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Preparation of Macroscopic Quantum-Interference States for a Collection of Trapped Ions Via a Single Geometric Operation

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Preparation of Macroscopic Quantum-Interference States for a Collection of Trapped Ions Via a Single Geometric Operationof,a,OF,for,FOR

    Preparation of Macroscopic

    Quantum-Interference States for a

    Collection of Trapped Ions Via a Single

    Geometric Operation

    Commun.Theor.Phys.(Beijing,China)53(2010)PP.920922

    ?ChinesePhysicalSocietyandIOPPublishingLtdVo1.53,No.5,May15,2010 PreparationofMacroscopicQuantum-InterferenceStatesforaCollectionofTrapped IonsViaaSingleGeometricOperation

    LINLiHua(林丽华)

    DepartmentofPhysicsandStateKeyLaboratoryBreedingBaseofPhotocatalysis,FuzhouUniversity,Fuzhou350002

    China

    (ReceivedJuly15,2009)

    Abstract?,edescribeaschemeforthegenerationofmacroscopicquantum

    interferencestatesforacollectionof

    trappedionsbyasinglegeometricphaseoperation.Intheschemethevibrationalmodeisdisplacedalongacircle?j

    theradiusproportionalt0thenumberofionsinacertaingroundelectronicstate.Forayeninteractbntime,the

    vibrationalmodereturnstotheoriginalstate,andtheionicsystemacquiresageometricphaseproportionaltothearea

    ofthecircle.evolvingfromacoherentstate9asuperpositionoftwocoherentstates.Theionsundergonoelectronic

    transitionsduringtheoperation.Takingadvantageoftheinherentfault

    tolerantfeatureofthegeometricoperation,our

    schemeisrobustagainstdecoherence.

PACSnumbers:42.50.Dv.42.50.Vk

    Keywords:macroscopicquantuminterferencestate,trappedion,geometricphase Overthepastfewyears,muchefforthasbeendirected totheso-calledSchr5dingercatstates.lji.e..superposi

    tionsofmacroscopicallydistinguishablequantumstates. Inquantumopticsthesestatesareusuallydefinedassu. perpositionsoftwocoherentstates.Thoughformedby quantumstatesclosesttotheclassicalones.suchsuper

    positionstatesmayexhibitvariousnonclassicalproper. ties.suchassqueezingandsubPoissonianstatistics.

    Thesesuperpositionstateshavebeenrealizedforacav

    ityfield[3]andthemotionofatrappedion.

    [4]Morere

    cently,therehasbeeninterestinanotherkindofsuper

    positionstates,associatedwithacollectionoftwo.1evel atoms.Ithasbeenshownthatsuperpositionsoftwo atomiccoherentstatesmayalsoexhibitnonclassicalprop

    erties,suchasoscillationsintheatomicpopulationand squeezedfluctuations.15]Recently,schemeshavebeenpro

    posedforthegenerationofsuchsuperpositionstatesin cavityQED.[5-8]

    Theabovementionedschemesarebasedondynamic evolution.Ontheotherhand,muchattentionhasbeen paidtogeometricoperation,apromisingapproachforthe implementationofbuilt..infault..tolerantquantumphase gatesandquantumstates.Comparedwiththedynamic gates,thegeometricgatesmayofferpracticaladvantages sincethephaseisdeterminedonlybythepatharea.insen

    sitivetothestartingstatedistributions,thepathshape, andthepassageratetotraversetheclosepath.Thus.

    geometricphasesmayberobustagainstdephasingand thefidelityofthegeometricgatesmightbesignificantly higherthanthatofthedynamicalones.asdemonstrated inarecentexperimentinthecontextoftrappedions.tl0] Inthispaperweproposeaschemeforthegeneration ofsuperpo8itionsofcollectiveatomiccoherentstatesin theiontrapsystem.Inourscheme,theionsaredrivenby twooif-resonantlaserfieldssothatthevibrationalmode isdisplacedalongacirclewiththeareadependingonthe numberofionsinacertaingroundelectronicstate.After asuitableinteractiontimethevibrationalmodereturns totheinitialstate.whiletheinternaldegreesoffreedom acquireageometricphaseproportionaltotheaxesofthe correspondingcircleandevolvefromacoherentstateto acatstate.Theschemehasintrinsicresistancetocertain errorssinceitisbasedongeometricoperation.Further

    more,itdoesnotinvovlesoif-resonantcarriertransition andthusdoesnotrequiretheRabifrequencytobemuch smallerthanthevibrationalfrequency,greatlyincreasing theoperationspeed.Finally.itdoesnotrequireindividual addressingoftheions.

    W_econsiderNidenticalthree-levelions.havingtwo groundstatesIe)andII9)andanexcitedstateIr),confined inalineartrap.Thetransitionle)-_+lr)isdrivenbytwo classicallaserfieldswithdetuningsAand?+15,with

    1beingthefrequencyofthecenter.of-massvibrational mode.TheHamiltonianforthissystemisgivenbv【】lJ

    NN

    H=.n+.?J+(?可+H.c.),j=lj=l

    (1)

where=1rA(ejl,=leJ)(I,,j=(1/2)(I)(}

    I.j)(.JI),aandaarethecreationandannihilationoper

    atorsforthevibrationalmode.1iSthefrequencyofthe ceI1terof-massvibrationalmode,andnandarethe transitionfrequencyanddipolematrixelementcharacter

    izingtheelectronictransition.=isthepositivepartof

    theclassicaldrivingfieldswithrespectivetothethion

    Ej=Ele-i[(wo-A)t-krrj

    +e一州.(?+ul)t-k2.rj}

    ,(2)

    whereElandk2(1=l,2)aretheamplitudeandwavevetc- torofthe/-thdrivingfield,respectively,andrjisthe positionoperatorofthe3thion.Underthecondition

    SupportedbyFundsfromtheStateKeyLaboratoryBreedingBaseofPhotocatalysis,Fuzhou

    University

    rE-mail:lhlinll@yahoo.corn.ca

    No.5PreparationofMacroscopicQuantum

    InterferenceStatesforaCollectionof']"rappedIonsViaaSingleGeometricOperation921

    ?》theionsdonotundergoelectronictransitions andtheeffectiveHamiltonianforthissystemisgivenby ?

    =n+?+Qek2r_

    J=1

    where

    +2etfctck.k'r+'

    2---x'-'},,

    Q=,Qz=2AEz,z

    Thesecondandthirdtermsinthelargebracketdescribe theexchangeofphotonsbetweenthetwolaserfieldsdue tothevirtualexcitationoftheelectronicstatesandthe

lasttermdescribestheStarkshiftsinducedbynonreso

    nantinteraction.[121Assumethatthewavevectordiffer

    encek2klisalongthetrapaxis,i.e.,thexaxis.Then wehave(k2k1)-rj=kxj,where=lk2klI.Therel

    ativedetuningofthetwolasersiscloseto,i.e.,u15.

    Inthiscasewecanneglectotherstretchvibrationalmodes

    andreplacekxjby(0+0t),[1a1whereistheLambDicke

    parameter.Iftheionsinitiallyhavenoprobabilitiesofbe

    ingpopulatedintheexcitedstatelr),theywillremainin

    thegroundstatesduringtheinteraction.Inthiscasethe

    Hamiltonianreducesto

    ?NCOO:wlata-coo+.?cmt?[+e+=

    1

    ++22

    (5)

    ConsiderthebehaviorofthetrappedionsintheLamb

    Dickeregime,where叼而《1,withbeingthemean

    phononnumberofthecenterof-massmode.Wecanex- pandtheHamiltoniantothefirstorderin

    {+Qc.s()

    

    (n+nt)一叩eu

    )+

    IntheinteractionpicturetheinteractionHamiltonianis

    givenby

    :

    N

    {Os(——5)t

    ei6t__

    ei)f]

.[e-i6t_e-i(2w1-5(el7

    AssumethatQcolandthuswecanneglectthe termsiTl(f~/2)ate[(2w~-5)++H.c.Thentheinterac tionHamiltonianreducesto =+Qcos(col1

    .e))fej)(ej

    As....

    s

    ....

    u

    ......

    m

    ........

    e

    ....——

    thateachionisinitiallyinthestate (1/,//.+1)(?IeJ)+l)).Thentheionsystemisinthe

    Blochstate[14]

    0))j?)=N((9)

    wherelk)isthesymmetricalDickestateswithkionsbeing

    inthestatele).[15]Set

    1-)1)6kl(

    ?

    =

    ?leb<e

    J=1

    Wlethenhave

    [b,b]=1,[?e,6]=,[?e,6]:6.(11)

    SetN1andNne,withebeingtheaveragehum

    berofionsbeingintheelectronicstateIe).Inthiscase

bandbtcanberegardedasthebosonicoperatorsand

    theDickestateI)correspondstotheFockstatewith quanta.JThentheatomiccoherentstateN)corre

    spondstothenormalcoherentstateofthebosonicsystem

    I)withoz=,/?f,i.e.,

    ),

    andtheHamiltoniancanberewrittenas

    Hi=——【2{+Q;

    4?+QCOS(co1a)t

    +teiat-ae-iSt)}bt6,

    (12)

    FortheinitialstateJ),duringtheinfinitesimalinter

    val[t,t+dt]theevolutionis

    J)l,(?))e--iHidtIk>lO()),

    -ex+acos(col--5)

    ×D(da))J,(?)),

    where

    d=77ed,

    andJ,())isthestateofthevibrationalstatecorre

    latedtotheinternalstatel)atthetimet.Therefore,

    thevibrationalmodeisdisplacedalongacirclewiththe

    radiusk,a/(2a1andtheangularfrequency.Theevolu

    tionduringtheinterval[0,r]is[] )expik{+col.-

    ~

    [sin(

    ×eiCkD(o~k)lk)l~(0))

    where

    一叼(eiSt-1)

    )7_])]

口一2

    +

    =

    

    

    

    r?J,L

    ??

    H11aVo1.53 922LINLi

    Ck=Im

    .d=--Im/o()ii.(1-eiSt')d

    =

    (细一1sn(18)

    I(0))istheinitialvibrationalstate.Withthechoice

    6T=27r.thevibrationalmodereturnstotheoriginal

    pointinthephasespace,i.e.,Otk=0,andreduces

    to

    ()..

    Thenweobtain

    ))xp)0))

    where

    =

    擘丁+4?'.QCOl——6sin(w1

    (19)

    (20)

    

    )7-.(21)

    Asthesystemisinitiallyinthecoherentstateof

    Eq.(12),afteraninteractiontime27r/thesystemevolves

    to

)2/2eik~exp)].(22)k=0''.

    Thevibrationalmodeisdisplacedalongdifferentcircles withradiiproportionaltok,butwiththesameangular frequency.Therefore,foreachDickestatesIk),thevi

    brationalmodereturnstotheoriginalstateandthesys

    temacquiresageometricphase西kproportionaltothearea

    ofthecorrespondingcircle.Weherehavediscardedthe stateofthevibrationalmodesinceitisdisentangledwith theelectronicdegreesoffreedom.Inthiscasetheevo? lutionofthesystemismathematicallyanalogoustothat ofalightfieldpassingthroughanamplitudedispersive medium.[16]Withthechoice

    ,Q,227r7r

    /l,

    i.e.,=Q,thestateI(7-))canberewrittenas[.

    =

    ak[ei~/4+(_)ke-iV/4)

    References

    =e/4lo~eikO)+e-iTr/4[(23)

    ThisisasuperpositionsoftwomacroscopicaUydistin- guishablequantumstatesforthecollectiveionsystem. Wnowgiveabriefdiscussionontheexperimental matters.ThehyperfinelevelsIF=1,m=1)and

    IF=2,m=2)ofs1/2of.Be+canactasthestatesle) and[10]respectively

    ,while/2lF=2,m=1)can

    actas1r).SetQ1=Q2=10Wl,=0.1,?=i00~i,and

    6=]/20.Thenwehave2=/2andthecondition

    ="Qcanbesatisfied.Thetimeneededtocompletethe

processisaboutT=2r/5=40/1.Wehaveneglected

    thecarriertransition.Theprobabilitythattheionsuri- dergothecarriertransitiontotheexcitedelectronicstate isgivenby(Q}+a~)/a=2x10_..Settheheatingrate tobeF=0.0001w1.[17]Thentheerrorinducedbyheating isontheorderofFT1.3X102.Withthesenonideal

    conditionsbeingconsidered,thefidelityisabout0.967. Inconclusion,wehavedescribedaschemeforthegen. erationofmacroscopicquantum-interferencestatesfora collectionoftrappedionsbyasinglegeometricphase operation.Intheschemeonegroundelectronicstate oftheionsiscoupledtothecenterof-massvibrational

    mode,andthevibrationalmodeisdisplacedalongaradius proportionaltothenumberofionsinthecorresponding groundelectronicstate.Foracertaininteractiontime,the vibrationalmodereturnstotheoriginalstate.withthe systemaqcuiringageometricphaseproportionaltothe squareofthenumberofionsinthecorrespondingground electronicstate.Theionsundergonoelectronictransi

    tionsduringtheoperation.Ourschemehastheinherent faulttolerantadvantageofgeometricoperation,opening propectivesfortheproductionofmacroscopicquantum- interferencestatesforcollectiveionsystems. E.Schr6dinger,Naturwissenschaft.23(1935)807. 2]Y.YiaandG.Gno,Phys.Lett.A136(1989)281;C.C Gerry,J.Mod.Opt.40(1993)1053.

    [3M.Brune,E.Hagley,J.Dreyer,X.Maitre,A.Maali,C ?nderlich,J.M.Ralmond,andS.Haroche,Phys.Rev Lett.77f1996)4887.

    [5]

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