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9RFRV::7W[\]8R8R9WLPH μV 3 W Q 3Q FRKHUHQWVWDWH)LJ Leibfriedetal.,Fig.40123n0123m00.10.20.30.4|ρ|0123n1QuantumStateManipulationOfTrappedAtomicIons†D.J.Wineland,C.Monroe,D.M.Meekhof,B.E.King,D.Leibfried,W.M.Itano,J.C.Bergquist,D.Berkeland,J.J.Bollinger,andJ.MillerIonStorageGroup,TimeandFrequencyDivisionNIST,Boulder,CO,80303,USAAbstract:Asinglelaser-cooledandtrappedBeionisusedtoinvestigatemethodsofcoherent9+quantum-statesynthesisandquantumlogic.Wecreateandcharacterizenonclassicalstatesofmotionincluding“Schrödinger-cat”states.Afundamentalquantumlogicgateisrealizedwhichusestwostatesofthequantizedionmotionandtwoioninternalstatesasqubits.Weexploresomeoftheapplicationsfor,andproblemsinrealizing,quantumcomputationbasedonmultipletrappedions.VwVo2cos(6Tt)1 x2 y2R2,2(1)1.INTRODUCTIONCurrently,amajorthemeinatomic,molecular,andopticalphysicsiscoherentcontrolofquantumstates.Thisthemeismanifestedinanumberoftopicssuchasatominterferometry,Bose-Einsteincondensationandtheatomlaser,cavityQED,quantumcomputation,quantum-stateengineering,wavepacketdynamics,andcoherentcontrolofchemicalreactions.Here,wereportrelatedtrapped-ionresearchatNISTwhichisdevotedprimarilytoachievingquantumcomputationbasedonaschemeproposedbyCiracandZoller[1].Asprogresstowardsthisgoal,wereportexperimentson(1)thedynamicsofatwo-levelatomicsystemcoupledtoharmonicatomicmotion,(2)thecreationandcharacterizationofnonclassicalstatesofmotionsuchas“Schrödinger-cat”superpositionstates,and(3)simplequantumlogicgateswhicharetheprecursorstoCiracandZoller'sion-trapquantumcomputer.2.TRAPPEDATOMICIONSBecauseoftheiroverallelectriccharge,atomicormolecularionscanbeconfinedbyparticulararrangementsofelectromagneticfieldsforrelativelylongperiodsoftimes(hoursorlonger)withrelativelysmallperturbationstotheirinternalenergylevelstructure.Forstudiesofionsatlowkineticenergy(1eV),twotypesoftraparecommonlyused-thePenningandPaultraps.ThePenningtrapusesacombinationofstaticelectricandmagneticfieldsforconfinement,andthePaul,orrftrapconfinesionsprimarilythroughponderomotivefieldsgeneratedbyinhomogeneousoscillatingelectricfields.Theoperationofthesetrapsisdiscussedinvariousreviews(forexample,seeRefs.2-4)andinarecentbook[5].Forbrevity,wediscussonetrapconfigurationwhichisusefulforthetopicsdiscussedinthispaper.InFig.1weshowschematicallya“linear”Paultrap.ThistrapisbasedontheonedescribedinRef.[6],whichisderivedfromtheoriginaldesignofPaul[7].Itisbasicallyaquadrupolemassfilterwhichispluggedattheendswithstaticelectricpotentials.AnoscillatingpotentialVcos(6t)isappliedbetweendiagonallyoppositerods,whicharefixedinaoTquadrupolarconfiguration,asindicatedinFig.1.Weassumethattherodsegmentsalongthez(horizontal)directionarecoupledtogetherwithcapacitors(notshown),sotherfpotentialisconstantasafunctionofz.NeartheaxisofthetrapthiscreatesanoscillatingpotentialoftheformwhereRisequaltothedistancefromtheaxistothesurfaceoftheelectrode[8].Thisgivesrisex(t)wX(1 (qx/2)cos6t)cos(7xt 1x),y(t)wY(1 (qy/2)cos6t)cos(7yt 1y),q0p 12m72i(x2 y2),3(2)(3)toa(harmonic)ponderomotivepotentialinthex-ydirectionwhich,forasingleion(orthecenter-of-mass(COM)motionofagroupofions),yieldsthemotionwhereq=-q 2qV/(mR6)=2 27/6,7=7=7=qV/( 2mR6),andmandqarexyoTxTxyoT222theionmassandcharge.Equations(2)arevalidforthetypicalconditionsthat q , q 1.Inxythisapproximation,ifweneglectthemicromotion(the(q/2)cos6ttermsinEq.(2)),theionibehavesasifitwereconfinedinapseudopotential0givenbypinwhichtheionoscillateswithsecularfrequency7.Toprovideconfinementalongthezidirection,staticpotentialsareappliedtotheendsegmentsoftherodsrelativetothecentralsegments.Nearthecenterofthetrap,thisprovidesastaticharmonicwellalongz.Thisnecessarilyweakensthewellintheradialdirection[6],butwewillassumethatthebindingintheradialdirectionislargecomparedtothatalongz,sothiseffectissmall.Figure1alsoshowsanimageofa“string”ofHgionswhichareconfinednearthezaxisofthetrapdescribedinRef.199+9.Thisstringofionsmightbeusedasaregisterinaquantumcomputer[1].Whenthiskindoftrapisinstalledinahigh-vacuumapparatus(pressure10Pa),ions-8canbeconfinedfordayswithminimalperturbationstotheirinternalstructure.Collisionswithbackgroundgasareinfrequent(themeantimebetweencollisionsistypicallymorethanafewminutes).EventhoughtheionsinteractstronglythroughtheirmutualCoulombinteraction,thefactthattheionsarelocalizednecessarilymeansthat E =0atthepositionsoftheions;therefore electricfieldperturbationsoccuronlyinsecondorder.Theseperturbationsaretypicallyverysmall[10].Magneticfieldperturbationsareimportant;however,ifweoperateatfieldswheretheenergyseparationbetweenlevelsisatanextremumwithrespecttofield,thecoherencetimeforsuperpositionstatesofthetwolevelscanbeverylong.Forexample,inaBe(Penningtrap)9+experimentoperatinginafieldof0.82T,thecoherencetimeofasuperpositionbetweenhyperfinestatesexceeded10minutes[11].Asdescribedbelow,wewillbeinterestedinexcitingcoherentlythequantizedmodesoftheionmotioninthetrap.Notsurprisingly,thecouplingtotheenvironmentisrelativelystrongbecauseoftheions'charge.On