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Non-equilibrium theory of charge qubit decoherence in the quantum point contact measurement

A non-equilibrium theory describing the charge qubit dynamics measured by a quantum point contact is developed based on Schwinger-Keldysh's approach. Using the real-time diagram technique, we derive the master equation to all orders in perturbation expansi

Non-equilibriumtheoryofchargequbitdecoherenceinthequantumpointcontact

measurement

a

Ming-TsungLeeaandWei-MinZhanga,b

NationalCenterforTheoreticalScience,Tainan,Taiwan70101,R.O.C.b

DepartmentofPhysicsandCenterforQuantumInformationScience,NationalChengKungUniversity,Tainan,Taiwan70101,R.O.C.

(Dated:April19,2008)

arXiv:0708.2581v2 [cond-mat.mes-hall] 19 Apr 2008

Anon-equilibriumtheorydescribingthechargequbitdynamicsmeasuredbyaquantumpointcontactisdevelopedbasedonSchwinger-Keldysh’http://doc.xuehai.netingthereal-timediagramtech-nique,wederivethemasterequationtoallordersinperturbationexpansions.Thenon-Markovianprocessesinthequbitdynamicsisnaturallytakenintoaccount.Thequbitdecoherence,inpartic-ular,thein uenceofthetunneling-electron uctuationinthequantumpointcontactwithalongertimecorrelation,isstudiedintheframework.WeconsidertheLorentzian-typespectraldensitytocharacterizethechannelmixtureoftheelectrontunnelingprocessesinducedbythemeasurementanddeterminethecorrelationtimescaleofthetunneling-electron uctuation.Theresultshowsthatasthequantumpointcontactiscastedwithanarrowerpro leofthespectraldensity,tunnelingelectronscanpropagatewithalongertimecorrelationandleadtothenon-Markovianprocessesofthequbitdynamics.Thequbitelectroninthechargequbitwillbedrivencoherently.Thequantumpointcontactmeasurementwiththeminimumdeviationoftheelectrontunnelingprocessespreventsthequbitstatefromthedecoherence.

PACSnumbers:03.65.Yz,73.23.-b,05.70.Ln,03.67.Lx

I.INTRODUCTION

Withtherapidprogressinnano-technology,thein-vestigationofquantumprocessesinsystemscouplingtoamesoscopicalmeasurementdevicebecomesaveryac-tiveresearch eldinrecentyears.[1,2,3,4]Notonlyitspracticalapplicationtoquantumcommunicationandquantumcomputation,[5]butalsothetheoreticalin-terestsinthemeasurement-inducedquantumdecoher-ence[3,6]haveattractedmuchattention.Inparticu-lar,intheinvestigationofthesolid-statequantumcom-puterwithchargequbits,thequantumpointcontact(QPC)hasbeenservedasanultrasensitiveelectrome-ter7,8,9,10].Intheliterature,themeasurementofchargequbitthroughtheQPChasbeentreatedbasedontheso-calledMarkovianapproximation,[11,12,14]inwhichthetimescaleofthequbitdynamicsisassumedmuchlargerthanthatofthetunneling-electroncorrela-tionintheQPC.ThistheoreticaltreatmentbasedontheMarkovianapproximationdescribesonlythequbitdy-namicsinthetime-asymptoticquasi-equilibriumstate.Mesoscopically,thequbitdecoherenceoccursinthetimescaleofthesameorderofthetunneling-electroncorrela-tiontimeintheQPC,wherethenon-Markoviandynam-icsofthequbitissigni cant.Thus,anon-equilibriumdescriptiontothequbitdynamicsismoredesired.Re-cently,theconsiderationsofthesolid-statesystempro-cessinginthenon-Markovianregimehaveindeedreceivedmoreattention.Forinstance,alocalelectronspincou-plingwithanuclearspinbaththroughtheFermicontacthyper neinteractioninwhichtheelectronspindynam-icsisinthetimescaleshorterthanthenucleardipole-dipolecorrelationtime[15],andelectrontransportsofin-teractingelectronsystemsmostlikelyinvolvethenon-

Markoviandynamics[16,17].Inourpreviouswork[14],thenon-equilibriume ectoftheQPConthequbitdy-namicswasstudiedbytreatingthefermienergy uctua-tionoftheQPCreservoirsperturbatively.Inthispaper,afullynon-equilibriumtheorydescribingthequbitde-coherencebytheQPCmeasurementisdevelopedusingSchwinger-Keldysh’sapproach[18].

Historically,theSchwinger-Keldysh’sapproachiswelldevelopedtosystematicallytreatthenon-equilibriumdynamicsofamany-body20,21]Forelec-trontransportsinnano-devices,thisapproachhasbeenusedtostudythetransportcurrent uctuationandthefullcountingstatisticsinthesingle-electron22],thecurrent uctuationinKondosystem[23,24,25],andthenoisespectrumintheQPCmeasurementofthechargeTotreatthenon-equilibriume ectoftheelectricalreservoir,literaturelythereal-timedi-agrammatictechniquewasconstructedtodiagrammati-callycalculatecorrelationfunctionsoftheelectricalreser-voirorderbyorder.[21,24]Fortheinvestigationofthenon-Markoviandynamicsinoursystem,analternativereal-timediagrammatictechniqueisdeveloped.Themasterequationforthechargequbitdynamicsisderivedandexpressedintermsofallordersirreduciblediagramstoallorders.Thenon-Markovianprocessesinthequbitdynamicscanbefullytakenintoaccount.Thee ectofthetimevariatedreservoir uctuationonthequbitdy-namicscanthenbeexplicitlystudiedinthisformulism.Inaddition,theassumptionoftheconstanthoppingamplitudeoftunnelingelectronsacrosstheQPCbarriertogetherwithaconstantdensityofstateoftheQPCreservoirsisusuallyutilizedtospecifytheQPCstruc-turerelatedtothetworeservoirsbandstructurecou-plingtothetwometalgates.Iteventuallyleadstothe

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