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Electronicheatcurrentfluctuationsinaquantumdot
épieux
AixMarseilleUniv,UniversitédeToulon,CNRS,CPT,Marseille,France
(Received27July2020;revised28December2020;accepted11January2021;published26January2021)
Thefluctuationsoftheheatcurrentinaquantumdotcoupledtoelectronreservoirsarecalculatedatfinite
frequency,voltage,
heatnoiseisexpressedasanintegralonenergycontainingfourcontributions,eachofwhichincludestrans-
missionamplitudes,electron-holepairdistributionfunctions,
arehighlightedanddiscussedforanequilibriumandanout-of-
withinthehightransmissionlimit,theheatnoiseiscloselyrelatedtotheradiativepowerspectrum,leadingtoan
out-of-equilibriumPlanck’.
DOI:.045427
[3,33–35],circuittheory[27],Tomonaga-
Luttingerliquidtheory[36,37],generalizedLindbladmaster
Inquantumdevices,theheatfluctuatesovertimefor
equations[2,12,28],mean-fieldmethodcoupledtoHartree-
severalreasons:thefirstoneisrelatedtothepresenceof
Fockapproximation[38],Boltzmann-Langevinapproach[9],
thermalagitationatfinitetemperature,thesecondonetothe
andinchwormquantumMonteCarlomethod[5].Thesystems
factthatthedeviceinteractswithitselectromagneticenvi-
inquestionareeithermolecularjunctions[34,38],quan-
ronmentbyemittingorabsorbingenergyviaphononsor
tumwires[31,36,39],mesoscopicconstrictions[21],quantum
photons,andthethirdonetotheprobabilisticnatureofparti-
dots[1,12,27,35,40–42],doublequantumdots[1,43],or
qubits[44].Intheseworks,thegeneratingfunction
heatfluctuationsprovidesavarietyofinformationonenergy
fortheheatfull-countingstatisticshasbeendetermined
dissipation[1],thepresenceoffinitecoherenceandentan-
[1–3,5,8,24,33,39,42–46],andthezero-frequencyheatnoise
glementinopenquantumsystems[2],andthehigher-order
hasbeencalculated[3–5,31,36,38,40,46–48]aswellasthe
cumulantsofchargecountingstatistics[3].Inaddition,they
symmetrizedfinite-frequencyheatnoise[21,22,32,34,35].
revealfeaturesthatarenotvisibleinthechargenoise[4],
Thenonsymmetrizedfinite-frequencyheatnoisehasalsobeen
suchasthesignatureofacrossoverfromCoulombblockade
calculatedforaquantumdot(QD),butonlyforsymmetrical
toKondophysicsinenergyfluctuations[5].Inthecaseof
couplingsbetweentheQDandtheelectronreservoirs[41].
on-demandsingle-electronsources,theheatfluctuateswhile
Theobjectiveofthepresentworkistwofold:first,togener-
thechargeemissionisnoiseless[6].Sofar,onlytempera-
alizethecalculationofnonsymmetrizedfinite-frequencyheat
turefluctuations[7],relatedtoenergyfluctuations[6,8,9],
noisetothecaseofasymmetricalcouplings,whichcandiffer
havebeenmeasured[10],butthereareseveralproposalsfor
byafactorofuptoten[49],bylookingatbothautocorrelators
themeasurementofheatfluctuations[8,9,11,12].Withthe
andcrosscorrelators,andsecond,tohighlightthemainfea-
fastprogressofheatmeasurementtechniquesinnanosys-
tems[13–16],itcanbeexpectedthatthiswillbepossiblein
contributiontotheheatnoiseisconsideredinthiswork.
Thepaperisorganizedasfollows:themodelandresultsare
initselfwellcontrolled[17],notablywiththeexperimental
,theequilibriumandout-of-equilibrium
confirmation[18]oftheexistenceofathermalconductance
heatnoisesare,respectively,,
quantum[19]andtheevidenceoftheheatCoulombblockade
.
effect[20].
Theissuesraisedbythesestudiesarealsooffunda-
fluctuation-dissipationtheoremtoheattransporthasbeen
addressed[21–23],aswellastheverificationofthefluctu-ThestandardHamiltoniandescribinganoninteractingQD
ationtheorem[12,24–28],whichisamicroscopicextensionconnectedtoleft(L)andright(R)reservoirsofelectronsis
,
heatexchangeinadrivenopenquantumsystemhavebeen
studied[29]aswellasthestatisticsofworkforatwo-††
H=εαccα+εdd
levelsysteminthepresenceofdissipation[30].Amongkαkk0
α=L,Rk∈α
thetheoreticalapproachesusedtostudytheheatfluctua-
tionsinquantumdevices,onecancitetheLandauer-Büttiker+†+..,
(VαkcαkdHc)(1)
formalism[31,32],nonequilibriumSchwinger-KeldyshGreenα=L,Rk∈α
2469-9950/2021/103(4)/045427(6)045427-1©2021AmericanPhysicalSociety
ÉPIEUXPHYSICALREVIEWB103,045427(2021)
γδε,ω=ε=
()appearinginthefinite-frequencyheatnoiseofEq.(2),settingh¯1,wheretαβ()
r2
iαβG(ε)isthetransmissionamplitude,Tαβ(ε)=|tαβ(ε)|isthetransmissioncoefficient,andEα(ε)=ε−μαisthedifferencebetween
theenergyεoftheparticleandthechemicalpotentialinthereservoirα.
γδ
Mαβ(ε,ω)γ=δ=Lγ=δ=Rγ=L,δ=Rγ=R,δ=L
α=Eε−ωεE2ε−ωEεEε−ω
LL()tLL()L2L()L()
22
β=L+E(ε)t∗(ε−ω)×T(ε)T(ε−ω)−Eε−ωt(ε)−Eε−ωt(ε−ω)
LLLLRLRL2LLL2LL
−Eε−ωε∗ε−ω2×Tε−ω×Tε
L2tLL()tLL()LR()LR()
α=E2ε−ωEε−ωεEε−ωEε
RR2R()tRR()R()R()
22
β=R×T(ε)T(ε−ω)+E(ε)t∗(ε−ω)−Eε−ωt(ε−ω)−Eε−ωt(ε)
LRLRRRRR2RRR2RR
−Eε−ωε∗ε−ω2×Tε×Tε−ω
R2tRR()tRR()LR()LR()
α=LEε−ωt(ε)t∗(ε−ω)Eε−ωt∗(ε)t(ε−ω)E(ε)t(ε)E(ε−ω)t∗(ε−ω)
R2LRLRL2LRLRLLLLLL
β=R×Eε−ωt∗(ε)t(ε−ω)×Eε−ωt(ε)t∗(ε−ω)−Eε−ωT(ε)−Eε−ωT(ε−ω)
L2LLLLR2RRRRL2LLL2LL
−E(ε−ω)t∗(ε)−E(ε−ω)t(ε)×E(ε−ω)t(ε−ω)×E(ε)t∗(ε)
LLLRRRRRRRRR
−Eεε−ω−Eε∗ε−ω−Eε−ωTε−ω−Eε−ωTε
L()tLL()R()tRR()R2RR()R2RR()
α=REε−ωt∗(ε)t(ε−ω)Eε−ωt(ε)t∗(ε−ω)E(ε)t∗(ε)E(ε−ω)t(ε−ω)
R2LRLRL2LRLRLLLLLL
β=L×Eε−ωt(ε)t∗(ε−ω)×Eε−ωt∗(ε)t(ε−ω)−Eε−ωT(ε)−Eε−ωT(ε−ω)
L2LLLLR2RRRRL2LLL2LL
−E(ε−ω)t(ε)−E(ε−ω)t∗(ε)×E(ε−ω)t∗(ε−ω)×E(ε)t(ε)
LLLRRRRRRRRR
−Eε∗ε−ω−Eεε−ω−Eε−ωTε−ω−Eε−ωTε
L()tLL()R()tRR()R2RR()R2RR()
††ω,
wherec(d),cαk(d)arethecreationandannihilationop-isapplicabletoanyfrequency,temperaturesTLR,voltage
αk
eratorsassociatedwiththereservoirα(respectively,theQD).V,andcouplingsL,.[41]
Theenergiesεαk,ε0,andVαkare,respectively,
theelectronsinthereservoirα,thediscreteenergyleveloftheexpressionsoftheelementsforthematrixMreducetothe
charge
QD,andthehoppingintegralbetweenthereservoirsandtheonesenteringintheexpressionofthechargenoiseSαβ(ω)
-ofRefs.[55,56],providedthatthefactorEα(ε)isreplaced
r−12
G(ε)=[ε−ε0+i(L+R)/2],whereα=2πρα|Vα|intotheexpressionoftheheatnoise:Eα(ε),theenergyof
isthecouplingbetweentheQDandthereservoirαassum-theelectroninthereservoirα;Eα(ε−h¯ω),theenergyofthe
ingthatthedensityofstatesραandVα≡Vαkareenergyholeinthereservoirα;andEα(ε−h¯ω/2),theaverageenergy
-
TheheatnoiseisdefinedastheFouriertransformofarerelatedtotheenergyexchangedwiththeelectromagnetic
thenonsymmetrizedcorrelatorofheatcurrentsattwodif-environmentsurroundingtheQDduringthevarioustransfer
heat∞−iωt
ferenttimes:Sαβ(ω)=−∞ Jα(t) Jβ(0)edt,
=−Sheatω
Jα(t)Jα(t)
isgivenby10innumber(forLL(),)andinvolvetransferof
=−H˙+μ˙H=ε†
[50–53]Jα(t)ααNα,whereαk∈ααkcαkcαkelectron-
is
theHamiltonianoftheuncoupledreservoirαandNα=locationoftheelectronandthehole,thenumberofpossible
†
k∈
α
-locatedintherightreservoir,thereisonlyoneprocess:P1.
frequencyheatnoiseisperformedusingthenonequilibriumWhentheelectronislocatedintheleft(right)reservoirand
[54]theholeintheright(left)reservoir,therearetwoprocesses:
∞P2andP3(P4andP5).Finally,whentheelectronandthehole
heat1γδeh
Sαβ(ω)=dεMαβ(ε,ω)fγ(ε)fδ(ε−h¯ω),arebothinitiallylocatedintheleftreservoir,therearefive
hγ,δ=,−∞
LRprocesses:
(2)eachofthesesetsofprocesses,onemusttakethequantum
e−1h
wherefγ(ε)={1+exp[(ε−μγ)/kBTγ]}andfδ(ε)=superpositionoftheprocesseshavingthesameinitialstate
e
1−fδ(ε)aretheFermi-Diracdistributionsfortheelectrons[54].Usingthissimplerule,onerecognizestheexpression
ofLL()
Tγare,respectively,thechemicalpotentialandthetemper-exchangedwiththeelectromagneticenvironmentduringthe
γδ
(ε,ω)areprocessesistheaverageenergyoftheelectron-holepair,.,
(ε−h¯ω/2),whentheelectronandtheholeboth
inEq.(2)variesaccordingtotheexperimentalconditions:makeanexcursioninthecentralpartoftheQD,or±h¯ω/2,
theyallhaveanequalweightatequilibrium,
LR
Mαβ(ε,ω).(2)isthecasefortheprocessesP3,P5,P8,andP10,whichhave
045427-2
ELECTRONICHEATCURRENTFLUCTUATIONSINA…PHYSICALREVIEWB103,045427(2021)
-Nyquist(a)heatand(b)chargenoisesatequi-
librium(eV=h¯ω=0andTL,R=T)asafunctionoftheQDlevel
energyε0atkBT=(.,T=)forseveralvaluesof
thecouplingsbetweenthereservoirsandtheQDtakensymmetrical:
L,R=(inmeV).
Scharge
,atequilibrium,thechargefluctuationsare
maximalwhentheQDenergylevelisalignedwiththechem-
icalpotentials,.,atε0=0whenμL,R=0,sincecharge
intheheatnoiseatε0=|ε0|,the
(blue)
spheresrepresentanelectron(hole)ofenergyε(ε−h¯ω).Theenergytozeroduetothefactthattheprobabilityforthechargetobe
releasedforeachprocessisindicatedontheleftside.
transferredthroughthedotvanishesathigh|ε0|.Whenthetwo
Sheatε≈±.
peaksinJNarepresent,theirpositionsare025kBTat
theparticularitytocontributetotheheatnoisewhereastheymost[54],inlinewithRef.[58]wheresuchadouble-peak
donotcontributetothechargenoise[55].structurehasbeenpredictedinthethermalconductanceof
Sheat
+
[54].OnefindsthattheconditionisLR
BeforeexploitingtheresultofEq.(2),onechecksthatitatequilibriumtheheatnoiseisproportionaltothethermal
givestheexpectedbehaviorforheatnoisewithinknownlim-conductance.
=0,symmetricalcouplingsL,R=
withT(ε)=2Gr(ε)Ga(ε),andusingtheopticaltheorem
-OF-EQUILIBRIUMHEATNOISE
thatholdsforanoninteractingQD,whichmeansthatone
hast(ε)+t∗(ε)=2T(ε),Eq.(2)leadsfortheautocorrelatorsTheheatnoiseissensitivetothefactthatthesystemis
(α=β)totheexpression[54]drivenoutofequilibriumeitherbyapplyingabiasvoltage,
∞atemperaturegradient,orbyconsideringthenoiseatfinite
heat12
Sαα(0)=dε(ε−μα)
h−∞
noisesintheleftreservoirasafunctionofthefrequencyat
ee2
×T(ε)[1−T(ε)]fα(ε)−fα(ε)
eheh
+T(ε)fα(ε)fα(ε)+fα(ε)fα(ε),(3)
inagreementwiththeresultsofRefs.[4,36,42].Theindex
αtakesthevalueRforα=LandthevalueLforα=R.
ThelastlineofEq.(3)correspondstotheequilibriumheat
Sheat
noiseJN(Johnson-Nyquist),whichcanbeexpressedasa
functionofthethermalconductanceKα=∂Jα/∂Tαbythe
Sheat=2+2
relationJNkBTLKLkBTRKR,inperfectagreementwith
Refs.[36,46,57].Oneisremindedthattheequilibriumcharge
Scharge
noiseJNisrelatedtotheelectricalconductancebyth
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