Detection of finite frequency photo-assisted shot noise with a resonant circuit

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by  Denis Chevallier

Detection offinitefrequency photo-assistedshotnoisewitharesonant circuit
D. Chevallier,1,2 T. Jonckheere,1 E. Paladino,3,4 G. Falci,3,4 and T. Martin1,2
1CentredePhysiqueThe´orique, UMR6207, Case907, Luminy, 13288 MarseilleCedex9, France
2Universite´ de la Me´diterrane´e, 13288 Marseille Cedex 9, France
3CentredePhysiqueThe´orique, Universite´ delaMe´diterrane´e, Case907, 13288 Marseille, France
4MATIS CNR-INFM, and D.M.F.C.I., Universita´ di Catania, 95125 Catania, Italy
Photo-assistedtransportthroughamesoscopicconductoroccurswhenanoscillatory(AC)voltageissuper-
0 posedtotheconstant(DC)biaswhichisimposedonthisconductor. Ofparticularinterestisthephotoassisted
1 shotnoise,whichhasbeeninvestigatedtheoreticallyandexperimentallyforseveraltypesofsamples. ForDC
0 biasedconductors, adetectionschemeforfinitefrequencynoiseusingadissipativeresonantcircuit,whichis
2 inductivelycoupledtothemesoscopicdevice,wasdeveloppedrecently.Wearguethatthedetectionofthefinite
frequencyphoto-assistedshotnoisecanbeachievedwiththesamesetup,despitethefactthattimetranslational
n
invarianceisabsenthere. Weshowthatameasureofthephoto-assistedshotnoisecanbeobtainedthroughthe
a
chargecorrelatorassociatedwiththeresonantcircuit,wherethelatterisaveragedovertheACdrivefrequency.
J
WetestourpredictionsforapointcontactplacedinthefractionalquantumHalleffectregime,forthecaseof
2
weakbackscattering. TheKeldyshelementsof thephoto-assisted noisecorrelator arecomputed. Forsimple
1
Laughlinfractions,themeasuredphoto-assistedshotnoisedisplayspeaksatthefrequencycorrespondingtothe

DCbiasvoltage,aswellassatellitepeaksseparatedbytheACdrivefrequency.
]
l
l
a PACSnumbers: 73.23.-b,72.70.+m,73.63.-b,
h
-
s
e I. INTRODUCTION
m
. Theunderstandingofthetransportpropertiesofnanoscaleconductorsatlowtemperatureshasknowntremendoussuccesses
t
a viaexperimentsinawiderangeofsystemsperformedforthemostpastinthestationaryregime. Correspondingly,theoretical
m modelling has allowed the description of these transport processes via scattering theory approaches as well as Hamiltonian
- formulations, in a fruitful dialogue with experimental investigations. Transport is first characterized by the average current
d flowingthroughconductors.Butfurtherinformationcanbegainedviathemeasurementandanalysisofthecurrentfluctuations1,2
n andmoregenerallyvia thehighercurrentmoments.3 Earlyinvestigationsof quantumtransportfocusedalmostexclusivelyon
o
the low frequencyregime. Few recent experimentshave probedquantumsystem on timescales comparablewith the electron
c
[ correlationtime,wherenewphysicaleffectsareexpected. Thepresentworkdealswiththedetectionofquantumnoiseatsuch
highfrequencies,whenbothaDCandaACbiasisimposedbetweenthesourceandthedrainofthemesoscopicsystem.
1
Indeed,highfrequencymeasurementscanmeanseveralthings. First,ifonlyaDCbiasisimposedonthesample,astationary
v
currentisgeneratedandhighfrequenciesrefertotheFouriercomponentofthecurrent-currentcorrelationfunctionintime.4–7
0
Second,highfrequenciescanbeinjectedasadriveonthemesoscopiccircuit,8–11 forinstancewhenanadditionalACdriveis
1
9 superposedtotheDCbias. Thelatereffectiscalledphoto-assisted(PA)transport: electronsundergoingtransmissionfromone
1 leadtoanotherareabletoabsorb/emit“photons”duringthisprocess. PAtransport,andinparticularPAnoisehasbeenstudied
. theoretically and experimentallyon several occasions for diffusive metals,5 tunnel junctions,12 normal metal/ superconductor
1
0 junctions13,14aswellasquantumpointcontacts.6ThenoisecaracteristicsthendisplayssomestructureatvaluesoftheDCbias
0 whicharemultipleoftheACdrivefrequency.
1 However,highrequencynoisedetectionrequiresspecialcare: conventional(low)frequencynoisedetectionsetupsareoften
: inadequate for such measurements, and one must often resort to on-chip detection schemes, or alternatively/equivalently to
v
i schemeswhereagoodconnectiontothemeasurementcircuitisachievedthroughadaptedimpedencelines.15Onchipdetectors
X
have allowed the detection of single electronstravellingthroughquantumdots. Such detectorsand the devicethey probe are
r partsof the same quantumsystem and mustbe treated onthe same footing. Theybear the peculiaritythatthe noise which is
a
measuredisanontrivialcombinationofnon-symmetrizednoisecorrelators.ForDCdrivensystemsthereareexistingproposals
todetecthighfrequencynoiseusingeithercapacitiveorinductivecouplingwithanon-chipcircuit.16
In a recenttheoreticalwork, a LC resonantcircuit, which was coupled inductivelyto the mesoscopic device circuitry, was
employed as a detector of both noise and higher current moments (third moment).17 The description of this generic detector
includeditselectromagneticenvironement,describedata bathofharmonicoscillatorswith theCaldeiraLegettmodel18. Pre-
dictionsweremadeontheroleofsuchadissipativeenvironmentandontherelevanceofthisharmonicdetectortocaptureon
highfrequencycurrentmoments. However,thisstudyconsideredthecaseofamesoscopicdeviceinastationaryregime(with
a DC bias only). Thehypothesisofa stationaryregimegreatlysimpliesthe analysisof the detectionprocessbecause of time
translationalinvariance.ThepresenceofanadditionalACvoltagedrivebreakssuchaproperty.
GiventheinterestinthestudyoftimedrivenmesoscopicsystemsandinparticularPAnoise,itseemsnecessarytoaddresshow
detectionwithanauxiliarycircuitcanbeachievedinsuchsituations. Thepurposeofthisworkistopresentahighfrequency
detection scheme for photoassisted noise, and to illustrate it with a calculation of photoassisted noise in a specific situation
2
where signatures of photoassisted transport are most dramatic. For devices composed of normal metal junctions as well as
superconducting/normalmetaljunctions, PA noiseexhibitssingularitiesatintegerratiosofthe DCvoltagewithrespectto the
AC frequency: the derivativeof this noise exibitsjumpsat such locations. On the otherhand, fora weaklypinchedquantum
pointcontactplaced in the fractionalquantumHall effectregime(FQHE),19–23 the PA noise divergeswhen the DC voltage –
multiplied by the filling factor – is a multiple of the AC frequency. This much stronger singularity is a motivation for us to
applyourmeasurementschemetothe FQHEsituation. We willshowthatasintheDCcase, themeasurednoisecapturesthe
responseofthemesoscopiccircuitattheresonantfrequencyoftheLCcircuit.ItexhibitsacentralpeakattheDCvoltage,which
issurroundedbysatellitepeaksshiftedbytheACfrequency.Thesepredictionshavethepotentialtobetestedinexperiments.
Thepaperisorganizedasfollows. InSec. IIwepresentthemodelfortheLCdetector. Wereviewtheresultsforthecharge
correlatoroftheDCcirsuitinSec. IIIandextendthisdiscussiontothePAsituation. Sec. IVisdevotedtothepresentationof
theQPCintheFQHEregimeanditscalculationofPAnoise. Plotsofthesequantitiesandofthemeasurednoisearediscussed
inSec. V.WeconcludeinSec. VI.
II. MODEL
The proposedsetup is the same as that presented in Ref. 17, exceptfor the fact that the voltage source on the mesoscopic
deviceis time dependent. A lead fromsuch deviceis inductivelycoupledto a resonantcircuit(capacitanceC, inductance ,
L
anddissipativecomponentR). Thesignalwhichcontainsinformationaboutthenoiseofthemesoscopiccircuitisencodedin
thetimecorrelationfunctionofthechargeonthecapacitor.
FIG.1:Mesoscopiccircuitiscoupledtoaresonantdissipativecircuit
Westartwiththedescriptionofthedetector. ThebasicHamiltonianwhichdescribesthedissipativeoscillatorcircuitreads:
H =H +H , (1)
osc 0 LC env
−
where
H =H +H (2)
0 LC env
is the Hamiltonianof the uncoupledsystem “LC oscillatorplusenvironment”,andH describesthe couplingbetween
LC env
thetwo. −
Fordissipativequantumsystems,itisconvenienttouseapathintegralformalism. Intheabsenceofdissipationandcoupling
tothemesoscopicdevice,theKeldyshactiondescribingthechargeoftheLC circuitreads:
1
SLC[q]= 2 dtdt′qT(t)G−01(t−t′)σzq(t′), (3)
Z
where
G−01(t−t′)=L[(i∂t)2−Ω2]δ(t−t′), (4)
isthe(inverse)Greenfunctionofanharmonicoscillator( isits“mass”),Ω=( C) 1/2istheresonantfrequencyofthecircuit,
−
qT = (q+,q ) is a two componentvectorwhichcontaiLnsthe oscillator coordLinateon the forward/backwardcontour,andσ
− z
3
isaPaulimatrixinKeldyshspace. DissipativeeffectsaretreatedwithintheCaldeira-Leggettmodel,wheretheenvironmentis
modeledbyasetofharmonicoscillators(bath)withfrequencies ω ;thecoordinateqiscoupledlinearlytothebathoscillators:
n
{ }
H =q λ x , (5)
LC env n n
−
n
X
withthecouplingconstantsλ .
n
ThepartitionfunctionoftheLC oscillatorplusbath,Z = q xeiS[q,x],hasanaction:
D D
1 R
S =SLC + 2 xTn ◦Dn−1◦σzxn−qT ◦σz λnxn , (6)
n n
X X
whereD 1(t) = M [(i∂ )2 ω2]δ(t)andthesymbol standsforconvolutionintime. Thebathdegreesoffreedomcanbe
integratedn−outin a stnandartd m−annner24. As a result, the G◦reen functionG of the LC circuit becomesdressed by its electronic
environment,
G−1 =G−01−Σ, (7)
withaself-energyΣ(t)=σ λ2D (t)σ . Intheremainerofthispaper,whenwementiontheLC circuit,itwillalsoimply
z n n n z
thepresenceofitssurroundingelectromagneticenvironment.
P
Next,weintroducetheinductivecouplingbetweenthemesoscopicdeviceandtheLC circuit,
H =αqI˙, (8)
int
where I˙ is the time derivative current operator25. This interaction is interpreted here as an external potential acting on the
oscillatorcircuit.TocalculatecorrelationfunctionsoftheLC circuitcoordinateq,weintroducethegeneratingfunctional,
1
[η,I]= qexpi qT G 1 q qTσ (αI˙ +η) , (9)
− z
Z D 2 ◦ ◦ − ◦
Z h i
whereηT = (η+,η )isatwo-componentauxiliaryfield. PerformingintegrationovertheLCoscillatorvariablesqresultsin
−
[η,I]=eiSeff[η,I] withaneffectiveaction(restoringintegrals):
Z
i
S [η,I] = dt dt(η(t)+αI˙(t))Tσ Gˇ(t t)
eff ′ z ′
−2 −
Z Z
σ (η(t)+αI˙(t)) . (10)
z ′ ′
×
i
III. CHARGECORRELATOR
By taking double derivatives of the Kelysh partition function with respect to the components of the spinor η, the charge
correlatorisobtained:
hqβ(t)qβ′(t′)i≡Zη−1[I]∂η(∂tβ2)Z∂ηη[I(t]β′) , (11)
′ η=0
where β,β 1 are indices specifying the upper/lower branch of the Keldysh contour. To leading order in the coupling
′
constantαbe≡twe±enthemesoscopiccircuitandthedetector17thiscanbeexpressedintermsofthecurrentderivativecorrelator:
Kβ1β2(τ ,τ )= T I˙(τ )β1I˙(τ )β2 , (12)
1 2 K 1 2
meso
D E
where the average ... represents a non equilibrium average containing information on the occupation of the reservoirs
h imeso
connectedtothesampleandonitsscatteringproperties.ThechargecorrelatorconsiststhenofaKeldyshmatrix:
TKqβ(t)qβ′(t′) =α2 dτ1dτ2 Gββ2(t−τ2)σzβ2β2Kβ2β1(τ2,τ1)σzβ1β1Gβ1β′(τ1−t′), (13)
D E Z βX1β2
where the integrandcontainsthe Green functionGββ′(t) of the LC circuit only. While this Green functionis a functionof a
singletimeargumentbecauseoftimetranslationalinvariance,thecurrentderivativecorrelatorKβ1β2(τ ,τ )isnotafunctionof
1 2
thedifferenceτ τ ifthebiasvoltageistimedependent.
1 2
−
4
A. DCVoltageonly
We recall the results obtained previously for the detection of finite frequency noise in the presence of time translational
invariance. Theinitialproposalof Ref. 25for a dissipationlessLC circuitwas to operaterepeatedtime measurementson the
chargeq.Thisallowstoconstructanhistogramforzerovoltage,yieldingthezerobiaspeakposition,itswidth,skewness,...Inthe
presenceofbias,thishistogramisshifted,andacquiresanewwidth,skewness,...Informationaboutallcurrentmomentsathigh
frequenciesisencodedinsuchhistograms. Here,however,weonlyfocusonthedetectionofnoise. InRef. 17,theinclusionof
dissipationduetotheelectromagneticenvironmentwasshowntobeessentialtoobtainafiniteresultforthemeasuringprocess.
There,expressionsfortheoffdiagonalKeldyshcomponentofthechargecorrelator T q (t)q+(t) = q(t t)q(0) were
K − ′ ′
h i h − i
derivedwith the helpof Eq. (13). Note that in thissituation, the currentderivativecorrelatorof Eq. (12) is a functionof the
differenceτ τ ,andthechargecorrelatorisaconvolutionproduct,whichexplainsitsdependenceont t only.
1 2 ′
− −
GoingtotherotatedKeldyshbasis(seeAppendixB)allowstorewritethechargefluctuationsatequaltime(t=t)as:
′
dω
δ q2 = α2 GR(ω) GK(ω)K+−(ω) (GR(ω) GA(ω))K−+(ω) , (14)
h i 2π { − − }
Z
withthethreeGreenfunctioncomponentsgivenby:
GR/A(ω)=[ (ω2 Ω2) isgn(ω)J(ω )] 1 , (15)
−
L − ± | |
and
GK =(2N(ω)+1)(GR(ω) GA(ω)), (16)
−
whereN(ω)istheBoseoccupationnumberoftheoscillatorandthebathspectralfunctionisdefinedas:
J(ω)=π λ2/(2M ω )δ(ω ω ). (17)
n n n − n
n
X
ThisspectralfunctionisattheoriginofthebroadeningfortheLC circuitGreenfunction.
The time derivativecorrelatorsK +,+ are related to the Fourier transformof the currentcurrentcorrelationfunctionsas
− −
K+ (ω)=ω2S+ (ω)andK +(ω)=ω2S +(ω),with
− − − −
S+−(ω)= dt I(0)I(t) eiωt , (18)
h i
Z
andS +(ω) = S+ ( ω)correspondingtotheresponsefunctionforemission/absorptionofradiationfrom/tothemesoscopic
− −
circuit25,26. Withthese−definitions,thefinalresultforthemeasurableexcessnoisereads:
δ q2 = 2α2 ∞ dωω2[χ (ω)]2
′′
h i 2π
Z0
S+ (ω)+N(ω)(S+ (ω) S +(ω)) , (19)
− − −
× −
whereχ (ω)=J(ω )/[ 2(ω2 Ω2)2+J2(ω )(cid:0)isthesusceptibilityofRef. 18,heregen(cid:1)eralizedtoarbitraryJ(ω ). Eq. (19)
′′
constitutesamesos|co|picLanalog−oftheradiatio|n|linewidthcalculation27: adissipativeLC circuitcannotyieldan|yd|ivergences
inthemeasurablenoise. Dissipationisessentialinthemeasurementprocess.
Eq. (19) indicates thatfor an infinitesimal line width, the integrandcan be computedat the resonantfrequencyΩ, and the
measurednoisetakestheformofRef. 25:
α2
q2 = γ 2 S+−(Ω)+NΩ(S+−(Ω)−S−+(Ω)) , (20)
L
(cid:10) (cid:11) (cid:8) (cid:9)
where the prefactor γ is defined assuming a strict Ohmic or Markovian damping (J(ω) = γω), which corresponds to a
L
memorylessbathwhichisconsistentwiththeadiabaticswitchingassumption,asdiscussedinRef. 17.
As an alternative to the measurement of the width of the charge distribution, one can imagine that the capacitor itself is
coupledto a measuringdevice (a single electron tunnelingdevice) which directly detects the Fouriertransform of the charge
correlator.28GiventhefactthatthechargecorrelatormatrixofEq. (13)isaconvolutionproductinthisstationarysituation,its
Fouriertransformtakethesimpleformofaproductofmatrices:
dt T qβ(t)qβ′(0) eiωt =α2 G˜(ω)σ K˜(ω)σ G˜(ω) ββ′ , (21)
K z z
h i
Z h i
whereG˜(ω)andK˜(ω)arerespectivelythematrixversionoftheLCGreen’sfunctionandofthecurrentderivativecorrelator.
Naturallythis will havesubstantialcontributionswhenboth K andG overlapsignificantly. Thisconstitutesa rathercompact
wayfordescribingthedetectionprocessinthecaseofaconstantbiasvoltage.
5
B. ACdriveandtemporalinvariance
Wenowturntothemainpointofthissection,whichistoaddresshowtodealwiththepresenceofanACvoltagesuperposed
totheDCone. ThetotalbiaspotentialV(t)whichisappliedtothemesoscopicdeviceisthusaperiodicfunctionoftimewith
periodτ =2π/ω . Westartbydefiningacorrelatork(T,t )fromthecurrentderivativecorrelatorofEq. (12):
AC ′′
t+t
K(t,t) k( ′,t t). (22)
′ ′
≡ 2 −
DefiningT =(t+t)/2,t =t t,thechargecorrelatorofEq. (13)isrewrittenas:
′ ′′ ′
−
TKqβ(t)qβ′(t′) =α2 dt1dt2 Gββ2(t′′−t2)σzβ2β2kβ2β1(T +t0,t2−t1)σzβ1β1Gβ1β′(t1). (23)
D E Z βX1β2
wheret =(t +t )/2 t /2.Next,wedefinetheaverageofthechargecorrelatorovertheperiodoftheACdrive11asfollows:
0 2 1 ′′
−
τ1 τdT qβ(t)qβ′(t′) =α2 dt1dt2 Gββ2(t′′−t2)σzβ2β2 τ dτTkβ2β1(T +t0,t2−t1)σzβ1β1Gβ1β′(t1). (24)
Z0 D E Z βX1β2 Z0
NotethatthelastintegraloverthevariableT isessentiallyaperiodaverageofthecorrelatork(T,t )withthevariableT shifted
′′
byt .InthepresenceofanACdrive,thisperiodaveragedoesnotdependontheshiftt ,becauseasafunctionofthevariableT,
0 0
k(T,t )containsonlyharmonicsofthedrivefrequencyω . Thishasbeennoticedinearlierworks.13,29,30 Forourpurposes,
′′ AC
itmeansthatwecansafelyreplacet by0. Asaresult,theperiodaveragedchargecorrelatortakestheformofaconvolution
0
productaswasthecasefortheconstantDCbias,anditthereforedependsonlyonthetimedifferencet t:
′
−
Qˆββ′(t t) 1 τdT qβ(t)qβ′(t)
′ ′
− ≡ τ
Z0 D E
= α2 dt1dt2 Gββ2(t′′−t2)σzβ2β2Kβ2β1(t2−t1)σzβ1β1Gβ1β′(t1), (25)
Z βX1β2
wherewedefinedtheperiodaveragedcorrelator:
1 τ
β2β1(t) dTkβ2β1(T,t). (26)
K ≡ τ
Z0
Finally,theaveragedchargecorrelatorcanbeexpressedintermsoftheFouriertransfromofboththeLCcircuitGreen’sfunction
andtheperiodaveragedcurrentcorrelator:
Qˆββ′(t′′)=α2 d2ωπe−iωt′′Gββ2(ω)σzβ2β2Kβ2β1(ω)σzβ1β1Gβ1β′(ω). (27)
βX1β2Z
This result is the exact analog of the DC formula Eq. 21, extended to and AC drive. In addition, at t = 0, this expression
′′
hasthesameformastheresultofRef. 17. We havethereforeidentifiedwhichquantity( )characterizestheinfluenceofthe
K
mesoscopiccircuitontheresponseoftheLCcircuit. Therefore,theprotocolformeasuringphotoassistedshotnoiseisthesame
as in the DC case providedone averagesthe response over the frequencyof the drive. This averagingprocedurerestores the
temporalinvarianceofthe chargecorrelator. Inthe followingsections, we will computethecurrentderivativecorrelatorsand
theirperiodaverageforaspecificsystem: aQPCplacedintheconditionsoftheFQHEwheretheelementarytransportprocess
isthePoissoniantransferofLaughlinquasiparticles.
IV. NONSYMMETRIZEDPHOTO-ASSISTEDNOISEINTHEFQHE
ThecalculationofthesymmetrizedphotoassistednoisehasbeencarriedoutinRef. 30. Hereweusethesamebasicmodel
and generalizethe calculationsof the noise correlatorto the full Keldyshmatrix elementsof this correlator. Next, we extract
fromthesethenoisederivativecorrelatorswhicharerelevantforthemeasurementprocess.
6
V(t)
Γ
I (t)
0 B
FIG.2:Quantumpointcontact
A. Modelforquasiparticlebackscattering
WeusetheTomonaga-Luttingerformalismtodescribetherightandleftmovingchiralexcitations.Intheabsenceoftunneling
betweenthetwoedges,theHamiltonianreads:
ν ~
H =( F ) ds(∂ φ )2, (28)
0 s r
4π
r Z
X
withr = +, forrightandleftmovers. Here,wefocussolelyontheweakbackscatteringregimebecauseitisalreadyknown
−
thatthePAshotnoiseexhibitssomestrongsingularities.ThebackscatteringofquasiparticlesisdescribedbytheHamiltonian:
HB(t)= A(ε)(t)[Ψ†+(t)Ψ (t)](ε) , (29)
−
ε
X
where A(ε)(t) is a tunneling amplitude which depends on the applied voltage via the Peierls substitution. Here the notation
ǫ = leavesanoperatorunchangedfor(ǫ = +)orspecifiesitsHermitianconjugate(ǫ = ). Ψ isthequasiparticleoperator
r
± −
whichisexpressedintermsofthebosonicchiralfieldφ :
r
1
Ψ (t)= ei√νφr(t) , (30)
r
√2πa
whereaisashortdistancecutoffandν isthefillingfactor(ν 1 isanoddintegertodescribeLaughlinfractions). Choosinga
−
timedependentvoltageintheformV(t)=V +V cos(ω t)resultsinatunnelingamplitude:
0 1 AC
e V
A(ε)(t)=Γ eiεω0texp(iε ∗ 1 sin(ω t)), (31)
0 ~ω AC
AC
wheree =νeandΓ isthebaretunnelingamplitude.ThebackscatteringcurrentisdeducedfromthebackscatteringHamilto-
∗ 0
nian:
ie
IB(t)= ~∗ εA(ε)(t)[Ψ†+(t)Ψ (t)](ε) . (32)
−
ε
X
B. Non-symmetrizednoise
ThegeneralexpressionfortheKeldyshcomponentsofthenoisecorrelatorintheHeisenbergrepresentationis:
Sββ′(t,t)= Iβ(t)Iβ′(t) I (tβ) I (tβ′) . (33)
′ h B B ′ i−h B ih B ′ i
SinceweareinterestedinPoissonianregimeonly,theproductofcurrentaveragescanbedroppedoutbecauseitcontributesto
higherorderinthebackscatteringHamiltonian.1Moreover,inthissecondordercalculationinthetunnelingamplitudeΓ ,there
0
isnodifferencebetweentheHeisenbergandinteractionpicture. Thenoisethenreads:
Sββ′(t,t′)=−(e∗)2 εε′A(ε)(t)A(ε′)(t′)hTK{[Ψ†+(tβ)Ψ−(tβ)](ε)[Ψ†+(t′β′)Ψ−(t′β′)](ε′)}i. (34)
εε′
X
This correlator is differentfrom zero only when ε = ε because of quasiparticle conservation. Replacing the quasiparticle
′
−
correlatorsbytheirbosonizedexpression,thenoiseisthenwrittenintermsofaproductoveraveragesofbosonicfields:
Sββ′(t,t′)= 4(πe∗2)a22 εA(ε)(t)A(−ε)(t′)hTKe−iε√νφ+(tβ)eiε√νφ+(t′β′)ihTKeiε√νφ−(tβ)e−iε√νφ−(t′β′)i. (35)
ε
X
7
Thefinalresultfortherealtimenoisecorrelatoristhen:
Sββ′(t,t′)= (e∗)2 e2νGββ′(t−t′)(A(t)A∗(t′)+A∗(t)A(t′)) , (36)
4π2a2
whereweintroducedthechiralgreenfunctionofthebosonicfields:
Gββ′(t,t)= T φ (tβ)φ (tβ′) 1 T φ (tβ)2 1 T φ (tβ′)2 . (37)
′ K r r ′ K r K r ′
h { }i− 2h { }i− 2h { }i
ThedoubleFouriertransformofthisquantity,whichwillallowtorelateittothenoisecorrelator,reads:
Sββ′(Ω1,Ω2)= dtdt′ei(Ω1t+Ω2t′)S(t,t′). (38)
Z Z
We nowspecifytheperiodicvoltagemodulation,whichallowsto write thetunnelingamplitudeintermsof aseriesofBessel
functionsJ :
n
+
A(t)=Γ ∞ ei(ω0+nωAC)tJ e∗V1 , (39)
0 n ~ω
n= (cid:18) AC(cid:19)
X−∞
whichgivestheFouriertransformofnon-symmetrisednoise:
Sββ′(Ω ,Ω ) = (e∗)2Γ20 +∞ +∞ J e∗V1 J e∗V1
1 2 2π2a2 n ω m ω
n= m= (cid:18) AC(cid:19) (cid:18) AC(cid:19)
X−∞ X−∞
dtdt′ei(Ω1t+Ω2t′)e2νGββ′(t,t′)cos(ω0(t t′)+ωAC(nt mt′)). (40)
× − −
Z Z
Next,itisconvenienttoperformachangeofvariableτ =t t andτ =t+t:
′ ′ ′
−
Sββ′(Ω ,Ω ) = 2(e∗)2Γ20 +∞ +∞ J e∗V1 J e∗V1
1 2 2π2a2 n ω m ω
n= m= (cid:18) AC(cid:19) (cid:18) AC(cid:19)
X−∞ X−∞
dτdτ′ei(Ω1−Ω2)τ/2ei(Ω1+Ω2)τ′/2e2νGββ′(τ)cos ω0+ n+mωAC τ + n−mωACτ′ .
× 2 2
Z Z (cid:18)(cid:18) (cid:19) (cid:19)
Usingstandardtrigonometricidentities,onecanwritethisexpressionasaproductofseparateintegralsoverτ andτ . Integrals
′
overτ containthe (zerotemperature)Green’sfunctionof the chiralfields andcan be expressedin termsofGamma function.
Theresulthastheform:
Sββ′(Ω ,Ω ) = 2(e∗)2Γ20 +∞ +∞ J e∗V1 J e∗V1
1 2 2π2a2 n ω m ω
n= m= (cid:18) AC(cid:19) (cid:18) AC(cid:19)
X−∞ X−∞
I (Ω +Ω ,ω)Iββ′(Ω Ω ,ω ,ω) I (Ω +Ω ,ω)Iββ′(Ω Ω ,ω ,ω) . (41)
1 1 2 2 1− 2 0 − 3 1 2 4 1− 2 0
h i
TheintegralsI ,Iββ′,I ,Iββ′ aredefinedandcomputedintheAppendix.Thefinalresultforthe4Keldyshmatrixelementsof
1 2 3 4
thenoisecorrelatoris:
Sβ−β(Ω1,Ω2)=2(4eπ∗)22aΓ220 +∞ +∞ Jn eω∗V1 Jm eω∗V1 Γ(π2ν)(νa )2ν×
n= m= (cid:18) AC(cid:19) (cid:18) AC(cid:19) F
X−∞ X−∞
(1 βsgn(Ω1+ω0+nωAC)) Ω1+ω0+nωAC 2ν−1δ(Ω1+Ω2+(n m)ωAC)
− | | −
h
+ (1 βsgn(Ω1 ω0 nωAC)) Ω1 ω0 nωAC 2ν−1δ(Ω1+Ω2 (n m)ωAC) , (42)
− − − | − − | − −
i
Sββ(Ω ,Ω )=2(e∗)2Γ20 +∞ +∞ J e∗V1 J e∗V1 π ( a )2ν e−βiπν
1 2 4π2a2 n ω m ω Γ(2ν) ν cos(πν)×
n= m= (cid:18) AC(cid:19) (cid:18) AC(cid:19) F
X−∞ X−∞
Ω1+ω0+nωAC 2ν−1δ(Ω1+Ω2+(n m)ωAC)+ Ω1 ω0 nωAC 2ν−1δ(Ω1+Ω2 (n m)ωAC) .
× | | − | − − | − −
h (43i)
8
WerecognizethatsincewearedealingwithsimpleLaughlinfractionsoftheFQHE,ν istheinverseofanoddintegerandall
KeldyshcomponentexhibitpowerlawsingularitieswhenthequantityΩ (ω +nω )vanishes.Asacheck,itispossibleto
2 0 AC
recoverfromthesecomponentsthepreviousresultforthesymmetrizedno±ise:30
1
S (Ω ,Ω ) = (S+ (Ω ,Ω )+S +(Ω ,Ω ))
sym 1 2 − 1 2 − 1 2
2
= 2(e∗)2Γ20 +∞ +∞ J e∗V1 J e∗V1 π ( a )2ν
4π2a2 n ω m ω Γ(2ν) ν
n= m= (cid:18) AC(cid:19) (cid:18) AC(cid:19) F
X−∞ X−∞
Ω1+ω0+nωAC 2ν−1δ(Ω1+Ω2+(n m)ωAC)
× | | −
h
+ Ω1 ω0 nωAC 2ν−1δ(Ω1+Ω2 (n m)ωAC) . (44)
| − − | − −
i
ItisalsousefultoknowthatthestandardpropertyofKeldyshGreen’sfunctions:
S++(Ω ,Ω )+S (Ω ,Ω )=S +(Ω ,Ω )+S+ (Ω ,Ω ) (45)
1 2 −− 1 2 − 1 2 − 1 2
appliesasitshouldforthedoubleFouriertransformexpressions.
C. Currentderivativecorrelators
TherelationbetweentheFouriercomponentsofthenoisecorrelatorcomputedintheprevioussectionandthecurrentderiva-
tivecorrelatorintroducedinSec. IIIreads:
Kββ′(Ω ,Ω )= Ω Ω Sββ′(Ω ,Ω ). (46)
1 2 1 2 1 2
−
Yet,weneedtorelatethenoisecorrelatorSββ′(Ω1,Ω2)tothecorrelatork(T,ω)andultimately,toitstimeaverage β2β1(ω).
K
Thisisachievedusingtherelation:
kββ′(T,ω)= dω1e−iω1TKββ′(ω1 +ω,ω1 ω). (47)
2π 2 2 −
Z
Sothefinalresultforthefouraveragednoisederivativecorrelatorsreads:
β β(ω)= 1 τ dTk+ (T,ω)= (e∗)2Γ20 +∞ J2 e∗V1 1 ( a )2νω2
K − τ − 4π2a2 n ω Γ(2ν) ν
Z0 n= (cid:18) AC(cid:19) F
X−∞
(1 βsgn(ω+ω0+nωAC)) ω+ω0+nωAC 2ν−1+(1 βsgn(ω ω0 nωAC)) ω ω0 nωAC 2ν−1
× − | | − − − | − − |
h (4i8)
ββ(ω)= 1 τdTk++(T,ω)= (e∗)2Γ20 +∞ J2 e∗V1 1 ( a )2ν e−βiπν ω2
K τ 4π2a2 n ω Γ(2ν) ν cos(πν)
Z0 n= (cid:18) AC(cid:19) F
X−∞
ω+ω0+nωAC 2ν−1+ ω ω0 nωAC 2ν−1 . (49)
× | | | − − |
h i
Tobecomplete,wecancomputealltheKeldyshelementintherotatedbasis. ThisisperformedinAppendixB. While the
advancedandretardedcontributiondonotbearinformationonthenonequilibriumnatureofthetransportprocessestakingplace
inthemesoscopicdevicesandthereforeinthedetector,theKeldyshcomponent:
dω
Q˜K =α2 G˜R(ω)˜R(ω)G˜K(ω)+G˜R(ω)˜K(ω)G˜A(ω)+G˜K(ω)˜A(ω)G˜A(ω) , (50)
2π K K K
Z h i
summarizessuchaninformationinacompactway. Werecallthatasanalternativetothemeasurementoftheequaltimecharge
correlator,Eqs. (27)and(50)arealsolikelytobemeasureddirectly(resolvedinfrequency)byanSETdevice.28
This completes the calculation of the current derivative correlators. In the following section we continue with the same
analysis as with the DC case17. That is we use the contour ordered elements of the charge correlator, in particular the +
−
componentevaluatedatequaltime:inSec.VweinserttheexpressionsforEqs. (48)-(49)anddiscusstheresults.
9
V. RESULTS
We now discuss the formulas obtained in Sec. IV. In all of the results below, we have checked that when the AC drive
frequencyissetto0,werecovertheDCresultsforthefinitefrequencynoiseatν = 1[1]andν = 1/3[31]. FortheQPCwe
focusonthevoltagedominatedregimewherethetemperatureistakentozerointhecurrentcorrelator,butitneverthelessenters
thedetectorresponse.
A. ExcessnoiseinthequantumHalleffect
We start with a discussion of the results for the non symmetrized excess PA noise. We show in Fig. 3 the curves for the
currentderivativecorrelator +(Ω)(seeeq. 48),whichisthequantitywhichenterstheexpressionofthemeasurednoise(the
−
K
chargecorrelator).Thisisdisplayedfortwodifferentvaluesofthefillingfactorν. WechooseforourmaininteresttheLaughlin
fraction ν = 1/3, which is in principle the easiest attainable Laughlin fraction of the FQHE in experiments, and ν = 1, the
integerquantumHalleffectcase, whichherealso correspondstothe noisecaracteristicsofa single channelnormaltunneling
junction.
HerewehavechosentheDCvoltagesothatthecentralfrequencyω νeV /~islargerthanthedrivefrequencyω ,and
0 0 AC
theamplitudeoftheACvoltage(ω =νeV /~)issuchthatω /ω =1≡
1 1 1 AC
For ν = 1/3 we find divergencesfor +(Ω) located at ω and at sidebands ω +nω . Sidebands with n = 1, 2
− 0 0 AC
arevisible. +(Ω)vanishesatzerofreqKuency. ForfrequencieslargerthanΩ = ω +2ω , thisnoisederivativeco±rrela±tor
− 0 AC
seems to beKnegligible. While the formulas for +(Ω) show a power law divergence, here one has to add a regularization
−
K
procedurebecausestricklyspeaking,thecalculationshavebeenperformedintheweakbackscatteringregime. Thismeansthat
thedifferentialconductanceassociatedwiththetunnelingcurrenthastobelowerthantheconductancequantum(otherwise,one
shouldexaminethecaseofthecrossovertothestrongbackscatteringregime). Thevalidityconditionofourresultshavebeen
previouslyderivedinEq.(24)ofRef. 30. Forourpurpose,itjustimpliesthatthefinitefrequencyPAnoisesaturatesatlocations
Ω=ω +nω .
0 AC
IntheintegerquantumHallcaseν = 1,nodivergencesarefoundfor +(Ω). Instead,singularitiesinthederivativeoccur
−
K
forω +nω ,andthecurrentderivativecorrelatorseemsagaintobenegligibleagainbeyondΩ=ω +2ω .
0 AC 0 AC
6
0.14
5 0.12
4 Ν=1(cid:144)3 0.10 Ν=1
L L
W W
0.08
+H 3 +H
- -
K K 0.06
2
0.04
1
0.02
0 0.00
0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5
W(cid:144)Ω W(cid:144)Ω
0 0
FIG.3: CurrentderivativecorrelatorforaQPCinthefractional(left)andinteger(right)quantumHalleffect(ω0 = 3ωAC,ω1 = ωAC and
K−+(Ω)isnormalizedbye∗IBω02)
However it is also interesting to plot +(Ω)/Ω2; in this way we have access to an “averaged” current correlator (noise)
−
becausethetermΩ2 in +(Ω)isinfacKtduetoderivativeoperatorsactingonthecurrentcorrelator. ThisisdepictedinFig. 4.
−
Forν = 1/3we againKfinddivergences(atthesame locationsasfor +(Ω)). Theonlynoticeabledifferencewiththe latter
−
K
curvesisthattheaveragednoisedoesnotvanishatzerofrequency. Ifoneignoresthesidebands,thecentralpeakremindsus
clearlyofthefinitefrequencynonsymmetrizednoisecomputedrecentlyforaQPCintheFQHE.31
Forν = 1thefinitefrequencynoiseagainexhibitsjumpsinitsderivativewithrespecttofrequency,butitsbehaviorislinear
betweentwosuccessivesingularities. Thus,forthisratiooffrequenciesω /ω > 1,theexcessnoisecharacteristicsresembles
0 1
the finite frequencynoise in the absence of an AC drive: the later is (essentially) linear for Ω < ω and vanishesbeyonthis.
0
YetthePAnoisedoesnotvanishatΩ = ω ,itshowsasingularityinitsderivativeatitslocation,togetherwithsingularitiesat
0
10
6 1.0
5 0.8
2W 4 2W
0.6
L(cid:144)W 3 Ν=1(cid:144)3 L(cid:144)W Ν=1
-+H -+H 0.4
K2 K
0.2
1
0 0.0
0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5
W(cid:144)Ω W(cid:144)Ω
0 0
FIG.4:Averagedcurrentcorrelator(sameparametersasinFig.3exceptthefactthatK−+(Ω)/Ω2isnormalizedbye∗IB)
Ω=ω +nω (n= 1isvisible).TonormalizethecurvesinFig. (3),Fig. (4)andinthefollowingsection,weusetheback
0 AC
scatteringcurrenttozer±oorderintheamplitudeofω ofthemodulationwichcorrespondstothepurestationnaryregime30:
1
e Γ2 a
IB(0) = 2π2a∗2Γ0(2ν)(ν )2νsgn(ω0)|ω0|2ν−1. (51)
F
B. MeasuredPAnoise
Inthissection,wedisplaycurvesforthechargecorrelatoratequaltimes. Weconsiderexcessquantities.Byexcess,wemean
thatthechargecorrelatoratzerovoltagehasbeensubtractedfromthechargecorrelatoratV ,V =0.
0 1
6
InFigs. 5,6,7and8weplotthesequantities:
dω
Qˆ−+(0)=α2 2πG−β2(ω)σzβ2β2Kβ2β1(ω)σzβ1β1Gβ1+(ω). (52)
βX1β2Z
with weak andstrongdissipation, low andhigh(detector)temperaturefortwo differentvaluesof the ratio ω /ω , which
1 AC
correspondstotheargumentoftheBesselfunctionsintheexpressionofthechargecorrelator. InthefollowingcurvesQ + is
−
alwaysnormalizedby 2/(α2e I ω2)anddissipationandtemperatureareinunitofω . ThefrequencyΩ=ω correspondsto
L ∗ B 0 0 0
thepositionsofthecentralpeak.
Westartwithω > ω . Inordertoresolvethesepeaks,itisnecessarythatthewidthoftheresonancelevelissmallerthan
0 AC
thespacingω betweenpeaks. Weobservethatbyvaryingω /ω ,therelativeamplitudesofthepeakscanbemodulated.
AC 1 AC
InFig. 6,thecurvescorrespondtoaratioω /ω = 2: wecanclearlyidentifythecentralpeakatΩ = ω butitissmaller
1 AC 0
thaninthecaseω /ω =1(inFig. 5). Inthissituationweidentifyveryclearlythefirstandthesecondsatellitepeaks,while
1 AC
thethirdone(n = 3)isvisiblebutwithalesserintensity. Therelativeamplitudeofthecentralpeakanditssatelliteistiedto
±
theoscillatorybehavioroftheBesselfunction.
Whenω /ω = 1,the0thorderBesselfunction,whichcorrespondstothecentralpeakhasalargeamplitude( 0.6). The
1 AC
≈
1storderBesselfunctionwhichcorrespondstothefirstsatellitepeak,hasasmalleramplitude( 0.2).ThethirdBesselfunction
≈
whichcorrespondsto the second satellite peak is almostzero. On the otherhandforω /ω = 2, the 0thand the 3rdorder
1 AC
Besselfunctionaresmallcomparedtoits1stand2ndordercounterparts,thusthecentralpeakissmallerthanthesatellites.
Next,wechooseω < ω inFig. 7and8. Thefinitefrequencyspectrumofchargefluctuationsdoesnotseemtodisplay
0 AC
anylongeracentralpeakwithequallyspacedsatellites.
InFig. 7,thecurvescorrespondtoaratioω /ω = 1andinFig. 8thecurvescorrespondtoaratioω /ω = 2. InFig.
1 AC 1 AC
7,thecurveshaveacentralpeakatfrequencyω ,asecondaryoneatω +ω andathirdoneatω +2ω . Howeverthere
0 0 AC 0 AC
appearpeaksatfrequencies ω +ω and ω +2ω : thiscorrespondsto thesatellites peaksofthe negativefrequency
0 AC 0 AC
− −
ω . We can explain this phenomena as the overlapping of two combs, centered at ω . In Fig. 8, the curves exhibit the
0 0
− ±
samephenomenabutthepeakshavedifferentrelativeamplitude,whichcanagainbeexplainedfromtheargumentoftheBessel
functions.Inthesedifferentsplots,wecanseeontheonehandtheeffectofdissipationwichreducesthenoiseandsmoothesthe