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Enhancing Photon Correlations through Plasmonic Strong Coupling R. Sáez-Blázquez,1 J. Feist,1 A. I. Fernández-Domínguez,1,∗ and F. J. García-Vidal1,2,† 1DepartamentodeFísicaTeóricadelaMateriaCondensadaandCondensedMatterPhysicsCenter(IFIMAC), Universidad Autónoma de Madrid, E- 28049 Madrid, Spain 2Donostia International Physics Center (DIPC), E-20018 Donostia/San Sebastián, Spain Weinvestigatethequantumstatisticsofthelightscatteredfromaplasmonicnanocavitycoupledtoa mesoscopicensembleofemittersunderlowcoherentpumping.Wepresentananalyticaldescriptionofthe intensitycorrelationstakingplaceinthesesystems,andunveilthefingerprintofplasmon-exciton-polaritons inthem.Ourfindingsrevealthatplasmoniccavitiesareabletoretainandenhanceexcitonicnonlinearities 7 evenwhenthenumberofemittersislarge.Thismakesplasmonicstrongcouplingapromisingroutefor 1 generatingnonclassicallightbeyondthesingleemitterlevel. 0 2 n Muchresearchattentionhasfocusedlatelyonnanocavi- a tiesforstrongcouplingapplications. Inthesedevices,the J interactionbetweensurfaceplasmons(SPs)andquantum 1 emitters(QEs)canbeintenseenoughtoyieldnewhybrid 3 D light-matterstates,theso-calledplasmon-exciton-polaritons (PEPs) [1]. PEPs involving macroscopic QE ensembles QE SP ] L ll have been reported in planar [2–4] and nanoparticle [5– QE QE a 7] geometries, and they have been used for controlling h r chemicalreactions[8,9]orenhancingcharge/energytrans- QE - SP s port[10,11]. Fromapurelyphotonicsperspective,room e SP m temperature PEP lasing has been recently reported [12]. QE nr QE However,inordertofullyharnessthepotentialofplasmonic . t cavities for optical applications, quantum nonlinearities a m mustberetained[13]. Thisisnotpossibleinmacroscopic ensembles,whichpresentcollectiveboson-likebehaviorat FIG. 1. A QE ensemble resonantly coupled to a generic plas- - moniccavity.Thesetofparameterscharacterisingthesystemare d pumpinglevelsbelowtheQEsaturationregime[14]. sketched.Therightinsetdepictsthetwo-levelQEmodel. n o Very recently, strong-coupling signatures in the power c [ spectrum of nano-gap metallic cavities filled with only a isthesolutionoftheLiouvillianequation few QEs have been reported [15, 16]. These experimen- 1 4v taaimlaidnvgatnoccelsahriafvyethbeeenneaarc-cfioemldpcaonniedditiboynsthyeioerldeitnicgalPeEfPfosratst (cid:126)i[ρˆ,Hˆ]+γ2SPLaˆ[ρˆ]+γQ2rELSˆ−[ρˆ]+γQ2nrE (cid:88)N Lσî[ρˆ] = 0, 6 the single emitter level [17]. However, the generation of i=1 9 nonclassical light through plasmonic strong coupling has (1) 8 whereaˆ,σˆ ,andSˆ− = (cid:80)N σˆ aretheannihilationopera- not been explored yet. In this Letter, we fill this gap by i i=1 i 0 torsfortheSPmode,thei-thQE,andtheensemblesuper- investigatingthequantumstatisticsofthephotonsscattered . 1 radiantstate,respectively. Thedampingassociatedtooper- byananocavitystronglycoupledtoamesoscopicemitter 0 ensemble(upto∼ 100QEs)undercoherentpumping. We atorOˆ isdescribedbystandardLindbladsuper-operators 7 1 develop an analytical description of the quantum optical LOˆ[ρˆ] = 2OˆρOˆ† −{Oˆ†Oˆ,ρ}. Intherotatingframe,the : propertiesofthesystemthatallowsustorevealthat,con- coherent dynamics is governed by the time-independent v trarytowhatisexpected,plasmoniccavitiesallowphoton Tavis-CummingsHamiltonian[18] i X correlationstosurviveinQEensemblesofconsiderablesize r understrongcouplingconditions. Hˆ = (cid:126)∆SPaˆ†aˆ+(cid:126)∆QESˆz +(cid:126)λ(Sˆ+aˆ+Sˆ−aˆ†)+ a +(cid:126)Ω (aˆ†+aˆ)+(cid:126)Ω (Sˆ++Sˆ−), (2) SP QE Figure1depictsthesystemunderstudy: N identicalQEs withtransitiondipolemomentµ andfrequencyω in- with∆ = ω −ω andSˆ = 1[Sˆ+,Sˆ−]. The QE QE QE/SP QE/SP L z 2 teractwiththenear-fieldE (thesameforallQEs)ofasin- QE-SPcouplingisλ = E ·µ ,whileΩ = E ·µ SP SP QE QE L QE gleSPmodeofenergyω supportedbyagenericnanocav- and Ω = E ·µ are the pumping amplitudes. Here, SP SP L SP ity. Both subsystems undergo radiative and nonradiative µ is the effective SP dipole moment. Once the steady- SP damping, with decay rates γ = γr +γnr , state densitymatrix is known, the first- and second-order QE/SP QE/SP QE/SP andarecoherentlydrivenbyalaserfieldE withfrequency correlationfunctionscanbecalculatedfromthescattered L ω . Thesteady-statedensitymatrixρˆforthehybridsystem far-fieldoperatoratthedetectorEˆ− ∝ µ aˆ†+µ Sˆ+. L D SP QE 2 obtainedfromourmodelisinaccordancewithmoresophis- ticateddescriptions[19]. ExactnumericalsolutionstoEquation(1)canbeobtained forstrongQE-SPcoupling. However,suchcalculationsare only possible for configurations involving very small QE ensembles[20],evenfarfromtheQEsaturationregime[21]. Inordertocircumventthislimitationandexplorephoton statistics in mesoscopic ensembles, we map Equation (1) into an effective non-Hermitian Hamiltonian [22] of the form γ γnr γr Hˆ = Hˆ −i(cid:126) SPaˆ†aˆ−i(cid:126) QESˆ −i(cid:126) QESˆ+Sˆ−(,3) eff 2 2 z 2 FIG.2.Zero-delaysecond-ordercorrelationfunctionversuslaser whereHˆ isgivenbyEquation(2). NotethatHˆ depends eff detuning for SP (black dash-dotted line) and QEs (color lines) onlyonthecollectivebrightstateoperatorsoftheQEsand uncoupled.Variousensemblessizesareshown,with(solid)and isindependentofthedarkstatesoftheensemble(superposi- without(dashed)theinclusionofQEnonradiativedecay,γnr . QE tionsofQEexcitationswhichdonotcoupletotheplasmon or external light), which means a drastic reduction in the Hilbert space for large N. Equation (3) results from ne- glecting the so-called refilling or feeding terms OˆρOˆ† in Before investigating photon correlations under strong theLindbladsuper-operatorsinEquation(1). Thisapproxi- coupling conditions, we consider first both SP and QE mateapproachcanbesafelyemployedintheregimeoflow subsystems uncoupled. For this purpose, we solve Equa- pumping,wherethegroundstate(noexcitationsinthesys- tion (1) numerically and compute the normalized zero- tem)canbeconsideredasareservoirwithpopulationequal delaysecond-ordercorrelationfunctioninthesteadystate to1. Inthislimit,wecansolvetheSchrödingerequation g(2)(0) = (cid:104)E−DE−DE+DE+D(cid:105)/(cid:104)E−DE+D(cid:105)2. We only consider forHˆ treatingthecoherentdriving,E ,asaperturbative low laser intensities, and study quantum correlations far eff L parameter[23]. fromthepumpingregimeinwhichQEsaturationbecomes Asweareinterestedinintensitycorrelations,wecanre- relevant. Figure 2 plots g(2)(0) as a function of the laser strictourperturbativetreatmentofEquation(3)tosecond detuningforanemptyplasmoniccavity(blackdash-dotted orderandtruncatetheHilbertspaceattheleveloftwoexci- line)andensemblesofdifferentnumberofemitters(color tations. Thisway,analyticalexpressionsforthescattered solidlines). TheparametersmodellingthesingleSPmode intensityandtheintensitycorrelationscanbeobtained. In are: ω = 3eV,γ = 0.1eVandµ = 19e·nm[5]. SP SP SP thefollowing,forsimplicity,wealsoassumethattheplas- Our calculations yield g(2)(0) = 1, as expected from the monicnear-field,E ,isparalleltothelaserfield,E (as, SP inherent bosonic character. The QE parameters are: SP L forexample,inparticle-on-mirrorcavities[15]). Moreover, ω = 3 eV, γr = 6 µeV (µ = 1 e·nm), and QE QE QE weonlyconsidertheoptimumconfigurationforstrongcou- γnr = 15 meV. These values correspond to low quan- QE pling, in which µ is aligned with E . The scattering tumyieldQEs,accountingfortheemissionquenchingthat QE SP intensity,I = (cid:104)E−E+(cid:105),isgivenwithinfirst-orderpertur- takesplaceindenseensembles[3,12]. Forcomparison,the D D correlationspectraforQEswithγnr = 0arealsoshown bationtheoryas QE (color dashed lines). In all cases, photon statistics is sub- (cid:12)(cid:12)η∆˜ +∆˜ /ηN −2λ(cid:12)(cid:12)2 Poissonian (g(2)(0) < 1), but the degree of antibunching I = (ηNµ Ω )2(cid:12) SP QE (cid:12) , (4) decreasesrapidlywiththeensemblesize. AsN increases, SP SP (cid:12)(cid:12) ∆˜SP∆˜QE−Nλ2 (cid:12)(cid:12) thesystembosonizesandthequantumcharacterofthescat- tered light is lost (note that g(2)(0) = 0.96 for N = 50). whereη = µQE/µSP = ΩQE/ΩSP,∆˜SP = ∆SP−iγSP/2 Neglectingnonradiativedampingonlyleadstoanextremely and∆˜QE = ∆QE−i(γQnrE+NγQrE)/2. Usingsecond-order narrowLorentzian-likeprofile,whichsuppressesantibunch- perturbation theory, the correlation function, g(2)(0), can ingexactlyatzerodetuning. Notethattheg(2)(0)behaviour beexpressedas g(2)(0) = (cid:12)(cid:12)(cid:12)(cid:12)1− 1 (cid:32) η∆˜SP−λ (cid:33)2 (∆˜QE+iNγQrE/2)[∆˜QE∆˜SP+(∆˜SP−λ/η)2−Nλ2] (cid:12)(cid:12)(cid:12)(cid:12)2. (cid:12) N η∆˜ +∆˜ /ηN −2λ (∆˜ +iγr /2)[∆˜ ∆˜ +∆˜2 −Nλ2]−2∆˜ (N −1)λ2(cid:12) (cid:12) SP QE QE QE QE SP SP SP (cid:12) (5) Note that Equation (5) yields g(2)(0) = (1 − 1/N)2 at λ = 0 and η → ∞, which recovers the flat correlation 3 FIG.3. ScatteringintensityI (a -d )andzero-delaysecond-ordercorrelationfunctiong(2)(0)(a -d )versuslaserfrequencyand 1 1 2 2 singleemittercooperativityforvariousQE-SPsystems.Intheupper(lower)panelsdotted(dashed)linesplotthePEPfrequencies(half frequencies)intheone-excitation(two-excitation)manifold. spectrainFigure2forlow-quality-factorQEensembles. bunchedemissiontakesplaceatlargercouplingstrengths Figure3rendersthefar-fieldintensity(toprow)andcor- andwithinbroaderspectraldomainsforallN. Remarkably, relations (bottom row) for a nanocavity filled with four there are spectral windows in which strong antibunching different QE ensembles: N = 1 (a), 5 (b), 25 (c) and 50 (g(2)(0) ≈ 0) takes place even for N = 50, whereas the (d). The horizontal and vertical axes correspond to laser emission from the uncoupled QE ensemble is essentially frequencyandQE-SPcouplingstrength,respectively. The classical(seeFigure2). ThisisthemainresultofthisLetter, latterisexpressedthroughthesingle-emittercooperativity, namelythatincomparisontotheuncoupledsubsystems,col- C = 2λ2/γ γ , with upper limit C = 2 (λ = 0.03 lectiveplasmonicstrongcouplingcansignificantlyenhance QE SP eV),wellbelowthecollectiveultra-strongcouplingregime. photoncorrelationsinmesoscopicPEPsystems. WerestrictourattentiontoQE-SPresonantcouplingand Bytakingadvantageofouranalyticalapproach,wecan consider the same parameters as in Figure 2. Although gainphysicalinsightintotheresultsshowninFigure3. The the quantitative results shown in Figure 3 depend on the intensitymapspresenttwoscatteringmaxima,whoseori- specificsofthesystem,wehavecheckedthatourfindings ginliesatthedenominatorofEquation(4). Itsvanishing andtheirfundamentalimplicationsremainvalidforawide conditionyieldsanalyticalexpressionsforthedispersionof rangeofrealisticconfigurations. thelower(LP)andupper(UP)PEPsinthefirstrung(one- Thecomplexg(2)(0)patternsinFigure3(a2)-(d2)reveal excitationmanifold)oftheTavis-Cummingsladd√er. These thatbothphotonbunchingandantibunchingtakeplacein PEPfrequencies,whichnaturallyincorporatethe N scal- the strong coupling regime. These panels also show that ingcharacteristicofcollectivestrongcoupling,areplotted themainquantumstatisticalfeaturesemergingatthesingle- in dotted lines in all top panels. Note that the intensity emitterlevel(whichareinqualitativeagreementwithrecent maximaoverlapwiththePEPdispersionbandsexceptfor experimental reports on semiconductor cavities [24, 25]) N = 1 and C (cid:46) 0.5. This region, also perceptible for aremostlyretainedasN increases. UptoN ∼ 25,photon N = 5atlowerC,fallswithintheweakcouplingregime, emissionisantibunchedwithinanarrowfrequencywindow whereFano-likeinterferencesbetweenSPandQEemission located at C (cid:46) 1, which implies that the single-emitter givesrisetosharpscatteringdips[26]. AsN increases,the cooperativitycanbeconsideredasthekeyparameterdeter- contrastbetweenUP(brighter)andLP(darker)scattering miningphotoncorrelationsinensemblescontainingupto peaksincreases. ByintroducingthePEPfrequenciesinthe severaltensofQEs. NotethatforverylargeN,antibunch- numerator of Equation (4), the origin of this asymmetry ingisalsoobservedforlargerC-values. Ontheotherhand, becomesclear. NeglectingQEandSPdamping,weobtain 4 √ I ∝ (1∓ Nη)2, wheretheupper(lower)signmustbe used for LP (UP). Thus, QE and SP dipole moments are antiparallelalongtheLPdispersion,whichdiminishesI as N approaches1/η2. Inasimilarwayasinthescatteredintensity,wecanex- pect that the vanishing of the denominator in the second term of Equation (5) could give rise to nonclassical light. At N = 1, the resonant frequencies emerging from this condition are equal to half the energies of the LP (upper sign)andUP(lowersign)inthesecondrungoftheJaynes- Cummings ladder [27]. For N > 1, the same condition leadstoacubicequation,asitaccountsfortheemergence ofthemiddlePEPsinthetwo-excitationmanifold(whose realhalf-frequencyisequaltoω ). Moreover,notice QE/SP the presence of the numerator of Equation (4) in the de- nominatorofthefirstfactorinEquation(5). Asdiscussed √ FIG.4.Maximum(top)andminimum(bottom)correlationfunc- above, thistermacquirestheform(1− Nη)attheLP tionasafunctionoftheQEensemblesizeforseveralvaluesofthe band. Therefore,wecaninferthatthedarkercharacterof singleemittercooperativity.Theinsetintheupperpanelshowthe LPsalsomakesthemmoresuitableforphotoncorrelations. mapofphotonpositive(yellow)andnegative(violet)correlations PEP half-frequencies in the two-excitation manifold are asafunctionofN andC. plotted in dashed lines in Figure 3(a )-(d ). The regions 2 2 of strong photon correlations do not occur exactly at one ofthepolaritonenergies,butslightlyabovetheLPdisper- ishesandreachesaminimumvalue,whichcorrespondsto sion. Thisindicatesthatphotoncorrelations,i.e.,significant thelowestg(2)(0)achievableforagivenN andanyC (or deviationsfromg(2)(0) = 1,donotoriginatefromtransi- viceversa). Itcanbeproventhatthisminimumcoincides tionsalongasinglePEPladder,butfromtheinterferencein withasharpdipinthepopulationoftheplasmonstate(writ- theemissioninvolvingdifferenthybridstates. Thisunder- tenasalinearcombinationofPEPs)inthetwo-excitation linesthecrucialrolethatstrongcouplingplays: Whileeach manifold. Inthelimitofvanishingη (whichisagoodap- PEPbyitselfisquasi-bosonic,thehybridizationachieved proximation for our problem at small N), this condition throughstrongcouplingnotonlymanifestsintheirmixed simplifiestoC = γQE+γSP (cid:39) 1. Figure4(bottom)shows 2γSP 2 light-mattercharacter,butalsoensuresthecoexistenceof thisminimumdevelopingwithincreasingcooperativityat multiple PEP modes separated by the Rabi splitting. It N ∼ 10 and reaching g(2)(0) = 0 at C = 0.5. Remark- is only the interference between the emission from these ably,thiszeroing(2)(0)shiftstolargerN forhighercoop- different but closely related modes that leads to strongly erativity,yieldingstrongphotonantibunchingatensemble nonclassicallightemission. sizesaslargeas100QEs. Therefore,asanticipatedinFig- Inordertoobtainageneralviewonthedegreeofbunch- ure3(d ),plasmonicstrongcouplingleadstotheemergence 2 ingandantibunchingattainablethroughQE-SPcoupling, ofquantumnonlinearitiesinlargeexcitonicsystems,which we evaluate Equation (5) at its spectral maxima and min- wouldpresentg(2)(0) (cid:39) 1whentheyarenotcoupledtothe ima. Figure 4 explore these extreme g(2)(0) values as a plasmonicnanocavity. functionofcooperativityandnumberofemitters. Theinset Toconclude,wehaveinvestigatedthecomplexphoton renders overlapping maps for Max[g(2)(0)] (yellow) and statisticsphenomenologythatemergesfromthestrongcou- Min[g(2)(0)](violet), andthetopandbottompanelsplot plingofamesoscopicensembleofquantumemittersand cutsofthesemapsforvariousC-values. Wecanidentify a single plasmon mode supported by a generic nanocav- threedomainsaccordingtothestatisticsofthescatteredpho- ity. We have presented an analytical method describing tons. ForsmallQEensemblesandlargeC, onlypositive the optical response of these systems under low-intensity correlationstakeplace,asinFigure3(a )-(c )forC > 1. In coherentillumination. Ourapproachprovidesinsightsinto 2 2 thisregime,Max[g(2)(0)]growswithincreasingcoupling therolethatboththeplasmon-exciton-polaritonladderand strengthanddevelopsamaximumatN ∼ 10forallC. For itstuningthroughthesingleemittercooperativityplayin very large N, a second domain is apparent. In this limit, the emission of strongly correlated (bunched and/or anti- PEPsbosonizeasthe1/N factorinEquation(5)governs bunched)light. Finally,ourresultsdemonstratetherobust- g(2)(0),yieldingmaximaandminimaapproaching1mono- ness of these compound systems against bosonization ef- tonicallyasthenumberofQEsincreases. Bothbunchedand fects,predictingstrongintensitycorrelationsatconsiderable antibunchedemissiontakesplace(withindifferentspectral ensemble sizes. Our theoretical findings demonstrate the windows)atintermediateN andC. Inthisthirddomain, feasibilityandestablishexperimentalguidelinestowardsthe positivecorrelationsdecaymonotonicallywithN,whereas realisationofnanoscalenonclassicallightsourcesoperating negative correlations are enhanced. 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