Polarization statistical properties of the emissio

arXiv:quant-ph/0407196v123Jul2004PolarizationstatisticalpropertiesoftheemissionfromthesinglemodeVertical-CavitySurface-EmittingLaserswiththeequallylivinglaserlevelsYu.M.GolubevandT.Yu.Golubeva∗V.A.FockPhysicsInstitute,St.PetersburgStateUniversity,ul.Ul’anovskaya1,198504St.Petersburg,StaryPetershof,RussiaE.Giacobino†LaboratoireKastlerBrossel,UniversitePierreetMarieCurie,F-75252ParisCedex05,France(Dated:February1,2008)Thefullquantum-statisticaltheoryoftheVertical-CavitySurface-EmittingLaser(VCSEL)intheformoftheLangevinequationsisconstructedforarbitraryrelationsbetweenthefrequencyparameters.ThesametheoreticaltreatmentasinRef.[1,2]areused.FordetailedanalysisthetheoryisappliedforlaserswithequallylivinglaserlevelsandonthisbasistheanalyticalexpressionsforthespectraldensitiesoftheStokesparameterfluctuationsareobtainedintheexplicitdependenceonthephysicalphenomena,includingthespin-flipandtheopticalanisotropy.Itisdemonstratedthearbitrarydistributionofelectronsbetweenthesub-levelsunderpumpingdoesnotrestrictapossibilitytoachievethenoisereductionbelowthequantumlimit.UndercomparisonwithphenomenologicaltreatmentRef.[3]itisshownthisapproachturnsouttobenotquitesatisfied.2I.INTRODUCTIONInthelastyearstherehasbeenanincreasedinteresttothepolarizationpropertiesoftheVCSELs.Thisinterestismotivatedinthefirstlinebythepotentialapplicationsofthistypeoflasersinthehigh-rateopticalcommunicationsRef.[4].ButthereisalsomorefundamentalreasonforunderstandingofthepolarizationbehaviorinVCSELs,namely,apossibilityofgeneratingtheintensity-squeezedlightusingthesub-PoissonianpumpoftheactivemediumRef.[5,6].Todate,squeezinginVCSELshasbeendemonstratedexperimentallyforboththesingle-modeoperationandinthemulti-transverse-moderegimeRef.[7].Nowinthedifferentscientificgroupstwomodelsoflaserarediscussedinthemain.Firstofthemiswiththeshortlylivingupperlevelandtheother-withtheequaledlifetimesforboththelevels.Theknownspin-fliptheory[8]waselaboratedforthefirsttimejustforthelattersystem.Neverthelessalotofitissuitableforthefirstonetoo.Asforstatisticalaspectsthelaserwiththeequallylivinglevelswasstudiedonlyphe-nomenologicallyRef.[3]indistinguishfromtheothersystemwhichhasbeenstudiedindetailswithinthelimitsofthequantumelectrodynamicsRef.[1,2].Ourmaingoalheretodevelopnotlessqualitativetheoryforthesystemwiththeequallylivinglevels.Thiswillallowusnotonlytowritethedifferentcorrectanalyticalexpressions,butalsotoestimatethepossibilitiesofthephenomenologicaltreatmentRef.[3].Thepaperisorganizedasfollows.InSec.IIthebasicequationsofthetheorywillbegivenintheadiabaticalapproximation.InSec.III,IVthefluctuationsoftheStokesparameterswillbeintroducedintoconsideration,bewrittentherespectivelinearizedequationsandtheirsolutions.InSec.Vthecomparisonofourresultswiththephenomenologicaloneswillbemade.Atlast,inSec.VIthepossibilitiesforthepolarizationsqueezingobservationwillbediscussed.∗Electronicaddress:[e-mail],[e-mail]†Electronicaddress:[e-mail]HETWO-LEVELLASERSWITHTWICEDEGENERATEDLEVELSThewellknownspin-flipVCSELtheoryRef.[8]isbasedontherepresentationaboutthesemiconductormediumasatwo-levelsystemwithatwicedegenerationofthelevels(seefig.1).Onepairofsub-levelsisconnectedbywavewiththeσ+-polarizationandtheother-withtheσ−one.Theupper”atomic”states|a±irelaxwiththerateγa,andthelower-withtherateγb.Forthedescriptionofoverturnsoftheelectronspins(thespin-flip)weintroduceintoconsiderationsomeincoherentprocesses|a+,b+i→|a−,b−iandbackwiththerespectiveratesγ(a,b)c.Wewillnotmentionallthedetails,howthemainequationsareproduced.TheycanbefoundinthepreviousworkRef.[1,2,9].Themainitemsareasfollows.Atfirst,theequationsfortheHermitianoperatorsofthefieldamplitudeswiththeσ±-polarizations,thepopulationsofthefoursub-levelsandthepolarizationsofthetwoactualtransitionsareconstructedonthebasisofthefullquantumtheoryforfieldandmatter.Theobtainedequations(theLangevin-Heizenbergones),first,areoperatorand,second,containtheoper-atorsources(theinhomogeneousterms)indistinguishfromtherespectivedynamicaltheory.Usuallyforwritingthecorrectcorrelationfunctionsoneusesso-calledEinsteinrelationshipsRef.[10].Next,tomakethemathematicalsituationmuchsimpler,onepassestoc-numberrepre-sentation:allthenormallyorderedoperatorscanbeconvertedontorespectivec-numberfunctionsRef.[9,10].Asaresulttheequations,calledtheLangevin’sequations,read:˙a±=−κa±−(κa+ıωp)a∓+gP±,(2.1)˙P±=−(γ⊥+iΔ)P±+gNa±−Nb±a±+FP±(t),(2.2)˙Na±=μa−γaNa±−γ(a)cNa±−Na∓−ga∗±P±+a±P∗±+Fa±(t),(2.3)˙Nb±=μb−γbNb±−γ(b)cNb±−Nb∓+ga∗±P±+a±P∗±+Fb±(t).(2.4)Herea±aretheσ±complexamplitudes.Theatomicc-numbervariablesarerepresentedbythepolarizationsP±andthepopulationsNa±andNb±.Onecanseethetheoryturnsouttobeverycomplicatedmathematically,becausegenerallyspeakingthereisasystemofthe12differentialequationsrelativetothe12variables.Thefrequencycoefficientshavethephysicalsenses:κisthespectralwidthofthecavity4lasermode,gistheatom-fieldcouplingconstant,μa,bisthemeanrateoftheincoherentpumptotheupper,lowerlaserlevel,γa,bisthementionedaboveconstantofdecayoftheupper