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Coherent-Entangled

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Coherent-Entangled

    Coherent-Entangled

    Commun.Theor.Phys.(Beijing,China)46(2006)pp.975982

    c?InternationalAcademicPublishersVol.46,No.6,December15,2006 Coherent-EntangledStateinThree-ModeandItsApplications FANHong-Yi1,2

    andWANGWen-Qin3

    1

    DepartmentofMaterialScienceandEngineering,UniversityofScienceandTechnologyofC

    hina,Hefei230026,China

    2

    DepartmentofPhysics,ShanghaiJiaoTongUniversity,Shanghai200030,China 3

    DepartmentofSpecialClassfortheGiftedYoung,UniversityofScienceandTechnologyofC

    hina,Hefei230026,China

    (ReceivedAugust3,2005)

    AbstractWeintroducethenewconceptofcoherent-entangledstate(CES).Byvirtueofthetec

    hniqueofintegration

    withinanorderedproductofoperatorsweintroduceanewkindofthree-modeCES|β,

    γ,x?,whichexhibitsboth

    propertiesofthecoherentstateandtheentangledstate.|β,

    γ,x?makesupanewquantummechancialrepresentation.

    Itsapplicationsinquantumopticsarealsopresented.

    PACSnumbers:03.65.-w,03.67.-a,42.50.Dv

    Keywords:coherent-entangledstate,IWOPtechnique

    1Introduction

    ItiswellknownthatDiracisthe?rstphysicistwho

    introducedrepresentationtheoryintoquantummecha-

nics,[1]

    whichnotonlylaidthefoundationofquantumme- chanicsformalism,butalsobroughtgreatconvenienceto studyingmiscellaneousproblemsinquantummechanics. Todayitisalmostimpossibletothinkaboutnewdevelop- mentsofquantumtheorywithouttheuseoftransparent anddeepformulationbasedonDirac'srepresentationthe- ory.Somerepresentationsareusednotonlyforexpand- ingphysicalstatesasmathematicalspacebasis,butalso possessphysicalmeaning,forexample,thecoherentstate |z?=exp{za?

    ?z?

    a}|0?[2,3]

    isanimportantconceptin

    quantumopticsandlaserphysics.Foranotherexample, therecentlyestablishedbipartiteentangledstateinthe two-modeFockspace,[4,5]

    |ε?=exp

    h

    ?

    |ε|

    2

    2

    +εa

    ?

    1

    ?ε?

    a

    ?

    2

+a

    ?

    1

    a

    ?

    2

    i

    |00?,

    ε=ε1+iε2,(1) isthecommoneigenstateoftwoparticles'relativecoordi-

    nateX1?X2andthetotalmomentumP1+P2,

    (X1?X2)|ε?=

    ?

    2ε1|ε?,

    (P1+P2)|ε?=

    ?

    2ε2|ε?,(2) whereXi=(ai+a

    ?

    i

    /

    ?

    2,Pi=(ai?a ?

    i

    /(i

    ?

    2),i=1,2, [ai,a

    ?

    j]=δij.Itre?ectsEinsteinPodolskyRosenquan- tumentanglement[6] explicitlybyreadingitsSchmidtde-

    compositioninthecoordinateeigenvectorspace|x?i,[7]

    |ε?=e?iε2ε1

    Z?

    ??

    dx|x?1

    ?

    ?

    ?

    ?x?

    ?

    2ε1

    E

    2

    ei

    ?

    2ε2x

    .(3)

    The|ε?statecanalsobeSchmidt-decomposedinmomen-

    tumeigenvectorspace|p?as |ε?=e

    iε1ε2/2

    Z?

    ??

    dp|p?1

    ?

    ?

    ?

?

    ?

    2ε2?p E

    2

    e

    ?i

    ?

    2ε1p .(4) CombiningEqs.(3)and(4)onecanseethatwhen

    onemeasurementshowsthatparticle1isin|x?1

    (or

    |p?1 )state,thenparticle2issimultaneouslycollapsedto?

    ?x?

    ?

    2ε1

    ?

    2

    (or

    ?

    ?

    ?

    2ε2?p ?

    2

    )state.Byusing|00??00|=:

    e?a

    ?

1

    a1?a

    ?

    2

    a2

    :,andtechniqueofintegrationwithinanor-

    deredproduct(IWOP)ofoperators[8]

    (foraReviewsee

    Refs.[9]and[10])wecanprovethecompleterelationand

    theorthonormalpropertyof|ε?,

    Z

    d2

    ε

    π

    |ε??ε|=

    Z

    d2

    ε

    π

    :e?|ε|

    2

    +εa

    ?

    1

    ?ε

    ?

    a

    ?

    2

    +a

?

    1

    a

    ?

    2

    +ε

    ?

    a1?εa2+a1a2?a

    ?

    1

    a1?a

    ?

    2

    a2

    :=1,

    ?ε|ε?

    ?=πδ(ε1?ε?

    1)δ(ε2?ε?

    2).(5)

    Allthequantummechanicalrepresentationspossessthecompleteness(orover-completenes

    s)relation.Choosingsuit-

    ablerepresentationisalsoessentialforsolvingdynamicproblems.AsDiracpointedoutinRef

    .[1]."Whenonehasa

    particularproblemtoworkoutinquantummechanics,onecanminimizethelaborbyusingare

    presentationinwhich

    therepresentativesofthemoreimportantabstractquantitiesoccurringinthatproblemareassi

    mpleaspossible",we

    believethatconstructingvariousentangledstaterepresentationswillbeusefulnotonlyintreat

    ingmanyproblemsin

    quantumoptics,[11]

    butalsomayopenup(explore)newresearchtopics.Thussearchingfornewphysicalapplicable

    quantummechanicalrepresentationsisoneoftheimportanttasksoftheoreticalphysicists.Aquestionthusnaturally

    arises:isthereanythree-modestate-vectorrepresentationwhichcanexhibitboththecoherentstate'spropertyand

    theentangledstate'sproperty?Theanswerisa?rmative.Inthisworkweproposeanewcoherent-entangledstatein

    three-modeFockspacewhichcanbeprovedtomakeupanewquantummechanicalrepresentation.Bynoticing

    [X1+X2+X3,a1?a2]=0,[X1+X2+X3,a2?a3]=0,(6)

    whereX3=(a3+a

    ?

    3

    /

    ?

    2,weknowthat(X1+X2+X3),(a1?a2),and(a2?a3)makeupacompletecompatible operatorset,sotheyshouldpossessacommoneigenvectorset,thenfourquestionsnaturallyfollow:whatistheexplicit

    formofthisstate?Doesitcomposeacompleteset?Howtoimplementitbyexperiments?Whatareitsapplications?

    Theworkisarrangedasfollows:inSec.2wederivesuchastatevectorinthree-modeFockspace.InSec.3wereveal

    976FANHong-YiandWANGWen-QinVol.46

    itscompletenessrelation,andinSec.4westudyitspartlyorthonormalproperty,whichexhibitsbothcoherentand

    entanglingbehavior.InSec.5wediscusshowtousebeamsplittertoproducesuchstates.InSec.6wepresentthe

    conjugatestateof|β,γ,x?(denotedas|σ,

    κ,p?),whosepropertiesarealsobrie?ydiscussed.InSecs.7and8wepresent someapplicationsofthenewstateinquantumoptics,includingderivingthegeneralizedWigneroperator(Sec.7)and

    three-modesqueezingoperator(Sec.8).InSecs.9and10areductionofthethree-modecoherent-entangledstateto

    two-modecaseismade,anditsapplicationincomposingageneralizedP-representationfortheoptical?eld,whichis

    twodi?erentlightmodes'superposition,ispresented.

    2Three-ModeCoherent-EntangledState

    Let|β,γ,x?besuchastatethatsatis?esthefollowingeigenvectorequations: 1

    3

    (X1+X2+X3)|β,γ,x?=

    1

    ?

    2

    x|β,γ,x?,(7)

    (a1?a2)|β,γ,x?=β|β,γ,x?,(8)

    (a2?a3)|β,γ,x?=γ|β,γ,x?,(9)

    whereβ=β1+iβ2andγ=γ1+iγ

    2arecomplexnumbersandxisreal.Theoperator(X1+X2+X3)/3canbe

    consideredasaquadratureofthreemodesofoptical?elds,while(a1?a2)and(a2?a3)mayrepresenttwo-light

    beams'subtraction.Inopticswhentwoormorelightbeamsaresuperposedtoproduceinterferencee?ects,thenthe

    totaloptical?eldateachpointinspaceandtimeistheopticalsumoropticalsubtractionofthe?eldscontributedby

    theseparatebeams.Aftermakingmanytrialswe?ndthat|β,

    γ,x?inthethree-modeFockspaceshouldbe

|β,γ,x?=exp

    ?

    ?1

    6

    (2|β| 2

    +2|γ| 2

    +βγ? +γβ? )?3 4

    x2

    +

    ?

    x+1 3

    (2β+γ)

    ?

    a?

    1

    +

    ?

    x+1 3

    (γ?β) ?

    a?

    2

    +

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