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[doc] Two Feasible Schemes for Preparing Cluster States with Ion-Trap Setup

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[doc] Two Feasible Schemes for Preparing Cluster States with Ion-Trap Setup

    Two Feasible Schemes for Preparing

    Cluster States with Ion-Trap Setup

    Commun.Theor.Phys.(Beijing,China)49(2008)PP.12171220 ;?ChinesePhysicalSocietyVo1.49,No.5,M15,2008

    ;TwoFeasibleSchemesforPreparingClusterStateswithIon-TrapSetup ;WANGXin-Wen.CAOShuai.andXIALiXin,.

    ;DepartmentofPhysics,HunanUniversityofScienceandEngineering,Yongzhou425100,China

    ;2CollegeofScience,

    ;SouthChinaAgriculturalUniversity,Guangzhou510642,China ;.DepartmentofPhysics,TsinghuaUniversity,Beijing100084,China ;(ReceivedMay8,2007;RevisedJuly5,2007)

    ;AbstractWepresenttwoschemesforpreparingclusterstateswithatomicqubitsaniontrapsystem.Inthefirst

    ;schemeanauxiliaryatomiclevelisneeded.While

    thesecondschemeanadditionalclassicaldrivenfieldisused,and ;themultiionclusterstatescanbegeneratedbyonestep.Boththeschemesareinsensitivetothermalmotionofthe

    ;ions.allthefacilitiesusedthemarewellwithinstateoftheart.

    ;PACSnumbers:03.67.Mn,03.65.Ud,42.50Vk,03.67.Lx

;Keywords:ion-trap,clusterstates,one-stepimplementation,controlledphas

    egates

    ;Thephenomenonofnonlocalcorrelations,orentangle- ;ment,betweenquantum-mechanicalparticlesisoneofthe ;moststrikingfeaturesofquantummechanics.Entangled ;statesnotonlyprovidepossibilitiestotestquantumme- ;chanicsagainstlocalhiddentheory,butalsohaveprac

    ;ticalapplicationsinquantuminformationprocessing.J

    ;Bipartiteentanglementiswellunderstood.whilemulti

    ;partiteentanglementisstillunderextensiveexploration. ;Recently,BriegelandRanssendorfintroducedanewclass ;ofmulti-qubitentangledstates,i…etheso-calledcluster

    ;states.Theclusterstateshavemanyinterestingfeatures. ;Forexample,theyhavehighpersistenceofentanglement ;andcanberegardedasanentanglementresourceforthe ;GHZstates,butemoreimmunetodecoherencethan ;GHZstates.Thusclusterstatesbecomeanimportant ;resourceinmanybranchesofphysics,especiallyinquan

    ;turninformation.Moreimportantly,ithasbeenshown ;thattheclusterstatescanbeusedtoimplementquan(

    ;turncommunication._4_aswellasconstituteauniversal ;resourceforso-calledone(wayquantumcomputationpro-

    ;ceedingonlybysingleparticlemeasurementsandlocal ;operations.[]

    ;Thepreparationofclusterstateshasattractedmuch ;attentionbecauseofitsuniquefeaturesandextensiveap

    ;plications.Andsomeschemesforpreparingclusterstates ;havebeenproposedinlinearopticssystems.0JcavityQED

    ;systems.[7latomicensembles.

    ;[8landquanturndot.

    ;[9]There

    ;arealsosomeexperimentalreportsontheobservationof ;clusterstates[10Janddemonstrationofthefeasibilityof ;theone-wayquantumcomputation[11Jwithlinearoptical ;elements.

    ;H=rataoj+{Qe

    ;Ontheotherhand,recentadvanceiniontrapshas ;openedtheprospectsforentanglementengineeringand ;quantuminformationprocessing.Lately,r~searchgroups ;haverealizedasixatomGHZstate.JandeightqubitW

    ;states[13Jinsuchasystem.However.multiqubitclus

    ;terstateshavenotbeendemonstratedinsuchasystem. ;Recently,Yang[14]andZheng[15]respectivelyproposeda ;theoreticalschemeforgenerationofclusterstateswith

;trappedions.InYang‟sscheme,2(N1)ioncluster

    ;statescanbeobtainedbyusingsidebandresonantexcita- ;tion,whileinZheng‟sscheme,

    ;wereusedforgeneratingfour

    ;twogeometricphasegates

    ;ionclusterstates.Inthis

    ;paper,wepresenttwoschemesforpreparingNionclus

    ;terstates.Inthefirstschemeanauxiliaryatomiclevelis ;needed.whileinthesecondschemeanadditionalclassical ;drivenfieldisused,andthemulti-ionclusterstatec3371be ;generatedbyonestep,whichisveryimportantinviewof ;thedecoherence.Bothofthemareinsensitivetoheating ;ofvibrationalmodeoftheions.whichisofimportance ;fromtheexperimentalpointofview.Theschemescanbe ;realizedwithpresentlyavailableion-traptechniques. ;,veconsiderNidenticalionswhichhavetheground ;statell9)andtheexcitedstatele)confinedinalineartrap. ;Anddrivingthetwoneighboringions.i.e.,theKthand

    ;(+1)thions,bytwoclassicalhomogeneouslaserswith ;frequencies0+//‟+and0d0//‟,where0d0isthefre-

    ;quencyofthetransitionfe)Hfl9)and//‟isthefrequency

    ;ofonecollectivevibrationalmode.Suppose,we

;canneglectothervibrationalmodes.Inthesecases,the

    ;Hamiltonianofthesystemisgivenby(forsimplicity,we

    ;let:11[16,17]

    ;K+1

    ;?eirh(at+a)e-i(wo+U+a)t+eirj2(at+a)e-i(wo-U-a)t]+H.C.)j=K ;(1)

    ;TheprojectsupportedbytheScientificResearchFundoftheEducationDepartmentofHunanProvinceunderGrantNo.06C354and

    ;theNatura1ScienceFoundationofHunanProvinceunderGrantNo.06JJ5015

    ;E.mail:xwwang0826@yahoo.com.ca

    ;?

    ;12

    ;

    ;1218WANGXinWen,CAOShuai,andXIALiXinVbl_49

    ;whereat(a)denotesthecreation(annihilation)operatorforthecollectivevibrationalmode,,,andare

    ;respectivelytheinversion,raising,andloweringoperatorsfortheions,andrh=,//(‰)/(2?m)(i=1,2)isthe

    ;Lamb

    -Dickeparameter,withkzbeingtheeffectivewavevectorofthei-thlaserfieldalongthedirectionofthestring

    ;ofion8andmbeingthemassofeachion.Weherehaveassumedthatbothfieldshavethesamephaseandintensity,

    ;andthattheion-laserinteractionstrengthcharacterizedbytheRabifrequency(2i8thesameforallionsfInapractical

    ;realization.onemightuseRamantransitionsbetweenlowlyingstatesofthe

    ionsduetotheirlongcoherencetime.By

    ;appropriateredefinitionofthesymbols.theformalismalsodescribesthisimplementation1.l8_

    ;Inpracticalrealizationtheeffectivewavevectorkicanbenearlyidenticalforthetwofieldsbymodulatingthe

    ;illuminativedirectionofthetwolasers,【驯andweassumel:2=

    .WhenQ,wediscardtherapidlyoscillating

    ;termsandobtaintheHamiltonianintheinteractionpicture, ;(at)Z+laZe-iStq_(at)ZaTMeTM

    ;UndertheLambDickeregime(i.e.,叼再《1,with

    ;Eq.(2)canbeapproximatedbytheexpansiontothefirst

    ;beingthevibrationalquantumnumber),theHamiltonianof ;orderin(assuming=/2),

    ;K+1

    ;HI=2?(+)(o+eist+oe)

    ;j=K

    ;Inthecaseof》【2

    ,theenergyconservingtransitionsarelCKeK+In)?_?IgKgK+ln)andlgKCK+tn)IeKgK+In).

    ;ThetransitionlCKeK+In)?_?IgKgK+ln)ismediatedbyIgKeK+tn

    1)andlCKgK+ln1).ThecontributionsoflgKeK+In

    ;1)areequaltothoseoflCKgK+ln

    1).ThecorrespondingRabifrequencyisgivenby2(2)/a.Sincethetransition

    ;pathsinterferedestructively,thereisnotransferofpopulationtostateswithdifferentvibrationalexcitationandthus

    ;theRabifrequencyisindependentofvibrationalquantumnumbers.JTheRabifrequencyforlgKCK+1n)?_?IeKgK+In),

    ;mediatedbyICKeK+ln1),andlgKgK+tn

    1),isalso2ta~)/a.ThentheeffectiveHamiltoniancanbedescribed ;bv[16.19]

    ;,K+1

    ;=

    ;?(IeJ)(eJl+Igj)(gj1)+(++H.c.)

    ;j=K

    ;where:22rl/a.Wenotethattheionicinternalstateevolutionoperator ;UKK+l()eie(5)

    ;isindependentofthephononnumberofthevibrationalmode,allowingittobeinathermalstate?

    ;Inordertopreparetheclusterstatesweintroduceanotherionicstatelf).Thetra

    nsitionfrequencybetweenthe ;statesIe)(I_9))andIf)ishighlydetunedandthusthestateIf)isnotaffectedduri

    ngtheinteractiondescribedby ;Eq.(4).Thenitiseasytoprovethat ;UK,

    ;K+l(t)l,_9+)=eiAt/I,_9+),(6) ;UK,

    ;K+l(t)leg+)=e-JAr[cos(At)leg+)isin(At)lge+t).(7)

    thionisinitiallyintheentangledstatewithothersubsystems(1/~/2)(;IftheK

    I)I)eK)l)),wherel)and

    ;I)arearbitrarynormalizedwavefunctionsofthesubsystems,andthe(K+1)thionisinitiallyinthestategK ;Afteraninteractiontimet=/(2),theionicstateevolvesinto

    ;I)…+=(ei~r/4If~)lg…)+)le…))),

    ;V

    ;andthestate

    ;1

    ;l),K+=(1)l)+lg)l))(1+)盯一lCK+I))

    ;canbeobtainedbytwoonequbitparticlerotations ;RK=(e-i=/l)(l_lg)(g1),

    ;where=I,)(,Ilg)(g

    ;R+l:1(1

;+)(+

    ;Vz

    ;e+)(f+l+lf+)(e+

    ;+Ie+)(e+I)(I,+)(_9+l+Ig+)(,+) ;(10)

    ;

    ;No.5TwoFeasib1eScheiilesforPreparingClusterStateswithIonnSetupl2l9

    ;Now,veshowhowan?一

    ionclusterstatecanbeprepared.AssumetheNionsareinitiallypreparedinthesta

    te

    ;(1/x/~)(1f1)lex))l2…).Thenwecan.btainan?一i.nclusterstate1)2,…,?byperfOrminghe.per. ;mentionedabove,oneverypairofneighboringionsinturn,i.e.,

    ;1..c2…,

    ;?=YfN-xjYfN--

    ;2]???1l)Ig3‟-)

    ;:

    ;1N-1

    ;(I,j)+(I,?).Ie?))(12)

    ;withtheconvention:1andYfK1=R+1RKUK, ;Kx(t=7r/2)(K=1,…,N1)?

    ;wenotethatthelusterstatesJCanalsbegeneratedwithoutusingadditionalionicstateI/).Consideringthat

    thand(;anotherclassicalresonant1asersimultaneouslyinteractswiththeK

    +1)thions,thesimplifiedinteraction

    ;pictureHamilt.nianunderther.tating-waveaPProximati.nis[2O

    ;K+1

    ;:g?(+)

    ;j=K

    ;where0isRabifrequencycharacterizingtheinteractionstrengthbetweentheionsandtheresonantlaser

    ;iseasytoprovethat

    ,]=0.Sotheoperationcanbeperformedafter,beforeorsimultaneously ;applicationof,andthetotalinternalstateevolutionoperatorofthetwoionscanbedescribedby

    ;‰(刊唧[K+I)(e1)_(“+…)]?

    ;field.It

    ;withthe

    ;IfthetwoioI1sareinthestatel9K9K+1),l9KeK+1),leK9K

    1)orLeKeK+1),undertheapplicationoftheunitary

    ;operation(14),theionicstateswillevolveintothefollowingstates, ;gKg+.)=e-JAr{cos(At)[cos(gt)Ig>isin()le)][cos(9)l

    )x+-isin(gt)le)K+.]

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