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Interfacing quantum optical and solid state qubits

We present a generic model of coupling quantum optical and solid state qubits, and the corresponding transfer protocols. The example discussed is a trapped ion coupled to a charge qubit (e.g. Cooper pair box). To enhance the coupling, and achieve compatibi

Interfacingquantumopticalandsolidstatequbits

12

L.Tian1,P.Rabl1,R.Blatt2andP.Zoller1

InstituteforTheoreticalPhysics,UniversityofInnsbruck,A–6020Innsbruck,AustriaInstituteforExperimentalPhysics,UniversityofInnsbruck,A–6020Innsbruck,Austria

Wepresentagenericmodelofcouplingquantumopticalandsolidstatequbits,andthecorre-spondingtransferprotocols.Theexamplediscussedisatrappedioncoupledtoachargequbit(e.g.Cooperpairbox).Toenhancethecoupling,andachievecompatibilitybetweenthedi erentexperimentalsetupsweintroduceasuperconductingcavityastheconnectingelement.

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00Signi cantprogresshasbeenmadeduringthelast2fewyearsinimplementingquantumcomputingpropos- talswithvariousphysicalProminentexam-cOplesarequantumopticalsystems,inparticulartrappedions[2]andatomsinopticallattices[3],andmorerecently 9solidstatesystems,suchasJosephsonjunctions[4,5]and quantumdots[6].Amongtherecentexperimentalhigh-1vlightsaredemonstrationof,or rststepstowardscoher-7entcontrolandreadoutofqubits,engineeringofentan-5glement,andsimplequantumalgorithms[7,8,9,0Basedonthisprogress,thequestionarisestowhatex-0tenthybridcanbedevelopedwiththegoal13ofcombiningadvantagesofvariousapproaches,while0stillbeingexperimentallycompatible.Attheheartof/thisproblemarequestionsofinterfacingquantum-opticalhpqubitsandsolidstatesystems:thiscanbeeitherina-formwherequantum-opticalqubitsareconnectedviaatnsolidstatedatabusasaroutetoscalablequantumcom-aputing,orbyreversiblytransferringaquantumopticalutosolidstatequbitsandcircuits.

:qvBelowwewilldiscussagenericphysicalmodelandicorrespondingprotocolforinterfacingaquantum-opticalXandsolidstatechargequbit.Onthequantumopticsside,rqubitsaretypicallyrepresentedbylonglivedinternala(nuclear)spinstatesofsingleatomsorThesesingleatomscanbetrappedandlasercooled.External elds,provideamechanismtomanipulatethequbit,aswellastoentanglethequbitwiththemotionalstateofthetrappedparticle.Inthecaseoftrappedionsthisallowsonetoconvertspin(thequbit)tochargesuperpo-sitions,eitherdynamically,e.g.bykickingwithalaser,orquasi-staticallybyapplyingspin-dependent(opticalormagnetic)potentials.Thisspintochargeconversionprovidesanaturalcapacitivecouplingtoasolidstatechargequbit,representede.g.byaJosephsonjunction[4]orachargeddoublequantumInsteadofdirectcouplingofthecharges,onecanintroduceauxiliaryele-ments,suchascavities.Thisservesthepurposeofallow-ingincreasedspatialseparationandmutualshieldingofthesystems,withthegoalofeasingexperimentalrequire-mentsforcoexistenceofthehybridqubits(e.g.trappingandlasermanipulationofatomsorions),whileenhanc-ingthecouplingstrengthincomparisonwithfreespace.Featuresofthishybridsystemarethesigni cantlydi er-enttimescalesoftheevolutionandcouplingofthetwo

qubits,and(incomparisonwithquantumopticalqubits)shortdecoherencetimeofthesolidstatesystems.Weex-ploitthisbydevelopingaprotocolofa“fastswap”withkickingbyshortpulsesonatimescalecomparabletothecharge-chargecouplingwhichismuchshorterthanthetrapperiod.Thishastheadditionalbene tofbe-ingahotgate,i.e.notrequiringcoolingtothemotionalgroundstate,andnotmakingaLamb-Dickeassumptionofstrongcon nement.

Modelforcoupledqubits.AHamiltonianforourcom-binedsystemhastheformHt=Hs+Hq+HintwithHamiltonianforthequantumopticalqubit

Hs=

p 2

x2mω2νx 2 +δ02

σs+eiδklx

+h.c. .(1)

thesolidstatechargequbit,

HEq=

z

2

σq

x

(2)

andinteractionterm

Hint= κ(t) xσq

z.

(3)

The rstterminHsdescribesthe1D-motionofachargedparticle(ion)intheharmonictrappingpoten-tialwithx thecoordinate,p xthemomentumandωνthetrappingfrequency.Apseudo-spinnotationwithPauli

operatorsσs

idescribestheatomicqubit.Physically,thequbitisrepresentedbytwoatomicgroundstatelevelswhicharecoupledbyalaserinducedRamantransitionwithRabifrequencyωR(t)anddetuningδ0.Transitionsbetweenthestatesareassociatedwithamomentumkickδklduetophotonabsorptionandemission,whichcouplesthequbittothemotionattheRabifrequencyωR(t)[15].TheHamiltonianforthesolidstatechargethegenericformwithσq

qubithas

Eparameters.ithePaulioperators,andzandExbeingtunableAHamiltonianofthisformisobtained,forexample,forasuperconductingchargequbit,i.e.asuperconductingislandconnectingwithalowcapacitanceandhighresistancetunneljunc-tion(seeFig.1(a)).Withaphase anditsconjugaten ,

theHamiltonianisHq=Ec( n+CgVg/2e)2

EJcos whereEJistheJosephsonenergyandEcisthecapaci-tiveenergywithEc EJ.ThegatevoltageVgcontrolsthequbitthroughthegatecapacitorCg.Thequbitforms

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