Non-classical properties and algebraic characteris

arXiv:quant-ph/9904027v229Nov1999NonclassicalpropertiesandalgebraiccharacteristicsofnegativebinomialstatesinquantizedradiationfieldsXiao-GuangWang∗Shao-HuaPanandGuo-ZhenYangCCAST(WorldLaboratory),P.O.Box8730,Beijing100080andLaboratoryofOpticalPhysics,InstituteofPhysics,ChineseAcademyofSciences,Beijing100080,P.R.China(February1,2008)AbstractWestudythenonclassicalpropertiesandalgebraiccharacteristicsofthenegativebinomialstatesintroducedbyBarnettrecently.Theladderopera-torformalismanddisplacementoperatorformalismofthenegativebinomialstatesarefoundandthealgebrainvolvedturnsouttobetheSU(1,1)LiealgebraviathegeneralizedHolstein-Primarkoffrealization.ThesestatesareessentiallyPeremolov’sSU(1,1)coherentstates.Werevealtheirconnectionwiththegeometricstatesandfindthattheyareexcitedgeometricstates.Asintermediatestates,theyinterpolatebetweenthenumberstatesandgeomet-ricstates.Wealsopointoutthattheycanberecognizedasthenonlinearcoherentstates.Theirnonclassicalproperties,suchassub-Poissoniandistri-butionandsqueezingeffectarediscussed.Thequasiprobabilitydistributions∗email:[e-mail]:D9123Section:OpticalPhysicsandQuantumOpticsRunningtitle:NegativebinomialstatesinquantizedradiationfieldsAcceptedinEPJDon22/11/991inphasespace,namelytheQandWignerfunctions,arestudiedindetail.Wealsoproposetwomethodsofgenerationofthenegativebinomialstates.PACSnumbers:42.50.Dv,03.65.Db,32.80.Pj,42.50.VkTypesetusingREVTEX2I.INTRODUCTIONSinceStoleretal.introducedthebinomialstates(BSs)[1],theso-calledintermediatestateshaveattractedconsiderableattentionofphysicistsinthefieldofquantumoptics.Afeatureofthesestatesisthattheyinterpolatebetweentwofundamentalquantumstates,suchasthenumber,coherentandsqueezedstates,andreducetothemintwodifferentlimits.Forinstance,theBSsinterpolatebetweenthecoherentstates(themostclassi-cal)andthenumberstates(themostnonclassical)[1-6],whilethenegativebinomialstates(NBSs)interpolatebetweenthecoherentstatesandgeometricstates[7-11].Anotherfeatureofsomeintermediatestatesisthattheirphotonnumberdistributionsaresomefamousdis-creteprobabilitydistributionsinprobabilitytheory:theBScorrespondstothebinomialdistribution,theNBStothenegativebinomialdistribution,thehypergeometricstate[12]tothehypergeometricdistribution,andthenegativehypergeometricstate[13]tothenegativehypergeometricdistribution.RecentlyBarnettintroducedanewdefinitionofNBS[14],|η,Mi=∞Xn=MCn(η,M)|ni=∞Xn=MnM!ηM+1(1−η)n−M#1/2|ni,(1)where|niistheusualnumberstate,0η≤1andMisanon-negativeinteger.TheyfindthattheNBS|η,MiandtheBShavesimilarpropertiesiftherolesofthecreationoperatora†andannihilationoperatoraareinterchanged.Thephotonnumberprobability|Cn(η,M)|2isassociatedwiththeprobabilitythatnphotonswerepresentgiventhatMarefoundandthattheprobabilityforsuccessfullydetectinganysinglephotonisη.Mixedstateswiththephotonnumberprobabilityincludethoseapplicabletophotodetectionandopticalamplification[15].TheBSisaintermediatenumber-coherentstateandtheoriginalNBSisaintermediategeometric-coherentstate.Onequestionnaturallyarisesthatifthereexistanintermediatestatewhichinterpolatesbetweenthenumberandgeometricstate.Inreality,thenewNBS3isjusttheintermediatenumber-geometricstate.Thisfactwillbeseeninthenextsec-tion.Thus,wehavethreeintermediatestateswhichinterpolatebetweentwoofthethreefundamentalstates(thenumber,coherentandgeometricstates).Inthepresentpaperweshallstudythenonclassicalpropertiesandalgebraiccharacteris-ticsofthenewNBS.Theladderoperatorformalism,displacementoperatorformalismandrelatedalgebraicstructurewillbeformulatedinsectionII.ItisinterestingthatthealgebraicstructureistheSU(1,1)LiealgebraviathegeneralizedHolstein-Primarkoffrealization.InsectionIII,wewillshowthattheNBScanbeviewedasexcitedgeometricstates,interme-diatenumber-geometricstatesandnonlinearcoherentstates.Thenonclassicalproperties,suchassub-PoissoniandistributionandsqueezingeffectwillbeinvestigatedindetailinsectionIV.TheQandWignerfunctionsarestudiedinSectionVandthetwomethodsofgenerationoftheNBSareproposedinsectionVI.AconclusionisgiveninsectionVII.II.LADDEROPERATORFORMALISM,DISPLACEMENTOPERATORFORMALISMANDALGEBRAICSTRUCTUREOFTHENEWNBSA.LadderoperatorformalismandalgebraicstructureItisknownthattheBSsarespecialSU(2)coherentstates[5,6]andtheoriginalNBSsarePerelomov’sSU(1,1)coherentstates[11]viathestandardHolstein-Primakoffrealizations.SoweexpectthatthealgebrainvolvedinthenewNBSisSU(1,1)Liealgebra.Itiseasytoevaluatethata†n|η,Mi=(M+n)!M!ηn#1/2|η,M+ni.(2)Inparticular,forn=1,wegeta†|η,Mi=M+1η!1/2|η,M+1i.(3)ThecreationoperatorraisestheNBS|η,Mito|η,M+1i.ThispropertyissimilartotheactionofthecreationoperatorontheFockstate|Mi,a†|Mi=√M+1|M+1i.(4)4Actually,inthelimitofη→1,theNBS|η,Mi−reducestothenumberstate|MiandEq.(3)naturallyreducestoEq.(4).ThekeyandinterestingpointisthatthereexistsanotheroperatorqˆN−MwhichalsoraisestheNBS|η,Mito|η,M+1i.FromEq.(1),thefollowingequationisdirectlyderivedasqˆN−M|η,Mi=1−ηη!1/2√M+1|η,M+1i.(5)ComparingEq.(3)andEq.(5),wegetqˆN−M|η,Mi=q1−ηa†|η,Mi.(6)MultiplyingthebothsidesoftheaboveequationbytheoperatorqˆN−Mfromleft,weobtain