obtains:
equation can be transformed into the form requiring that
theintegrandinEq.(21)isconstantintime: (cid:104)Q(cid:105)(dis) = (cid:15)i tanh (cid:15)i (cid:2)1− 1tanh2 (cid:15)i (cid:3).
4τΓ 2k T 3 2k T
dJ (cid:15)˙ (cid:112) m B B (41)
=0, i.e. =− 4k TJ.
dt (cid:112)Γ ((cid:15))cosh((cid:15)/2k T) B Weseethatintherelevantlimitoflargeinitialenergy(cid:15) (cid:29)
Σ B i
(36) k T,non-optimizeddissipativeheat(41)islargerthanthe
B
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10 DmitriAverinetal.:ReversibleMaxwell’sdemon
optimizedheat(40)byalargefactor(cid:15)i/3kBT.Thisshows [10]E. Lutz and S. Ciliberto, Phys. Today 68, Issue 9, p. 30
thattheoptimizationprocedurecanplayquiteanimportant (2015);D.V.Averin,ibid.69,Issue8,p.12(2016).
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Wecanalsofindexplicitlythetimedependenceofthe J.S.Tsai,IEEETrans.Appl.Supecond.19,961(2009).
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[13]A.Be´rut,A.Arakelyan,A.Petrosyan,S.Ciliberto,R.Dil-
withtheNIStunnelingrate(39),weobtain
lenschneider,andE.Lutz,Nature483,187(2012).
t/τ =1−tanh((cid:15)(t)/2T)/tanh((cid:15) /2k T). [14]Y.Jun,M.Gavrilov,andJ.Bechhoefer,Phys.Rev.Lett.113,
i B
190601(2014).
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of the optimum adiabatic ramp that minimizes the dissi- J. Goold, K. Modi, D.O. Soares-Pinto, and L.C. Ce´leri,
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1960),Ch.13.
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B
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[20]B.Ku¨ng,C.Ro¨ssler,M.Beck,M.Marthaler,D.S.Golubev,
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4 Conclusions In this work, we have discussed the
(2012).
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[21]O.-P.Saira,Y.Yoon,T.Tanttu,M.Mo¨tto¨nen,D.V.Averin
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