Enhancing Photon Correlations through Plasmonic Strong Coupling

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by  R. Sáez-Blázquez

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 Lσˆi[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. Min[g(2)(0)] dimin- beyondthesingle-emitterlevel.
5
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