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Wigner Distribution Function and Husimi Function of a Kind of Squeezed Coherent State

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Wigner Distribution Function and Husimi Function of a Kind of Squeezed Coherent Stateof,a,A,and,Kind,kind,coher

    Wigner Distribution Function and Husimi Function of a Kind of Squeezed Coherent

    State

    Commun.Theor.Phys.(Beijing,China)47(2007)PP.427-430

    ?InternationalAcademicPublishersV_01.47,No.3,March15,2007

    WignerDistributionFunctionandHusimiFunctionofaKindofSqueezedCoherent State

    FANHong-YiandLIUShu-Guang

    DepartmentofMaterialsScienceandEngineering,UniversityofScienceandTechnologyofChina,Hefei230026,China

    AbstractWefindanewx-parametersqueezedcoherentstatelP,g)Krepresentation,whichpossesseswell-behaved

    features,j.e.,itsWignerfunction'smarginaldistributioninthe''q-direction"andinthe'

    direction"istheGaussian

    formexp{(g,

    g)),andexp{(p-V)./),respectively.Basedonthis,theHusimifunctionofIv,g)Kisa/soobtained,

    whjchisaGaussianbroadenversionoftheWignerfunction.ThelP,g)Kstateprovidesagoodrepresentativespacefor

    studyingvariouspropertiesoftheHusimioperator.

    PACSnumbers:03.65.w,03.65.Ca,42.50.Dv

    Keywords:Wignerfunction,Husimifunction,squeezedcoherentstate,IWOPtechnique 1Introduction

    Coherentstates[1,2andsqueezedstates[3,

    aretwomajortopicsinquantumoptics.Whilethecoherentstatemakes

    thecoordinate-momentumuncertaintyrelationminimum,thesqueezedstatemakesthequan

tumfluctuationofone

    quadratureoffieldlessthanthatofthecoherentstateattheexpansethattheanotherquadrature'sfluctuationis

    morethanthatofthecoherentstate.Squeezedstateshavebeenwidelyusedinopticalcommunicationandweaksignal

    detection.Inthelastdecade.squeezedstateshasbeenconnectedtotheentangledstatestightly.Inthisworkweshall

    introduceanewkindofsqueezedcoherentstate,denotedasIV,q)K,whichprovidesagoodrepresentativespacetor

    studyingvariouspropertiesoftheHusimioperator.TheWignerfunctionandtheHusimifunctionofIP,q)areeasily

    derivedinthisapproach.

    2NewKindofSqueezedCoherentState

    ThenewstatethatweintroduceinF0dspaceis

    lP,q)()e{2(i+)+v~

    at

    (il

    2(i+1.)I(1)

    whereisequivalenttoasqueezingparameterasonecanseeshortlylater,[0,0t]=1,010)=0.(Forareviewof

    squeezedstates,werefertoRefs.3and4).Inparticular,when=1,

    IpIq)=exp{+p)+1(q+ip).)10),(2)

    whichisjustthecanonicalcoherentstate.【】

    ThusIP,q)Kisitsnon-trivialgeneralization.Usingthenormallyordered formofthevacuumstateprojector

    10)(0I=:e.'.:,.

    andthetechniqueofintegrationwithinanorderedproduct(IWOP)ofoperators[,.

    wecanexpresstheprojector

    where

IpIq),gI=:{q2+(.2+.t2)_p2

    

    .

    t.+

    1--~av/2q(a~a*)1i(.))

    =:{()一而1(p-P)):,

    Q=,P=00t

    (3)

    (4)

    ———

    1.TheprojectBupportedbytheSpecializedReBearchFundfortheDoctorialProgresBofHig

    herEducationofChinaunderGran No.20040358019

    FANHong-YiandLIUShu-GuangVl01.47 Itthenfollowsthecompletenessrelation

    dpdql(p,q

    Usingtheovercompletenessrelationofthecoherentstate

    d2z

    )(I=1,

    andthefollowingformula dZz

    exp(fflzl+?++,z2+9}.=

    whoseconvergentconditionis Re(e+f+g)<0,

    Wecancalculatetheoverlap ,

    gIp,g)=.,d

    

    Zz

,gI)(Ip,g)

    =

    d2z

    (0Iexp{2(1+)

    ,

    e

    ['

    Re()<.

    g.+csq'-ip')+

    ×(leXp{++12(1+).)

    1

    2(1+)

    10)

    .

    =e

    {(gg)l(p,-V)+gpg)+il-npg,)), whichisnon-orthogona1.Especially,whenP=Pandq=q,

    ,alp,g)=1,

    SOlp'g)Kiscapableofmakingupanewquantummechanicalrepresentation?Dueto

    .Ilp)g)p,=[(ip)+.t]Ip', thusIv,g)Kistheeigenvectorof,,

    (acoshr-atsinhr)Iv}g)=(g+)Ig), .

    (5)

    (6)

    (7)

    (8)

    (9)

    (10)

    (11)

(12)

    where(1a)/(1+)

    tanhristhesqueezingparameter,coshr=(1+)/2.InthissenseIp,g)Kisasqueezed

    coherentstate. 3WignerFunctionofIP,)

    Recal1.thattheWigneroperatorinthecoherentstaterepresentationis[71

    ?(,)=fd2Zllr2+)(Ie:.:' UsingEqs.(13)and(7)weevaluatetheWignerfunctionofIp,g)K,:

    qlpIg)=/rzIv}g)e:':' =

    ./e{p2oxp{V~(nqip)1+(+Z)+1,( +V~

    T

    (sq+ip)(十一

    )+1

    .

    -

    n(十一))eaZ?-za'(0I+)(I.) 1r

    eXpt

    12

    +2

    

    ~2q2

    

    -

    4-p2+V~(tcZq

    -

    iP)+

V~(sZq

    +iv)

    (+)1+2.I)

    }

    +)

    (13)

    (14)

NO.3WignerDistributionFunctionandHusimiFunctionofaKindofSqueezedCoherentSta

    te429

    Letting

    =

    (q+ip)

    equation(14)becomes(writing?(Q,Ot)-??,q))

    ?

    (pl,qll?,g)lp,q)=exp{(q'-q)(pp))

    =

    ,qf?,q,)IP,).

    (15)

    (16)

    When=1,equation(16)reducestotheWignerfunctionofthecoherentstate.FromEq.(16)wes

    eethattheWigner

    functionofnewstatefP,ql>Kpossessesthewell-behavedfeatureinthesensethatitsmargi

    naldistributioninthe"q

    direction"isaGaussianformexp{(q

    q)),whileitsmarginaldistributioninthe''p-direction"isexp{pl--p)/).

    ThisgoodfeaturenaturallyleadstothedefinitionofHusimifunction.

    4HusimiFunctionofIP,q)

    InordertoovercometheappearanceofnegativevalueofWigenrdistributionprobability,the

    Husimidistributionis

    introducedinRef.8].BasedonthisideainRef.9]9theso-calledHusimioperator.?^(p,q)(orthegeneralizedWigner

    operator)isintroducedviathefollowingintegration: /dp'dq'?I,q1)exp{(ql_q)一一p))=?^,g),

    wheretheexponentialisjusttheoneinEq.(16),SOwehave

    7r

    //dpdq?(p,q)(P,ql?(p,qP,q)=/kh(p,q).

    UsingthenormallyorderedformoftheWigneroperator ?,g)::e-(Qq).(pp).:,

    ,I

    ,

    toperformtheintegrationinEq.(17)andusingEq.(4)wecanobtain

    Ig)=-exp{(a+at/~一一面a--af)):

    =

    lP,q)(p,q1.

    SotheHusimioperatorisjusttheprojectorJP,g),.FromEq.(10)wesee '

    TrAh(p,q)=1.

    Thusequation(18)canbeexpressedas

    7r

    //dpdq?,g),gJ?(J,,g,)JP,g)=Jp,g),gJ.

    Usingequation(10)weknow

    7r

    //dpdq((p,ql?,q)Ip,q))=1.

    NowusingEqs.(20)and(9)wecanimmediatelyderivetheHusimifunctionoffPql>,

    pll

    }qlAh(p,q)lP,=(p,P,q)(p,qlP,ql

    =exp

    {(q-g)(pp))

    =

TrAh(p,q)Ah(P,g,).

    OrwecanuseEq.(18)tocalculate (p,qI/kh(p,q)lIp,q)=27r//dpdq(p,q"I?(p,q)Ip",q)(p,ql?(p,q)Ip,q) =exp

    {(q-q)1(iollp)),

    (17)

    (18)

    (19)

    (20)

    (21)

    (22)

    (23)

    (24)

    (25)

    whichstatesthattheHusimifunctionisaGaussianbroadenversionoftheWignerfunction.Co

    mparingtheWigner

    function(16)withtheHusimifunction(25)wesee

    (p,,q,l?,q)Ip',q1):(,,,,/,I?^,q)l,,/,,q,)(26)

    FANHong-YiandLIUShu-GuangV_ol_47 TMsisanothergoodfeatureofthestateIP,g) 5SqueezingGeneratedfromIp,q)Rpresentation

    UsingEqs.(1),(3),and(24)andtheIWOPmethod['.wehave =

    //dpdqI,vq)kk,qI

    :

    {ng(+?V']at~v/

    p2-'t-"

    

    /v/'2a?#

一鱼

    1-t-to

    ,

    /+(a2-4-at2t.):

    where

    

    ()+(02-a?a):

    :e

    []:e{[]0f0):exp【——————一Jxpt【——一一J'

    ×

    exp[.2](27)

    L=(1+v2)(1+).(28)

    Inparticular,when=1/,wesee

    gg

    >I=[]

    

    {[]0f0):exp[一者.],(2g)

    whicllisjustthenormallyorderedsingle-modesqueezingoperator,.soelefthandsideofEq.(29)manifestlyshows thatthesqueezingiscausedfromP_?即,q_?glq.

    Insummarwehaveintroducedthe~-parametersqueezedcoherentstateIP,g)whoseWignerf

    unction''marginal

    distributioninhe"口一direction,,andinthe''p-direction''istheGaussianformexp{(g-q)),andexp{(p-V)/)

    respectively.Basedonthis,theHusimifunctionofIP,g)isalsoobtainedwhichisaGaussianbr

    oadenversionofthe

    Wignerfunction.ThustheIP,g),cstateprovidesagoodrepresentativespaceforstudyingvario

    uspropertiesofthe

    Husimioperator.WebelievethattheIP,g),crepresentationisanimportantoneforquantumstat

    istics?

Rferences

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    [2J.R.KlauderandB.S.Skagerstam,CoherentStates, WorldScientiiic,Singapore(19s5).

    Seee.g.,G.M.D'Ariano,M.G.Rassetti,J.Katriel,and A.I.Solomon.SqueezedandNonclassicafLight,eds.P. TombesiandE.R.Pike.Plenum,NewYDrk(1989)P.301; vBuzek,J.Mod.Opt.37(1990)303;R.Loudonand P.L.Knight,J.Mod.Opt.34(1987)709;Foraveryrecent review,seeV.V.Dodonov,J.Opt.B:QuantumSemiclass. Opt.4f20021R1.

    [4]M.Orszag,QuantumOptics,Springer-Verlag,Berlin (2000);w.0IfgangP.Schleich,QuantumOpticsinPhase Space,Wiley.Vch,Berlin(2001);M.O.ScullyandM.S. Zubairy,QuantumOptics,CambidgeUniversityPress, Cambridge(1997).

    5H.Y.Fan,J.Opt.B:QuantumSemiclass.Opt.5(2003) R147;H.Y.FanandJ.R.Klauder,Phys.Rev.D35(1987) 1831.

    61H.Y.Fan,Int.J.Mod.Phys.B18(2004)1387. 71H.Y.FanandT.N.Ruan,Commun.Theor.Phys.(Bei- jing,China)3(1984)345;2(1983)1563;H.Y.Fanand H.R.Zaidi,Pl!ys.Lett.'A124(1987)343.

    [8K.Husimi,Proc.Phys.Math.Soc.Jpn.22(1940)264. [9]H.Y.FanandY.L.Yang,Phys.Lett.A353(2oo6)439.

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