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New Approach for Calculating Tomograms of Quantum States by Virtue of the IWOP Technique

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New Approach for Calculating Tomograms of Quantum States by Virtue of the IWOP Techniqueiw,IW,of,new,the,for,The

    New Approach for Calculating Tomograms of Quantum States by Virtue of the IWOP

    Technique

    Commun.Theor.Phys.(Beijing,China)4T(2007)PP.431-436

    @InternationalAcademicPublishersVo1.47,No.3,March15,2007

    NewApproachforCalculatingTomogramsofQuantumStatesbyVirtueoftheIWoP Technique.

    FANHong-Yi,andWANGJiSuo.'

    1DepartmentofPhysics,ShanghaiJiaoTongUniversity,Shanghai200030,China. 2DepartmentofMaterialScienceandEngineering,UniversityofScienceandTechnologyofChina,Hefei230026,China

    3DepartmentofPhysics,LiaochengUniversity,Liaocheng252059,China fReceivedOctober31,2005;RevisedAugust15,2006)

    AbstractWeshowthatfortomographicapproachthereexisttwomutualconjugatequantumstateslp,,7-)and

    lz,,z,)namedtheintermediatecoordinate-momentumrepresentation),andthetwoRadontransformsoftheWigner

    operatorarejustthepure-statedensitymatriceslp),r,r(Plandl)^,^,(lrespectively.Asaresult,thetomogram

    ofauantumstatescanbeconsideredasthemodule-squareofthestates'wave/'unctioninthesetworepresentations.

    Throughoutthepaperwefullyemploythetechniqueofintegrationwithinanorderedproductofoperators./nthisway

    weestablishanewconvenientformalismofquantumtomogram.

    PACSnumbers:03.65.w.42.50.Dv

    Keywords:tomogram,intermediatecoordinatemomentumrepresentation

1Introduction

    Inrecentyears,tomogramapproach,whichiscon

    nectedwithitspropertytobeastandardpositiveproba- bilitydistributionfunctiondescribingthequantumstate inquantumstatisticsandquantumoptics,hasbrought muchinterestsofphysicists.Asiswel1known.ray

    oropticaltomographicimagingtechniques(classicalto- mography)derivetwodimensionaldatafromathree-

    dimensionalobjecttoobtainasliceimageoftheinter- nalstructureandthushavetheabilitytopeerinside theobjectnoninvasively.Inthecontextofphasespace theoryofquantumstatisticsVogelandRisken[Jpointed outtheprobabilitydistributionfortherotatedquadra- turephaseXo兰【atexp(iO)+aexp(-iO)]/,//2canbe

    expressedintermsofWignerfunction,andthereverse isalsotrue(namedasVogel-Riskenrelation),i.e.,one canobtaintheWignerdistributionbytomographicinver- sionofasetofmeasuredprobabilitydistributions,P0(x0), ofthequadratureamplitude.Inquantumopticstheory, aX+rPandarerealnumbers,X=(at+a)/,/,2,

    P=i(ata)/,//2)representsallpossiblelinearcom

    binationofquadraturesXandPoftheoscillatorfield modeaandcanbemeasuredbythehomodynemeasure- mentjustbyvaryingthephaseofthelocaloscillator.The averageoftherandomoutcomesofthemeasurement,ata givenlocaloscillatorphase,isconnectedwiththemarginal distributionofWignerfunction,thusthehomodynemea- surementoflightfieldpermitsthereconstructionofthe Wignerfunction[2ofaquantumsystembyvaryingthe

phaseofthelocaloscillator.Smithey.BeckandRaymer[3

    alsopointedoutthatoncethedistributionP0(xo)isob

    tained,onecanusetheinverseRadontransformationfa- miliarintomographicimagingtoobtaintheWignerdis- tributionanddensitymatrix.Thustomographicapproach ofquantumtheoriesoffersadescriptionintermsoftomo- graphicprobabilities.

    AsDiracpointedout,"Whenonehasaparticu-

    larproblemtoworkoutinquantummechanics,one canminimizethelabourbyusingarepresentationin whichtherepresentatiVesofthemoreimportantabstract quantitiesoccurringinthatproblemareassimpleas possible".【】InthispaperbasedonthefactthattheRadon transform(withtheintegraltransformationkernelbeing (p一一rp)or(一一p))ofWignerfunction

    W(,P)isthequantumtomogramofadensitymatrix P,whichisapositivefunctionofthreevariables(,,) or(,,p),weshallshowthatfortomographicapproach thereexisttwomutualconjugatequantummechanical representationsIp)andI).(namedtheintermedi

    atecoordinatemomentumrepresentation),andtheRadon transformoftheWigneroperatorisjustthepurestate

    densitymatricesIp)r.rIorI).,(I.Asaresult, thetomogramofquantumstatescanbeconsideredasthe module-squareofthestates'wavefunctionsintheinter- mediatecoordinatemomentumrepresentation,andcanbe directlyandconciselyobtained.InthiswaytheRadon transformofWignerfunctionscanbeformulatedmuch simplerandmoreelegantly.Throughoutthepaperwe TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantNo.104

75056andtheSpecializedResearchFund

    fortheDoctorialProgressofHigherEducationunderGrantNo.20040358019

FANHong-YiandWANGJiSuoV61.47

    shallemploythetechniqueofintegrationwithinanor

    deredproduct(IWOP)ofoperators.5,6

    Ourworkisarrangedasfollows:InSec.2weintroduce theexpliciteigenvectorJp).1.oftheoperatorTP+aX, andshowlp).r.1.(plisaRadontransformoftheWigner operator.InSec.3wediscusstheinverseofthisRadon transform..

    InSec.4weconstructl)A.representation

    whichisconjugateto).1-when7-=1,InSec.5

    wederivetomogramsofsomequantumstatesincluding numberstate,squeezednumberstate.InSec.6wediscuss therelationshipbetweenHusimifunctionandtomogram ofaquantumstate.

    2IntermediateRepresentationIp),rfor

    QuantumTomogram

    Givenadensitymatrixp,oneconstructsthecorre- sP0ndingWignerfunctionasTrfA(p,),whereA(p,) dxIdpl5--0.~/--=

    1

    7r7-

    1

    7r7-

    2

    istheWigner

    ,

    operator,inIrepresentationofthecoot- dinateX,itisexpressedas[

    d

    

    v

    eipV)((1)

    UsingI)=7r/exp-/2+v~xat0/2]l0)andthe. IWOPtechniqueandthenormallyorderedformofthe

    vacuumstateprojector

    0)(0l=:e.'.:,(2)

    wehaveperformedtheintegrationinEq.(1)andderived

    thenormallyorderedformof?,),f

    ?(,p)=:e(.).(pP).:.(3)

    UsingEq.(3)theRadontransformoftheWigneroperator

    canbeconciselycalculated, dxdp'5(p,Tp,1:ex,-x).(pP).:

    dx:

    dx:

    ?

    P2

    e(z-x).(p)/.r-P).

    (+)X+opr一竽)

    U2X2

    T2

    +2):rJ

    e

    [_+iv~pativ~pa

    +2

    Tia0]:,.f1

    dx'dp'5--0.XI--=

    Notingequation(2)wecandecomposetheright-handsideofEq.(5)as

    (7-P+

    dxdp0.XtTp)A(,P)=lp)""(pl,

    whereweintroducedanewstatevector

    )=[]exp[+iv+/2Pat+.t]J0)

    )

    (4)

    (5)

    (6)

    (7)

    Thusequation(5)showsthattheRadontransformoftheWigneroperatorisjustthepure-stated

    ensitymatrix

    lp)(p1.Itthenfollows

    广oo

    (l//dxdp一一印)?(,p,)l)=(lp)(pl),--

    OO

    sothequantumtomogramofthestatevectorl

    ),theRadontransformofWignerfunction,isthesquare thewavefunctionofI)inthe

    ,1.(pIrepresentation,whichisapositivefunctionofthreevariables.

    NowwecanprovethatIp)"satisfiesthecompletenessrelation dJ)叩叩J=,f./:eXpt0-2+T2(7-P+)):=l

    (8)

    moduleof

    (9)

    ?一?啪

    +

NO.3NewApproachforCalculatingTomogramsofQuantumStatesbyVirtueofthe1wOPTe

chnique433

    Operatingaonlp)",wehave

    [(+i)口一(丁一i)at]lp),=iv~p[p),,

    whichisequivalentto

    (1.P+)lp)=Plp),

    SOlp),ristheeigenvect.r.f(7.P+),so';wenameitm.mentum-p.sitionintermediaterepresenta

    .

    ti.n?

    3Inverseof17~donTransform,

    FromRef.8wehavederivedwleylorderingformoftheWigneroperat~r

    A(,P)=::((PP)

    (10)

    (11)

    (12)

    where:denotesWeylordering,inRef.8wealsointroducedthetechniqueofintegrationwithinWeylordered

    product(IWWOP)ofoperators.noEqs.(6)and(12)WecanderiveWeylorder.

    ingoflp)l,

    l:=:dx'dp'(p—二叩:)-P):=

    ::(p一一7.P).__

    UsingEq?(13)andtheIWWOPt~chniquewe.ee dpdadrI)=1/.dpd训丁一-rP).-e…卅

    _

    1/ddr::ei(+rPz,rp)::

    :::()::=A(,P),

    SOequation(14)istheinversetransformationofEq:(6).Furthermore?from)and(9)wehave

    dp

    ,

    le-lSp=e5'+rP)=::e--is(aX+'rP):: =

    .)_P)::e

    =dx'dp'A(,).

    ConsideringitasaFouriertransform,SOitsinversetransformis

    where

    ?(,P)=1ff

    dp,ff,le-iS'(p,/……

    s,

    ,

    :s,cos=smIp

    (13)

    (14)

    (15)

    (16)

    (17)

    Thusfromthemeasurementresultof(lp),l)=l,rl)lthetomogramofl),onecancalculatethe

    Wignerfunction(lA(,P)l).

    4IntermediateRepresentationl).i, Whenweperformtheintegration(anotherRadontransform)fortheWigneroperatoras

    /r./ddp5(x-Ax"-)?(,,p)=:f_L+b,2

    UsingEq.(2)wedecomposetherighthandsideofEq.(18)andobtain

    ?dx'dp'5(X--.~X,一?(XI,p1)=l

    (+P)):(18)

    (19)

    FANHong?YiandWANGJi?SuoVb1.47 where

    )=[(+)]-i/4exp[22(+)+

    iv

whichistheeigenvectorof(AX+P),

    (AX+vP)l).=l)

    t+]l0)l0,

    l).isanotherintermediaterepresentation.UsingEq.(12)wederivetheWeylorderedformofl),

    ,(l,

    .

    l=dx'dp'5(X--~X!--::L)LP)::

    =::5(AXP).''

    Thecompletenessrelationofl).is

    /dxl),,(l=1.

    TheinversetransformofEq.

    Byobserving

    (19)is

    p

    1dxdAdve".

    leiK?=ei~(AX+uP)=::ei~(AX+uP)::

    =-P)..e",)

    dx'dp'?(,,p,)e-i-(z+p,)j

    andconsideringitasaFouriertransform,SOitsinversetransformis

    ,p)=1lnle,/-ps),

    where

    ,

    =

    Ifonedemandsthatmutuallyconjugatebetween(AX+vP)and(rP+aX),whichmeans

    VO")=i=[X,P, [(+vP),(rP+)=i(r

    thenweshouldposethecondition

    5TomogramsofSomeQuantumStates

    r=1.

    Foranystatel)usingEq.(19)wecanestablishtherelationshipbetweenitsWigner

    modulesquareofthewavefunctionofl)intheintermediaterepresentation(l,

5(x-~z'-(,,p.

    (20)

    (21)

    (22)

    (23)

    (24)

    (25)

    (26)

    (27)

    (28)

    (29)

    functionW(,P)andthe (30)

    Sincethea~lontransformofWignerfunctionisthequantumtomogramofadensitymatrix,SO

    wecandirectlyobtain the,

    quantumtomo~amofapurestatedensitymatrixl)(lbyexplicitlywritingl(

    l).l..Forexample,when l)(lisanun-normalizedcoherentstate,.

    l)=exp[za*]10),thenitsquantumtomogramis

    Using

    (l).l=[(+)]-1/2exp{2+,

    (?)=dn

    le)(p[e+

    e2I,It=0

    (31)

    (32)

No.3

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